Question

Section 1: Finite Element Derivation and Validation In this section of the report you will develop your own Finite Element method for 1-dimensional axial loading. The governing equation for displacement, u is Poisson's Equation: อั1 where E is the modulus of elasticity, A(a) is the cross-sectional area as a function of length, and q(x) is the loading distribution as a function of length. The weak form of this equation with 0 1. Starting from the weak form of the governing equation, using the Galerkin method with two elements, with boundary conditions u(0)-0 and uz(L) = 0, and without specified external loads, derive a solution in the form: where K is the stiffness matrix, U is the global displacement vector and Q is the global load vector. Use linear basis functions in your derivation and assume that the area is constant across each element. Be concise in the presentation of your working (use phrases like "col- lecting terms", and "similarly, the next integral is ..." to reduce the lines of working in your report). (15%) 2. Using your solution for Question 1, expand your solution to an arbitrary number of elements and show how this result is achieved. (10%)

Answer 1

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