w = 50(2 - x)2 kN/m EA = constant W 2 L/2 L/2 k * Figure 3: 3-noded bar element Figure 3 shows a uniform 3-node bar with

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W 50 2 X 2 Kn M Ea Constant W 2 L 2 L 2 K Figure 3 3 Noded Bar Element Figure 3 Shows A Uniform 3 Node Bar With 1 (60.34 KiB) Viewed 26 times

w = 50(2 - x)2 kN/m EA = constant W 2 L/2 L/2 k * Figure 3: 3-noded bar element Figure 3 shows a uniform 3-node bar with Young's modulus E, cross-sectional area A and length L = 2 m. The bar is fixed at point 3 and carries a distributed axial load w = 50 (2- 2)kN/m along its full length. (a) Derive the finite element formulation for the bar under the given load. [3 marks) (b) Determine the displacements at points 1 and 2. (3 marks (C) If the yield stress for the bar is Oy = 280 MPa, calculate the minimum cross-sectional area to prevent the structure from failure (yield). [4 marks] The shape functions (equations 1) and the stiffness matrix (equation 2) for the 3-node bar element are as follows: Ni=2 () - 3 (†) +1, = -3 . N2 = -4 4 ()* +4 (ů) N3 = 2 = 2 () - () (1) [k] EA 3L [ 7 -8 1 -8 16 - 8 1 -8 7 (2)