data/chunks/chunk_a71f59bd_21.jsonl

{"pdf_file": "Design of Structural Elements_ Concrete, Steelwork, Masonry and Timber Designs to British Standards and Eurocodes, 3rd Edition   ( PDFDrive ).pdf", "page_num": 397, "prompt": "Analyze this page with diagrams, tables, formulas, calculations, and explanations.", "response": "[NATIVE TEXT]\nEurocode 3: Design of steel structures\n376\nChapter 1: General\nChapter 2: Basis of design\nChapter 3: Materials\nChapter 4: Durability\nChapter 5: Structural analysis\nChapter 6: Ultimate limit state\nChapter 7: Serviceability limit state\nAnnex A – Method 1: Interaction factors\nAnnex B – Method 2: Interaction factors\nAnnex AB – Additional design provisions\nAnnex BB – Buckling of components of building\nstructures\nThe purpose of the following discussion is prin-\ncipally to highlight the main differences between\nEC 3, EC 3–5, EC 3–8 and BS 5950: Part 1.\nA number of comparative design studies are also\nincluded to illustrate the new design procedures.\n9.2 Structure of EC 3\nApart from the major re-packaging exercise referred\nto above, the organisation of material within indi-\nvidual sub-parts is also quite different to that used\nin BS 5950, as exemplified by the above contents\nlist for EC 3. Thus, whereas BS 5950 contains\nseparate sections on the design of different types of\nelements in building structures, e.g. beams, plate\ngirders, compression members, etc., EC 3, on the\nother hand, divides the material on the basis of\nstructural phenomena, e.g. tension, compression,\nbending, shear and buckling, which may apply to\nany element. This change has largely been imple-\nmented in order to avoid duplication of design\nrules and also to promote a better understanding\nof structural behaviour.\n9.3 Principles and Application\nrules (Cl. 1.4, EC 3)\nAs with the other structural Eurocodes and for the\nreasons discussed in section 7.5.2, the clauses in\nEC 3 have been divided into Principles and Applica-\ntion rules. Principles comprise general statements,\ndefinitions, requirements and models for which no\nalternative is permitted. Principles are indicated\nby the letter P after the clause number. The Appli-\ncation rules are generally recognised rules which\nfollow the statements and satisfy the requirements\ngiven in the Principles. The absence of the letter\nP after the clause number indicates an application\nrule. The use of alternative application rules to\nthose recommended in the Eurocode is permitted\nprovided it can be shown that the alternatives are\nat least equivalent and do not adversely affect other\ndesign requirements. It is worth noting, however,\nthat if an alternative application rule is used the\nresulting design will not be deemed Eurocode\ncompliant (Fig. 9.1).\n9.4 Nationally Determined\nParameters\nLike the other structural Eurocodes, Eurocode 3\nallows some parameters and design methods to be\ndetermined at the national level. Where a national\nchoice is allowed this is indicated in a note in the\nnormative text under the relevant clause. The note\nmay include the recommended value of the par-\nameter, or preferred method, etc., but the actual\nvalue, methodology, etc., to be used in a particular\nMember State country is/will be given in the appro-\npriate National Annex. The recommended values\nof these parameters and design methods/procedures\nare collectively referred to as Nationally Determined\nParameters (NDPs). The NDPs determine various\naspects of design but perhaps most importantly\nthe level of safety of structures during execution\n(i.e. construction/fabrication) and in-service, which\nremains the responsibility of individual nations.\nThe UK National Annexes to EC 3 and EC 3–5\nwere published in 2008 and 2005 respectively. The\nNational Annex to EC 3–8 is still awaited. Thus\nthe discussion and worked examples in sections 9.10\nand 9.11 on, respectively, beam and column design,\nare based on the material in EC 3, EC 3–5 and\nthe NDPs in the accompanying National Annexes\n6.2.4 Compression\n(1)P\nThe design value of the compression force NEd at\neach cross-section shall satisfy:\nN\nN\nEd\nc,Rd\n  ,\n≤10\n(6.9)\n(2)\nThe design resistance of the cross-section for\nuniform compression Nc,Rd should be determined as\nfollows:\nN\nAf\nc,Rd\ny\nM\n= γ\n0\nfor class 1, 2, or 3 cross-sections\n(6.10)\nN\nA f\nc,Rd\neff y\nM\n= γ\n0\nfor class 4 cross-sections\n(6.11)\nFig. 9.1 Example of Principle and Application rules (Cl.\n6.2.4, EC 3, p49).\n\n[TABLES]\n[[{\"0\": \"\", \"1\": \"for class 1, 2, or 3 cross-sections(6.10)Nc,Rdγ\"}, {\"0\": \"structures\", \"1\": \"M0\"}, {\"0\": \"The purpose of the following discussion is prin-\", \"1\": \"A feff yN= for class 4 cross-sections(6.11)\"}, {\"0\": \"cipally  to  highlight  the  main  differences  between\", \"1\": \"c,RdγM0\"}, {\"0\": \"EC  3,  EC  3–5,  EC  3–8  and  BS  5950:  Part  1.\", \"1\": \"Fig. 9.1 Example of Principle and Application rules (Cl.\"}, {\"0\": \"A  number  of  comparative  design  studies  are  also\", \"1\": \"6.2.4, EC 3, p49).\"}, {\"0\": \"included to illustrate the new design procedures.\", \"1\": \"\"}, {\"0\": \"9.2 Structure of EC 3\", \"1\": \"provided  it  can  be  shown  that  the  alternatives  are\"}, {\"0\": \"\", \"1\": \"at least equivalent and do not adversely affect other\"}, {\"0\": \"Apart from the major re-packaging exercise referred\", \"1\": \"design  requirements.  It  is  worth  noting,  however,\"}, {\"0\": \"to above, the organisation of material within indi-\", \"1\": \"that  if  an  alternative  application  rule  is  used  the\"}, {\"0\": \"vidual sub-parts is also quite different to that used\", \"1\": \"resulting  design  will  not  be  deemed  Eurocode\"}, {\"0\": \"in  BS  5950,  as  exemplified  by  the  above  contents\", \"1\": \"compliant (Fig. 9.1).\"}, {\"0\": \"list  for  EC  3.  Thus,  whereas  BS  5950  contains\", \"1\": \"\"}, {\"0\": \"separate sections on the design of different types of\", \"1\": \"\"}, {\"0\": \"elements  in  building  structures,  e.g.  beams,  plate\", \"1\": \"9.4 Nationally Determined\"}, {\"0\": \"girders, compression members, etc., EC 3, on the\", \"1\": \"Parameters\"}, {\"0\": \"other  hand,  divides  the  material  on  the  basis  of\", \"1\": \"\"}, {\"0\": \"structural  phenomena,  e.g.  tension,  compression,\", \"1\": \"Like  the  other  structural  Eurocodes,  Eurocode  3\"}, {\"0\": \"bending,  shear  and  buckling,  which  may  apply  to\", \"1\": \"allows some parameters and design methods to be\"}, {\"0\": \"any  element.  This  change  has  largely  been  imple-\", \"1\": \"determined at the national level. Where a national\"}, {\"0\": \"mented in  order  to  avoid  duplication  of  design\", \"1\": \"choice is allowed this is indicated in a note in the\"}, {\"0\": \"rules  and  also  to  promote  a  better  understanding\", \"1\": \"normative text under the relevant clause. The note\"}, {\"0\": \"of structural behaviour.\", \"1\": \"may  include  the  recommended  value  of  the  par-\"}, {\"0\": \"\", \"1\": \"ameter,  or  preferred  method,  etc.,  but  the  actual\"}, {\"0\": \"\", \"1\": \"value, methodology, etc., to be used in a particular\"}, {\"0\": \"9.3 Principles and Application\", \"1\": \"Member State country is/will be given in the appro-\"}, {\"0\": \"rules (Cl. 1.4, EC 3)\", \"1\": \"priate  National  Annex.  The  recommended  values\"}, {\"0\": \"\", \"1\": \"of these parameters and design methods/procedures\"}, {\"0\": \"As with the other structural Eurocodes and for the\", \"1\": \"are collectively referred to as Nationally Determined\"}, {\"0\": \"reasons  discussed  in  section  7.5.2,  the  clauses  in\", \"1\": \"Parameters (NDPs). The NDPs determine various\"}, {\"0\": \"EC 3 have been divided into Principles and Applica-\", \"1\": \"aspects  of  design  but  perhaps  most importantly\"}, {\"0\": \"tion rules. Principles comprise general statements,\", \"1\": \"the  level  of  safety  of  structures  during  execution\"}, {\"0\": \"definitions, requirements and models for which no\", \"1\": \"(i.e. construction/fabrication) and in-service, which\"}, {\"0\": \"alternative is  permitted.  Principles  are indicated\", \"1\": \"remains the responsibility of individual nations.\"}, {\"0\": \"by the letter P after the clause number. The Appli-\", \"1\": \"The UK National Annexes to EC 3 and EC 3–5\"}, {\"0\": \"cation  rules  are  generally  recognised  rules  which\", \"1\": \"were published in 2008 and 2005 respectively. The\"}, {\"0\": \"follow the statements and satisfy the requirements\", \"1\": \"National  Annex  to  EC  3–8  is  still  awaited.  Thus\"}, {\"0\": \"given  in  the  Principles.  The  absence  of  the  letter\", \"1\": \"the discussion and worked examples in sections 9.10\"}, {\"0\": \"P after the clause number indicates an application\", \"1\": \"and 9.11 on, respectively, beam and column design,\"}, {\"0\": \"rule.  The  use  of  alternative  application  rules  to\", \"1\": \"are  based  on  the  material  in  EC  3,  EC  3–5  and\"}, {\"0\": \"those  recommended  in  the  Eurocode  is  permitted\", \"1\": \"the NDPs in the accompanying National Annexes\"}], [{\"0\": \"Eurocode 3: Design of steel structures\", \"1\": \"\"}, {\"0\": \"Chapter 1: General\", \"1\": \"6.2.4 Compression\"}, {\"0\": \"Chapter 2: Basis of design\", \"1\": \"(1)PThe design value of the compression force NEd at\"}, {\"0\": \"Chapter 3: Materials\", \"1\": \"each cross-section shall satisfy:\"}, {\"0\": \"Chapter 4: Durability\", \"1\": \"\"}, {\"0\": \"Chapter 5: Structural analysis\", \"1\": \"N NEd ≤ 10(6.9)\"}, {\"0\": \"Chapter 6: Ultimate limit state\", \"1\": \"c,Rd\"}, {\"0\": \"Chapter 7: Serviceability limit state\", \"1\": \"(2)The design resistance of the cross-section for\"}, {\"0\": \"Annex A – Method 1: Interaction factors\", \"1\": \"uniform compression Nc,Rd should be determined as\"}, {\"0\": \"Annex B – Method 2: Interaction factors\", \"1\": \"follows:\"}, {\"0\": \"Annex AB – Additional design provisions\", \"1\": \"Af\"}, {\"0\": \"Annex BB – Buckling of components of building\", \"1\": \"yfor class 1, 2, or 3 cross-sections(6.10)N= c,Rdγ\"}, {\"0\": \"structures\", \"1\": \"M0\"}, {\"0\": \"The purpose of the following discussion is prin-\", \"1\": \"A feff yN= for class 4 cross-sections(6.11)\"}, {\"0\": \"cipally  to  highlight  the  main  differences  between\", \"1\": \"c,RdγM0\"}, {\"0\": \"EC  3,  EC  3–5,  EC  3–8  and  BS  5950:  Part  1.\", \"1\": \"Fig. 9.1 Example of Principle and Application rules (Cl.\"}, {\"0\": \"A  number  of  comparative  design  studies  are  also\", \"1\": \"6.2.4, EC 3, p49).\"}, {\"0\": \"included to illustrate the new design procedures.\", \"1\": \"\"}, {\"0\": \"9.2 Structure of EC 3\", \"1\": \"provided  it  can  be  shown  that  the  alternatives  are\"}, {\"0\": \"\", \"1\": \"at least equivalent and do not adversely affect other\"}, {\"0\": \"Apart from the major re-packaging exercise referred\", \"1\": \"design  requirements.  It  is  worth  noting,  however,\"}, {\"0\": \"to above, the organisation of material within indi-\", \"1\": \"that  if  an  alternative  application  rule  is  used  the\"}, {\"0\": \"vidual sub-parts is also quite different to that used\", \"1\": \"resulting  design  will  not  be  deemed  Eurocode\"}, {\"0\": \"in  BS  5950,  as  exemplified  by  the  above  contents\", \"1\": \"compliant (Fig. 9.1).\"}, {\"0\": \"list  for  EC  3.  Thus,  whereas  BS  5950  contains\", \"1\": \"\"}, {\"0\": \"separate sections on the design of different types of\", \"1\": \"\"}, {\"0\": \"elements  in  building  structures,  e.g.  beams,  plate\", \"1\": \"9.4 Nationally Determined\"}, {\"0\": \"girders, compression members, etc., EC 3, on the\", \"1\": \"Parameters\"}, {\"0\": \"other  hand,  divides  the  material  on  the  basis  of\", \"1\": \"\"}, {\"0\": \"structural  phenomena,  e.g.  tension,  compression,\", \"1\": \"Like  the  other  structural  Eurocodes,  Eurocode  3\"}, {\"0\": \"bending,  shear  and  buckling,  which  may  apply  to\", \"1\": \"allows some parameters and design methods to be\"}, {\"0\": \"any  element.  This  change  has  largely  been  imple-\", \"1\": \"determined at the national level. Where a national\"}, {\"0\": \"mented in  order  to  avoid  duplication  of  design\", \"1\": \"choice is allowed this is indicated in a note in the\"}, {\"0\": \"rules  and  also  to  promote  a  better  understanding\", \"1\": \"normative text under the relevant clause. The note\"}, {\"0\": \"of structural behaviour.\", \"1\": \"may  include  the  recommended  value  of  the  par-\"}, {\"0\": \"\", \"1\": \"ameter,  or  preferred  method,  etc.,  but  the  actual\"}, {\"0\": \"\", \"1\": \"value, methodology, etc., to be used in a particular\"}, {\"0\": \"9.3 Principles and Application\", \"1\": \"Member State country is/will be given in the appro-\"}, {\"0\": \"rules (Cl. 1.4, EC 3)\", \"1\": \"priate  National  Annex.  The  recommended  values\"}, {\"0\": \"\", \"1\": \"of these parameters and design methods/procedures\"}, {\"0\": \"As with the other structural Eurocodes and for the\", \"1\": \"are collectively referred to as Nationally Determined\"}, {\"0\": \"reasons  discussed  in  section  7.5.2,  the  clauses  in\", \"1\": \"Parameters (NDPs). The NDPs determine various\"}, {\"0\": \"EC 3 have been divided into Principles and Applica-\", \"1\": \"aspects  of  design  but  perhaps  most importantly\"}, {\"0\": \"tion rules. Principles comprise general statements,\", \"1\": \"the  level  of  safety  of  structures  during  execution\"}, {\"0\": \"definitions, requirements and models for which no\", \"1\": \"(i.e. construction/fabrication) and in-service, which\"}, {\"0\": \"alternative is  permitted.  Principles  are indicated\", \"1\": \"remains the responsibility of individual nations.\"}, {\"0\": \"by the letter P after the clause number. The Appli-\", \"1\": \"The UK National Annexes to EC 3 and EC 3–5\"}, {\"0\": \"cation  rules  are  generally  recognised  rules  which\", \"1\": \"were published in 2008 and 2005 respectively. The\"}, {\"0\": \"follow the statements and satisfy the requirements\", \"1\": \"National  Annex  to  EC  3–8  is  still  awaited.  Thus\"}, {\"0\": \"given  in  the  Principles.  The  absence  of  the  letter\", \"1\": \"the discussion and worked examples in sections 9.10\"}, {\"0\": \"P after the clause number indicates an application\", \"1\": \"and 9.11 on, respectively, beam and column design,\"}, {\"0\": \"rule.  The  use  of  alternative  application  rules  to\", \"1\": \"are  based  on  the  material  in  EC  3,  EC  3–5  and\"}, {\"0\": \"those  recommended  in  the  Eurocode  is  permitted\", \"1\": \"the NDPs in the accompanying National Annexes\"}, {\"0\": \"376\", \"1\": \"\"}], [], []]", "page_type": "mixed", "image": "Design of Structural Elements_ Concr_a6902759_p0397.png"}
{"pdf_file": "Design of Structural Elements_ Concrete, Steelwork, Masonry and Timber Designs to British Standards and Eurocodes, 3rd Edition   ( PDFDrive ).pdf", "page_num": 398, "prompt": "Analyze this page with diagrams, tables, formulas, calculations, and explanations.", "response": "[NATIVE TEXT]\n377\nBasis of design\nwhereas the material in section 9.12 on connec-\ntions is based solely on the contents of EC 3–8.\n9.5 Symbols (Cl. 1.6, EC 3)\nThe symbols used in EC 3 are extremely schematic\nbut a little cumbersome. For example, Mpl,y,Rd\ndenotes the design plastic moment of resistance\nabout the y–y (major) axis. Symbols such as this\nmake many expressions and formulae much longer\nand seem more complex than they actually are.\nA number of symbols which are used to identify\nparticular dimensions of Universal sections have\nchanged in Eurocode 3 as discussed in section 9.6.\nFor example, the elastic and plastic moduli about\nthe major axis are denoted in Eurocode 3 by the\nsymbols Wel,y and Wpl,y respectively rather than Zxx\nand Sxx as in BS 5950. Likewise, the elastic and\nplastic moduli about the minor axis are denoted in\nEC 3 by the symbols Wel,z and Wpl,z, respectively,\nrather than Zyy and Syy as in BS 5950.\n9.6 Member axes (Cl. 1.6.7, EC 3)\nMember axes for commonly used steel members\nare shown in Fig. 9.2. This system is different to\nthat used in BS 5950. Thus, x–x is the axis along\nthe member, while axis y–y is what is termed in\nBS 5950 as axis x–x, the major axis. The minor axis\nof bending is taken as axis z–z in EC 3. Considering\nsteel design in isolation, this change need not have\nhappened. It was insisted on for consistency with\nthe structural Eurocodes for other materials, e.g.\nconcrete and timber.\n9.7 Basis of design\nLike BS 5950, EC 3 is based on the limit state\nmethod and for design purposes principally con-\nsiders two categories of limit states: ultimate and\nserviceability. A separate (third) category of dura-\nbility is also mentioned in clause 4 of EC 3 which\ncovers the limit states of corrosion, mechanical wear\nand fatigue. Of these, corrosion is generally viewed\nas the most critical issue for steel building struc-\ntures and involves taking into account the expected\nexposure conditions, the design working life of the\nstructure, the composition and properties of the\nsteel, structural detailing and protective measures,\namongst other factors (clause 2.4, EN 1990).\nThe terms ‘ultimate limit state’ (ULS) and ‘ser-\nviceability limit state’ (SLS) apply in the same way\nas we understand them in BS 5950. Thus, ultimate\nlimit states are those associated with collapse, or\nwith other forms of structural failure which may\nendanger the safety of people while serviceability\nlimit states concern states beyond which specified\nservice criteria, for example the functioning of the\nstructure or member, the comfort of people and\nappearance of the structure, are no longer met\n(clauses 3.3 and 3.4 of EN 1990).\nFig. 9.2 Member definitions and axes used in EC 3 (NB symbols in parentheses denote symbols used in BS 5950).\n\n[TABLES]\n[[{\"0\": \"\", \"1\": \"Basis of design\"}, {\"0\": \"whereas  the  material  in  section  9.12  on  connec-\", \"1\": \"steel design in isolation, this change need not have\"}, {\"0\": \"tions is based solely on the contents of EC 3–8.\", \"1\": \"happened.  It  was  insisted  on  for  consistency  with\"}, {\"0\": \"\", \"1\": \"the  structural  Eurocodes  for  other  materials,  e.g.\"}, {\"0\": \"\", \"1\": \"concrete and timber.\"}, {\"0\": \"9.5 Symbols (Cl. 1.6, EC 3)\", \"1\": \"\"}, {\"0\": \"The symbols used in EC 3 are extremely schematic\", \"1\": \"9.7 Basis of design\"}, {\"0\": \"but  a little  cumbersome.  For  example,  Mpl,y,Rd\", \"1\": \"Like  BS  5950,  EC  3  is  based  on  the  limit  state\"}, {\"0\": \"denotes  the  design  plastic  moment  of  resistance\", \"1\": \"method  and  for  design  purposes  principally  con-\"}, {\"0\": \"about  the  y–y  (major)  axis.  Symbols  such  as  this\", \"1\": \"siders  two  categories  of  limit  states:  ultimate  and\"}, {\"0\": \"make many expressions and formulae much longer\", \"1\": \"serviceability.  A  separate  (third)  category  of  dura-\"}, {\"0\": \"and seem more complex than they actually are.\", \"1\": \"bility is also mentioned in clause 4 of EC 3 which\"}, {\"0\": \"A number of symbols which are used to identify\", \"1\": \"covers the limit states of corrosion, mechanical wear\"}, {\"0\": \"particular  dimensions  of  Universal  sections  have\", \"1\": \"and fatigue. Of these, corrosion is generally viewed\"}, {\"0\": \"changed in Eurocode 3 as discussed in section 9.6.\", \"1\": \"as  the  most  critical  issue  for  steel  building  struc-\"}, {\"0\": \"For  example,  the  elastic  and  plastic  moduli  about\", \"1\": \"tures and involves taking into account the expected\"}, {\"0\": \"the  major  axis  are  denoted  in  Eurocode  3  by  the\", \"1\": \"exposure conditions, the design working life of the\"}, {\"0\": \"symbols Wel,y and Wpl,y respectively rather than Zxx\", \"1\": \"structure,  the  composition  and  properties  of  the\"}, {\"0\": \"and  Sxx  as  in  BS  5950.  Likewise,  the  elastic  andplastic moduli about the minor axis are denoted in\", \"1\": \"steel, structural detailing and protective measures,\"}, {\"0\": \"EC  3  by  the  symbols  Wel,z  and  Wpl,z,  respectively,\", \"1\": \"amongst other factors (clause 2.4, EN 1990).\"}, {\"0\": \"rather than Zyy and Syy as in BS 5950.\", \"1\": \"The terms ‘ultimate limit state’ (ULS) and ‘ser-\"}, {\"0\": \"\", \"1\": \"viceability limit state’ (SLS) apply in the same way\"}, {\"0\": \"\", \"1\": \"as we understand them in BS 5950. Thus, ultimate\"}, {\"0\": \"9.6 Member axes (Cl. 1.6.7, EC 3)\", \"1\": \"limit  states  are  those  associated  with  collapse,  or\"}, {\"0\": \"\", \"1\": \"with  other  forms  of  structural  failure  which  may\"}, {\"0\": \"Member  axes  for  commonly  used  steel  members\", \"1\": \"endanger  the  safety  of  people  while  serviceability\"}, {\"0\": \"are  shown  in  Fig.  9.2.  This  system  is  different  to\", \"1\": \"limit  states  concern  states  beyond  which  specified\"}, {\"0\": \"that used in BS 5950. Thus, x–x is the axis along\", \"1\": \"service criteria, for example the functioning of the\"}, {\"0\": \"the  member,  while  axis  y–y  is  what  is  termed  in\", \"1\": \"structure  or  member,  the  comfort  of  people  and\"}, {\"0\": \"BS 5950 as axis x–x, the major axis. The minor axis\", \"1\": \"appearance  of  the  structure,  are  no longer  met\"}, {\"0\": \"of bending is taken as axis z–z in EC 3. Considering\", \"1\": \"(clauses 3.3 and 3.4 of EN 1990).\"}], [], []]", "page_type": "mixed", "image": "Design of Structural Elements_ Concr_a6902759_p0398.png"}
{"pdf_file": "Design of Structural Elements_ Concrete, Steelwork, Masonry and Timber Designs to British Standards and Eurocodes, 3rd Edition   ( PDFDrive ).pdf", "page_num": 399, "prompt": "Analyze this page with diagrams, tables, formulas, calculations, and explanations.", "response": "[NATIVE TEXT]\nEurocode 3: Design of steel structures\n378\nThe actual serviceability limit states relevant to\nsteel building structures are outlined in section 7\nof EC 3. The code provides very few details, how-\never, as many SLS are not material specific and for\nrelevant design guidance the designer must turn to\nclause A1.4 of EN 1990. The principal serviceability\nlimit states recommended in EC 3 are listed below:\n1. Deformations, e.g. vertical and horizontal\ndeflections which cause damage to finishes or\nnon-structural elements such as partitions and\ncladding, or adversely affect the comfort of\npeople or appearance of the structure.\n2. Dynamic effects, e.g. vibration, oscillation or\nsway which causes discomfort to the occupants\nof a building or damage to its contents.\nBased on the recommendations given in clauses\n3.2 and 3.3 of EN 1990, the ultimate limit states\nrelevant to steel structures and components are:\n(1) static equilibrium of the structure\n(2) rupture, excessive deformation of a member,\ntransformation \nof \nthe \nstructure \ninto \na\nmechanism\n(3) instability induced by second order effects, e.g.\nlack of fit, thermal effects, sway\n(4) fatigue\n(5) accidental damage, e.g. exposure to fire, explo-\nsion and impact.\nAs can be appreciated, these are roughly similar\nto the ultimate limit states used in BS 5950\n(Table 4.1) although there are slight changes in the\nterminology.\nIn the context of elemental design, the designer\nis principally concerned with the ultimate limit state\nof rupture (i.e. strength) and the serviceability limit\nstate of deflection. EC 3 requirements in respect of\nthese two limit states will be discussed later for\nindividual element types.\nIn a similar way to BS 5950, clauses 1.5.3 and\n5.1 of EC 3 distinguishes between three types of\nframes used for building structures and associated\njoint behaviour:\n(a) simple frames in which the joint are not\nrequired to resist moments, i.e. the joints are\npinned and by implication the structure is fully\nbraced.\n(b) continuous frames in which the joints are rigid\n(for elastic analysis) or full strength (for plastic\nanalysis) and can transmit all moments and\nforces.\n(c) semi-continuous frames in which the actual\nstrength of the joints is taken into account.\n9.8 Actions (Cl. 2.3, EC 3)\nActions is Eurocode terminology for loads and\nimposed deformations. Permanent actions, G, are\nall the fixed loads acting on the structure, including\nfinishes, immovable partitions and the self weight\nof the structure. Variable actions, Q, include the\nimposed, wind and snow loads. Clause 2.3.1 of\nEC 3 recommends that the values of characteristic\npermanent, Gk, and variable, Qk, actions should be\ntaken from relevant parts of Eurocode 1: Actions on\nstructures. Guidance on determining the design value\nof actions and combination of actions is given in\nEN 1990: Basis of structural design. These documents\nand topics are briefly discussed in section 8.5 of\nthis book. As noted there, the design value of an\naction (Fd) is obtained by multiplying the repre-\nsentative value (Frep) by the appropriate partial safety\nfactor for actions (γf):\nFd = γ fFrep\n(9.1)\nTable 9.3 shows the relevant partial safety factors\nfor the ultimate limit state of strength. Other safety\nfactors will apply in other design situations. For\nexample, the partial factors for the ultimate limit\nstates of equilibrium are shown in Table 8.6. In\nequation 9.1, Frep is generally taken as the charac-\nteristic value of a permanent or variable action (i.e.\nFk). Assuming that the member being designed is\nsubjected to one or more permanent actions and\none variable action only, i.e. load combination 1 in\nTable 9.3, the partial safety factor for permanent\nactions, γG can conservatively be taken as 1.35 and\nfor the variable action, γQ as 1.5. The correspond-\ning values of the safety factors in BS 5950 are 1.4\nand 1.6 respectively. As discussed in section 8.5.3,\nit is possible to improve structural efficiency by\nusing expressions 6.10a and 6.10b of EN 1990\n(respectively load combination 3(a) and 3(b)/3(c)\nin Table 9.3) to estimate the design values of actions\nbut the value of 1.35 for γG is conservative and\nused throughout this chapter.\n9.9 Materials\n9.9.1 DESIGN STRENGTHS (CL. 2.4.3, EC 3)\nThe design strengths, Xd, are obtained by dividing\nthe characteristic strengths, Xk, which can be taken\nas nominal values in calculations, by the material\npartial safety factor, γm, i.e.\n    \nX\nX\nd\nk\nm\n  \n= γ\n(9.2)\n\n[TABLES]\n[[{\"0\": \"Eurocode 3: Design of steel structures\", \"1\": \"\"}, {\"0\": \"The  actual  serviceability  limit  states  relevant  to\", \"1\": \"9.8 Actions (Cl. 2.3, EC 3)\"}, {\"0\": \"steel  building  structures  are  outlined  in  section  7\", \"1\": \"Actions is  Eurocode terminology for loads  and\"}, {\"0\": \"of EC 3. The code provides very few details, how-\", \"1\": \"imposed  deformations.  Permanent  actions,  G,  are\"}, {\"0\": \"ever, as many SLS are not material specific and for\", \"1\": \"all the fixed loads acting on the structure, including\"}, {\"0\": \"relevant design guidance the designer must turn to\", \"1\": \"finishes,  immovable  partitions  and  the  self  weight\"}, {\"0\": \"clause A1.4 of EN 1990. The principal serviceability\", \"1\": \"of  the  structure.  Variable  actions,  Q,  include  the\"}, {\"0\": \"limit states recommended in EC 3 are listed below:\", \"1\": \"imposed,  wind  and  snow  loads.  Clause  2.3.1  of\"}, {\"0\": \"1. Deformations, e.g. vertical and horizontal\", \"1\": \"EC 3 recommends that the values of characteristic\"}, {\"0\": \"deflections  which  cause  damage  to  finishes  or\", \"1\": \"permanent, Gk, and variable, Qk, actions should be\"}, {\"0\": \"non-structural  elements  such  as  partitions  and\", \"1\": \"taken from relevant parts of Eurocode 1: Actions on\"}, {\"0\": \"cladding,  or  adversely  affect the  comfort  of\", \"1\": \"structures. Guidance on determining the design value\"}, {\"0\": \"people or appearance of the structure.\", \"1\": \"of  actions  and  combination  of  actions  is  given  in\"}, {\"0\": \"2. Dynamic  effects,  e.g.  vibration,  oscillation  or\", \"1\": \"EN 1990: Basis of structural design. These documents\"}, {\"0\": \"sway which causes discomfort to the occupants\", \"1\": \"and  topics  are  briefly  discussed  in  section  8.5  of\"}, {\"0\": \"of a building or damage to its contents.\", \"1\": \"this  book.  As  noted  there,  the  design  value  of  an\"}, {\"0\": \"Based  on  the  recommendations  given  in  clauses\", \"1\": \"action  (Fd)  is  obtained  by  multiplying  the  repre-\"}, {\"0\": \"3.2  and  3.3  of  EN  1990,  the  ultimate  limit  states\", \"1\": \"sentative value (Frep) by the appropriate partial safetyfactor for actions (γ\"}, {\"0\": \"relevant to steel structures and components are:\", \"1\": \"f):\"}, {\"0\": \"(1)static equilibrium of the structure\", \"1\": \"(9.1)Fd = γf Frep\"}, {\"0\": \"(2)rupture,  excessive  deformation  of  a  member,\", \"1\": \"Table 9.3 shows the relevant partial safety factors\"}, {\"0\": \"transformation of the structure into a\", \"1\": \"for the ultimate limit state of strength. Other safety\"}, {\"0\": \"mechanism\", \"1\": \"factors  will  apply  in  other  design  situations.  For\"}, {\"0\": \"(3)instability induced by second order effects, e.g.\", \"1\": \"example,  the  partial  factors  for  the  ultimate  limit\"}, {\"0\": \"lack of fit, thermal effects, sway\", \"1\": \"states  of  equilibrium  are  shown  in  Table  8.6.  In\"}, {\"0\": \"(4)fatigue\", \"1\": \"equation 9.1, Frep is generally taken as the charac-\"}, {\"0\": \"(5)accidental damage, e.g. exposure to fire, explo-\", \"1\": \"teristic value of a permanent or variable action (i.e.\"}, {\"0\": \"sion and impact.\", \"1\": \"Fk).  Assuming  that  the  member  being  designed  is\"}, {\"0\": \"As can be appreciated, these are roughly similar\", \"1\": \"subjected  to  one  or  more  permanent  actions  and\"}, {\"0\": \"to the  ultimate limit states  used in  BS  5950\", \"1\": \"one variable action only, i.e. load combination 1 in\"}, {\"0\": \"(Table 4.1) although there are slight changes in the\", \"1\": \"Table  9.3,  the  partial  safety  factor  for  permanent\"}, {\"0\": \"terminology.\", \"1\": \"actions, γG can conservatively be taken as 1.35 and\"}, {\"0\": \"In the context of elemental design, the designer\", \"1\": \"for the variable action, γQ as 1.5. The correspond-\"}, {\"0\": \"is principally concerned with the ultimate limit state\", \"1\": \"ing values of the safety factors in BS 5950 are 1.4\"}, {\"0\": \"of rupture (i.e. strength) and the serviceability limit\", \"1\": \"and 1.6 respectively. As discussed in section 8.5.3,\"}, {\"0\": \"state of deflection. EC 3 requirements in respect of\", \"1\": \"it  is  possible  to  improve  structural  efficiency  by\"}, {\"0\": \"these  two  limit  states  will  be  discussed  later  for\", \"1\": \"using  expressions  6.10a  and  6.10b  of  EN  1990\"}, {\"0\": \"individual element types.\", \"1\": \"(respectively  load  combination  3(a)  and  3(b)/3(c)\"}, {\"0\": \"In  a  similar  way  to  BS  5950,  clauses  1.5.3  and\", \"1\": \"in Table 9.3) to estimate the design values of actions\"}, {\"0\": \"5.1  of  EC  3  distinguishes  between  three  types  of\", \"1\": \"but  the  value  of  1.35  for  γG  is  conservative  and\"}, {\"0\": \"frames used for building structures and associated\", \"1\": \"used throughout this chapter.\"}, {\"0\": \"joint behaviour:\", \"1\": \"\"}, {\"0\": \"(a)simple frames in  which the joint  are  not\", \"1\": \"9.9 Materials\"}, {\"0\": \"required  to  resist  moments,  i.e.  the  joints  are\", \"1\": \"\"}, {\"0\": \"pinned and by implication the structure is fully\", \"1\": \"9.9.1 DESIGN STRENGTHS (CL. 2.4.3, EC 3)\"}, {\"0\": \"braced.\", \"1\": \"The design strengths, Xd, are obtained by dividing\"}, {\"0\": \"(b)continuous frames in which the joints are rigid\", \"1\": \"the characteristic strengths, Xk, which can be takenas  nominal  values  in  calculations,  by  the  material\"}, {\"0\": \"(for elastic analysis) or full strength (for plastic\", \"1\": \"partial safety factor, γ\"}, {\"0\": \"analysis)  and  can  transmit  all  moments  and\", \"1\": \"m, i.e.\"}, {\"0\": \"forces.\", \"1\": \"\"}], [{\"0\": \"Eurocode 3: Design of steel structures\", \"1\": \"\"}, {\"0\": \"The  actual  serviceability  limit  states  relevant  to\", \"1\": \"9.8 Actions (Cl. 2.3, EC 3)\"}, {\"0\": \"steel  building  structures  are  outlined  in  section  7\", \"1\": \"Actions is  Eurocode terminology for loads  and\"}, {\"0\": \"of EC 3. The code provides very few details, how-\", \"1\": \"imposed  deformations.  Permanent  actions,  G,  are\"}, {\"0\": \"ever, as many SLS are not material specific and for\", \"1\": \"all the fixed loads acting on the structure, including\"}, {\"0\": \"relevant design guidance the designer must turn to\", \"1\": \"finishes,  immovable  partitions  and  the  self  weight\"}, {\"0\": \"clause A1.4 of EN 1990. The principal serviceability\", \"1\": \"of  the  structure.  Variable  actions,  Q,  include  the\"}, {\"0\": \"limit states recommended in EC 3 are listed below:\", \"1\": \"imposed,  wind  and  snow  loads.  Clause  2.3.1  of\"}, {\"0\": \"1. Deformations, e.g. vertical and horizontal\", \"1\": \"EC 3 recommends that the values of characteristic\"}, {\"0\": \"deflections  which  cause  damage  to  finishes  or\", \"1\": \"permanent, Gk, and variable, Qk, actions should be\"}, {\"0\": \"non-structural  elements  such  as  partitions  and\", \"1\": \"taken from relevant parts of Eurocode 1: Actions on\"}, {\"0\": \"cladding,  or  adversely  affect the  comfort  of\", \"1\": \"structures. Guidance on determining the design value\"}, {\"0\": \"people or appearance of the structure.\", \"1\": \"of  actions  and  combination  of  actions  is  given  in\"}, {\"0\": \"2. Dynamic  effects,  e.g.  vibration,  oscillation  or\", \"1\": \"EN 1990: Basis of structural design. These documents\"}, {\"0\": \"sway which causes discomfort to the occupants\", \"1\": \"and  topics  are  briefly  discussed  in  section  8.5  of\"}, {\"0\": \"of a building or damage to its contents.\", \"1\": \"this  book.  As  noted  there,  the  design  value  of  an\"}, {\"0\": \"Based  on  the  recommendations  given  in  clauses\", \"1\": \"action  (Fd)  is  obtained  by  multiplying  the  repre-\"}, {\"0\": \"3.2  and  3.3  of  EN  1990,  the  ultimate  limit  states\", \"1\": \"sentative value (Frep) by the appropriate partial safetyfactor for actions (γ\"}, {\"0\": \"relevant to steel structures and components are:\", \"1\": \"f):\"}, {\"0\": \"(1)static equilibrium of the structure\", \"1\": \"(9.1)Fd = γf Frep\"}, {\"0\": \"(2)rupture,  excessive  deformation  of  a  member,\", \"1\": \"Table 9.3 shows the relevant partial safety factors\"}, {\"0\": \"transformation of the structure into a\", \"1\": \"for the ultimate limit state of strength. Other safety\"}, {\"0\": \"mechanism\", \"1\": \"factors  will  apply  in  other  design  situations.  For\"}, {\"0\": \"(3)instability induced by second order effects, e.g.\", \"1\": \"example,  the  partial  factors  for  the  ultimate  limit\"}, {\"0\": \"lack of fit, thermal effects, sway\", \"1\": \"states  of  equilibrium  are  shown  in  Table  8.6.  In\"}, {\"0\": \"(4)fatigue\", \"1\": \"equation 9.1, Frep is generally taken as the charac-\"}, {\"0\": \"(5)accidental damage, e.g. exposure to fire, explo-\", \"1\": \"teristic value of a permanent or variable action (i.e.\"}, {\"0\": \"sion and impact.\", \"1\": \"Fk).  Assuming  that  the  member  being  designed  is\"}, {\"0\": \"As can be appreciated, these are roughly similar\", \"1\": \"subjected  to  one  or  more  permanent  actions  and\"}, {\"0\": \"to the  ultimate limit states  used in  BS  5950\", \"1\": \"one variable action only, i.e. load combination 1 in\"}, {\"0\": \"(Table 4.1) although there are slight changes in the\", \"1\": \"Table  9.3,  the  partial  safety  factor  for  permanent\"}, {\"0\": \"terminology.\", \"1\": \"actions, γG can conservatively be taken as 1.35 and\"}, {\"0\": \"In the context of elemental design, the designer\", \"1\": \"for the variable action, γQ as 1.5. The correspond-\"}, {\"0\": \"is principally concerned with the ultimate limit state\", \"1\": \"ing values of the safety factors in BS 5950 are 1.4\"}, {\"0\": \"of rupture (i.e. strength) and the serviceability limit\", \"1\": \"and 1.6 respectively. As discussed in section 8.5.3,\"}, {\"0\": \"state of deflection. EC 3 requirements in respect of\", \"1\": \"it  is  possible  to  improve  structural  efficiency  by\"}, {\"0\": \"these  two  limit  states  will  be  discussed  later  for\", \"1\": \"using  expressions  6.10a  and  6.10b  of  EN  1990\"}, {\"0\": \"individual element types.\", \"1\": \"(respectively  load  combination  3(a)  and  3(b)/3(c)\"}, {\"0\": \"In  a  similar  way  to  BS  5950,  clauses  1.5.3  and\", \"1\": \"in Table 9.3) to estimate the design values of actions\"}, {\"0\": \"5.1  of  EC  3  distinguishes  between  three  types  of\", \"1\": \"but  the  value  of  1.35  for  γG  is  conservative  and\"}, {\"0\": \"frames used for building structures and associated\", \"1\": \"used throughout this chapter.\"}, {\"0\": \"joint behaviour:\", \"1\": \"\"}, {\"0\": \"(a)simple frames in  which the joint  are  not\", \"1\": \"9.9 Materials\"}, {\"0\": \"required  to  resist  moments,  i.e.  the  joints  are\", \"1\": \"\"}, {\"0\": \"pinned and by implication the structure is fully\", \"1\": \"9.9.1 DESIGN STRENGTHS (CL. 2.4.3, EC 3)\"}, {\"0\": \"braced.\", \"1\": \"The design strengths, Xd, are obtained by dividing\"}, {\"0\": \"(b)continuous frames in which the joints are rigid\", \"1\": \"the characteristic strengths, Xk, which can be takenas  nominal  values  in  calculations,  by  the  material\"}, {\"0\": \"(for elastic analysis) or full strength (for plastic\", \"1\": \"partial safety factor, γ\"}, {\"0\": \"analysis)  and  can  transmit  all  moments  and\", \"1\": \"m, i.e.\"}, {\"0\": \"forces.\", \"1\": \"X\"}, {\"0\": \"(c)semi-continuous  frames in  which  the  actual\", \"1\": \"k(9.2)X  =dγ\"}, {\"0\": \"strength of the joints is taken into account.\", \"1\": \"m\"}, {\"0\": \"378\", \"1\": \"\"}], [], []]", "page_type": "mixed", "image": "Design of Structural Elements_ Concr_a6902759_p0399.png"}
{"pdf_file": "Design of Structural Elements_ Concrete, Steelwork, Masonry and Timber Designs to British Standards and Eurocodes, 3rd Edition   ( PDFDrive ).pdf", "page_num": 400, "prompt": "Analyze this page with diagrams, tables, formulas, calculations, and explanations.", "response": "[NATIVE TEXT]\n379\nMaterials\nTable 9.3\nLoad combinations and partial safety/combination factors for the ultimate limit state of\nstrength\nLimit state/Load combination\nLoad Type\nPermanent, Gk\nImposed, Qk\nWind, Wk\nUnfavourable\nFavourable\nUnfavourable\nFavourable\nStrength:\n1. Permanent and variable\n1.35/1.35ξ\n1.0\n1.5\n0\n–\n2. Permanent and wind\n1.35/1.35ξ\n1.0\n–\n–\n1.5\n3. Permanent, imposed and wind\n(a)\n1.35\n1.0\n1.5ψ0,1\n0\n1.5ψ0,2\n(b)\n1.35/1.35ξ\n1.0\n1.5\n0\n1.5ψ0\n(c)\n1.35/1.35ξ\n1.0\n1.5ψ0\n0\n1.5\nBut this is in fact not used in EC 3. Instead,\nEC 3 divides the cross-section resistance, Rk, by a\npartial safety factor to give the design resistance,\nRd for the member, i.e.\n    \nR\nR\nd\nk\nM\n  \n= γ\n(9.3)\nwhere γM is the partial safety factor for the\nresistance. Thus, γM in EC 3 is not the same as γm\nin BS 5950. Here γm is applied to material strength;\nγM is applied to members or structures and takes\naccount of both material and modelling/geometric\nuncertainties.\n9.9.2 NOMINAL STRENGTHS (CL. 3.2, EC 3)\nTable 9.4 shows the steel grades and associated\nnominal values of yield strength and ultimate\ntensile strength for hot rolled steel sections. The steel\ngrades are the same as those mentioned in BS 5950\nexcept grade S235 which is popular in Continental\ncountries but not in the UK and was therefore\nnot referred to in BS 5950. Note that the values of\nthe yield strength shown in the table have been\nobtained from the product standard (EN 10025–2)\nand not from EC 3, as recommended in clause 2.4\nof the National Annex.\n9.9.3 PARTIAL FACTORS, γM (CL. 6.1, EC 3)\nValues of the partial factor γM (section 9.9.1)\napplied to characteristic values of resistance to\nobtain design resistances are given in clause 6.1\nof EC 3. The factor γM assumes different values\ndepending on the type of resistance being verified\nas indicated below:\nTable 9.4\nNominal values of yield strength fy and ultimate tensile\nstrength fu for hot rolled structural steel to EN 10025–2 (Table 3.1,\nEC 3)\nNominal\nNominal thickness of the element, t (mm)\nsteel\ngrade\nt ≤ 16 mm\n16 mm < t ≤ 40 mm\nfy(N/mm2)\nfu(N/mm2)\nfy(N/mm2)\nfu(N/mm2)\nS235\n235\n360\n235\n360\nS275\n275\n410\n265\n410\nS355\n355\n470\n345\n470\nS450\n440\n550\n430\n550\n\n[TABLES]\n[[{\"0\": \"Materials\"}, {\"0\": \"Table 9.3Load combinations and partial safety/combination factors for the ultimate limit state of\"}, {\"0\": \"strength\"}, {\"0\": \"Limit state/Load combinationLoad Type\"}, {\"0\": \"Permanent, GkImposed, Q kWind, Wk\"}, {\"0\": \"UnfavourableFavourableUnfavourableFavourable\"}, {\"0\": \"Strength:\"}, {\"0\": \"1. Permanent and variable1.35/1.35ξ1.01.50–\"}, {\"0\": \"2. Permanent and wind1.35/1.35ξ1.0––1.5\"}, {\"0\": \"3. Permanent, imposed and wind\"}, {\"0\": \"01.5ψ(a)1.351.01.5ψ0,10,2\"}, {\"0\": \"(b)1.35/1.35ξ1.01.501.5ψ0\"}, {\"0\": \"(c)1.35/1.35ξ1.01.5ψ01.50\"}, {\"0\": \"But  this  is  in  fact  not  used  in  EC  3.  Instead,tensile strength for hot rolled steel sections. The steel\"}, {\"0\": \"grades are the same as those mentioned in BS 5950EC 3 divides the cross-section resistance, Rk, by a\"}, {\"0\": \"partial  safety  factor  to  give  the  design  resistance,except grade S235 which is popular in Continental\"}, {\"0\": \"countries  but  not  in  the  UK  and  was  thereforeRd for the member, i.e.\"}, {\"0\": \"not referred to in BS 5950. Note that the values ofR\"}, {\"0\": \"kthe  yield  strength  shown  in  the  table  have  been(9.3)R  =dγ\"}, {\"0\": \"Mobtained from the product standard (EN 10025–2)\"}, {\"0\": \"and not from EC 3, as recommended in clause 2.4γwhere is the  partial safety factor for theM\"}, {\"0\": \"of the National Annex.resistance. Thus, γM in EC 3 is not the same as γm\"}, {\"0\": \"in BS 5950. Here γm is applied to material strength;\"}, {\"0\": \"γM  is  applied  to  members  or  structures  and  takes9.9.3PARTIAL FACTORS, γM (CL. 6.1, EC 3)\"}, {\"0\": \"account of both material and modelling/geometricValues  of the  partial factor  γ(section  9.9.1)M uncertainties.\"}, {\"0\": \"applied to  characteristic  values  of  resistance to\"}, {\"0\": \"obtain  design  resistances  are  given  in  clause  6.1\"}, {\"0\": \"of  EC  3.  The  factor  γ9.9.2 NOMINAL STRENGTHS (CL. 3.2, EC 3)M  assumes  different  values\"}, {\"0\": \"Table  9.4  shows  the  steel  grades  and  associateddepending  on  the  type  of  resistance  being  verified\"}, {\"0\": \"nominal  values  of  yield strength  and  ultimateas indicated below:\"}], [{\"0\": \"Table  9.4  shows  the  steel  grades  and  associateddepending  on  the  type  of  resistance  being  verified\"}, {\"0\": \"strength  and  ultimateas indicated below:\"}, {\"0\": \"Table 9.4Nominal values of yield strength fy and ultimate tensile\"}, {\"0\": \"strength fu for hot rolled structural steel to EN 10025–2 (Table 3.1,EC 3)\"}, {\"0\": \"NominalNominal thickness of the element, t (mm)\"}, {\"0\": \"steel\"}, {\"0\": \"gradet ≤ 16 mm16 mm < t ≤ 40 mm\"}, {\"0\": \"fy(N/mm2)fu(N/mm2)fy(N/mm2)fu(N/mm2)\"}, {\"0\": \"S235235360235360\"}, {\"0\": \"S275275410265410\"}, {\"0\": \"S355355470345470\"}, {\"0\": \"S450440550430550\"}], [], [], [{\"\": \"\"}]]", "page_type": "mixed", "image": "Design of Structural Elements_ Concr_a6902759_p0400.png"}
{"pdf_file": "Design of Structural Elements_ Concrete, Steelwork, Masonry and Timber Designs to British Standards and Eurocodes, 3rd Edition   ( PDFDrive ).pdf", "page_num": 401, "prompt": "Analyze this page with diagrams, tables, formulas, calculations, and explanations.", "response": "[NATIVE TEXT]\nEurocode 3: Design of steel structures\n380\nresistance of cross-sections\nγM0 = 1.00\nresistance of member to instability\nγM1 = 1.0\nresistance of cross-section to fracture γM2 = 1.10\nThe γM factors for joints are given in EC 3–8 as\ndiscussed in Section 9.13.\n9.9.4 MATERIAL COEFFICIENTS (CL. 3.2.5, EC 3)\nThe following coefficients are specified in EC 3\nModulus of elasticity, E = 210,000 N mm−2\nShear Modulus,\n    \nG\nF\n  (   )  \n \n=\n+\n≈\n−\n2 1\n81000\n2\nν\nN mm\n(9.4)\nPoisson’s ratio, ν = 0.3\nCoefficient of linear thermal expansion, α = 12 ×\n106 per deg C\nDensity, ρ = 7850 kg m−3\nNote that the value of E is slightly higher than\nthat specified in BS 5950 (205 kN mm−2).\n9.10 Classification of\ncross-sections (Cl. 5.5, EC 3)\nClassification has the same purpose as in BS 5950,\nand the four classifications are identical:\nClass 1 cross-sections: ‘plastic’ in BS 5950\nClass 2 cross-sections: ‘compact’ in BS 5950\nClass 3 cross-sections: ‘semi-compact’ in BS 5950\nClass 4 cross-sections: ‘slender’ in BS 5950\nClassification of a cross-section depends upon\nthe proportions of each of its compression elements.\nThe highest (least favourable) class number should\nbe quoted for a particular section.\nAppropriate sections of EC 3’s classification\nare given in Table 9.5. Comparison with Table 5.2\nof EC 3 shows that the symbols are slightly differ-\nent. This change has been carried to prevent any\nconfusion as to which dimension is to be used to\nclassify sections as EC 3 uses the same symbols to\ndefine different section dimensions. Comparison\nwith the corresponding limits given in BS 5950\nfor rolled sections (Table 4.4) shows that the c*/tw\n(= d/t) ratios are slightly more onerous for webs,\nand c/tf (= (B − tw − 2r)/2T) ratios are perhaps\nslightly less onerous for outstand flanges. It should\nbe noted that in EC 3 the factor ε is given by\nε = (235/fy)0.5\n(9.5)\nnot (275/py)0.5 as in BS 5950.\nFor Class 4 sections effective cross-sectional\nproperties can be calculated using effective widths\nTable 9.5\nMaximum width-to-thickness ratios\nfor compression elements (Table 5.2, EC3)\nType of element\nClass of element\nClass 1\nClass 2\nClass 3\nOutstand flange\n    \nc\ntf\n≤ 9ε\n    \nc\ntf\n≤ 10ε\n    \nc\ntf\n≤ 14ε\nfor rolled\nsection\nWeb with\n    \nc\nt\n*\nw\n≤ 72ε\n    \nc\nt\n*\nw\n≤ 83ε\n    \nc\nt\n*\nw\n≤ 124ε\nneutral axis at\nmid depth,\nrolled section\nWeb subject to\n    \nc\nt\n*\nw\n≤ 33ε\n    \nc\nt\n*\nw\n≤ 38ε\n    \nc\nt\n*\nw\n≤ 42ε\ncompression,\nrolled sections\nfy\n235\n275\n355\nε\n1\n0.92\n0.81\nc* = d; c = (b − tw − 2r)/2\nof plate which in some cases put the member\nback into Class 3. The expressions for calculating\neffective widths of hot rolled sections are given in\nEC 3–5.\n9.11 Design of beams\nAs noted at the outset, the information needed\nfor beam design is no longer in one place but\ndispersed in EC 3, EC 3–5 and EN 1990. To ease\nunderstanding of this material, as in Chapter 4 of\nthis book, we will consider the design of fully later-\nally restrained and unrestrained beams separately.\nThus, the following section (9.11.1) will consider the\ndesign of beams which are fully laterally restrained.\nSection 9.11.2 will then look at Eurocode 3: Part 1’s\nrules for designing beams which are not laterally\ntorsionally restrained.\n9.11.1 FULLY LATERALLY RESTRAINED\nBEAMS\nGenerally, such members should be checked for\n(1) resistance of cross-section to bending ULS\n(2) resistance to shear buckling ULS\n(3) resistance to flange-induced buckling ULS\n(4) resistance of the web to transverse forces ULS\n(5) deflection SLS.\n\n[TABLES]\n[[{\"0\": \"Eurocode 3: Design of steel structures\", \"1\": \"\"}, {\"0\": \"γresistance of cross-sectionsM0 = 1.00\", \"1\": \"Table 9.5 Maximum width-to-thickness ratios\"}, {\"0\": \"γresistance of member to instabilityM1 = 1.0\", \"1\": \"for compression elements (Table 5.2, EC3)\"}, {\"0\": \"resistance of cross-section to fracture γM2 = 1.10\", \"1\": \"\"}, {\"0\": \"The γM factors for joints are given in EC 3–8 as\", \"1\": \"Type of elementClass of element\"}, {\"0\": \"discussed in Section 9.13.\", \"1\": \"Class 1Class 2Class 3\"}, {\"0\": \"9.9.4 MATERIAL COEFFICIENTS (CL. 3.2.5, EC 3)\", \"1\": \"cccOutstand flange≤ 9ε≤ 10ε≤ 14ε\"}, {\"0\": \"The following coefficients are specified in EC 3\", \"1\": \"for rolledt ft ft f\"}, {\"0\": \"Modulus of elasticity, E = 210,000 N mm−2\", \"1\": \"section\"}, {\"0\": \"Shear Modulus,\", \"1\": \"c t* wc t* wc t* wWeb with≤ 72ε≤ 83ε≤ 124ε\"}, {\"0\": \"F−2=≈(9.4)81000N mmG\", \"1\": \"neutral axis at\"}, {\"0\": \"ν+2 1 )\", \"1\": \"mid depth,\"}, {\"0\": \"\", \"1\": \"rolled section\"}, {\"0\": \"Poisson’s ratio, ν = 0.3\", \"1\": \"Web subject to≤ 33ε≤ 38ε≤ 42ε\"}, {\"0\": \"Coefficient of linear thermal expansion, α = 12 ×\", \"1\": \"c t* wc t* wc t* wcompression,\"}, {\"0\": \"106 per deg C\", \"1\": \"rolled sections\"}, {\"0\": \"Density, ρ = 7850 kg m−3\", \"1\": \"235275355fy\"}, {\"0\": \"Note  that  the  value  of  E  is  slightly  higher  than\", \"1\": \"ε10.920.81\"}, {\"0\": \"that specified in BS 5950 (205 kN mm−2).\", \"1\": \"\"}, {\"0\": \"\", \"1\": \"c* = d; c = (b − t w − 2r)/2\"}, {\"0\": \"9.10 Classification of\", \"1\": \"\"}, {\"0\": \"cross-sections (Cl. 5.5, EC 3)\", \"1\": \"\"}, {\"0\": \"Classification has the same purpose as in BS 5950,\", \"1\": \"\"}, {\"0\": \"and the four classifications are identical:\", \"1\": \"of  plate  which in  some  cases  put the  member\"}, {\"0\": \"Class 1 cross-sections: ‘plastic’ in BS 5950\", \"1\": \"back into Class 3. The expressions for calculating\"}, {\"0\": \"Class 2 cross-sections: ‘compact’ in BS 5950\", \"1\": \"effective  widths  of  hot  rolled  sections  are  given  in\"}, {\"0\": \"Class 3 cross-sections: ‘semi-compact’ in BS 5950\", \"1\": \"EC 3–5.\"}, {\"0\": \"Class 4 cross-sections: ‘slender’ in BS 5950\", \"1\": \"\"}, {\"0\": \"Classification  of  a  cross-section  depends  upon\", \"1\": \"9.11 Design of beams\"}, {\"0\": \"the proportions of each of its compression elements.\", \"1\": \"As  noted  at  the  outset,  the  information  needed\"}, {\"0\": \"The highest (least favourable) class number should\", \"1\": \"for  beam  design is  no longer in  one  place  but\"}, {\"0\": \"be quoted for a particular section.\", \"1\": \"dispersed in EC 3, EC 3–5 and EN 1990. To ease\"}, {\"0\": \"Appropriate sections  of  EC  3’s  classification\", \"1\": \"understanding of this material, as in Chapter 4 of\"}, {\"0\": \"are given in Table 9.5. Comparison with Table 5.2\", \"1\": \"this book, we will consider the design of fully later-\"}, {\"0\": \"of EC 3 shows that the symbols are slightly differ-\", \"1\": \"ally  restrained  and  unrestrained  beams  separately.\"}, {\"0\": \"ent.  This  change  has  been  carried  to  prevent  any\", \"1\": \"Thus, the following section (9.11.1) will consider the\"}, {\"0\": \"confusion  as  to  which  dimension  is  to  be  used  to\", \"1\": \"design of beams which are fully laterally restrained.\"}, {\"0\": \"classify sections as EC 3 uses the same symbols to\", \"1\": \"Section 9.11.2 will then look at Eurocode 3: Part 1’s\"}, {\"0\": \"define  different  section  dimensions.  Comparison\", \"1\": \"rules  for  designing  beams  which  are  not  laterally\"}, {\"0\": \"with  the  corresponding  limits  given  in  BS  5950\", \"1\": \"torsionally restrained.\"}, {\"0\": \"for rolled sections (Table 4.4) shows that the c*/t w(=  d/t)  ratios  are  slightly  more  onerous  for  webs,\", \"1\": \"\"}, {\"0\": \"and  c/t f  (=  (B  −  t w  −  2r)/2T )  ratios  are  perhaps\", \"1\": \"9.11.1FULLY LATERALLY RESTRAINED\"}, {\"0\": \"slightly less onerous for outstand flanges. It should\", \"1\": \"BEAMS\"}, {\"0\": \"be noted that in EC 3 the factor ε is given by\", \"1\": \"Generally, such members should be checked for\"}, {\"0\": \"(9.5)ε = (235/fy)0.5\", \"1\": \"(1)resistance of cross-section to bending ULS\"}, {\"0\": \"\", \"1\": \"(2)resistance to shear buckling ULS\"}, {\"0\": \"not (275/py)0.5 as in BS 5950.\", \"1\": \"(3)resistance to flange-induced buckling ULS\"}, {\"0\": \"For  Class  4 sections  effective  cross-sectional\", \"1\": \"(4)resistance of the web to transverse forces ULS\"}, {\"0\": \"properties  can  be  calculated  using  effective  widths\", \"1\": \"(5) deflection SLS.\"}, {\"0\": \"380\", \"1\": \"\"}], [], []]\n\n[FORMULAS]\nE = 210; ν = 0", "page_type": "mixed", "image": "Design of Structural Elements_ Concr_a6902759_p0401.png"}
{"pdf_file": "Design of Structural Elements_ Concrete, Steelwork, Masonry and Timber Designs to British Standards and Eurocodes, 3rd Edition   ( PDFDrive ).pdf", "page_num": 402, "prompt": "Analyze this page with diagrams, tables, formulas, calculations, and explanations.", "response": "[NATIVE TEXT]\n381\nDesign of beams\n9.11.1.1 Resistance of cross-sections-bending\nmoment (Cl. 6.2.5, EC 3)\nWhen shear force is absent or of a low value, the\ndesign value of the bending moment, MEd, at each\nsection should satisfy the following:\n    \nM\nM\nEd\nc,Rd\n  .\n≤1 0\n(9.5)\nwhere Mc,Rd is the design resistance for bending\nabout one principal axis taken as follows:\n(a) the design plastic resistance moment of the\ngross section\nMpl,Rd = Wp1fy/γM0\n(9.6)\nwhere Wpl is the plastic section modulus, for\nclass 1 and 2 sections only;\n(b) the design elastic resistance moment of the gross\nsection\nMel,Rd = Wel,min fy/γM0\n(9.7)\nwhere Wel,min is the minimum elastic section\nmodulus for class 3 sections;\n(c) the design local buckling resistance moment\nof the gross section\nMc,Rd = Weff,min fy/γM0\n(9.8)\nwhere Weff,min is the minimum effective section\nmodulus, for class 4 cross-sections only;\n(d) the design ultimate resistance moment of the net\nsection at bolt holes Mu,Rd, if this is less than\nthe appropriate values above. In calculating this\nvalue, fastener holes in the compression zone\ndo not need to be considered unless they are\noversize or slotted and provided that they are\nfilled by fasteners. In the tension zone holes do\nnot need to be considered provided that\n    \nA\nf\nA f\nf,net\nu\nM2\nf\ny\nM0\n0 9\n.\n  \nγ\nγ\n≥\n(9.9)\nConsideration of fastener holes in bending is a\ndifference from BS 5950.\n9.11.1.2 Resistance of cross-sections – shear\n(Cl. 6.2.6, EC 3)\nThe design value of the shear force VEd at each\ncross-section should satisfy the following\n    \nV\nV\nEd\nc,Rd\n  .\n≤1 0\n(9.10)\nwhere Vc,Rd is the design shear resistance. For\nplastic design Vc,Rd is taken as the design plastic\nshear resistance, Vpl,Rd, given by\nVpl,Rd = Av( fy/  3)/γM0\n(9.11)\nwhere Av is the shear area, which for rolled I and\nH sections, loaded parallel to the web is\nAv = A − 2btf + (tw + 2r)tf ≥ ηhwtf\n(9.12)\nwhere\nA\ncross-sectional area\nb\noverall breadth\nr\nroot radius\ntf\nflange thickness\ntw\nweb thickness\nhw\ndepth of the web\nη\nconservatively taken as 1.0\nEquation 9.12 will generally give a slightly higher\nestimate of the shear area than in BS 5950.\nFastener holes in the web do not have to be\nconsidered in the shear verification.\nShear buckling resistance for unstiffened webs\nmust additionally be considered when\n    \nh\nt\nw\nw\n  \n> 72 ε\nη\n(9.13)\nWhen the changed definition of ε is taken into\naccount, this value is more onerous than the corres-\nponding value in BS 5950.\nFor a stiffened web (clause 5.1, EC 3–5), shear\nbuckling resistance will need to be considered when\n    \nh\nt\nk\nw\nw\n  \n> 31\nη ε\nτ\n(9.14)\nwhere kτ is the buckling factor for shear and is\ngiven by\nfor a/hw < 1 (Fig. 9.3)\nkτ = 4 + 5.34(hw/a)2\n(9.15)\nfor a/hw ≥ 1\nkτ = 5.34 + 4(hw/a)2\n(9.16)\n9.11.1.3 Resistance of cross-sections-bending\nand shear (Cl. 6.2.8, EC 3)\nThe plastic resistance moment of the section is\nreduced by the presence of shear. When the design\nvalue of the shear force, VEd, exceeds 50 per cent of\nthe design plastic shear resistance, Vpl,Rd, the design\nresistance moment of the section, Mv,Rd, should be\ncalculated using a reduced yield strength taken as\n(1 − ρ)fy for the shear area only\n(9.17)\nwhere\n    \nρ  \n  \n=\n−\n\n\n\n\n2\n1\n2\nV\nV\nEd\npl,Rd\n(9.18)\n\n[TABLES]\n[[{\"0\": \"\", \"1\": \"Design of beams\"}, {\"0\": \"9.11.1.1 Resistance of cross-sections-bending\", \"1\": \"(9.11)M0Vpl,Rd = Av( fy /   3 )/γ\"}, {\"0\": \"moment (Cl. 6.2.5, EC 3)\", \"1\": \"where  Av  is  the  shear  area,  which  for  rolled  I  and\"}, {\"0\": \"When  shear  force  is  absent  or  of  a  low  value,  the\", \"1\": \"H sections, loaded parallel to the web is\"}, {\"0\": \"design value of the bending moment, MEd, at each\", \"1\": \"\"}, {\"0\": \"section should satisfy the following:\", \"1\": \"(9.12)A v = A − 2bt f + (t w + 2r)t f ≥ ηh wt f\"}, {\"0\": \"M MEd(9.5) ≤ 1 0\", \"1\": \"where\"}, {\"0\": \"\", \"1\": \"Across-sectional area\"}, {\"0\": \"c,Rd\", \"1\": \"boverall breadth\"}, {\"0\": \"where  Mc,Rd  is  the  design  resistance  for  bending\", \"1\": \"rroot radius\"}, {\"0\": \"about one principal axis taken as follows:\", \"1\": \"flange thicknesstf\"}, {\"0\": \"(a)the  design  plastic  resistance  moment  of  the\", \"1\": \"web thicknesstw\"}, {\"0\": \"gross section\", \"1\": \"depth of the webhwη\"}, {\"0\": \"\", \"1\": \"conservatively taken as 1.0\"}, {\"0\": \"(9.6)Mpl,Rd = Wp1fy /γM0\", \"1\": \"Equation 9.12 will generally give a slightly higher\"}, {\"0\": \"where  Wpl  is  the  plastic  section  modulus,  for\", \"1\": \"estimate of the shear area than in BS 5950.\"}, {\"0\": \"class 1 and 2 sections only;\", \"1\": \"Fastener  holes  in  the  web  do  not  have  to  be\"}, {\"0\": \"(b)the design elastic resistance moment of the gross\", \"1\": \"considered in the shear verification.\"}, {\"0\": \"section\", \"1\": \"Shear  buckling  resistance  for  unstiffened  webs\"}, {\"0\": \"(9.7)Mel,Rd = Wel,min fy/γM0\", \"1\": \"must additionally be considered when\"}, {\"0\": \"where  Wel,min  is  the  minimum  elastic  section\", \"1\": \"h tε ηw(9.13)  > 72\"}, {\"0\": \"modulus for class 3 sections;\", \"1\": \"w\"}, {\"0\": \"(c)the  design  local  buckling  resistance  moment\", \"1\": \"When  the  changed  definition  of  ε  is  taken  into\"}, {\"0\": \"of the gross section\", \"1\": \"account, this value is more onerous than the corres-\"}, {\"0\": \"(9.8)Mc,Rd = Weff,min fy /γM0\", \"1\": \"ponding value in BS 5950.\"}, {\"0\": \"\", \"1\": \"For a stiffened web (clause 5.1, EC 3–5), shear\"}, {\"0\": \"where Weff,min is the minimum effective section\", \"1\": \"buckling resistance will need to be considered when\"}, {\"0\": \"modulus, for class 4 cross-sections only;\", \"1\": \"\"}, {\"0\": \"(d)the design ultimate resistance moment of the net\", \"1\": \"h twεk  > 31(9.14)τη\"}, {\"0\": \"section  at  bolt  holes  Mu,Rd,  if  this  is  less  thanthe appropriate values above. In calculating this\", \"1\": \"w\"}, {\"0\": \"value,  fastener  holes  in  the  compression  zone\", \"1\": \"where  kτ  is  the  buckling  factor  for  shear  and  is\"}, {\"0\": \"\", \"1\": \"given by\"}, {\"0\": \"do  not  need  to  be  considered  unless  they  areoversize  or  slotted  and  provided  that  they  are\", \"1\": \"for a/hw < 1 (Fig. 9.3)\"}, {\"0\": \"filled by fasteners. In the tension zone holes do\", \"1\": \"(9.15)kτ = 4 + 5.34(h w/a)2\"}, {\"0\": \"not need to be considered provided that\", \"1\": \"for a/hw ≥ 1\"}, {\"0\": \"A fA0 9.ffyf, netu≥\", \"1\": \"(9.16)kτ = 5.34 + 4(h w/a)2\"}, {\"0\": \"(9.9)  γγM2M0\", \"1\": \"\"}, {\"0\": \"Consideration  of  fastener  holes  in  bending  is  a\", \"1\": \"9.11.1.3 Resistance of cross-sections-bending\"}, {\"0\": \"difference from BS 5950.\", \"1\": \"and shear (Cl. 6.2.8, EC 3)\"}, {\"0\": \"\", \"1\": \"The  plastic  resistance  moment  of  the  section  is\"}, {\"0\": \"9.11.1.2 Resistance of cross-sections – shear\", \"1\": \"reduced by the presence of shear. When the design\"}, {\"0\": \"(Cl. 6.2.6, EC 3)\", \"1\": \"value of the shear force, VEd, exceeds 50 per cent of\"}, {\"0\": \"The  design  value  of  the  shear  force  VEd  at  each\", \"1\": \"the design plastic shear resistance, Vpl,Rd, the design\"}, {\"0\": \"cross-section should satisfy the following\", \"1\": \"resistance moment of the section, M v,Rd, should becalculated using a reduced yield strength taken as\"}, {\"0\": \"V VEd(9.10) ≤ 1 0\", \"1\": \"(9.17)(1 − ρ)fy for the shear area only\"}, {\"0\": \"c,Rd\", \"1\": \"where\"}, {\"0\": \"is the  design  shear  resistance.  Forwhere  Vc,Rd\", \"1\": \"2V2\"}], [{\"0\": \"\", \"1\": \"Design of beams\"}, {\"0\": \"9.11.1.1 Resistance of cross-sections-bending\", \"1\": \"(9.11)M0Vpl,Rd = Av( fy /   3 )/γ\"}, {\"0\": \"moment (Cl. 6.2.5, EC 3)\", \"1\": \"where  Av  is  the  shear  area,  which  for  rolled  I  and\"}, {\"0\": \"When  shear  force  is  absent  or  of  a  low  value,  the\", \"1\": \"H sections, loaded parallel to the web is\"}, {\"0\": \"design value of the bending moment, MEd, at each\", \"1\": \"\"}, {\"0\": \"section should satisfy the following:\", \"1\": \"(9.12)A v = A − 2bt f + (t w + 2r)t f ≥ ηh wt f\"}, {\"0\": \"M MEd(9.5) ≤ 1 0\", \"1\": \"where\"}, {\"0\": \"\", \"1\": \"Across-sectional area\"}, {\"0\": \"c,Rd\", \"1\": \"boverall breadth\"}, {\"0\": \"where  Mc,Rd  is  the  design  resistance  for  bending\", \"1\": \"rroot radius\"}, {\"0\": \"about one principal axis taken as follows:\", \"1\": \"flange thicknesstf\"}, {\"0\": \"(a)the  design  plastic  resistance  moment  of  the\", \"1\": \"web thicknesstw\"}, {\"0\": \"gross section\", \"1\": \"depth of the webhwη\"}, {\"0\": \"\", \"1\": \"conservatively taken as 1.0\"}, {\"0\": \"(9.6)Mpl,Rd = Wp1fy /γM0\", \"1\": \"Equation 9.12 will generally give a slightly higher\"}, {\"0\": \"where  Wpl  is  the  plastic  section  modulus,  for\", \"1\": \"estimate of the shear area than in BS 5950.\"}, {\"0\": \"class 1 and 2 sections only;\", \"1\": \"Fastener  holes  in  the  web  do  not  have  to  be\"}, {\"0\": \"(b)the design elastic resistance moment of the gross\", \"1\": \"considered in the shear verification.\"}, {\"0\": \"section\", \"1\": \"Shear  buckling  resistance  for  unstiffened  webs\"}, {\"0\": \"(9.7)Mel,Rd = Wel,min fy/γM0\", \"1\": \"must additionally be considered when\"}, {\"0\": \"where  Wel,min  is  the  minimum  elastic  section\", \"1\": \"h tε ηw(9.13)  > 72\"}, {\"0\": \"modulus for class 3 sections;\", \"1\": \"w\"}, {\"0\": \"(c)the  design  local  buckling  resistance  moment\", \"1\": \"When  the  changed  definition  of  ε  is  taken  into\"}, {\"0\": \"of the gross section\", \"1\": \"account, this value is more onerous than the corres-\"}, {\"0\": \"(9.8)Mc,Rd = Weff,min fy /γM0\", \"1\": \"ponding value in BS 5950.\"}, {\"0\": \"\", \"1\": \"For a stiffened web (clause 5.1, EC 3–5), shear\"}, {\"0\": \"where Weff,min is the minimum effective section\", \"1\": \"buckling resistance will need to be considered when\"}, {\"0\": \"modulus, for class 4 cross-sections only;\", \"1\": \"\"}, {\"0\": \"(d)the design ultimate resistance moment of the net\", \"1\": \"h twεk  > 31(9.14)τη\"}, {\"0\": \"section  at  bolt  holes  Mu,Rd,  if  this  is  less  thanthe appropriate values above. In calculating this\", \"1\": \"w\"}, {\"0\": \"value,  fastener  holes  in  the  compression  zone\", \"1\": \"where  kτ  is  the  buckling  factor  for  shear  and  is\"}, {\"0\": \"\", \"1\": \"given by\"}, {\"0\": \"do  not  need  to  be  considered  unless  they  areoversize  or  slotted  and  provided  that  they  are\", \"1\": \"for a/hw < 1 (Fig. 9.3)\"}, {\"0\": \"filled by fasteners. In the tension zone holes do\", \"1\": \"(9.15)kτ = 4 + 5.34(h w/a)2\"}, {\"0\": \"not need to be considered provided that\", \"1\": \"for a/hw ≥ 1\"}, {\"0\": \"A fA0 9.ffyf, netu≥\", \"1\": \"(9.16)kτ = 5.34 + 4(h w/a)2\"}, {\"0\": \"(9.9)  γγM2M0\", \"1\": \"\"}, {\"0\": \"Consideration  of  fastener  holes  in  bending  is  a\", \"1\": \"9.11.1.3 Resistance of cross-sections-bending\"}, {\"0\": \"difference from BS 5950.\", \"1\": \"and shear (Cl. 6.2.8, EC 3)\"}, {\"0\": \"\", \"1\": \"The  plastic  resistance  moment  of  the  section  is\"}, {\"0\": \"9.11.1.2 Resistance of cross-sections – shear\", \"1\": \"reduced by the presence of shear. When the design\"}, {\"0\": \"(Cl. 6.2.6, EC 3)\", \"1\": \"value of the shear force, VEd, exceeds 50 per cent of\"}, {\"0\": \"The  design  value  of  the  shear  force  VEd  at  each\", \"1\": \"the design plastic shear resistance, Vpl,Rd, the design\"}, {\"0\": \"cross-section should satisfy the following\", \"1\": \"resistance moment of the section, M v,Rd, should becalculated using a reduced yield strength taken as\"}, {\"0\": \"V VEd(9.10) ≤ 1 0\", \"1\": \"(9.17)(1 − ρ)fy for the shear area only\"}, {\"0\": \"c,Rd\", \"1\": \"where\"}, {\"0\": \"is the  design  shear  resistance.  Forwhere  Vc,Rd\", \"1\": \"2V2\"}, {\"0\": \"plastic  design  Vc,Rd  is  taken  as  the  design  plastic\", \"1\": \"  Ed=−(9.18)ρ    1V\"}, {\"0\": \"shear resistance, Vpl,Rd, given by\", \"1\": \"pl,Rd\"}], [], []]\n\n[FORMULAS]\nRd = Wp1fy/γM0; Rd = Wel; Rd = Weff; Rd = Av( fy/  3)/γM0; kτ = 4 + 5", "page_type": "mixed", "image": "Design of Structural Elements_ Concr_a6902759_p0402.png"}
{"pdf_file": "Design of Structural Elements_ Concrete, Steelwork, Masonry and Timber Designs to British Standards and Eurocodes, 3rd Edition   ( PDFDrive ).pdf", "page_num": 403, "prompt": "Analyze this page with diagrams, tables, formulas, calculations, and explanations.", "response": "[NATIVE TEXT]\nEurocode 3: Design of steel structures\n382\nFS\nFS\nFS\nV1,S\nV2,S\nVS\nhw\nsS\nsS\nsS\nc\na\nType (a)\nType (b)\nType (c)\nFig. 9.3 Different types of forces applied to a web and associated buckling coefficients (Fig. 6.1, EC 3–5).\nk\nh\na\nF\nw\n    \n=\n+\n\n\n\n\n6\n2\n2\nk\nh\na\nF\nw\n  .   \n=\n+\n\n\n\n\n3 5\n2\n2\nk\ns\nc\nh\nF\ns\nw\n    \n  \n  \n=\n+\n+\n\n\n\n≤\n2\n6\n6\nThus, for rolled I and H sections the reduced\ndesign resistance moment for the section about the\nmajor axis, My,v,Rd, will be given by\nMy,v,Rd = fy(Wpl,y − ρAv\n2/4tw)/γM0 ≤ My,c,Rd\n(9.19)\n9.11.1.4 Shear buckling resistance\n(Cl. 6.2.6, EC 3)\nAs noted above the shear buckling resistance of\nunstiffened beam webs has to be checked when\n    \nh\nt\nw\nw\n  \n> 72 ε\nη\n(9.20)\nClause 2.4 of the National Annex to EC 3–5\nrecommends a value of 1.0 for η for all steel grades\nup to and including S460. For standard rolled\nbeams and columns this check is rarely necessary,\nhowever, as hw/tw is almost always less than 72ε and\nhas therefore not been discussed in this section.\n9.11.1.5 Flange-induced buckling (Cl. 8, EC 3–5)\nTo prevent the possibility of the compression flange\nbuckling in the plane of the web, EC 3–5 requires\nthat the ratio hw/tw of the web should satisfy the\nfollowing criterion:\n    \nh\nt\nk E\nf\nA\nA\nw\nw\nyf\nw\nfc\n  \n≤\n(9.21)\nwhere\nAw\nis the area of the web = (h − 2tf)tw\nAfc\nis the area of the compression flange = btf\nfyf\nis the yield strength of the compression flange\nThe factor k assumes the following values:\nPlastic rotation utilised, i.e. class 1 flanges: 0.3\nPlastic moment resistance utilised, i.e. class 2\nflanges: 0.4\nElastic moment resistance utilised, i.e. class 3 or\nclass 4 flanges: 0.55\n9.11.1.6 Resistance of the web to transverse\nforces (Cl. 6, EC 3–5)\nEC 3–5 distinguishes between two types of forces\napplied through a flange to the web:\n(a) forces resisted by shear in the web (Fig. 9.3,\nloading types (a) and (c)).\n(b) forces transferred through the web directly to\nthe other flange (Fig. 9.3, loading type (b)).\nFor loading types (a) and (c) the web is likely to\nfail as a result of\n(i) crushing of the web close to the flange accom-\npanied by yielding of the flange, the combined\neffect sometimes referred to as web crushing\n(ii) localised buckling and crushing of the web\nbeneath the flange, the combined effect some-\ntimes referred to as web crippling (Fig. 9.4a).\nFor loading type (b) the web is likely to fail as a\nresult of\n(i) web crushing\n(ii) buckling of the web over most of the depth of\nthe member (Fig. 9.4b).\nProvided that the compression flange is adequately\nrestrained in the lateral direction, the design resist-\nance of webs of rolled beams under transverse forces\ncan be determined in accordance with the recom-\nmendations contained in clause 6 of EC 3–5.\nHere it is stated that the design resistance of\nwebs to local buckling is given by\n    \nF\nf\nL\nt\nRd\nyw\neff\nw\nM1\n  \n=\nγ\n(9.22)\nwhere\nfyw\nis the yield strength of the web\ntw\nis the thickness of the web\nγM1\nis the partial safety factor = 1.0\nLeff\nis the effective length of web which resists\ntransverse forces = χFly\n(9.23)\n\n[TABLES]\n[[{\"0\": \"\", \"1\": \"\", \"2\": \"Type (b)\", \"3\": \"\", \"4\": \"\"}], [{\"0\": \"\", \"1\": \"\", \"2\": \"Type (a)\", \"3\": \"\", \"4\": \"\"}], [{\"0\": \"\", \"1\": \"\"}, {\"0\": \"\", \"1\": \"Type (c)\"}, {\"0\": \"\", \"1\": \"\"}, {\"0\": \"\", \"1\": \"FS\"}], [], [], [{\"\": \"\", \"None\": \"\"}, {\"\": \"\", \"None\": \"\"}], [{\"\": \"\", \"None\": \"\"}, {\"\": \"\", \"None\": \"\"}], [{\"\": \"c s\\nS\"}, {\"\": \"\"}]]\n\n[FORMULAS]\nange = btf", "page_type": "mixed", "image": "Design of Structural Elements_ Concr_a6902759_p0403.png"}
{"pdf_file": "Design of Structural Elements_ Concrete, Steelwork, Masonry and Timber Designs to British Standards and Eurocodes, 3rd Edition   ( PDFDrive ).pdf", "page_num": 404, "prompt": "Analyze this page with diagrams, tables, formulas, calculations, and explanations.", "response": "[NATIVE TEXT]\n383\nFS\nsS\n45°\nDesign of beams\nLoading types (a) and (c) (Fig 9.3)\nLoading type (b) (Fig 9.3)\nFig. 9.4 Web failure.\nin which\nχF\nis the reduction factor due to local buckling\ncalculated as discussed below\nly\nis the effective loaded length, appropriate to\nthe length of the stiff bearing ss. According\nto clause 6.3 of EC 3–5, ss should be taken\nas the distance over which the applied load\nis effectively distributed at a slope of 1.1 as\nshown in Fig. 9.5, but ss ≤ h w.\nReduction factor, χF\nAccording to clause 6.4 the reduction factor χF is\ngiven by\n    \nχF\nF\n  .   \n=\n≤\n0 5\n1\nd\n(9.24)\nwhere\n      \nd\nB\nF\ny w\nyw\ncr\n  \n=\nt f\nF\n(9.25)\nin which\n    \nF\nk E t\nh\ncr\nF\nw\n3\nw\n  .\n= 0 9\n(9.26)\nFor webs without longitudinal stiffeners kF\nis obtained from Fig. 9.3 and ly is obtained as\nfollows.\nEffective loaded length, ly\nAccording to clause 6.5 for loading types (a) and\n(b) in Fig. 9.3, the effective loaded length, ly, is\ngiven by\nBy = ss + 2tf    1  \n  \n+\n+\n(\n)\nm\nm\n1\n2  ≤ a\n(9.27)\nwhere\n    \nm\nf b\nf\nt\n1\nyf\nf\nyw w\n  \n=\n(9.28)\nand if dF > 0.5\n    \nm\nh\nt\n2\nw\nf\n  .\n=\n\n\n\n\n0 02\n2\n(9.29)\nor if dF ≤ 0.5, m2 = 0\nFor loading type (c) ly is taken as the smallest\nvalue obtained from equations 9.30 and 9.31 as\nfollows\n        \nly\ne\nf\n1\ne\nf\n  \n \n  \n  \n=\n+\n+ \n\n\n\n+\nB\nB\nt\nm\nt\nm\n2\n2\n2\n(9.30)\n      B\nB\ny\ne\nf\n1\n  \n \n  \n=\n+\n+\nt\nm\nm2\n(9.31)\nFig. 9.5 Length of stiff bearing (based on Fig. 6.2,\nEC 3–5).\n\n[TABLES]\n[[{\"0\": \"Loading types (a) and (c) (Fig 9.3)\", \"1\": \"Loading type (b) (Fig 9.3)\"}, {\"0\": \"Fig. 9.4 Web failure.\", \"1\": \"\"}, {\"0\": \"45°\", \"1\": \"in which\"}, {\"0\": \"FS\", \"1\": \"\"}, {\"0\": \"\", \"1\": \"t hw3(9.26)F = 0 9k EcrF\"}, {\"0\": \"\", \"1\": \"w\"}, {\"0\": \"\", \"1\": \"For  webs  without longitudinal stiffeners kF\"}, {\"0\": \"\", \"1\": \"is  obtained  from  Fig.  9.3  and is  obtained  asly\"}, {\"0\": \"sS\", \"1\": \"follows.\"}, {\"0\": \"Fig. 9.5 Length of stiff bearing (based on Fig. 6.2,\", \"1\": \"Effective loaded length, lyAccording  to  clause  6.5  for  loading  types  (a)  and\"}, {\"0\": \"EC 3–5).\", \"1\": \"(b)  in  Fig.  9.3,  the  effective  loaded  length,  ly,  is\"}, {\"0\": \"\", \"1\": \"given by\"}, {\"0\": \"in which\", \"1\": \"++ ≤ a(9.27)1  m  m()12By = ss + 2t f\"}, {\"0\": \"χis the reduction factor due to local bucklingF\", \"1\": \"where\"}, {\"0\": \"calculated as discussed below\", \"1\": \"\"}, {\"0\": \"is the effective loaded length, appropriate toly\", \"1\": \"fbyff(9.28)m  =\"}, {\"0\": \"the length of the stiff bearing ss. According\", \"1\": \"1ftyw w\"}, {\"0\": \"to clause 6.3 of EC 3–5, ss should be takenas the distance over which the applied load\", \"1\": \"and if d F > 0.5\"}, {\"0\": \"is effectively distributed at a slope of 1.1 as\", \"1\": \"2\"}, {\"0\": \"shown in Fig. 9.5, but ss ≤ h w.\", \"1\": \" h t w=(9.29)m  0 022\"}, {\"0\": \"\", \"1\": \"f\"}, {\"0\": \"Reduction factor, χF\", \"1\": \"or if d F ≤ 0.5, m 2 = 0\"}, {\"0\": \"According  to  clause  6.4  the  reduction  factor  χF  is\", \"1\": \"For  loading  type  (c)  l y  is  taken  as  the  smallest\"}, {\"0\": \"given by\", \"1\": \"value  obtained  from  equations  9.30  and  9.31  as\"}, {\"0\": \"0 5(9.24)χ=≤    1F\", \"1\": \"follows\"}, {\"0\": \"dF\", \"1\": \"2Bm\"}, {\"0\": \"where\", \"1\": \"  1e(9.30)l=+++B   t    m2yef2t\"}, {\"0\": \"Btf\", \"1\": \"f\"}, {\"0\": \"y wywd(9.25)  =FF\", \"1\": \"=++B   tm  (9.31)m2\"}, {\"0\": \"cr\", \"1\": \"yef1      B\"}], [], []]\n\n[FORMULAS]\nm2 = 0", "page_type": "mixed", "image": "Design of Structural Elements_ Concr_a6902759_p0404.png"}
{"pdf_file": "Design of Structural Elements_ Concrete, Steelwork, Masonry and Timber Designs to British Standards and Eurocodes, 3rd Edition   ( PDFDrive ).pdf", "page_num": 405, "prompt": "Analyze this page with diagrams, tables, formulas, calculations, and explanations.", "response": "[NATIVE TEXT]\nEurocode 3: Design of steel structures\n384\nwhere\n      \nBe\nF\nw\n2\nyw\nw\ns\n=\n≤\n+\n \n    \nk Et\nf\nh\ns\nc\n2\n(9.32)\nInteraction between shear force and bending\nmoment\nAccording to clause 7 of EC 3–5, where the web is\nalso subject to bending the combined effect should\nsatisfy the following\nη2 + 0.8η1 ≤ 1.4\n(9.33)\nwhere\n    \nη\nγ\n1\n1 0\n  \n  .\n=\n≤\nM\nf W\nEd\ny\npl\nM0\n(9.34)\n    \nη\nγ\n2\n1 0\n  \n  .\n=\n≤\nF\nf\nL t\nEd\nyw\neff w\nM1\n(9.35)\n9.11.1.7 Deflections (Cl. 7, EC 3)\nClause 7 of EC 3 highlights the need to check the\nSLS of deflection but provides few details on the\nsubject other than to refer the designer to Annex\nA1.4 of EN 1990. Clause 7.2.1 of EC 3 makes\nclear, however, that the designer is responsible for\nspecifying appropriate limits for vertical deflections,\nwhich should be agreed with the client.\nSeveral vertical deflections are defined in EN\n1990. However, like BS 5950, the National Annex\nto EC 3 recommends that checks on the vertical\ndeflections, δ, under unfactored imposed loading\nshould be carried out and suggests that in the\nabsence of other limits the limits shown in Table\n9.6 may be used.\nDESIGN BENDING MOMENT\nDesign action (FEd) = (γGgk + γQqk) × span\n= (1.35 × 8 + 1.5 × 6)8 = 158.4 kN\nDesign bending moment (MEd) = \n    \nF l\nEd\nkNm\n8\n158 4\n8\n8\n158 4\n  \n.     \n.  \n=\n×\n=\nSTRENGTH CLASSIFICATION\nFlange thickness, tf = 11.5 mm, steel grade S275. Hence from Table 9.4, fy = 275 N mm−2\nExample 9.1 Analysis of a laterally restrained beam (EC 3)\nCheck the suitability of 356 × 171 × 51 kg m−1 UB section in S275 steel loaded by uniformly distributed loading\ngk = 8 kN m−1 and qk = 6 kN m−1 as shown below. Assume that the beam is fully laterally restrained and that the\nbeam sits on 100 mm bearings at each end. Ignore self weight of beam.\nTable 9.6 Recommended vertical deflection\nlimits (Clause 2.24 of National Annex to EC 3)\nConditions\nDeflection limits\nCantilevers\nLength/180\nBeams carrying plaster\nSpan/360\nor other brittle finish\nOther beams\nSpan/200\n(except purlins\nand sheeting rails)\nPurlins and sheeting rails\nTo suit the characteristics\nof particular cladding\n\n[TABLES]\n[[{\"0\": \"Eurocode 3: Design of steel structures\"}, {\"0\": \"whereA1.4  of  EN  1990.  Clause  7.2.1  of  EC  3  makes\"}, {\"0\": \"w2clear, however, that the designer is responsible fork Et\"}, {\"0\": \"F=≤+specifying appropriate limits for vertical deflections,(9.32)sc     Bes2fh\"}, {\"0\": \"ywwwhich should be agreed with the client.\"}, {\"0\": \"Several  vertical  deflections  are  defined  in  EN\"}, {\"0\": \"1990. However, like BS 5950, the National AnnexInteraction between shear force and bending\"}, {\"0\": \"to  EC  3  recommends  that  checks  on  the  verticalmoment\"}, {\"0\": \"deflections,  δ,  under  unfactored  imposed  loadingAccording to clause 7 of EC 3–5, where the web is\"}, {\"0\": \"should  be  carried  out  and  suggests  that in  thealso subject to bending the combined effect should\"}, {\"0\": \"absence  of  other  limits  the  limits  shown  in  Tablesatisfy the following\"}, {\"0\": \"9.6 may be used.η(9.33)2 + 0.8η1 ≤ 1.4\"}, {\"0\": \"whereTable 9.6 Recommended vertical deflection\"}, {\"0\": \"MEdlimits (Clause 2.24 of National Annex to EC 3)η=≤(9.34)    1 01\"}, {\"0\": \"f WyplConditionsDeflection limits\"}, {\"0\": \"γM0\"}, {\"0\": \"CantileversLength/180FEdη=≤(9.35)    1 0\"}, {\"0\": \"2Beams carrying plasterSpan/360fLtyweff w\"}, {\"0\": \"or other brittle finishγ\"}, {\"0\": \"M1Other beamsSpan/200\"}, {\"0\": \"(except purlins9.11.1.7 Deflections (Cl. 7, EC 3)\"}, {\"0\": \"and sheeting rails)Clause 7 of EC 3 highlights the need to check the\"}, {\"0\": \"Purlins and sheeting railsTo suit the characteristicsSLS  of  deflection  but  provides  few  details  on  the\"}, {\"0\": \"of particular claddingsubject  other  than  to  refer  the  designer  to  Annex\"}, {\"0\": \"Example 9.1 Analysis of a laterally restrained beam (EC 3)\"}, {\"0\": \"Check  the  suitability  of  356  ×  171  ×  51  kg  m−1  UB  section  in  S275  steel  loaded  by  uniformly  distributed  loading\"}, {\"0\": \"gk  =  8  kN  m−1  and  qk  =  6  kN  m−1  as  shown  below.  Assume  that  the  beam  is  fully  laterally  restrained  and  that  thebeam sits on 100 mm bearings at each end. Ignore self weight of beam.\"}, {\"0\": \"DESIGN BENDING MOMENT\"}, {\"0\": \"Design action (FEd) = (γGgk + γQqk) × span\"}, {\"0\": \"= (1.35 × 8 + 1.5 × 6)8 = 158.4 kN\"}, {\"0\": \"×F158 48  lEd==    158 4 kNmDesign bending moment (MEd) =\"}, {\"0\": \"88\"}, {\"0\": \"STRENGTH CLASSIFICATION\"}, {\"0\": \"Flange thickness, tf = 11.5 mm, steel grade S275. Hence from Table 9.4, fy = 275 N mm−2\"}, {\"0\": \"384\"}], [], []]\n\n[FORMULAS]\n8 = 158; tf = 11", "page_type": "mixed", "image": "Design of Structural Elements_ Concr_a6902759_p0405.png"}
{"pdf_file": "Design of Structural Elements_ Concrete, Steelwork, Masonry and Timber Designs to British Standards and Eurocodes, 3rd Edition   ( PDFDrive ).pdf", "page_num": 406, "prompt": "Analyze this page with diagrams, tables, formulas, calculations, and explanations.", "response": "[NATIVE TEXT]\n385\nDesign of beams\nSECTION CLASSIFICATION\nε = (235/fy)0.5 = (235/275)0.5 = 0.92\n    \nc\ntf\n  \n.\n.   .\n  \n    .\n  .\n=\n=\n<\n=\n×\n=\n71 9\n11 5\n6 25\n9\n9\n0 92\n8 28\nε\n(where c = (b − tw − 2r)/2 = (171.5 − 7.3 − 2 × 10.2)/2 = 143.8/2 = 71.9 mm)\n      \nc\nt\n*\nw\n  \n.\n.\n  \n.   \n  \n  .\n  \n.\n=\n=\n<\n=\n×\n=\n312 3\n7 3\n42 8\n72\n72\n0 92\n66 2\nε\nHence from Table 9.5, 356 × 171 × 51 UB belongs to Class 1.\nRESISTANCE OF CROSS-SECTION\nBending moment\nSince beam section belongs to class 1, plastic moment of resistance, Mpl,Rd, is given by\n    \nM\nW f\npl,Rd\npl,y y\nM0\n  \n  \n  \n  \n.\n=\n=\n×\n×\nγ\n895\n10\n275\n1 00\n3\n= 246.1 × 106 Nmm\n= 246.1 kNm > MEd (= 158.4 kNm)\nOK\nShear\nDesign shear force, VEd, is\n    \nV\nF\nEd\nEd\nkN\n  \n  \n.   \n.  \n=\n=\n=\n2\n158 4\n2\n79 2\nFor class 1 section design plastic shear resistance, Vpl,Rd, is given by\nVpl,Rd = A v(fy/  3 )/γM0\nwhere Av = A − 2btf + (tw + 2r)tf ≥ ηhwtw = 1.0 × (355.6 − 2 × 11.5) × 7.3 = 2428 mm2\n= 64.6 × 102 − 2 × 171.5 × 11.5 + (7.3 + 2 × 10.2)11.5\n= 2834 mm2\nOK\nHence, Vpl,Rd = 2834(275/  3 )/1.00\n= 540371 N = 540.4 kN > VEd (= 79.2 kN)\nOK\nBending and shear\nAccording to EC 3 the theoretical plastic resistance moment of the section, i.e. Mpl,Rd, is reduced if\nVEd > 0.5Vpl,Rd\nBut in this case\nVEd = 0\n(at mid-span)\nHence, no check is required.\nSHEAR BUCKLING RESISTANCE\nAs \n    \nh\nt\nw\nw\n  \n.\n.\n  \n.   \n  \n  .\n.\n  \n.\n=\n=\n<\n=\n×\n=\n332 6\n7 3\n45 6\n72\n72\n0 92\n1 0\n66 24\nε\nη\nExample 9.1 continued\n\n[TABLES]\n[[{\"0\": \"Example 9.1 continued\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"SECTION CLASSIFICATION\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"ε = (235/fy)0.5 = (235/275)0.5 = 0.92\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"c t. .71 9==<ε=×=    6 25  9  9  0 92\", \"1\": \"8 28\", \"2\": \"\"}, {\"0\": \"11 5f\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"(where c = (b − tw − 2r)/2 = (171.5 − 7.3 − 2 × 10.2)/2 = 143.8/2 = 71.9 mm)\", \"2\": \"\"}, {\"0\": \"c t* w312 3==<ε=×    42 8  72  72  0 92\", \"1\": \"=\", \"2\": \"66 2\"}, {\"0\": \"7 3\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"Hence from Table 9.5, 356 × 171 × 51 UB belongs to Class 1.\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"RESISTANCE OF CROSS-SECTION\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"Bending moment\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"Since beam section belongs to class 1, plastic moment of resistance, Mpl,Rd, is given by\", \"2\": \"\"}, {\"0\": \"3××W f895  10  275y==\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"M    pl,Rdγ1 00M0\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"= 246.1 × 106 Nmm\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"= 246.1 kNm > MEd (= 158.4 kNm)\", \"1\": \"\", \"2\": \"OK\"}, {\"0\": \"Shear\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"Design shear force, VEd, is\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"F158 4Ed===V      79 2 kNEd\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"22\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"For class 1 section design plastic shear resistance, Vpl,Rd, is given by\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"M0Vpl,Rd = A v(fy/   3 )/γ\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"where A v = A − 2btf + (tw + 2r )t f ≥ ηhwtw = 1.0 × (355.6 − 2 × 11.5) × 7.3 = 2428 mm2\", \"2\": \"\"}, {\"0\": \"= 64.6 × 102 − 2 × 171.5 × 11.5 + (7.3 + 2 × 10.2)11.5\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"= 2834 mm2OK\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"Hence, Vpl,Rd = 2834(275/   3 )/1.00\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"OK= 540371 N = 540.4 kN > VEd (= 79.2 kN)\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"Bending and shear\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"According to EC 3 the theoretical plastic resistance moment of the section, i.e. Mpl,Rd, is reduced if\", \"2\": \"\"}, {\"0\": \"VEd > 0.5Vpl,RdBut in this case\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"(at mid-span)VEd = 0\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"Hence, no check is required.\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"SHEAR BUCKLING RESISTANCE\", \"1\": \"\", \"2\": \"\"}], [], []]\n\n[FORMULAS]\n.5 = 0; .8/2 = 71", "page_type": "mixed", "image": "Design of Structural Elements_ Concr_a6902759_p0406.png"}
{"pdf_file": "Design of Structural Elements_ Concrete, Steelwork, Masonry and Timber Designs to British Standards and Eurocodes, 3rd Edition   ( PDFDrive ).pdf", "page_num": 407, "prompt": "Analyze this page with diagrams, tables, formulas, calculations, and explanations.", "response": "[NATIVE TEXT]\nEurocode 3: Design of steel structures\n386\nExample 9.1 continued\nwhere\nhw = h − 2tf = 355.6 − 2 × 11.5 = 332.6 mm\nNo check on shear buckling is required.\nFLANGE-INDUCED BUCKLING\nAw, area of web = (h − 2tf)tw = (355.6 − 2 × 11.5) × 7.3 = 2428 mm2\nAfc, area of compression flange = btf = 171.5 × 11.5 = 1972.2 mm2\n    \nh\nt\nk E\nf\nA\nA\nw\nw\nyf\nw\nfc\n  \n.   \n  .   \n  \n.   \n.\n=\n≤\n=\n×\n×\n=\n45 6\n0 3\n210\n10\n275\n2428\n1972 2\n254 2\n3\nOK\nHence, no check is required.\nWEB BUCKLING\nAt an unstiffened end support (i.e. load type (c) in Fig. 9.3)\n    \nk\ns\nc\nh\nF\ns\nw\n    \n  \n  \n=\n+\n+\n\n\n\n≤\n2\n6\n6\n  \n=\n+\n+\n\n\n\n=\n   \n  \n.\n  .\n2\n6 100\n0\n332 6\n3 80\n    \nF\nk E t\nh\ncr\nF\nw\nw\n  .\n  .   .\n  \n  \n  \n.\n.\n  \n N\n=\n=\n×\n×\n×\n×\n=\n0 9\n0 9\n3 80\n210\n10\n7 3\n332 6\n840024\n3\n3\n3\n    \nm\nf b\nf t\n1\nyf f\nyw w\n  \n  \n  \n.\n  .\n  \n.\n=\n=\n×\n×\n=\n275\n171 5\n275\n7 3\n23 5\nAssuming d > 0.5\n    \nm\nh\nt\n2\nw\nf\n  .\n  .\n  \n.\n.\n  \n.\n=\n\n\n\n\n=\n× \n\n\n\n=\n0 02\n0 02\n332 6\n11 5\n16 73\n2\n2\n      \nBe\nF\nw\n2\nyw w\ns\nmm\n  \n      \n    \n \n=\n≤\n+\n=\n+\n=\nk Et\nf h\ns\nc\n2\n100\n0\n100\n  \n=\n×\n×\n×\n×\n×\n=\n .\n  \n  \n  .\n  \n  \n.\n  \n.  \n3 80\n210\n10\n7 3\n2\n275\n332 6\n232 5\n3\n3\nmm\nHence le = 100 mm\n      \nB\nB\nB\ny\ne\nf\n1\ne\nf\n  \n  \n  \n  \n=\n+\n+ \n\n\n+\nt\nm\nt\nm\n2\n2\n2\n    \n=\n+\n+ \n\n\n\n+\n=\n \n  \n.\n.   \n.\n  \n.   \n100\n11 5 23 5\n2\n100\n11 5\n16 7\n2\n217.3 mm\n      B\nB\ny\ne\nf\n1\n  \n  \n  \n=\n+\n+\nt\nm\nm2\nBy = 100 +   11 5 23 5\n16 7\n.\n.   \n.\n+\n = 172.9 m\n\n[TABLES]\n[[{\"0\": \"\", \"1\": \"Eurocode 3: Design of steel structures\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\"}, {\"0\": \"\", \"1\": \"Example 9.1 continued\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\"}, {\"0\": \"where\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"hw = h − 2tf = 355.6 − 2 × 11.5 = 332.6 mm\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\"}, {\"0\": \"No check on shear buckling is required.\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\"}, {\"0\": \"\", \"1\": \"FLANGE-INDUCED BUCKLING\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\"}, {\"0\": \"\", \"1\": \"Aw, area of web = (h − 2tf)tw = (355.6 − 2 × 11.5) × 7.3 = 2428 mm2\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\"}, {\"0\": \"\", \"1\": \"Afc, area of compression flange = btf = 171.5 × 11.5 = 1972.2 mm2\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\"}, {\"0\": \"w\", \"1\": \"=\", \"2\": \"\", \"3\": \"≤\", \"4\": \"\", \"5\": \"E\", \"6\": \"\", \"7\": \"w\", \"8\": \"=\", \"9\": \"\", \"10\": \"×\", \"11\": \"×210\", \"12\": \"310\", \"13\": \"2428\", \"14\": \"=\", \"15\": \"\", \"16\": \"\"}, {\"0\": \"h tw\", \"1\": \"\", \"2\": \"45 6\", \"3\": \"\", \"4\": \"k\", \"5\": \"f\", \"6\": \"yf\", \"7\": \"A Afc\", \"8\": \"\", \"9\": \"0 3\", \"10\": \"\", \"11\": \"275\", \"12\": \"\", \"13\": \"1972 2\", \"14\": \"\", \"15\": \"254 2\", \"16\": \"OK\"}, {\"0\": \"Hence, no check is required.\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\"}, {\"0\": \"WEB BUCKLING\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\"}, {\"0\": \"\", \"1\": \"At an unstiffened end support (i.e. load type (c) in Fig. 9.3)\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\"}], [], []]\n\n[FORMULAS]\n11.5 = 332; 11.5 = 1972", "page_type": "mixed", "image": "Design of Structural Elements_ Concr_a6902759_p0407.png"}
{"pdf_file": "Design of Structural Elements_ Concrete, Steelwork, Masonry and Timber Designs to British Standards and Eurocodes, 3rd Edition   ( PDFDrive ).pdf", "page_num": 408, "prompt": "Analyze this page with diagrams, tables, formulas, calculations, and explanations.", "response": "[NATIVE TEXT]\n387\nDesign of beams\nExample 9.1 continued\nHence, ly = 172.9 mm\n      \nd\nB\nF\ny w yw\ncr\n  \n  \n.   .   \n   .\n  .\n=\n=\n×\n×\n=\n>\nt f\nF\n172 9\n7 3\n275\n840024\n0 6428\n0 5\nOK\n    \nχF\nF\n  .   \n.\n.\n  .\n  \n=\n=\n=\n<\n0 5\n0 5\n0 6428\n0 778\n1\nd\nOK\nLeff = χFly = 0.778 × 172.9 = 134.5 mm\n    \nF\nf L t\nV\nRd\nyw eff w\nM1\nEd\nkN\nkN\n  \n  \n  \n.   .  \n.\n  \n  \n \n  \n (  \n.  \n)\n=\n=\n×\n×\n×\n=\n>\n=\n−\nγ\n275\n134 5\n7 3\n1 00\n10\n270\n79 2\n3\nOK\nDEFLECTION\nThe maximum bending moment due to working load is:\n    \nM\ng\nq L\nmax  (\n  \n)   (   )\n  \n \n=\n+\n=\n+\n=\nk\nk\nkNm\n2\n2\n8\n8\n6 8\n8\n112\nElastic resistance:\n    \n(\n)   \n  \n  \n  \n.\n  \n  \n \n  \nmax\nM\nW\nf\nM\nc,Rd el\nel,y y\nM0\nNmm\n=\n=\n×\n×\n=\n×\n>\nγ\n796\n10\n275\n1 05\n199\n10\n3\n6\nHence deflection can be calculated elastically.\nDeflection due to variable loading (qk) = 6 kNm−1 = 6 Nmm−1\n    \nδ\nω\n  \n  \n  \n  \n  (   \n)\n  \n  \n  \n   \n=\n×\n=\n×\n×\n×\n×\n×\n×\n=\n5\n384\n5\n384\n6\n8\n10\n210\n10\n14200\n10\n11\n4\n3 4\n3\n4\nL\nEI\nmm\nAssuming the beam supports brittle finishes the maximum permissible deflection is\n  \nspan\nmm\n360\n8\n10\n360\n22\n3\n    \n  \n \n  \n=\n×\n=\n> δ\nOK\nDESIGN BENDING MOMENT\nDesign action (FEd) = (FEd)udl + (FEd)pl\n= (γGgk + γQqk) × span + γQQk\n= (1.35 × 6 + 1.5 × 6)6 + 1.5 × 25\n= 102.6 + 37.5 = 140.1 kN\nExample 9.2 Design of a laterally restrained beam (EC 3)\nSelect and check a suitable beam section using S 235 steel to support the loads shown below. Assume beam is fully\nlaterally restrained and that it sits on 125 mm bearings at each end.\n\n[TABLES]\n[[{\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"Design of beams\"}, {\"0\": \"Example 9.1 continued\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"Hence, ly = 172.9 mm\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"dF\", \"1\": \"=\", \"2\": \"\", \"3\": \"B\", \"4\": \"tfy w yw\", \"5\": \"=\", \"6\": \"172 9\", \"7\": \"××  7 3\", \"8\": \"275\", \"9\": \"=\", \"10\": \"\", \"11\": \"0 6428\", \"12\": \"\", \"13\": \"> 0 5\", \"14\": \"\", \"15\": \"\", \"16\": \"OK\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"Fcr\", \"5\": \"\", \"6\": \"\", \"7\": \"840024\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"χF\", \"1\": \"=\", \"2\": \"0 5\", \"3\": \"\", \"4\": \"=\", \"5\": \"0 5\", \"6\": \"=\", \"7\": \"<0 778  1\", \"8\": \"OK\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"d\", \"3\": \"F\", \"4\": \"\", \"5\": \"0 6428\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"\", \"1\": \"L eff = χ\", \"2\": \"\", \"3\": \"Fl y = 0.778 × 172.9 = 134.5 mm\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"FRd\", \"1\": \"=\", \"2\": \"f\", \"3\": \"Lyw\", \"4\": \"teff\", \"5\": \"275w=\", \"6\": \"×\", \"7\": \"×134 57 3\", \"8\": \"×\", \"9\": \"10\", \"10\": \"−3\", \"11\": \"=\", \"12\": \"270\", \"13\": \"kN\", \"14\": \">\", \"15\": \"V\", \"16\": \"Ed\", \"17\": \"(\", \"18\": \"=\", \"19\": \"79 2\", \"20\": \"kN )\", \"21\": \"OK\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"γ\", \"4\": \"M1\", \"5\": \"\", \"6\": \"\", \"7\": \"1 00\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"DEFLECTION\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"The maximum bending moment due to working load is:\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"=\", \"7\": \"2+(g  q)Lkk\", \"8\": \"=\", \"9\": \"8(\", \"10\": \"+\", \"11\": \"6 8\", \"12\": \"2=\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"M\", \"6\": \"max\", \"7\": \"8\", \"8\": \"\", \"9\": \"\", \"10\": \"8\", \"11\": \"\", \"12\": \"\", \"13\": \"112\", \"14\": \"\", \"15\": \"kNm\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"Elastic resistance:\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"W f=\", \"6\": \"y=\", \"7\": \"×  79610\", \"8\": \"3×\", \"9\": \"275\", \"10\": \"\", \"11\": \"=\", \"12\": \"\", \"13\": \"×\", \"14\": \"6\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \">\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"(M\", \"3\": \"c,Rd\", \"4\": \")el\", \"5\": \"γM0\", \"6\": \"\", \"7\": \"\", \"8\": \"1 05\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"199\", \"13\": \"10\", \"14\": \"\", \"15\": \"\", \"16\": \"Nmm\", \"17\": \"\", \"18\": \"\", \"19\": \"M\", \"20\": \"max\", \"21\": \"\"}, {\"0\": \"Hence deflection can be calculated elastically.\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"Deflection due to variable loading (qk) = 6 kNm−1 = 6 Nmm−1\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"δ\", \"3\": \"=\", \"4\": \"\", \"5\": \"ω5×\", \"6\": \"4L=\", \"7\": \"5×\", \"8\": \"\", \"9\": \"6\", \"10\": \"× (\", \"11\": \"8\", \"12\": \"×10\", \"13\": \"34)\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"=\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"EI384\", \"6\": \"\", \"7\": \"384\", \"8\": \"×210\", \"9\": \"10\", \"10\": \"3\", \"11\": \"×\", \"12\": \"14200\", \"13\": \"\", \"14\": \"×\", \"15\": \"10\", \"16\": \"4\", \"17\": \"\", \"18\": \"\", \"19\": \"mm11\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"Assuming the beam supports brittle finishes the maximum permissible deflection is\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"span\", \"7\": \"×810  =\", \"8\": \"3=\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"> δ\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"360\", \"7\": \"360\", \"8\": \"\", \"9\": \"22\", \"10\": \"mm\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"OK\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"Example 9.2 Design of a laterally restrained beam (EC 3)\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"Select and check a suitable beam section using S 235 steel to support the loads shown below. Assume beam is fully\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"laterally restrained and that it sits on 125 mm bearings at each end.\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"DESIGN BENDING MOMENT\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"Design action (FEd) = (FEd)udl + (FEd)pl\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}, {\"0\": \"= (γ\", \"1\": \"Ggk + γ\", \"2\": \"\", \"3\": \"\", \"4\": \"Qqk) × span + γ\", \"5\": \"\", \"6\": \"\", \"7\": \"QQk\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\", \"21\": \"\"}], [], []]\n\n[FORMULAS]\nly = 172; 172.9 = 134; 25\n= 102; 37.5 = 140", "page_type": "mixed", "image": "Design of Structural Elements_ Concr_a6902759_p0408.png"}
{"pdf_file": "Design of Structural Elements_ Concrete, Steelwork, Masonry and Timber Designs to British Standards and Eurocodes, 3rd Edition   ( PDFDrive ).pdf", "page_num": 409, "prompt": "Analyze this page with diagrams, tables, formulas, calculations, and explanations.", "response": "[NATIVE TEXT]\nEurocode 3: Design of steel structures\n388\nExample 9.2 continued\nDesign bending moment \n    \nM\nF\nl\nF\nl\nEd\nEd udl\nEd pl\nkNm\n  (\n)\n  (\n)\n  \n.     \n.     \n.  \n=\n+\n=\n×\n+\n×\n=\n8\n4\n102 6\n6\n8\n37 5\n6\n4\n133 2\nSECTION SELECTION\n(Refer to Resistance of cross-sections: bending moments)\nAssume suitable section belongs to class 1\nHence, the minimum required plastic moment of resistance about the major axis (y–y), Wpl,y, is given by\n    \nW\nM\nf\npl,y\npl,Rd\nM0\ny\n  \n≥\nγ\nPutting Mpl,Rd = MEd and assuming fy = 235 N mm2 (Table 9.4) gives\n    \nW\nM\nf\npl,y\npl,Rd\nM0\ny\nmm\ncm\n  \n  \n.   \n .\n  \n.   \n \n  \n.  \n≥\n=\n×\n×\n=\n×\n=\nγ\n133 2\n10\n1 00\n235\n566 8\n10\n566 8\n6\n3\n3\n3\nFrom the steel table (Appendix B), try 356 × 171 × 45 kg m−1 UB (Wpl,y = 774 cm3)\nCHECK STRENGTH CLASSIFICATION\nFlange thickness (tf) = 9.7 mm\nHence, from Table 9.4 fy = 235 N mm−2 as assumed\nCHECK SECTION CLASSIFICATION\nε = (235/fy)0.5 = (235/235)0.5 = 1.0\n    \nc\nt f\n  \n.\n.\n= 71 85\n9 7  = 7.41 < 9ε = 9 × 1.0 = 9\n(where c = (b − tw − 2r)/2 = (171.0 − 6.9 − 2 × 10.2)/2 = 143.7/2 = 71.85 mm)\n      \nc\nt\n*\nw\n  \n.\n.\n= 312 3\n6 9  = 45.3 < 72ε = 72 × 1.0 = 72\nHence from Table 9.5, section belongs to class 1 as assumed.\nRESISTANCE OF CROSS-SECTION\nBending\nPlastic moment of resistance, Mpl,Rd, of 356 × 171 × 45 UB is given by\n    \nM\nW f\npl,Rd\npl,y y\nM0\n  \n  \n  \n  \n.\n=\n=\n×\n×\nγ\n774\n10\n235\n1 00\n3\n= 181.9 × 106 Nmm = 181.9 kNm > MEd (= 133.2 kNm)\nOK\nShear\nDesign shear force, VEd, is\n    \nV\nF\nEd\nEd\nkN\n=\n=\n=\n \n  \n.   \n \n2\n140 1\n2\n70\n\n[TABLES]\n[[{\"0\": \"\", \"1\": \"Eurocode 3: Design of steel structures\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"Example 9.2 continued\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"Design bending moment\", \"1\": \"M\", \"2\": \"E d\", \"3\": \"=\", \"4\": \"(FE d\", \"5\": \")udl\", \"6\": \"l\", \"7\": \"××(F)l102 6637 56    E dpl+=+=       133 2\", \"8\": \"kNm\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"8\", \"6\": \"\", \"7\": \"484\", \"8\": \"\"}, {\"0\": \"SECTION SELECTION\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"(Refer to Resistance of cross-sections: bending moments)\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"Assume suitable section belongs to class 1\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"Hence, the minimum required plastic moment of resistance about the major axis (y–y), Wpl,y, is given by\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"γMpl,RdM0W  ≥\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"pl,yfy\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"Putting Mpl,Rd = MEd and assuming fy = 235 N mm2 (Table 9.4) gives\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"≥\", \"3\": \"M\", \"4\": \"γpl,Rd\", \"5\": \"M0\", \"6\": \"\", \"7\": \"6××133 2  10 1 0033==×\", \"8\": \"3=\"}, {\"0\": \"W\", \"1\": \"pl,y\", \"2\": \"\", \"3\": \"\", \"4\": \"fy\", \"5\": \"\", \"6\": \"\", \"7\": \"566 8  10 mm235\", \"8\": \"566 8 cm\"}, {\"0\": \"\", \"1\": \"From the steel table (Appendix B), try 356 × 171 × 45 kg m−1 UB (Wpl,y = 774 cm3)\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"CHECK STRENGTH CLASSIFICATION\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"Flange thickness (tf) = 9.7 mm\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"Hence, from Table 9.4 fy = 235 N mm−2 as assumed\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"CHECK SECTION CLASSIFICATION\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"ε = (235/fy)0.5 = (235/235)0.5 = 1.0\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"c t\", \"7\": \". .= 71 85 = 7.41 < 9ε = 9 × 1.0 = 9  9 7\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"f\", \"8\": \"\"}, {\"0\": \"(where c = (b − tw − 2r )/2 = (171.0 − 6.9 − 2 × 10.2)/2 = 143.7/2 = 71.85 mm)\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"c t\", \"6\": \"* w\", \"7\": \"= 312 3  = 45.3 < 72ε = 72 × 1.0 = 726 9\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"Hence from Table 9.5, section belongs to class 1 as assumed.\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"RESISTANCE OF CROSS-SECTION\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"Bending\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"Plastic moment of resistance, Mpl,Rd, of 356 × 171 × 45 UB is given by\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"=\", \"2\": \"\", \"3\": \"W f\", \"4\": \"y=\", \"5\": \"\", \"6\": \"774\", \"7\": \"3××  10  235\", \"8\": \"\"}, {\"0\": \"Mpl,Rd\", \"1\": \"\", \"2\": \"\", \"3\": \"γM0\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"1 00\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"= 181.9 × 106 Nmm = 181.9 kNm > MEd (= 133.2 kNm)\", \"6\": \"\", \"7\": \"\", \"8\": \"OK\"}, {\"0\": \"Shear\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"Design shear force, VEd, is\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"F140 1Ed===V      kN70Ed\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"22\", \"8\": \"\"}, {\"0\": \"388\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}], [], []]\n\n[FORMULAS]\n.5 = 1; .0 = 9; .7/2 = 71; .0 = 72", "page_type": "mixed", "image": "Design of Structural Elements_ Concr_a6902759_p0409.png"}
{"pdf_file": "Design of Structural Elements_ Concrete, Steelwork, Masonry and Timber Designs to British Standards and Eurocodes, 3rd Edition   ( PDFDrive ).pdf", "page_num": 410, "prompt": "Analyze this page with diagrams, tables, formulas, calculations, and explanations.", "response": "[NATIVE TEXT]\n389\nDesign of beams\nFor class 1 section, design plastic shear resistance, Vpl,Rd, is given by\nVpl,Rd = A v    ( /\n)\nfy\n3 /γM0\nwhere Av = A − 2btf + (tw + 2r)tf > ηhwtw = 1.0 × 332.6 × 6.9 = 2295 mm2\n= 57 × 102 − 2 × 171.0 × 9.7 + (6.9 + 2 × 10.2)9.7\n= 2647.4 mm2\nOK\n(hw = h − 2tf = 352.0 − 2 × 9.7 = 332.6 mm)\nHence, Vpl,Rd = 2647.4  (\n/\n)\n235\n3 /1.00\n= 359192 N = 359 kN > VEd (= 70 kN)\nOK\nBending and shear\nVEd = 0 at mid-span section, hence the moment of resistance of the section is unaffected.\nSHEAR BUCKLING RESISTANCE\nAs \n    \nh\nt\nw\nw\n  \n.\n.\n  \n.   \n  \n  .\n.   \n=\n=\n<\n=\n×\n=\n332 6\n6 9\n48 2\n72\n72\n1 0\n1 0\n72\nε\nη\n, no check on shear buckling is required.\nFLANGE INDUCED BUCKLING\nAw, area of web = (h − 2tf)tw = (352 − 2 × 9.7) × 6.9 = 2294.9 mm2\nAfc, area of compression flange = btf = 171 × 9.7 = 1658.7 mm2\n    \nh\nt\nk E\nf\nA\nA\nw\nw\nyf\nw\nfc\n  \n.\n.\n  \n.   \n  .   \n  \n.\n.   \n.\n=\n=\n≤\n=\n×\n×\n=\n332 6\n6 9\n48 2\n0 3\n210\n10\n235\n2294 9\n1658 7\n315 3\n3\nOK\nHence, no check is required.\nWEB BUCKLING\nAt an unstiffened end support\n    \nk\ns\nc\nh\nF\ns\nw\n    \n  \n  \n=\n+\n+\n\n\n\n≤\n2\n6\n6\n  \n=\n+\n+\n\n\n\n=\n   \n  \n.\n  .\n2\n6 125\n0\n332 6\n4 25\n    \nF\nk E t\nh\ncr\nF\nw\nw\nN\n  .\n  .   .\n  \n  \n  \n.\n.\n  \n \n=\n=\n×\n×\n×\n×\n=\n0 9\n0 9\n4 25\n210\n10\n6 9\n332 6\n793370\n3\n3\n3\n    \nm\nf b\nf t\n1\nyf f\nyw w\n  \n  \n  \n.\n  .\n  \n.\n=\n=\n×\n×\n=\n235\n171 0\n235\n6 9\n24 8\nAssuming dF > 0.5\n    \nm\nh\nt\n2\nw\nf\n  .\n  .\n  \n.\n.\n  \n.\n=\n\n\n\n\n=\n× \n\n\n\n=\n0 02\n0 02\n332 6\n9 7\n23 5\n2\n2\nExample 9.2 continued\n\n[TABLES]\n[[{\"0\": \"Example 9.2 continued\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"For class 1 section, design plastic shear resistance, Vpl,Rd, is given by\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"/3 /γfyM0Vpl,Rd = A v    (\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"where A v = A − 2btf + (tw + 2r )tf > ηhwtw = 1.0 × 332.6 × 6.9 = 2295 mm2\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"\", \"1\": \"= 57 × 102 − 2 × 171.0 × 9.7 + (6.9 + 2 × 10.2)9.7\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"OK= 2647.4 mm2\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"(hw = h − 2t f = 352.0 − 2 × 9.7 = 332.6 mm)\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"235/3 /1.00Hence, Vpl,Rd = 2647.4  (\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"= 359192 N = 359 kN > VEd (= 70 kN)\", \"1\": \"OK\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"Bending and shear\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"VEd = 0 at mid-span section, hence the moment of resistance of the section is unaffected.\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"SHEAR BUCKLING RESISTANCE\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"h tε η. .332 61 0w==<=×=    48 2  72  72    72As\", \"1\": \", no check on shear buckling is required.\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"6 91 0w\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"FLANGE INDUCED BUCKLING\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"Aw, area of web = (h − 2tf)tw = (352 − 2 × 9.7) × 6.9 = 2294.9 mm2\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"Afc, area of compression flange = btf = 171 × 9.7 = 1658.7 mm2\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"E332 6ww==≤\", \"1\": \"=\", \"2\": \"×\", \"3\": \"210\", \"4\": \"×\", \"5\": \"10\", \"6\": \"3\", \"7\": \"2294 9\", \"8\": \"=\"}, {\"0\": \"h tA Ak    48 2  f6 9wyffc\", \"1\": \"0 3\", \"2\": \"\", \"3\": \"\", \"4\": \"235\", \"5\": \"\", \"6\": \"\", \"7\": \". .1658 7\", \"8\": \"OK  315 3\"}, {\"0\": \"Hence, no check is required.\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"WEB BUCKLING\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}, {\"0\": \"At an unstiffened end support\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\"}], [], []]\n\n[FORMULAS]\n9.7\n= 2647; 9.7 = 332; Rd = 2647; 6.9 = 2294; .7 = 1658", "page_type": "mixed", "image": "Design of Structural Elements_ Concr_a6902759_p0410.png"}
{"pdf_file": "Design of Structural Elements_ Concrete, Steelwork, Masonry and Timber Designs to British Standards and Eurocodes, 3rd Edition   ( PDFDrive ).pdf", "page_num": 411, "prompt": "Analyze this page with diagrams, tables, formulas, calculations, and explanations.", "response": "[NATIVE TEXT]\nEurocode 3: Design of steel structures\n390\nExample 9.2 continued\n      \nBe\nF\nw\nyw w\ns\nmm\n  \n      \n    \n \n=\n≤\n+\n=\n+\n=\nk Et\nf h\ns\nc\n2\n2\n125\n0\n125\n  \n=\n×\n×\n×\n×\n×\n=\n \n  \n.  \n4.25  210  10   6.9\n2  235  332.6\nmm\n3\n2\n271 8\nHence le = 125 mm\n      \nB\nB\nB\ny\ne\nf\n1\ne\nf\n  \n  \n  \n  \n=\n+\n+ \n\n\n+\nt\nm\nt\nm\n2\n2\n2\n    \n=\n+\n+ \n\n\n\n+\n=\n \n  .\n.   \n.\n  \n.   \n125\n9 7 24 8\n2\n125\n9 7\n23 5\n2\n262.8 mm\n      B\nB\ny\ne\nf\n1\n  \n  \n  \n=\n+\n+\nt\nm\nm2\nBy = 125 +   9 7 24 8\n23 5\n.\n.   \n.\n+\n = 192.4 mm\nHence, ly = 192.4 mm\n      \nd\nB\nF\ny w yw\ncr\n  \n  \n.   .   \n   .\n  .\n=\n=\n×\n×\n=\n>\nt f\nF\n192 4\n6 9\n235\n793370\n0 627\n0 5\nOK\n    \nχF\nF\n  .   \n.\n.\n  .   \n=\n=\n=\n<\n0 5\n0 5\n0 627\n0 8\n1\nd\nOK\nLeff = χFly = 0.8 × 192.4 = 153.9 mm\n    \nF\nf L t\nV\nRd\nyw eff w\nM1\nEd\nkN\nkN\n  \n  \n  \n.   .  \n.\n  \n  \n.  \n  \n (  \n \n)\n=\n=\n×\n×\n×\n=\n>\n=\n−\nγ\n235\n153 9\n6 9\n1 05\n10\n237 6\n70\n3\nOK\nDEFLECTION\nDeflection due to variable uniformly distributed loading qk = 6 kN m−1 = 6 Nmm−1 and variable point load, Qk =\n25000 N is\nHence, \n    \nδ2\n4\n3\n5\n384\n48\n  \n  \n=\n+\nq L\nEI\nQ L\nEI\nk\nk\n  \n=\n×\n×\n×\n×\n×\n×\n×\n+\n×\n×\n×\n×\n×\n×\n×\n \n    (   \n)\n  \n  \n  \n  \n  \n  \n  (   \n)\n  \n  \n  \n  \n5\n6\n6\n10\n384\n210\n10\n12100\n10\n25\n10\n6\n10\n48\n210\n10\n12100\n10\n3 4\n3\n4\n3\n3 3\n3\n4\n= 4 mm + 4.4 mm\n  \n=\n<\n=\n=\n×\n=\n\n\n\n\n  \n  \n  \n    \n  \n \n8\n6\n10\n360\n16\n3\nmm\nallowable\nspan\n360\nmm\nOK\n\n[TABLES]\n[[{\"0\": \"\", \"1\": \"Eurocode 3: Design of steel structures\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}, {\"0\": \"\", \"1\": \"Example 9.2 continued\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"k Et=\", \"7\": \"2w≤+\", \"8\": \"\", \"9\": \"=\", \"10\": \"+\", \"11\": \"=\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"Be\", \"6\": \"2f\", \"7\": \"s shyww\", \"8\": \"c\", \"9\": \"125\", \"10\": \"\", \"11\": \"0\", \"12\": \"125\", \"13\": \"mm\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"4.25 =\", \"7\": \"×× 210\", \"8\": \"10\", \"9\": \"×\", \"10\": \"6.9\", \"11\": \"2=\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"×2  235\", \"8\": \"×\", \"9\": \"332.6\", \"10\": \"\", \"11\": \"\", \"12\": \"271 8\", \"13\": \"\", \"14\": \"mm\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}, {\"0\": \"Hence le = 125 mm\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"Bm\", \"8\": \"\", \"9\": \"2\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"=BB  ye\", \"6\": \"+  t\", \"7\": \" 1+  f2t\", \"8\": \" e\", \"9\": \"+\", \"10\": \"m2\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"f\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"24 8\", \"8\": \"\", \"9\": \"125\", \"10\": \"2\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"=\", \"6\": \"+125\", \"7\": \"9 7 2\", \"8\": \" +\", \"9\": \"9 7\", \"10\": \" \", \"11\": \"+ 23 5\", \"12\": \"=\", \"13\": \"262.8 mm\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"=B  ye      B\", \"6\": \"+  t\", \"7\": \"+m  f1m2\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"By = 125 +   9 7\", \"7\": \"24 8\", \"8\": \"+\", \"9\": \"23 5\", \"10\": \"= 192.4 mm\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}, {\"0\": \"Hence, ly = 192.4 mm\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}, {\"0\": \"d\", \"1\": \"= F\", \"2\": \"\", \"3\": \"B\", \"4\": \"tfy w yw\", \"5\": \"=\", \"6\": \"192 4\", \"7\": \"××  6 9\", \"8\": \"235\", \"9\": \"=\", \"10\": \"0 627\", \"11\": \">\", \"12\": \"0 5\", \"13\": \"\", \"14\": \"OK\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"Fcr\", \"5\": \"\", \"6\": \"\", \"7\": \"793370\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}, {\"0\": \"χ\", \"1\": \"= F\", \"2\": \"\", \"3\": \"0 5\", \"4\": \"=\", \"5\": \"0 5\", \"6\": \"= 0 8\", \"7\": \"<  1OK\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"dF\", \"4\": \"\", \"5\": \"0 627\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}, {\"0\": \"\", \"1\": \"L eff = χ\", \"2\": \"\", \"3\": \"\", \"4\": \"Fly = 0.8 × 192.4 = 153.9 mm\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}, {\"0\": \"FRd\", \"1\": \"=\", \"2\": \"f\", \"3\": \"Lyw\", \"4\": \"teffw\", \"5\": \"235=\", \"6\": \"×\", \"7\": \"×153 96 9\", \"8\": \"×\", \"9\": \"−310\", \"10\": \"=\", \"11\": \"237 6\", \"12\": \"kN\", \"13\": \">\", \"14\": \"V\", \"15\": \"Ed\", \"16\": \"(\", \"17\": \"=\", \"18\": \"70\", \"19\": \")kN\", \"20\": \"OK\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"γ\", \"4\": \"M1\", \"5\": \"\", \"6\": \"\", \"7\": \"1 05\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}, {\"0\": \"DEFLECTION\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}, {\"0\": \"25000 N is\", \"1\": \"Deflection  due  to  variable  uniformly  distributed  loading  qk  =  6  kN  m−1  =  6  Nmm−1  and  variable  point  load,  Q k  =\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}, {\"0\": \"4q L=+δ\", \"1\": \"Q L\", \"2\": \"3\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}, {\"0\": \"5 3Hence,     284EI\", \"1\": \"48\", \"2\": \"EI\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\", \"15\": \"\", \"16\": \"\", \"17\": \"\", \"18\": \"\", \"19\": \"\", \"20\": \"\"}], [], []]\n\n[FORMULAS]\nly = 192; 192.4 = 153", "page_type": "mixed", "image": "Design of Structural Elements_ Concr_a6902759_p0411.png"}
{"pdf_file": "Design of Structural Elements_ Concrete, Steelwork, Masonry and Timber Designs to British Standards and Eurocodes, 3rd Edition   ( PDFDrive ).pdf", "page_num": 412, "prompt": "Analyze this page with diagrams, tables, formulas, calculations, and explanations.", "response": "[NATIVE TEXT]\n391\nDesign of beams\nExample 9.3 Design of a cantilever beam (EC 3)\nA cantilever beam is needed to resist the loading shown below. Select a suitable UB section in S 275 steel to satisfy\nbending and shear criteria only assuming full lateral restraint.\nDESIGN BENDING MOMENT AND SHEAR FORCE\nDesign action (FEd) = γGgk + γQqk\n= 1.35 × 500 + 1.5 × 350 = 1200 kN\nDesign bending moment at A, MEd, is\n    \nM\nF l\nEd\nEd\nkNm\n  \n  \n  .   \n \n=\n=\n×\n=\n2\n1200\n1 5\n2\n900\nDesign shear force at A, VEd, is\nVEd = FEd = 1200 kN\nSECTION SELECTION\nSince the cantilever beam is relatively short and subject to fairly high shear forces, the bending capacity or the shear\nstrength of the section may be critical and both factors will need to be considered in order to select an appropriate\nsection.\nHence, plastic moment of section (assuming beam belongs to class 1), Wpl, must exceed the following to satisfy\nbending:\n    \nW\nM\nf\npl,y\npl,Rd\nM0\ny\nmm\ncm\n  \n  \n  \n .\n  \n  \n \n  \n \n≥\n=\n×\n×\n=\n×\n=\nγ\n900\n10\n1 00\n265\n3396\n10\n3396\n6\n3\n2\n3\nThe shear area of the section, A v, must exceed the following to satisfy shear:\n    \nA\nV\nf\nν\nγ\n  \n( /\n)\n  \n  \n .\n(\n/\n)\n  \n \n≥\n=\n×\n×\n=\npl,Rd\nM0\ny\nmm\n3\n1200\n10\n1 00\n265\n3\n7843\n3\n2\nwhere A v = A − 2btf + (tw + 2r)tf > ηhwtw\nHence, try 610 × 305 × 149 kg m−1 UB section\nCHECK STRENGTH CLASSIFICATION\nFlange thickness (tf) = 19.7 mm\nHence from Table 9.4, fy = 265 N mm−2 as assumed\nCHECK SECTION CLASSIFICATION\nε = (235/fy)0.5 = (235/265)0.5 = 0.94\n    \nc\nt f\n  \n.\n.\n= 129 95\n19 7  = 6.59 < 9ε = 9 × 0.94 = 8.46\n\n[TABLES]\n[[{\"0\": \"Design of beams\"}, {\"0\": \"Example 9.3 Design of a cantilever beam (EC 3)\"}, {\"0\": \"A cantilever beam is needed to resist the loading shown below. Select a suitable UB section in S 275 steel to satisfy\"}, {\"0\": \"bending and shear criteria only assuming full lateral restraint.\"}, {\"0\": \"DESIGN BENDING MOMENT AND SHEAR FORCE\"}, {\"0\": \"Design action (FEd) = γGgk + γQqk\"}, {\"0\": \"= 1.35 × 500 + 1.5 × 350 = 1200 kN\"}, {\"0\": \"Design bending moment at A, MEd, is\"}, {\"0\": \"Fl1 5Ed==×=900M    1200     kNmEd\"}, {\"0\": \"22\"}, {\"0\": \"Design shear force at A, VEd, is\"}, {\"0\": \"VEd = FEd = 1200 kN\"}, {\"0\": \"SECTION SELECTION\"}, {\"0\": \"Since the cantilever beam is relatively short and subject to fairly high shear forces, the bending capacity or the shear\"}, {\"0\": \"strength of the section may be critical and both factors will need to be considered in order to select an appropriate\"}, {\"0\": \"section.\"}, {\"0\": \"Hence, plastic moment of section (assuming beam belongs to class 1), Wpl, must exceed the following to satisfybending:\"}, {\"0\": \"6γ××M900  10 1 00pl,RdM0323≥==×=\"}, {\"0\": \"W      3396  10 mm  3396 cmpl,yf265y\"}, {\"0\": \"The shear area of the section, A v, must exceed the following to satisfy shear:\"}, {\"0\": \"3γ××V   1200101 00pl,RdM02≥==\"}, {\"0\": \"ν       A7843mm(/)(/)f32653y\"}, {\"0\": \"where A v = A − 2btf + (tw + 2r)tf > ηhwtwHence, try 610 × 305 × 149 kg m−1 UB section\"}, {\"0\": \"CHECK STRENGTH CLASSIFICATION\"}, {\"0\": \"Flange thickness (tf) = 19.7 mm\"}, {\"0\": \"Hence from Table 9.4, fy = 265 N mm−2 as assumed\"}, {\"0\": \"CHECK SECTION CLASSIFICATION\"}, {\"0\": \"ε = (235/fy)0.5 = (235/265)0.5 = 0.94\"}], [], []]\n\n[FORMULAS]\n.5 = 0; .94 = 8", "page_type": "mixed", "image": "Design of Structural Elements_ Concr_a6902759_p0412.png"}
{"pdf_file": "Design of Structural Elements_ Concrete, Steelwork, Masonry and Timber Designs to British Standards and Eurocodes, 3rd Edition   ( PDFDrive ).pdf", "page_num": 413, "prompt": "Analyze this page with diagrams, tables, formulas, calculations, and explanations.", "response": "[NATIVE TEXT]\nEurocode 3: Design of steel structures\n392\nExample 9.3 continued\n(where c = (b − tw − 2r)/2 = (304.8 − 11.9 − 2 × 16.5)/2 = 259.9/2 = 129.95 mm)\n      \nc\nt\n*\nw\n  \n.\n.\n= 537 2\n11 9  = 45.14 < 72ε = 72 × 0.94 = 67.68\nHence from Table 9.5, section belongs to class 1 as assumed.\nRESISTANCE OF CROSS-SECTION\nBending\nPlastic moment of resistance of 610 × 305 × 149 UB about the major axis (y-y) is given by\n    \nM\nW f\npl,Rd\npl,y y\nM0\n  \n  \n  \n \n.\n=\n=\n×\n×\nγ\n4570\n10\n265\n1 00\n3\n= 1211 × 106 Nmm = 1211 kNm > MEd (= 900 kNm)\nOK\nShear\nFor Class 1 section, design plastic shear resistance, is given by\nVpl,Rd = Av(fy/  3 )/γM0\nwhere Av = A − 2btf + (tw + 2r)tf > ηhwtw = 1.0 × 570.2 × 11.9 = 6785.38 mm2\n= 190 × 102 − 2 × 304.8 × 19.7 + (11.9 + 2 × 16.5)19.7\n= 7875.4 mm2\nOK\n(hw = h − 2tf = 609.6 − 2 × 19.7 = 570.2 mm)\nHence, Vpl,Rd = 7875.4   (\n/\n)\n265\n3 /1.00 = 1204.9 × 103 N\n= 1204.9 kN > VEd (= 1200 kN)\nOK\nBending and shear\nSince\nVEd = 1200 kN > 0.5Vpl,Rd = 0.5 × 1204.9 = 602.5 kN\nthe section is subject to a ‘high shear load’ and the design resistant moment of the section (about the major axis)\nshould be reduced to My,V,Rd which is given by\n    \nM\nW\nA\nt f\ny,V,Rd\npl,y\nw\n2\nw\ny\nM0\n  (\n  \n/\n)   (\n  \n .\n  \n.\n/   \n. )\n.\n=\n−\n=\n×\n−\n×\n×\nρ\nγ\n4\n4570\n10\n0 984\n6785 38 4\n11 9 265\n1 00\n3\n2\n= 958.8 × 106 Nmm = 958.8 kNm > MEd (= 900 kNm)\nOK\nwhere\nρ = (2VEd/Vpl,Rd − 1)2 = (2 × 1200/1204.9 − 1)2 = 0.984\nAw = hwtw = 570.2 × 11.9 = 6785.38 mm2\nSelected UB section, 610 × 305 × 149, is satisfactory in bending and shear.\n\n[TABLES]\n[[{\"0\": \"Eurocode 3: Design of steel structures\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"Example 9.3 continued\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"(where c = (b − tw − 2r )/2 = (304.8 − 11.9 − 2 × 16.5)/2 = 259.9/2 = 129.95 mm)\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"c t* w= 537 2   = 45.14 < 72ε = 72 × 0.94 = 67.6811 9\", \"2\": \"\"}, {\"0\": \"Hence from Table 9.5, section belongs to class 1 as assumed.\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"RESISTANCE OF CROSS-SECTION\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"Bending\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"Plastic moment of resistance of 610 × 305 × 149 UB about the major axis (y-y) is given by\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"3××W f4570  10 265y==\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"M    pl,Rdγ1 00M0\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"= 1211 × 106 Nmm = 1211 kNm > MEd (= 900 kNm)\", \"2\": \"OK\"}, {\"0\": \"Shear\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"For Class 1 section, design plastic shear resistance, is given by\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"M0Vpl,Rd = A v(fy/   3 )/γ\", \"2\": \"\"}, {\"0\": \"where A v = A − 2btf + (tw + 2r)t f > ηhwtw = 1.0 × 570.2 × 11.9 = 6785.38 mm2\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"= 190 × 102 − 2 × 304.8 × 19.7 + (11.9 + 2 × 16.5)19.7\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"= 7875.4 mm2OK\", \"2\": \"\"}, {\"0\": \"(hw = h − 2tf = 609.6 − 2 × 19.7 = 570.2 mm)\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"Hence, Vpl,Rd = 7875.4   (\", \"1\": \"265/3 /1.00 = 1204.9 × 103 N\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"OK= 1204.9 kN > VEd (= 1200 kN)\", \"2\": \"\"}, {\"0\": \"Bending and shear\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"Since\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"VEd = 1200 kN > 0.5Vpl,Rd = 0.5 × 1204.9 = 602.5 kN\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"the section is subject to a ‘high shear load’ and the design resistant moment of the section (about the major axis)\", \"2\": \"\"}, {\"0\": \"should be reduced to My,V,Rd which is given by\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"ρ γw232−×−××4(W  A/t)f4570100 9846785 384(     / pl,ywy==\", \"2\": \"11 9 265\"}, {\"0\": \"\", \"1\": \"M    y,V,Rd1 00M0\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"OK= 958.8 × 106 Nmm = 958.8 kNm > MEd (= 900 kNm)\", \"2\": \"\"}, {\"0\": \"where\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"ρ = (2VEd/Vpl,Rd − 1)2 = (2 × 1200/1204.9 − 1)2 = 0.984\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"A w = hwtw = 570.2 × 11.9 = 6785.38 mm2\", \"2\": \"\"}, {\"0\": \"Selected UB section, 610 × 305 × 149, is satisfactory in bending and shear.\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"392\", \"1\": \"\", \"2\": \"\"}], [], []]\n\n[FORMULAS]\n.9/2 = 129; .94 = 67; Rd = Av(fy/  3 )/γM0; 11.9 = 6785; 19.7\n= 7875; 19.7 = 570; Rd = 7875; Rd = 0; 1204.9 = 602; 2 = 0; 11.9 = 6785", "page_type": "mixed", "image": "Design of Structural Elements_ Concr_a6902759_p0413.png"}
{"pdf_file": "Design of Structural Elements_ Concrete, Steelwork, Masonry and Timber Designs to British Standards and Eurocodes, 3rd Edition   ( PDFDrive ).pdf", "page_num": 414, "prompt": "Analyze this page with diagrams, tables, formulas, calculations, and explanations.", "response": "[NATIVE TEXT]\n393\nDesign of beams\nExample 9.4 Design of a beam with stiffeners (EC 3)\nThe figure shows a simply supported beam and cantilever with uniformly distributed loads applied to it. Using grade\nS 275 steel and assuming full lateral restraint, select and check a suitable beam section.\nDESIGN SUMMARY\nPermanent action = 200 kN m−1\nVariable action = 100 kN m−1\nGrade S 275 steel\nFull lateral restraint\nULTIMATE LIMIT STATE\n3 load cases:\n(a) 1.35Gk + 1.5Qk on span 1\n1.35Gk + 1.5Qk on span 2\n(b) 1.35Gk + 1.5Qk on span 1\n1.0Gk + 0.0Qk on span 2\n(c) 1.0Gk + 0.0Qk on span 1\n1.35Gk + 1.5Qk on span 2\nStatic equilibrium (refer to Table 8.6)\nOverturning moment = (1.1 × 200 + 1.5 × 100)2.5 × 1.25 = 1156.25 kNm (load case (c))\nRestorative moment = (0.9 × 200)5 × 2.5 = 2250 kNm > 1156.25 kNm\nOK\nSHEAR AND BENDING\nShear force and bending moment diagrams for the 3 load cases are shown below.\n\n[TABLES]\n[[{\"0\": \"\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"S 275 steel and assuming full lateral restraint, select and check a suitable beam section.\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\"}], [{\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\", \"10\": \"\", \"11\": \"\", \"12\": \"\", \"13\": \"\", \"14\": \"\"}], [{\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\"}, {\"0\": \"\", \"1\": \"\", \"2\": \"\", \"3\": \"\", \"4\": \"\", \"5\": \"\", \"6\": \"\", \"7\": \"\", \"8\": \"\", \"9\": \"\"}], [], []]\n\n[FORMULAS]\n1.25 = 1156", "page_type": "mixed", "image": "Design of Structural Elements_ Concr_a6902759_p0414.png"}
{"pdf_file": "Design of Structural Elements_ Concrete, Steelwork, Masonry and Timber Designs to British Standards and Eurocodes, 3rd Edition   ( PDFDrive ).pdf", "page_num": 415, "prompt": "Analyze this page with diagrams, tables, formulas, calculations, and explanations.", "response": "[NATIVE TEXT]\nEurocode 3: Design of steel structures\n394\nExample 9.4 continued\nSECTION SELECTION\nFor class 1 section\n    \nW\nM\nf\npl,y\npl,Rd\nM0\ny\nmm\ncm\n  \n  \n  \n .\n  .\n  \n \n  \n \n≥\n=\n×\n×\n=\n×\n=\nγ\n1313\n10\n1 00\n265\n4 955\n10\n4955\n6\n6\n2\n3\nTry 762 × 267 × 173 UB section\nSTRENGTH CLASSIFICATION\nFlange thickness = 21.6 mm, steel grade S 275. Hence from Table 9.4, fy = 265 N mm−2.\nSECTION CLASSIFICATION\nε = (235/fy)1/2 = (235/265)1/2 = 0.94\n    \nc\nt f\n  \n.\n.\n= 109 7\n21 6  = 5.08 < 9ε = 9 × 0.94 = 8.46\n(where c = (b − tw − 2r)/2 = (266.7 − 14.3 − 2 × 16.5)/2 = 219.4/2 = 109.7 mm)\n    \nc\nt\n∗=\nw\n  \n.\n.\n685 8\n14 3  = 47.9 < 72ε = 72 × 0.94 = 67.7\nHence from Table 9.5, section belongs to class 1 as assumed.\nRESISTANCE OF CROSS-SECTIONS\nShear resistance\nFor class 1 section, design plastic shear resistance, is given by\nVpl,Rd = Av    ( /\n)\nfy\n3 /γMO\nwhere Av = A − 2btf + (tw + 2r)tf > ηhwtw = 1.0 × 718.8 × 14.3 = 10278.8 mm2\n= 220 × 102 − 2 × 266.7 × 21.6 + (14.3 + 2 × 16.5)21.6\n= 11500.2 mm2\nOK\n(hw = h − 2tf = 762 − 2 × 21.6 = 718.8 mm)\nHence, Vpl,Rd = 11500.2  (\n/\n)\n265\n3 /1.00 = 1759.5 × 103 N\n= 1759.5 kN > VEd (= 1313 kN)\nOK\nBending and shear\nVEd = 1313 kN = 0.746Vpl,Rd > 0.5Vpl,Rd\nHence, beam is subject to HIGH SHEAR LOAD\n    \nM\nW\nA\nt f\ny,V,Rd\npl,y\nw\n2\nw\ny\nM0\nmm\n  (\n  \n/\n)   (\n  \n .\n  \n.  \n/   \n. )\n.\n=\n−\n=\n×\n−\n×\n×\nρ\nγ\n4\n6200\n10\n0 243\n10278 8\n4\n14 3265\n1 00\n3\n= 1524 × 106 Nmm = 1524 kNm > MEd (= 1313 kNm)\nOK\n\n[TABLES]\n[[{\"0\": \"Eurocode 3: Design of steel structures\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"Example 9.4 continued\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"SECTION SELECTION\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"For class 1 section\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"6γ××M1313  10 1 00pl,RdM0623≥==×=\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"W      4 955  10 mm  4955 cmpl,yf265y\", \"2\": \"\"}, {\"0\": \"Try 762 × 267 × 173 UB section\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"STRENGTH CLASSIFICATION\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"Flange thickness = 21.6 mm, steel grade S 275. Hence from Table 9.4, fy = 265 N mm−2.\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"SECTION CLASSIFICATION\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"ε = (235/fy)1/2 = (235/265)1/2 = 0.94\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"c t= 109 7   = 5.08 < 9ε = 9 × 0.94 = 8.4621 6\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"f\", \"2\": \"\"}, {\"0\": \"(where c = (b − tw − 2r )/2 = (266.7 − 14.3 − 2 × 16.5)/2 = 219.4/2 = 109.7 mm)\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"c t∗685 8=   = 47.9 < 72ε = 72 × 0.94 = 67.7\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"14 3w\", \"2\": \"\"}, {\"0\": \"Hence from Table 9.5, section belongs to class 1 as assumed.\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"RESISTANCE OF CROSS-SECTIONS\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"Shear resistance\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"For class 1 section, design plastic shear resistance, is given by\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"/3 /γfyMOVpl,Rd = A v    (\", \"2\": \"\"}, {\"0\": \"where A v = A − 2btf + (tw + 2r)tf > ηhwtw = 1.0 × 718.8 × 14.3 = 10278.8 mm2\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"= 220 × 102 − 2 × 266.7 × 21.6 + (14.3 + 2 × 16.5)21.6\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"= 11500.2 mm2OK\", \"2\": \"\"}, {\"0\": \"(hw = h − 2tf = 762 − 2 × 21.6 = 718.8 mm)\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"Hence, Vpl,Rd = 11500.2  (\", \"1\": \"265/3 /1.00 = 1759.5 × 103 N\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"OK= 1759.5 kN > VEd (= 1313 kN)\", \"2\": \"\"}, {\"0\": \"Bending and shear\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"VEd = 1313 kN = 0.746Vpl,Rd > 0.5Vpl,Rd\", \"2\": \"\"}, {\"0\": \"Hence, beam is subject to HIGH SHEAR LOAD\", \"1\": \"\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"ρ γw23−×−××(  /4)WAtf(6200  10 0 243  10278 8 mm/4  14 3 265pl,ywy==\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"My,V,Rd1 00M0\", \"2\": \"\"}, {\"0\": \"\", \"1\": \"= 1524 × 106 Nmm = 1524 kNm > MEd (= 1313 kNm)\", \"2\": \"OK\"}, {\"0\": \"394\", \"1\": \"\", \"2\": \"\"}], [], []]\n\n[FORMULAS]\n.94 = 8; .4/2 = 109; .94 = 67; 14.3 = 10278; 21.6\n= 11500; .6 = 718; Rd = 11500", "page_type": "mixed", "image": "Design of Structural Elements_ Concr_a6902759_p0415.png"}
{"pdf_file": "Design of Structural Elements_ Concrete, Steelwork, Masonry and Timber Designs to British Standards and Eurocodes, 3rd Edition   ( PDFDrive ).pdf", "page_num": 416, "prompt": "Analyze this page with diagrams, tables, formulas, calculations, and explanations.", "response": "[NATIVE TEXT]\n395\nDesign of beams\nwhere\nρ = (2VEd/Vpl,Rd − 1)2 = (2 × 1313/1759.5 − 1)2 = 0.243\nAw = hwtw = 718.8 × 14.3 = 10278.8 mm2\nShear buckling resistance\nAs will be seen later, the web needs stiffeners. Hence, provided c∗/tw for the web is less than \n    \n31\nη ε\nτk  (clause 5.1,\nEC 3–5) no check is required for resistance to shear buckling.\nkτ = 5.34 (assuming stiffeners needed only at supports)\nε = 0.94\nη = 1.2\nHence, \n    \n31\n31\n1 2\n0 94\n5 34\n56 1\n48 0\nη ε\nτk\nc t\n  .   .\n  \n.\n  \n.   \n/\n (  \n. )\n=\n×\n×\n=\n> ∗\n=\nw\nOK\nFlange-induced buckling\n    \nh\nt\nk E\nf\nA\nA\nw\nw\nyf\nw\nfc\n  \n.\n.\n  \n.   \n  .   \n  \n  \n.\n=\n=\n≤\n=\n×\n×\n=\n718 8\n14 3\n50 3\n0 3\n210\n10\n265\n10279\n5761\n317 6\n3\nwhere\nAw, area of web = (h − 2tf)tw = (762 − 2 × 21.6) × 14.3 = 10279 mm2\nAfc, area of compression flange = btf = 266.7 × 21.6 = 5761 mm2\nHence, no check is required.\nResistance of web to transverse forces\nThe web capacity at supports A and C needs checking but by inspection it would seem that the web resistance at C\nis almost certainly more critical. Therefore only the calculations for this support are presented.\nExample 9.4 continued\nStiff bearing length, ss ≈ 2tf + tw + r = 2 × 36.6 + 21.5 + 24.1 = 118.8 mm\nSince the load is applied through the flange and is resisted by shear forces in the web (i.e. load type (a) in Fig. 9.3),\nkF is given by\n    \nk\nh\na\nF\nw\n    \n=\n+\n\n\n\n\n6\n2\n2\n\n[TABLES]\n[[{\"0\": \"\", \"1\": \"\"}, {\"0\": \"\", \"1\": \"\"}, {\"0\": \"\", \"1\": \"\"}, {\"0\": \"\", \"1\": \"\"}, {\"0\": \"\", \"1\": \"\"}, {\"0\": \"\", \"1\": \"\"}, {\"0\": \"\", \"1\": \"\"}], [], []]\n\n[FORMULAS]\n2 = 0; 14.3 = 10278; ε = 0; 24.1 = 118", "page_type": "mixed", "image": "Design of Structural Elements_ Concr_a6902759_p0416.png"}