A 1651 times 30 CHS manufactured by Smorgon Steel Tube Mills

A 165.1 times 3.0 CHS manufactured by Smorgon Steel Tube Mills is built-in at the base and is experiencing a variety of loads. It experiences 2 applied twisting moments and an unknown axial tension. The torques are applied at mid-height and at the end, while the tension is applied at the free end. At the support, there is a point on the outer surface of the CHS which is under consideration. The Smorgon Steel Tube Mills product catalogue gives the following geometric properties for this section A = 1530 mm^2, I = 5.02 times 10^6 mm^4, Z = 60.8 times 10^3 mm^3, J = 10.0 times 10^6 mm^4. Draw the BMD, TMD and AFD and determine the internal actions that apply at the cross-section at the support. At the point under consideration, determine the stress due to the torsion. Is this a normal stress or a shear stress? Use von Mises\' yield criterion and given that f_y = 400 MPa, determine the maximum value of P that will cause yielding due to the combined stresses caused by P and the applied torques. Draw a diagram that shows the stresses acting at this point, draw the Mohr\'s circle, and establish the principal stresses for this point based on the value of P calculated in (c). How would the answers to (b) and (c) above change if the point under consideration was on the inside surface of the CHS? Explain the answer.

Solution

Using index notation prove the following identities among vectors A,B,C,D a- (AXB)X(CXD)=[A.(CXD)]B-[B.(CXD]A