ENGIN5302 MODELLING & SIMULATION

PRE-EXAMINATION TEST

Instructions: this test contains 2 questions. A question may have additional sub-questions. Answer all questions. Both questions have equal weighting. The total time allowed is 1 hour and 15 minutes.

QUESTION 1 (5 MARKS)

Figure 1 shows an assembly of springs subjected to forces F2, F3, F4 and F5. The node numbers are shown beside the gray filled circles. Node 1 is constrained from motion. The stiffness of the individual springs are shown in the figure as ki, where i = 1, 2, 3, 4 and 5. Determine the following:

The global assembled system of equations in terms of F2, F3, F4 and F5; and k1, k2, k3, k4 and k5 Apply the boundary conditions and reduce the global assembled system of equations obtained in part (a)

Note: You do not have to solve for the displacements of the system

QUESTION 2 (10 MARKS)

Figure 2 shows an assembly of two trusses modelled with the finite element method with node numbers shown beside the filled circles and element numbers in superscripted brackets ( ). Nodes 2 and 3 are pin-supported. A force F=0.1 kN is applied at node 1. If the lengths of elements 1 and 2 are L(1)=1.2m and L(2)=0.6m, determine the displacement at node 1.

Assume that both elements 1 and 2 are made of the same material with Youngs modulus E=100 kPa and have the same cross sectional area A=0.01m2.

Figure 2. Truss system subjected to force F.

Useful formulas

Stiffness matrix of a spring element

Stiffness matrix of a truss element with arbitrary orientation

Sample Test

Question 1 (5 marks)

For the seepage problem shown in Figure 1, use the finite element method to calculate the hydraulic heads at points P and Q. Use the following data: k1=0.042 mm/s, k2=0.61 mm/s, k3=0.053 mm/s and k4=0.21 mm/s, L1=220 mm, L2=51 mm, L3=260 mm and L4=36 mm. The known hydraulic heads hC=284 mm, hB=65 mm, hA=36 mm. The cross-sectional area of all the soil sections are A=0.2 mm2.

QUESTION 2 (5 MARKS)

Figure 4. Truss system subjected to force PThe system of spring elements shown in Figure 4 is subjected to force P. The node numbers are shown next to the filled circles. The stiffness of each spring is shown in Figure 4. Obtain the global system of finite element equations using spring elements, apply the boundary conditions and reduce the system to the simplest form representing only the active degrees-of-freedom. You do not need to solve for the displacements.