A masonry gate with loads in the x-y plane is modelled using two triangular and two...

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A masonry gate with loads in the x-y plane is modelled using two triangular and two quadrilateral 2D plane stress elements (thickness equal to 200mm) as shown in Figure Q2. The Young modulus E-20,000MPa and the Poisson's ratio vis equal to 0.1. 0.5m 0.5m 2 3 A y 0.5m 4 5 0.5m Figure Q2 (a) Given the boundary conditions in Figure Q2 and the nodal displacements U4 = 1mm, us = -1mm, v4 = -2mm and vs = -2mm, calculate the internal displacement vector u at the middle point between nodes 4 and 5 in this case. In the calculations use element defined by nodes 2, 3, 4 and 5 (use local nodes 1, 2, 3 and 4 for global nodes 2, 3, 4 and 5 respectively). [5] (b) Calculate the first column of the kinematic matrix B for element 1-2-5 (adopt local nodes 1, 2 and 3 for nodes 1, 2 and 5 respectively). [8] (c) Calculate element K₁1,10 in the 11th row and 10th column of the structure stiffness matrix knowing that the horizontal reactions due to a vertical displacement of -1mm in node 4 are p₁ = 20 kN, p2 = -30 kN, p3 = -10 KN and that the horizontal reactions due to a vertical displacement of -10 mm in node 5 are p₁ = 120 kN, p2 = -230 KN, p3= -150KN. [7] (d) One Gauss Point is adopted for each of the elements shown in Fig. Q2. Apply the Maxwell's Rank Test (c + nr ≤ d ) to assess whether the structure stiffness matrix is singular? Does the structure pass this test and are the results conclusive? [5] A masonry gate with loads in the x-y plane is modelled using two triangular and two quadrilateral 2D plane stress elements (thickness equal to 200mm) as shown in Figure Q2. The Young modulus E-20,000MPa and the Poisson's ratio vis equal to 0.1. 0.5m 0.5m 2 3 A y 0.5m 4 5 0.5m Figure Q2 (a) Given the boundary conditions in Figure Q2 and the nodal displacements U4 = 1mm, us = -1mm, v4 = -2mm and vs = -2mm, calculate the internal displacement vector u at the middle point between nodes 4 and 5 in this case. In the calculations use element defined by nodes 2, 3, 4 and 5 (use local nodes 1, 2, 3 and 4 for global nodes 2, 3, 4 and 5 respectively). [5] (b) Calculate the first column of the kinematic matrix B for element 1-2-5 (adopt local nodes 1, 2 and 3 for nodes 1, 2 and 5 respectively). [8] (c) Calculate element K₁1,10 in the 11th row and 10th column of the structure stiffness matrix knowing that the horizontal reactions due to a vertical displacement of -1mm in node 4 are p₁ = 20 kN, p2 = -30 kN, p3 = -10 KN and that the horizontal reactions due to a vertical displacement of -10 mm in node 5 are p₁ = 120 kN, p2 = -230 KN, p3= -150KN. [7] (d) One Gauss Point is adopted for each of the elements shown in Fig. Q2. Apply the Maxwell's Rank Test (c + nr ≤ d ) to assess whether the structure stiffness matrix is singular? Does the structure pass this test and are the results conclusive? [5]

Answer rating: 100% (QA)

a For element 2345 local nodes 1234 the shape functions are N1 0251x1y N2 0251x1y N3 0251x1y N4 0251... View the full answer