Question: Force F has a magnitude of 600 lbf acting 60 degrees CCW from the positive x-axis. Determine the scalar components of F in the x and y directions.

Answer: Fx = 300 lbf, Fy = 519.6 lbf


Question: Force F has a magnitude of 600 lbf acting 60 degrees CCW from the positive x-axis. A new coordinate system is created by rotating the x- and y-axes 65 degrees clockwise. Determine the scalar components of F in the x' and y' directions.

Answer: Fx' = -344.1 lbf, Fy' = 491.5 lbf


Question: Find the magnitude and direction of force F which has scalar components: Fx = -2 N and Fy = -6 N

Answer: 6.325 N, 251.6 degrees


Question: Vector R has vector components A and B where A has a magnitude of 4 kN and points 35 degrees CW from the positive x-axis, and B has a magnitude of 6 kN pointing 85 degrees CCW from the positive x-axis. Find the magnitude and direction of force of the resultant R.

Answer: 5.3 kN, 44.1 degrees CCW from the positive x-axis


Question: Determine the resultant of three forces A, B and C, with scalar components of A (-6.26, 12.5 lbf), B (35.4, -35.4 lbf) and C (-36.8, 27.6 lbf).

Answer: 9.06 lbf, 148.3 degrees


Question: Three forces act on a particle. Force A is 600 N acting in the negative y direction. Force B acts 15 degrees CCW from the positive y-axis. Force C acts 60 degrees CCW from the positive x-axis. Determine the magnitudes of forces B and C.

Answer: B = 424.3 N, C = 219.6 N


Question: A small massless ring is located at (0, 0m). A is located at (0, -4m), B is located at (-2, 6m) and C is located at (4, 3m). A force of 400 N is applied to the ring in the direction towards C. The ring is held in equilibrium by two inextensible cables, one connected to point A and one connected to point B. Determine the tension in cable A and cable B.

Answer: Cable A = 1200 N, Cable B = 1012 N


Question: A mass of 10 kg is held in equilibrium by two cables, cable A and cable B. The mass is located at (0, 0m). Point A is located at (-4.25, 3.5m) and point B is located at (4.5, 3.5m). Determine the tension in cable A and cable B.

Answer: Cable A = 79.36 N, Cable B = 77.61 N


Question: A 2 lb cylinder rests in a trough with smooth frictionless surfaces. The left side of the trough is angled at 35 degrees from the horizontal. The right side of the trough is angled at 60 degrees to the horizontal. Determine the reactions on the cylinder from the left side and the right side of the trough.

Answer: Left side = 1.74 lb, right side = 1.15 lb


Question: Cable ABC has a length of 8 ft and the cable supports a weight of 100 lbf riding on a frictionless pulley of negligible radius located at B. The cable is supported at A (0, 0 ft) and C (6.5, 2.5 ft). Determine the horizontal distance from A of the pulley and the tension in the cable, required for equilibrium.

Answer: Distance = 1.51 ft, tension = 85.8 lbf


Question: A 320 mm x 550 mm steel plate 35 mm thick is suspended from a point P located 350 mm above the center of the plate (in the z direction). Three cables support the plate at A, B and D. The four corners of the plate are E located at (0, 0mm), C located at (0, 550mm), B located at (320mm, 550mm) and A located at (320, 0mm). Point D is located 165 mm from E along the plate edge EC. The density of steel is 7970 kg per cubic metre. Determine the tension in cable A, cable B and cable D.

Answer: Cable A = 97.6 N, cable B = 227.8 N, cable D = 275.4 N


Question: A system of cables are suspended by connections at B, E and D. A weight of 75 lbf is located at A and a weight of 55 lbf is located at C. Point B is located at the left end of the system of cables, and cable B to A forms an angle of 10 degrees to the horizontal where A is below and to the right of B. Cable A to C forms an angle of 15 degrees to the horizontal and C is above and to the right of A. Two cables connect to C, a cable from C to E which forms an angle of 70 degrees to the horizontal where E is above and to the right of C, and cable C to D which is in line with the horizontal and D is to the right of C. Determine the tensions in cables AB, AC, CE and CD.

Answer: Cable AB = 171.4 lbf, cable AC = 174.8 lbf, cable CE = 106.7 lbf, cable CD = 132.3 lbf


Question: Two cylinders are placed in a 14 ft wide container. The cylinder that sits on the bottom of the container has a radius of 5 ft and weighs 3750 lbf. The second cylinder sits on top of the other cylinder and has a radius of 4.5 ft and weighs 3000 lbf. Assuming smooth contact surfaces determine the magnitude at the contact point between the two cylinders.

Answer: 3406 lbf


Question: Point A is located at (-6, -2m), point B is located at (8, -4m) and point C is located at (0, 6m). A force of magnitude 40 kN acts along the line passing through points A and B. Determine to moment of the force about point C, with CCW moments positive.

Answer: 350.7 kN.m


Question: A 16 in long control rod is connected to the ground at B and extends in a direction 100 degrees CCW from the positive x-axis to its other end at A. A force is applied at the end of the rod at A, in a direction of 45 degrees CW from the line of action of the rod. If the force creates a CW moment of 893.8 lbf.in about point B, calculate the magnitude of the force.

Answer: 79 lbf


Question: An L-shaped bracket ABC is pinned at A (0,0). Point B is located at (-12, 0cm) and point C is located at (-12, 6cm). A cable extends from point B to a support at point D (-15, 11cm). A force of 45 N is applied to the bracket at B, in a direction 50 degrees CW from the positive x-axis. Determine the moment of the force about A and the tension in the cable CD required for equilibrium.

Answer: MA = 413.7 N.cm, Cable CD = 57.43 N


Question: A rigid bent bar ABCD is pinned at B. The points are located at: A (0,0), B (1, 0ft), C (1, 1ft), D (4, 1ft). A force of 25 lbf is applied at A in a direction of 60 degrees CW from the positive x-axis. A force of 25 lbf is applied at C in a direction of 150 degrees CW from the positive x-axis. A force of 25 lbf is applied at D. Determine the direction (CW from the positive x-axis) of the force at D such that the sum of the moments about B are zero.

Answer: 128.35 degrees


Question: A solid rectangle ABCD has corners A (0, 0m), B (0, 4m), C (10, 4m) and D (10, 0m). Two force couples act on the rectangle. One couple consists of 100 N acting in the negative x-axis direction at A and 100 N acting in the positive x-axis direction at C. The other couple consists of 35 N acting at B in a direction 120 degrees CW from the positive x-axis and 35 N acting at D in a direction 60 degrees CCW from the positive x-axis. Determine the magnitude of the resulting moment, with positive in the CW direction.

Answer: 26.9 N.m


Question: An 8 m long cantilever beam has two vertical downwards forces applied, a force of 120 kN applied at the end of the cantilever and a force of 75 kN located at 5.2 m from the end of the cantilever. Determine the single equivalent resultant force and its location from the left end of the beam.

Answer: 195 kN located 6 m from the left end


Question: An L-bracket has its corner located at C (0, 0 in), its horizontal leg extends 5 in to end A in the positive x direction and its vertical leg extends 4 in to end B in the positive y direction. A vertical downwards force of 150 lbf is applied at C. A force of 160 lbf is applied at A in a direction 60 degrees CCW from the positive x-axis. Replace the forces at A and C with an equivalent force-couple system acting at point B (force angle CCW from the positive x-axis and moment in the CCW direction).

Answer: 80.8 lbf acting at -8.14 degrees, 1010 lbf.in


Question: A force couple systems acting at the origin (0, 0m) consists of a force of 90 N in a direction 20 degrees CCW from the positive x-axis and a couple of 369 N.m CW. Replace the force couple with a single force which has the same effect. Specify the magnitude and direction (CCW from the positive x-axis) of the resultant force, and the x coordinate of the point on the axis through which the line of action of the force passes.

Answer: 90 N at 20 degrees, passing through the x-axis at -12 m


Question: The system consists of a rigid linkage. All dimensions are in inches. Point B is a pin support located at the origin, B = (0, 0). From point B, a rigid member extends upward and to the left to point A. Point A is located 24 in to the left of B and 20 in above B, giving A = (−24, 20). A cable connects point A to point C. Point C is a pin support located directly above point B at a vertical distance of 24 in, so C = (0, 24). A CCW moment M is applied at point B. Given that the tension in cable AC is 10 lbf, determine the magnitude of moment M.

Answer: 236.7 lbf.in


Question: A 75 kg pole is being raised by pulling a cable. Point A is a pin support located on the ground. The member AB is inclined upward from A at an angle of 70 degrees CCW from the positive x-axis. At the end of the pole at point B a cable is attached in order to raise the pole, and the cable extends towards the left creating an angle between the cable and the member AB of 74 degrees. Determine the tension in the cable.

Answer: 130.9 N


Question: The left end of a 1 m cantilever beam OABC is fixed at point O, which is located at O = (0, 0). Point A lies on the beam 0.2 m to the right of O. Point B lies a further 0.3 m to the right of A. Point C is located 0.5 m to the right of B, at the right end of the beam. An upward vertical force acts at point A. At point C, a force is applied in the direction 140 degrees CW from the positive x-axis. A CW moment is applied at B. Determine the reactions at O when A = 100 N, B = 325 N.m and C = 110 N.

Answer: MO = 375.7 N.m CCW, Ox = 84.3 N, Oy = -29.3 N


Question: A 20 ft long beam ABC supports a frictionless pulley with radius 6 in at point B, located 13 ft from point A at the left end. Point A is a pin support. Point C is the right end of the beam, where the beam is supported by a roller on an inclined surface that makes an angle of 25 degrees with the horizontal. Point D lies on a vertical wall directly above A, and a cable runs from D through the pulley at B below which a weight of 150 lbf is supported. The cable BD is inclined such that it makes an angle of 65degrees with the vertical wall at point D. Determine the reactions at A and C.

Answer: Reaction C = 62.1 lbf at 115 degrees, Ax = 162.2 lbf, Ay = 30.3 lbf


Question: A rigid V-shaped two-member linkage is pinned at A. From point A, one rigid member extends upward and to the left at an angle of 20 degrees above the horizontal, terminating at point B. A vertical downward force acts at point B. From point A, a second rigid member extends upward and to the right at an angle of 30 degrees above the horizontal, terminating at point C. At point C, a force is applied downward and slightly to the right, perpendicular to member AC. Member AB is 22 in long and AC is 22 in long. Determine the maximum force that can be applied at B if the magnitude of the reaction at A is not to exceed 950 lbf, and the force C that corresponds to this equilibrium state.

Answer: B = 507 lbf, C = 476.5 lbf


Question: A car is parked on a 15 degree slope, with rear wheels locked by the parking brake. The vehicle dimensions (relative to when it is sitting horizontal) are: the wheelbase (distance between the front and rear wheels) is 2579 mm, the mass is 1291 kg, and the location of the center of gravity is 722 mm behind and 664 mm above the front wheel. Find the components of the forces acting on each of the front and rear wheels, parallel and perpendicular to the roadbed.

Answer: Front parallel = 0 N, front perpendicular = 3982 N, rear parallel = 1639 N, rear perpendicular = 2134 N


Question: A triangular plate ABC is secured by a pin at A and a roller at C. The interior angles of the triangle at the base are both 65 degrees. At the top point B a force is applied in the direction 60 degrees CCW from edge AB. Determine the reactions at the pin and the roller when the plate is subjected to a 500 lbf load at B.

Answer: C = 512.3 lbf, Ax = 286.8 lbf, Ay = 102.7 lbf


Question: A truss is pinned at the left side at A, and sits on a roller at the right side at D. The bottom horizontal chord AFED consists of members AF, FE and ED which are all 2 m in length. The top horizontal chord is BC, where B is located 2.6 m above F and C is located 2.6 m above E. The truss has other members AB, BF, BE, EC and CD. Loads in the positive x direction are located at B of 3 kN and C of 5 kN. Vertical downloads loads of 1 kN are applied at F and also at E. Determine the reaction at D.

Answer: D = 4.5 kN upwards


Question: A horizontal beam ABC is loaded with a force of 150 kN at B in the direction 30 degrees CW from the positive x-axis. Distance AB is 0.4 m and distance BC is 1.6 m. Left end A is pinned and right end C rests on a roller. Determine the magnitudes of the resultant reactions at A and C.

Answer: A = 143.1 kN, C = 15 kN


Question: A cantilever truss is connected to a wall with pinned joints at E and vertically above E at D. The horizontal bottom chord of the truss extends a distance of X m from E to C, then a further 2X m from C to A. The diagonal top chord extends right and downwards from D to B, then B to A. The angle between EC and internal member DC is 60 degrees. The angle between CA and internal member CB is 60 degrees. The angle between CA and BA is 30 degrees. Vertical downwards forces are 400 kN applied at A, 950 kN applied at B and 750 kN applied at C. Use the method of joints to find the forces in all members. Indicate tension as positive and compression as negative.

Answer: AB = 800 kN, AC = -693 kN, BC = -823 kN, BD = 1275 kN, CD = 1689 kN, CE = -1949 kN


Question: A triangle truss has horizontal bottom chord members AC and CD, and is pinned at A at the left and sits on a roller at D at the right. A vertical internal member extends vertically upwards from C to B. The angle between members AB and AC is 30 degrees. The angle between members CD and BD is 30 degrees. A vertical downwards force of 625 kN is applied at C. A force of 150 kN is applied at B in a direction 70 degrees CW from the positive x-axis. Determine the forces in all members. Indicate tension as positive and compression as negative.

Answer: BD = -796 kN, CD = 689 kN, AC = 689 kN, BC = 625 kN, AB = -736 kN


Question: A symmetric triangular truss is pinned at the left side at A. The horizontal bottom chord consists of members extending from A to C, from C to E, from E to G and from G to H. Members AC, CE, EG and GH are all 6 ft long. The truss sits on a roller at H. Joint B is located 5 ft vertically above C. Joint D is located 10 ft vertically above E. Joint F is located 5 ft vertically above G. The other members of the truss are AB, BD, DF, FH, CB, EB, ED, EF and GF. Vertical downwards loads are applied as 500 lbf at A, 500 lbf at H, 400 lbf at B, 400 lbf at D and 400 lbf at F. Using the method of joints, determine the forces in all members. Indicate tension as positive and compression as negative.

Answer: AB = -937 lbf, AC = 720 lbf, BC = 0, BD = -625 lbf, BE = -312 lbf, CE = 720 lbf, DE = 400 lbf, DF = -625 lbf, EF = -312 lbf, EG = 720 lbf, FG = 0, FH = -937.2 lbf, GH = 720 lbf


Question: A cantilever truss connects to a wall with pins at F and G, and F is 6.5 ft vertically above G. The top chord extends horizontally from F to D, from D to B and from B to A. Members FD, DB and BA are each 5 ft long. The diagonal bottom chord extends from G right and upwards towards A, including members GE where joint E is vertically below D, EC where joint C is vertically below B and CA. Other internal members are FE, DE, DC and BC. A force of 325 lbf is applied at A in a direction 100 degrees CW from the positive x-axis. Determine the forces in members DB, DC and EC. Indicate tension as positive and compression as negative.

Answer: DB = 682 lbf, DC = 0, EC = -805 lbf


Question: A triangular Howe truss has overall dimensions of 36 ft wide and 9 ft high. The truss is pinned at A at the left and sits on a roller at G at the right. The horizontal bottom chord consists of members from left to right of AH, HJ, JK, KL, LM, MG and each of these members is 6 ft long. Joint B is located vertically above H. Joint C is located vertically above J. Joint D is located 9 ft vertically above K. Joint E is located vertically above L. Joint F is located vertically above M. Top chords extend right and upwards from A towards D, and left and upwards from G to D, and include members AB, BC, CD, DE, EF and FG. Internal members are HB, JB, JC, KC, KD, KE, LE, LF and MF. Vertical downwards forces applied are 250 lbf at C, 100 lbf at H and 350 lbf at J. Determine the forces in members DE, KE and KL. Indicate tension as positive and compression as negative.

Answer: DE = -484 lbf, KE = 0, KL = 433 lbf


Question: A rectangular Pratt truss has overall dimensions of 16 m wide and 4.5 m high. The top and bottom chords are both horizontal. The truss is pinned at A at the left and sits on a roller at G at the right. The bottom chord consists of members from A to H, from H to J, from J to K and from K to G. Members AH, HJ, JK and KG are each 4 m long. Joint B is located 4.5 m vertically above A. The top chord consists of members B to C, C to D, D to E and E to F. Members BC, CD, DE and EF are each 4 m long. Vertical members are AB, HC, JD, KE and GF and are each 4.5 m long. Diagonal members are HB, JC, JE and KF. Vertical downwards loads are 300 N at H and 150 N at K. A horizontal load of 425 N is applied at F in the positive x-axis direction. Determine the forces in members DE, JE and JK. Indicate tension as positive and compression as negative.

Answer: DE = 12.5 N, JE = 210 N, JK = 273 N


Question: A trussed frame is connected to a wall at A and B. B (0, 0m) is a roller joint. A (0, 6m) is a pin joint. Joint F is located at (6, 6m) and joint C is located at (10, 10m). Truss members include AB, members from A right and upwards towards C and members from B right and upwards towards C. Joint D is at the midpoint between A and C, and creates two equal length members AD and DC. Joint E is at the midpoint between B and F and creates two equal length members BE and EF. Member FC connects F to C. Internal members are AE, AF and FD. A cable connects to the wall 4 m vertically above A and runs through a frictionless pulley at C, and there is a mass suspended vertically from the cable below C. Determine the maximum mass that may be supported by the cable without exceeding 60 kN in tension or 85 kN in compression in any member of the frame.

Answer: 6120 kg


Question: An A-frame consists of three members. Member AC is pinned at A (0, 0m) and connected to member BC at C (2, 4m) with a pin joint. Member BC is supported by a roller at B (4, 0m). Member DF is 1.5m long and is connected to AC at D (1.5, 3m) and BC at E (2.5, 3m) with pin joints. Member DF cantilevers out from E a distance of 0.5m to free end F. A vertical downwards load of 110 N is applied at F. Determine the planar components of the forces acting on member AC at joints A, D and C.

Answer: Ax = 0 N, Ay = 27.5 N, Dx = 82.5 N, Dy = 55 N, Cx = -82.5 N, Cy = -82.5 N


Question: A frame consists of two members AB and BC. The left member AB is pinned at A (0, 0m). The right member is pinned at C (8, 2m). The two members are pin connected at B (3, 4m). A vertical downwards force of 250 N is applied at B. Determine the scalar components of the reactions at pins A and C.

Answer: Ax = 144.2 N, Ay = 192.3 N, Cx = -144.2 N, Cy = 57.7 N


Question: A frame consists of two members AB and BC. The left member AB is pinned at A (0, 0ft). The right member is pinned at C (8, 3ft). The two members are pin connected at B (5, 3ft). A horizontal force of 175 lbf in the positive x direction is applied at the midpoint of AB. Determine the scalar components of the reactions at pins A and C.

Answer: Ax = -118.8 lbf, Ay = -18.8 lbf, Cx = -56.3 lbf, Cy = 18.8 lbf


Question: A frame consists of two members AB and BC. The left member AB is pinned at A (0, 0ft). The right member is pinned at C (8, 0ft). The two members are pin connected at B (4, 5ft). A horizontal force of 475 lbf in the negative x direction is applied at the midpoint of AB. A CCW moment of 800 lb.ft is applied at the midpoint of BC. Determine the scalar components of the reactions at pins A and C.

Answer: Ax = 436.3 lbf, Ay = 248.4 lbf, Cx = 38.8 lbf, Cy = -248.4 lbf


Question: A frame consists of two members AB and BC. The left member AB is pinned at A (0, 0m). The right member is pinned at C (10, -1m). The two members are pin connected at B (5, 3m). A horizontal force of 75 N in the negative x direction is applied at the midpoint of AB. A CW moment of 100 N.m is applied at the midpoint of BC. Determine the scalar components of the reactions at pins A and C.

Answer: Ax = 44.6 N, Ay = 4.29 N, Cx = 30.4 N, Cy = -4.29 N


Question: Given the function (y = -1.5.x^2 + 2.x + 2.5), determine the area under the curve from the y-axis to the point where the curve crosses the positive x-axis. The x- and y- coordinates are in inches.

Answer: 5.03 in^2


Question: Use integration with horizontal strips to determine the area of the spandrel bounded by the function (y = h - k.x^0.5) and the lines y = h and x = b, and also find the coordinates of its centroid. Note: b and h are constants. Give your answers for the area and the centroid in terms of b and h.

Answer: Area = (b.h / 1.5), centroid x coordinate = (1.5.b / 2.5), centroid y coordinate = (2.5.h / 4)


Question: Determine the coordinates of the centroid of the shaded area bounded by the y-axis, a parabola passing through the origin (0, 0m) and point A, and a straight line passing through points A and B. Point A is located at (9, 3m) and B is located at (0, 9m).

Answer: (3.2, 3.7m)


Question: A 16 ft long beam is supported by a pin at A located 1 ft from the left end, and a roller at B located 2 ft from the right end. A downwards vertical uniformly distributed load of 20 lbf/ft is applied between 7 ft from the left end and 4 ft from the right end. The beam weight is negligible. Determine the reactions at A and B.

Answer: Ax = 0 lbf, Ay = 34.6 lbf, By = 65.4 lbf


Question: A 15 m long beam is supported by a pin at the left end A (0, 0m) and a roller at the right end B (15, 0m). Point C is located at (1, 0m) and D is located at (12, 0m). A downwards vertical distributed load is applied to the beam and varies linearly between 45 kN/m at C to 60 kN/m at D. The beam self weight is 15 kN/m. Determine the reactions at A and B.

Answer: Ax = 0 kN, Ay = 429.7 kN, B = 372.8 kN


Question: A 20 m long beam is supported by a pin at the left end A (0, 0m) and a roller at the right end B (20, 0m). Point C is located at (6, 0m), D is located at (12, 0m) and E is located at (19, 0m). A downwards vertical distributed load is applied to the beam and varies linearly between 0 kN/m at C to 60 kN/m at D, then remains constant at 60 kN/m to E. The beam weight is negligible. Determine the reactions at A and B.

Answer: Ax = 0 kN, Ay = 184.5 kN, By = 415.5 kN


Question: An 18 m long beam supports a vertical downwards load that varies uniformly from 50 N/m at the left end to 150 N/m at the right end. The beam is pinned at A (0, 0m) and sits on a roller at B (10, 0m). Determine the internal shear and bending moment at a section passing through point C (6, 0m).

Answer: V = -490 N, M = -1640 N.m


Question: A beam BC is 4 ft long and is pinned at the right end C (4, 0ft). At the left end B (0, 0ft) the beam is pin connected to member AB. A is located to the left and upwards from B, creating an angle of 25 degrees with the x-axis. Beam BC supports a vertical downwards uniformly distributed load of 15 lbf/ft. Determine the internal forces at point D located at the midlength of BC.

Answer: V = 0 lbf, N = 64.3 lbf, M = 30 lbf.ft


Question: A 12 m long beam is pinned at the left end A and has a roller at the right end B. Vertical downwards forces are 500 kN at 8 m from A, and 650 kN at 10 m from A. Find the shear and bending moment equations as function of x for the three ranges: 0m < x < 8m, 8m < x < 10m, 10m < x < 12m.

Answer: (0m < x < 8m: V = 275 kN, M = 275.x kN.m), (8m < x < 10m: V = -225 kN, M = 275.x - 500.(x - 8)), (10m < x < 12m: V = -875 kN, M = 275.x - 500.(x - 8) - 650.(x - 10))


Question: A 20 m long beam is pinned at the left end A and has a roller at the right end B. Vertical downwards forces are 165 kN at 2 m from A, and 15 kN at 9 m from A. Find the shear and bending moment diagrams, expressed as a series of points at critical locations (x distance from the left end, value).

Answer:
V (m, kN): (0, 0), (0, 157), (2, 157), (2, -8.3), (9, -8.3), (9, -23.3), (20, -23.3), (20, 0)
M (m, kN.m): (0, 0), (2, 314), (9, 256), (0, 0)


Question: A 6 ft long cantilever beam is fixed at the left end A. Vertical downwards forces are 140 lbf at 3 m from A, and 240 lbf at 6 m from A. A CCW moment of 2340 lbf.ft is applied at 4 m from A. Find the shear and bending moment diagrams, expressed as a series of points at critical locations (x distance from the left end, value).

Answer:
V (ft, lbf): (0, 0), (0, 380), (3, 380), (3, 240), (6, 240), (6, 0)
M (ft, lbf.ft): (0, 0), (0, 480), (3, 1620), (4, 1860), (4, -480), (6, 0)


Question: A 20 m long beam is pinned at the left end A and has a roller at the right end B. Vertical downwards uniformly distributed loads are: 200 kN/m from A to 9 m from A, and 180 kN/m from 15 m from A to 20 m from A. Find the shear and bending moment diagrams, expressed as a series of points at critical locations (x distance from the left end, value).

Answer:
V (m, kN): (0, 0), (0, 1508), (9, -293), (15, -293), (20, -1192), (20, 0)
M (m, kN.m): (0, 0), (7.5, 5681), (9, 5468), (15, 3713), (20, 0)


Question: A 20 m long beam is pinned at the left end A and has a roller at the right end B. Vertical downwards loads are 400 kN at 2 m from A and a uniformly distributed load of 300 kN/m from 12 m from A to 17 m from A. Find the shear and bending moment diagrams, expressed as a series of points at critical locations (x distance from the left end, value).

Answer:
V (m, kN): (0, 0), (0, 773), (2, 773), (2, 373), (12, 373), (17, -1128), (20, -1128), (20, 0)
M (m, kN.m): (0, 0), (2, 1545), (12, 5270), (13.2, 5501), (17, 3383), (20, 0)


Question: A 18 m long beam is pinned at the left end A and has a roller at the right end B. Vertical downwards linearly varying distributed loads are 0 kN/m at A varying linearly to 25 kN/m at 9 m from A, then varying linearly to zero at B. Find the shear and bending moment diagrams, expressed as a series of points at critical locations (x distance from the left end, value).

Answer:
V (m, kN): (0, 0), (0, 113), (9, 0), (18, -113), (18, 0)
M (m, kN.m): (0, 0), (9, 675), (18, 0)


Question: A 20 m long beam is pinned at the left end A and has a roller at the right end B. The beam is subjected to a vertical downwards symmetric distributed parabolic loading w(x), given by the equation w(x) = -2.x^2 + 40.x kN/m. Find the maximum values of shear and moment and their locations in the beam.

Answer: V = 1333 kN at x = 0 m and x = 20 m, M = 8333 kN.m at x = 10 m


Question: A 20 ft long beam is pinned at the left end A and has a roller at the right end B. The beam is subjected to a vertical downwards distributed parabolic loading w(x), given by the equation w(x) = - 0.007.x^3 - 0.03.x^2 + 3.x lbf/ft. Find the bending moment function in terms of x.

Answer: M = (x^4)/400 + (x^5)/2857 - (x^3)/2 + 124.x


Question: A 220 kN horizontal force P is applied to a box with weight 200kN resting on a surface with an incline of 20 degrees from the horizontal. The line of action of P passes through the center of gravity of the box. The box is 4 m wide and 3 m tall, and the static and kinetic coefficients of friction between the box and the surface are 0.7 and 0.57, respectively. Determine the magnitude of the friction force acting on the box.

Answer: 138.3 kN


Question: A block weighs 500 N and is resting on an incline of 30 degrees from the horizontal. The coefficient of friction between the block and the inclined surface is 0.4 . A force P is applied to the box. The line of action of P passes through the center of gravity of the box and is inclined 65 degrees from the horizontal. Determine the maximum value of P before the block starts to move up the incline (i.e. impending upwards motion), and the minimum value of P required to prevent the block from moving down the incline (i.e. impending downward motion).

Answer: Maximum = 404 N, minimum = 130 N


Question: A block with mass 10 kg rests on a horizontal surface. A block with mass 35 kg rests on top of the 10 kg block, and is restrained from moving horizontally. The coefficient of friction between all surfaces is 0.2. A horizontal force P is applied to the 10 kg block with a line of action through its center of gravity. Determine the smallest force P required to start the 10 kg block moving.

Answer: 157 N


Question: A 2D object is defined by 5 vertices in the x-y space with straight lines between them. The vertices are: (-1, 0cm), (-7, 0cm), (0, 7cm), (1, 7cm), (1, 6cm). Calculate the moment of inertia and radius of gyration with respect to the x-axis.

Answer: Ix = 278.4 cm^4, kx = 3.3 cm


Question: A square has vertices in the x-y space of (0, 6in), (-4, 6in), (-4, 10in) and (0, 10in). Calculate the moments of inertia of the object about both the x- and y-axes.

Answer: Ix = 1045 in^4, Iy = 85.3 in^4


Question: A 2D object is defined by 6 vertices in the x-y space with straight lines between them. The vertices are: (0,0ft), (0, 9ft), (1, 9ft), (1, 3ft), (4, 3ft), (4, 0ft). Calculate the moments of inertia of the object about both the x- and y-axes.

Answer: Ix = 270 ft^4, Iy = 66 ft^4


Question: A circle has a diameter of 8 in and is positioned in the x-y space such that its centre is located at (5, 4in). Determine the moment of inertia and radius of gyration about the y-axis.

Answer: Iy = 1458 in^4, ky = 5.4 in


Question: A rectangle has vertices in the x-y space of (0, 0cm), (0, 8cm), (10, 8cm), (10, 0cm). Inside this rectangle is a rectangular void with vertices of (1, 6cm), (1, 7cm), (9, 7cm), (9, 6cm). Determine the moment of inertia of the composite shape about the horizontal neutral axis of the composite shape.

Answer: 370.4 cm^4


Question: A 2D object is defined by 8 vertices in the x-y space with straight lines between them. The vertices are: (0, 2ft), (7, 2ft), (7, 8ft), (8, 8ft), (8, -8ft), (7, -8ft), (7, -2ft), (0, -2ft). Calculate the moment of inertia of the object about the vertical neutral axis of the object.

Answer: 278.6 ft^4


Question: Two steel channel sections are placed back to back to each other with a distance of 120 mm separating them. They are placed upright such that their large dimension (depth) is in line with the vertical y-axis. A built up beam is constructed by welding two plates to these channels, symmetrically placed, one on the top and one on the bottom. The plates are both 300 mm wide and 35 mm thick. The channel sections have properties: Area = 4740 mm^2, depth = 305 mm, flange width = 77.5 mm, Ixx = 59900000 mm^4, Iyy = 1850000 mm^4, and the horizontal distance to the section centroid is 17.1 mm. Determine the moment of inertia and radius of gyration of the built up beam with respect to the x- and y-axes.

Answer: Ix = 728800000 mm^4, kx = 154.6 mm, Iy = 217600000 mm^4, ky = 84.5 mm


Question: A pedestal has a triangular cross section. If it is subjected to a compressive force P of 500 lb, specify the x and y coordinates for the location of point P(x, y), where the load must be applied on the cross section, so that the average normal stress is uniform. Compute the stress. Vertices of the triangular cross section: Vertex 1: (0 in., 0 in.), Vertex 2: (0 in., 9 in.), Vertex 3: (12 in., 3 in.)

Answer: x = 4in, y = 4in, stress = 9.26psi


Question: A metal rod specimen failed in a tension test at an angle of 52 degrees (to the longitudinal axis) when the axial load was 19.80 kip. If the diameter of the specimen is 0.5 in., determine the average normal and average shear stress acting on the area of the inclined failure plane. Also, what is the average normal stress acting on the cross section when failure occurs?

Answer: Average normal stress on cross section: 100.9 ksi, Average normal stress on inclined failure plane: 62.7 ksi, Average shear stress on inclined failure plane: 48.9 ksi


Question: Rods AB and BC have diameters of 4 mm and 6 mm, respectively. The geometry: Point A: Pin support on left, Point B: Ring connection where both rods meet, Point C: Pin support on right, Rod AB: Horizontal, connects A to B, Rod BC: Inclined at theta = 60 degrees above horizontal, connects B to C. If a vertical downwards load of 8 kN is applied to the ring at B, determine the average normal stress in each rod if theta = 60 degrees.

Answer: AB = 368MPa, BC = 327MPa


Question: The bars of the truss each have a cross-sectional area of 1.25 in squared. Geometry: Joint A: (0 ft, 0 ft), Joint B: (4 ft, 3 ft), Joint C: (8 ft, 3 ft), Joint E: (4 ft, 0 ft), Joint D: (8 ft, 0 ft), Joint C: Pinned to wall, Joint D: Pinned to wall. Members: AB, AE, ED, EB, BC, BD. Force P = 8 kip acting downward at joint A, Force 0.75P = 6 kip acting downward at joint E. Determine the average normal stress in each member due to the loading P = 8 kip. State whether the stress is tensile (T) or compressive (C).

Answer: AB=10.7ksi T, AE=8.53ksi C, ED=8.53ksi C, EB=4.8ksi T, BC=23.5ksi T, BD=18.7ksi C


Question: A beam and hanger assembly is used to support a distributed loading of w = 0.8 kip/ft. Beam: Pinned to the wall at C (0, 0ft) and extends horizontal for 6ft. Hanger rod AB: Pinned to the wall at B (0, 3ft) and connected to the beam with a bolt at A (4, 0ft). Uniformly distributed load w = 0.8 kip/ft acting downward over the 6 ft beam length. Determine the average shear stress in the 0.40 in. diameter bolt at A and the average tensile stress in rod AB, which has a diameter of 0.5 in. If the yield shear stress for the bolt is 25 ksi, and the yield tensile stress for the rod is 38 ksi, determine the factor of safety with respect to yielding in each case.

Answer: Bolt: shear stress=23.9ksi, FOS=1.05. Rod: stress=30.6ksi, FOS=1.24


Question: A rod BC is made of steel having an allowable tensile stress of 155 MPa and supports the left end of beam AB. Joint A: (4.5, 0 m) right end, pinned support, Joint B: (0, 0 m) left end, connection point on beam for rod BC, Joint C: Above point B, upper connection of rod to support. Distributed downwards load increases from zero at B to w = 15 kN/m at (3, 0m) then decreases to zero at A. Determine its smallest diameter so that it can support the load.

Answer: 11.1 mm


Question: A thin strip of rubber has an unstretched length of 15 in. If it is stretched around a pipe having an outer diameter of 5 in., determine the average normal strain in the strip.

Answer: 0.0472 in/in


Question: A bar CA has length L m and is supported at its right end A by a wire AB. At the left end the bar is attached to a wall by a pin at C. The wire is attached to the wall at B which is L m vertically above C. When the wire AB is unstretched it makes an angle of theta = 45 degrees below the horizontal at B. If a load is applied to the bar AC which causes the angle to change to theta = 47 degrees, determine the normal strain in the wire.

Answer: 0.0343 m/m


Question: Two wires are connected together at A with a ring. Wire CA: Connects C to A, length = 300 mm, oriented at 30 degrees above horizontal (measured from horizontal at A). Wire BA: Connects B to A, length = 300 mm, oriented at 30 degrees below horizontal (measured from horizontal at A). B and C are connections to a wall and C is vertically above B. A horizontal force P is applied at A. If the force P causes point A to be displaced horizontally 2 mm away from the wall, determine the normal strain developed in each wire.

Answer: Strain in both wires = 0.0058 mm/mm


Question: A right-angled triangular plate ABC has vertices: A (0, 0mm), B (0, 800mm), C (800, 0mm). Edge BC is fixed, and its apex A is given a displacement of 5 mm in the direction perpendicular to edge BC. Determine the shear strain at A.

Answer: 0.0088 rad


Question: A rectangular plastic block is glued at its top and bottom to rigid plates. Point A: Bottom-left corner (0, 0). Point D: Bottom-right corner. Point B: Top-left corner (0, 2in). Point C: Top-right corner. A horizontal force P is applied to the top plate in the positive x direction, and causes the material to deform so that its sides are described by the equation y = 3.56x^(1/4). Determine the shear strain in the material at its corners A and B.

Answer: A = 0, B = 0.199 rad


Question: The stress - strain diagram for an elastic fibre has coordinates: (0 in/in, 0 psi), (2 in/in, 11 psi), (2.25 in/in, 55 psi). Determine the modulus of elasticity and estimate the modulus of toughness and modulus of resilience.

Answer: modulus of elasticity = 5.5psi, modulus of toughness = 19.25psi, modulus of resilience = 11 psi


Question: The stress-strain (s-e) diagram for a bone is described by the equation e = 0.45(10^-6)s + 0.36(10^-12)s^3, where s is in kPa. Determine the modulus of toughness and the amount of elongation of a 200 mm long region just before it fractures if failure occurs at e = 0.12 mm/mm.

Answer: 613MPa, 24mm


Question: A specimen is originally 1 ft long, has a diameter of 0.5 in., and is subjected to a force of 500 lb. When the force is increased to 1800 lb, the specimen elongates 0.9 in. Determine the modulus of elasticity for the material if it remains elastic.

Answer: 88.3ksi


Question: A bar BC has length 10 ft and is supported at its right end B by a wire AB with diameter 0.2 in made of A-36 steel. At the left end the bar is attached to a wall by a pin at C. The wire is attached to the wall at A which is vertically above C. Wire AB makes an angle of 30 degrees above the horizontal at B. A downwards uniformly distributed load is applied to the full length of the rod BC. Determine the magnitude of the distributed load if the end B is displaced by 0.75 in downwards.

Answer: 0.246kip/ft


Question: An aluminum block has a rectangular cross section and is subjected to an axial compressive force of 8 kip. If the 1.5 in. side changed its length to 1.500132 in., determine Poisson's ratio and the new length of the 2 in. side. The modulus of elasticity is 10(10^3) ksi.

Answer: Poissons ratio = 0.33, side length = 2.00018 in


Question: An 8 mm diameter brass rod has a modulus of elasticity of 100 GPa. If it is 3 m long and subjected to an axial load of 2 kN, determine its elongation. What is its elongation under the same load if its diameter is 6 mm?

Answer: For 8mm diameter = 1.19mm, for 6mm diameter = 2.12m


Question: An assembly consists of a steel rod CB of length 3 m and an aluminum rod BA of length 2 m, each having a diameter of 12 mm. The rod is fixed at C and free at A and is subjected to axial loadings at A and at B: 6 kN acting at B in the negative x-direction, 18 kN acting at end A in the positive x-direction. Determine the displacement at the free end A. Modulus of elasticity (steel) is 200 GPa and Modulus of elasticity (aluminum) is 70 GPa.

Answer: 6.14mm


Question: A truss is made of three A-36 steel members AB, BC and CA, each having a cross-sectional area of 400 mm squared. Joint A: (0 m, 0 m), Joint C: (1.4 m, 0 m), Joint B: (0.8 m, 0.8 m). Joint A is pinned and joint C is a roller. Forces: Force P acting downward at joint B, force 5 kN acting horizontally to the right at joint B. Determine the magnitude P required to displace the roller to the right 0.2 mm.

Answer: 21.7kN


Question: A linkage is made of three pin-connected A-36 steel members DA, CA and AB, each having a cross-sectional area of 0.730 in squared. Joint D: (0 ft, 0 ft), Joint C: (6 ft, 0 ft), Joint A: (3 ft, -4 ft), Joint B: (3 ft, -10 ft). Joints C and D are pinned. A vertical force of P = 50 kip is applied to the end B of member AB. Determine the vertical displacement of point B.

Answer: 0.281in


Question: A 500 mm long steel pipe is filled with concrete and subjected to a compressive force of 80 kN. The steel pipe outer diameter is 80 mm and inner diameter is 70 mm. Modulus of elasticity (steel) is 200 GPa, Modulus of elasticity (concrete) is 24 GPa. Determine the stress in the concrete and the steel due to this loading.

Answer: steel stress = 48.8MPa, concrete stress = 5.85MPa


Question: A 10 mm diameter steel bolt is surrounded by a bronze sleeve. The outer diameter of this sleeve is 20 mm, and its inner diameter is 10 mm. The assembly is subjected to a compressive load P. The yield stress for the steel is 640 MPa and the bronze is 520 MPa. Modulus of elasticity (steel) is 200 GPa, Modulus of elasticity (bronze) is 100 GPa. Determine the magnitude of the largest elastic load P that can be applied to the assembly.

Answer: 126kN


Question: A tapered member is fixed connected at its ends A and B and is subjected to a downwards load P. The member is 2 in thick and 60 in long. The height tapers linearly from 6 in at A to 3 in at B. Determine the distance from A of the load position and its greatest magnitude, if the allowable normal stress for the material is 4 ksi.

Answer: 28.9in, 60.4kip


Question: A post is made from 6061-T6 aluminum and has a diameter of 50 mm and length 0.5 m. It is fixed supported at A and B and at its center C there is a coiled spring with stiffness 200 MN/m attached to a rigid collar. If the spring is originally uncompressed, determine the compression of the spring when a load of P = 50 kN is applied to the collar at C.

Answer: 0.039mm


Question: A thermo gate consists of two 6061-T6 aluminum plates that have a width of 15 mm and are fixed supported at their ends. One plate is 600 mm length, the other is 400 mm length, and both plates have thickness 10 mm. If the gap between them is 1.5 mm when the temperature is 25 degrees C, determine the temperature required to just close the gap. The coefficient of thermal expansion is 24(10^-6)/degrees C.

Answer: 87.5 degrees C


Question: A member is to be made from a steel plate that is 0.25 in thick. If a 1 in hole is drilled through its center, determine the approximate width of the plate so that it can support an axial force of 3350 lb. The allowable stress is 22 ksi.

Answer: 2.49in


Question: A 0.25 in diameter steel rivet having a temperature of 1500 degrees F is secured between two plates such that at this temperature it is 2 in long and exerts a clamping force of 250 lb between the plates. Determine the approximate clamping force between the plates when the rivet cools to 70 degrees F. Assume that the heads of the rivet and the plates are rigid and use a coefficient of thermal expansion of 8(10^-6)/degrees F, and modulus of elasticity of 29(10^3) ksi.

Answer: 16.5kip


Question: A copper pipe has outer diameter 40 mm and inner diameter 37 mm. It is fixed at a wall at A and the axis is the longitudinal axis, positive away from the wall. Three torques are applied, sequentially moving away from the wall: -30 Nm, 20 Nm then -80 Nm. Determine the maximum shear stress in the pipe.

Answer: 26.7MPa


Question: A steel tube having an outer diameter of 2.5 in. is used to transmit 350 hp when turning at 27 rev/min. Determine the inner diameter of the tube if the allowable shear stress is 10 ksi.

Answer: 2.48in


Question: Two steel 20 mm diameter steel shafts are connected using a brass coupling. If the yield point for the steel is 100 MPa and for the brass is 250 MPa, determine the required outer diameter of the coupling so that the steel and brass begin to yield at the same time when the assembly is subjected to a torque T. Assume that the coupling has an inner diameter of 20 mm.

Answer: 21.9mm


Question: The drive shaft of an automobile is made of a steel having an allowable shear stress of 8 ksi. If the outer diameter of the shaft is 2.5 in. and the engine delivers 200 hp to the shaft when it is turning at 1140 rev/min, determine the minimum required thickness of the shaft's wall.

Answer: 0.174in


Question: A 0.75 in. diameter shaft for an electric motor develops 0.5 hp and runs at 1740 rev/min. Determine the torque produced and compute the maximum shear stress in the shaft. The shaft is 6 in long and supported by ball bearings at its ends.

Answer: 1.51lbft, 219psi


Question: The propellers of a ship are connected to an A-36 steel shaft that is 60 m long and has an outer diameter of 340 mm and inner diameter of 260 mm. If the power output is 4.5 MW when the shaft rotates at 20 rad/s, determine the maximum torsional stress in the shaft and its angle of twist.

Answer: 44.3MPa, 0.2085rad


Question: An 8 mm diameter A-36 bolt is screwed tightly into a block at A with a wrench. The wrench is a rigid bar that extends 150 mm each side, such that the force couple is separated by 300 mm. The wrench connects to the top of the bolt which is located 80 mm above the block at A. Determine the couple forces F that should be applied to the wrench so that the maximum shear stress in the bolt becomes 18 MPa. Also, compute the corresponding displacement of each force F needed to cause this stress.

Answer: 6.03N, 0.72mm


Question: A turbine and shaft is arranged as follows: Point B: Turbine connection (x = 0), Point C: First gear location (x = 3 m), Point D: Second gear location (x = 7 m), Point E: Journal bearing (x = 9 m). The turbine develops 150 kW of power, which is transmitted to the gears such that C receives 70 percent and D receives 30 percent. If the rotation of the 100 mm diameter A-36 steel shaft is 800 rev/min., determine the absolute maximum shear stress in the shaft and the angle of twist of end E of the shaft relative to B. The journal bearing at E allows the shaft to turn freely about its axis.

Answer: 9.12MPa, 0.0102rad


Question: An A-36 bolt is tightened within a hole such that the reactive torque on the shank AB can be expressed by the equation t = (kx squared) N.m/m, where x is in meters. If a torque of 50 N.m is applied to the bolt head, determine the constant k and the amount of twist in the 50 mm length of the shank. Assume the shank has a constant radius of 4 mm.

Answer: 1.2(10^6)N/m^2, 0.0622rad


Question: A beam AB has the following features: Point A: Left bearing support (x = 0), Downwards load of 800 N at x = 125 mm, Downwards load of 800 N at x = 725 mm, Point B: Right bearing support (x = 800 mm). The bearings at A and B exert only vertical reactions on the shaft. Express the shear and moment in the shaft as a function of x within the region 125 mm < x < 725 mm.

Answer: V = 15.6N, M = 15.6x + 100 Nm


Question: A solid rectangular member has dimensions of 12 in and 6 in and is to be used to resist an internal bending moment of 2 kip.ft. Determine the maximum stress in the member if the moment is applied (a) about the x axis, (b) about the y axis. The larger dimension is aligned with the y axis.

Answer: Stress x = 167psi, stress y = 333psi


Question: A steel I-section has flanges 6 in wide and a web 6 in deep, both 0.25 in thick, such that the total depth is 6.5 in. The allowable stress of the steel is 24 ksi. Determine the largest moment the section can resist if the moment is applied (a) about the x axis, (b) about the y axis. The web is aligned with the y axis and the section is symmetric about the x and y axes.

Answer: Mx = 20.8 kip.ft, My = 6 kip.ft


Question: An unequal flange I-section has a top flange 5 in wide, bottom flange 3 in wide and web 8 in deep. All elements are 1 in thick, such that the section is 10 in deep. The web is aligned with the y axis and the section is symmetric about the y axis. The section is subjected to a moment of 15 kip.ft. Determine the resultant forces from the bending stresses in the top and bottom flanges.

Answer: Force top = 17.7kip, Force bottom = 13.7kip


Question: A beam AB has the following features: Point A: Left bearing support (x = 0), Downwards load of 800 N at x = 125 mm, Downwards load of 800 N at x = 725 mm, Point B: Right bearing support (x = 800 mm). The bearings at A and B exert only vertical reactions on the shaft. The cross-section is solid circular with a diameter of 50 mm. Determine the maximum stress in the beam.

Answer: 9.05MPa


Question: A 4.5 m beam rests on supports that are 1.5 m apart, and has 1.5 m overhangs on both sides of these supports. Loads are P acting downward at left end (x = 0) and P acting downward at right end (x = 4.5 m). The beam has a rectangular cross section 150 mm wide and 250 mm deep. Determine the largest load P that can be supported on its overhanging ends so that the bending stress does not exceed 10 MPa.

Answer: P = 10.4kN


Question: A wood beam is reinforced with bonded steel plates at its top and bottom, with dimensions: wood core 200 mm wide and 300 mm deep, steel plates both 200 mm wide and 20 mm thick. Determine the maximum bending stress developed in the wood and steel if the beam is subjected to a bending moment of M = 5 kN.m. Modulus of elasticity (wood) is 11 GPa, Modulus of elasticity (steel) is 200 GPa.

Answer: stress wood = 3.26MPa, stress steel = 3.7MPa


Question: A T-section has a top flange 8 in wide and a web 10 in deep, both elements are 2 in thick, such that the total depth is 12 in. Determine the plastic section modulus and the shape factor about the neutral axis parallel to the flange.

Answer: modulus = 114 in^3, shape factor = 1.78


Question: A 20 ft long beam is subjected to a downwards uniformly distributed load W over its full length. It is made of an elastic-plastic material with a yield stress of 30 ksi, and has a solid square cross-section that is 8 in x 8 in. Determine the load W which causes the maximum moment to be (a) the largest elastic moment (b) the largest plastic moment.

Answer: elastic = 4.27kip/ft, plastic = 6.4kip/ft


Question: A 16 ft long reinforced concrete beam is pinned at the left and rests on a roller at the mid-length. A downwards uniformly distributed load W acts over its full length. The cross-section is rectangular, 10 in wide and 20 in deep. Two 0.75 in diameter steel reinforcing bars are located 2.5 in below the top surface. Determine the maximum uniformly distributed load W that can be supported by the beam if the allowable tensile stress for the steel is 28 ksi, and the allowable compressive stress for the concrete is 3 ksi. Assume the concrete cannot support a tensile stress. Modulus of elasticity (steel) is 29(10^3) ksi, Modulus of elasticity (concrete) is 3.6(10^3) ksi.

Answer: 1.03kip/ft


Question: A T section has a flange 14 in wide and 3 in thick, and a web 6 in deep and 6 in thick. The overall depth is 9 in and the section is symmetric about the vertical axis. The section is subjected to a vertical shear of 10 kip, determine the maximum shear stress in the beam. Also, compute the shear-stress jump at the flange-web junction AB.

Answer: Max stress = 276 psi, stress jump = 156 psi


Question: A T section has a flange 14 in wide and 3 in thick, and a web 6 in deep and 6 in thick. The overall depth is 9 in and the section is symmetric about the vertical axis. The section is subjected to a vertical shear of 10 kip, determine the vertical shear force resisted by the flange.

Answer: 3.05kip


Question: A cruciform section is 350 mm wide and 350 mm deep and the elements are all 50 mm thick. The section is symmetric about both horizontal and vertical axes. The section is subjected to a vertical shear of 130 kN. Determine the resultant shear force developed in the vertical segment element.

Answer: 50.3kN


Question: A solid steel rod has a radius of 1.25 in. If it is subjected to a shear force of 5 kip determine the maximum shear stress.

Answer: 1.36ksi


Question: A horizontal cantilever beam is 800 mm long and fixed at the left end A. Point B is 250 mm from A, point C is 500 mm from A, point D is 800 mm from A at the free end. Downwards loads are 2 kN at C and 4 kN at D. The cross-section is an inverted T section, with the flange at the bottom. The web is 70 mm deep, the flange is 50 mm wide, both elements are 20 mm thick, and the total height is 90 mm. Determine the maximum shear stress acting in the section at point B.

Answer: 4.85MPa


Question: An unequal flange I-section has a top flange 10 in wide, bottom flange 6 in wide and web 12 in deep. The web is 1 in thick and the flanges are 1.5 in thick, such that the section is 15 in deep. The web is aligned with the y axis and the section is symmetric about the y axis. The cross-section is fabricated from 3 timber boards that are nailed together. The vertical shear force is 5 kip and each nail can support a shear force of 500 lb. Determine the spacing of the nails used to hold the top and bottom flanges to the web.

Answer: top spacing = 1.42in, bottom spacing is 1.69in


Question: A thin-walled rectangular hollow section is 100 mm wide and 200 mm deep. There is a slit along its side located at the mid-height of the right web. Determine the horizontal location of the shear centre as the horizontal distance from the centreline of the left web.

Answer: 70mm


Question: A spherical gas tank has an inner radius of 1.5 m. If it is subjected to an internal pressure of 300 kPa, determine its required thickness if the maximum normal stress is not to exceed 12 MPa.

Answer: 18.8mm


Question: A gas pipe line is supported every 20 ft by concrete piers and also lays on the ground. If there are rigid retainers at the piers that hold the pipe fixed, determine the longitudinal and hoop stress in the pipe if the temperature rises 60 degrees F from the temperature at which it was installed. The gas within the pipe is at a pressure of 600 lb/in^2. The pipe has an inner diameter of 20 in. and thickness of 0.25 in. The material is A-36 steel, use a coefficient of thermal expansion of 6.6(10^-6)/degrees F, and modulus of elasticity of 29(10^3) ksi.

Answer: longitudinal = 11.5ksi, hoop = 24ksi


Question: A plate with negligible weight is fixed at its top, is 200 mm wide, 800 mm deep and 10 mm thick. A force is applied at the bottom of the plate at the mid-thickness. Determine the shortest distance to the edge of the plate at which the force can be applied so that it produces no compressive stress at a section 300 mm from the top support.

Answer: 66.7mm


Question: A steel I-section has flanges 4 in wide and a web 6 in deep, both 0.5 in thick, such that the total depth is 7 in. The web is aligned with the y axis and the section is symmetric about the x and y axes. The section is the cross-section of a simply supported 16 ft long beam, with downwards loads located from the left end A of: 500 lb 2 ft from A, 2500 lb 6 ft from A and 3000 lb 10 ft from A. Determine the stress components at a point 4 ft from A, located at the centre width and the top surface of the top flange.

Answer: longitudinal stress = -9.41 ksi (C), shear stress = 0ksi


Question: A bar having a square cross section of 30 mm by 30 mm is 2 m long and is held upward. If it has a mass of 5 kg/m, determine the largest angle measured from the vertical, at which it can be supported before it is subjected to a tensile stress along its axis near the grip.

Answer: 0.286 degrees


Question: A rectangular road sign is fixed to a 6 m tall post, and has horizontal dimension 2 m (along the y axis) and vertical dimension 1 m (z axis). The sign is attached to the post along one of its vertical edges, such that it cantilevers out from the top 1 m of the post. The sign is subjected to a horizontal uniform wind loading of 1.5 kPa (in the x direction). The circular post has a diameter of 100 mm. The x and y axes are defined such that the origin is at the centre of the circular cross-section, and the y axis aligns with the y axis of the sign. Determine the stress components in the post a distance of 2 m above the ground, at the section locations where the section crosses the x and y axes.

Answer: stress in MPa: +x-axis (50, 0) normal stress=106.9 (T), shear stress=15.28; -x axis (-50, 0) normal stress=-106.9 (C), shear stress=15.28; +y axis (0, 50) normal stress=0, shear stress=14.77; -y axis (0, -50) normal stress=0, shear stress=14.77


Question: An element is subjected to a normal y stress of 300 psi compression and a shear stress of 950 psi (positive upwards on the right side). Determine the equivalent state of stress if the element is rotated 30 degrees clockwise.

Answer: s_x' = -897.7 psi (compression), s_y' = 597.7 psi (tension), t_x'y' = 604.9 psi


Question: The state of stress on an element is: Normal stress s_y = 60 MPa (compressive), Normal stress s_x = 45 MPa (tensile), Shear stress t_xy = 30 MPa (positive acting upward on right face). Determine the principal stresses and the maximum in-plane shear stress and their orientations.

Answer: s_1 = 53MPa, s_2 = -68MPa, orientation_1 = 14.9 degrees, orientation_2 = -75.1 degrees, t_max-in-plane = 60.5MPa, orientation = -30.1 degrees


Question: A plane a-a is horizontal and a plane b-b interests it at an angle of 60 degrees clockwise from plane a-a. The stresses along planes a-a and b-b are 45MPa to the right and 25MPa upwards and leftwards, respectively. Determine the normal stress on plane b-b and the principal stresses.

Answer: s_b = 121MPa, s_1 = 126MPa, s_2 = -16.1MPa


Question: A wooden block will fail if the shear stress acting along the grain is 550 psi. The grain is oriented 58 degrees CCW from the positive x axis. If the normal stress s_x = 400 psi, determine the necessary compressive stress s_y that will cause failure.

Answer: -824 psi


Question: A rod has a circular cross section with a diameter of 2 in. It is subjected to a torque of 12 kip.in. and a bending moment M. The greater principal stress at the point of maximum flexural stress is 15 ksi. Determine the magnitude of the bending moment.

Answer: 8.73kip.in


Question: A solid rod with radius 25 mm is fixed at the wall and subjected to a torque about the longitudinal axis positive away from the wall of +45 Nm, a moment of 300 Nm causing compression to the top surface of the rod, and a downward force at the free end of 800 N. Determine the principle stresses acting at a point at the top surface a distance of 450 mm from the free end.

Answer: s_1 = 5.5MPa, s_2 = -0.611MPa


Question: A cylindrical pressure vessel has an inner radius of 1.25 m and a wall thickness of 15 mm. It is made from steel plates that are welded along the 45 degrees seam. Determine the normal and shear stress components along this seam if the vessel is subjected to an internal pressure of 8 MPa.

Answer: normal = 500MPa, shear = 167MPa


Question: A differential element on a bracket is subjected to plane strain that has the following components: e_x = 150(10^-6), e_y = 200(10^-6), gamma_xy = -700(10^-6). Determine the equivalent in-plane strains on an element oriented at an angle of theta = 60 degrees counterclockwise from the original position.

Answer: e_x = -116(10^-6), e_y = 466(10^-6), gamma_xy = 393(10^-6)


Question: The state of strain at a point on a wrench has components e_x = 120(10^-6), e_y = -180(10^-6), gamma_xy = 150(10^-6). Determine the in-plane principal strains and the maximum in-plane shear strain and average normal strain.

Answer: e_1 = 138(10^-6), e_2 = -198(10^-6), gamma_max = 335(10^-6), e_ave = -30(10^-6)


Question: A steel bar has dimensions 15 in x 2 in x 0.5 in and is subjected to a 500 lb tensile load in the direction of the 15 in dimension. Determine the maximum shear strain. E = 29(10^3) ksi, Poisson's ratio is 0.3.

Answer: 22.4(10^-6)


Question: A 45 degrees strain rosette is mounted on a machine element. The following readings are obtained from each gauge: e_upwards = 650(10^-6), e_diagonal = -300(10^-6), e_left = 480(10^-6). Determine the in-plane principal strains and the maximum in-plane shear strain and associated average normal strain.

Answer: e_1 = 1434(10^-6), e_2 = -304(10^-6), gamma_max = 1738(10^-6), e_ave = 565(10^-6)


Question: A bar of copper alloy is loaded in a tension machine and it is determined that e_x = 940(10^-6) and sigma_x = 14 ksi, sigma_y = 0, sigma_z = 0. Determine the modulus of elasticity and the dilatation of the copper. Poisson's ratio = 0.35.

Answer: E = 14.9(10^3)ksi, dilation = 0.282(10^-3)


Question: A thin-walled cylindrical pressure vessel is 3 m long and has an inner radius of 0.5 m and a thickness of 10 mm, and is filled with air having an internal gauge pressure of 15 MPa. Determine the strains in the circumferential and longitudinal directions, and the increase in both the diameter and the length. E = 200 GPa, Poisson's ratio = 0.3.

Answer: e_circum = 3.188(10^-3), e_long = 0.75(10^-3), diameter increase = 3.19mm, length increase = 2.25mm


Question: An aluminium alloy with yield stress 37 ksi is to be used for a solid shaft that transmits 40 hp at 2400 rev/min. Using a factor of safety of 2 with respect to yielding, determine the smallest diameter shaft that can be used based on maximum shear stress theory.

Answer: 0.833 in


Question: A continuous beam is made of timber that has an allowable bending stress of 6.5 MPa and an allowable shear stress of 500 kPa. The beam is 8 m long and rests on supports that are 4 m apart, with 2 m long overhangs at both ends. Determine its cross section dimensions if it is to be rectangular and have a height-to-width ratio of 1.25.

Answer: 211mm wide and 264mm deep


Question: A 4 m long steel cantilever beam is fixed at the left end and has downwards loads each of magnitude P that are located 2 m from A and 4 m from A. The cross-section is a T section with dimensions: flange width 150 mm, web depth 150 mm, both elements are 15 mm thick, such that the total height is 165 mm. Determine the maximum loads P that can be safely supported on the beam if the allowable bending stress is 170 MPa and the allowable shear stress is 95 MPa.

Answer: 2.9kN


Question: A steel I-section has flanges with unknown width and a web 7 in deep, both 0.5 in thick, such that the total depth is 8 in. The web is aligned with the y axis and the section is symmetric about the x and y axes. The section is used as a 20 ft long simply supported beam with loads of 6 kip acting downward at 6 ft from left end and 8 kip acting downward at 14 ft from left end. If the maximum bending stress is not to exceed 22 ksi, determine the required width of the flanges.

Answer: 5.86in


Question: Two pulleys of radius 0.5 ft are attached to an 8 ft long shaft and loaded. The ends of the shaft are supported by bearings that exert only horizontal and vertical forces on the shaft. The dimensions are: Point A: x = 0 (left bearing), Point D: x = 1 ft (first pulley), Point C: x = 6 ft (second pulley), Point B: x = 8 ft (right bearing). The loads on each side of the pulley at D are downwards 150 lb and 250 lb. The loads on each side of the pulley at C are 150 lb and 250 lb horizontal and perpendicular to the longitudinal axis of the shaft. Determine the required diameter of the shaft if the allowable shear stress is 12 ksi.

Answer: 1.46in


Question: A 500 mm long tubular shaft has an inner diameter of 15 mm. The shaft is supported by bearings that are 200 mm apart, such that the shaft has 150 mm overhangs from the bearings on each side. The bearings exert only horizontal and vertical forces on the shaft. Both shaft ends have a gear attached with radius 100 mm. The load on the gear at one end is 500 N horizontal and perpendicular to the longitudinal axis of the shaft, and the other end has a load on the gear of 500 N downwards. Determine the outer diameter if the allowable shear stress is = 70 MPa.

Answer: 20.8mm


Question: A simply supported beam is 12 ft long and has a downwards uniformly distributed load of 80 lb/ft over the central 8 ft of the beam. Determine the maximum deflection of the beam as a function of the constant EI.

Answer: 18.8/EI kip.ft^3 downwards


Question: A 60 in long shaft has dimensions: A (x = 0) left end, B (x = 12 in), C (x = 36 in) and D (x = 60 in) right end. The shaft is supported by bearings at B and D. The bearings exert only horizontal and vertical forces on the shaft. Downwards loads are 50 lb at A and 80 lb at C. Determine the slope of the shaft at A as a function of the constant EI.

Answer: -1920/EI lb.in^2


Question: A 6 m long cantilever beam is fixed at the left end and has dimensions: A (x = 0) fixed end, B (x = 3 m), C (x = 4.5 m) and D (x = 6 m) free end. Downwards loads are: a uniformly distributed load of 2 kN/m from A to B, 4 kN at C and 6 kN at D. The beam is rectangular 200 mm wide and 400 mm deep, and made of timber with E = 12 GPa. Determine the deflection at the free end D.

Answer: 51.7mm downwards


Question: A 16 ft long simply supported beam has downwards uniformly distributed load of 6 kip/ft over the left half (8 ft) and a CCW moment of 5 kip.ft at the right end. The beam is a steel section with I = 82.8 in^4 and E = 29,000 ksi. Determine the deflection at the centre of the beam.

Answer: 1.9in


Question: A 200 mm long cantilever beam is fixed at the left end and supported by a spring at the right end. The spring has a stiffness of k = 2 N/mm. The beam is rectangular 5 mm wide and 10 mm deep. A downwards load of 50 N is applied at the right end. The steel is A-36 and E is 29,000 ksi. Determine the deflection at the right end.

Answer: 1.5mm


Question: A beam has length (a + L) and has dimensions: D (x = 0) left end, A (x = a), C (x = a + L/2) and B (x = a + L). The beam is simply supported at B and rests on a roller at A. Downwards loads of magnitude P are located at both D and C. Determine the value of a (as a proportion of L) such that the deflection at C is equal to zero. EI is constant. Use the moment-area theorems.

Answer: a = L/3


Question: A 8 m long continuous beam has supports: the left end rests on a roller (A), the centre of the beam rests on a roller (B) and the beam is pinned at the right end (C). A downwards uniformly distributed load of 8 kN/m is applied over the full length. Use discontinuity functions to determine the reactions at the supports. EI is constant.

Answer: Ay = 12kN, By = 40kN, Cy = 12kN


Question: A 4 m long shaft is supported by 3 bearings at A (x = 0), B (x = 2) and C (x = 4). A downwards force of 200 N is applied at the centre of segment AB (x = 1). The bearings exert only vertical reactions on the shaft. Determine these reactions using the moment-area theorems.

Answer: Ay = 81.3N, By = 138N, Cy = -18.8N


Question: An 18 in long solid circular rod from A-36 steel has pinned ends and is subjected to an axial compressive load of 4 kip. Determine the smallest diameter of the rod such that it does not buckle.

Answer: 0.551in


Question: A W12 x 87 structural A-36 steel column has a length of 12 ft. If its bottom end is fixed supported while its top is free, and it is subjected to an axial load of P = 380 kip, determine the factor of safety with respect to buckling. Section Properties (W12 x 87): A = 25.6 in^2, I_x = 740 in^4, I_y = 241 in^4 (controls)

Answer: 2.19


Question: An A-36 steel pipe has an outer diameter of 2 in. and a thickness of 0.5 in. The pipe has a height of 12 ft and it is held in place by a guywire. The guywire connects from the top of the pipe to a ground anchor, where the horizontal distance to the guywire anchor is 5 ft. Assume that the ends of the pipe are pin connected. Determine the largest horizontal force P that can be applied to the top of the pipe without causing the pipe to buckle.

Answer: 4.23kip


Question: A truss is made from A-36 steel bars, each of which has a solid circular cross section. The joints are located at: Joint A: (0, 0 ft), Joint B: (8, 0 ft), Joint D: (8, 3 ft), Joint C: (4, 3 ft). Members are: AB, AC, BC, CD. Joints B and D are pinned. A downwards force of 10 kip is applied at A. Determine the diameter of member AB that will prevent this member from buckling. The members are pin supported at their ends.

Answer: 1.72in


Question: A steel bar AB of a frame is pin-connected at its ends. Joints are C (0,0 m), A (3, 4 m), B (3, -2 m). Supports B and C are pinned, and members AC and AB are pinned at A. A horizontal force of 18 kN is applied at A. The bar AB is rectangular 50 mm wide and 100 mm deep, with properties of E_st = 200 GPa, yield stress = 360 MPa. Determine the factor of safety with respect to buckling.

Answer: 2.38


Question: A steel W8 x 15 column has a height of 8 ft and is fixed at the base and free at the top. The cross-section is oriented such that the web of the section is aligned with the y axis. An axial compressive force is applied at the top, located on the cross section 1 in along the y axis below the section centroid. Determine the maximum load that can be applied. A = 4.44 in^2, I_x = 48.0 in^4, I_y = 3.41 in^4, r_x = 3.29 in., depth = 8.11 in.

Answer: 26.5 kip


Question: A solid aluminium rod with diameter 200 mm is fixed at its base, free at the top, with a height of L m. An axial compressive force of 200 kN is applied eccentrically to the centroid of the cross section by 5 mm. Determine the greatest height L of the rod such that it does not buckle. E = 72 GPa, yield stress = 410 MPa.

Answer: 8.34m


Question: An A-36 steel shaft with radius 30 mm is subjected to two torques of 4 kN.m and 3 kN.m separated by 0.5 m and acting in opposite directions. The shaft extends 0.5 m away from each torque, such that the rod has total length 1.5 m. Determine the torsional strain energy.

Answer: 26.2 J


Question: An A-36 steel beam is simply supported and 10 m long. A downwards uniformly distributed load of 1.5 kN/m is applied to the full length, and an axial tension force of 15 kN is applied at the end with a roller. A = 2300 mm^2, I = 9.5(10^6) mm^4. Determine the total strain energy.

Answer: 496 J


Question: A solid circular concrete column has a height of 5 ft and radius of 12 in. The column contains six 1 in. diameter steel reinforcing rods. If the column supports a load of 300 kip, determine the strain energy in the column. E_st = 29(10^3) ksi, E_c = 3.6(10^3) ksi.

Answer: 0.129 ft.kip


Question: An A-36 steel 25 ft long cantilever beam is fixed at its left end. A downwards load of 8 kip is applied at 15 ft from the fixed end. Determine the displacement beneath the load using work and energy methods. I = 250 in^4.

Answer: 2.15in


Question: A steel cable having a diameter of 0.4 in. wraps over a drum and is used to lower an elevator having a weight of 800 lb. The elevator is 150 ft below the drum and is descending at the constant rate of 2 ft/s when the drum suddenly stops. Determine the maximum stress developed in the cable when this occurs. E = 29(10^3) ksi, yield stress = 50 ksi.

Answer: 20.3ksi


Question: A steel chisel has a diameter of 0.5 in. and a length of 10 in. It is struck by a hammer that weighs 3 lb, and at the instant of impact it is moving at 12 ft/s. Determine the maximum compressive stress in the chisel, assuming that 80% of the impacting energy goes into the chisel. E = 29(10^3) ksi, yield stress = 100 ksi.

Answer: 43.6MPa


Question: An A-36 steel bolt is required to absorb the energy of a 2 kg mass that falls 30 mm. If the bolt has a diameter of 4 mm, determine its required length so the stress in the bolt does not exceed 150 MPa.

Answer: 850mm