Consider the bar in Figure 2, which has cross-sectional area A = 1-10-3 m, modulus of...

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Consider the bar in Figure 2, which has cross-sectional area A = 1-10-3 m², modulus of elasticity E = 1-10¹¹ N/m², and length 1 m. The bar is fixed to a wall on its left hand side. Along the left half of the bar, x=[0 m, 0.5 m], there is a constant distributed force of 1(r) = 10 kN/m. Along the right half of the bar, x=[0.5 m, 1 m], there is a constant distributed force of 1(x) = 20 kN/m. 0m 1 10 kN/m 1 0.5 m 0.5 m Figure 2: Bar domain with varying distributed forces. a) Given the finite element mesh in Figure 3, consisting of two elements of equal size L = 0.5 m, sketch the associated basis functions of the mesh, and write the approximation of the displacement function u(r) as an expansion of these basis functions. Use u₁, ₂ and us to denote the expansion coefficients for each node. 2 20 kN/m 2 X 0.5 m 1m Figure 3: Mesh of 2 elements. Elements are numbered with underlines. b) The general form of the element stiffness matrix system, with nodes indexed by i and j, is given as, - F0 AE [11] {u} = {√ N₁(2)(a)dx} + {Nj(1)ƒ(1) - N,(0) FO (2) where F0 and f(1) denote boundary forces at positions z = 0 and r = 1, respectively. Form the two basis functions for element 2, and evaluate the right hand side vector of the matrix system 2 to form the local system of equations for element 2. Then use the local system for element 1 given by (3) AE AF [17]{}={250 500 FO). - to form and solve the global system of equations for u₁, ₂ and us. Consider the bar in Figure 2, which has cross-sectional area A = 1-10-3 m², modulus of elasticity E = 1-10¹¹ N/m², and length 1 m. The bar is fixed to a wall on its left hand side. Along the left half of the bar, x=[0 m, 0.5 m], there is a constant distributed force of 1(r) = 10 kN/m. Along the right half of the bar, x=[0.5 m, 1 m], there is a constant distributed force of 1(x) = 20 kN/m. 0m 1 10 kN/m 1 0.5 m 0.5 m Figure 2: Bar domain with varying distributed forces. a) Given the finite element mesh in Figure 3, consisting of two elements of equal size L = 0.5 m, sketch the associated basis functions of the mesh, and write the approximation of the displacement function u(r) as an expansion of these basis functions. Use u₁, ₂ and us to denote the expansion coefficients for each node. 2 20 kN/m 2 X 0.5 m 1m Figure 3: Mesh of 2 elements. Elements are numbered with underlines. b) The general form of the element stiffness matrix system, with nodes indexed by i and j, is given as, - F0 AE [11] {u} = {√ N₁(2)(a)dx} + {Nj(1)ƒ(1) - N,(0) FO (2) where F0 and f(1) denote boundary forces at positions z = 0 and r = 1, respectively. Form the two basis functions for element 2, and evaluate the right hand side vector of the matrix system 2 to form the local system of equations for element 2. Then use the local system for element 1 given by (3) AE AF [17]{}={250 500 FO). - to form and solve the global system of equations for u₁, ₂ and us.

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a The basis functions for this mesh are N1x 1 for 0 ... View the full answer