1. A thick-walled cylinder with 0.12 m internal diameter and 0.20 rn external diameter is fabricated of a material whose elastic limit is 350 MPa and Poisson's ratio v = 0.28. The cylinder is subjected to an internal pressure five times greater than the external pressure. Calculate the allowable internal pressure according to:
a) the maximum shear stress theory, and
b) the energy of distortion theory
2. A two-dimensional strain field is given by a„ -c(- I 8x2±42y2) Ey -c(6x2-30y2) y„y =6bxy where h and c are nonzero constants.
a) What is the relationship between b and c for this field to satisfy the strain compatibility conditions?
b) Determine the displacements u(x,y) and v(x,y) corresponding to this field of strain.
3. A thin square plate of one meter by one meter is subjected to a state of plane stress represented by uniform normal stresses cr„ and ay as illustrated below. All other stresses are zero. The two stresses shown cause the plate to elongate by 0.4 mm in the x direction and by 0.1 mm in the y direction. If it is known that ay is equal to 100 MPa and E is equal to 80 GPa and that all deformations are in the linear-elastic range, determine
a) σx
b) the Poisson's ratio for the material from which the square is made, and
c) the strain in the thickness direction (z-direction).
4. Using an energy method, determine the displacement of point 13 of the beam shown below. Take E = 200 GPa, I = 185x106 mm4.
5. Determine the magnitude and direction (up or down) of the force P applied at point A of the beam below if the displacement at A was not to exceed 2 mm (down). Take E = 200 GPa, I = 750x106=n4, L=4 m.
6. A bar of solid circular cress-section of 50 mm diameter is subjected to a torque, T, and an axial tensile load, P. A rectangular strain gauge rosette attached to the surface of the bar gives the following strain readings: Eco =800 l0-6, EA5 = -200x10-6 and &9.0 = -400x10-6 with the 0 degree gauge being aligned with the axial direction of the bar. If Young's modulus, E, for the bar is 80 GPa and Poisson's ratio, v, is 0.28, calculate the values of T-and P.
7. A pin-ended column of height 2_0 m has a circular cross-section of diameter 60 nim, wall thickness 2.0 mm and is converted to an open section by a narrow longitudinal slit; the ends of the column are free to warp. If the column is made from materials with E =50 GPa and G = 1.5 GPa, determine the values of axial load which would cause the column to buek[e
a) pure bending mode
b) pure torsion mode
8. Calculate the forces in the members FO, GD and CD of the truss shown in the figure below using the principle of virtual work, All horizontal and vertical members are ltn long.
9. The steel compression strut BC of the frame ABC in the figure below is of tubular cross section with an outer diameter of 55 mm and a wall thick:ness of 7 nun.
a) Determine the factor of safely against elastic buckling of BC if the 50,000 N load shown below is applied at the mid point between A and B. Let E = 200 GPa and σyielding. =320 MPa,
b) What is the wall thickness that BC can have if the buckling safety factor was 1.5?
10. The state of plane stress shown below is defined by the following stresses:
σx =210 MPa σy= 70 MPa and τxy = -120 MPa
a) Show this state of stress on a properly constructed Mohr's circle.
b) Will the above stress condition cause yielding according to the maximum shear stress theory? Assume σyielding = 290 MPa.
c) Determine σx. and τxy, on an element rotated 60 degrees clockwise from the x-axis.
11. A three element rosette is mounted on a thin steel specimen with a Young's modulus of 190 GPa and a Poisson's ratio of 0.3. The rosette provides the following readings along the 0, 60 and 120 degree directions respectively:
ε0=300 μ ε60=1500 μ ε120=600 μ
a) From these readings, calculate the strains εx' , εy and γxy, along the +45 degree direction.
b) Determine the principal strains ε1 and ε2 and the principal directions.
c) Using the generalized Hooke's law, calculate σ, , σy and τxy
12. A force P = 30 KN is applied at joint B of the four-member structure below, at a 450 angle from the horizontal line. Each member has a cross section area A = 100 mm2 and a modulus of elasticity E = 75 GPa. Use an energy method of your choice to determine the member forces Fl to F4 and the corresponding stre8ses and strains, A331.1MC linear elastic behaviour of the members.
13. The figure below shows a steel ring of 250 mm mean radius and a uniform rectangular section of 85 mm wide and 20 mm thick. A rigid bar is fitted horizontally as shown. Assuming an allowable stress of 230 MPa, determine the maximum tensile force P that can be carried by the ring.
14. A 7000 N force is applied horizontally at joint B of the three-element, pin-joined truss shown below. Cross section area for all members is 8.0 cm2 and modulus is E = 80 GP a. Determine the horizontal displacement u and the vertical displacement v at joint B
15. The rods 1, 2, and 3 shown below are welded together, mounted between two rigid walls and subjected to the two forces shown at points B and C. The rods are all of the same length, namely L = 1 in. Rods 1 and 3 are made from a material with E = 70)(109 Pa. Rod 2 is made from a material with E 120x109 Pa. The cross sections are given by: A1= A3 = 40x101 mm2' and A2 = 80x103 mm2. Determine the displacements of points B and C.
16. An aluminum alloy bar of solid square cross-section (creidi,E 40 ksi) is subjected to a compressive axial force of magnitude P = 48 x 103 lb and a torque T = 13 x 103 lb.in as shown in the figure below. This member is to be designed in accordance with the maximum-shear-stress criterion of failure, with a safety factor of 2.
a. What is the minimum allowable cross-sectional dimension b? provide your answer to the nearest 0,1 in.
b. What would your answer be if the Von-Mises stress criterion is used.
17. Under a given load, the 2 rn by 1 m by 1.5 rn parallelepiped shown below is deformed by movement of corner point A to a new location A' with coordinates (1,9975, 0.9991, 1.4988). If the displacement field is given by: u =c ixyz v c2xyz w = c3xyz
a. Determine Ex, ey, Txy, y, and yyz
b. Evaluate the normal strain in the direction of line AB
c. Calculate the shear strain for perpendicular lines AB and AC.
18. A thick-walled cylinder with 0.25 m internal diameter and 0.45 m external diameter is fabricated of a material whose elastic limit is 325 M13a. Let v = 030.
a) Determine for an external pressure po = 0 the maximum internal pressure to which the cylinder may be subjected without exceeding the elastic limit,
b) Determine for an internal pressure pi -= 0 the maximum external pressure to which the cylinder may be subj ected without exceeding the elastic limit.
19. A triaxial state of stress is schematically shown below (only one component of each stress is shown).
a) Show this state of stress on a properly constructed Mohr circle
b) Determine the principal stresses corresponding to this state of stress
c) Determine the maximum shear stress.
20. Two uniform linearly elastic rods are welded together at B, and the resulting two-segment rod is attached to rigid supports at A and C. Rod (1) has a modulus E1 = 220 MPa, cross-sectional area A1= 5 cm2, length L1 .= 150 cm., and coefficient of thermal expansion a1= 9 x 10-b1°C. Rod (2) has a modulus E2 = 120 E13a, cross-sectional area A2 = 8 cm2, length L2 = 110 cm, and coefficient, of thermal expansion a2 =15 x 10-6PC.
Determine the axial stresses in the rods if the temperature of both is raised by 40 °C. h) Determine whether joint B moves to the right or left and by how much?
21. The beam cross section shown below has a variable wall thickness as shown and is subjected to a constant vertical shear force of 2200 N and a torque of 500 N,nt
a) Determine and plot the flexural shear flow in the two flanges and the web.
b) Locate the shear centre of the beam.
22. Using Castigliano's second theorem, determine the slope at the point of support C and the displacement of point B of the simply supported beam shown below. Take.E = 210 GPat I= 250x106 mm4, L 2.5 m
23. The steel compression strut BC of the frame ABC in the figure below is a steel tube with an outer diameter of 55 mm and a wall thickness of 6 ram. Determine the factor of safety against elastic buckling of BC if a distributed load is applied as shown below. Let E = 205 GPa and ayklding 6r 350 NIPa.
24. The thin-walled open section shown below is subjected to a vertical upward force of 55000 N acting through the shear center.
a) Find the shear flow distribution in the thin walls of the section. All of the walls have a thickness of 8 mm. All ,the dimensions shown are in mm and are to the medians of the flanges and webs.
b) Locate the shear center relative to the vertical web.
