University of MN Aerospace structures questions

1) Find the displacement function 𝑢𝑧 (𝑧) by solving the one dimensional equilibrium equation and applying boundary conditions. 𝜌, 𝑓 = 𝑔𝜌𝐴 𝑙 (𝑙𝑒𝑛𝑔𝑡ℎ) 𝐴, 𝑔 𝐸 z 2) The displacement field for a deformed elastic body is : 𝑢𝑥 (𝑥, 𝑦, 𝑧) = 𝑠𝑦 𝑢𝑦 (𝑥, 𝑦, 𝑧) = 𝑠𝑥 𝑢𝑧 (𝑥, 𝑦, 𝑧) = 0 a) Compute individualized components of the symmetrical strain tensor. i.e. compute 𝑒𝑥𝑥 (𝑥, 𝑦, 𝑧), 𝑒𝑥𝑦 (𝑥, 𝑦, 𝑧), 𝑒𝑥𝑧 (𝑥, 𝑦, 𝑧), 𝑒𝑥𝑦 (𝑥, 𝑦, 𝑧), 𝑒𝑦𝑦 (𝑥, 𝑦, 𝑧), 𝑒𝑦𝑧 (𝑥, 𝑦, 𝑧), 𝑒𝑥𝑧 (𝑥, 𝑦, 𝑧), 𝑒𝑦𝑧 (𝑥, 𝑦, 𝑧), 𝑒𝑧𝑧 (𝑥, 𝑦, 𝑧) 𝑒𝑥𝑥 𝑒 [𝑒𝑥𝑦 𝑒𝑥𝑧 𝑒𝑥𝑦 𝑒𝑦𝑦 𝑒𝑦𝑧 𝑒𝑥𝑧 𝑒𝑦𝑧 ] 𝑒𝑧𝑧 b) Compute the total change of the angles (i.e. the engineering shear stress). Give your answer in terms of s. (90 degree angles, rotated 45 degrees) y D A B C x