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Assume that we have a tapered beam with rectangular cross-section, according to Figure 1. The beam material has a constant modulus of elasticity (Young's modulus), E and is assumed to be linear elastic. The height of the cross-section is varying linearly from x = 0 to x = L according to the function H(x) = (10L – 9x). The beam has a constant thickness, t, and a fixed-end at point A. The beam is (101 - loaded with a constant point load P in the axial direction at point B. Assume in the following, for simplicity, that this can be regarded as a one-dimensional problem, except in problem 3 where it regarded as a 2-dimensional problem. L/3 L/3 L/3 с Section C-C 10h |H(x) ht X,u 2H C H(x=L/6) H(x=3L/6) 1 u1 2 u2 u3 TH(x=5L/6) 4 14 e 1 e2 Figure 1: Axially loaded tapered beam (upper) and finite element approximation by bar elements (lower).
Problem 1: Find an analytical expression for the displacement u(x) of the tapered beam due to the point force applied at B. The following relations are given: du(x) Geometric relation: € = where the one-dimensional strain e is equal to the derivative of the dx displacement u(x) with respect to x. Constitutive relation: 0 = Ee, where the one-dimensional stress o is equal to the elastic modulus times the one-dimensional strain (i.e. Hooke's law). The stress is also equal to the applied axial force divided Р by the cross-sectional area, o = A(x)' = and x = L (i.e. put in values for x in the 2L Calculate the analytical displacements at x = 3 x = expression for u(x)). '