# Chapter5 - stress solution for the same infinite plate...

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AE 321 – Practice Problems Chapter 5: Problem Formulation 1. Find all boundary conditions on all surfaces of the hollow circular cylinder (inner radius a, outer radius b) shown below in both (i) Cartesian coordinates and (ii) cylindrical coordinates, where P , P 0 and P i are normal tractions and q a shear traction. Note: the loading shown on the cross sectional figure on the right hand side acts over the entire length L and over the entire circumference. 2. Assume that in addition to the internal body force density f i there exists an internal body moment density w i ( x 1 ,x 2 ,x 3 ) such that the moment about the origin caused by this was ! M w = w i ! e i dV V ! In this situation how will the equations of motion and the symmetry of the stress tensor be affected and why? P L P i P o q Loading on cylindrical surfaces

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3. Given the solution to the problem of an infinite (i.e. very large) plate with a hole subjected to far field tension illustrated below on the left, how would you calculate the
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Unformatted text preview: stress solution for the same infinite plate subjected to the shear loading q shown on the right? 4. A linear elastic rod is subjected to a shear traction σ rz ( θ ) on the lateral boundary that is independent of coordinate z. Also σ r θ = σ rr =0 on that boundary. The two ends are traction free. (a) State the boundary conditions of this problem in cylindrical coordinates . (b) Assume that the solution to such a problem is independent of z (i.e. ∂ / ∂ z=0) and that u=v=0 and w=w(x,y). Simplify the field equations in Cartesian coordinates . State your answer in terms of the function w. Assume that there are no body forces. Do not solve for w in the resulting equations. Note: Such a problem is called anti-plane shear and forms the counter part to plane strain/stress problems, i.e. a shear traction in the axial direction applied on the lateral surface. ! ! q q x y z...
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