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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).
L 3L Problem 2: Approximate the beam with three linear finite elements (trusses) according to the lower part of Figure 1. The point load is applied at node 4. Assume a constant rectangular cross-section of 5L each element and use the height values H(x), x = 5,x = and x = to find the area of each x = element. The elastic modulus is assumed to be constant for all elements and the length of each element is Le = 1/3. = Use the principle of stationary (minimum) potential energy to derive the global stiffness matrix for the three-bar-element model of the beam. The total potential energy II is equal to the elastic potential energy Ue of the system minus the mechanical work of the applied forces, W. = = Use the global stiffness matrix to solve the static problem [K]{d} = {P} for the unknown nodal displacements. Here [K] is the global stiffness matrix, {d} = {U1 U2 Uz u4}T is the column vector of nodal displacements and {P} = {P] P2 P3 P4}" is the column vector of applied nodal forces (external and reaction forces). Hint! It can be a quite tedious process to find an analytical expression for {d} and therefore you can directly enter the numerical values below and solve for the displacements. You can use the program developed in Problem 3 (or by hand). Use the numerical values E = 200 GPa, h = 0.06 m, t = 0.05 m, L = 3.0 m, P = 500 kN and calculate the magnitude of x-displacement at the nodes. Compare with the analytical solution calculated using the expressions you derived in problem 1 (by inserting the same numerical values) at the same x- positions. Convert the calculated displacements to element strains and stresses by utilizing the geometrical and constitutive relations given above (formulate expressions for strain and stress before you put in the numerical values).