Modelling & Simulation in Engineering Assignment Questions

Question 1 - For the seepage flow problem shown in Figure 1, if the hydraulic heads h_A = 300 mm, h_B = 200 mm and h_C = 100 mm, use the finite element method to find the hydraulic head h at the intersection where the flow branches into k₂ and k₃ i.e. point P in the figure. Determine also the nodal rate of water supply in each element. Assume the following properties and geometric parameters. The permeability of the soils are k₁ = 0.2 mm/s; k₂ = 0.05 mm/s and k₃ = 0.0004 mm/s. The cross sectional areas of the soil sections are the same. They are circular with diameter d = 100 mm. The lengths of the soil sections are the same with L = 80 mm.

Question 2 - For the truss system shown in Figure 2, determine the displacements of node 2. The node numbers are indicated near the filled circles. The element numbers are shown in brackets. The cross sectional area of each element is the same A = 0.2 m². Assume the Young's modulus to be E = 200 GPa.

Question 3 - Consider the beam shown in Figure 3 and modelled using two beam elements and subjected to the distributed load and point load F shown. The beam is fixed at the left and propped at the right end. If E = 210 GPa and I_z = 5 × 10^-6m⁴ determine the displacements and slopes at nodes 2 and 3.

Question 4 - A mesh of 2 constant strain triangular (CST) elements is shown in Figure 4 with the element numbers (in brackets), nodal numbers and dimensions. Nodes 1, 3 and 4 are xed. Assume plane stress condition with thickness in the out of plane direction t = 1. Let Young's modulus E = 70 GPa and Poisson's ratio ν = 0. Determine the following: 1. The displacements of node 2. The stresses in elements 1 and 2.

Question 5 - Consider the 4-node iso-parametric elements shown in Figure 5 with corresponding node numbers and coordinates. Nodes 1, 3 and 4 are fixed. Assume plane stress condition with thickness in the out of plane direction t = 1. Let Young's modulus E = 70 GPa and Poisson's ratio ν = 0. The stiffness matrices are to be calculated by using 1-point Gauss integration. The integration point is located at the local coordinates (ξ, η) = (0.0, 0.0). Assume that the out of plane thickness t = 1 m.

1. Determine the displacements of node 2 and the stress in the element.

2. Compare the results obtained in this Question with Question 4. Comment on the similarities or differences if any.