Determine (analytically) the transformed state of stress on an element at the sa

Question

Determine (analytically) the transformed state of stress on an element at the same point oriented 40 degree clockwise with respect to the element shown. Sketch the results on the rotated element. (ii) Determine (analytically) principal stresses and orientation of principal plane. Sketch the results on rotated element. (iii) Determine (analytically) maximum in-plane shear stresses and orientation of maximum shear plane. Sketch the results on rotated element. (iv) Verify the results on Mohr's Circle.

Explanation / Answer

solution:

1) here if given point is rotated by a=-40 cw then stresses in element is given by

sx'=sx+sy/2+(sx-sy/2)cos2a+txysin2a=120+80/2+40/2cos80-40sin80=64.08 mpa

sy'=sx+sy/2-(sx-sy/2)cos2a-txysin2a=135.91 mpa

txy'=-(sx-sy/2)sin2a+txycos2a=26.64 mpa

2) here principle stresses in rotated plane is given by

s1,2sx+sy/2+-((sx-sy/2)^2+txy^2)^.5

on putting value we get

s1=144.72 mpa

s2=55.28 mpa

where principle stress same for any position for same element

principle plane angle=tan2*ap=2txy'/sx'-sy'=2*26.64/64.08-135.91

ap=-36.56 degree and it is vary with position

4) maximum shear stress is

txymax=((sx-sy/2)^2+txy^2)^.5=44.72 mpa

priciple shear plane angle=tan2*as=-(sx'-sy'/2txy')

as=8.44 degree

where as=ap(+-)45

5) this result can be evaluated over mohr cicle of radius

r=s1-s2/2

and plotting all value over s abd txy axis

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