Problem 3: Mathematics of Element Formulation The meshed model of a bath is shown in Figure CW2-2. The bath is made of steel of Young's Modulus, E = 210 GPa. The mesh is based on 2D quadrilateral element. We are interested in developing the element formulation for our bespoke four-node quadrilateral element whose isoparametric natural coordinates are given in Figure CW2-2. For the quadrilateral element: A. Show derivations of the relevant element shape functions, Ni where i = 1, 2, 3, 4, needed to describe the element formulation for the quadrilateral element. [40 marks] B. The plot of the four shape functions [20 marks] C. Derive the resulting strain-displacement matrix, B of this element type. [20 marks] D. Using the B-matrix above and material properties, derive the applicable stiffness matrix, K that will be used within the finite element solver for this

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Y [-0.5,0.5] n 3 [0.25,0.75] X 1 2 [-0.2,-0.8] [0.5,-0.5] Y [-0.5,0.5] n 3 [0.25,0.75] X 1 2 [-0.2,-0.8] [0.5,-0.5]