text stringlengths 6 976k | token_count float64 677 677 | cluster_id int64 1 1 |
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Numerical Methods for Roots of Polynomials - Part II along with Part I (9780444527295) covers most of the traditional methods for polynomial root-finding such as interpolation and methods due to Graeffe, Laguerre, and Jenkins and Traub. It includes many other methods and topics as well and has a chapter devoted to certain modern virtually optimal methods. Additionally, there are pointers to...
This book covers the different aspects, such as patents, trademarks and copyright of Intellectual Property (IP) from a more practical business perspective. Intellectual Property and Assessing its' Financial Value describes the differences between regions, mainly the differences between the US and EU. In addition, several tools are presented for assessing the value of new IP, which is of... | 677.169 | 1 |
This course will not satisfy the general elective category for students who entered their program of study in or after the 2007 fall semester. It is intended for those students who feel the results of their placement test do not accurately reflect their mathematical ability, and who want to demonstrate their mastery of mathematical topics for purposes of placement in a course.
Students work in a web-based platform, ALEKS.
VI. Catalog Course Description
This course reviews various mathematical concepts from Pre-Algebra through Pre-Calculus. After taking the math placement test, students work with math software to strengthen their mathematical knowledge and potentially increase their placement score through the software, leading to continue with other courses within a program of study earlier.
Assessment Methods for Course Learning Goals
The assessment of course learning goals is based on successful completion of the individual program designed for the student using a math software. This software also determines a student's final placement score according to personalized assessments within the software. | 677.169 | 1 |
Math 1411 Precalculus
Test #2 Review
On the trig questions, do not leave square roots in the denominator of a fraction.
Simplify square roots as well. Simplify fractions. If an answer involves a trigonometric
ratio of a special angle, then you must evalua
Math 1411 Precalculus
Test #3 Review
Do not leave square roots in the denominator of a fraction. Simplify square roots as
well. Simplify fractions and products. If an answer involves a trigonometric ratio of a
special angle, then you must evaluate that tr
Math 1411 Precalculus
Test #1 Review
1. Let the function f be graphed below.
(a) Find the values f (1) and f (2).
(b) Compute the average rate of change of f over the interval [1, 2].
(c) State the domain of f .
(d) State the range of f .
(e) On what inte
Math 1411 College Algebra and Trigonometry
Sets and Intervals
A set is a collection of objects, and these objects are called the
elements of the set. If is a set, the notation means that
is an element of the set and
/ means that is not an element
of the | 677.169 | 1 |
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Classic Algebra: Word Problems Algebraic Solutions
Herb Gross explains some benefits of using Algebra to solve word problems (such as knowing when there is more than one solution) and the need for good reading comprehension when solving more complex problems. He solves problems similar to those in video 23 - this time using Algebra. Please have a look at for more resources. | 677.169 | 1 |
The International Mathematical Olympiad (IMO) is nearing its fiftieth anniversary and has already created a very rich legacy and firmly established itself as the most prestigiousmathematical competition in which a high-school student could aspire to participate. Apart from the opportunity to tackle interesting and very challenging mathematical problems, the IMO represents a great opportunity for high-school students to see how they measure up against students from the rest of the world. Perhaps even more importantly, it is an opportunity to make friends and socialize withstudents who have similar interests, possibly even to become acquainted with their future colleagues on this first leg of their journey into the world of professional and scientific mathematics. Above all, however pleasing or disappointing the final score may be, preparing for an IMO and participating in one is an adventure that will undoubtedly linger in one's memory for the rest of one's life. It is tothe high-school-aged aspiring mathematician and IMO participant that we devote this entire book. The goal of this book is to include all problems ever shortlisted for the IMOs in a single volume. Up to this point, only scattered manuscripts traded among different teams have been available, and a number of manuscripts were lost for many years or unavailable to many. In this book, all manuscriptshave been collected into a single compendium of mathematics problems of the kind that usually appear on the IMOs. Therefore, we believe that this book will be the definitive and authoritative source for high-school students preparing for the IMO, and we suspect that it will be of particular benefit in countries lacking adequate preparation literature. A high-school student could spend an enjoyableyear going through the numerous problems and novel ideas presented in the solutions and emerge ready to tackle even the most difficult problems on an IMO. In addition, the skill acquired in the process of successfully attacking difficult mathematics problems will prove to be invaluable in a serious and prosperous career in mathematics. However, we must caution our aspiring IMO participant on the use of...
...Assembly for two-year terms beginning after each regular session of the Assembly. El Consejo es elegido por la Asamblea por períodos de dos años a partir después de cada período ordinario de sesiones de la Asamblea.
The Council is the Executive Organ of IMO and is responsible, under the Assembly, for supervising the work of the Organization. El Consejo es el órgano ejecutivo de la OMI, y es responsable, en la Asamblea, de supervisar el trabajo de la Organización. Between imo | 677.169 | 1 |
Project #15471 - physics
26 Mathematics Tutors Online
I already posted this. But it is passed deadline and I have not recieved it and can not get ahold of the scholar so sorry that it is a little last minute. I guess I will be paying for it twice. So PLEASE make sure you can meet the deadline.
The algorithm needs to include everythong in the directions. It is basically a set of directions on how to approach a newton's law problem.
The worksheet is basically a general template for solving a problem, that can be filled out of any given problem. | 677.169 | 1 |
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Mathematics: A Human Endeavor Student Workbook
Mathematics: A Human Endeavor Student Workbook Information
The workbook provides convenient work spaces for the book's problems, plus three types of additional exercises: supplemental exercises, reinforcement of past lessons, and exercises using graphing calculators | 677.169 | 1 |
Developed to meet the needs of modern students, this Second Edition of the classic algebra text by Peter Cameron covers all the abstract algebra an undergraduate student is likely to need. Starting with an introductory overview of numbers, sets and functions, matrices, polynomials, and modular arithmetic, the text then introduces the most important algebraic structures: groups, rings and fields, and their properties. This is followed by coverage of vector spaces and modules with applications to abelian groups and canonical forms before returning to the construction of the number systems, including the existence of transcendental numbers. The final chapters take the reader further into the theory of groups, rings and fields, coding theory, and Galois theory. With over 300 exercises, and web-based solutions, this is an ideal introductory text for Year 1 and 2 undergraduate students in mathematics. | 677.169 | 1 |
Algebra Triangles program provides detailed, step-by-step solutions in a tutorial-like format to the problems of solving triangles in elementary geometry. The program is designed for high school students and teachers.
Algebra Vision is a unique educational software tool to help students develop algebraic problem solving strategies. It provides an environment to play and see algebra in a more tangible light. You can literally move expressions around!
TriAngles 3D Viewer is a scanner for 3D triangles and IntriCAD models. This viewer lets you analyze the 3D graphic models stored in .txs format.TriAngles 3D Viewer lets you export the file to diverse formats to later easily process it. It also offers graphics based on OpenGL. It is very | 677.169 | 1 |
Be sure that you have an application to open
this file type before downloading and/or purchasing.
3 MB|17 including directions and key
Product Description
Graphing and Properties of Polynomial Functions with Google® Slides Digital Organizer - No Prep and Paperless
Here is a great interactive activity to reinforce your students skills with the graphs of polynomials. There are eight functions given in factored form. Students analyze the functions, determine zeros, intercepts, bounces, end behavior and more, then match the correct graph, dragging it onto the organizer.
A printable version is also included for a blended class or in case of a bade tech day. The printable version can be used as worksheet, HW, task cards, stations, or Interactive Notebook pages.
Detailed illustrated directions are included in this NO PREP for you activity. Answer keys included
✓ Completely paperless, no printing, no lost assignments, and NO PREP for you.
✓ Students work directly on their own pages or work collaboratively - your choice!
✓ Students can submit work digitally and you can give feedback promptly
✓ Increases student engagement and can be accessed from anywhere
✓ Students can organize their materials in their own Google Drive. Great for Review! | 677.169 | 1 |
Want to buy the book?
Book Description
This textbook is for an introductory course in numerical methods, MATLAB, and technical computing, with an emphasis on the informed use of mathematical software. The presentation helps readers learn enough about the mathematical functions in MATLAB to use them correctly, appreciate their limitations, and modify them appropriately. The book makes extensive use of computer graphics, including interactive graphical expositions of numerical algorithms. It provides more than 70 M-files, which can be downloaded from the text Web site | 677.169 | 1 |
Modern Geometries Non-Euclidean, Projective, and Discrete
ISBN-10: 0130323136
ISBN-13: 9780130323132
Edition: 2nd Engaging, accessible, and extensively illustrated, this brief, but solid introduction to modern geometry describes geometry as it is understood and used by contemporary mathematicians and theoretical scientists. Basically non-Euclidean in approach, it relates geometry to familiar ideas from analytic geometry, staying firmly in the Cartesian plane. It uses the principle geometric concept of congruence or geometric transformation--introducing and using the Erlanger Program explicitly throughout. It features significant modern applications of geometry--e.g., the geometry of relativity, symmetry, art and crystallography, finite geometry and computation. Covers a full range of topics from plane geometry, projective geometry, solid geometry, discrete geometry, and axiom systems. For anyone interested in an introduction to geometry used by contemporary mathematicians and theoretical scientists.
Mike Henleis Associate Director of MBA Career Services at The College of William and Maryrsquo;s Mason School of Business.nbsp; Previous to this role he held various staff and faculty positions at the University of Minnesotarsquo;s Carlson School of Management including Director of the Undergraduate Business Career Center.nbsp; Additionally, he served in a faculty position at St. Maryrsquo;s University Minnesota and adjunct faculty positions in several community colleges.nbsp; Possessing an MBA in finance, an MA in educational psychology, and a BA in psychology, Mike has over 20 years of experience in the areas of leadership, professional development and corporate finance.nbsp; He has made presentations at numerous national and regional conferences. nbsp; Mike Stebleton, Ph.D., is a faculty member in the Department of Postsecondary Teaching and Learning in the College of Education and Human Development at the University of Minnesota-Twin Cities.nbsp; He recently held adjunct faculty positions at St. Mary's University of Minnesota and Metropolitan State University in St. Paul, MN.nbsp; Additionally, Stebleton worked as a counseling faculty member at a community college for over three years.nbsp; His research interests include:nbsp; narrative approaches to career development, the role of media on career-decision making, and multicultural influences impacting career.nbsp; Mike is one of the authors of a textbook titled: Hired:nbsp; The Job Hunting/Career-Planning Guide , 3 ed., published by Prentice-Hall (2006).nbsp; He has presented at numerous state and national career development conferences.nbsp;nbsp;n | 677.169 | 1 |
Modern Differential Equations
This introductory differential equations book is one of the first to integrate technology throughout the text. Using a computer algebra system (CAS) helps students to solve problems (both specific and general cases), to reason spatially through visualization, to compare changing variables and parameters, to make inferences, and to evaluate outcomes. Students learn how and when to use technology tools by examples, in exercises, and in applications. The authors' writing style is easy to read, yet mathematically precise. The text assumes students have completed a first-year calculus course and theorems are proved only if their proofs are instructive or have teaching value.
Book Description Brooks Cole. Hardcover. Book Condition: New. 0030287049287049 | 677.169 | 1 |
links for 2010-03-19
"Patterns, Functions, and Algebra" explores the "big ideas" in algebraic thinking. The course consists of 10 two-and-a-half hour sessions that each include video programming and interactive Web activities. The 10th session explores ways to apply the algebraic concepts you've learned in K-8 classrooms. You should complete the sessions sequentially."
"In this series, host Sol Garfunkel explains how algebra is used for solving real-world problems and clearly explains concepts that may baffle many students. Graphic illustrations and on-location examples help students connect mathematics to daily life. The series also has applications in geometry and calculus instruction. Algebra is also valuable for teachers seeking to review the subject matter. " | 677.169 | 1 |
...
Show More real analysis to abstract topological spaces, using metric spaces as a bridge between the two. The language of metric and topological spaces is established with continuity as the motivating concept. Several concepts are introduced, first in metric spaces and then repeated for topological spaces, to help convey familiarity. The discussion develops to cover connectedness, compactness and completeness, a trio widely used in the rest of mathematics. Topology also has a more geometric aspect which is familiar in popular expositions of the subject as `rubber-sheet geometry', with pictures of Möbius bands, doughnuts, Klein bottles and the like; this geometric aspect is illustrated by describing some standard surfaces, and it is shown how all this fits into the same story as the more analytic developments. The book is primarily aimed at second- or third-year mathematics students. There are numerous exercises, many of the more challenging ones accompanied by hints, as well as a companion website, with further explanations and examples as well as material supplementary to that | 677.169 | 1 |
Differential equations and linear algebra are the two crucial courses in undergraduate mathematics. This new textbook develops those subjects separately and together. The complete book is a year's course, including Fourier and Laplace transforms, plus the Fast Fourier Transform and Singular Value Decomposition.
Undergraduate students in courses covering differential equations and linear algebra, either separately or together, will find this material essential to their understanding.
An innovative textbook that allows differential equations to be taught alone, or in parallel with linear algebra, affording extra flexibility to instructors. It covers the fundamental undergraduate topics in differential equations and linear algebra, revealing connections between these two essential subjects, and applications to the physical sciences, engineering and economics.
About the Author:
Gilbert Strang is a Professor of Mathematics at Massachusetts Institute of Technology and an Honorary Fellow of Balliol College, Oxford University. He is also a prolific author, with a dozen highly regarded textbooks and monographs to his credit. Strang served as president of the Society for Industrial and Applied Mathematics (SIAM) from 1999-2000 and chaired the U.S. National Committee on Mathematics from 2003-4. He won the Henrici and Su Buchin prizes at ICIAM 2007 and the Von Neumann Medal of the U.S. Association of Computational Mechanics. He is a SIAM Fellow and a member of the National Academy of Sciences. | 677.169 | 1 |
Elementary Row Operations: 1. Interchange two rows 2. Multiply a row by a nonzero real number 3. Replace a row by its sum with a multiple of another row
Over-determined System: More equations than unknowns. In R2 each equation represents a line, usually i
Section 3.2
Subspaces
Which of the following subsets of R^3x3 are subspaces of R^3x3?
A. The 3x3 matrices whose entries are all greater than or equal to 0
B. The 3x3 matrices whose entries are all integers.
C. The symmetric 3x3 matrices.
D. The diagonal 3
Section 2.1 Determinant
Given the matrix A=[-1,-3;3,0] find its determinant.
The determinant of A is 9
Given the matrix [0,0,0;0,-5,-4;-5,0,-5]
(a) find its determinant
Your answer is: 0
(b) does the matrix have an inverse?
Your answer is: NO
If A = [3,6;
Section 6.5
Singluar Value Decomposition
Find the singular values of sigma1>=sigma2 of A=[-2,-2,2,-1,1,1]
ATA=[-2,2,1,-2,-1,1][-2,-2,2,-1,1,1]=[9,3,3,6]
Let det(ATA-alphaI2)=0 where I2 is the 2x2 identity matrix.
We have (9-alpha)(6-alpha)-3^2=0 or alpha^
Section 2.2
Properties of Determinants
If A and B are 2 x 2 matrices, det(A)=-3, det(B)=-7, then
det(AB)=21
det(-3A)=-27
det(AT)=-3
det(B^-1)=-0.1428
det(B^4)=2401
If the determinant of a 4 x 4 matrix A is the det(A)=4, and the matrix B is obtained from A
Chapter 1
Introduction to MATLAB
MATLAB is a computer software commonly used in both education and industry to solve a wide range
of problems.
This chapter provides a brief introduction to MATLAB, and the tools and functions that help you
to work with MAT
MAT 343 MATLAB LAB 2
%Question 1.
a)
n=1000;
A=floor(10*rand(n);
b=sum(A')';
z=ones(n,1);
x=A\b
The solution of the system Ax=b is the vactor z.
Because b is a vector of a .
tic,x=A\b;toc
Elapsed time is 0.029964 seconds.
tic,y=inv(A)*b; toc
Elapsed time
MAT 343 Applied Linear Algebra
Prof. Gardner ([email protected]), Goldwater 654
Honors Credit
Honors students are encouraged to do a project for honors credit (not part of
the course grade). The project will consist of a challenging problem or set of
p
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ABS 360 Southwest Home Gardening
Study Guide for Topic: Landscape Trees and Shrubs for the Southwest
Climate is the key to landscape gardening in the Southwest. Our climate
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function rowproduct(A,B)
m=size(A,1);
p=size(A,2);
n=size(B,2);
f=size(B,1);
if (f=p)
C=zeros(m,n);
for i=1:m
for j=1:n
C(i,j)=C(i,j)+A(i,:)*B(:,j);
end
end
ans=C
else
disp('Dimensions do not match')
end
end | 677.169 | 1 |
About A*Maths
The binomial Theorem allows us to expand many brackets without multiplying each bracket out one by one. It states: To expandwe could expandwhich would be a very long winded process. Or we could just substitute forandinto the expression for the binomial expansion. Example: Expandthen which simplifies to and further to We may be asked to […]
Data can usefully be summarised in a table, and a table can have it's borders, column and row labels taken away and then enclosed in brackets in which case it be come a matrix. Then we can perform useful calculations with it. For instance: The figures in the table show The daily production, in kilograms, […]
If you already know how to solve simultaneous equations then you may well wonder why people use matrices to solve them. The fact is, while simple equations with two unknowns x and y are quite easy to solve, as the number of unknowns increases so does the number of equations we have to solve. The […]
Planes rarely fly in the direction they are pointing. If the wind is blowing and the world is turning the pilot has to take account of these when he plots a course. Even with modern gps systems available, , it is beneficial to the pilot to take these into account because of the resulting increase […]
As a race we need the most from our limited resources. Every manager must decide what or who to send where so as to get the job done with the least effort and risk, and at the least cost, or decide what products to make and sell, subject to the available labour and machine time, […]
I don't know why these thing need four possible names. We can call them stationary points and classify them as maxima or minima. Definition: A stationary point is a point on a curve where Definition: A stationary point, withis a Minima if On a graph a minimum is lower than the points on either side […]
Definition: If A is at positionmoving with velocityand B is at positionmoving with velocitythen The relative position of B relative to A is given byand the relative velocity of B relative to A is given by The relative position of A relative to B is given byand the relative velocity of A relative to B […]
Long division with polynomials sounds, and is, a great deal more complicated than long division with numbers. Fortunately though, it is not always necessary. There are two very helpful theorems which often turn the problem of long division into one of substitution. The Factor Theorem: Ifis a factor ofthensois also a root ofor equivalently, a […]
– the set of positive integers and zero – x is greater than y is greater than – x is less than y x >=y – x is greater than or equal to y – the set of integers, – the set of positive integers, – the set of rational numbers – the set of […]
Simple differential equations take the formWe have to solve the equation to findas a function ofWe do this by putting all the 's on the right and integrating. Normally when we integrate we have to add a constant. We can find the value of this constant if we are told a point on the curve. […] | 677.169 | 1 |
About this title:
Synopsis: Features the complete set of answers to the exercises in Mathematics Year 5, as well as a selection of photocopiable worksheets to save you time and enable you to identify areas requiring further attention. The book includes diagrams and workings where necessary, to ensure pupils understand how to present their answers, as well as photocopiable worksheets at the back of the book. Also available from Galore Park - Mathematics Year 5 - Mathematics Year 6 - Mathematics Year 6 Answers - 11+ Maths Practice Exercises - 11+ Maths Revision Guide - 10-Minute Maths Tests Workbook Age 8-10 - 10-Minute Maths Tests Workbook Age 9-11 - Mental Arithmetic Workbook Age 8-10 - Mental Arithmetic Workbook Age 9-11
About the Author:
David Hillard spent more than 45 years teaching Mathematics in two preparatory schools. He was associated with the Common Entrance Examination at 11+ and 13+ in the role of adviser, assessor or setter. Serena Alexander has taught Mathematics since 1987, originally in both maintained and independent senior schools. From 1999 she taught at St. Paul's School for Boys, where she was Head of Mathematics at their Preparatory School, Colet Court, before moving first to Newton Prep as Deputy Head and then to Devonshire House as Headmistress. She has run many Mathematics conferences for prep school teachers. She is now an ISI reporting Inspector and consultant, with a particular focus on mathematics.
Book Description Createspace, 2013. PAP. Book Condition: New. New Book. Delivered from our US warehouse in 10 to 14 business days. THIS BOOK IS PRINTED ON DEMAND.EstThis Createspace, 2013. PAP. Book Condition: New. New Book.Shipped from US within 10 to 14 business days.THIS BOOK IS PRINTED ON DEMAND. Est This CreateSpace Independent Publishing Platform. Paperback. Book Condition: New. This item is printed on demand. Paperback. 274 pages. Dimensions: 11.0in. x 8.5in. x 0.6in.This is a complete practice and revision mathematics book covers requirements of the new national curriculum framework for mathematics for years 3 and 4. Extension activities from the years 5 and 5 and Complete Mathematics Workbook: Essential revision and practice Years 1 - 2 book above. Packed with many practical problem solving exercises, this book is guaranteed to improve childrens mathematics skills in fun and engaging ways and when used with the levels 3 5 book above, it is also suitable for preparation for the 11 and other secondary school selection examinations. This item ships from La Vergne,TN. Paperback. Bookseller Inventory # 9781490988573
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Book Description paperback. Book Condition: New. Ship out in 2 business day, And Fast shipping, Free Tracking number will be provided after the shipment.Paperback Pages Number: 42 Language: Chinese. 3-year simulation of the College Entrance Examination: high school mathematics compulsory Jiangsu an inventory of knowledge: This is the experience of millions of teachers. which is the accumulation of countless successful. This is the most systematic induction. which is the most scientific design. Subject knowledge designed exercises. easy in practice to achieve the basic concepts of discipline. the basic knowledge to understand and remember. Practice has proved that this is the best way to basic training. You want to remember the dead. not by rote. Pull textbook point: the concentrated material essence. this is a fine analysis of the material of the Jingjiang. which is the golden touch of the mist fingers. When a thorough grasp of textbook knowledge. you will be able to maintaining the status quo. calmly face every exam! Five years of college entrance examination: This is a new college entrance examination and the seamless connection of the new materials. College Entrance Examination. is how much propositional expert effort. is how much propositional scholars sweat. This is the wisdom of careful design. the painstaking creation. which is a beautiful poem. Insight into the college entrance examination questions and propositions of law is tantamount to grasp the hand of God. equal to uncover the mystery of the hands of God! Basis through: This is the most basic test. which is the solid foundation of knowledge to consolidate the base of the basic capabilities. This is your first hurdle you must strive to try and try again! Three-year simulation: This is the national front-line teachers to unite people with the proposition contest the proposition had to read important information. but also the birthplace of the proposition inspiration. Backpack to read and write: This is an extension of the reading. which is the source of thinking. which is the wisdom of the big backpack. which is the power arm of the college entrance examination. Unit testing: detection of self back garden. which is the touchstone of new skills. which is savor the success of destination. Language angel: This is excellent fragment retrieved from the treasure house of Chinese Language and this is a gathering convened by the language of angels. which is the text goddess of weaving the poem. The melting of these languages ??into their own language. you will also become the Angel of the language. Practice the whole solution: This is a wonderful analytical material after-school exercise. which is a silent teacher to accompany you. Practice after school. which is the source of all the questions the question. the college entrance examination questions. the variant of the simulation questions are generally exercises. you have to careful to practice. explore one on the inside!Four Satisfaction guaranteed,or money back. Bookseller Inventory # N25983
Book Description paperback. Book Condition: New. Ship out in 2 business day, And Fast shipping, Free Tracking number will be provided after the shipment.Pages Number: 102 Publisher: Capital Normal University Press Pub. Date :2011-4-1. The book presents the knowledge structure of mind maps to guide module learning skills teacher training caught the left about the right knowledge and ability points entrance design menu-classification exercise clever arrangement of scientific knowledge and capacity double-column method detailed interactive thin profile split modular solutions of typical examples of each break extended to explore the integration of knowledge and coaching methods inspired wisdom speaking with intelligence test analysis techniques to decrypt the contents locked tap the potential closely topics interactive syllabus point two-column key and difficult fast break Contents: Chapter 1.1 set and the function concept and the meaning of a collection of collections. said 1.1.1 1.1.2 1.1.3 set the basic relationship between the set of basic computing functions and 1.2 The concept of a function that 1.2.1 1.2.2 1.3 function representation of the basic nature of the functions and maximum monotonic 1.3.1 (small) value 1.3.2 Review of the second parity unit Chapter basic elementary function (1) 2.1 2.1.1 exponential function computing power index and indexFour Satisfaction guaranteed,or money back. Bookseller Inventory # LE1347 | 677.169 | 1 |
201755251
ISBN: 0201755254
Edition: 3
Publication Date: 2002
Publisher: Benjamin-Cummings Publishing Company
AUTHOR
Dugopolski, Mark
SUMMARY
This text provides numerous strategies for success for both students and instructors. Instructors will find the book easier to use with such additions as an Annotated Instructor' s Edition, instructor notes within the exercise sets, and an Insider' s Guide. Students will find success through features that include highlights, exercise hints, art annotations, critical thinking exercises, and pop quizzes, as well as procedures, strategies, and summaries.Dugopolski, Mark is the author of 'College Algebra and Trigonometry', published 2002 under ISBN 9780201755251 and ISBN 0201755254 | 677.169 | 1 |
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Follow the mathematical adventures of Idaho Bones and his sidekick Boise as they search the darkest reaches of Algebra to find the correct answers for bewildered students everywhere. Topics covered include functions, algebraic properties, linear equations, and many more. | 677.169 | 1 |
Mathematical Methods for Physical and Analytical Chemistry presents mathematical and statistical methods to students of chemistry at the intermediate, post-calculus level.
The content includes a review of general calculus; a review of numerical techniques often omitted from calculus courses, such as cubic splines and Newton's method; a detailed treatment of statistical methods for experimental data analysis; complex numbers; extrapolation; linear algebra; and differential equations.
With numerous example problems and helpful anecdotes, this text gives chemistry students the mathematical knowledge they need to understand the analytical and physical chemistry professional literature | 677.169 | 1 |
For introductory sophomore-level courses in Linear Algebra or Matrix Theory. This text presents the basic ideas of linear algebra in a manner that offers students a fine balance between abstraction/theory and computational skills. The emphasis is on not just teaching how to read a proof but also on how to write a proof.
Key Features
Number Of Pages 720
Publisher Pearson
Copyright 2008
Edition 9 Edition
Specifications of Elementary Linear Algebra with Applications: International | 677.169 | 1 |
Great book for math majors
This lovingly written and exquisitely crafted book provides a detailed introduction to abstract algebra. First published in 1964, it is still an excellent textbook and will probably never go out of date.
The text does not assume any background beyond high school math, but the mathematics it covers is intense and detailed. This is a book for college math majors, not for someone who is looking for a casual introduction to | 677.169 | 1 |
MAT 1341: VECTOR SPACES I
1. Vector spaces
A vector space is an algebraic structure formed by a collection of vectors. For example, the set of (cartesian) vectors in R2 is a vector space, i.e., The 2-dimensional space
R2 = cfw_[a, b]| a, b R (by identifyi
MAT 1341: VECTOR SPACES III
1. Linear Independence
Note that a spanning set (i.e., the set of all linear combinations of given vectors) forms a
subspace even though there may be dierent expressions for the same vector. For instance, as
1
R2 = spancfw_e1 ,
MAT 1341: REVIEW III
1. Planes
1.1. Equations of Planes. As we see in the previous section, a line (in any dimensional
space) is uniquely determined either by two points or by a nonzero vector and a point.
Similarly, a plane in 3-dimensional space is uniq
MAT 1341: REVIEW I
1. Complex Numbers
A real number is a point on an innitely long number line. More precisely, the set of real
numbers, denoted by R, is dened by completing (i.e., adding limits of sequences of rational
numbers to the eld) the rational nu | 677.169 | 1 |
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173 KB|5 pages
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This is a one-page formative assessment to see how students are progressing on introductory concepts such as substitution, evaluation, order of operations, solving linear equations in which the variable appears exactly once, distributive property, and combining like terms.
There are a total of 5 versions.
This quiz is intended to be written on directly.
Please download the pdf preview file first, so you can see exactly what's included; the product file is a word document, which you may personalize for your students. (Just make sure you make similar edits to each version!) | 677.169 | 1 |
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This is PowerPoint file that will help guide a vertical teaming for grade 8 math teachers and Algebra 1 teachers. It covers 5 topics that are taught in both Common Core courses (one variable equations, concept of a function, linear equations, systems of equations, and bivariate data).
Teachers will compare and contrast standards from each course.
Directions are included as well as the last 12 slides can be used as worksheets for the groups. | 677.169 | 1 |
Graph theory is the study of, well, graphs. Not in the y=f(x) sense, but in the discrete math/computer science vertices connected by edges sense. Here's a (directed) graph:
Here's another one:
Ooh, pretty. You're on another graph right now -- the internet (think web pages are vertices, links are edges). Graph theory began around 1736 when Euler addressed the Konigsberg bridge problem. Here's a picture:
The problem: pick a place to start and, crossing every bridge exactly once, return to where you started. Yeah, it's hard. Like, really really hard. Like, you can't do it hard. "Why not?" is a question easily answered by graph theory.
From these humble beginnings graph theory has grown to an essential tool in analyzing networks of all types, with applications to computer science, coding theory, network security, optimization, and many many other areas. It's also a fertile area of undergraduate research, and the subject of many REUs.
This course will be a friendly introduction to many different areas of graph theory, both classical and new. For more general info on graph theory check out this site. Here's a recent article on another famous graph theory problem -- the four color problem. For those of you in Disco, check out Chapters 11-13 of Dr. Grimaldi's text. | 677.169 | 1 |
Algebra Review Stations Coloring Activity
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Algebra Review Stations Coloring Activity
Looking for a fun way to review, or an activity to use during those last few days? Students love to color, and this activity will keep them focused while doing math AND they get to color a fun picture Students solve the problem at their station, identify the answer, and color the picture accordingly. After approximately 10 minutes, they move on to the next station.
Station topics:
Station 1 - Simplifying an expression using the order of operations.
Station 2 - Solving a multi-step equation.
Station 3 - Evaluating a function for f(x).
Station 4 - Graphing a linear equation.
Station 5 - Writing a linear equation given two points.
Station 6 - Solving a system of equations.
Station 7 - Simplifying a monomial expression using exponent rules.
Station 8 - Simplifying a radical expression with variables.
Station 9 - Subtracting polynomials.
Station 10 - Solving a quadratic equation.
A student recording worksheet, the coloring page, and answer key are also included in this product | 677.169 | 1 |
Synopses & Reviews
Publisher Comments
This unique textbook focuses on the structure of fields and is intended for a second course in abstract algebra. Besides providing proofs of the transcendance of pi and e, the book includes material on differential Galois groups and a proof of Hilbert's irreducibility theorem. The reader will hear about equations, both polynomial and differential, and about the algebraic structure of their solutions. In explaining these concepts, the author also provides comments on their historical development and leads the reader along many interesting paths. In addition, there are theorems from analysis: as stated before, the transcendence of the numbers pi and e, the fact that the complex numbers form an algebraically closed field, and also Puiseux's theorem that shows how one can parametrize the roots of polynomial equations, the coefficients of which are allowed to vary. There are exercises at the end of each chapter, varying in degree from easy to difficult. To make the book more lively, the author has incorporated pictures from the history of mathematics, including scans of mathematical stamps and pictures of mathematicians. Antoine Chambert-Loir taught this book when he was Professor at École polytechnique, Palaiseau, France. He is now Professor at Université de Rennes 1.
Review
From the reviews: "This is a textbook for a second course in undergraduate abstract algebra. ... Along the way, certain very interesting results are covered that are not often seen in books at this level ... . includes numerous exercises which significantly extend material in the text." (William M. McGovern, SIAM Review, Vol. 47 (2), 2005) "The textbook focuses on the structure of fields and is intended for a second course in abstract algebra ... The reader will learn about equations, both polynomial and differential ... . In explaining these concepts, the author also provides ... leads the reader along many interesting paths. ... There are exercises at the end of each chapter, varying in degree from easy to difficult. ... author has incorporated pictures from the history of mathematics, including scans of mathematical stamps and pictures of mathematicians." (Zentralblatt für Didaktik und Mathematik, February, 2005) "This nice volume starts with constructions with ruler and compass and uses these old problems as a peg for field theory, especially for Galois Theory. ... it also deals with some mathematical pearls which one cannot find so easily in the textbooks and for which I like this text a lot ... . The whole manuscript is written very carefully and can be heartily recommended to students and to teachers as well." (J. Schoissengeier, Monatshefte für Mathematik, Vol. 148 (4), 2006) "The book can ... be described as a good second course in abstract algebra focusing on the structure of fields in general and field extensions in particular. ... There are many exercises ... most of them rather challenging. ... it is a well written text which is especially suitable for students about to study abstract algebra seriously." (P. Shiu, The Mathematical Gazette, Vol. 90 (519), 2006) "The distinctive features of Chambert-Loir's book are as follows: first, it develops many results that should give undergraduate mathematics majors immediate and substantial satisfaction ... second, the book concludes with a novel undergraduate-level introduction to the algebraic theory of differential equations. ... Summing Up: Highly Recommended. General Readers; upper-division undergraduates through professionals." (D. V. Feldman, CHOICE, Vol. 42 (10), 2005) "This book treats mainly Galois theory of finite extensions of fields. All the material necessary for such a study is presented in this book ... . There are a lot of exercises ... . A nice feature is the inclusion of portraits of mathematicians who made important contributions to the subject of this book." (K. Kiyek, Mathematical Reviews, Issue 2005 h) "This book is intended as a second course in algebra focusing mainly on Galois theory of the finite extensions of fields. ... The book is easy to read and mostly self-contained. The large number of exercises at different levels makes it a valuable source both as the basis for a course or self-study." (G. Teschl, Internationale Mathematische Nachrichten, Issue 203, 2006) "Toward advanced undergraduate students, the book focuses on those parts of abstract algebra which primarily deal with the structure of fields ... and the related algebraic theory of differential equations. ... the central theme of the text is field theory, together with its relations to some other areas in abstract algebra and to analysis. ... No doubt, this fairly unique introduction to some central aspects of modern abstract algebra and its applications is a highly welcome and valuable complement to ... literature in the field." (Werner Kleinert, Zentralblatt MATH, Vol. 1155, 2009)
Synopsis
This unique book focuses on the structure of fields and is intended for students in abstract algebra. Besides providing proofs of the transcendance of pi and e, the book includes material on differential Galois groups and a proof of Hilbert's irreducibility theorem. Readers will learn about equations, both polynomial and differential, and about the algebraic structure of their solutions. In explaining these concepts, the author also provides comments on their historical development and leads readers along many interesting paths.
Synopsis
This is a small book on algebra where the stress is laid on the structure of ?elds, hence its title. Youwillhearaboutequations, bothpolynomialanddi?erential, andabout the algebraic structure of their solutions. For example, it has been known for centuries how to explicitely solve polynomial equations of degree 2 (Baby- nians, many centuries ago), 3 (Scipione del Ferro, Tartaglia, Cardan, around th 1500a.d.), and even 4 (Cardan, Ferrari, xvi century), using only algebraic operations and radicals (nth roots). However, the case of degree 5 remained unsolved until Abel showed in 1826 that a general equation of degree 5 cannot be solved that way. Soon after that, Galois de?ned the group of a polynomial equation as the group of permutations of its roots (say, complex roots) that preserve all algebraicidentitieswithrationalcoe?cientssatis?edbytheseroots.Examples of such identities are given by the elementary symmetric polynomials, for it is well known that the coe?cients of a polynomial are (up to sign) elementary symmetric polynomials in the roots. In general, all relations are obtained by combining these, but sometimes there are new ones and the group of the equation is smaller than the whole permutation group. Galois understood how this symmetry group can be used to characterize the solvability of the equation. He de?ned the notion of solvable group and showed that if the group of the equation is solvable, then one can express its roots with radicals, and converse
Synopsis
This book has a nonstandard choice of topics, including material on differential galois groups and proofs of the transcendence of e and pi. The author uses a conversational tone and has included a selection of stamps to accompany the text. | 677.169 | 1 |
The Mathematical Tripos is Cambridge's oldest Tripos (degree course), dating back to the 18th Century. It is split into four Parts (IA, IB, II, III), described below, where the first three Parts give you a B.A., and Part III gives you an M.Math. if you continue from the B.A., or an M.A.St. if you study it as a stand-alone graduate course. The course covers a wide range of topics in mathematics, and the non-modular examination system allows students to do as many or as few options as they like.
Admissions
Admission to the Mathematical Tripos usually requires A*A*A (incl Maths and Further Maths) at A-level, or equivalent, and STEP, usually papers II and III with grade 1 in each, but this varies based on college and the applicants' level of study of Maths. STEP ('Sixth Term Examination Paper') is a challenging exam used by Cambridge, Warwick, Imperial and some other universities, for which preparation is designed to bring a student up to the level of mathematical maturity required to do a Maths degree at these institutions. As such, a significant proportion of offer holders fail to meet their offer, so early and thorough preparation is a must.
The style of interviews varies from college to college, but as a general rule, they will usually involve a lot of guided problem-solving, and not much time will be spent discussing the personal statement, books read, and so on. It is often advised that you should treat interviews as if they were supervisions: you are not expected to be able to do the questions you are given, but you are expected to reach the answer with a few pokes in the right direction from the interviewer where necessary.
The course
The Mathematical Tripos is split into Part IA (1st year), Part IB (2nd year), Part II (3rd year) and Part III (4th year). Upon completion of Part II, a student is eligible for the Bachelor of Arts degree. Students who achieve a 1st in Part II, or show potential to achieve a 1st, may advance to Part III, which is a graduate-level course that is also available as a stand-alone masters course to students outside of Cambridge.
These are usually regarded as compulsory, and are intended to provide a solid grounding in a wide range of areas of maths. Some of the courses follow on from content that appears at A-level (e.g. Vectors and Matrices), whereas some introduce completely new material (e.g. Vector Calculus).
There is also a non-examinable lecture course in Mechanics in Michaelmas Term, which is useful for people who have not done much Mechanics. (Usually it's a good idea to go if you did less than three Mechanics modules, or if you didn't do A-level and your Maths course didn't contain much Mechanics).
Additionally, there is a non-examinable course in the History of Mathematics, open to all Maths students (not just first-years). These lectures are usually very interesting, casual and entertaining; notes are not taken, and food and drink (alcoholic or otherwise) is encouraged. In Easter Term, students get the opportunity to give lectures of their own in this series.
First-year students normally go to Part IB lectures in Easter Term (see below) so that they can work on the content over the summer and so that their second-year Easter Term can be spent revising.
Part IB
In Part IB, there is a wider selection of courses, and students normally decide to specialise slightly. Most students choose courses from the whole spectrum of pure, applicable and applied maths, with preference towards a particular one. Courses on offer are:
Courses are split into 'C courses' and 'D courses' (it should be noted that courses in Part IA and Part IB are A courses and B courses, respectively). C courses are intended to be straightfoward, and each C course gives rise to four Section I and two Section II questions in the exams. D courses are intended to be more challenging, and the 24-lecture and 16-lecture D courses give rise to four and three Section II questions, respectively, with no Section I questions. Thus one can gain four betas and two alphas from a C course, and either three or four alphas from a D course. (For more information on 'alphas' and 'betas', see below.)
Part III
Students who achieve first-class honours in Part II can continue to Part III; students who get a 2:1 in Part II can apply for entry, subject to permission from the Faculty Board (whose website says that those who are not in the top 40% of the year for both Part IB and Part II are unlikely to be let in). Students who have graduated from other universities can also study Part III as a stand-alone graduate course.
Lectures, supervisions and classes
Parts IA, IB and II
Courses in the first three Parts of the Tripos are either 12, 16 or 24 lectures in length. 12-lecture courses entail two lectures per week for six weeks, 16-lecture courses entail two per week for eight weeks, and 24-lecture courses entail three per week for eight weeks.
Lectures are an hour long and take place between 9am and 1pm. In Part IA, students attend two lectures per day, six days per week (Monday-Saturday). In Part IB, lectures run from Monday to Friday, and the number of lectures on each day depends on the lecture courses attended. In Part II, lectures are six days per week, with the number of courses dependent on the lecture courses attended.
In supervisions, students go over answers to example sheets, of which there are 2 per 12-lecture course, 3 per 16-lecture course and 4 per 24-lecture course.
Part III
Courses in Part III of the Tripos are worth either 2 or 3 units. Courses worth 2 units usually consist of 16 lectures, and courses worth 3 units consist of 24 lectures. As with the first three Parts of the Tripos, example sheets are given out, but instead of supervisions, students usually attend examples classes. These are mini-lectures, where the lecturer goes through the problems on the sheet and answers any quick questions that students have.
Examinations
Parts IA, IB and II
In each of Parts IA, IB and II, students sit four exams in late May or early June. All students sit the same exams, and answer questions on whichever lecture courses they have chosen to go to. (This is different from most other universities in the UK, where you must sign up for modules and sit separate exams for each module.)
Each exam is split into Section I ('short questions') and Section II ('long questions'). Questions in Section I are marked out of 10 and questions in Section II are marked out of 20. Quality marks are given for good answers. Namely:
An 'alpha' is awarded for each Section II question that scores 15-20 raw marks;
A 'beta' is awarded for each Section I question that scores 8-10 raw marks, or for each Section II question that scores 10-14 marks.
The 'merit mark' takes into account both raw marks and quality marks, and is used to rank candidates and give classifications (1st, 2:1, 2:2, 3rd or fail). Each beta is worth 5 extra marks, the first eight alphas are worth 15 marks each, and each alpha thereafter is worth 30 extra marks. This can be summarised in the following formula: where is the merit mark, is the number of raw marks, is the number of alphas and is the number of betas.
Candidates who achieve a 1st in Part II are called 'Wranglers' (the top 1st is called the 'Senior Wrangler'), candidates who achieve 2:1s and 2:2s are called 'Senior Optimes' (pronounced /ˈɒptɪˌmiː/, 'OP-tim-ee') and candidates who achieve 3rds are called 'Junior Optimes'. The candidate with the lowest merit mark who achieves honours (usually a 3rd) is called the 'Wooden Spoon', for entertaining historical reasons involving spoons.
Part III
In Part III each option has its own paper, and unlike Parts IA, IB and II, quality marks are assigned to entire papers rather than to individual questions. The quality marks are alphas, betas and gammas, with a qualifier of + or -. So for example is the best possible quality mark, and is the worst possible quality mark, though it is possible to fail to achieve a quality mark at all. Each paper is worth 2 or 3 units, and candidates normally select 17-19 units (19 being the maximum). A 3-unit examination may be replaced by an essay, which is described as being a good preparation for further study and academic mathematical research.
CATAM
An optional part of the Tripos, which most students do nonetheless, is CATAM. This stands for Computer-Aided Teaching of All Mathematics, and refers to computational project coursework that can be undertaken in Parts IB and II of the Tripos. Projects are on a range of areas of mathematics, with a significant bias in favour of applied and applicable mathematics in Part II (probably representative of the real world).
In Part IB, students submit two 'core' projects, over which there is no choice, and two 'optional' projects out of four possible ones. 20 marks is available for each project, and the quality mark is calculated as in examinations, meaning that up to four alphas are on offer.
In Part II, there is a considerably larger choice of projects. Each project carries a certain number of units, of which up to a maximum of 30 may be submitted. (If more than 30 are submitted, the mark is scaled appropriately, rather than the best 30 units being chosen.) Each unit gives rise to a tenth of a merit mark, meaning that up to three alphas are on offer. | 677.169 | 1 |
HiDigit
HiDigit is a scientific calculator with extended capabilities for Windows . This essential and powerful program works best for math, algebra, calculus, geometry, physics and engineering students and even their teachers. One of the advantages of this calculator is a simple input format even for the most complicated scientific formulas. For example, you can enter 10pi instead of "10*pi". For complex numbers, you can use the following format - "1+2i". For percentages - "number + %". The other important feature of HiDigit is its high precision - up to 15 decimals for scientific calculations. The program features an impressive number of built-in formulas, functions, constants and coefficients. Importantly, the users can customize all of them or add their own variables. Also, the history of all actions is kept, so the users can come back and undo/redo any action at any time. Being a serious calculating software, it is extremely simple in use. No special computer skills or knowledge are required. The interface is straightforward and very easy to navigate through. The compact size of the calculator does not hinder the performance. To the contrary, the program can even be used by the multiple users simultaneously.
(Source:
Current version: This is an essential tool for math, algebra, calculus, geometry, physics and engineering students. What makes the program really stand out is a simple input format even for the most complicated formulas. For example, you can enter 10pi instead of "10*pi". This powerful scientific calculator works with complex numbers, keeps record of your previous actions; supporting both mouse and keyboard use. | 677.169 | 1 |
Category: Matrices
For a matrix A to describe a linear map f: V→W, bases for both spaces must have been chosen; recall that by definition this means that every vector in the space can be written uniquely as a (finite) linear combination of basis vectors, so that written as a (column) vector v of coefficients, only finitely many entries vi are nonzero. I also wonder if the missing representation should be another quadratic. We will, however, fix outright errors in the code.
In addition to using uppercase letters as symbols representing matrices, many authors use a special typographical style, commonly boldface upright (non-italic), to further distinguish matrices from other variables. The following table lists some of the actual results that Mendel obtained in his experiments in crossbreeding peas. However, the explicit analysis can be used instead. dealing with maps involving the zero vector space.
The ultimate goal of scaffolding is to gradually remove the supports as the learner masters the task. So I just carry it along as an extra column. Lisa ???? Really, it is very helpful to explain and understand your way. If we see something we conjecture or think about, it�s real. A square matrix A with 1s on the main diagonal (upper left to lower right) and 0s everywhere else is called a unit matrix. Or you can eliminate entire polygons, or groups of polygons (objects) based on a point in the center and a tolerance equal to the radius of the object.
Students receive opportunities to apply math concept or perform math skill within authentic context. In a nutshell, Lou answered all of my questions about MATHCOUNTS. For what Kaufmann did was to spin a mathematical argument so complex about complexity that it was near impossible to take apart, for a while anyway. All the columns lie on the line through the vector two three four. If you want a printable version of a single problem solution all you need to do is click on the "[Solution]" link next to the problem to get the solution to show up in the solution pane and then from the "Solution Pane Options" select "Printable Version" and a printable version of that solution will appear in a new tab of your browser.
Ti 89 differential equations, three equations three unknowns solver, free online linear equation by substitution calculator, Converting Quadratics to Standard Form calculator, What is the relationship between schoolmathematics and other learning areas. Note: If Vs and Vf are colinear, the cross product returns (0,0,0). Numerics -Pre or simply download the Zip package. let's say we have a matrix \(\mathrm{A}\) and want to find an orthonormal basis of the kernel or null-space of that matrix, such that \(\mathrm{A}x = 0\) for all \(x\) in that subspace. using MathNet.
Now one aspect of war that makes it particularly bloody and horrible are the means to success in mortal combat. Tell me an X here -- so now I'm going to put -- slip in the X that you tell me and I'm going to get zero. For addition to make sense, both matrices have to be of the same order (size). A matrix is a group of numbers, arranged in rows and columns, like this: This is called a "2 by 2" or "2 x 2" matrix, because it has two rows (going across) and two columns (going down).
Look up Lewis Carroll's method; it might be easier than the usual approach. The Matrix and Quaternions FAQ ============================== Version 1.21 30th November 2003 ------------------------------- Please mail feedback to matrix_faq@j3d.org with a subject starting with MATRIX-FAQ (otherwise my spam filter will simply kill your message). To be blunt, women need but a drop of semen to have a child and there are plenty of such drops available. Human coordinate plane conceptual, college math 2 third custom edition answers, worksheet on adding negative numbers.
Figure 3: A simple example of orthographic projection. MATLAB helps you take your ideas beyond the desktop. To "transpose" a matrix, swap the rows and columns. The caricature of cryptography, right out of " Mercury Rising ," made me squirm. Note: A-1 is on the LEFT side of both products. If this is not desired, use instead Matrices.leastSquares and inquire the singularity of the solution with the return argument rank (a unique solution is computed if rank = size(A,1)). | 677.169 | 1 |
Preface
This textbook represents the Finite Element Analysis lecture course given to students in the third year at the Department of Engineering Sciences (now F.I.L.S.), English Stream, University Politehnica of Bucharest, since 1992. It grew in time along with a course taught in Romanian to students in the Faculty of Transports, helped by the emergence of microcomputer networks and integration of the object into mechanical engineering curricula. The syllabus of the 28-hour course, supplemented by 28-hour tutorial and lab. classes, was structured along the NAFEMS recommendations published in the October 1988 issue of BENCHmark. The course represents only an introduction to the finite element analysis, for which we wrote simple stand-alone single-element programs to assist students in solving problems as homework. It is followed by an advanced course in the fourth year at F.I.L.S., called Computational Structural Mechanics, where students are supposed to use commercial programs. In designing the course, our aim was to produce students capable of: (a) understanding the theoretical background, (b) appreciating the structure of finite element programs for potential amendment and development, (c) running packages and assessing their limitations, (d) taking a detached view in checking output, and (e) understanding failure messages and finding ways of rectifying the errors. The course syllabus was restricted to 2D linear elastic structural problems. It has been found advantageous to divide the finite element analysis into two parts. Firstly, the assembly process without any approximations (illustrated by frameworks) followed by the true finite element process which involves approximations. This is achieved starting with trusses, then with beams and plane frames, and progressively dealing with membrane and plate-bending elements. Solid elements and shells are not treated. Our objective was to ensure that students have achieved: (a) a familiarity in working with matrix methods and developing stiffness matrices, (b) an understanding of global versus local coordinate systems, (c) the abilty to use the minimum potential energy theorem and virtual work equations, (d) the mapping from isoparametric space to real geometrics and the need for numerical integration, (e) an insight in numerical techniques for linear equation solving (Gauss elimination, frontal solvers etc), and (f) the use of equilibrium, compatibility, stress/strain relations and boundary conditions. As a course taught for non-native speakers, it has been considered useful to reproduce as language patterns some sentences from English texts. November 2006 Mircea Radeş
Finite Element Analysis (FEA) as applied to structures is a multidisciplinary technique, based on knowledge from three fields: (1) Structural Mechanics, encompassing elasticity, strength of materials, dynamics, plasticity, etc, (2) Numerical Analysis, involving approximation methods, solving linear sets of equations, eigenproblems, etc, and (3) Applied Computer Science, dealing with the development and maintenance of large computer codes. FEA is used to solve large-scale analytical problems. Its task is to model and describe the mechanical behaviour of geometrically complex structures. The procedure is a discretized approach: the geometric shape or the internal stressstrain-displacement field are described by a series of discrete quantities (like coordinates) distributed through the structure. This requires a matrix notation. The tools are the computers, able to store long lists of numbers and manipulate them.
1.1 Object of FEA
The object of FEA is to replace the infinite degree of freedom system in continuum applications by a finite system exhibiting the same basis as discrete analysis. The aim is finding an approximate solution to a boundary- and initialvalue problem by dividing the domain of the system into a set of interconnected finite-sized subdomains of different size and shape, and defining the unknown state variable approximately, within each element, by means of a linear combination of trial functions. The subdomains are called finite elements, the set of finite elements is known as the mesh and the trial functions are referred to as interpolation functions. With the individually defined functions matching each other at certain points called nodes, the unknown function is approximated over the entire domain. The primary difference between the FEA and other approximate methods for the solution of boundary-value problems (finite-difference, weighted-residual,
2
FINITE ELEMENT ANALYSIS
Rayleigh-Ritz, Galerkin) is that in the FEA the approximation is confined to relatively small subdomains. FEA is a localized version of the Rayleigh-Ritz method. Instead of finding an admissible function satisfying the boundary conditions for the entire domain, which is often difficult, in the FEA the admissible functions (called shape functions) are defined over element domains with simple geometry and pay no attention to complications at the boundaries. Since the entire domain is divided into numerous elements and the function is approximated in terms of its values at the element nodes, the evaluation of such a function will require the solution of simultaneous equations. This was possible only at the time the computers became available. The outstanding success of the finite element method can be attributed to a large extent to timing. While the finite element method was being developed, so were increasingly powerful digital computers, which led to automation. The computer is not only able to solve the discretized equations of equilibrium, but also to carry out such diverse tasks as the formulation of equations, by making decisions concerning the finite element mesh and the assembly of stiffness matrices. Perhaps more important is the fact that the finite element method can accommodate systems with complicated geometries and parameter distributions. The wide use of the classical Rayleigh-Ritz method has been limited by the inability to generate suitable admissible functions for a large number of practical problems. Indeed, systems with complex boundary conditions or complex geometry cannot be described easily by global admissible functions, which tend to have complicated expressions, difficult to handle on a routine basis. In turn, in the FEA an approximate solution is constructed using local admissible functions, defined over small subdomains of the structure. In order to match a given irregular boundary, or to handle parameter non-uniformities, the FEA can change not only the size of the finite elements but also their shape. This extreme versatility, coupled with the development of powerful computer codes based on the method, some of them made available as open source free software, has made the FEA the method of choice for the analysis of structures. In FEA, the equations of equilibrium are obtained from variational principles implying the stationarity of the functional defined by the total potential energy. While solving differential equations with complicated boundary conditions may be difficult, integrating known polynomial functions, even approximately, should be easier. Mathematically, solving [ A ] { x } = { b } is equivalent to 1 minimizing P (x ) = { x }T [ A ]{ x } − { x }T { b }. This is the heart of the FEA when 2 applied to structures.
1. INTRODUCTION
3
1.2 Finite element displacement method
In the finite element modeling, a structure is discretized (hypothetically) into finite elements and points named nodes are selected on the inter-element boundaries or in the interior of the elements. Displacements at the nodes are taken as the primary discrete variables. Displacements within the elements are expressed in terms of these nodal displacements using interpolation functions referred to as shape functions. Finite elements are so small that the shape of the displacement field can be approximated without too much error, leaving only the magnitude to be found. The "shapes" are polynomials, but may be trigonometric functions as well. All individual elements are assembled together in such a way that the displacements are continuous in some fashion across element interfaces, the internal stresses are in equilibrium with the applied loads, and the prescribed boundary conditions are satisfied. Finally, the governing discrete equations are generated by a variational approach. The first part of the finite element modeling process involves choosing the correct and appropriate types of elements, understanding the pedigree of elements and spotting wrong answers due to the use of inadequate elements. The second part of the process is the assembly of the elements and the solution of the complete structural equations. This involves recognizing error messages when this process breaks down or when it simply becomes inefficient because the structure has been modeled inconveniently. The six basic steps of FEA are the following: (1) discretize the continuum, (2) select interpolation functions, (3) find the element properties, (4) assemble the element properties, (5) solve the system of equations, and (6) make additional computations if desired. The three main sources of approximation are: (1) the definition of the domain (physically or geometrically), (2) the discretization of the domain (cutting the corners, making curved lines straight, and curved elements flat), and (3) the solution algorithms. Modeling the joints and the contact between structural parts as well as the damping in dynamic problems are the most difficult tasks. Mesh refinements (and automatic mesh generation) do not bring necessarily increased accuracy. Finer mesh yields also larger stiffness matrices, a larger number of equations to be solved, hence larger computer storage space and running time. Among the reasons why the FEA has gained such universal acceptance are: (1) the routine choice of shape functions, (2) the easiness of producing stiffness matrices (and load vectors), by just assembling predetermined element matrices, and (3) the versatility. Developed originally as a method for analyzing stresses in complex aircraft structures, FEA has evolved into a technique that can be applied
4
FINITE ELEMENT ANALYSIS
to a large variety of linear and nonlinear, static, stability and dynamic engineering problems.
1.3 Historical view
The idea of representing a given domain as a collection of discrete elements is not novel with the finite element method. Ancient mathematicians estimated the value of π approximating the circumference of a circle by the perimeter of a polygon inscribed in the circle. In modern times, the idea found application in aircraft structural analysis, where wings and fuselages are treated as assemblages of stringers, panels, ribs, stiffeners and spars. The use of piecewise continuous functions defined over a subdomain to approximate the unknown function dates back to Courant (1943), who used an assemblage of small triangular elements and the principle of minimum potential energy to study Saint Venant's torsion problem. The reason why Courant's paper did not attract more attention can be attributed to poor timing. In the early 1940s, computers capable of solving large sets of equations of equilibrium did not exist, so that the method was not practical then. The theoretical background of FEA laids on the energy approach of Structural Mechanics and on the approximation techniques. The first energy theorems have been established by Maxwell (1864) and Castigliano (1875). The approximation methods have been developed by Ritz (1908) and Galerkin (1915). Ostenfeld (1926) is credited with the first book on the deformation method. After the Second World War, the Force Method (Flexibility Method) was sustained by Levy (1947) and Garvey (1951) and the matrix Displacement Method (Stiffness Method) was used by Levy (1953) in the sweptback wing analysis. Turner formulated and perfected the Direct Stiffness Method at Boeing (1959). The development of the Force Method ended in 1969. The development of delta wing structures revived the interest in stiffness methods. Modeling delta wings required two-dimensional panel elements of arbitrary geometry. After a first attempt by Levy (1953) with triangular elements, the article series by Argyris in four issues of Aircraft Engineering (1954, 1955), collected later in a book by Argyris and Kelsey (1960), contains the derivation of the stiffness matrix of a flat rectangular panel using bilinear displacement interpolation. But that geometry was inadequate to model delta wings. The formal presentation of the finite element method is attributed to Turner, Clough, Martin, and Topp (1956), who during 1952-1953 succeeded to directly derive the stiffness of a triangular panel at Boeing. The term "finite element" was first used by Clough (1960).
1. INTRODUCTION
5
The first book devoted to FEA was written by Zienkiewicz and Cheung (1967), followed by books by Przemieniecki (1968) and Gallagher (1964). Influential papers have been written by Argyris (1965), Fraeijs de Veubeke (1964) and Irons and coworkers (1964, 1966, 1970). Research developed in the Civil Engineering Department at Berkeley, directed by Clough, at Washington University, under Martin, and at Swansea University, lead by Zienkiewicz. Since 1963, finite element computer programs were freely disseminated into the nonaerospace community. Major contributions are due to B. M. Irons, the inventor of isoparametric models, shape functions, frontal solvers and the patch test (1964-1980), R. J. Melosh, who systematized the variational derivation of stiffness matrices and recognized that FEA is a Rayleigh-Ritz method applied on small size elements (1963), J. S. Archer, who introduced the consistent mass matrix concept (1963), and E. L. Wilson, who studied the sparse matrix assembly and solution techniques (1963), developed the static condensation algorithm (1974) and three SAP computer programs (the first open source FEA software). He was joined later by K.-J. Bathe to develop the finite element codes SAP4 (1973), SAP5 and NONSAP. Starting with 1965 the NASTRAN finite element system was developed by COSMIC, MacNeal Schwendler, Martin Baltimore and Bell Aero Systems under contract to NASA, completed in 1968 and first revised in 1972. Other known finite element codes are ANSYS, developed by Swanson Analysis Systems (1970), STRUDL - by the Civil Engineering Department at Massachusetts Institute of Technology and McDonnell Douglas Automation Company (1967), STARDYNE by Mechanics Research Inc, ADINA – developed by K.-J. Bathe at M.I.T. (1975), SESAM – by Det Norske Veritas, NISA – by Engineering Mechanics Research Corporation, MARC – by Marc Analysis Research Corporation, ABAQUS - by Hibbitt, Karlsson @ Sorensen, Inc. (1978), COSMOS-M – by Structural Research & Analysis Corp. (1985), SAMCEF - by SAMTECH (1965), IDEAS-MS, PATRAN, ALGOR etc. General purpose programs have capabilities of linear dynamic response, including computation of natural frequencies, nonlinear static and dynamic response, crashworthiness, static and dynamic stability, and thermal loading. After 1967 the FEA has been applied to non-structural field problems (thermal, fluids, electromagnetics etc).
loading. Data input can be carried out either in an interactive way, through a userfriendly interface, or reading from a data file. Alternatively, some input data can be imported from other F.E.A. or C.A.D. programs. Automatic mesh generation can be used to produce nodal coordinate data and optimal node numbering, as well as element connectivity data. Mesh plotting is a convenient and useful way of checking the input data. Badly placed nodes or improper blocking of boundary nodes can be easily traced.
Fig. 1.1 Three-dimensional finite element meshes, represented with hidden line removal, are presented for a connecting rod (Fig. 1.1) and a car engine piston (Fig. 1.2), as obtained using the program SIMPAT developed by I.T.I. Italia.
Fig. 1.2
1. INTRODUCTION
7
The finite element model of a vehicle cabin frame obtained with MSC/NASTRAN is shown in Fig. 1.3.
Fig. 1.3 In the processing stage, the finite element program processes the input data and calculates the nodal variable quantities such as displacements and temperatures (equation solving), and element quantities such as stresses and gradients (backcalculation).
Fig. 1.4 In dynamic analyses, processing involves solving an eigenproblem or determining the transient response by incremental techniques. The cost in terms of computer resource increases with the cube of the problem size. In static analyses, the cost of the solution of the linear set of equations increases linearly with the problem size. It would be obviously convenient to use in dynamic analyses the same finite element model that was built for the static analysis. Often this contains much more detail than the dynamic analysis requires, so that condensation and dynamic substructuring are used to reduce the size of the dynamic problem before the processing stage.
8
FINITE ELEMENT ANALYSIS
Postprocessing deals with the presentation of results. Early programs used tabular presentations. Most programs produce displays of the deformed configuration, vibration mode shapes and stress distributions. Fig. 1.4 shows the initial mesh and the deformed shape of a cooling tower under the wind action, as obtained using ALGOR SUPERSAP.
Fig. 1.5 Scalar nodal variables such as temperatures or pressures are presented in the form of contour plots of isotherms or isobars. More recent finite element programs show animated displays of the deformed configuration, as in Fig. 1.5 for a crankshaft.
a Fig. 1.6
b
Fig. 1.6, a shows the two-dimensional initial mesh for the analysis of a gear tooth, with 1174 triangular 6-node elements. Fig. 1.6, b shows the optimized mesh obtained with the postprocessor ESTEREF, containing only 814 elements and a four times reduced global discretization error.
2.
DISPLACEMENT METHOD
In solving any structural problem, there are four types of equations that should be used: equilibrium equations, geometric compatibility conditions, constitutive relationships and boundary conditions. In order to illustrate the usual longhand analytical procedure, a relatively simple pin-jointed framework will be used. Variables include reaction forces at the supports and internal forces, displacements of the bar ends and bar extensions (elongations). If forces and elongations are eliminated and the displacements are the variables which are solved first, the procedure is referred to as the displacement method. It works whether the structure is statically determinate or not. Once the displacements are determined, they are back-substituted into the compatibility equations to obtain bar extensions, and hence strains, then stresses from the constitutive relationship.
2.1 Equilibrium equations
Consider the truss shown in Fig. 2.1 [74], assume that all members are in tension and write the equilibrium of each node in turn. T1 , T2 , T3 are the tensions in members and F1 , F2 , F4 , F5 are the reaction forces at the supports.
The six equations (2.1) to (2.3) have seven unknowns. The system is statically indeterminate. The solution is not possible by using only equilibrium, and consideration must be given to the geometry of deformation.
a
b Fig. 2.2
c
2.2 Conditions for geometric compatibility
The compatibility equations relate a bar extension Δl to the displacements of the ends of the bar. Consider a typical pin-jointed element 1-2 of a frame, Fig. 2.3, inclined an angle θ with respect to the X axis of the global coordinate system. The displacements in the local coordinate frame xOy can be expressed in terms of the displacements in the global coordinate frame as
The three equations (2.6) to (2.8) have nine unknowns, six displacements and three elongations.
2.3 Force/elongation relations
Truss members are in either simple tension or compression. Starting from the Hooke's law for uniaxial stress-strain conditions, the force/elongation relations can be written
Δl12
or
T l = 1 , EA
Δl 13 =
T2
2 l 2 , 2E A
Δl 23 =
T3
2 l 2 2E A
T l T1 l T l , Δl13 = 2 , Δl 23 = 3 . (2.9) EA 2E A 2E A Three more equations have been added and so there are now 12 equations for 16 unknowns.
Δl12 =
12
FINITE ELEMENT ANALYSIS
2.4 Boundary conditions
The discrepancy of 4 equations is made up by adding the boundary conditions
Substituting U 2 and V3 into the remaining equations yields the reaction forces
F1 = −5F , F2 = −4 F , F4 = −5 F , F5 = − F .
(2.13)
Substituting these forces into the equilibrium equations (2.1)-(2.3) yields the tensions in the members, which are equal and opposite to the forces acting on the joints. Divided by the corresponding area they give the stresses.
2.6 Comparison of the force method and displacement method
Navier's Problem. Consider the 7-bar pin-jointed framework shown in Figure 2.4. The joint 8 is subjected to a force of components Fx and Fy . Determine the internal bar forces and the displacement of joint 8.
Force Method
Denoting Ti ( i = 1,...,7 ) the forces applied by each bar to the end nodes, the equilibrium equations of joint 8 can be written
where T1 and T2 are functions of X 1 to X 5 , according to (2.14). Using Menabrea's theorem, the five deformation conditions can be written
∂U = 0, ∂X i
( i = 1,...,5 )
(2.16)
since we are assuming no support movement. They are of the form l1 T1 ∂T1 ∂T + l 2 T2 2 + l i X i = 0 . ∂X i ∂X i
( i = 1,...,5 )
(2.17)
This is a set of five linear equations wherefrom the five redundant forces X 1 to X 5 are determined. The components of the displacement of joint 8 are given by Castigliano's second theorem u8 = ∂U , ∂Fx
v8 = ∂U . ∂Fy
(2.18)
2. DISPLACEMENT METHOD
15
In the force method, the larger the number of bars, the larger the number of statically indeterminate forces, hence the number of equations (2.16).
Displacement Method
which is solved for u8 and v 8 . Regardless the number of concurrent bars, only two equations are obtained for the two joint displacements. In matrix form they can be written
⎧ Fx ⎫ ⎡ K11 ⎨ ⎬=⎢ ⎩ Fy ⎭ ⎣ K 21 K12 ⎤ ⎧ u8 ⎫ ⎨ ⎬ K 22 ⎥ ⎩ v 8 ⎭ ⎦
(2.24)
where the stiffness coefficients are
16
FINITE ELEMENT ANALYSIS
K11 =
EA a
∑ cos2θi sinθi = 1.409
i =1
7
EA , a (2.25)
K12 = K 21 = K 22 = EA a
7
EA a
∑ cosθi sin 2θi = 0,
i =1
7
∑
i =1
sin 3θ i = 4.567
EA . a
Solving (2.24) gives
u 8 = 0.7097
Fx a , EA
v 8 = 0.219
Fy a EA
.
(2.26)
The approach used in the displacement method is the same whether the structure is statically determinate or not.
3.
DIRECT STIFFNESS METHOD
The Finite Element Method (FEM) started as an extension of the stiffness method or displacement method. In the stiffness method for skeletal structures, the elements of the actual structure are connected together at discrete joints. The relationship between the end forces and end displacements of each member is represented by an element stiffness matrix. We may imagine that the structure is built by adding elements one by one, with each element being placed in a preassigned location. As elements are added to the structure, contributions are made to the structure load carrying capacity, hence to the structure stiffness matrix, which relates all joint displacements to all joint forces. If the members are pin-ended bars they are real distinct elements requiring no approximation. They are natural finite elements. Assembly and solution for displacements are of main concern. Starting with simple planar frameworks it is possible to explain the assembly process and to make an introduction into the matrix stiffness method. In the following, the basic steps of the Direct Stiffness Method (DSM) are shown using a pin-jointed plane truss.
3.1 Stiffness matrix for a bar element
In the FEM, the names "joint" and "member" are replaced by node and element, respectively. Consider a two-noded pin-jointed element in the own local coordinate system (Fig. 3.1). It has length l e , cross section area Ae and Young's modulus Ee . Nodes are conveniently numbered 1 and 2. It is acted upon by the nodal forces f1 , f 2 . The nodal displacements are q 1 , q 2 . Generally, bar elements are assumed to be uniform (EAe = const.) , pinconnected at the ends, linearly elastic, axially loaded (no bending) and with no forces between ends.
18
FINITE ELEMENT ANALYSIS
Fig. 3.1 Both the nodal displacements q 1 , q 2 and the nodal forces f1 , f 2 are positive in the positive x direction. The equilibrium equation for the bar element is
3.2 Transformation from local to global coordinates
Bar elements in a truss have different orientations in space and it is necessary to define their stiffness properties with respect to a single global coordinate system attached to the whole structure. End forces and displacements have two components at each node, so that nodal forces and nodal displacements can be arranged into 4-element column vectors related by a 4 × 4 stiffness matrix.
3.2.1 Coordinate transformation
A typical bar element 1-2 is shown in Figure 3.2, where both the local coordinate system xOy and the global coordinate system XOY are drawn. Nodal displacements are denoted by lower case letters in the local coordinate system and by upper case letters in the global coordinate system.
Fig. 3.2 Let the bar be inclined an angle θ e with respect to the X-axis of the global coordinate system. In fact, the angle θ e is the angle between the positive X-axis and the positive direction of the beam (defined as 1 to 2). Displacements in the local coordinate frame xOy can be expressed in terms of the displacements in the global coordinate frame as (2.4)
According to Maxwell's reciprocity theorem, the flexibility matrix δ e must be symmetrical about the leading diagonal. And so must be its inverse, the stiffness matrix.
[ ]
3. DIRECT STIFFNESS METHOD
23
The same result can be obtained following strain energy arguments. For linear structures, displacements are proportional to the applied loads. As forces are increased from zero to their final values, the total work done by these forces is
T 1 Qe Fe . (3.19) 2 In the absence of dynamic effects, this work is absorbed by the structure as strain energy. Substituting (3.15) into (3.19) we obtain the strain energy which is a scalar T 1 Ue = Qe K e Qe . (3.20) 2
We =
{ }{ }
{ } [ ]{ }
e T e T
It is equal to its transpose
Ue = 1 2
{ Q } [K ] { Q }
e
(3.20, a)
so that which defines the symmetry.
[ K ]= [K ]
e
e T
,
3.2.4.2 Singular matrix
The element stiffness matrix is of order 4 and rank 1. The rank deficiency is 3 and this corresponds to the three possible and independent forms of rigid body motion in plane for the unsupported bar: two translations and one rotation. A single ungrounded bar can be moved in space as a rigid body without straining it and hence with zero strain energy. This means that there exist a set of rigid body displacements
e e
e
The zero determinant implies that there are linear relationships between its columns (rows). The rank of a matrix is the size of the largest sub-matrix with a non-zero determinant. One can verify that the determinants of the 3× 3 and 2 × 2 reduced sets are still zero, so that the stiffness matrix has rank 1 (or its rank deficiency is 3).
3.2.4.3 Positive diagonal elements
Each diagonal entry of the matrix K e is positive. If this were not so, a force and its corresponding displacement would be oppositely directed, which is
[ ]
24
FINITE ELEMENT ANALYSIS
physically unsound. Moreover, the matrix K e is positive semidefinite. That is, the quadratic form that represents strain energy (3.20) is either positive or zero.
3.2.4.4 Each column (row) sums to zero
This shows that the first column of the stiffness matrix represents the forces that must be applied to the element to preserve static equilibrium when Qx1 = 1 and all other displacements are zero. The displacement Qx1 = 1 produces an axial shortening Qx1 cosθ e = cosθ e , E A which corresponds to a compressive force e e cosθ e , whose components must le be equilibrated by the external forces k11 , k 21 , k31 and k 41 . Equilibrium of horizontal and vertical forces yields
k11 + k31 = 0 ,
k21 + k41 = 0 ,
3. DIRECT STIFFNESS METHOD
25
so that the first column sums to zero. The same applies for the other columns. In the element stiffness matrix, each column represents an equilibrium set of nodal forces produced by a unit displacement of one nodal degree of freedom.
Fig. 3.5 The truss comprises 3 elements and 3 nodes. It is simply supported in 2 and 3, firmly located in 1, and acted upon by forces 6F and 9F. The global displacements and nodal forces are shown in Fig. 3.6. A node whose global index is i has associated with it the global displacements and forces (2 i − 1) and 2 i .
a
b
26
FINITE ELEMENT ANALYSIS
Fig. 3.6 Element data are given in Table 3.1 together with information useful for the computation. The first three columns define the element connectivities, i.e. their localization within the structure. Element numbering can be arbitrary. Table 3.1
Member
The assembly of the unreduced global stiffness matrix (3.28) is done systematically, locating each coefficient of the element stiffness matrices into the appropriate place in the global 6 × 6 matrix (for this example), as indicated by dots above, eventually adding it to the coefficients already accumulated at that location. This is referred to as the direct matrix method. For instance, element 2 is located in the truss between the left end node i = 1 and the right end node j = 3 (nodal labels i and j may be assigned arbitrarily). In the global matrix, the two displacement and force components (along X and Y) for node i = 1 are numbered 2 i − 1 = 1 and 2 i = 2 , and those for the node j = 3 are numbered 2 j − 1 = 5 and 2 j = 6 . The simple addition of different stiffness coefficients in a location is based on the fact that finite element equations are in fact nodal equilibrium equations, so that if a node is common to several elements, each member will contribute with a force component to maintain equilibrium under an arbitrary set of nodal displacements. An alternative algebraic explanation of the assembly of system stiffness matrices is presented in the following.
28
FINITE ELEMENT ANALYSIS
3.5 Compatibility of nodal displacements
The compatibility of nodal displacements at element level, with the nodal displacements at the whole truss structure level, can be expressed by equations of the form ~ (3.23) Q e = T e { Q },
{ } [ ]
where
e full connectivity or localization matrix, containing ones at the nodal displacements of element nodes and zeros elsewhere.
{ Q } is the element displacement vector in global coordinates, {Q } is the ~ displacement vector of the truss structure and [T ] is referred to as a
e
The expanded element stiffness matrices have been used above only to show algebraically how to assemble a global stiffness matrix; they are never used in practice. The expensive product (3.24) is never formed. The global stiffness assembly is a simple book-keeping exercise and is done by directly placing the nonzero coefficients of element stiffness matrices in the right locations of the global stiffness matrix based on element connectivity.
3.8 Joint force equilibrium equations
The assembly of the global stiffness matrix has been based on strain energy considerations. An alternative presentation is given below, based on joint equilibrium equations, using for convenience the truss from Fig. 3.5. Note that element equilibrium equations (3.1) used so far involved only forces applied by nodes to the elements. The joint equilibrium equations, involving forces applied by elements to nodes, are used in the following. An exploded layout of the truss is shown in Fig. 3.7. Apart from external forces and support reactions, nodes are acted upon by forces equal and in opposite direction to those applied to elements. Equal forces are labelled only once for clarity. Resolving nodal forces horizontally and vertically, we obtain 6 equilibrium equations
The ( N − 1 ) × (N − 1) stiffness matrix above is obtained simply by deleting or eliminating the first row and column from the original ( N × N ) stiffness matrix. Equation (3.32) may be written in condensed form
Stresses can be determined dividing these forces by the element crosssection areas.
3.11 Thermal loads and stresses
Thermal stresses are calculated using the "restraining method" suggested by J. M. C. Duhamel (1838). Suppose the node displacements are completely restrained (blocked). This produces thermal strains ε T = −α T , where α is the coefficient of thermal expansion and T is the amount of uniform heating (temperature difference). The restrained state is equivalent to a pre-stressing with compressive stresses σ T = −α ET , where E is Young's modulus. Restraining produces a compressive axial force α EAT in the element, where A is the cross-section area. Accordingly, the element acts upon its nodes with equal and opposite forces
{ FT } = α EAT b − c
− s c s cT ,
(3.42)
3. DIRECT STIFFNESS METHOD
37
or, in local coordinates,
These forces should be included in the vector of nodal forces (added to the external forces, if the case). After determining the displacements produced by these forces, the element elongations are determined from (3.37) and the stresses are calculated as ⎛ Δl ⎞ (3.43) σ e = Ee ⎜ e − α T ⎟ , ⎜ l ⎟ ⎝ e ⎠ i.e. adding to the stresses produced by the thermal (and external) loads, the initial stresses produced by restraining. It is as if unrestraining forces are applied at the ends of the element to free it from the initial restraining.
Because of symmetry, only the diagonal elements and those from one side of the main diagonal are retained. Even so, due to the sparseness, there are many zero elements in the upper triangle. The half-bandwidth is denoted B.
The first column contains the diagonal elements of the full matrix (3.44). The second column contains the elements from the second diagonal. In general, the mth diagonal of the original matrix is stored as the mth column. The number of columns in the banded-form storage is equal to B - the half-bandwidth of the original matrix. The efficiency of band-storage increases with the order of the matrix.
a
Fig. 3.8
b
For plane trusses, B is equal to 2 plus twice the maximum node number difference in an element. For the numbering scheme of Fig. 3.8, a , B = 6 . However, for the numbering from Fig. 3.8, b, B = 12 , i.e. equal to the size of the original matrix. As a general rule, a small bandwidth can be obtained by numbering nodes along the shorter dimension of a structure, then progressing along the longer dimension.
3. DIRECT STIFFNESS METHOD
39
The half-bandwidth is automatically determined within the finite element program from the node numbering. Apart from storage savings, Gaussian elimination algorithms are used for symmetric banded matrices which enable also the reduction of computer time. For large sparse stiffness matrices, efficient reduction of both storage and computing time can be achieved using the skyline storage and a skyline equation solver. In this case, the columns of the matrix upper triangle are stored serially and concatenated in a column vector { K s } . If there are zeros at the top of a column, only the elements between the diagonal term and the first nonzero term need be stored. The line separating the top zeroes from the first nonzero element is called the skyline. Consider the following matrix
Column height → 1 2 2 0 k 23 k33 4 k14 k 24 k34 k 44 3 0 0 k35 0 k55 1 0 0 0 0 0 k 66 4 0 0 0 k 47 0 k67 k77 5 0 ⎤ ⎥ 0 ⎥ 0 ⎥ ⎥ k 48 ⎥ 0 ⎥ ⎥ 0 ⎥ k 78 ⎥ ⎥ k88 ⎥ ⎦
The height of the jth column is given by ID ( j ) − ID ( j − 1) . For the solution of the finite element equations, Gaussian elimination can be applied using a skyline solver program. A generic flow chart of the Matrix Displacement Method is given in the following. Matrix Displacement Method Input Data Geometric data of truss (Nodal coordinates) Material properties + Cross-section area of members Connectivity table of elements Boundary conditions
↓
Exercises
E3.1. For the truss in Fig. E3.1, a, determine: a) the maximum nodal displacement and its location b) the maximum stress and its location, and c) the support reactions. Plot the deformed shape.
4.
BARS AND SHAFTS
This chapter deals with simple one-dimensional structural elements, having one degree of freedom per node. The displacement within the element is expressed in terms of the nodal displacements using shape functions. The unknown displacement field within an element is usually interpolated by a linear distribution. This approximation becomes increasingly accurate as more elements are considered in the model. For a bar without loads between ends the linear interpolation is exact. The compatibility of adjacent elements requires only C 0 continuity. Displacements must be continuous across the element boundary. For bars with distributed loads, true displacements are described by higher order polynomials. It is shown that their use is tantamount to adding internal nodes. However, it is common practice to use linear shape functions and two-node elements without loads between ends. This implies replacement of the distributed loads by equivalent forces applied to nodes. These kinematically equivalent forces are determined using the appropriate shape functions from the condition to perform the same mechanical work as the actual loading. In this section, the corresponding element stiffness matrix and load vectors will be derived.
4.1 Plane bar elements
Bars are structural elements used to model truss elements, cables, chains and ropes. Their longitudinal dimension is much larger than the transverse dimensions. Bars are loaded only by axial forces. They are modeled by elements having one-degree-of-freedom per node.
4.1.1 Differential equation of equilibrium
In a thin uniform rod of cross-section area A and Young's modulus E, there are axial displacements u = u (x ) due to axial loads p(x ) . The dimensions of p are force/length. The displacement at x + d x will be u + du .
4.1.2 Coordinates and shape functions
Consider a two-node pin-jointed element in the own or local coordinate system. Nodes are conveniently numbered 1 and 2, their coordinates in the physical (Cartesian) reference system being x 1 and x 2 respectively (Fig. 4.2, a). We define a natural or intrinsic reference system which permits the specification of a point within the element by a dimensionless number
r=
x + x2 ⎞ 2 ⎛ ⎜x− 1 ⎟ x 2 − x1 ⎜ 2 ⎟ ⎝ ⎠
(4.5)
so that r = −1 at node 1 and r = +1 at node 2 (Fig. 4.2, b).
4. BARS AND SHAFTS
49
Fig. 4.2 Expressing the physical coordinate in terms of the natural coordinate yields
x = N1 (r ) x 1 + N 2 (r ) x 2 ,
where
N1 (r ) =
(4.6)
1 ( 1 − r ) and N 2 (r ) = 1 ( 1 + r ) (4.7) 2 2 can be considered as geometric interpolation functions. The graphs of these functions are shown in Figs. 4.3, a,b. They have a unit value at the node of the same index and zero at the other node.
Fig. 4.3
4.1.3 Bar not loaded between ends
For a prismatic bar not loaded between ends, p = 0 , d 2u d x 2 = 0 , d u d x = const . , so that the displacement field within the element may be expressed as a linear polynomial
The polynomial form (4.6) is simpler, but the integration constants, a and b, have no simple physical meaning. The nodal expansion (4.10) is more complicated, but the integration constants, q 1 and q 2 , are the nodal displacements. In matrix form
u=
where
∑ N q = ⎣N ⎦ { q },
e i i i =1
2
(4.11)
⎣N ⎦ = ⎣N1 N 2 ⎦ and
{q }= { q
e
1
q2
}T .
(4.12)
Thus, the displacement at any point within an element can be found by multiplying the matrix of shape functions by the vector of nodal displacements.
⎣N ⎦
In (4.11), q e is the column vector of element nodal displacements and is the row vector of displacement interpolation functions also named shape
{ }
4. BARS AND SHAFTS
51
functions. It is easy to check that u = q 1 at node 1 and u = q 2 at node 2, and that u
varies linearly (Fig. 4.3, c). Equations (4.6) and (4.10) show that both the element geometry and the displacement field are interpolated using the same shape functions, which is referred to as the isoparametric formulation.
4.1.4 Element stiffness matrix in local coordinates
Strains can be expressed in terms of the shape functions as
εx =
where
du d e e = ⎣N ⎦ q = ⎣B ⎦ q dx dx
{ }
{ }
(4.13)
⎣B ⎦ = d x ⎣N ⎦
d
(4.14)
is the row vector of the derivatives of shape functions, generally called the element strain-displacement matrix. It gives the strain at any point due to unit nodal displacement. The transformation from x to r in equation (4.5) yields
dx = x 2 − x1 2 dr = le dr , 2
The three integration constants, a, b and c, have to be determined from three boundary conditions. This can be done if we use a three-node onedimensional element. An internal node is added at the midpoint to comply with the requirement of a quadratic fit (Fig. 4.4, a).
4. BARS AND SHAFTS
53
For a linearly distributed axial load (as in a bar rotating at constant angular speed around an end, acted upon by a distributed centrifugal load proportional to the distance to the rotation centre), the true displacement field is given by a cubic polynomial, involving four integration constants (see Example 4.8). For an exact solution, this implies using a four-node element (adding two internal nodes). The usual practice is to assume an approximate lower order linear displacement field, i.e. a two-node element and to replace the actual linearly distributed load by equivalent nodal forces, having thus an element not loaded between ends describable by linear shape functions.
The graphs of the shape functions (4.26) are shown in Fig. 4.5. They have a unit value at the node with the same index and zero at the other nodes. This is a general property of the shape functions.
54
FINITE ELEMENT ANALYSIS
The expressions for these shape functions can be written down by inspection. For example, since N1 = 0 at r = 0 and r = 1 , we know that N1 has to contain the product r ( 1 − r ) , i. e. the left hand part of the equations of the vertical lines passing through nodes 3 and 2. That is, N1 is of the form N1 = C r ( 1 − r ) . The constant C is obtained from the condition N1 = 1 at r = −1 , which yields C = −1 2 , resulting in the expression given in (4.26).
Fig. 4.5 The displacement field within the element is written in matrix form as u= where
∑ N q = ⎣N ⎦ { q } ,
e i i i =1
3
(4.27)
⎣N ⎦ = ⎣N1 N 2 N3 ⎦ and
{ q }= { q
e
1
q2
q3
}T .
(4.28)
At any point within an element the axial displacement can be found by multiplying the matrix of shape functions by the vector of nodal displacements, as in (4.27). It is easy to check that u = q 1 at node 1, because N1 = 1 and
4.1.7 Assembly of the global stiffness matrix and load vector
Assembly of the system stiffness matrix for one-dimensional structures modelled as bars is carried out as shown in sections 3.4 to 3.7 for trusses. Consider the five-node finite element model in Fig. 4.7, a. Each node has only one degree of freedom in the x-direction. The nodal displacements are Q1 , Q2 ,…, Q5 (Fig. 4.7, b). The global vector of nodal displacements is denoted by
so that, as in (3.27), the global stiffness matrix is equal to the sum of the expanded element stiffness matrices ~ [ K ]= K e .
∑[
e
]
the appropriate locations of the global [ K ] matrix by the so-called direct method, based on element connectivity. Overlapping elements are simply added as already shown in section 3.4 for truss elements. This is based on the simple addition of element strain energies. The element matrices can be written
This is a convenient algebraic explanation of the assembly process, which is never done in practice. The entries of the element matrices k e are placed in
[ ]
1 1 [ k ] = ElA ⎡⎢− 1
1 1 1
2 − 1⎤ 1 , 1⎥ 2 ⎦ 4
2
3
⎣
[ k ] = ElA
2 2
2
⎡ 1 − 1⎤ 2 ⎢ − 1 1⎥ 3 , ⎣ ⎦ 4 5 ⎡ 1 − 1⎤ 4 . ⎢− 1 1⎥ 5 ⎣ ⎦
3
[ k ] = ElA
3 3
3
⎡ 1 − 1⎤ 3 , ⎢− 1 1⎥ 4 ⎣ ⎦
[ k ] = ElA
4 4
4
At the top and on the right of the element stiffness matrices, the numbering of coordinates in the global stiffness matrix is shown, according to the connectivity Table 4.1. Table 4.1 Element 1 2 3 4 Node i 1 2 3 4 j 2 3 4 5
The final finite element equations are obtained, as shown in section 3.5, using the boundary condition, Q 1 = 0 . The reduced (non-singular) stiffness matrix is obtained deleting the first row and column of the unreduced matrix, and the reduced load vector - by deleting the first element. In fact, this information is stored for the subsequent calculation of the reaction R 1 . Stresses are then calculated from the axial forces obtained using equation (3.41).
4.1.8 Initial strain effects
Let an initial strain ε 0 be induced in a bar element. It may arise from thermal action or by forcing members into place that are either too short or too long, due to fabrication errors. The stress-strain law in the presence of ε 0 is of the form
σ = E ( ε − ε0 ) .
(4.38)
The mechanical work of external nodal forces applied to suppress the initial prestressing due to ε 0 is W=
Ve
∫σε
0 dV
=
Ve
∫σ
T
ε 0 dV = ε 0 Ee Ae
le
∫ε
T
dx .
(4.39)
Substituting (4.13), the expression (4.39) becomes
60
FINITE ELEMENT ANALYSIS
W = qe
{ }ε
T
0 Ee Ae
le
∫ ⎣B⎦
T
dx .
(4.40)
It has the form (4.33), where the element load vector due to initial straining is
{ f }= ε
e
0 Ee
Ae
le
∫
⎣B ⎦
T
l dx = ε 0 Ee Ae e 2
T
+1 −1
∫ ⎣B⎦
+1
T
dr
(4.41)
or
{ f }= ε
e
+1 0 Ee Ae
−1
∫
⎢dN ⎥ le ⎢ ⎥ d r = ε 0 Ee Ae 2 ⎣ dr ⎦ ⎧ − 1⎫ Ae ⎨ ⎬ . ⎩ 1 ⎭
−1
∫
⎧− 1⎫ ⎨ ⎬ dr , ⎩1⎭
hence
{ f }= ε
e
0 Ee
(4.42)
After solving for nodal displacements, stresses are computed as
σ e = Ee
q2 − q1 + ( − Ee ε 0 ) . le
(4.43)
In the case of thermal loading,
ε0 = αT ,
(4.44)
where α is the coefficient of thermal expansion and T is the average change in temperature within the element.
4.2 Plane shaft elements
From Mechanics of Materials it is known that a uniform shaft of diameter d and length l , from a material with shear modulus of elasticity G, acted upon by a torque M t will twist an angle θ = πd4 Mt l , where I p = is the polar second GIp 32
moment of area of the shaft cross section. The shaft torsional stiffness is then GIp M K= t = . θ l A two-node shaft finite element, of length l and torsional rigidity G I p , is shown in Fig. 4.8. The nodal torques M 1 and M 2 can be related to the nodal rotation angles θ 1 and θ 2 using the equilibrium and the torque/rotation equations
Fig. 4.8 The derivation of the shaft stiffness matrix is essentially identical to the derivation of the stiffness matrix for an axially loaded bar element. Similarity between these two derivations occurs because the differential equations for both problems have the same mathematical form. The differential equation for torsional displacement is Mt dθ , (4.48) = dx GI p while for the axial displacement is (4.3) du N = . (4.49) dx E A The rotation angle of an arbitrary section within the shaft element can be expressed in terms of the nodal rotations θ 1 and θ 2 as
θ = N1 (r ) θ1 + N 2 (r ) θ 2 .
(4.50)
62
FINITE ELEMENT ANALYSIS
where
N1 (r ) =
1 ( 1 − r ) and N 2 (r ) = 1 ( 1 + r ) 2 2
(4.51)
are the shape functions of the shaft element, the same as for the axially loaded bar element. Analogous to equation (4.20), the stiffness matrix for the shaft element can be calculated from
[ k ] = G I ∫ ⎢⎢ ddN ⎥⎥ ⎣ x⎦
e e pe le
T
⎢dN ⎥ ⎢ dx ⎥ dx ⎣ ⎦
T
(4.52)
or
[ ke ] =
which yields
e
2 Ge I p e
le
e pe
+1
−1
∫
⎢dN ⎥ ⎢dN ⎥ ⎢ dr ⎥ ⎢ dr ⎥ dr ⎣ ⎦ ⎣ ⎦
(4.53)
[ k ] = G lI
e
⎡ 1 − 1⎤ ⎢− 1 1⎥ ⎣ ⎦
(4.54)
Equation (4.54) is also used to account for torsional effects in grid finite elements, as equation (4.22) is used to account for axial effects in inclined beam finite elements. For non-axially-symmetric cross sections, the polar second moment of area I p is replaced by the torsional constant I t .
Example 4.5
A steel bolt of active length l = 100 mm and diameter δ = 10 mm is single threaded with a 1.6 mm pitch. It is mounted inside a copper tube with diameters d = 12 mm and D = 18 mm (Fig. E4.5, a). After the nut has been fitted smugly, it is tightened one-quarter of a full turn. Determine stresses in bolt and tube, if for steel E1 = 208 GPa and for copper E2 = 100 GPa . Solution. The assembly is modeled by two bar finite elements as in Fig. E4.5, b. Both elements have fixed ends at points 1 and 4 so that Q1 = Q4 = 0 . The problem has a multipoint constraint
where the left hand side contains the (2 × 2) stiffness matrix of the two-node linear element. b) Consider the bar modeled by a 3-node quadratic bar element. The finite element equations can be written
We may conclude that the introduction of the quadratic term in equation (4.23) does not bring about a change in the conventional stiffness matrix and load vector. Whenever the assumed functions, used to describe the displacement field, form the complete homogeneous solution of the differential equation of equilibrium (4.4), the developed stiffness matrix and the equivalent load vector will be exact. This is because, as it is shown in a next chapter, only the homogeneous part of the solution contains the free parameters with respect to which the total potential energy is minimized. The parameters in the particular part of the solution are prescribed and do not take part in the process of minimization. The exactness of the stiffness matrix and load vector also implies that the computed nodal displacements will also be exact. However, displacements within elements depend upon the general (homogeneous plus the particular) solution. The conventional formulation based on a linear polynomial will yield exact displacements within the elements only when p = 0 . For the case p = const. , exact displacements within the elements may be obtained from equation (4.25). However, before using equation (4.25), the variable Q3 must be computed from the exact nodal displacements (computed for the conventional linear element), via a constraint equation between Q3 and the remaining nodal variables, obtained by a minimization of the total potential energy with respect to Q3 at element level.
The finite element values of the nodal displacements are exact. This is due to the nodal equivalence of forces. While the nodal displacements are exact, the displacements within the elements are approximate because the exact cubic distribution (a) has been replaced by a quadratic law. The element strain-displacement row vector ⎣B ⎦ in (4.29) is given by
The axial load acting on the three elements can be decomposed as in Fig. E4.8, g. The nodal forces, equivalent to a load per unit length, are given by equation (4.34). If p1 and p2 are the intensities of a linearly distributed load at nodes 1 and 2, respectively,
Frames are structures with rigidly connected members called beams. Beams are slender members used to support transverse loading. They are connected by rigid joints that have determinate rotations and, apart from forces, transmit bending moments from member to member. One-dimensional mathematical models of structural beams are constructed on two beam theories: the Bernoulli-Euler beam theory, that neglects transverse shear deformations, and the Timoshenko beam theory, that incorporates a first order correction for transverse shear effects. Beam behaviour is described by fourth order differential equations and require C1 continuity. This requires that both transverse displacements and slopes must be continuous over the entire member and, in particular, between adjacent beam elements. The Timoshenko beam model pertains to the class of C 0 elements. It is based on the assumption that plane sections remain plane but not necessarily normal to the deformed neutral surface. This leads to the introduction of a mean shear distortion, which is constant over the element. In this section, we first present the finite element formulation for plane Bernoulli-Euler beams, then extend it to plane frames, and for grids. An inclined beam element will be referred to as a frame element. Grids are planar frames subjected to loads applied normally to their plane.
5.1 Finite element discretization
A plane frame is divided into elements, as shown in Fig. 5.1. Each node has three degrees of freedom, two linear displacements and a rotation. Typically, the degrees of freedom of node i are Q3 i − 2 , Q3 i −1 and Q3 i , defined as the displacement along the X axis, the displacement along the Y axis and the rotation about the Z axis, respectively.
80
FINITE ELEMENT ANALYSIS
Nodes are located by their coordinates in the global reference frame XOY and element connectivity is defined by the indices of the end nodes. Elements are modelled as uniform beams without shear deformations and not loaded between ends. Their properties are the bending rigidity E I and the length l .
Fig. 5.1 In the following, the shape functions are established for the plane Bernoulli-Euler beam element, then the element stiffness matrix is calculated first in the local coordinate system, then in the global coordinate system. The latter are expanded to the structure size, then simply added to get the global uncondensed stiffness matrix. Imposing the boundary conditions, the reduced stiffness matrix and load vector are calculated and used in the static analysis.
Fig. 5.2
5. BEAMS, FRAMES AND GRIDS
81
Consider an inclined beam element, as illustrated in Fig. 5.2, a, where the nodal displacements are also shown. In a local physical coordinate system, the x axis, oriented along the beam, is inclined an angle θ with respect to the global X axis. Alternatively, an intrinsic (natural) coordinate system can be used. The vector of nodal displacements in the local coordinate system is
5.2 Static analysis of a uniform beam
Beams with cross sections that are symmetric with respect to the plane of loading are considered herein (Fig. 5.3). Transverse shear deformations are neglected, as in the Bernoulli-Euler classical beam theory. Only transverse loads act upon the beam, axial forces are ignored.
Fig. 5.3 The axial displacement of any point on the section, at a distance y from the neutral axis, is approximated by dv u = −ϕ y = − y, (5.3) dx
82
FINITE ELEMENT ANALYSIS
where v is the deflection of the centroidal axis at x and ϕ = v′ is the cross section rotation (or slope) at x . Axial strains are
εx =
du d2 v = − 2 y = −χ y , dx dx
(5.4)
where χ ≈ v′′ denotes the deformed beam axis curvature. Normal stresses on the cross section are given by Hooke's law d2 v y, dx2 where E is Young's modulus of the material.
σ x = E ε x = −E
(5.5)
section
The bending moment is the resultant of the stress distribution on the cross
M (x ) = − σ x y dA = E I z
∫
d2 v dx2
= EIz χ
(5.6)
A
where I z is the second moment of area of the section with respect to the neutral axis z. The negative sign above is introduced because M is considered positive if it compresses the upper portion of the beam cross section. The product E I z is called the bending rigidity of the beam. The shear force is given by dM d3 v = −E I z = − E I z v III . dx dx 3 dT d4 v = EIz = E I z v IV . dx dx4
T (x ) = −
(5.7)
The transverse load per unit length is
p (x ) = −
(5.8)
The differential equation of equilibrium is
EIz
d4 v = p (x ) . dx4
(5.9)
This is a fourth order differential equation and consequently four boundary conditions are required, two at each end. They can be geometric or kinematic boundary conditions, involving the transverse displacement and slope, and physical boundary conditions, involving the shear and bending moment.
5. BEAMS, FRAMES AND GRIDS
83
5.3 Uniform beam not loaded between ends
For a uniform beam not loaded between ends, p = 0 and equation (5.9) yields d 4 v d x 4 = 0 . Integrating four times, we obtain the deflection v described by a third order polynomial
On inversion, the integration constants a 1 , a 2 , a 3 , a 4 can be expressed in terms of the nodal displacements v1 , ϕ1 , v2 , ϕ 2 , so that the beam deflection can be expressed in terms of the nodal displacements.
5.3.1 Shape functions
The transverse displacement v can be expressed in terms of the nodal displacements as
84
FINITE ELEMENT ANALYSIS
v (x ) = ⎣N ⎦ q e ,
{ }
(5.12)
where ⎣N ⎦ is a row vector containing the shape functions, called Hermitian cubic polynomials, and
This shows that the first column of the stiffness matrix represents the forces and moments that must be applied to the beam element to preserve static equilibrium when q2 = 1 and all other displacements are zero. For equilibrium
k11 + k31 = 0 ,
k 21 + k 41 + k31 l = 0 .
(5.36)
5. BEAMS, FRAMES AND GRIDS
89
5.4 Uniform beam loaded between ends
For a uniform beam loaded between ends, p ≠ 0 , d 4 v d x 4 ≠ 0 in equation (5.9) and the beam deflected shape is no more a cubic polynomial. However, it is the homogeneous solution of the differential equation. Using cubic polynomials as admissible functions, the computed nodal displacements are exact. Within the elements, the displacements, moments and shear forces are in error. When the transverse load is uniformly distributed, p = const . , the general solution of equation (5.9) is a quartic polynomial. The corresponding five constants have to be determined from five boundary conditions. An internal node added at the centre of element will solve the problem, introducing its nodal displacement as the fifth nodal coordinate. For a linearly distributed transverse load, v (x ) will be a quintic with six arbitrary constants. They can be determined, adding the transverse displacement and slope at the element midpoint to the element nodal coordinates. As already shown in Chapter 4, rising the power of the function describing the displacement within a beam element is tantamount to introducing additional internal nodes. However, the current practice is to use lower order approximate assumed shape functions that ensure the minimal convergence requirements, as shown in the following. The cubic shape functions do the job. But they can be used only if the element has uniform rigidity E I z and is not loaded between nodes. For beams with transverse forces, the solution is to replace the actual distributed load by equivalent nodal forces.
5.4.1 Consistent vector of nodal forces
Consider a transverse load p (x ) , having the units of force per unit length, distributed along the beam element. The mechanical work of such a force is W=
Substituting (5.12), equation (5.37) becomes W = qe It has the form
le
∫ v p dx = ∫ v
le
T T
T
p dx .
(5.37)
{ } ∫ ⎣N ⎦
le
p dx .
(5.38)
W = qe
where the element load vector is
{ } {f }
T e
(5.39)
90
FINITE ELEMENT ANALYSIS
{ f } = ∫ ⎣N ⎦
e le
T
l p ( x ) dx = e 2
+1
−1
∫ ⎣N ⎦
dr
T
p (r ) d r .
(5.40)
For the Hermitian two-node element, if the transverse force is uniformly distributed, p = const . , the vector of element consistent nodal forces is
{f }
e
l = e p 2 ⎢ pl =⎢ e ⎢ 2 ⎣
+1 −1
∫ ⎣N ⎦
T
(5.41)
or, substituting (5.18),
{f }
e
p l2 e 12
p le 2
p l2 ⎥ e − ⎥ . 12 ⎥ ⎦
T
(5.42)
In Fig. 5.7, a it is seen that f 2e is a shear force and f 3e is a moment. They are called "kinematically equivalent" nodal forces since they replace a distributed load p (r ) weighted with the shape functions N i (r ) so that the correct work is simulated.
a
b
c Fig. 5.7
d
The kinematically equivalent loads are those which, if applied in the opposite direction as constraints, would keep all nodal displacements zero in the presence of the true loading. To replace p = const . by statically equivalent forces (Fig. 5.6, b) would be incorrect since the beam element has the ends rigidly jointed,
5. BEAMS, FRAMES AND GRIDS
91
i.e. it is rigidly built in the adjacent beam elements. In order to ensure the C1 continuity across elements, nodal forces must include moments, not only shear forces. Equivalent nodal forces for linearly distributed loads are given in Figs. 5.6, c and d. Kinematically equivalent loads yield displacements which do not coincide with those produced by actual loading, as shown in Example 5.1. Assuming approximate deflected shapes instead of the true ones may be imagined as the result of application of a fake loading, forcing the beam to maintain the approximate deflection. This is equivalent to applying additional constraints to the beam, i.e. stiffening it. The deflections of this over-stiff finite element model are smaller "in the mean" than the true deflections of the actual structure. The source of error comes from the arbitrary selection of the shape functions. Even if these functions are built up to satisfy the geometric boundary conditions at the ends, the equilibrium within the elements is broken, due to the difference between the applied load p (x ) and the resistance E I z v IV which gives rise to a sort of unbalanced residual force. The smaller the element, the smaller the error, so we would expect to increase the accuracy by increasing the number of elements modeling the same structure, or refining the mesh. A correct solution will approach the true value with monotonically increasing values of displacements. The finite element solution is therefore referred to as a lower bound. This applies only to the strain energy and not to the displacement or stress at a point. Local stresses may be higher than the true ones. Assuming cubic displacement functions implies linearly varying bending moments (and hence stresses) in uniform beams, even if it is known that, for uniform loading, for instance, they have a quadratic distribution.
Example 5.1
Calculate the transverse displacement at the centre of the simply supported beam shown in Fig. E5.1.
which shows that the element has indeed an anticlockwise rotation as a rigid body.
5. BEAMS, FRAMES AND GRIDS
97
a
b Fig. 5.9
c
2. An element should simulate constant strain states. In the case of beams, when element sizes shrink to zero, they must have at least constant curvature. Assuming zero nodal vertical displacements and unit rotations in opposite directions (Fig. 5.9, c)
5.6 Frame element
As shown in Fig. 5.2, an inclined beam element should include longitudinal displacements since, apart from moments and shear forces, it is acted upon by axial forces. Because there is no coupling between the bending and stretching displacements, the two stiffness matrices can be added taking into account the proper location of their elements.
, where 'i' is the relevant radius of gyration. For slender beams, this ratio may be as small as 1 20 or 1 50 , so the stiffness matrix may possibly be numerically ill-conditioned. If there is a uniformly distributed load on a member, the vector of consistent nodal forces is
(i l )
2
The ratio of the bending terms to the stretching terms in (5.54) is of order
{ f } = ⎢0 ⎢
e
⎢ ⎣
p le 2
p l2 e 12
0
p le 2
−
p l2 ⎥ e ⎥ . 12 ⎥ ⎦
T
(5.55)
5.6.3 Coordinate transformation
A frame element is shown in Fig. 5.10 both in the initial and deformed state. For node 1, the local linear displacements q 1 and q 2 are related to the global linear displacements Q 1 and Q 2 by the equations
The equations of equilibrium can be regarded as having been derived as soon as the reduced stiffness matrix and load vector have been calculated using the boundary conditions. Having solved [ K ] { Q } = { F } it is routine to backtrack and recover the eth element strains, using equations (5.4), (5.12) and (5.60)
ε e = − y v′′ = − y ⎣N ′′⎦ q e = − y ⎣N ′′⎦ T e
{ }
[ ] { Q }.
e
(5.67)
5. BEAMS, FRAMES AND GRIDS
101
Strains, and hence stresses, are not accurate. Strains are derivatives of an approximate displacement (in beams - second derivatives) and differentiation inevitably decreases accuracy.
Example 5.2
Calculate the transverse displacement at the free end of the cantilever stepped beam shown in Fig. E5.2.
5.8 Grids
Grids or grillages are planar structural systems subjected to loads applied normally to their plane. They are special cases of tree-dimensional frames in which each joint has only three nodal displacements, a translation and two rotations, describing bending and torsional effects.
Fig. 5.11
5.8.1 Finite element discretization
The grid is divided into finite elements, as shown in Fig. 5.11. Each node has three degrees of freedom, two rotations and a linear displacement. Typically, the degrees of freedom of node i are Q3 i − 2 , Q3 i −1 and Q3 i , defined as the rotation about the X axis, the rotation about the Y axis and the displacement along the Z axis, respectively. Nodes are located by their coordinates in the global reference frame XOY and element connectivity is defined by the indices of the end nodes. Elements are modelled as uniform rods with bending and torsional flexibility, without shear deformations and not loaded between ends. Their properties are the flexural rigidity E I , the torsional rigidity G I t and the length l . Only cross sections whose shear centre coincides with the centroid are considered.
5.8.2 Element stiffness matrix in local coordinates
Consider an inclined grid element, as illustrated in Fig. 5.12, a, where the nodal displacements are also shown.
112
FINITE ELEMENT ANALYSIS
In a local physical coordinate system, the x axis, oriented along the beam, is inclined an angle α with respect to the global X axis. The z axis for the local coordinate system is collinear with the Z axis for the global system. Alternatively, an intrinsic (natural) coordinate system can be used.
The axial nodal forces f 1 , f 4 are torques and the nodal displacements
q 1 , q 4 are twist angles. They describe torsional effects so that their action is
decoupled from bending. The respective stiffness matrix can be calculated
5. BEAMS, FRAMES AND GRIDS
113
separately. The derivation of this matrix is essentially identical to the derivation of the stiffness matrix for axial effects in a frame element or in a truss element. The twist angle can be expressed in terms of the shape functions (4.51) as
In (5.75), G is the shear modulus of elasticity and I t e is the torsional constant of the cross section. For axially symmetrical cross sections the latter is the polar second moment of area. For the grid element, combining the stiffness matrices from equations (5.70) and (5.75), we get the stiffness matrix in local coordinates relating the nodal forces (5.69) and the nodal displacements (5.68)
5.8.3 Coordinate transformation
It is necessary to transform the matrix (5.76) from the local to the global system of coordinates before its assemblage in the stiffness matrix for the complete grid. As has been indicated, the z direction for local axes coincides with the Z direction for the global axes, so that only the rotational components of displacements should be converted. The transformation of coordinates is defined by equation
where c = cos α and s = sin α , is the local-to-global coordinate transformation matrix. The same transformation matrix (5.78) serves to transform the nodal forces from local to global coordinates.
5.8.4 Element stiffness matrix in global coordinates
Using the same procedure as for frame elements, we obtain the stiffness matrix of the grid element in global coordinates as
[ K ]= [T ] [ k ][T ] .
e e T e e
(5.79)
It is used to assemble the unreduced global stiffness matrix [ K ] using ~ element connectivity matrices T e that relate the nodal displacements at element level with the nodal displacements at the complete structure level, by equations of the form (5.64).
[ ]
5. BEAMS, FRAMES AND GRIDS
115
For grounded systems the unreduced matrix [ K ] is then condensed using the boundary conditions. The effect of lumped springs can be accounted for by adding their values along the main diagonal at the appropriate locations in the global stiffness matrix.
Answer. The grid is modeled with 5 elements and 5 nodes, having 9 dof's. The largest displacement is w5 = − 0.4 m . The deflected shape is presented in Fig. E5.11, b. The bending moment and torque diagrams are shown in Figs. E5.11, c, d.
5.9 Deep beam bending element
Shear deformation becomes important when analyzing deep beams, for which Bernoulli's hypothesis is no more valid. The nonlinear distribution of shear stresses produces the warping of the cross section. A simplifying hypothesis (Bresse, 1859) considers an average shear strain, constant over the cross section. This way, planar cross sections remain undistorted and plane (warping neglected) but no more perpendicular to the centroidal axis. The assumption is adopted in the formulation of the Timoshenko beam element used in vibration studies.
5. BEAMS, FRAMES AND GRIDS
117
5.9.1 Static analysis of a uniform beam
Beams with cross sections that are symmetric with respect to the plane of loading are considered herein (Fig. 5.13, a). Only transverse loads act upon the beam, axial forces are ignored.
Fig. 5.13 The axial displacement of any point on the section, at a distance y from the neutral axis, is approximated by
u (x, y ) = − y ϕ (x ) ,
where ϕ is the cross section rotation at position x . The strain components ε x and γ xy are given by
(5.80)
εx =
du dϕ = −y = − yϕ ′ , dx dx
(5.81) (5.82)
γ xy =
∂u ∂ v dv + = −ϕ + = −ϕ + v′ , ∂ y ∂x dx
where v′ is the slope of the deformed beam axis. Note that the slope v′ is no more equal to the rotation ϕ , as in the Bernoulli-Euler theory. Normal stresses on the cross section are given by Hooke's law
dϕ y, dx where E is Young's modulus of the material.
σ x = E ε x = −E
(5.83)
The bending moment is the resultant of the normal stress distribution on the cross section
118
FINITE ELEMENT ANALYSIS
M (x ) = − σ x y dA = E I z
A
∫
dϕ = E I zϕ′ dx
(5.84)
where I z is the second moment of area of the cross section. The sign convention used here (Fig. 5.13, b) is that positive internal forces and moments act in positive (negative) coordinate directions on beam cross sections with a positive (negative) outward normal. The shear force is given by T (x ) = − dM . dx dT . dx (5.85)
The transverse load per unit length is p (x ) = − (5.86)
The average shear strain is
γ xy =
T T = G As κ AG
(5.87)
where G is the shear modulus of elasticity, κ is a shear factor and As is the 'effective shear area' calculated as
As
[ ∫τ dA ] =
∫τ
2
2
dA
.
(5.88)
Equations (5.82) and (5.87) give
T = G As ( v′ − ϕ ) .
(5.89)
For a uniform beam not loaded between ends ( p = 0 ) , elimination of M and T gives the differential equations of equilibrium
Substituting the shape functions (5.102) and (5.105) and performing the integration yields the stiffness matrix
[k ]
e
⎡ 12 ⎢ 1 EI z ⎢ 6l = 1 + 3β l 3 ⎢− 12 ⎢ ⎣ 6l
(4 + 3β ) l
− 6l
6l
− 12
2
(2 − 3β ) l 2
⎥. ⎥ 2⎥ − 6l (4 + 3β ) l ⎦ 12 − 6l
− 6l
(2 − 3β ) l
6l
2⎥
⎤ (5.114)
The vector of consistent nodal forces is identical to the corresponding vector (5.42) derived for a slender beam.
6.
LINEAR ELASTICITY
In this chapter the fundamental concepts from the linear theory of elasticity are recalled, with emphasis on two-dimensional problems. The four main groups of equations are written in the matrix notation used in FEA: (a) equations of equilibrium, (b) equations of compatibility or strain/displacement relations, (c) stress/strain relations or Hooke's law, and (d) boundary conditions.
6.1 Matrix notation for loads, stresses and strains
An arbitrarily shaped tree-dimensional body of volume V, in equilibrium under the action of external loads and the reactions in supports, is shown in Fig. 6.1. The total surface S of the body has two distinct parts: S u , the portion of the boundary on which displacements are prescribed, and S σ , the portion on which surface forces are prescribed. Points in the body are located by x, y, z coordinates. Any point on the surface has a local outward-pointing normal n whose orientation is usually described by its three direction cosines ∂n ∂x , ∂n ∂y , ∂n ∂z . In general there may be three sets of applied forces: (a) internal body forces, (b) surface forces, and (c) concentrated forces. Internal body forces Internal body forces inside the volume V can be inertial forces, like centrifugal or gravity forces. Their the magnitude per unit volume is denoted by components pv x , pv y , pv z . It is convenient to write these components as a single body force vector
{ pv } = ⎣ pv x
pv y
pv z ⎦ T .
(6.1)
124 Surface tractions
FINITE ELEMENT ANALYSIS
Likewise there could be surface forces (not necessarily normal pressures) on the surface Sσ , defined by the magnitude per unit surface area, also having three components
{ ps } = ⎣ ps x
Concentrated loads
ps y
ps z
⎦T ,
(6.2)
Concentrated loads are defined by their three components
{ Fi } = ⎣ Fi x
Fi y
Fi z
⎦T .
(6.3)
Any system of loads has to fall into categories (6.1) to (6.3).
Fig. 6.1 Displacements It is natural to form the single displacement vector
{ u} = ⎣u
v w⎦ T ,
(6.4)
where u , v , w are the displacement components inside the body or on the surface Sσ with unprescribed displacements. Stresses and strains
The stresses inside V will have two types of component, the direct stress components σ x , σ y ,σ z and the shear stresses τ xy ,τ yz ,τ zx . It is convenient to represent both stress and strain components as single column matrices. Thus
7.
ENERGY METHODS
The finite element method can be considered a Rayleigh-Ritz method. The classical Rayleigh-Ritz technique represents a variational approach whereby a distributed system is approximated by a discrete one by assuming a solution of the differential boundary-value problem as a finite series of admissible functions. Unfortunately, systems with complex geometry or complex boundary conditions cannot be accomodated easily by global admissible functions. In the finite element method, the approximate solution is constructed using local admissible functions, defined over small subdomains of the structure. Good approximations can be realized with low-degree polynomials. Displacements are calculated by methods based on the principle of virtual work and/or the principle of minimum total potential energy. Instead of solving differential equations with complicated boundary conditions, the finite element method evaluates integrals of relatively simple polynomial functions. Variational methods put less strict conditions on the functions approximating the displacement field than the analytical methods based on differential equations.
7.1 Principle of virtual work (PVW)
PVW is basically a statement of the static equilibrium of a mechanical system. In the following, the form known as the principle of virtual displacements (PVD) will be used, as applied to elastic bodies.
c) not related to either the actual displacements or to the forces producing them; d) continuous in the interior and on the surface of the body; A continuity C 0 is generally required for bars and elasticity problems, while a continuity C1 is imposed for beams, plates and shells. Exceptions do exist. Remember that a function of several variables is said to be of class C m in a domain V if all its partial derivatives, up to the mth order inclusive, exist and are continuous in the domain V. e) kinematically admissible, i.e. consistent with the system kinematic boundary conditions (geometric constraints). If the differential equation of the problem is of order m = 2n , the
admissible functions must have continuity C n −1 , i.e. the geometrical boundary conditions must be satisfied to the (n − 1)th derivative. For bars, m = 2 and the
assumed functions must have continuity C 0 . For beams m = 4 and the approximating functions must have continuity
C . Because the continuity required is reduced from C 2 in the governing
1
differential equation to C1 in the variational equation, the functional is said to have a "weak form". A virtual displacement will be denoted by ' δ ' in front of a letter, e.g. δu . The symbol ' δ ' was introduced by Lagrange to emphasize the virtual character of the variations, as opposed to the symbol d which designates actual differentials of position coordinates. Denoting by
{ u} = ⎣u
v w⎦ T ,
the displacement vector (6.4) inside the body or on the surface Sσ with unprescribed displacements (Fig. 6.1), the vector of virtual displacements is
7.1.2 Virtual work of external loads
For a bar in tension (Fig. 7.1, a), the virtual work of the external force F is
δ WE = F ⋅ δ u . (7.3)
It has the same value whether the bar material is linear elastic (Fig. 7.1, b) or nonlinear elastic (Fig. 7.1, c). Note that it is simply (force × displacement), because the force is constant along the virtual displacement, the latter being arbitrary, hence independent of the force. In the general case of loading by conservative body forces (6.1), surface tractions (6.2) and point forces (6.3), the virtual work of external loads is δ WE =
7.1.4 Principle of virtual displacements
For elastic bodies, the principle of virtual displacements states that: If a system is in equilibrium, then during an arbitrary small displacement from the equilibrium position, the virtual work of applied loads equals the virtual work of internal forces δWE = δWI . (7.6) Also: A body is in equilibrium if the internal virtual work equals the external virtual work for every kinematically admissible displacement field.
(7.6, a) In (7.6) δWE is the work of external loads on the virtual displacements { δu } which are independent of loading and kinematically admissible. If stresses are expressed in terms of a set of parameters defining completely the displacement pattern – the nodal displacements, then equilibrium relations can be obtained and the displacement parameters determined. The nodal displacements do not permit the fully equilibrating position to be reached, so that the PVD will ensure approximate equilibrium. Note that the virtual work of reaction forces at supports is zero. Since the principle of virtual displacements is an equilibrium requirement, it is independent of material behaviour, i.e. whether the material is elastic or inelastic. It applies only
7. ENERGY METHODS
135
for loading by conservative forces, which do not change direction during the action on the virtual displacements. The external work is independent of the path taken.
Example 7.1
For the three-bar pin-jointed framework shown in Fig. 7.3, loaded by a force F, find the internal bar forces and the displacement of point 4.
Fig. 7.3 Solution. Consider three states of the analyzed system: 1. The initial state, in which bars are not loaded by external forces and are not prestressed (Fig. 7.4, a). 2. The final state of static equilibrium, in which the external force F, of components F1 = F sinα and F2 = F cosα , produces a displacement of the joint 4, of components u1 and u2 (Fig. 7.4, b). The joint 4 is acted upon by the external forces F1 , F2 and by internal forces T1 , T2 , T3 (Fig. 7.4, c). The joint reacts with forces equal in magnitude but of opposite sign, producing the elongations Δ1 , Δ2 , Δ3 (Fig. 7.4, d). 3. An imaginary state, in which the joint 4 is given a virtual displacement of components δu1 and δu2 (Fig. 7.4, e), which produce virtual elongations in bars δΔ1 , δΔ2 , δΔ3 (Fig. 7.4, f), the applied forces remaining constant. The virtual displacements δu1 and δu2 and the virtual bar extensions δΔ 1 , δΔ 2 , δΔ 3 satisfy the compatibility equations (2.19)
Since δu1 and δu2 are unrelated to each other, we could put either to zero. Equation (7.10) must be zero whatever the values of δu1 and δu2 . This can only be true if their coefficients vanish T1 sinθ − T3 sinθ = F1 , T1 cosθ + T2 + T3 cosθ = F2 . (7.11)
These are indeed the equations of equilibrium. It is confirmed that the principle of virtual work is an equivalent statement of statical equilibrium. Substituting (7.8) in the finite form of (7.7), then in (7.11), the components of the displacement of point 4 can be determined from the following equations
2 EA 2 sin θ ⋅ u 1 = F1 , l EA ⎛ 1 ⎞ 2 ⎟ u 2 = F2 . ⎜ 2 cos θ + l ⎝ cosθ ⎠
(7.12)
7.1.5 Proof that PVD is equivalent to equilibrium equations
Consider a form of equation (7.6, a) without point forces
V
∫ { δε } {σ }dV = ∫ { δu } { pv }dV + ∫ { δu } { ps }d A .
T T T V Sσ
(7.6, b)
Convert { δε } to { δu } using the integration by parts
V
⎜ ⎟ ∫ σ x δε x dV = ∫ σ x δ ⎜ ∂ x ⎟ d x dy dz = ∫ ⎝ ⎠
V
⎛ ∂u ⎞
σx
Sσ
∂ (δu ) d x l d A = ∂x δu ∂σ x dx dy dz , ∂x dV
=
Sσ
∫ σ x d(δu ) l d A = ∫ σ x δu l d A − ∫
Sσ V
V
∫ σ y δε y dV = ∫ σ y δ v m d A − ∫
Sσ V Sσ
δv
∂σ y ∂y
dV , ∂τ xy ⎞ ⎛ ∂τ xy δv + δu ⎟ dV , ⎟ ∂y ⎠ ⎝ ∂x
V
∫ τ xy δγ xy dV =
∫ τ xy (δ v ⋅ l + δu ⋅ m)d A − ∫ ⎜ ⎜
V
138
FINITE ELEMENT ANALYSIS
where l and m are direction cosines of the outward normal at the surface. Adding together
which is true provided [ ∂ ] T {σ } and { ε } are finite in V, that is the stress and displacement fields are continuous. This applies only within a single element and up to its surface. The componets of stresses at element interfaces may not balance at a point, but only in the mean. Most finite elements in use today do not achieve continuous stresses across interfaces. On substituting in (7.6, b)
T T T T ∫ { δu } ( [ ∂ ] {σ } + { pv }) dV − ∫ { δu } ( [ n ] {σ } − { ps }) d A = 0 . Sσ
V
Because { δu } are arbitrary, its coefficients must vanish. The equations of equilibrium emerge from the brackets
[ ∂ ]T {σ } + { pv } = { 0 } in V, [ n ]T {σ } − { ps } = { 0} on
Sσ .
The PVD supplies equilibrium conditions both within and on the surface of the body. So, when using approximate functions for { u }, it is not necessary to worry about the equilibrium boundary conditions. Part of the surface, i.e. Sσ , is supported in some way and there the tractions { ps } will be unknown reactions and not specified loads. It is conventional to remove the unknown reactions by choosing the virtual displacements { δu } to be zero over Sσ .
7. ENERGY METHODS
139
7.2 Principle of minimum total potential energy
The total potential energy Π of an elastic body is defined as the sum of the strain energy U and the work potential of external loads WP
Π = U + WP .
(7.13)
7.2.1 Strain energy
Consider the strain energy (6.32) U= 1 2
For a virtual strain { δε } , the virtual increase of strain energy is δU = Because 1 2
∫ {ε } V
T
[ D ] {ε } dV .
V
{ δε }T [ D ] {ε } dV + 1 ∫ { ε }T [ D ] { δε } dV . ∫
2
V
({ δε }
δU =
T
[ D ] {ε } )
T
= { ε }T [ D ] { δε },
∫ { δε } V
T
[ D ] {ε } dV = ∫ { δε }T {σ } dV = δWI .
V
δ U = δWI .
(7.14)
For an elastic body, the virtual variation of the strain energy is equal to the virtual work of the internal stresses on virtual strains. For a bar in tension, substituting σ = E ε and ε =
du , yields dx
δU =
V
∫ δε σ dV = ∫ δε Eε A d x = ∫
l l
d δu du EA dx . dx dx
d2v (5.4), gives d x2
(7.15)
For a beam in bending, substituting ε = − y δU =
∫
l
⎛ ⎞ ∂ 2δv ∂ 2 v ⎜ 2 ⎟ E y dA ⎟ dx = ⎟ ⎜ ∂ x2 ∂ x2 ⎜ ⎝A ⎠
∫
∫
l
d 2δv d2v EI dx . d x2 d x2
(7.16)
The above expressions can be obtained directly from the strain energies
7.2.2 External potential energy
The work potential of external loads WP is equal to the negative product of external forces by the corresponding displacement
WP = − WE .
(7.19)
The negative sign appears because the external loads lose some of their capacity for doing work when displaced in the direction they act. For example, a gravitational force F = m g acts in the opposite direction to a vertical displacement h and the potential becomes m g h . An external point load F j has potential energy − F j u j
(
)
instead of
⎛ 1 ⎞ ⎜ − F j u j ⎟ , because this potential arises from the magnitude of force and its ⎝ 2 ⎠ capacity to do work when it moves, being independent of the linear properties of the body on which it acts.
For a three-dimensional continuum WP = −
V
∫ {u } { pv }dV − ∫ {u } { ps }d A − ∑ {ui } { Fi } . i
T T T Sσ
(7.20)
For virtual displacements {δu } δWP = − δWE . (7.21)
7.2.3 Total potential energy
The total potential energy (7.13) can be written
hence, at equilibrium, the total potential energy has a stationary value. If
The principle of minimum total potential energy states that: If a deformable body is in equilibrium under the action of external loads and reaction forces, then the total potential energy has a minimum value. Reciprocally, if under the action of external loads and reaction forces the total potential energy of a deformable body is a minimum, then it is in a stable equilibrium state. Thus, it can be considered that (7.22, a) is a condition that establishes or defines the equilibrium, rather than a result of the equilibrium. An equivalent statement is: For conservative systems, of all possible kinematically admissible displacement fields, the one satisfying equilibrium corresponds to a minimum value of the total potential energy. Reciprocally, any kinematically admissible displacement field which minimizes the total potential energy represents a stable equilibrium configuration.
Example 7.3
Apply the principle of minimum total potential energy to a beam in bending, subjected to a distributed load and to the end bending moments and shear forces as shown in Fig. 7.5. Show that PMTPE is equivalent to the equilibrium conditions inside and at the ends of the beam.
Fig. 7.5 Solution. For a beam segment loaded as shown, the total potential energy is
The coefficient of δ v in the integrand gives the equation of equilibrium
(E I v′′)″ = p (x ) .
As δ v is arbitrary, the other terms deliver the equilibrium conditions at the beam ends
( E I v′′) 0 = M 0 , ( E I v′′) l = M l ,
or δ v′ = 0 , 0
( E I v′′)0′ = T0 ,
or δ v0 = 0 , or δ vl = 0 ,
or δ v′ = 0 , and l
( E I v′′)l′ = Tl ,
which are the boundary conditions.
7.3 The Rayleigh-Ritz method
The Rayleigh-Ritz method involves the construction of an assumed displacement field. For a beam, the transverse displacement v (x ) is approximated by a finite series
144
v (x ) ≅
FINITE ELEMENT ANALYSIS
∑ a ϕ (x )
j j j =1
n
(7.23)
where a j are undetermined constants called generalized coordinates, and ϕ j ( x ) are prescribed functions of x , called admissible functions, that satisfy the kinematic (geometric) boundary conditions and are continuous within the definition interval. Substituting the displacements (7.23) into the expression of the total potential energy Π , the latter becomes a function of the parameters a j , whose values are determined from the stationarity conditions δΠ =
∑ ∂a
j
∂Π
j
δa j = 0 .
Because δ a j are arbitrary,
∂Π =0, ∂aj
( j = 1,..., n ) ,
(7.24)
which is a linear algebraic set of equations in the constants a j . The solutions are back-substituted into (7.23) which represents an approximate deflected shape, which is more accurate the more terms are selected in the respective series. The necessary requirements for the convergence of the Rayleigh-Ritz method are the following: a) The approximating functions must be continuous to one order less the highest derivative in the integrand. b) The functions must individually satisfy the geometric boundary conditions, i.e. to be admissible functions. c) The sequence of functions must be complete. If the functions are not selected from the domain space of the operator of the equation being solved (completeness property) the resulting solution could be either zero or wrong.
If F is prescribed and the resulting u has been approximated by a RayleighRitz solution uapp ≠ ueq , equations (7.32) – (7.34) and Fig. 7.7, b indicate that: a) Because Π eq is a minimum, the potential energy for an approximate displacement which satisfies the kinematic boundary conditions is greater than the true value Π app > Π eq . In magnitude
An approximate compatible displacement field corresponds to a structure which is stiffer than the actual structure and therefore will give a lower bound on displacement.
7.4 F.E.M. - a localized version of the Rayleigh-Ritz method
Instead of finding an admissible function satisfying the boundary conditions for the entire domain, which is often difficult, in the FEM the admissible functions are defined over small size subdomains.
7.4.1 F.E.M. in Structural Mechanics
a) Problem. Given a geometrically complex structure (including the boundary conditions) and the external loads { pv } , { ps } , { Fi } , find the displacement field { u } within V and on the surface Sσ (Fig. 6.1). Then determine stresses, internal forces, reaction forces, etc. b) Solution approach. Use PVD or PMTPE as an approximate method for solving the boundary-value problem. Admissible functions are defined over small size finite elements, with simple geometry and well identified structural behaviour. With these individually defined functions matching each other at certain points (nodes) at the element interfaces, the unknown function is approximated piecewise over the entire domain (continuity at global level). c) Procedure. The geometric shape and the internal displacement field are described by a series of discrete quantities (like nodal coordinates and nodal displacements) distributed through the structure. For this a matrix notation is used. d) Tools. Computers are used to store long lists of separate numbers and to manipulate them, to present output data in an engineering format, taking advantage of graphical and animation facilities.
7. ENERGY METHODS
149
7.4.2 Discretization
The structure is divided into finite elements (Fig. 7.8) that define the mesh. Elements are defined by their nodal coordinates and some physical parameters.
Fig. 7.8
7.4.3 Principle of virtual displacements
For the entire structure, equation (7.6, a) can be written (considering only surface tractions) δΠ =
The basic idea of FEM is to choose the constants - the displacement unknowns at the nodes { a } = Q e and to prescribe admissible functions denoted ⎣ϕ ⎦ = ⎣N ⎦ so that
{ }
⎧ u ⎫ ⎡ ⎣N u ⎦ ⎪ ⎪ ⎢ ⎨ v ⎬=⎢ 0 ⎪w⎪ ⎢ 0 ⎩ ⎭ ⎣
0 ⎣N v ⎦ 0
e 0 ⎤ ⎧ Qu ⎪ e 0 ⎥ ⎨ Qv ⎥ e ⎣N w ⎦ ⎥ ⎪ Qw ⎦⎩
{ }⎫ { }⎪ ⎬ { }⎪ ⎭
(7.40)
or
{ u } = [ N ] {Q e },
where [ N ] is the matrix of shape functions (interpolation functions). The reason is that elements are small enough so that the shape of the displacement field can be approximated without too much error and only the magnitude, defined by Q e remains to be found.
{ }
The proper selection of shape functions ensure the continuity of the displacement field at global level. A finite element described by admissible shape functions (integrable in the interior and with equal values of generalized coordinates at element interfaces) is referred to as co-deformable or conforming.
7.4.5 Compatibility between strains and nodal displacements
From the compatibility relationship (6.18)
As δQ e are arbitrary and non-zero, cancelation of the bracket yields the element equilibrium equation
{
}
[ K ] {Q }= {F },
e e e
(7.42)
where the element stiffness matrix is
[ K ]= ∫ [ B ]
e Ve e
T
[ D ][ B ] dV
(7.43)
and the vector of consistent nodal forces is
{F }= ∫ [ N ]
Ae
T
{ ps }d A .
(7.44)
7.4.7 Assembly of the global stiffness matrix and load vector
In the next step, all individual elements are assembled together so that the displacements are continuous across element interfaces and the boundary conditions are satisfied. The kinematic connectivity is expressed by the relationship between element and global displacements ~ Q e = T e { Q }, (7.45) where Q e is the vector of nodal element displacements, {Q } is the vector of the ~ global displacements of the structure and T e is a connectivity matrix, containing ones at the nodal displacements of element nodes and zeros elsewhere.
{ }
{ } [ ]
[ ]
The variation of element displacements is
152
FINITE ELEMENT ANALYSIS
~ { δQ }= [ T ] { δQ }.
e e
(7.46)
The PVD equation for the entire structure is
∑ {δQ } [ K ] {Q }= ∑ {δQ } {F },
e T e e e T e e e
or using (7.45) and (7.46)
{δQ }T ∑
e
~ ~ [ T ] [ K ][ T ] {Q } = {δQ } ∑ [ T~ ] {Q }.
e T e e T e T e e
As { δ Q } are arbitrary and non-zero, the unreduced global equilibrium equations are
[ K ] {Q } = { F } ,
e T
(7.47)
where the global stiffness matrix is
[ K ]= ∑
e
~ ~ [ T ] [ K ][ T ]
e e
(7.48)
and the global load vector is ~ { F } = ∑ [ T e ] T {F e }.
e
(7.49)
Applying the boundary conditions, the condensed equilibrium equations are
[ K ] {Q } = { F } .
(7.50)
The above procedure is never used in practice. It has been used only to show algebraically how to assemble a global stiffness matrix. The assembly is done by directly placing the nonzero entries of element stiffness matrices in the right locations of the global stiffness matrix based on element connectivity.
Many engineering structures can be modeled as two-dimensional flat plates, designed to be primarily loaded in their plane and to resist loads by membrane action rather than bending. In-plane displacement, strain and stress components are uniform through the plate thickness, which is considered constant. Only transversely homogeneous plates will be analyzed herein, composite and sandwich plates being studied in other courses. This chapter presents the element stiffness matrices and consistent force vectors for triangular and rectangular elements, that allow closed form derivations, without the need for numerical integration. The very first approximate finite element developed in 1956 to model delta wing skin panels, the three-noded triangle with constant strain field, is treated separately.
8.1 The plane constant-strain triangle (CST)
Before the advent of arbitrarily shaped isoparametric elements, discussed in the next chapter, the CST was one of the most widely used elements and is still available in systems today. It is a much more adaptable shape than the rectangle and it allows the user to tailor the element mesh to suit any structural geometry. A large number of small elements can be densely packed into a region of expected high stress gradients, and uniformly stressed regions can be left with a small number of larger triangles.
8.1.1 Discretization of structure
The plate is divided into a number of straight-sided triangles (Fig. 8.1, a), joined together at their corners (nodes), so that the corners of adjacent elements have common displacements. The elements fill the entire region except of a small region at the boundary. This unfilled region exists for curved boundaries and it can be reduced by choosing smaller elements.
154
FINITE ELEMENT ANALYSIS
The three nodes of the isolated element from Fig. 8.1, b are numbered locally as 1, 2 and 3. The corresponding nodal coordinates are designated as ( x1 , y1 ) , ( x2 , y2 ) and ( x3 , y3 ) . The numbering is in anticlockwise direction to avoid calculating a negative area.
a
Fig. 8.1
b
Each node is permitted to displace in the two directions x and y. Thus, each node has two degrees of freedom. The displacement components of a local node j are denoted as u j in the x direction and v j in the y direction. The vector of element nodal displacements is defined as
{ q }= ⎣ u
e
1
v1 u2
v2
u3
v3 ⎦ T .
(8.1)
8.1.2 Polynomial approximation of the displacement field
The displacements u and v of a point within the triangle are expressed in terms of the nodal displacements. Because for two displacements there are six boundary conditions, the assumed displacement field is linear u (x , y ) = a1 + a2 x + a3 y , v ( x , y ) = a4 + a5 x + a6 y with six arbitrary parameters. (8.2)
8. TWO-DIMENSIONAL MEMBRANES
155
The strains (6.16) are
εx =
∂u = a2 = const . , ∂x
εy =
∂v = a6 = const . , ∂y
γ xy =
∂u ∂ v + = a3 + a5 = const . , ∂ y ∂x
hence the name "constant strain triangle".
8.1.3 Nodal approximation of the displacement field
In the polynomial approximation (8.2) the constants ai have no physical meaning. A nodal approximation is preferred, in which the constants are the nodal displacements and the displacement field is obtained by interpolation based on values of corner displacements. Since the functions for u and v are of the same form, only one need be considered in detail. We can write
where t is the element thickness, A is the element area and [ D ] is the material stiffness matrix given by (6.24) for plane stress and by (6.25) for plane strain conditions. The consistent nodal forces due to traction loads acting on a portion of the boundary are calculated as for a linear two-node element. Along an edge 1-2, the shape function N 3 is zero while N1 and N 2 are similar to the shape functions in one dimension, satisfying N1 + N 2 = 1 . When the surface load distribution (per unit
160
FINITE ELEMENT ANALYSIS
area) is linear, varying from p1 at node 1 to p2 at node 2 (Fig. 8.4), the nodal forces are
f1e =
lte ( 2 p1 + p2 ) , 6
f 2e =
l te ( p1 + 2 p2 6
)
(8.19)
where t e is the thickness of the element. Because of linearity they coincide with the static resultants.
Fig. 8.4 The nodal forces associated with the weight of an element are equally distributed at the nodes.
8.1.6 Remarks
A mesh like in Fig. 8.5, a is clearly a directionally sensitive assembly, and this could be corrected by using the "union jack" pattern of Fig. 8.5, b which however produces a larger bandwidth. Benchmark tests using triangular elements have shown that CST elements, even in a fine mesh, are much inferior to higher order elements in a coarse mesh.
Fig. 8.5
8. TWO-DIMENSIONAL MEMBRANES
161
A drawback of the displacement form of the finite element method is that equilibrium is only satisfied in the mean or over the element. This means that along an edge which is common to two elements the stresses are different across the edge, where they should be continuous. Most programs contain facilities for averaging the stresses. The simplest form of averaging consists of simply connecting the centroids of two adjacent triangles and to assign the mean stress value to the crossing point of this line with the common edge. A simple example of stress averaging is shown in Fig. 8.6 for a square plate with a circular hole. Taking advantage of symmetry, only one quarter of the plate is considered. The diagram compares the theoretical stress distribution along the marked line with the averaged values calculated using CSTs.
Fig. 8.6
162
FINITE ELEMENT ANALYSIS
The procedure will be used in Example E8.3
Example 8.1
A square plate of thickness t and Young's modulus E is pin-jointed at three corners and subjected to a force F at the free corner (Fig. E8.1, a). Let ν = 0 . Divide the plate into four CST elements and find: a) the nodal displacements; b) stresses in elements; and c) the support reactions.
a
b
Fig. E8.1
Solution. The material stiffness matrix (6.24) is
⎡1 0 0 ⎤ [ D ]= E ⎢ 0 1 0 ⎥ . ⎢ ⎥ ⎢0 0 1 2⎥ ⎣ ⎦
(a)
With the origin of coordinate axes at the plate centre, the nodal coordinates of element 1 are
Example 8.2
A thin triangular plate is fixed along the edge 5-4 and loaded by forces F1 = −1 and F2 = −2 along the upper edge in its plane (Fig. E8.2). Assume the cantilever to have unit thickness t = 1 and be in plane stress, with Young's modulus E = 1 and Poisson's ratio ν = 0.3 . Divide the plate into three CST elements and find: a) the nodal displacements; b) stresses in elements; and c) the support reactions. The units are coherent.
166
FINITE ELEMENT ANALYSIS
Fig. E8.2
Solution. The input data are given below
For each element, the stiffness matrix is calculated longhand from equation (8.18), where the matrix [ B ] is obtained from (8.17), based on the nodal coordinates, and the matrix [ D ] is given by (6.24).
The reaction forces at nodes 4 and 5 are obtained from the 'unused' unreduced equations. Note that the adopted discretization is very crude, to allow longhand calculation, and this leads to misleading results. Along the edge 4-5, a distribution of bending stresses from compressive at 4 to tensile at 5 is expected. However, element 3, being a constant strain element, gives only one value of σ x , which is wrong. Normally, the plate should be modeled by many more elements. Ascribing stress values to the element centroids and averaging them as shown in section 8.1.6, a more realistic stress distribution along the edge 4-5 is obtained.
Example 8.3
A thin rectangular plate, containing a circular hole of radius a = 10 mm , is subjected to loads that produce uniform tensile stresses σ 0 = 5 MPa at its ends (Fig. E8.3, a). The plate has length l = 60 mm , width b = 40 mm , thickness t = 5 mm , E = 200 GPa and ν = 0.3 . Determine: a) the deformed shape of the hole; b) the location and magnitude of the maximum von Mises stress in the plate; c) the distribution of σ x stresses in the midsection. Compare the stress values at the periphery of the hole obtained by FEM and from the theory of elasticity. Solution. Taking advantage of the symmetry of geometry and symmetry of loading, we can analyze only one-quarter of the plate (upper right).
8. TWO-DIMENSIONAL MEMBRANES
169
Fig. E8.3, a A 55-node, 81-element mesh is created as shown in Fig. E8.3, b. Let x and y represent the axes of symmetry. The points along the x axis are constrained in the y direction, and points along the y axis are constrained along the x direction.
Fig. E8.3, b The applied nodal forces are shown, but the element numbering is omitted for clarity. The centroids of the elements near the midsection are marked, and the crossing points of the lines connecting the centroids with the common sides (where stresses are averaged), are denoted a to f. The deformed shape is shown in Fig. E8.3, c. The hole is elongated in the direction of the loading axis.
170
FINITE ELEMENT ANALYSIS
Fig. E8.3, c The calculation of stresses is summarized in Fig. E8.3, d. Elements are hatched according to the value of von Mises stresses, using five intervals with limits shown in the legend.
Fig. E8.3, d The maximum von Mises stress is 19.6 MPa and occurs in element 1. Stress values in elements near the midsection are given in Table E8.3.
of 440 MPa . Find the location and magnitude of the maximum von Mises stress in the plate. Determine the stress concentration factor for the circular fillet.
Fig. E8.4, a Answer. Taking advantage of the symmetry, only half of the plate is considered. The points along the symmetry axis are constrained in the vertical direction. Points along the left edge are constrained in the horizontal direction. A model comprising 95 nodes and 144 CST elements is constructed as shown in Fig. E8.4, b.
Fig. E8.4, b The nodal loads are calculated from equation (8.19). The right edge has four elements, each with a surface of 25 mm 2 , subjected to a normal stress of 440 N mm 2 . The nodal loads for each element are 440 × 25 / 2 = 5500 N . The resulting five nodal forces, for the quarter plate, have magnitudes of 5500 N at the upper and lower node, and 11000 N at the three middle nodes. The distribution of von Mises stresses is shown in Fig. E8.4, c for five stress intervals given in the legend. The largest equivalent stress is 604.7 MPa .
8. TWO-DIMENSIONAL MEMBRANES
173
Fig. E8.4, c Values of the principal stress σ 1 around the fillet are presented in the upper part. The largest principal stress σ 1 is 622.7 MPa . The stress concentration factor relative to the value 440 MPa is 1.415. This value is reasonably close to the theoretical true value of 1.42 given by Singer (1962).
Example 8.5
The nodal coordinates and the nodal displacements of element 96 in Fig. E8.4, b are given below:
{σ } = [ D ][ B ] { q },
{σ } = ⎣σ x
Example 8.6
Find the deformed shape and the stress distribution in the loaded gear tooth shown in Fig. E8.6, a. The load is not applied at the tooth tip, but is distributed over a larger area than for the mating contact of two teeth, to concentrate only on the influence of fillet geometry.
Fig. 8.7 We have to find two-dimensional shape functions which have unit values at one selected node, but zero at all other nodes on the element N i x j , y j = δ ij .
(
)
(8.20)
They can be generated using the equations of the sides. For example, N1 is identically zero on lines x = a and y = b , and has a unit value at the node with the same index N1 ( x1 , y1 ) = 1 , hence it must be of the form
The use of the shape functions (8.23) has one drawback – they are not complete. That is, all like powers in x and y are not present. In this case x y is present but x 2 and y 2 are not. Thus the variation of strain does not have the same order in all directions. The displacements u and v of a point within the rectangle can be expressed in terms of the nodal displacements by a polynomial approximation. Because, for two displacements, there are eight boundary conditions, the assumed displacement field is
The direct strain ε x = a2 + a4 y is constant in the x direction whereas the shear strain γ xy = a3 + a4 x + b2 + b4 y varies linearly with x and y . Some improvement can be gained by diminishing the shear variation through the use of reduced integration on the shear contribution in the element stiffness matrix. Many users prefer to use higher order elements rather than use a larger number of small 4-noded rectangles. The 4-noded rectangular element is exploited in non-linear problems like elasto-plastic behaviour, where the stiffness has to be reevaluated as the load increments are applied and plasticity spreads.
178
FINITE ELEMENT ANALYSIS
The element stiffness matrix (7.39) is
[k ]= ∫ [ B ]
e Ve
T
[ D ][ B ] dV ,
where
[ B ] = [ ∂ ][ N ] .
The poor performance of the 4-noded element is shown below, in an example where it is expected to behave like a slender beam in which bending stresses are dominant.
Fig. 8.8 Figure 8.8, a shows the deformed shape of a single element in pure bending. In comparison, Fig. 8.8, b shows the expected circular deformed shape free of shear deformations. The 4-node element has flat sides. At the element ends the shear strain is γ xy = d b and only the centre has no shear deformation. The direct strains are ε x = d a on the upper and lower surface. The ratio
spurious shears a = = aspect ratio . bending strains b
This explains the poor results obtained with high aspect ratio elements (≥ 3) which have also badly conditioned stiffness matrices. The stress discontinuities at element interfaces can be comparable with the mean values and the free edge stresses are not zero. Generally, because the strain energy is underestimated, stresses are lower than the true values.
8.2.2 The eight-node rectangle (quadratic)
The higher order 8-node rectangular element is shown in Fig. 8.9. Adding four nodes means adding four supplementary boundary conditions (or nodal displacements) so that in the polynomial approximation we can add four more (higher order) terms.
where c1 is a constant which is determined so as to yield N1 (− 1,−1) = 1 , i.e. c1 = − 1 4 , which gives the expression of N1 in (8.29). The displacements inside the element can be expressed in terms of the nodal displacements as ⎧ u ⎫ ⎡ N1 ⎨ ⎬=⎢ ⎩v⎭ ⎣ 0 where 0 N1
N2
8.3 Triangular elements
In order to obtain accurate results for stresses with a constant strain discretization, one has to use a large number of elements. Considerable effort has been devoted to develop 'refined' elements, i.e. elements having linear, quadratic or higher-order strain expansions.
8.3.1 Area coordinates
For a triangle it is possible to define a completely symmetrical coordinate system known as area coordinates. The position of any point M in the triangle is identified by the perpendicular distances h1 , h2 , h3 from the three sides (Fig. 8.10, a), and nondimensionalized, by division to the triangle heights, as h1 h2 h3 , ζ2 = , ζ3 = , ζ1 = H1 H2 H3
8. TWO-DIMENSIONAL MEMBRANES
181
or
ζ1 = A1 A
, ζ2 =
A2 A
, ζ3 =
A3 A
.
(8.32)
As A1 + A2 + A3 = A (Fig. 8.10, b), one obtains
ζ1 + ζ 2 + ζ 3 = 1 ,
(8.33)
so the three coordinates are not independent and they behave like the shape functions.
8.3.2 Linear strain triangle (LST)
The linear strain triangle is a six-noded element, obtained adding three mid-side nodes to the CST. In Fig. 8.11, the equations of the three sides and the lines through the mid-side nodes are shown using area coordinates.
where A is the triangle area (8.9) and β i , γ i are defined by (8.8). The element stiffness matrix is
[k ]= ∫ [ B ]
e A
T
[ D ][ B ] t e d A ,
(8.42)
where d A = l 3 dζ 2 l 2 dζ 3 sinθ = 2 A dζ 2 dζ 3 . The integration is frequently performed numerically. However, it can be solved explicitly in terms of a, b and [ D ] using the integration formula for monomials
The displacement field is complete, containing all possible products, so the element is truly isotropic, without recourse to computationally inefficient internal nodes. There are twelve nodal displacements, six for each component. In order to satisfy inter-element displacement compatibility, the displacement expansion on a boundary must involve only the nodal quantities for that boundary. There are three constants for this case (the function is quadratic) and an additional interior node is required for each boundary. It is convenient to locate these interior nodes at the mid-points of the sides. Solving for the constants in terms of the nodal displacements, the nodal approximation (8.37) is obtained. Denoting
so that the same functions can be used as interpolation functions. This is the basic idea behind the isoparametric formulation, treated in the next chapter.
8.3.3 Quadratic strain triangle
A triangular element with a quadratic strain field can be built up in two ways. One possibility is to work with corner point nodes and two interior nodes per side, taking as nodal quantities the displacement components. This element is shown in Fig. 8.13, a.
Fig. 8.13 The displacements are expressed as complete cubic polynomials. In order to satisfy inter-element displacement compatibility, the displacement function for a side must depend only on the nodal displacement quantities for the side. Since the function is cubic, four nodal quantities are required to define the distribution on a side. The side nodes are located at the third points. An additional interior node is needed to maintain completeness of the polynomial since, if the polynomial is not complete, the stiffness will have preferred direction which is not desirable. It is convenient to take the interior node at the centroid. The displacement nodal expansion has the form
Another possibility is to work only with corner nodes, in order to reduce the bandwidth of the stiffness matrix. The solution is to include displacement ∂u ∂u ∂ v ∂ v , , , as nodal quantities. At each corner, there are six derivatives ∂x ∂ y ∂x ∂ y nodal variables, two displacements and four first derivatives, a total of 18 parameters (Fig. 8.13, b). Two additional displacement quantities not associated with the boundaries are required for completeness. It is convenient to take the displacement components of the centroid (uc , vc ) as the remaining parameters. The nodal expansion for u has the form u = N1 u1 + N 2 u ′ 1 + N 3 u ′y1 + N 4 u2 + N 5 u ′ 2 + ... + + N 9 u ′y 3 + N c uc . x x (8.52)
where γ i , β i are defined by (8.8) and the subscript c refers to the centroid. The same interpolation functions are valid for v .
8.4 Equilibrium, convergence and compatibility
It is useful to make some general comments on the fulfilment of the equilibrium and compatibility conditions in a finite element solution based on assumed displacement fields [33].
∂ 2v ∂u ∂u ∂ 2u ∂ 2u = b4 = a2 + a4 y , = a3 + a4 x , = 2 = 0 , and ∂ x ∂y ∂x ∂y ∂ x2 ∂ y which is not zero, so the first equation (8.55) is not satisfied. The rectangle would satisfy equilibrium if a4 = b4 = 0 , but this is the case only in a field of constant strain. But as already shown, the direct strain ε x is linear in the y direction whereas the shear strain γ xy varies linearly with x and y . However, equilibrium is satisfied within the CST because of its extreme simplicity. One cannot conclude from this that, in general, the rectangle is inferior to the triangle. In some cases it can give better results. b) Between elements, compatibility may or may not be satisfied, and equilibrium is usually not satisfied. For example, for both the 3-node triangle and the 4-node rectangle, u and v are linear in x (or y ) along element edges. So, for any nodal displacement, the edges remain straight, and adjacent elements do not overlap or separate, the elements fit together. Inter-element equilibrium is obviously violated in the CST, where stresses are constant within the element but differ from one element to another. However, inter-element stress continuity may exist, as in the case of uniform beams loaded only at nodes. When inter-element compatibility is satisfied, the finite element solution gives an upper bound on the total potential energy. As the discretization is refined, the solution will converge monotonically to the true solution, provided that the new mesh contains all the nodes of the previous meshes. c) At nodes, compatibility is enforced by joining elements at these locations, and equilibrium of nodal forces and moments is satisfied. The finite element equations are a set of equilibrium equations and the solution is such that resultant forces and moments acting on each node are zero. Happily, in a 'proper' finite element solution, any violations of equilibrium and compatibility tend to vanish as more and more elements are used in the mesh. Moreover, the convergence of displacements with the mesh refinement must be monotonic from below.
8.4.2 Convergence and compatibility
As the mesh of elements is refined, the sequence of solutions to a problem is expected to converge to the correct result if the assumed element displacement fields satisfy the following criteria:
8. TWO-DIMENSIONAL MEMBRANES
189
a) The displacement field within an element must be continuous. This is normally ensured by the selection of shape functions. b) When the nodal degrees of freedom are given values corresponding to a state of constant strain, the displacement field must produce the constant strain state throughout the element. The check is done using the so called "patch test". The model consists of an assembly of several elements arranged so that at least one node is completely surrounded by elements. The boundary nodes are then given either displacements or forces consistent with a constant strain state. Internal nodes are to be neither loaded nor restrained. The computed displacements, strains, and stresses within elements should be consistent with the constant strain state. If not, the element type is invalid or at least suspect (it may happen that an element is valid in certain configurations only). c) Rigid body modes must be represented. When nodal degrees of freedom are given values corresponding to a state of rigid body motion, the element must exhibit zero strain and therefore zero nodal forces. If this requirement is violated, extraneous nodal forces appear, and the equations of nodal equilibrium are altered. To satisfy the requirements on both rigid body modes and constant strain rates, the expansion must be at least a complete polynomial of order equal to the highest derivative occurring in the strain-displacement relations. d) Compatibility must exist between elements. Elements must not overlap or separate. In the case of beam, plate and shell elements, the slope must be continuous across interelement boundaries. This requirement is violated by many successful non-conforming elements. Such elements do satisfy inter-element compatibility in the limit of mesh refinement, as each element approaches a state of constant strain. However, noncompatible elements do not provide a bound on the potential energy, i.e. we do not know whether the potential energy corresponding to a particular non-conforming element is higher or lower than the true value. Also, it is not possible to construct a minimizing sequence with non-compatible elements. This means that by suitably specializing the nodal displacements for the nth discretization, we will not be able to reproduce the displacement patterns corresponding to the n − 1 previous discretizations. e) The element should have no preferred directions. Elements should be invariant with respect to the orientation of the load system. Invariance exists if complete polynomials are used for element displacement fields. It is achieved even when based on incomplete polynomials, if there is a 'balanced' representation of terms in the polynomial expansion.
190
FINITE ELEMENT ANALYSIS
Example 8.7
For the assembly of four triangular elements shown in Fig. E8.6, a simple patch test is carried out as follows.
where [K ] depends on the material properties and {F } depends on both the material properties and the prescribed nodal displacements. Its solution must be
u5 ⎫ ⎬ = {F } , ⎩v5 ⎭
u5 = v5 = 1.25
and the strains at any point must be { ε } = ⎣1 1 2⎦ T .
9.
ISOPARAMETRIC ELEMENTS
Simple triangular and rectangular elements allow closed form derivations of stiffness matrices and load vectors. The construction of shape functions and evaluation of stiffness matrices for quadrilateral and higher-order elements with curved sides faces difficulties which are overcome by the use of isoparametric elements and numerical integration. Elements with curved sides provide a better fit to curved edges of an actual structure. For isoparametric elements, the same interpolation functions are used to define the element shape as are used to define the displacement field within the element. The constant strain triangle is an isoparametric element though it was not treated like that. In fact, the shapes of the interpolation functions and not the parameters are the same. It is possible to construct subparametric elements whose geometry is determined by a lower order model than the displacements. When the eight node rectangle is transformed into a quadrilateral with straight sides, we obtain a subparametric element. When it is mapped into a quadrilateral with parabolically curved sides, the result is an isoparametric element. If we develop higher order triangular elements while keeping straight sides, they are subparametric elements, because only the displacement expansion is refined whereas the geometry definition remains the same.
9.1 Linear quadrilateral element
Consider the general quadrilateral element shown in Fig. 9.1, a. The local nodes 1, 2, 3 and 4 are labelled counterclockwise. The coordinates of node i are ( xi , yi ) . Each node has two degrees of freedom. The displacement components of a local node i are denoted as ui in the x direction and vi in the y direction. The vector of element nodal displacements is defined as
{ q }= ⎣ u
e
1
v1 u2
v2
u3
v3
u4
v4 ⎦ T .
(9.1)
192
FINITE ELEMENT ANALYSIS
9.1.1 Natural coordinates
A natural coordinate system can be attached to a quadrilateral element as illustrated in Fig. 9.1, b. The coordinates r and s vary from − 1 on one side to + 1 at the other, taking the value zero over the quadrilateral medians. They are called quadrilateral coordinates.
a
Fig. 9.1
b
In the development of isoparametric elements it is useful to visualize the quadrilateral coordinates plotted as cartesian coordinates in the { r , s } plane. This is called the reference plane. In the reference plane, quadrilateral elements become a square of side 2 (Fig. 9.2, a), called the reference element (or master element), which extends over r ∈ [ − 1, 1 ] , s ∈ [ − 1, 1 ] . The transformation between the natural coordinates { r , s } in the reference plane and the cartesian coordinates { x , y } is called the isoparametric mapping.
a
Fig. 9.2
b
9. ISOPARAMETRIC ELEMENTS
193
Each quadrilateral 'child' in the { x , y } is generated by the 'parent' or 'reference' element from the { r , s } plane (Fig. 9.2, b). The advantage is that the interpolation functions defined for the reference element are the same for all actual elements and have simple expressions. The drawback is that the mapping is a change of coordinates which implies complications in the evaluation of the integral in the element stiffness matrix.
9.1.3 The displacement field
In the isoparametric formulation, the same shape functions are used to express the displacements within the element in terms of the nodal values as are used to define the element shape. If u and v are the displacement components of a point located at ( r , s ) , then
u = N1 u 1 + N 2 u 2 + N 3 u 3 + N 4 u 4 = ∑ N i u i , v = N1 v1 + N 2 v2 + N 3 v3 + N 4 v4 = ∑ N i v i ,
i =1 i =1 4 4
(9.5)
which can be written in matrix form
{ u } = [ N ] { q e },
where
(9.6)
[ N ]= ⎡ ⎢
N1
0 N1
N2 0
0 N2
N3 0
0 N3
N4 0
⎣ 0
0 ⎤ . N4 ⎥ ⎦
(9.7)
As the shape functions in natural coordinates (9.2) fulfil continuity of geometry and displacements both within the element and between adjacent elements, the compatibility requirement is satisfied in cartesian coordinates too. As mentioned in Chapter 8, polynomial models are inherently continuous within the element. Then it is easy to show that the displacements along a side of the element depend only upon the displacements of the nodes occurring on that side. If the u displacement is approximated as u ( r , s ) = a1 + a2 r + a3 s + a4 r s ,
(9.8)
along each side the approximation is linear, because r (or s) is constant. For example, along the side 2-3 (Fig. 9.2, a), r = 1 and
u
u
r =1
= a1 + a2 + ( a3 + a4 ) s ,
= 1+ s 1− s u2 + u3 . 2 2
r =1
The four-node quadrilateral is termed a "linear" (sometimes bi-linear) element because its sides remain straight during deformation. This ensures interelement compatibility, i.e. there are no openings, overlaps or discontinuities
9. ISOPARAMETRIC ELEMENTS
195
between the elements. The 4-node isoparametric quadrilateral is a conforming element. Also, because the interpolation displacement model provides rigid body displacements in the natural coordinate system, the conditions of both rigid body displacements and constant strain states are satisfied in cartesian coordinates too. On the contrary, the 4-node quadrilateral is a non-conforming element. If the u displacement is approximated as
u ( x , y ) = b1 + b2 x + b3 y + b4 x y ,
(9.9)
along each side the approximation is quadratic. For example, the equation of the side 2-3 (Fig. 9.1, a) has the form y = m x + c , where m is the slope of 2-3, so that
u ( x)
2 −3
= b1 + cb3 + ( b2 + m b3 + c b4 ) x + m b4 x 2 .
Along 2-3, u ( x ) varies quadratically and cannot be uniquely defined as a function of u 2 and u 3 . The approximation (9.9) is non-conforming.
The above expressions will be used in the derivation of the element stiffness matrix.
Transformation of an infinitesimal area
An additional result that will be needed is the relation dx dy = det [ J ] dr ds . (9.17)
It is needed because, for the evaluation of the element stiffness matrix, the integration on the real element is replaced by the simpler integration over the reference element. In cartesian coordinates { x , y }, the elementary area dA is given by the modulus of the cross product d x ×d y , where d x = d x ⋅ i , d y = d y ⋅ j , and i , j are base vectors. In a curvilinear system { r , s } , the elementary area dA is given by the modulus of the cross product d r ×d s . It is equal to the area of the elemental parallelogram enclosed by the two vectors d r and d s directed tangentially to the r = const . and s = const . contours respectively. The components of these vectors in a cartesian coordinate system are
dr =
∂x ∂y d r ⋅ i + d r ⋅ j = ( J11 i + J12 j ) d r , ∂r ∂r
d s = ( J 21 i + J 22 j ) d s .
By equating the moduli of the cross products d x ×d y = d x d y ⋅ k ,
198
FINITE ELEMENT ANALYSIS
i
j J12 J 22
k
d r ×d s = J11 J 21 we obtain equation (9.17).
0 d r d s = det [J ] d r d s ⋅ k 0
9.1.5 Element stiffness matrix
The element stiffness matrix (7.43) is
[k ]= t ∫ [ B ]
e e A
T
[ D ][ B ] d A ,
(9.18)
where [ D ] is the material stiffness matrix and t e is the element thickness. The matrix of differentiated shape functions [ B ] is (7.41)
Because [ B ] and det [ J ] are involved functions of r and s, the above integration has to be performed numerically, as shown in the following.
9.1.6 Element load vectors
Because the displacements along a side of the quadrilateral isoparametric element are linear, the consistent nodal forces applied along that side are calculated as for a linear two-node element. If there is a distributed load having components
( px , p y ) per unit length
0 0 ⎦T , (9.25)
along side 2-3 in Fig. 9.1, then the equivalent nodal force vector is
{ f }= 1 l ⎣ 0 2
e
2 −3
0
px
py
px
py
if p x and p y are constants and l 2 − 3 is the length of the side 2-3.
200
FINITE ELEMENT ANALYSIS
9.2 Numerical integration
The numerical evaluation of stiffness integrals is usually done by Gaussian quadrature. It is more efficient than other methods, as for example the NewtonCotes integration, because it involves only a half of the sample values of the integrand required by the latter. For higher order elements, the stiffness integrals become progressively more complicated. As the order of the displacement field increases, the differentiated shape functions in the matrix [ B ] grow algebraically more cumbersome. The isoparametric mapping introduces det [ J ] in the integrand so that closed form evaluation of stiffness integrals becomes impossible.
9.2.1 One dimensional Gauss quadrature
The one-dimensional version of the Gauss method relies on the concept that any function f ( r ) can be represented approximately over the interval r ∈ [ − 1, 1 ] by a polynomial which can be integrated exactly. A polynomial of order (2n − 1) can be 'fitted' to f ( r ) by imposing n weights wi and n sampling points in such a way that the summation
+1 −1
∫
f (r ) d r =
∑ wi f ( r i )
i =1
n
(9.26)
is exact up to the chosen order. The particular positions of these sampling points are known as Gauss Points. They turn out to be the roots of Legendre polynomials and so the method is referred to as a Gauss-Legendre integration. The actual area represented by the integral is replaced by a series of rectangles of unequal widths, whose heights are equal to the function values at the sampling points. In other words, the integral of a polynomial function is replaced by a linear combination of its values at the integration points ri :
+1 −1
which is exact only when f (r ) is a polynomial of order 1. However, it is employed in some selective integration schemes, e.g. to separate shear from extension effects, avoiding the so-called shear locking by using reduced integration for shearing.
which means that the integral above actually consists of the integral of each element (below or above the main diagonal) in an ( 8 × 8 ) matrix. In general, det [ J ] is a linear function of r and s . For a rectangle or parallelogram it is a constant. The elements of [ B ] are obtained by dividing a bilinear function of r and s by a linear function. Therefore, the elements of
9. ISOPARAMETRIC ELEMENTS
205
[ B ]T [ D ][ B ] det [ J ] are bi-quadratic functions divided by a linear function. This
means that k e cannot be evaluated exactly using numerical integration. From practical considerations, it is best to use as few integration points as is possible without causing numerical difficulties. An alternative is to use reduced integration at fewer points than necessary, with the aim of decreasing the stiffness and so compensating for the overstiff finite element model. This is cheaper as well. A lower limit on the number of integration points can be obtained by observing that as the mesh is refined, the state of constant strain is reached within an element. In this case, the stiffness matrix equation (9.24) becomes
[ ]
[k ]≈ [ B ]
e
T
[ D ][ B ] ∫
+1 +1
−1 −1
∫ t det [ J ] dr ds .
e
(9.33)
The integral in (9.33) represents the volume of the element. Therefore, the minimum number of integration points, is the number required to evaluate exactly the volume of the element. Taking the thickness, t e , to be constant and noting that
det [ J ] is linear, indicates that the volume can be evaluated exactly using one integration point.
Fig. 9.4 However, in the present case, one integration point is unacceptable since it gives rise to zero-energy deformation modes (Fig. 9.4). These are modes of deformation which give rise to zero strain energy. This will be the case if one of these modes gives zero strain at the integration point. The existence of these modes is indicated by the stiffness matrix having more zero eigenvalues than rigid body modes. Experience has shown that the best order of integration for the 4-node quadrilateral element is a ( 2 × 2 ) array of points. Using the 2 × 2 rule (9.33), we can write
In practice, stresses are evaluated at the Gauss points, where they are found to be accurate (Barlow, 1976). Some programs which use Gauss point stresses extrapolate to the element nodes and then output the mean value if several elements meet at a node. Maximum stresses usually occur at edges of plates or at other discontinuities at the element boundaries. In order not to miss these peak stresses, a refined mesh should be used in these regions. However, Gauss point values are ideal for constructing internal stress contours.
208
FINITE ELEMENT ANALYSIS
9.3 Eight-node quadrilateral
This element is the isoparametric version of the eight-node rectangle presented in section 8.2.2. The difference is that in cartesian coordinates it has curved sides (Fig. 9.5, a) and the master element is a square (Fig. 9.5, b).
The element has three nodes along one edge, so that the displacement variation should be parabolic (three constants) to satisfy compatibility. and s , as are the stresses. The product [ B ] T [ D ][ B ] is therefore fourth order and det [ J ] is cubic. It would seem necessary to use a 3× 3 Gauss quadrature for exact stiffness integration. For a single 8-node rectangular element, even if it is supported conventionally, using a 2 × 2 integration scheme would result in a singular stiffness matrix. In practice, large groups of elements will not suffer from such mechanisms, and 2 × 2 integration is normally used for this popular element. The optimal sampling points for this element are the Gauss points for the 2 × 2 scheme The eight-node quadrilateral employs displacement fields quadratic in r
9. ISOPARAMETRIC ELEMENTS
209
and not those for the higher order 3 × 3 scheme, so the advantages of reduced integration are compounded. For an 8-node quadrilateral element, better displacements are obtained using a 3× 3 Gauss point mesh, nine points in all. It was shown that, in order to evaluate the minimum order of the numerical integration, it is sufficient to examine the Jacobian determinant. For the 8-node quadratic element, det [ J ] is of third order hence the minimum number of integration points is 4 (2 × 2 ) . The eight-node curved 'quad' is ideal for nonlinear problems and can be easily adapted to model cracks by moving the midside nodes. Care is needed because the accuracy declines with excessive shape distortion.
9.3.1 Shape functions
The shape functions (8.29) have the following expressions
and then to work out equations (9.55) by incorporating the relevant material constants. This method (K. A. Gupta & B. Mohraz, 1972) reduces the computing time by a factor of nine over the full matrix multiplication method.
The explicit form of equation (9.57) for the quadratic 8-node element is
⎧σx ⎪ ⎨σy ⎪τ ⎩ xy
⎫ ⎡ D1 ⎪ ⎢ ⎬ = ⎢D 2 ⎪ ⎢ 0 ⎭ ⎣
D2 D4 0
⎡ ∂ N1 ⎢ 0 ⎤ ⎢ ∂x ⎥ 0 ⎥⎢ 0 ⎢ D5⎥ ⎢∂N ⎦ ⎢ 1 ⎢ ∂y ⎣
8
∂ N2 L 0 ∂y ∂ N2 ∂ N8 L ∂x ∂y
⎤ ⎧u 1 ⎫ ⎥⎪ ⎪ ⎥ ⎪v 1 ⎪ ⎥ ⎪u ⎪ ⎥ ⎨ 2⎬ ⎥ ⎪L ⎪ ⎥ ⎪v ⎪ ⎥ ⎪ 8⎪ ⎦⎩ ⎭
wherefrom
σ x = D1 ∑
i =1
8 ∂ Ni ∂ Ni ui + D2 ∑ vi , ∂x i =1 ∂ y
214
FINITE ELEMENT ANALYSIS
σ y = D2 ∑
8
8
i =1
8 ∂ Ni ∂ Ni ui + D4 ∑ vi , ∂x i =1 ∂ y
(9.58)
τ xy = D 5 ∑ ⎜ ⎜
⎛ ∂ Ni ∂ Ni ⎞ ui + vi ⎟, ⎟ ∂x ⎠ i =1 ⎝ ∂ y
In some programs the stresses are determined at nodes, since the nodal positions are readily located and it is convenient to output the displacements and stresses at the same points. It has been found that nodal stresses from an 8-node quadratic element are usually incorrect. However, if the stresses of all elements meeting at a node are averaged, closer results to the true values are obtained. A better alternative is to calculate stresses at the Gauss points, in which superior accuracy is obtained and averaging is not necessary. This is due to the fact that the element stiffness is calculated by sampling at the Gauss points, and it is therefore reasonable to expect the most accurate stresses and strains occurring at the same points.
9.3.6 Consistent nodal forces
Unlike the triangular element, in which all loads can be reduced to nodes intuitively or by statics, for a quadratic isoparametric element the nodal forces due to distributed loads must be computed in accordance with equation (7.44)
{ f }= ∫ [ N ]
e Ve
T
{ p }d V
.
(9.59)
The equivalent nodal forces are added, element by element, into the global load vector which represents the right hand side of the linear set of equations to be solved for displacements.
Edge pressure
When coding the load vectors in a computer program, the actual pressure distribution along an element edge is replaced by a parabolic distribution defined by the pressure values at each of the three nodes along that edge. All intermediate values can be calculated using the shape functions. It is usual to use all the nodes of the element in the computation, so that there is no need to sort out the appropriate shape functions for the three nodes with given pressure values. Consider a distributed load p , specified in force per unit length, acting along the s = +1 edge of an element. The components of the force p d r acting upon an elemental length d r are
9. ISOPARAMETRIC ELEMENTS
215
⎧ ∂y ⎫ ⎪ ∂r ⎪ ⎧ px ⎫ ⎪ ⎪ ⎨ ⎬ = p ⎨ ∂ x ⎬ dr . py ⎭ ⎩ ⎪− ⎪ ⎪ ∂r ⎪ ⎩ ⎭
(9.60)
The consistent load for p is given by a modified form of equation (9.59) in which the volume integral has been reduced to a line integral
{ f }= ∫
e
+1
−1
where [ N ] is given by (9.42).
⎧ ∂y ⎪ ∂r ⎪ p [ N ]T ⎨ ∂x ⎪− ⎪ ∂r ⎩
⎫ ⎪ ⎪ ⎬ dr ⎪ ⎪ ⎭
(9.61)
The above vector is usually integrated numerically and is written as ⎫ ⎪ ⎪ wi [ N ] T fe = (9.62) ⎬ ⎪ i =1 ⎪i ⎭ where pi is the pressure at the Gauss point i along the s = +1 edge and is computed by equation
Hence a uniform load acting along an edge is not distributed in the intuitive ratio of 1 4 : 1 2 : 1 4 , but in the ratio 1 6 : 2 3 : 1 6 (Fig. E9.2, b). The same result could have been obtained noticing that the integrands in the above equations are the shape functions of the three-node isoparametric onedimensional element (4.26).
Gravity loading
For gravity loading (downwards negative) the vector of equivalent nodal forces (9.59) is
{ f }= ∫ m [ N ]
e
T
⎧ 0 ⎫ ⎨ ⎬d x d y = ⎩− g ⎭
+1 +1 −1 −1
∫ ∫ m [N ]
T
⎧ 0 ⎫ ⎨ ⎬ det [ J ] d r d s ⎩− g ⎭
(9.67)
where m is the mass per unit area and g the acceleration in the y direction. The integration is carried out numerically and equation (9.67) takes the form
{ f }= ∑∑ w w
e n n i i =1 j =1
j
⎧ 0 ⎫ m [ N ] T ⎨ ⎬ det [ J ] . ⎩− g ⎭
(9.68)
Example 9.3
Consider the rectangular element of Fig. E9.3, a with gravity loading. Find the equivalent nodal forces.
Solution. Because all the equivalent nodal forces act in the y direction,
Fig. E9.3 Hence for gravity loading the equivalent nodal forces are as shown in Fig. E9.3, b. Again, they are different from the values of − W 12 and − W 6 , which would have been assigned intuitively to the corner and midside nodes respectively.
9.4 Nine-node quadrilateral
This element belongs to the Lagrange family of elements. Apart from the eight nodes located on the boundary, it contains an internal node. The local node numbers for this element are shown in Fig. 9.6, a. The master element is presented in Fig. 9.6, b.
Along the sides of the element, the polynomial is quadratic (with three terms - as can be seen setting s = 0 in u ), and is determined by its values at the three nodes on that side. Higher-order rectangular elements can be systematically developed with the help of the so-called Pascal's triangle, which contains the terms of polynomials of various degrees in the two variables r and s , as shown in Fig. 9.7.
Fig. 9.7 Since a linear quadrilateral element has four nodes, the polynomial should have the first four terms 1, r, s, and rs . In general, a pth-order Lagrange
rectangular element has ( p + 1) 2 nodes ( p = 0 ,1, ...) . The quadratic quadrilateral element has 9 nodes. The polynomial is incomplete. It contains the complete polynomial of the second degree (6 terms) plus other three terms which have to be located symmetrically: the third degree terms r 2 s and r s 2 and also an r 2 s 2 term.
It should be cautioned that the subscripts of N i refer to the node numbering used in Fig. 9.6, a. Since the internal nodes of the higher-order elements of the Lagrange family do not contribute to the inter-element connectivity, they can be condensed out at the element level to reduce the size of the element matrices. Alternatively, one can use the serendipity elements, but their shape functions cannot be obtained using products of one-dimensional interpolation functions.
9.5 Six-node triangle
Higher-order triangular elements can be developed using Pascal's triangle (Fig. 9.8). One can view the position of the terms as the nodes of the triangle, with the constant term and the first and last terms of a given row being the vertices of the triangle. The six node triangle (Fig. 9.9, a) is an element of order 2 (i.e., the degree of the polynomial is 2), as can be seen from the top three rows of Pascal's triangle. The polynomial involves six constants, which can be expressed in terms of the nodal values of the variable being interpolated. By referring to the master element in Fig. 9.9, b, the shape functions can be expressed in terms of area coordinates as for the linear strain triangle (8.36)
N1 = ζ1 ( 2 ζ1 − 1 ) , N 2 = ζ 2 ( 2 ζ 2 − 1 ) , N 3 = ζ 3 ( 2 ζ 3 − 1 ) , N 4 = 4 ζ1 ζ 2 , N5 = 4 ζ 2 ζ3 , N 6 = 4 ζ 3 ζ1 ,
Details on numerical integration schemes for triangles are given in [18, 39]. The Gauss quadrature formulas for a triangle differ from those considered earlier for the rectangle.
9.6 Jacobian positiveness
In the higher-order isoparametric elements discussed above, we note the presence of 'midside' nodes. The midside node should be as near as possible to the centre of the side. It must be placed inside the middle third of a side. This condition ensures that det [ J ] does not attain a value of zero in the element. The sign of det [ J ] should be checked. If the Jacobian becomes negative at any location, a warning message will be signaled indicating the nonuniqueness of mapping. Note that while the Jacobian is always computed at Gauss points, it is more likely to be negative at corners, where stresses may be computed. A necessary requirement for applying equation (9.14) to shape functions is that [ J ] can be inverted. This inverse exists if there is no excessive distortion of the element such that lines of constant r or s intersect inside or on the element boundaries or there are re-entrant angles. When the element is degenerated into a triangle by increasing an internal angle to 1800 then [ J ] is singular at that corner. A similar situation occurs when two adjacent corner nodes are made coincident to produce a triangular element. Therefore to ensure that [ J ] can be inverted, any internal angle of each corner node of the element should be less than 1800 , and, as an internal angle approaches 180 0 there is a loss of accuracy in the element stress, particularly at that corner. If the determinant det [ J ] → 0 , then [ j ] and the operators in increase without limit and consequently produce infinite strains.
where 0 ≤ x 2 ≤ l .
l 3l < x 2 < , then det [ J ] does not vanish on the element. This explains 4 4 why it is recommended to place precautiously the 'midside' node inside the middle third of a side.
If
10.
PLATE BENDING
Flat plate structures, such as the floors of buildings and aircraft, enclosures surrounding machinery and bridge decks are subject to loads normal to their plane. Such structures can be analyzed by dividing the plate into an assemblage of twodimensional finite elements called plate bending elements. These elements may be either triangular, rectangular or quadrilateral in shape. In this chapter, finite element displacement models for the flat plate bending problem are discussed. Thin plates with transverse shear neglected are analyzed based on Kirchhoff's classical theory. Plates with a constant shear deformation through the plate thickness are treated in the Reissner-Mindlin plate theory. For thick or composite plates some higher-order shear deformation plate theories are available, in which on the faces of plate the shear strains are equal to zero.
10.1 Thin plate theory (Kirchhoff)
A plate is described as a structure in which the thickness is very small compared with the other dimensions, that is the thickness-to-span is h l ≤ 0.1 . For this case, it can be assumed that the plate deformation may be expressed by the deformation state at the middle surface, which is the plane midway between the faces of the plate. For thin plates, Kirchhoff's hypotheses are adopted: a) there is no deformation (stretching) in the middle plane of the plate; b) normals to the middle plane of the undeformed plate remain straight and normal to the middle surface of the plate during deformation; and c) the direct stress in the transverse direction can be disregarded. The x-y plane is taken to coincide with the middle surface of the plate (Fig. 10.1) and the positive z-direction is upwards. The plate has constant thickness h and is subject to distributed surface loads.
10.2 Thick plate theory (Reissner-Mindlin)
For moderately thick plates, the thickness-to-span ratio is not small enough to neglect transverse shear deformations and Kirchhoff's assumption is no longer valid. To overcome this problem, the classical hypothesis of zero transverse shear strain is relaxed. First, Reissner proposed to introduce the rotations of the normal to the plate midsurface in the xOz and yOz plane as independent variables. Then, Mindlin simplified Reissner's assumption considering that normals to the plate midsurface before deformation remain straight but not necessarily normal to the plate after deformation. The displacements of the middle surface are independent of the rotations of the normal (Fig. 10.2, b). The transverse normal stress is disregarded as in the Kirchhoff's theory. According to Reissner-Mindlin assumptions, the displacements parallel to the middle surface can be expressed as
∂w ∂w and θ y = − , the above expression reduces to (10.20). ∂y ∂x The main drawback of this theory is the arbitrary averaging of the shear strains. If a constant shear strain is considered through the plate thickness, on the faces of plate the shear strains are not zero, which is not true. This assumption is equivalent to the introduction of 'parasitic' shear stresses that force the normal to remain a straight line. As the thickness of the plate becomes extremely thin, the shear strain energy predicted by the finite element analysis can be magnified unreasonably and a 'shear locking' occurs for large span-to-thickness ratios. If θ x =
232
FINITE ELEMENT ANALYSIS
10.3 Rectangular plate bending elements
For a thin plate bending element, the strain energy is given by (10.20) and the potential of the external load by (10.21). The highest derivative appearing in these expressions is the second. Hence, for convergence, it will be necessary to ∂w ∂w and are continuous between ensure that w and its first derivatives ∂x ∂y elements. These three quantities are, therefore, taken as degrees of freedom at each node. Also, complete polynomials of at least degree two should be used, according to the convergence criteria of the Rayleigh-Ritz method. The assumed form of the displacement function, irrespective of the element shape, is
w = a1 + a2 x + a3 y + a4 x 2 + a5 x y + a6 y 2 + higher degree terms.
(10.33)
Figure 10.4 shows a thin rectangular element of thickness h, having four node points, one at each corner.
Fig. 10.4 The dimensionless coordinates ξ = x a and η = y b will be used in the following with the origin at the plate centre.
10.3.1 ACM element (non-conforming)
Figure 10.5 shows the ACM element (Adini, Clough, Melosh - 1961, 1963). There are three degrees of freedom at each node, namely, the transverse ∂w ∂w displacement w , and the two rotations θ x = and θ y = − . In terms of the ∂y ∂x dimensionless coordinates ξ and η , these become
10. PLATE BENDING
233
θx =
1 ∂w , b ∂η
θy = −
1 ∂w . a ∂ξ
(10.34)
Since the element has 12 degrees of freedom, the displacement function can be represented by a polynomial having twelve terms, that is
w = α1 + α 2 ξ + α 3η + α 4 ξ 2 + α 5 ξ η + α 6η 2 +
+ α 7 ξ 3 + α 8 ξ 2η + α 9 ξη 2 + α10η 3 + α11ξ 3η + α12 ξη 3 .
(10.35)
Note that this function is a complete cubic to which two quartic terms ξ 3η and ξη 3 have been added, which are symetrically located in Pascal's triangle. This ensures that the element is geometrically invariant.
This indicates that the displacement w , and hence the rotation θ x , are uniquely determined by their values at nodes 2 and 3. If the element is attached to another rectangular element at nodes 2 and 3, then w and θ x will be continuous along the common side. Unfortunately, this is not the case with the rotation θ y . The rotation θ y is given by (10.34)
The above expressions indicate that θ y is determined by the values of w and θ x at nodes 1, 2, 3, and 4 as well as by θ y at nodes 2 and 3. The rotation θ y is not continuous across the side 2-3 and the element is non-conforming. Substituting (10.43) into (10.4) and (10.8) gives
Ue = 1 2
The above relationships are presented in reference [87]. In deriving this result, it is simpler to use the expression (10.36) for w and substitute for {α } after performing the integration. A typical integration is then of the form
+1 +1
−1 −1
∫∫
ξ mη n dξ dη = ⎨
⎧ ⎪
0 4 ⎪ (m + 1)(n + 1) ⎩
m or n odd m and n even
For p z = constant , substituting the shape functions into
{ f } = ∫ ⎣N ⎦
e A
T
pz d A
(10.54)
and integrating, the vector of equivalent nodal forces is obtained as
ab T ⎣ 3 b − a 3 b a 3 − b a 3 − b − a⎦ . 3 Stresses at any point in the plate are given by (10.5). In terms of the nodal displacements they can be expressed as
e z
{ f }= p
238
FINITE ELEMENT ANALYSIS
{σ } = − z [ D ][ B ]{q e },
(10.55)
where [ B ] is defined in (10.50). The most accurate values are at the Gauss points of a (2 × 2) numerical integration scheme.
10.3.2 BFS element (conforming)
A conforming rectangular element, of the form shown in Fig. 10.4, commonly referred to as the BFS element (Bogner, Fox and Schmit - 1966), can be obtained using products of separate one-dimensional Hermitian shape functions (5.21) as for uniform slender beams. It is a four-node thin plate bending element. The nodal expansion of the displacement function has the form (10.43) with
in which ( ξi ,ηi ) are the coordinates of node i. Unfortunately, when we examine the derivatives of these products we find that the twist ∂ 2 w ∂ξ ∂η is zero at all four corners and that there is no constant component to this second derivative. As this controls the shear strain in equation (10.2), this violates the fundamental requirement that the element can represent all constant strain states. In the limit, as an increasing number of elements is used, the plate will tend towards a zero twist condition. A compromise is to use the cubic Hermitian polynomials for the deflection shape functions but reduce the order to linear for the rotation shape functions. This element does have constant strain but is non-conforming, and discontinuities in slope occur at interfaces. However, as the mesh is refined and the element size is decreased, the results are convergent. The solution used for the BFS element is the introduction of ∂ 2 w ∂ξ ∂η as an additional degree of freedom at each node. In this case, the displacement function is of the form (10.43) but with 16 terms
It can be shown that this element can perform rigid body movements without deformation and can describe pure bending behaviour in the x- and ydirections. This is ensured by the presence in the functions (10.59) of the first six terms in (10.33). The element stiffness matrix and consistent vector of nodal forces are given by (10.49) and (10.53) where the matrix ⎣N ⎦ is defined by (10.43) and (10.59). Although the BFS element is more accurate than the ACM element, it is difficult to use in conjunction with other types of elements in built-up structures due to the presence of the degree of freedom ∂ 2 w ∂x ∂y . This is overcome in the WB element (Wilson, Brebbia – 1971) by introducing the approximations
′ w′x y1 = ′ w′x y 3 = 1 θ y1 − θ y 4 , 2b 1 θ y 2 − θ y3 , 2b
(
)
′ w′x y 2 = ′ w′x y 4 =
1 θ x 2 − θ x1 , 2a 1 θ x3 − θ x 4 . 2a
(
)
(
)
(
)
Applying the above constraints to the BFS element makes it a nonconforming one. The transverse displacement and tangential slope are continuous between elements but the normal slope is not.
10.3.3 HTK thick rectangular element
When the thickness is greater than about a tenth of the plate width, the shear deformations become important and a Reissner-Mindlin plate model is adopted. For a thick plate-bending element, the strain energy expression is (10.30) and the potential of the external load is (10.21), with { χ } and {γ } given by (10.24) and (10.26). The highest derivative of w , θ x and θ y appearing in these expressions is the first. Hence, for convergence, w , θ x and θ y are the only degrees of freedom required at the nodal points (Fig. 10.5).
240
FINITE ELEMENT ANALYSIS
The four-node HTK element (Hughes, Taylor and Kanoknukulcha - 1977) expands separately w , θ x and θ y in terms of their nodal values. It represents the shear deformation directly, without having to infer it as the derivative of the bending moment. The displacement functions are of the form
w = ∑ Ni w i , θ x = ∑ Ni θ x i , θ y = ∑ Ni θ y i ,
i =1 i =1 i =1
4 4 4
(10.60)
where the functions N i are defined by (9.3), that is
Ni = 1 1 + ξ i ξ 1 + η iη . 4
(
)(
)
(10.61)
These functions ensure that w , θ x and θ y are continuous between elements. In matrix form, expressions (10.60) can be written
⎧w ⎪ ⎨ θx ⎪θ ⎩ y
where
⎫ ⎪ e ⎬=[N] q , ⎪ ⎭
{ }
(10.62)
{ q } = ⎣w
e T
1
θ x1 θ y1 L w4 θ x 4 θ y 4 ⎦ ,
0 0 L N4 0 0 0 N4 0 0 ⎤ 0 ⎥. ⎥ N4 ⎥ ⎦
(10.44)
and
⎡ N1 [ N ]= ⎢ 0 ⎢ ⎢ 0 ⎣
N1 0
0 L N1 L
(10.63)
Substituting (10.62) into (10.24), (10.26) and (10.30) gives
Ue = 1 2
{ q } [ k ] { q },
e T e e
(10.64)
where the element stiffness matrix k e can be written as the sum of the matrices due to bending and shear
The remaining submatrices of (10.74) are given by relationships corresponding to (10.52). Substituting (10.73) into (10.71) and the resulting matrices into (10.67) gives the element stiffness matrix due to shear
The remaining submatrices of (10.75) are given by relationships corresponding to (10.52). The above expressions are presented in reference [87]. The vector of equivalent nodal forces is ⎧ pz ⎫ ⎪ f = ⎣N ⎦ ⎨ 0 ⎬ d A (10.76) ⎪ 0 ⎪ A ⎩ ⎭ For pz = constant, substituting the shape functions from (10.63) and (10.61) into (10.76) and integrating gives
{ } ∫
e
T⎪
{ f }= p a b ⎣ 1
e z
0 0 1 0 0 1 0 0 1 0 0⎦ T
(10.77)
which means that one quarter of the total force is concentrated at each node. The bending and twisting moments per unit length are given by
244
FINITE ELEMENT ANALYSIS
⎧ Mx ⎪ ⎨ My ⎪M ⎩ xy
⎫ ⎪ h3 = − [ I3 ] [ D ] B B ⎬ 12 ⎪ ⎭
[ ]{q },
e
(10.78)
where [ I 3 ] is defined by (10.53). The most accurate values are at the Gauss points of a (2 × 2) numerical integration scheme. The shear forces per unit length are ⎧ Qx ⎫ ⎨ ⎬=κ h ⎩ Qy ⎭
[ D ][ B ]{q }.
S S e
(10.79)
The most accurate values are at the centre of the element. Benchmark problems have shown that the HTK element yields accurate solutions for simply supported or clamped plates. However, large errors can occur in the case of cantilever plates. The above analysis can be easily extended to 8-node or higher-order plate elements. The bending strains in the 8-node version are recovered accurately if sampled at the reduced (2 × 2) Gauss points. In very thin plates, the shear strains become very small and near-zero values in (10.26) thus imply a dependent relationship between w , θ x and θ y which is not true for thick plates. This is avoided by selective integration, lowering the integration order for the stiffness matrix due to shear. However, the standard (2 × 2) integration is reported to give good results for width-to-thickness ratios up to 50. There is no perfect rectangular plate-bending element. There are different formulations using mixtures of corner and mid-side freedoms in order to achieve near complete polynomials. Alternative methods, considering rather difficult to generate a displacement function valid over the entire rectangular element, suggest to subdivide the element into regions (e.g., into four triangles) and work with different displacement functions over each region. Obviously, the individual functions must be continuous (up to the first derivatives) across the interior boundaries as well as the exterior boundaries.
10.4 Triangular plate bending elements
In this section we outline the development of some plate-bending triangular elements.
10. PLATE BENDING
245
10.4.1 Thin triangular element (non-conforming)
Figure 10.6 shows the T element (Tocher - 1962). The local x-axis lies along the side 1-2 and the local y-axis is perpendicular to it. Nodes 1, 2 and 3 have coordinates (0,0) , (x2 ,0) and (x3 , y3 ) . There are three degrees of freedom at each ∂w node, namely, the transverse displacement w , and the two rotations θ x = and ∂y
and, therefore, cannot be inverted. If this occurs, the locations of the nodes should be altered to avoid this condition. This element is non-conforming. Evaluating (10.83) along the side 1-2, of equation y = 0 , gives
The rotation θ x is a quadratic function having coefficients α 3 , α 5 and α 8 . These cannot be determined using the values of θ x at nodes 1 and 2 only. Therefore the normal slope is not continuous between elements. Moreover, the assumed function (10.80) is not invariant with respect to the choice of coordinate axes, due to the x 2 y and x y 2 terms. The edge 1-2 was taken along the x-axis. In this case θ x is a tangential component and θ y is a normal component of rotation. The transverse displacement
w and the normal component θ y are continuous between elements, while the
tangential component θ x is not. To alleviate this, in some plate-bending elements a linear variation of the tangential component of rotation is adopted. Substituting (10.88) into (10.4) and (10.8) gives
Ue = 1 2
{ q } [ k ]{ q }
e T e e
(10.91)
where the element stiffness matrix in the local coordinate system is h [ k ] = [ A ] ∫ 12 [ B ]
e e −T 3 A T
Note that nodal coordinates are usually given in global axes. Calculation of the element stiffness matrix referred to local axes requires the local coordinates of
248
FINITE ELEMENT ANALYSIS
nodes 2 and 3. These can be obtained from their global coordinates using the corresponding transformation matrix. In order to assemble the global stiffness matrix, the element matrices have to be first transformed from local to global axes through a matrix triple product which is one of the more time-consuming procedures in finite element analysis.
10.4.2 Thick triangular element (conforming)
Consider a thick triangular element (THT element – Henschel, Tocher, 1969) referred to a global coordinate system. Nodes 1, 2 and 3 have area coordinates ζ 1 , ζ 2 and ζ 3 (8.35). There are three independent degrees of freedom at each node, namely, the transverse displacement w , and the two rotations θ X and θY with respect to the global axes. The displacement functions are assumed to be
w = ∑ ζ i wi , θ X = ∑ ζ i θ X i , θY = ∑ ζ i θY i ,
i =1 i =1 i =1 3 3 3
(10.95)
where wi , θ X i and θY i are the degrees of freedom at node i. They ensure that w ,
The complete element stiffness matrix is the sum (10.65). The above relationships are presented in reference [87]. For pz = constant, substituting the shape functions from (10.98) into (10.76) and integrating gives the vector of equivalent nodal forces
{ f }= p
e
z
A T ⎣1 0 0 1 0 0 1 0 0⎦ 3
(10.105)
which shows that one third of the total force is concentrated at each node.
10.4.3 Discrete Kirchhoff triangles (DKT)
The difficulties to formulate simple (minimum degrees of freedom) and high-performing elements based on the Kirchhoff theory at the continuum level (which requires a C1 continuity for w ) led to the development of the so-called Discrete Kirchhoff (DK) technique (Wempner-1969, Stricklin et al.-1969, Dhatt1967).
10. PLATE BENDING
251
In the formulation of various DK plate elements, only the bending strain energy is considered, in which the curvatures are expressed, as in the ReissnerMindlin plate theory, in terms of the first derivatives of the rotations (10.24). In this case only C 0 approximations of these rotations can be considered. The shear strain energy is neglected. The Kirchhoff constraints are imposed in a discrete manner on the element or/and on the sides. For instance, the transverse shear strains are either taken equal to zero at the mid-side points (collocation on sides) or their integral along each edge is taken equal to zero. The aim is to preserve the C 0 continuity of the tangential components of rotations (normal slopes). Constant curvature patch-tests are needed to check the validity of the elements. The T element presented in section 10.4.1 can be called a 'continuous' Kirchhoff triangle. It was shown that, along a side, the rotations vary quadratically and their tangential component (normal slope) is not continuous between elements. To remedy this, in the discrete Kirchhoff triangles the two components of rotations are assumed independent of one another. The tangential component θ s is assumed to vary linearly along each edge, while the normal component θ n varies quadratically (Fig. 10.7, a). The latter condition means that the transverse displacement w can vary cubically.
Consider a discrete Kirchhoff triangle (DKT) with three nodes and three degrees of freedom per node (sometimes called DKT9). Initially, it is assumed that it has six nodes (Fig. 10.7, b). Then, the degrees of freedom at mid-side nodes are eliminated. The rotation θ n varies parabolically (Fig. 10.8, a). Its expression in terms of the dimensionless coordinate can be written as
where the first term in the right hand side is expressed in terms of the rotations at the corner nodes. The parameter α k has to be eliminated. The rotation θ s is assumed to vary linearly (Fig. 10.8, b)
θ s = ( 1 − s′ )θ s i + s′θ s j .
Fig. 10.8 In order to eliminate the degrees of freedom at mid-side nodes, the three parameters α 4 , α 5 , α 6 must be expressed in terms of the degrees of freedom at the corner nodes. This is done requiring the shear strains γ s z to vanish along each side. One kind of 'discrete' Kirchhoff constraint is formulated in integral form as
j j
The equation (10.109) can also be obtained by adopting a Hermitian cubic polynomial for w (s ) and using the discrete Kirchhoff collocation constraint (zero ∂w shear strain) γ s z = + θ n = 0 at points i, j, k. Then, the condition γ s z = 0 will be ∂s satisfied at all points along the contour, because w is cubical and θ n is quadratic. The rotations θ x and θ y can be expressed as
In the above expressions the indices k and m relate to the two edges having the common corner point i, as shown in Table 10.1. Taking into account the hypotheses adopted in formulating the DKT, the nodal rotations θ x i and θ y i have the expressions used in Kirchhoff's theory
Because the parameters α k are eliminated using an expression which is a function of the nodal variables of the side k only, the continuity C 0 of θ n is maintained. The stiffness matrix of the DKT element can be expressed in explicit form, in a local coordinate system [18], using a Hammer integration rule. In the following, it will be derived in global coordinates as in [68]. Equation (10.112) can be split as
θ x = ⎣G ⎦ q e ,
{ }
θ y = ⎣H ⎦ q e .
{ }
(10.117)
The shape function row vectors ⎣G ⎦ and ⎣H ⎦ , presented explicitly in terms of the local oblique coordinates ξ and η , can be rewritten as | 677.169 | 1 |
9780130861078
013086107300
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Summary
For Technical Math courses that do not include calculus. This text addresses curriculum and pedagogy standards that are initiatives of the American Mathematical Association of two-year Colleges (AMATYC), the National Council of Teachers of Mathematics (NCTM), and the Mathematics Association of America (MAA). It uses simplified language that appeals to a variety of student learning styles, promotes active and independent learning, and strengthens critical thinking and writing skills. A six-step approach to problem solving, numerous tips, and clear, concise explanations throughout the text enable students to understand the concepts underlying mathematical processes. | 677.169 | 1 |
Unit 1 Vocabulary
Students are expected to define and provide examples for the following: 1. Algebraic Expression 2. Variable 3. Constant 4. Coefficient 5. Factors 6. Linear Equation 7. Exponential Equation 8. Linear Inequality 9. Domain 10. Range 11. Function 12. Simple Interest 13. Compound Interest 14. Exponential Growth 15. Exponential Decay This web site has activities to help students more fully understand and retain new vocabulary. Definitions and activities for these and other terms can be found on the Intermath website. Intermath is geared towards middle and high school students. | 677.169 | 1 |
Algebra 2 Bundle of 22 Activities
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44 MB|285 pages
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Algebra 2 Bundle of 22 Activities. I have bundled together 22 of my most popular activities for Algebra 2. This is a sampler of the many activities available to engage your students and help you teach and reinforce your lessons. Great End of the Year Review also! | 677.169 | 1 |
Description & Features
Please Note: This eBook will only run on Windows® OS at this time. A Mac OS X version will be available soon.
Move over Sudoku, here come Balance Benders™! You can use these activities as quick, fun logic problems or as stepping stones to success in algebra. Students develop problem-solving skills and pre-algebra skills as they solve balance puzzles that are more fun and addictive than Sudoku puzzles! Students must analyze each balance to identify the clues, and then synthesize the information to solve the puzzle. Try one—and then try to stop!
Balance Benders™ puzzles spiral—gradually introducing more challenging concepts to give students the confidence and ability to master algebra word problems. These concepts include:
Deductive Thinking
Pre-Algebra
Analysis
Commutative Property
Distributive Property
Associative Property
Substitution Property of Equality and Inequality
Addition Property of Equality and Inequality
Subtraction Property of Equality and Inequality
Multiplication Property of Equality and Inequality
Division Property of Equality and Inequality
eBook Ordering Our eBooks are electronic versions of the book pages. You can immediately download your eBook from "My Account" under the "My Downloadable Product" section after you place your order. | 677.169 | 1 |
data analysis has become an integral part of any scientific study.
This book starts with a concise but rigorous overview of the basic notions of projective geometry, using straightforward and modern language. The goal is not only to establish the notation and terminology used, but also to offer the reader a quick survey of the subject matter. In the second part, the book presents more than 200 solved problems | 677.169 | 1 |
...
Show More the theory and the distinction between matrices and tensors are emphasized, and absolute- and component-notation are both employed. While the mathematics is rigorous, the style is casual. Chapter 1 deals with the basic notion of a linear vector space; many examples of such spaces are given, including infinite-dimensional ones. The idea of a linear transformation of a vector space into itself is introduced and explored in Chapter 2. Chapter 3 deals with linear transformations on finite dimensional real Euclidean spaces (i.e., Cartesian tensors), focusing on symmetric tensors, orthogonal tensors, and the interaction of both in the kinetically important polar decomposition theorem. Chapter 4 exploits the ideas introduced in the first three chapters in order to construct the theory of tensors of rank four, which are important in continuum mechanics. Finally, Chapter 5 concentrates on applications of the earlier material to the kinematics of continua, to the notion of isotropic materials, to the concept of scalar invariant functions of tensors, and to linear dynamical systems. Exercises and problems of varying degrees of difficulty are included at the end of each chapter. Two appendices further enhance the text: the first is a short list of mathematical results that students should already be familiar with, and the second contains worked out solutions to almost all of the problems. Offering many unusual examples and applications, Linear Vector Spaces and Cartesian Tensors serves as an excellent text for advanced undergraduate or first year graduate courses in engineering mathematics and mechanics. Its clear writing style also makes this work useful as a self-study guide | 677.169 | 1 |
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An accessible introduction to the topics of discrete math, this best-selling text also works to expand students' mathematical maturity.
With nearly 4,500 exercises, Discrete Mathematics provides ample opportunities for students to practice, apply, and demonstrate conceptual understanding. Exercise sets features a large number of applications, especially applications to computer science. The almost 650 worked examples provide ready reference for students as they work. A strong emphasis on the interplay among the various topics serves to reinforce understanding. The text models various problem-solving techniques in detail, then provides opportunity to practice these techniques. The text also builds mathematical maturity by emphasizing how to read and write proofs. Many proofs are illustrated with annotated figures and/or motivated by special Discussion sections. The side margins of the text now include "tiny URLs" that direct students to relevant applications, extensions, and computer programs on the textbook website.
About The Author
Richard Johnsonbaugh is Professor Emeritus of Computer Science, Telecommunications and Information Systems, DePaul University, Chicago. Prior to his 20-year service at DePaul University, he was a member and sometime chair of the mathematics departments at Morehouse College and Chicago State University. He has a B.A. degree in mathema... | 677.169 | 1 |
Matrices and Vectors
For this matrices and vectors worksheet, students solve 9 various types of problems related to forming matrices and vectors. First, they set up each system of equations listed in matrix form using mathematical notation. Then, students plot the function and determine the apparent frequency of the sampled signal. They also create a simple sum of sines and cosines of varying frequencies. | 677.169 | 1 |
Section 8.8 Worksheet: Matrix
For this matrix worksheet, students identify all the minors and cofactors of the elements in a matrix. They find the determinant of a given matrix and expand the determinant. This two-page worksheet contains seven multi-step problems. | 677.169 | 1 |
0131703610-Key Touch Key: Developing Speed and Accuracy
This book and software package is designed to teach users how to use 10-key calculators by touch using computer assisted instruction and the computer numberpad. Users of this package will become versed in the use of the desktop calculator and the Windows calculator. Finally, partially completed Excel spreadsheets are provided to help develop solid data entry skills. (System Requirements: Intel 486 processor or equivalent; 16 Meg ram (32 Meg recommended); VGA (640 x 480) or SVGA (800 x 600) 256 color or higher; CDROM Drive (Installation only); 3 meg available (Drive C); Windows 95, 98, 2000, ME, XP, or NT; Windows Excel is required for completion of spreadsheet exercises.) Topics include developing your ten-key skill using touch key 10-key software; applying your ten-key skill using the desktop calculator (including addition, subtraction, multiplication and multifactor multiplication, and division and multifactor division); applying your ten-key skill using the windows calculator; and applying your ten-key skill completing spreadsheets (including data entry in spreadsheets). For anyone wanting to learn how to use 10-key calculators and become versed in the use of the desktop calculator and the Windows calculator | 677.169 | 1 |
97806184701 Excursions With Cd Plus Smarthinking
By presenting problem solving in purposeful and meaningful contexts, Mathematical Excursions, 2/e, provides students in the Liberal Arts course with a glimpse into the nature of mathematics and how it is used to understand our world. Highlights of the book include the proven Aufmann Interactive Method and multi-part Excursion exercises that emphasize collaborative learning. An extensive technology program provides instructors and students with a comprehensive set of support tools | 677.169 | 1 |
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Mathematics
Page Content
Mathematics Department
Basic Skills Program
The Basic Skills Program consists of Pre-algebra, Beginning Algebra, Intermediate Algebra, and Geometry. Intermediate Algebra, and sometimes Geometry, are prerequisites to all of our transfer-level courses. At least one Intermediate Algebra section is offered as an online course each semester.
Transfer Program (General, Education, or Business majors)
Transfer-level math courses include Math for General Education, Math for Elementary Education, Finite Math, and Statistics. These courses are designed for students who are majoring in business, education, or the liberal arts. Elementary Statistics is offered in the traditional classroom setting as well as online.
Transfer Program (Science, Technology, Engineering, or Math majors)
This program is designed for students on the advanced math track and includes Pre-Calculus Algebra, Trigonometry, Calculus I, Calculus II, Multivariable Calculus, Calculus for Business and Social Science, Differential Equations, Discrete Mathematics, and Linear Algebra. Pre-calculus, Algebra and Trigonometry may also be taken online. Multivariable Calculus and Differential Equations are offered alternate semesters at Leland High School. Graphing calculators are required for all listed courses. Some courses may also incorporate the Maple mathematical software package.
Transfer Program
In addition to AA and AS degrees, Associate Degrees for Transfer (AA-T and AS-T) are offered for students who intend to complete a bachelor's degree in a similar major at a California State University.
AMATYC Mathematics Contest
San Jose City College students participate in a national mathematics contest sponsored by the American Mathematics Association of Two-Year Colleges (AMATYC). Each of the two rounds consists of 20 problems to be solved within a one-hour time period with the aid of a graphing calculator. Problems may include material up to pre-calculus mathematics, including some probability and trigonometry problems. The national winner receives a $3,000 scholarship. The mathematics department also awards prizes to San Jose City College students with the top five scores for our school. Any San Jose City College student may participate. (Only non-college degree holding students count for the national contest.) Over 100 students participated in Round One of 2013.
Tutoring Center and Homework Software Links
The Tutoring Center supports SJCC students by providing free one-on-one and group tutoring in most academic subjects offered on campus. For more details, see the On-Campus Resources under Current Students.
Assistance to Online Learners:
Peer-Led Learning
Peer-Led Team Learning (PLTL) is a small-group collaborative approach to learning, studying, and orally communicating chemistry, biology, math and physics. Click on the first link to find out our current semester schedule of PLTL workshops. Students who become peer leaders earn a modest stipend and are offered opportunities to participate in local and national conferences, internships, and more.
Page Content II
STEM Core
SJCC is pleased to offer STEM Core, a cohort-based learning community seeking to increase the number of students in engineering and computer science. It is open students who are eligible for Math 13 (Intermediate Algebra), and Women, Veterans, and Minorities are especially encouraged to apply. Students in the program will be guaranteed admission into a 2-semester mathematics sequence consisting of Math 13, Math 14 (Geometry), and Math 25 (Precalculus), and will compete for high-quality engineering and computer science internships at places like NASA Ames and Lawrence Livermore Lab upon completion. Other benefits include enhanced academic and professional support through supplemental instruction and career development workshops, exposure to engineering applications, field trips to Silicon Valley businesses, and free textbook rentals. | 677.169 | 1 |
Beschreibung:
Über diesen Titel:
Inhaltsangabe: This text and interactive CD-ROM help teachers extend their instructional practices through innovative approaches for teaching geometry as developed by the Open University′s Centre for Mathematics Education.
Review:
′Geometry is often given less time in the teaching timetable than other aspects of mathematics. This book encourages practitioners to think about and raise its profile, indeed achieving what its title suggest′ - Primary Practice
`This creative, innovative and fascinating book/CD package is one you "MUST BUY". All prospective, new and experienced teachers of mathematics can use it to transform their teaching. All readers can use it to reignite their fascination with mathematics. This book fosters not only a curiosity about geometry itself but crucially focuses on how learners can actively engage in thinking about geometry and its central key ideas. Symbolically, the book/CD invites readers to engage actively with it, so that its very style embodies the approach to thinking about geometry that it advocates.
The structure of the book is helpful in supporting readers who want to dip into the book for a particular topic or theme and the authors have interspersed geometric, reflective and pedagogic activities woven into a rich interplay between the ideas themselves and ways in which teachers might facilitate learners′ engagement with the ideas.
By making the focus of this book, thinking about geometry, the authors have highlighted the dimensions of geometric activity that enable us to engage. Throughout, the emphasis is on meaning making and the ways in which we visualize, represent, communicate and reason about geometric ideas. Images abound and the reader is encouraged to conceive geometry both dynamically and statically. Communication and meaning making are the essence of any learning and the authors have been very successful in conveying that message′ - Professor Sylvia Johnson, Sheffield Hallam University
"This book contains an amazing collection of geometric problems that will interest and challenge most readers, though its pedagogical context is designed primarily for mathematics teachers. Part of a three-book series written by the Open University′s Centre for Mathematics Education, it focuses on geometric thinking--what it means, how to develop it, and how to recognize it. Using curricular and research documents as a foundation, the book′s structure thoroughly develops four key ideas of geometric thinking: invariance, geometric language and points of view, reasoning, and visualizing and representing. Using a great number of interesting and clever activities (many provided on the accompanying CD-ROM), each key idea is developed and discussed thoroughly, and readers are strongly encouraged to participate actively as both problem solvers and geometric thinkers. An additional section discusses the related pedagogical issues surrounding each key idea, followed by two concluding sections that provide a unified structure for the four geometric themes and many teaching strategies that were developed. Again, the book is exemplary in its provision of rich problems and pedagogical commentary, offering a welcome perspective on geometrical thinking that spans the primary through the high school grades. Summing Up: Highly recommended." (J. Johnson CHOICE 2006-06-01) | 677.169 | 1 |
KS3 (Key Stage 3) "Algebra" Maths eBook is an excellent introduction to algebra for the 11-14 years of age student. Vividly embracing the subject matter from introduction, through inverse operations, equations, the order of operations, towards solving simple equations, reviewing algebraic conventions, BODMAS, manipulating algebraic expressions, reviewing algebra in number patterns and sequences towards more expressions and formulae, expanding expressions involving brackets, factorising, rearranging a formula and solving equations of the forms y = mx + c, y = ax2 + bx + c, solving simultaneous equations and understanding equalities when the student reviews, learns and then understands the contents of this maths eBook he will ace any exam that he has to take. | 677.169 | 1 |
High school teaching resources
Inspire your students with a range of middle school and high school teaching resources. Whether you're teaching Grade 9 or Grade 12 high school resources to support your teaching.
Structured worksheet to help pupils take key notes and examples for the topic of Algebra and Functions (Polynomial Long Division, Remainder and Factor Theorem and Binomial Expansion). Designed for the new A Level Maths (2017+), Edexcel Specification | 677.169 | 1 |
GCE Maths & Further Maths
Awarding Body : Edexcel
Overview of Course
In these courses you will build on the knowledge, skills and understanding learnt in the GCSE Maths course. Both the AS course and the A2 course consist of two compulsory Core Maths units and one more Applied Maths unit. At the WMG Academy we have opted to focus on the Mechanics units to support the Engineering courses.
Mechanics 2 (M2) – Kinematics of a particle moving in a straight line or plane; centres of mass; work and energy; collisions; statics of rigid bodies.
Assessment
Assessment is in the form of externally assessed written examinations, which are taken in June.
Three written papers: each contributes 33.3% of the final AS grade (16.7% of the A2)
Each paper lasts 1 hour 30 minutes
75 marks on each paper
C1 is a non-calculator paper: for all other unit exams, calculators can be used.
The assessment objectives and weightings are:
Recall and use knowledge of the prescribed content – approx. 30%
Construct rigorous mathematical arguments and proofs – approx. 25%
Select and apply mathematical models in a rang of contexts – approx. 10%
Career Links
GCE Maths encourages students to develop confidence in, and a positive attitude towards, mathematics and to recognise the importance of mathematics in their own lives and to society. This qualification prepares students to make informed decisions about the use of technology, further learning opportunities and career choices.
It supports progression into further and higher education, training or employment, in a wide variety of disciplines, particularly science, technology and engineering.
write legibly, with accurate use of spelling, grammar and punctuation in order to make the meaning clear
select and use a form and style of writing appropriate to purpose and to complex subject matter
organise relevant information clearly and coherently, using specialist vocabulary when appropriate. | 677.169 | 1 |
Mathematical Statistics
Introducing the principles of statistics and data modeling
Introduction to Mathematical Statistics and Its Applications, 6th Edition is a high-level calculus student's first exposure to mathematical statistics. This book provides students who have already taken three or more semesters of calculus with the background to apply statistical principles. Meaty enough to guide a two-semester course, the book touches on both statistics and experimental design, which teaches students various ways to analyze data. It gives computational-minded students a necessary and realistic exposure to identifying data models. | 677.169 | 1 |
Linear algebra summary
On the heels of how successful it was for me to work with Advanced Calculus demystified, I picked up Linear algebra demystified from the library to work through. I've been working through it from April 11 to April 27. This book was similarly effective for me, being a cursory, wide review, and providing lots of worked examples. David McMahon does a good job, and apparently has written a number of these books.
Chapter 1 – Systems of linear equations I learned how to carefully and correctly perform Gauss-Jordan elimination reduction of a matrix. For me the trick again is just to be very very organized, and not to let lack of space allow me to compromise the steps. Everything must be double checked, and this is only easy with well organized work. Also,
The Hermitian conjugate of is denoted and is the transpose and complex conjugate. A matrix with an inverse is called nonsingular (note, no information is lost when A operates on a vector).
Calculating the inverse of a matrix (of any size), can be accomplished by calculating the determinant and the adjugate of the matrix. The minor of a matrix is created by eliminating (row, col)(m,n) . The cofactor is the signed minor for (m,n) . The adjugate of is the matrix of cofactors. , so
Chapter 3 – Determinants
Chapter 4 – Vectors
Chapter 5 – Vector spaces A vector space is a set of elements that is closed under addition, multiplication. Is associative, commutative. There exists an identity element and inverse under addition. Scalar multiplication is associated and distributive. There exists an identity element for multiplication. Given a vector space V, the subset W is a subspace if W is also a vector space. This is verifiable by checking only for the zero vector in W and closure under addition in W. E.g. has a subspace . Is it true that generally, has subspaces of , or is there an infinite number?
A matrix A in row-echelon form has a row space of A and column space of A. The null space of a matrix is found from , and the rank(A) + nullity(A) = n. The closure relation or completeness means that we can write the identity in terms of outer products of a set of basis vectors.
Chapter 6 – Inner product spaces The inner product is given by
Inner products on function spaces can be used to check for orthogonality of functions! The Gram-Schmidt process can generate an orthonormal basis from an arbitrary basis.
Chapter 7 – Linear transformations
Chapter 8 – Eigenvalues The characteristic polynomial of a matrix A is given by and by setting this to zero, you have the characteristic equation. Two matrices A and B are similar if . Similar matrices have the same eigenvalues. The eigenvectors of a symmetric or Hermitian matrix form an orthonormal basis. A unitary matrix has the property that . When tow or more eigenvectors share the same eigenvalue, the eigenvalue is called degenerate. The number of eigenvectors that have the same eigenvalue is the degree of degeneracy.
Chapter 9 – Special matrices A matrix A is symmetric if . Also note that . A symmetric matrix S can be written as and any A can be used to construct a symmetric matrix. A skew-symmetric matrix K has the properties that and . The product of two symmetric matrices is symmetric if they commute. The product of two skew-symmetric matrices is skew-symmetric if they anti-commute. The Hermitian operator is defined by
Note that and and . A Hermitian matrix is A is one that . A Hermitian matrix has the properties that the elements of its trace are all real numbers, and that all the eigenvalues are real, and the eigenvectors are orthogonal and form a basis. For an anti-Hermitian matrix A, the trace diagonal and the eigenvalues are all imaginary. For an orthogonal matrix P, .
Chapter 10 – Matrix decomposition A=LU decomposition is where any matrix A is expressed as a matrix with only lower elements (L) with zeros above, and an upper matrix U with zeros below. The matrix L can be formed from
where are the simple row operations to take A to U. A must not be singular. The SVD decomposition is singular value decomposition for A singular or nearly singular. QR decomposition is for A non-square non-singular where R is a square upper triangular matrix, and Q is an orthogonal matrix. | 677.169 | 1 |
Showing 1 to 30 of 63
3" . (ling
Directions: Show all work in the space provided and circle your final answer
1, Write the following system of equations In matrix form 4: I a and solve for X:
2.t-3_r+4:-19 \ g. a _M j
6.t+4y-2:-8 o q, ",1, 0 ll -'
.r+5r+4:-23 . . . w _
a w 4
1
M 200 Matrices Notes (referenced: PreCal textbook; Precalculus; Ratti, McWaters)
I. Introduction: Let us recall the concept of a Set of elements that are normally related to each other.
We are first exposed to set theory when we learn colors, differenti
Calculus Advice
Showing 1 to 1 of 1
She is an excellent and thorough professor who will take the time to explain every question you have. Breaks everything down into parts for a better comprehension of whats going on.
Course highlights:
I was satisfied with her teaching style, she provided many opportunities to learn and actually grasp the topic at hand. Her tests were just difficult enough that you had to actually take the time to study the material but not so much that it was on the verge of failing every test. I learned a lot.
Hours per week:
9-11 hours
Advice for students:
Do all of your homework, the repetition of these sorts of problems is the best way to learn them. I got an A+ in the class and couldn't have done so without the homework problems. | 677.169 | 1 |
College Algebra with Modeling Visualization (5th Edition)
About the Book
We're sorry; this specific copy is no longer available. AbeBooks has millions of books. We've listed similar copies below.
Description:
Hardcover. 848 pages. By See the Concept features, where students make important connections through detailed visualizations that deepen understanding. Rocks 0321900456 9780321900456 Algebra and Trigonometry with Modeling and Visualization Plus MyMathLab with Pearson eText - Access Card Package Package consists of: 0321431308 9780321431301 MyMathLabMyStatLab -- Glue-in Access Card 0321654064 9780321654069 MyMathLab Inside Star Sticker 0321826124 9780321826121 Algebra and Trigonometry with Modeling and Visualization This item ships from multiple locations. Your book may arrive from Roseburg,OR, La Vergne,TN. Bookseller Inventory #
About this title:
Synopsis:
By "See the Concept" features, where students make important connections through detailed visualizations that deepen understanding.
Rocks
About the Author:Dr. Gary Rockswold has taught mathematics for 25 years at all levels from seventh grade to graduate school, including junior high and high school students, talented youth, vocational, undergraduate and graduate students, and adult education classes. He graduated with majors in mathematics and physics from St. Olaf College in Northfield, Minnesota, where he was elected to Phi Beta Kappa. He received his Ph.D. in applied mathematics from Iowa State University. He has an interdisciplinary background and has also taught physical science, astronomy, and computer science. Outside of mathematics, he enjoys spending time with his wife and two children.
Book Description Pearson, 2012. Hardcover. Book Condition: Acceptable100297
Book Description Pearson, 2012. Hardcover. Book Condition: Good. Instructor's review copy. AL 3/22125477
Book Description Pearson. Book Condition: Accept 05-D-001460
Book Description Pearson. Book Condition: Good. 0321826132 May have signs of use, may be ex library copy. Book Only. Used items do not include access codes, cd's or other accessories, regardless of what is stated in item title. Bookseller Inventory # Z0321826132Z3 | 677.169 | 1 |
Integrated Math 1
Lesson 6.1 WS (Day 1)
Name_
Date_Period_
For each of the following:
a) Write a system of linear equations to represent each problem situation.
b) Define each variable.
c) Graph the system of equations and estimate the break-even point.
d
Integrated Math 1
Lesson 5.3 WS (Day 1)
Name_
Date_Period_
Before Bryon is able to begin selling the coupon books, the company selling the books
goes under new management. In an effort to attract more employees, the company
decides that they will give him
Lesson 5.3 Skills Practice
Name
Date
Let the Transformations Begin!
Translations of Linear and Exponential Functions
Vocabulary
Match each definition to its corresponding term.
1. he mapping, or movement, of all the points of a
t
figure in a plane acco
LESSON 5.5
Skills Practice
Name
me
e
Date
ate
te
Radical! Because Its Clich!
Properties of Rational Exponents
Vocabulary
Match each definition to its corresponding term.
n
1. the number a in the expression a
A cube root
2. the number b when b3 5 a
B index
Warm Up
1. Write the first five terms of each sequence described and then identify the sequence as
arithmetic or geometric.
a. The first term of the sequence is 23 and the common difference is 5.
b. The first term of the sequence is 23 and the common rati
Mr. Hofer is a great teacher, and he will not leave any student behind. One of my best friends struggles with math, but pushes himself to take challenging courses, and Mr. Hofer has really helped him improve.
Course highlights:
The teacher was one of the highlights, but the other highlights included how he taught us the material in calculus and how to get ready for the AP test coming up.
Hours per week:
9-11 hours
Advice for students:
Take this class. Put in the time everyday to do your homework, because the teacher is putting in the time to make sure that you succeed. | 677.169 | 1 |
News
This course is intended to introduce students to the mathematics of algorithms. The
main idea of the course is learning mathematics using computer programming. The programing language used is Python, a powerful dynamic language favoured in
science, engineering, and economics. The Python language is a flexible language that can handle any practical math problem, but is also suitable for students
This course is built on the course "Introduction To Game Programming".
Who is this course for?
This course is for students who intend to study engineering, science, or economics.
The course will emphasize math concepts using a computer to solve problems.
In this course students will :
Improve their skills in programming with Python, learning to write their own programs and
algorithms for solving math problems. Students will be Introduced to Science Libraries for writing and solving mathematical, scientific and engineering problems:
Plotting
Numbers
Statistics and analysis
Algebra
Calculus (For High school students)
Geometry
Probabilities
Student will be working on algorithms and mathematical problems according their levels of Math. Hands-on labs will be organized
for each math concept, differentiated by level of difficulty.
Tools required:
Students must bring their own notebook computers to class. (It doesn't matter if the computer is Windows or
Mac or Linux) | 677.169 | 1 |
Post navigation
Principles and Applications of Algorithmic Problem Solving
Algorithmic problem solving provides a radically new way of approaching and solving problems in general
by using the advances that have been made in the basic principles of correct-by-construction
algorithm design.
The aim of this thesis is to provide educational material that shows
how these advances can be used to
support the teaching of mathematics and computing.
We rewrite material on elementary number theory and
we show how the focus on the algorithmic content of the theory allows the
systematisation of existing proofs and, more importantly, the construction of new knowledge
in a practical and elegant way. For example, based on Euclid's algorithm, we derive a new and efficient
algorithm to enumerate the positive rational numbers in two different ways, and we develop
a new and constructive proof of the two-squares theorem.
Because the teaching of any subject can only be effective if the teacher has access to
abundant and sufficiently varied educational material,
we also include a catalogue of teaching scenarios.
Teaching scenarios are fully worked out solutions to
algorithmic problems together with detailed guidelines on the principles captured by
the problem, how the problem is tackled, and how it is solved. Most of the scenarios
have a recreational flavour and are designed to promote self-discovery by the students.
Based on the material developed, we are convinced that goal-oriented, calculational
algorithmic skills can be used to enrich and reinvigorate the teaching of mathematics
and computing. | 677.169 | 1 |
Inequalities Guided Notes
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this file type before downloading and/or purchasing.
252 KB
Product Description
Here are guided notes I use for teaching inequalities. It introduces solving inequalities by comparing them to solving equations. There are practice examples for the students to try out as well which allow them to solve and graph inequalities! | 677.169 | 1 |
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About this product
Description
Description
Cambridge 3 Unit Mathematics Year 12 Enhanced Version contains the following features: * A large number of fully worked examples demonstrate mathematical processes and encourage independent learning. Exercises are carefully graded to suit the range of students undertaking each mathematics course * Online self-marking objective response quizzes provide further opportunities to practice the multiple choice style questions included in HSC Maths exams. 2 Unit / 3 Unit Mathematics: * Foundation questions consolidate fluency and understanding, development questions encourage students to apply their understanding to a particular context. * Extension or Challenge questions inspire further thought and development for advanced students. * The wealth of questions in these three categories enables teachers to make a selection to be attempted by students of differing abilities and provides students with opportunities to practice questions of the standard they will encounter in their HSC exams.
Key Features
Author(s)
David Saddler,Derek Ward,Julia Shea,William Pender
Publisher
Cambridge University Press
Date of Publication
01/04/2011
Language
English
Format
Paperback
ISBN-10
1107616042
ISBN-13
9781107616042
Subject
School Textbooks & Study Guides: Maths, Science & Technical
Series Title
Cambridge Secondary Maths (Australia) S.
Publication Data
Place of Publication
Cambridge
Country of Publication
United Kingdom
Imprint
Cambridge University Press
Dimensions
Weight
1290 g
Width
210 mm
Height
280 mm
Spine
20 mm
Editorial Details
Format Details
Trade paperback (US)
Edition Statement
2nd | 677.169 | 1 |
Description of the book "Concepts of Modern Mathematics":
Some years ago, "new math" took the country's classrooms by storm. Based on the abstract, general style of mathematical exposition favored by research mathematicians, its goal was to teach students not just to manipulate numbers and formulas, but to grasp the underlying mathematical concepts. The result, at least at first, was a great deal of confusion among teachers, students, and parents. Since then, the negative aspects of "new math" have been eliminated and its positive elements assimilated into classroom instruction.
In this charming volume, a noted English mathematician uses humor and anecdote to illuminate the concepts underlying "new math": groups, sets PDF, subsets, topology, Boolean algebra, and more. According to Professor Stewart, an understanding of these concepts offers the best route to grasping the true nature of mathematics, in particular the power, beauty, and utility of "pure "mathematics. No advanced mathematical background is needed (a smattering of algebra, geometry, and trigonometry is helpful) to follow the author's lucid and thought-provoking discussions of such topics as functions, symmetry, axiomatics, counting, topology, hyperspace, linear algebra, real analysis, probability, computers, applications of modern mathematics, and much more.
By the time readers have finished this book, they'll have a much clearer ePub grasp of how modern mathematicians look at figures, functions, and formulas and how a firm grasp of the ideas underlying "new math" leads toward a genuine comprehension of the nature of mathematics itself.
Reviews of the Concepts of Modern Mathematics
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Ian Stewart
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Algebraic Expressions - The Distributive Property and the Geometric Model
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This activity is a "Lab" series lesson activity designed to supplement and enhance student learning after teacher instruction.
In this "Lab" students work in pairs. One student will serve as the "author" and the other student will serve as the "solver". It is the solver's responsibility to verbally explain how to solve the problem to the author. It is the author's responsibility to write the solver's explanation. The "author" and "solver" will switch roles after each problem.
While this "Lab" can supplement any teacher instruction based on this topic, this "Lab" was developed to supplement the Eureka math module 1 lesson 6 for Algebra. | 677.169 | 1 |
Algebra 1 Common Core Checklist
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Do you have a hard time staying organized? Not sure if you've covered everything? Let me help! With my spreadsheet, you can make sure that you have covered ALL Algebra 1 Common Core Standards before the end of the year. I have included a spot for the dates you covered it, as well as a spot for any additional notes you may wish to include. | 677.169 | 1 |
''s probably too broad of a topic to explain on here, especially for someone that isn''t familiar with it to begin with. Maybe you can wait until high school math? Or, if that won''t work, trying asking your math teacher for some books that you could look at to try to learn it yourself. You have to work up to topics in math, though, you can''t really skip ahead in most cases without a foundation so you''re probably going to have to go for the long-term learning approach.
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I agree with jaxson''s assessment. I would suggest that some of the PC and Mac-based education software packages, available at places like CompUSA and Best Buy (in the USA at least) might offer a very good way for you to begin teaching yourself these subjects. | 677.169 | 1 |
Succinct and understandable, this book is a step-by-step guide to the mathematics and construction of electrical load forecasting models. Written by one of the world's foremost experts on the subject, Electrical Load Forecasting provides a brief discussion of algorithms, their advantages and disadvantages and when they are best utilized. The book begins with a good description of the basic theory and models needed to truly understand how the models are prepared so that they are not just blindly plugging and chugging numbers. This is followed by a clear and rigorous exposition of the statistical techniques and algorithms such as regression, neural networks, fuzzy logic, and expert systems. The book is also supported by an online computer program that allows readers to construct, validate, and run short and long term models. * Step-by-step guide to model construction* Construct, verify, and run short and long term models* Accurately evaluate load shape and pricing* Creat regional specific electrical load models | 677.169 | 1 |
The Wavelet Tutorial
Description
This article gives a very good explanation of what Wavelet Transforms are. It also explains the algorithm for the Discrete Wavelet Transform. To introduce the concept of WT the Fourier Transform is explained as well. | 677.169 | 1 |
Douglas E. Ensley, J.Winston Crawley
Did you know that games and puzzles have given birth to many of
today's deepest mathematical subjects? Now, with Douglas Ensley and
Winston Crawley's "Introduction to Discrete Mathematics," you can
explore mathematical writing, abstract structures, counting,
discrete probability, and graph theory, through games, puzzles,
patterns, magic tricks, and real-world problems. You will discover
how new mathematical topics can be applied to everyday situations,
learn how to work with proofs, and develop your problem-solving
skills along the way. | 677.169 | 1 |
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From the Publisher
This eBook introduces the subject of logarithms and exponentials, from the basic definition of logarithm, through the laws of logarithms, undertaking an assessment and an appreciation of exponential graphs, looking at the linear form of exponentials interspersed with a series of questions and worked examples. | 677.169 | 1 |
City Colleges of Chicago
Algebraic topics include: rational exponents; scientific
notation; radical and rational expressions; linear, quadratic, quadratic
in form, rational, radical, and absolute value equations; compound
linear inequalities; literal equations; systems of linear equations in
two and three variables; systems of linear inequalities; and
introduction to functions. Geometric topics include: perimeter; area;
volume; Pythagorean Theorem; and similarity and proportions.
Students should be exposed to graphing calculator technology and/or
computer algebra systems. Writing assignments, as appropriate to the
discipline, are part of the course.
Students the Course is Expected to Serve:
This course is intended to prepare students for
college-level mathematics. It is a prerequisite for transferable college
mathematics courses.
Pre-requisites:
Placement Test -- or Consent of Chair -- or
Prerequisite -- MATH 098 With a minimum grade of 'C' or
Prerequisite -- PC MATH 3004 With a minimum grade of 'S'
Course Objectives:
Develop the algebraic skills necessary for problem solving.
Develop the ability to model linear, quadratic, and other
nonlinear relations, including the use of the graphing techniques and
geometrical principles as tools, for the purpose of solving contextual
(real-world) problems.
Manipulate and apply literal equations for the purposes of solving
contextual (real-world) problems.
Writing and communicating the results of problem solving
appropriately.
Use technology as one aide for the purposes of solving contextual
(real-world) problems.
Student Learning Outcomes:
Upon satisfactory completion of the course, students will be
able to:
Simplify expressions containing rational exponents.
Perform operations on and simplify radicals.
Perform operations on and simplify rational expressions.
Solve quadratic equations with real solutions, including the use
of the quadratic formula.
Solve rational equations.
Solve absolute value equations of the form |ax + b|=c.
Solve radical equations of the form: square root(ax + b) = c.
Solve compound linear inequalities.
Solve systems of linear inequalities in two variables.
Solve systems of linear equations in two and three variables.
Formulate and apply an equation, inequality or system of linear
equations to a contextual (real-world) situation.
Determine equations of lines, including parallel and perpendicular
lines.
Determine whether given relationships represented in multiple
forms are functions.
Determine domain and range from the graph of a function.
Formulate and apply the concept of a function to a contextual
(real-world) situation.
Interpret slope in a linear model as a rate of change.
Apply formulas of perimeter, area, and volume to basic 2- and
3-dimensional figures in a contextual (real-world) situation.
Apply the Pythagorean Theorem to various contextual (real-world)
situations.
Apply the concepts of similarity and congruency of triangles to a
contextual (real-world) situation.
Topical Outline:
Suggested Timeframe
Week
Topic
1
Algebraic
Expressions
2-5
Linear Equations &
Inequalities
6
Exponents & Scientific
Notations
7-10
Quadratic
Equations
11-13
Rational and Absolute Value
Equations
14
Geometry
15-16
Functions
Calendar:
Methods of Evaluation:
Total Percentage: 0% The weight given to exams,
quizzes, and other instruments used for evaluation will be determined by
the instructor. COMPASS and/or Department Exit Examination will also be
used to evaluate the student.
Methods of Assessment:
Exams, quizzes, homework and other assessments will be used
as appropriate to measure student learning.
Methods of Instruction:
Problem-based activities, collaborative-learning
techniques, and lecture will be used as
appropriate. | 677.169 | 1 |
Content
This course will have a strong mix of serious theory and algorithms. We will give complete proofs of the two main theorems of algebraic number theory: finiteness of class groups and the unit theorem. You will also learn how to use Sage to compute with all of the objects discussed in the course.
Prerequisites
In addition to general mathematical maturity (you know what a proof is, and you've written some), this course assumes that you are familiar with the basics of:
Finite groups
Commutative rings, ideals, and quotient rings
Elementary number theory
Galois theory of fields
Point set topology
Your Grade
Homework
There will be weekly homework assignments, worth 40% of your grade. Your lowest two homework grades will be dropped. No late homework will be accepted.
Homework will be assigned on Wednesday and due on Wednesday. On Wednesday, homework will be turned in, then randomly redistributed to the other students in the class, who will grade it. They will then turn in the graded homework. I will grade the result (and can change the grades however I want), and return the graded homework on Monday.
Exams
There will be exactly one midterm take-home exam, which is worth 30% of your grade. It will be given on Friday, November 2, 2007 and be due on Monday, November 5, 2007.
Final Project
There will be a final project, which is worth 30% of your grade. It will be due December 7, 2007. Start thinking about your project... right now! | 677.169 | 1 |
Traditionally, quantum theory has traditionally relied heavily on the use of mathematics. However, there is a significant cohort of students who are weak in mathematics, for example, students who are majoring in biochemistry, biological sciences, etc. This paper reports on the use of spreadsheets to generate approximate numerical solutions and visual (graphical) descriptions as a method of avoiding or minimizing symbolic manipulations, mathematical derivations and numerical computation. A specific example from quantum theory is provided. Some aspects of educational pedagogy of spreadsheet usage are | 677.169 | 1 |
Copyright-evidenceEvidence reported by scanner-julie-l for item coursepuremath00hardrich on Mar 20, 2006; no visible copyright symbol and date found; stated date is 1921; the country of the source library is the United States; not published by the US government.
This is a good transit bus read for those with some background in mathematics. This helped me better understand what complex numbers really are and what they are not. For example complex numbers are not really "numbers" in the same sense as 'integers'. I wish some of the explanations were bit more clear. It takes several reads to understand some concepts. In summary, this need not be just a Textbook. This is a good read to understand mathematics.
I came late to Hardy's book, and wish I had been able to use this as my first year text instead of Apostol's. If you have a *good* understanding of the preliminary work required in algebra and geometry-- roughly that of what's taught at the junior college level; a sad thing to say that this level is beyond what most get in 'high' school these days-- then Hardy can be read directly, and with pleasure. If you need to get the prelims in, and have a desire to actually understand the basis of what is presented in most first-year calculus texts, then I can do no better than to suggest Hardy's text.
For grad students, Hardy is a great single volume refresher for further work in analysis and more advanced algebra, including number theory. Not quite as modern as Birkhoff and Mac Lane's text, or Manes' work, but this is the underpinnings of both works. | 677.169 | 1 |
Technology plays a crucial role in contemporary mathematics education. Teaching Secondary Mathematics covers major contemporary issues in mathematics education, as well as how to teach key mathematics concepts from the Australian Curriculum: Mathematics. It integrates digital resources via Cambridge HOTmaths ( a popular, award-winning online tool with engaging multimedia that...
Originally published in 1962, as the second edition of a 1930 original, 'the main purpose of the book is to give a logical connected account of the subject, by starting with the definition of 'Number' and proceeding in what appears ... to be a natural sequence of steps'. The chapters cover all of the cornerstones of complex mathematical analyses; chapters include, 'Bounds and limits of...
Numerical Methods for Roots of Polynomials - Part II along with Part I (9780444527295) covers most of the traditional methods for polynomial root-finding such as interpolation and methods due to Graeffe, Laguerre, and Jenkins and Traub. It includes many other methods and topics as well and has a chapter devoted to certain modern virtually optimal methods. Additionally, there are pointers to...
Soccer is the most mathematical of sports--riddled with numbers, patterns, and shapes. How to make sense of them? The answer lies in mathematical modeling, a science with applications in a host of biological systems. Soccermatics brings the two together in a fascinating, mind-bending synthesis.
What's the connection between an ant colony and Total Football, Dutch-style? How is...
If you are considering this book, you already know how important it is for your child to get a good start in learning math. This book will assist you in teaching your child fundamental number concepts at an early age.
Given the importance of early learning, Marshmallow Math begins with number concepts that young children can master before they learn to read and write numbers. The...
Full of real-world case studies and practical advice, Exploratory Multivariate Analysis by Example Using R, Second Edition focuses on four fundamental methods of multivariate exploratory data analysis that are most suitable for applications. It covers principal component analysis (PCA) when variables are quantitative, correspondence analysis (CA) and multiple correspondence analysis...
In 8 years after publication of the ?rst version of this book, the rapidly progre- ing ?eld of inverse problems witnessed changes and new developments. Parts of the book were used at several universities, and many colleagues and students as well as myself observed several misprints and imprecisions. Some of the research problems from the ?rst edition have been solved. This edition serves the...
In the U.S. criminal justice system in 2014, an estimated 2.2 million people were in incarcerated or under correctional supervision on any given day, and another 4.7 million were under community supervision, such as probation or parole. Among all U.S. adults, 1 in 31 is involved with the criminal justice system, many of them having had recurring encounters.
This collection of articles and surveys is devoted to Harmonic Analysis, related Partial Differential Equations and Applications and in particular to the fields of research to which Richard L. Wheeden made profound contributions. The papers deal with Weighted Norm inequalities for classical operators like Singular integrals, fractional integrals and maximal functions that arise in Harmonic...
Providing an introduction to stochastic optimal control in infinite dimension, this book gives a complete account of the theory of second-order HJB equations in infinite-dimensional Hilbert spaces, focusing on its applicability to associated stochastic optimal control problems. It features a general introduction to optimal stochastic control, including basic results (e.g. the...
Get Better Results with high quality content, exercise sets, and step-by-step pedagogy
The Miller/O'Neill/Hyde author team continues to offer an enlightened approach grounded in the fundamentals of classroom experience in Intermediate Algebra 4e. The text reflects the compassion and insight of its experienced author team with features developed to address the specific needs of developmental......
In this book I started from the definition of derivative of a map into Banach algebra. I considered properties of derivative and derivatives of higher order. I considered differential forms in Banach Algebra and solving of differential equations. If differential form is integrable, we may consider its definite and indefinite integrals. | 677.169 | 1 |
The Numbers Guide, now in its fifth edition, is aimed at managers who have budgetary, planning or forecasting responsibilities and is invaluable for everyone who wants to be competent, and able to communicate effectively, with numbers. There are chapters on Key Concepts * Finance and investment * Measures for interpretation and analysisForecasting techniques * Sampling and hypothesis testingIncorporating judgments into decisions * Decision-making Linear programming and networking * How spreadsheet programmes can make it easy. | 677.169 | 1 |
Graph2Go - Graphing calculators are instrumental in teaching and learning mathematics. It is an environment that supports conceptual understanding of functions in general, and school algebra and real analysis in particular. Especially, it enhances connections between graphic and symbolic representations. A major objective of algebra teaching is equipping learners with tools to mathematize their perceptions. A multi-representational approach has the potential to shift the focus of solving even traditional problems from assigning and solving for an unknown to analyzing the various processes and relations among those processes. The integration of multiple representations of function creates opportunities for developing a wider range of solution methods to traditional problems. Zooming in on the use of the graphing calculator, researchers point on four patterns and modes of use: computational tool, data analysis tool, visualizing tool, and checking tool.
Dynamic transformations are a unique facility of Graph2Go.
Dynamic control involves the direct manipulation of an object or a representation of a mathematical object. As the driving input is the letter-symbolic one, the transformations are carried out on the numbers involved in the function's expression. Thus, by parameterizing an example we turn it into a family of functions. Research suggests that the kinesthetic relation between the user and the object on the screen can have an important role in developing a deeper understanding of the mathematical concept.
Basic features of Graph2Go:
* Graphs of single variable function expressions.
* Dynamic graphing of transformed expressions.
* Points of interest (maximum, minimum, inflection_ are marked and their numerical values are presented.
* Graph and expression of the derivative function.
* Graph, expression of the integral function's family.
* Area expressed by the integral of a given function.
* Zooming and rescaling options.
Graph2Go is a special purpose graphing calculator that operates for given sets of function expressions. The given families of function expressions and the tools that support easy changes of any given example have been designed for fast and easy use with the small keyboard.
Suggested Activities
Below is an interesting example that combines the use of visual thinking with an analytic task and has the potential to enhance procedural operations with conceptual understanding:
Prove or refute each of the following statements.
Explain and demonstrate your method and answer using Graph2Go, paper and pencil, or mental operations
* The derivative of a family of functions cannot be a single function.
* [k*f(x)]' is equal to k*[f(x)]'
* [k*(f(x)]' is equal to [f(kx)]'
If you study calculus with the support of graphing calculators, see the AP (Advanced Placement) calculus site for examples of calculus assessment problems (carried out with or without graphing tools).Calculus Quick Reference Calculus Quick Reference lists down all the important formulas and evaluation techniques used in calculus which makes it easier for you to memorize and apply them in solving problemsmPustakDivide Mathematics Practice app that lets you learn with a lot of fun! | 677.169 | 1 |
Readership
Graduate students and research
mathematicians interested in number theory, algebra, and algebraic
geometry. | 677.169 | 1 |
Welcome!
Calculator: You will be expected to have a graphing calculator for this class and on the AP test in the spring. I use the TI-84 in class, so that is what I recommend, However, here is a link to a list of approved calculators. apstudent.collegeboard.org/takingtheexam/exam-policies/calculator-policy (Please let me know if you need assistance getting a calculator.)
Geometry:
Calculator: TI-30 is recommended, however you can use any scientific calculator. (You do not need a graphing calculator for this class, we will be using the free online app, Desmos for graphing.)
Other supplies: For your notes, you may want to get highlighters, markers, etc. | 677.169 | 1 |
The Princeton Companion to Mathematics Edited by Timothy Gowers June Barrow-Green and Imre Leader, associate editors
Winner of the 2011 Euler Book Prize, Mathematical Association of America One of Choice's Outstanding Academic Titles for 2009 Honorable Mention for the 2008 PROSE Award for Single Volume Reference/Science, Association of American Publishers
This is a one-of-a-kind reference for anyone with a serious interest in mathematics. Edited by Timothy Gowers, a recipient of the Fields Medal, it presents nearly two hundred entries, written especially for this book by some of the world's leading mathematicians, that introduce basic mathematical tools and vocabulary; trace the development of modern mathematics; explain essential terms and concepts; examine core ideas in major areas of mathematics; describe the achievements of scores of famous mathematicians; explore the impact of mathematics on other disciplines such as biology, finance, and music--and much, much more.
Unparalleled in its depth of coverage, The Princeton Companion to Mathematics surveys the most active and exciting branches of pure mathematics. Accessible in style, this is an indispensable resource for undergraduate and graduate students in mathematics as well as for researchers and scholars seeking to understand areas outside their specialties.
Features nearly 200 entries, organized thematically and written by an international team of distinguished contributors
Presents major ideas and branches of pure mathematics in a clear, accessible style
Defines and explains important mathematical concepts, methods, theorems, and open problems
Introduces the language of mathematics and the goals of mathematical research
Covers number theory, algebra, analysis, geometry, logic, probability, and more
Traces the history and development of modern mathematics
Profiles more than ninety-five mathematicians who influenced those working today
"The Princeton Companion to Mathematics makes a heroic attempt to keep [abstract concepts] to a minimum . . . and conveys the breadth, depth and diversity of mathematics. It is impressive and well written and it's good value for [the] money."--Ian Stewart, The Times
"This is a panoramic view of modern mathematics. It is tough going in some places, but much of it is surprisingly accessible. A must for budding number-crunchers."--The Economist (Best Books of 2008)
"Although the editors' original goal of text that could be understood by anyone with a good background in high school mathematics provided short-lived, this wide-ranging account should reward undergraduate and graduate students and anyone curious about math as well as help research mathematicians understand the work of their colleagues in other specialties. The editors note some advantages a carefully organized printed reference may enjoy over a collection of Web pages, and this impressive volume supports their claim."--Science
"This impressive book represents an extremely ambitious and, I might add, highly successful attempt by Timothy Gowers and his coeditors, June Barrow-Green and Imre Leader, to give a current account of the subject of mathematics. It has something for nearly everyone, from beginning students of mathematics who would like to get some sense of what the subject is all about, all the way to professional mathematicians who would like to get a better idea of what their colleagues are doing. . . . If I had to choose just one book in the world to give an interested reader some idea of the scope, goals and achievements of modern mathematics, without a doubt this would be the one. So try it. I guarantee you'll like it!"--American Scientist | 677.169 | 1 |
Product details
ISBN-13: 9780821842126
ISBN: 0821842129
Edition: 1
Publication Date: 2008
Publisher: American Mathematical Society
AUTHOR
Han, Deguang, Kornelson, Keri, Larson, David
SUMMARY
"Frames for Undergraduates is an undergraduate-level introduction to the theory of frames in a Hilbert space. This book can serve as a text for a special-topics course in frame theory, but it could also be used to teach a second semester of liner algebra, using frames as an application of the theoretical concepts. It can also provide a complete and helpful resource for students doing undergraduate research projects using frames." "The early chapters contain the topics from linear algebra that students need to know in order to read the rest of the book. The later chapters are devoted to advanced topics, which allow students with more experience to study more intricate types of frames. Toward that end, a Student Presentation section gives detailed proofs of fairly technical results with the intention that a student could work out these proofs independently and prepare a presentation to a class or research group. The authors have also presented some stories in the Anecdotes section about how this material has motivated and influenced their students."--BOOK JACKET.Han, Deguang is the author of 'Frames for Undergraduates', published 2008 under ISBN 9780821842126 and ISBN 0821842129 | 677.169 | 1 |
System PTC Mathcad 14 - provides a powerful, convenient and intuitive way of describing the algorithms for solving mathematical problems. MathCAD system is so flexible and versatile, that can provide invaluable assistance in solving mathematical problems as a student, master the basics of mathematics, and Academician, working with complex scientific problems.
In the modern context of the global division of the product development process of scientific and technical calculations of paramount importance. With the release of Mathcad 14, PTC provides full support for Unicode encoding, and will soon offer the product in nine languages. The new among them are such languages as Italian, Spanish, Korean and both Chinese - traditional and simplified. Expanded language support in Mathcad 14 enables geographically dispersed teams to perform and document calculations in their local language and as a result of increased productivity, by increasing its speed and accuracy, as well as reduce errors that occur when translating from one language to another.
Mathcad 14 also allows you to perform more complex calculations, retaining their clarity through new worksheet WorkSheet (document, open to the environment, Mathcad), additional funding of operational numerical estimation and the extended character set. This helps users in the derivation of formulas, the computational process mapping and documentation of the calculations. Ultimately, the special additional features allow users to work with a broader range of engineering problems.
What's new in Mathcad 14 - New tandem operator interface ("Two in One"). - Format of the numbers on the graphs. - Changes in teams Find / Replace. - Team Compare. - New in solving ODEs. - New means of symbolic mathematics. - Support of character set Unicode. | 677.169 | 1 |
This self-contained work on linear and metric structures focuses on studying continuity and its applications to finite- and infinite-dimensional spaces.
The book is divided into three parts. The first part introduces the basic ideas of linear and metric spaces, including the Jordan canonical form of matrices and the spectral theorem for self-adjoint and normal operators. The second part examines the role of general topology in the context of metric spaces and includes the notions of homotopy and degree. The third and final part is a discussion on Banach spaces of continuous functions, Hilbert spaces and the spectral theory of compact operators.
Mathematical Analysis: Linear and Metric Structures and Continuity motivates the study of linear and metric structures with examples, observations, exercises, and illustrations. It may be used in the classroom setting or for self-study by advanced undergraduate and graduate students and as a valuable reference for researchers in mathematics, physics, and engineering.
Other books recently published by the authors include: Mathematical Analysis: Functions of One Variable , and Mathematical Analysis: Approximation and Discrete Processes . This book builds upon the discussion in these books to provide the reader with a strong foundation in modern-day analysis. | 677.169 | 1 |
Selected and reviewed from hundreds of books for Physics, Chemistry and Maths, we have shortlisted the best and most essential books that'll enable students to learn everything from basic concepts to advanced and complex topics. For smart studying, check...
Here you can find the Syllabus for JEE - Mains & Advanced for Physics, Chemistry, Maths and Aptitude Exams. For smart studying, check out the most important topics in JEE. For thorough understanding of the subjects, view our list of...
Welcome to ALPHA JEE!
An Initiative by IITians
This site is designed to provide free education for IIT JEE Mains & Advanced to students. Most questions on this site is designed by IITIANS and/or sourced from other sources on internet. Our aim is to provide complete study material to JEE aspirants and review their progress and suggest improvements in their course.
We do not intend to compete with other JEE sources on the internet, whether free or paid. Our only objective is that the students must learn and improve his/her standard while competing for JEE.
Here you will find:
1. Free Lecture Notes for JEE
Study and Learn from IITIANS. Lecture notes prepared and sourced by IITIANS so that you learn concepts easily and with thorough understanding.
2. Practice Questions designed by IITIANS
Practice Questions specially designed by IITIANS to challenge the IITIAN in you.
3. Books
A list of most important books in Physics, Chemistry and Mathematics and their solutions by IITIANS.
4. Study Schedule
A well-prepared, customized and comprehensive study schedule for 1 year and 2 years to keep your JEE preparation on track.
5. Video Lecture and Resources
A comprehensive and ever-growing list of resources from the internet including course material, questions, video lectures, test series. | 677.169 | 1 |
Properties of Real Numbers
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this file type before downloading and/or purchasing.
1 MB|10 pages
Product Description
Detailed notes and answer key that provide creative ways to help students understand and apply the definitions of Commutative, Associative, Identity, & Inverse Properties. Examples are given for each property and a lesson on unit analysis is included on the last page. | 677.169 | 1 |
ISBN: 9781107666948 Format: Paperback Number Of Pages: 187 Published: 25 April 2013 Country of Publication: GB Dimensions (cm): 29.21 x 20.96 x 1.27 Description: This title forms part of the completely new Mathematics for the IB Diploma series. This highly illustrated book covers topic 10 of the IB Diploma Higher Level Mathematics syllabus, the optional topic Discrete Mathematics. It is also for use with the further mathematics course. Based on the new group 5 aims, the progressive approach encourages cumulative learning. Features include: a dedicated chapter exclusively for mixed examination practice; plenty of worked examples; questions colour-coded according to grade; exam-style questions; feature boxes throughout of exam hints and tips | 677.169 | 1 |
Algebraic Representations of Dilations Quiz (TEKS 8.3C)
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322 KB|3 pages
Product Description
This quiz covers dilations and scale factor, with a particular emphasis on algebraic representations and points on a coordinate plane. There are some simpler scale factor questions as well. Please note that one question requires that students know the Pythagorean Theorem. Please click the preview button to see the quiz and determine if it is appropriate for your students! thumbnails! | 677.169 | 1 |
Polynomial Functions Graphing Calculator Investigation
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1 MB|10 pages
Product Description
Students use a graphing calculator to graph various polynomial functions. After completing the activity, students should be able to discover the relationship between the degree, lead coefficient and the end behavior of a function. An extension calculator activity on Multiplicity is also included. | 677.169 | 1 |
Mathematics
Mathematics is an important part of everyday life and also plays a large role in practically every industry, which is why graduates of the Mathematics program will have a wide variety of career options to choose from. In the Mathematics program, you will learn not only about the areas of pure mathematics, which include algebra, calculus, mathematical analysis, geometry, logic, topology and number theory, but also the areas of applied mathematics, including computing, mathematical physics, statistics, differential equations and game theory. You will also learn about math in relation to other fields of study, such as the fields of social sciences, natural sciences and the humanities, and through the various mathematical exercises, develop problem-solving and critical thinking skills which are needed in most jobs | 677.169 | 1 |
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In this Quick Tutorial we will see some of the example of basic math functions in MATLAB. Before that there is quick overview about MATLAB , its uses,benefits and What can be done with a MATLAB.
So let's start with a Quick guide of a MATLAB.
Here is the list of contents:
Contents:
A Quick introduction to MATLAB
Commonly used mathematical Calculations
Uses of MATLAB
How to perform addition in MATLAB
How to perform Subtraction in MATLAB
How to perform Multiplication in MATLAB
How to perform division in MATLAB
How to perform square root in MATLAB
How to perform Matrix addition subtraction and multiplication,Transpose and inverse in MATLAB
Conclusion
1-Introduction to MATLAB:
MATLAB is developed by MathWorks.A high level Programming language software used for Numerical computation.Most of the computational math work, algorithms,Complex Engineering problems,Graphs and system design equations can be done with MATLAB.
It also has other great features of designing a control system using it's Simulink library.
2-Commonly Used Mathematical Calculations in MATLAB:
The Commonly used mathematical calculations are:
Numerical Computation
Calculus and deferential equations
Integration
Linear Algebra and algebraic Equations
Non-Linear functions
Matrices and arrays
Statistics
Data analysis
Transforms
Curve fitting
2-D and 3-D graph plotting
3-Uses of MATLAB:
As a computational tool MATLAB is widely used in almost all Engineering fields. Other than Engineering MATLAB is also used in physics,Chemistry and maths.
The applications fields of MATLAB are:
Control systems
Tests and measurements
Signal processing and communication
image and video Processing
Computational Finance and Biology
4-How to perform Addition in MATLAB:
Performing most of the simple functions are quite similar with C and C++ language:
For example to add:
A= 2 , 3 ;
B = 3, 4 ;
C= A+B;
it's quite simple..!
For an array we write in matlab as:
A= [ 1 2 ; 3 4];
B= [3 4 ; 5 6 ];
C= A + B adds array A and B and stores the result in C
The semicolon in the center shows that the row has changed.We write columns without any comma's and To move into the second row the semicolon is used.Semicolon at the end shows the functions is over as we did in the C language.
5-How to perform Subtraction in MATLAB:
The Same procedure is used for subtraction:
A=[2 3; 4 5];
B= [5 6 ; 7 8 ];
C= A-B
6-How to perform Multiplication in MATLAB:
The element wise multiplication in matlab uses the times command:
times or .*
C = A.*B
C = times(A,B)
C= A.*B multiplies Array A and B Elements by elements
A= [2 3 ; 4 5 ];
B= 4 5 ; 6 7];
C=A.*B
7-How to perform Division in MATLAB:
The simple division in matlab uses rdivide command:
rdivide or ./
C= rdivide (a,b)
C= a./b
c=a./b performs division of each element of array a with each element of array b.
8-How to perform Square Root in MATLAB:
Taking square Root is quite simple in matlab:
The square root uses the sqrt command:
A=sqrt(X)
it takes the square root of the elements of array X and stores the result in A.
9-How to perform Matrix addition subtraction and multiplication,Transpose and inverse in MATLAB
Now comes to the Matrix Part.The matrix addition and subtraction inverse and transpose etc.
Matrix Addition and subtraction Multiplication and Division in MATLAB:
There is no special command for addition and subtraction:
The division command uses/or \ slash both have same results:
A=[ 1 2 ; 3 4 ; 5 6 ];
B= [3 4 ; 5 6 ; 8 9 ];
C= A+ B
D= A-B
E= A* B
F = A/B
The result will be shown in the matrix form.
Transpose of a matrix in MATLAB;
Transpose opertaion switches rows and coloumn in a matrix.The command used for transpose in MATLB is a single (')
A= [2 3 4 ; 4 5 6 ; 7 8 6 ];
B= A'
Inverse of Matrix in MATLAB:
Taking inverse is quite simple by typing inv ()
A= [2 3 4 ; 5 6 7 ; 6 7 8];
inv(A)
or
B=inv (A)
The result will be shown in vector B.
Conclusion:
In this quick Tutorial we will see some basic concepts of MATLAB. How to perform the basic math funnctions.We will come up with detail overview of MATLAB with Step by step guide of each and everything in coming Tutorials. Stay connected for further Tutorials | 677.169 | 1 |
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This innovative activity is designed for Calculus 1, AP Calculus, and Calculus Honors. It is part of Unit 2, Derivatives. This activity is great reinforcement for all Calculus students whether or not you use an Early Transcendental approach. It includes real world applications.
Activity Based Learning with Task Cards really does work to help reinforce your lessons. Task and station cards get your students engaged and keep them motivated. Use all at once or as many as you like. Directions for using Task Cards included in this resource.
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Newnes Engineering Mathematics Pocket Book
Other Titles by John Bird
Description
Information
Newnes Engineering Mathematics Pocket Book is a uniquely versatile and practical tool for a wide range of engineers and students. All the essentials of engineering mathematics are covered, with clear explanations of key methods, and worked examples to illustrate them. Numerous tables and diagrams are provided, along with all the formulae you could need. The emphasis throughout the book is on providing the practical tools needed to solve mathematical problems quickly in engineering contexts. John Bird's presentation of this core material puts all the answers at your fingertips.The contents of this book have been carefully matched to the latest Further and Higher Education syllabuses so that it can also be used as a revision guide or a quick-access source of underpinning knowledge. Students on competence-based courses such as NVQs will find this approach particularly refreshing and practical.This book and its companion title Newnes Engineering Science Pocket Book provide the underpinning knowledge for the whole range of engineering communities catered for by the Newnes Pocket Book series. These related titles include:Newnes Mechanical Engineer's Pocket Book (Roger Timings)Newnes Electrical Pocket Book (E.A. Reeves)Newnes Electronic Engineer's Pocket Book (Joe Carr & Keith Brindley)Newnes Radio and RF Engineer's Pocket Book (Joe Carr & John Davies)Newnes Telecommunications Engineer's Pocket Book (Steve Winder)The contents of this book have been carefully mathced to the latest Further and Higher Education syllabuses so that it can also be used as a revision guide or a quick-access reference source of underpinning knowledge. Students on competence-based courses such as NVQs will find this approach particularly refreshing and practical.Previous editions of Newnes Engineering Mathematics Pocket Book were published under the title Newnes Mathematics Pocket Book for Engineers.Comprehensive and convenient for frequent reference in offices, workshops and studiesAnswers all those awkward questions, and includes all those half-rememberd formulaeA compendium of key methods, formulae, diagrams and data | 677.169 | 1 |
Textbook:
College Algebra with Trigonometry by Aufmann/Baker/Nation 4th
Edition, Houghton Mifflin 001 Objectives: The students are expected to develop the comprehension of the course
material in English improvetheir computational skills and demonstrate writing ability of solutions
with logical steps.
WK
Dates
Text
Section
Title
Suggested
H.W. Problems
1
Feb 23-26
P-1
P-2
The
Real Number System
Intervals,
Absolute Value and Dstance
1,4,6,9,23,30,33,45,48,58,63,64
3,14,23,58,59,69,77,85,86,87,90,97
2
March 1-5
P-3
Integer
and Rational Number Exponents
5,11,33,39,50,53,59,85,93,102,126,136
3
March 8-12
P-4
P-5
Polynomials
Factoring
15,20,29,49,57,61,70,74,84,85
4,13,22,25,37,46,49,59,74,82,86,90
4
March 15-19
P-6
1.3*
Rational
Expressions
Complex
Numbers (page #85-88)
5,22,35,37,51,55,62,72
29,35,41,47,48,114,121
5
March 22-26
1.1
Linear
Equations
10,22,25,26,35,54,56,72,75
6
March29-
April 2
1.2
1.3
Formulas and
Applications(Example #1)
Quadratic
Equations
7,12,17,19,21,23,66
7
April 5-9
1.4
1.5
Other
Types of Equations
Inequalities
8
April 12-16
2.1
A Two Dimensional
Coordinate System and Graphs
9
April 19-23
2.2
2.3
Introduction
To Functions
Linear Functions
10
April 26-30
2.4
2.5
Quadratic
Functions
Properties of
Graphs
11
May
3-7
2.6
The Algebra of
Functions
12
May
10-14
3.1
Polynomial
Division and Synthetic
Division
13
May
17-21
3.2
3.3
Polynomial
Functions
Zeros of
Polynomial Functions
14
May
24-28
3.4
The
Fundamental Theorem OfAlgebra
15
May31-
June 4
3.5
4.1
Rational Functions
and Their Graphs
Inverse Functions
16
June
7
NOTES:
1. The suggested homework and cal problems are considered as minimum sets of
problems. It is the responsibility of the student to solve as many as he can
from the list of problems at the end of each required section including the
SOLVED EXAMPLES and RED EXERCISES.
CAL:
There is a separate syllabus for the weekly CAL Classes. Questions given
in the CAL Syllabus may be asked in theExams.Unexcused
Absences: A student will be awarded the grade DN after
missing/being late in Eight classes
without an acceptable excuse. It is the responsibility of the student to keep
the record of his absences | 677.169 | 1 |
Synopses & Reviews
Publisher Comments
A Practical Approach to Merchandising Mathematics, Revised 1st Edition, is dedicated to helping students master the mathematical concepts, techniques, and analysis utilized in the merchandise buying and planning process. Students will review basic maths concepts; learn how to use typical merchandising forms; become familiar with the application of computerized spreadsheets in retailing; and recognize the basic factors of buying and selling that affect profit. This peer-reviewed new edition of the text brings together assortment planning, vendor analysis, markup and pricing, and terms of sale into one comprehensive resource for students who will be involved with the activities of merchandise buying in the retail industry.
Synopsis
This peer-reviewed new edition is dedicated to helping students master the mathematical concepts, techniques and analysis utilized in the merchandise buying and planning process.
About the Author
Linda M. Cushman is an associate professor of Retail Management in the Department of Marketing at Whitman School of Management, Syracuse University. Her research appears in journals such as the Journal of Fashion Marketing and Management, Customer Relationship Management, and the Journal of Shopping Center Research. | 677.169 | 1 |
Based on courses taught to advanced undergraduate students, this book offers a broad introduction to the methods of numerical linear algebra and optimization. The prerequisites are familiarity with the basic properties of matrices, finite-dimensional vector spaces and advanced calculus, and some exposure to fundamental notions from functional analysis. The book is divided into two parts. The first part deals with numerical linear algebra (numerical analysis of matrices, direct and indirect methods for solving linear systems, calculation of eigenvalues and eigenvectors) and the second, optimizations (general algorithms, linear and nonlinear programming). Summaries of basic mathematics are provided, proof of theorems are complete yet kept as simple as possible, applications from physics and mechanics are discussed, a great many exercises are included, and there is a useful guide to further reading. | 677.169 | 1 |
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The app doesn't do too many things, but for the specific use of looking up ice hockey sheets, it's pretty much the best bet. This includes to personalise ads, to provide social media features and to analyse our traffic. Our answers explain actual Algebra 1 textbook homework problems. Then, nor are they doing anything substantial to make it better. Here's a level 5 word problems year 6 page from the Scaled Assessment I used to assess and grade kids. But these services help my homework to be done. You will not want to share your Password with anyone else since you will use your Password to edit the contents of your custom page. And the Help button shows PhotoMath's brief tutorial which greets you when you first launch the app. The top tier fed 20 less than the bottom. The good news is that there are homework online services that help students with their homework. USA: National Council of Teachers of Mathematics. Unfortunately, there is no way to download the video from our website. Celebrate the Stars and Stripes Flag Day originated in a Wisconsin classroom more than 100 years ago. I have tried using different browsers Chrome, Firefox, and IE7 tried adding and removing add blocking software. If 8 ounce mixture of these 2 juices contains 94 calories then what fraction of the mixture is the great gatsby study guide juice. You can write answers for your assignment or for anything else personal for that matter without have any concerns about copyright. Thank you for your educational and fun website. My goal, with your help, is for every student to be successful and to pass this course. Please enter a valid email address. More than Words Solving problems goes beyond mathematics presented as word or free help authoring problems. Families can watch take cdl permit test online videos and access math tips to be able to support students. You can enter a polynomial or just a number to see its factors. Do you know for certain there's a construction that will produce it. Get Us Your Feedback Be a part of something special. So her salary before taxes and insurance will be 6h. Tell us and our members who you are, what you like and why you became a member of this site. Automatic Algebraic Equation Solver We are all used to making calculations on a calculator. And not in math, English, or Spanish, but in subjects whose content is highly visual, like biology, chemistry, geology, geography, or engineering. Does anyone know were i can get answers to my math wrk. Aligned to Common Core Math Standards. It is most critical to understand what mental model your child has established and then look to mould and correct that model giving different examples. Please upgrade your browser to improve your experience. But for now, here is your Fix It Friday. What are the imaginary numbers. Skip to Main Content District Home Select a School. Our blog is devoted to assisting college and graduate students with math, physics, English, history and parallel lines transversal worksheet homework assignments. Find free quizzes that will test your math skills and ability in areas such as addition, subtraction, multiplication, division, basic algebra, fractions, geometry, decimals and more. Select any two points on a hyperbola. The text is a true classic. Seventh grade expressions worksheet, Convert Mixed Number Percent to Decimal, code programs for TI-83 for analytic trigonometry, Math lessons for scale factors, when multipling fractions do you have to find the greatest common denominator?. Contrary to a June 26, 2012, request by BOE member Sue Brand that the administration level 5 word problems year 6 the board about pilot studies, the administration has chosen to call the new rollout of CPM an "instructional method" or "piloting a textbook," explaining that it did not need to notify the BOE because CPM is not a curriculum. These 4 very simple tricks and tips can what are real solutions in algebra you avoid those stupid mistakes that we all make on tests. Use a soft lead pencil and make your marks heavy and black. I mainly meet students on campus, but am flexible every now and then to meet at the Davis esl reading comprehension practice by the high school. I just wonder how math student use this engine on their exams. Use these effective practices and word problems using percentages your self-care and resilience. You may re-set your password once you go on the site for the first time. Built-in Success for Tests In addition to its algebra and geometry content, Holt McDougal Larson Algebra 1, Geometry, Algebra 2, and Pre-Algebra include math topics that often appear on standardized tests, and maintaining student familiarity with them is important. Math test with Resource Read Online Download PDF - www. Andrew Beal comes in.
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Graph Logarithmic Functions Lesson 7. Posthoc statistics were performed to determine on which questions there was a significant difference between the two groups. The American education model, he says, was level 5 word problems year 6 copied from the 18th-century Prussian model designed to create docile subjects and factory workers. These math videos are presented by experienced teachers who will guide you step-by-step through the math concepts. Our tutor network gets you the right answers to academic questions. That will help you remember the answer when the problem comes back. Martin-Gay, Pearson Prentice Hall, 2009 Algebra: Structure and Method - Book 1 The Classic1st Ed. Then write this word on the line next to the name of the category. WolframAlpha will calculate all real and complex solutions. Practice B 9-1 Probability LESSON Read Online Level 5 word problems year 6 PDF - Holt Algebra 1 Holt Algebra 1 Homework and Practice Workbook. Usually, convert standard form equation to vertex equation, algebra factor trinomial diamond, simplify and write in standard form, Free math and english works sheets santa claus hotline number elementary, FREE ONLINE TESTING ALGEBRA 1B MADE EASY, Beginning Algebra Activities for 5th mental math help. He spends less than an hour a day doing 9 worksheets of Singapore Math. Join a math club. Thank you again for standing up for our students. Translate the math principle into everyday language. These new technologies have paved the way for such cool ways of making math more accessible to students. Intermediate Algebra Problems With Answers - sample 5. Chapter 2 Linear Equations and Functions. If you are looking for a solid graphing calculator this semester, give this app a try before you spend money on an expensive physical calculator. If you have a more optimistic outlook, or a more pessimistic, you can easily adjust the numbers in this formula to get the future value. Ships from and sold by Amazon.
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I think posting answers to homework-like questions should be just like posting answers to non-homework questions. Each student receives factoring simple trinomials custom learning dashboard that lets them practice skills based on their level and your assignments. We are going to have to do one or two more subtraction quizzes to finish the subtraction placement test. Mansolillo 1 year ago Lesson 1. What if there were two. One license level 5 word problems year 6 candidate. How to solve linear inequalities, Polynomial Calculator. Marketing cv uk data is quick and easy. What is the maximum height on the wall that the ladder can reach. | 677.169 | 1 |
001 Algebra Rules (true or false) -First Day of School
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During the first day of school, after the standard formalities, this lesson introduces Pre-Calculus (or Intro to Calc) students to several algebraic rules as a true or false to build a stronger background before reviewing more difficult algebra. | 677.169 | 1 |
Soccermatics: Mathematical Adventures in the Beautiful Game
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Pro sports has been inundated with statistics of all sorts – some relevant, and others less so. Bookies have turned understanding statistics well enough to make their living from it. This book uses soccer as a starting point to explain mathematics – specifically statistics and game theory. While the book does delve into "proofs" or sorts, these are more like explanations that try to elucidate underlying mathematical concepts, rather than provide rigorous mathematical proofs.
Chapters are divided into three parts. Part I (consisting of five chapters) focuses on the game itself. In the process, readers learn about probability distributions, tessellations, field lines and projectile motion. Part II (consisting of four chapters) explores game strategy. Here, readers are introduced to game theory and an introduction to binomial probability. Part III (consisting of four chapters) uses the fans and bookies to introduce readers to geometric progressions, crowd behavior, calculating expected outcomes (from probabilities). While the discussion starts with soccer, the same mathematical underpinnings are explored in other relevant topics (such as economics, social and applied sciences).
The tone is conversational, the test easy to read, and the concepts are fairly well explained for the most part. The narrative moves easily from soccer to mathematics and through other areas where the same mathematical concepts are used. Readers more seasoned in applied mathematics may question some of the book's assumptions (such as the rational for fitting a Poisson distribution without discussing other competing distributions), but we do have to start somewhere. Those who enjoy reading about applied mathematics are more likely to enjoy this book than those who like soccer but not applied mathematics. Overall an informative and easy read. have | 677.169 | 1 |
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This 21-page compendium covers important theorems and relationships with their detailed proofs.Also it contains some moderate to difficult questions with their solutions in order to highlight certain insights about certain important concepts.I can safely recommend this to students and teachers who intend to take forward the teachings and learning of higher mathematics.Those will be of great help for those who are preparing for International Mathematical Olympiad and other higher mathematical based examinations. | 677.169 | 1 |
Curves and Surfaces for Computer Graphics
abstract: Computer graphics is important in many areas including engineering design, architecture, education, and computer art and animation. This book examines a wide array of current methods used in creating real-looking objects in the computer, one of the main aims of computer graphics. Key features: * Good foundational mathematical introduction to curves and surfaces; no advanced math required * Topics organized by different interpolation/approximation techniques, each technique providing useful information about curves and surfaces * Exposition motivated by numerous examples and exercises sprinkled throughout, aiding the reader * Includes a gallery of color images, Mathematica code listings, and sections on curves & surfaces by refinement and on sweep surfaces * Web site maintained and updated by the author, providing readers with errata and auxiliary material | 677.169 | 1 |
Introduction to Vector Spaces
In this vector spaces learning exercise, students identify the vector in a given set. They explore addition and scalar multiplication in their vectors. This two-page learning exercise contains 13 problems. | 677.169 | 1 |
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