1 FINAL EXAM MECH 4175/5175: Finite Element Analysis Fall 2020 Instructions: This exam has three problems. Work the problems out on your own paper, and then upload a pdf of your work to Canvas by 5:30...

Finite Element Analysis.See attached file. Please complete this as soon as possible!


1 FINAL EXAM MECH 4175/5175: Finite Element Analysis Fall 2020 Instructions: This exam has three problems. Work the problems out on your own paper, and then upload a pdf of your work to Canvas by 5:30 p.m. on Wednesday, December 9. Problem 1 (14 points): 1-D Elements, Shape Functions, and Boundary Conditions An axial bar element spanning the distance from x1 to x2 is shown below. (a) Find the two shape functions as functions of x. (b) Given that u1 = +0.01 and u2 = -0.02, write a linear expression for u as a function of x. Use the shape functions found in (a). (c) For each of the four separate sets of boundary conditions listed below, state whether the set can be successfully implemented in the bar element. If any set of conditions is not valid for this element, explain why. (1) u1 = 0 at node 1; a concentrated force of 5 N in the x-direction at node 2. (2) u1 = 0 at node 1; a concentrated force of 5 N in the y-direction at node 2. (3) u1 = +0.05 at node 1; u2 = +0.06 at node 2. (4) u1 = +0.05 at node 1, u2 = 0.0 at node 2; a 5N concentrated force in the x-direction at node 1. 2 Problem 2 (16 points): 2-D Triangle Element for Steady-State Heat Transfer A 2-D triangle element (one d.o.f. per node) used for analysis of steady-state heat transfer is pictured below. (a) Find the element’s three interior shape functions as functions of x and y. (b) Given that T1 = 100, T2 = 50, and T3 = 120, write an expression for T as a function of x and y. Use the shape functions found in (a). (c) Each of the contours (bold black lines) labeled as d1 – d6 on the element pictured below has a constant temperature. Find the temperature associated with each contour. 3 Problem 3 (20 points): 2-D Triangular Element for Plane Stress A 2-D triangular element with two degrees of freedom per node is pictured below. Answer each question below using the element. (a) Find the area bounded by the element’s three sides. (b) Find the three shape functions (N1(x,y), N2(x,y), and N3 (x,y)) for the interior of the element. (c) Find the strain-displacement matrix B for the interior of the element. (d) Given the set of nodal displacements { ?1 ?1 ?2 ?2 ?3 ?3} = { 0 0 0.1 0.1 0.05 −0.1} , find the displacement vector �⃑� = { ? ? } at the point (x=2, y = 3). (e) Given the set of nodal displacements { ?1 ?1 ?2 ?2 ?3 ?3} = { 0 0 0.1 0.1 0.05 −0.1} , find the strain vector ? = { ?? ?? ??? } for the element.