ENGR 214 Recitation Exercise, April 24, 2000 (axial bars, torsion and beam bending)
1. Consider the axial bar shown below:
Bar 1 and 2 have the same E and L. Area of bar 1 is A and bar 2 has area of 2A (i.e., twice that bar 1). An axial load of F is applied at points A and B.
a) Solve for the axial force in each bar in terms of applied force F and given bar properties. Show all steps.
b) Determine the axial displacement at points A and B.
c) Determine the axial stress in each bar and indicate whether tension or compression.
2. The horizontal bar is rigid (does not deform), is pinned at its left end, and is pushed upward by the 1,000 lbf force at point C. Bar 1 is made of structural steel and has a cross-sectional area of 0.5 sq. in. Bar 2 is made of aluminum and has a cross-section area of 0.3 sq. in.
a) Determine the force in each bar.
b) Determine the vertical displacement of point C.
c) Determine the axial stress and strain in each bar.
3. Consider a solid cylindrical bar as shown
The diameter of the left bar is D and diameter of the right bar is D/2 (half of left bar). Torques of T=5,000 in-lb are applied at points A and B. Assume the material has a shear modulus G=11.5 million psi.
a) Determine the angle of twist at points A and B in terms of the parameter D.
b) Determine the value of D so that the angle of twist at B is equal to 1 degree.
c) Determine the smallest value of D so that the shear stress does not exceed 30,000 psi in either bar.
4. Same as problem 3 except a distributed torque of 500 in-lb/in replaces the torque at point A as shown below.
The diameter of the left bar is D and diameter of the right bar is D/2 (half of left bar). Assume the material has a shear modulus G=11.5 million psi.
a) Determine the angle of twist at points A and B in terms of the parameter D.
b) Determine the value of D so that the angle of twist at B is equal to 1 degree.
c) Determine the smallest value of D so that the shear stress does not exceed 30,000 psi in either bar.
5. Consider a simply supported beam with uniform distributed normal loading p (force/length) as shown below.
a) Determine the shear and moment equations V(x) and M(x).
b) Sketch (accurately) the shear and moment diagrams. Indicate the values of V and M at points A, B, C and D on the diagrams.
c) Determine the magnitude and location (relative to point A) of the maximum moment.
d) Determine the maximum tensile bending stress and its location.
e) Determine the maximum shear stress and its location.
6. Consider a cantilever beam of length L which has a uniform distributed loaded of p over a distance of “a” as shown below.
Assume L=20 in, a=10 in, b=0.4 in, h=1 in, E=10 million psi, and p=20 lb/in.
a) Determine the equation for the transverse deflection assuming x from the left end.
b) Evaluate the deflection at the end of the beam (x=L).
c) Determine the normal stress at the top surface of the beam at x=5 in. Is it tensile or compressive stress?
d) Where is the normal stress a maximum? What is the magnitude?
e) Determine the maximum shear stress and its location.