Chemical Engineering - Mechanics of Solids and Structures II

Full exam

Test on 18/07/2018 Student:______________________________________________________ Id. Number:______________ Please note that: correct solution of exercise 1 is mandatory to pass the exam. Exercise 1 Consider the structural scheme depicted in Figure 1(a) for a very high distillation column. The structure is subject to the uniform distributed horizontal load q, which schematizes the wind forces, applied on the continuous vertical beam ABC. The axial and bending stiffness of the cross sections are denoted as EA and EI, respectively. By neglecting the shear deformability of the structural elements and assuming that EA = EI/L 2, 1. sketch the diagrams of the internal forces (axial force, shear force, bending moment) in the structure. Figure 1. Exercise 2 Consider the cross section depicted in Figure 2. The section is subjected to: (a) a shear force V = 1 kN (defining with the horizontal axis an angle equal to 30°) applied to the centroid G, and (b) to a bending moment M = 0.5 kNm. 1. Evaluate the position of the centroid G. 2. Evaluate the position of the shear center C 3. Disregarding the effect of the torsional moment ensuing from the shear force V being applied in G, evaluate and sketch the distributions of tangential and normal stresses along the beam cross section centerline. Figure 2 Exercise 3 By applying Prandtl’s membrane analogy to a shaft with a rectangular cross section of sides b and a (with b>>a) derive the expression for the shear stresses due to torsion.