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ME 422 Winter 2010-2011 Do all 4 problems Name Problem Problem Problem Problem TOTAL 1 2 3 4 (20) (30) (30) (20) (100) 1 Problem 1 (20 points): The transient stress analysis shown below has the governing equation KU + MÜ = F (1) 2 1 H P sin(ωt) 3 4 L (a) Once we impose the boundary conditions, we have the “global” system, Equation (1). What size is the global mass matrix for this problem? (i) 8×8 (ii) 4×4 (iii) 2×2 (iv) Other (specify) (b) Find the global mass matrix. Set up all integrals as completely as possible, but do not integrate. 2 H T=0 Problem 2 (30 points): Consider two physical systems, as shown in the diagram below. System I is a static stress analysis problem with the bar fixed to the wall and a pressure load at the tip. System II is a steady heat transfer problem with a zero temperature at one end and a distributed heat flux at the tip. P H Q System II System I Because of symmetry, we can model the top half of each of the systems with four-node isoparametric elements, as shown below. 2 2 1 4 3 1 4 3 System I System II (a) How many unknowns will there be in the global (after boundary conditions are applied) K matrix for System I? Label them on the element. (b) How many unknowns will there be in the global (after boundary conditions are applied) K matrix for System II? Label them on the element. (c) The shape functions for the element used for System I will be: (Circle one choice) (i) the same as those used for the element in System II (ii) different than those used for the element in System II (iii) We don’t have enough information to answer this question (d) The Jacobian for the element used for System I will be: (Circle one choice) (i) the same as that used for the element in System II (ii) different than that used for the element in System II (iii) We don’t have enough information to answer this question 3 (e) What is the global (after boundary conditions are applied) forcing vector for System I? (f) What is the global (after boundary conditions are applied) forcing vector for System II? 4 Problem 3 (30 points): Suppose that we have two triangular elements used to represent a plate in a transient plane stress analysis. v =L/20 1 1 2 L u1=L/10 b a 4 3 L Using ∆t=0.1 second, after 10 time steps the displacements are u1 = L/10 and v1 = L/20. E, ν, and L are given. (a) What are the stresses at point a at t=1 second? (b) What are the stresses at point b at t=1 second? 5 6 Problem 4 (20 points): Recall the tapered bar from the first exam. x=0 Base Area, A 0 x X=L Tip Area, A L F The weak form for that problem was Z o L ∂δ ∂δ EA dx = δ L FL + δ 0 F0 ∂x ∂x Create a finite element model of this system using a single truss finite element. What is the solution for the displacement of the tip due to the applied force? Write your answers in terms of A0 , AL , L, E, and F . 7