A compression bar, having a square cross section of side length b, must support a compressive load of magnitude W, uniformly distributed over the bar cross section. The bar is made of two segments connected by a glued joint along plane pq, which is at an angle α to the vertical

Question

Consider the rotated coordinate frame (x1,y1) defined, as indicated in the figure, with the coordinate axis y1 normal to the plane of the joint.

Remember that, in the stress-transformation relationships, the angle θ from the x-axis to the x1-axis is defined positive counterclock-wise so, for this problem, θ=−(π2−α). Also keep in mind the positive convention for σy1 and τx1y1

Use the stress-transformation relationships for the plane stress state in the x−y plane to obtain symbolic expressions for the normal and shear stress components acting on the joint, σy1 and τx1y1, respectively. Express your answer in terms of W, b, and α.

In your symbolic expression, type α as "alpha".

σy1= ?

τx1y1= ?

Now take the magnitude of the compressive load to be W=1000N and the angle of the joint to be α=35∘.

The bar is constructed of a polymer for which the maximum allowable magnitude of stress in compression is σmax=1.1MPa. Also, the joint will fail under excessive levels of shear stress: take the maximum allowed magnitude for the shear stress along the joint to be τmax=0.5MPa.

If you want to avoid both failure of the polymer in compression and failure of the joint in shear, determine the numerical value, in units of cm, to the third decimal digit, of the minimum side length b of the bar:

bmin= ?