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Question 1 (7 marks)

Figure 1. Two beam elements subjected to distributed load q and concentrated couple moment M.

Figure 1 shows two beam elements subjected to a distributed load q and a concentrated couple moment M. Use the finite element method to solve for the rotation at node 1 and 2 and the displacement at node 2. The flexural rigidity of the beam is EI = 2 × 104  kNm2 , the distributed load q = 20 kN/m, the moment M = 3.6 kNm. The lengths L1  =  16 m and L2  =  12 m.

Question 2 (5 marks)

Consider the seepage flow problem in Figure 2. The element numbers are shown in brackets. The nodes numbers are shown beside the black circles. Use the finite element method, solve for the hydraulic heads in nodes 2, 3 and 4. The following are the properties of the soil columns: k (1)  = 0.25mm/s;    k (2)   = 0. 10mm/s;     k (3)  = 0.75mm/s;    k (4)   = 0.3mm/s;     A(1)  = 100 mm2 ;  A(2)  = 200 mm2 ;  A(3)   = 250 mm2 ;  A(4)  = 50mm/s.  The  length  of  the  soil columns are the same in all elements L  = 100 mm. The known hydraulic heads at nodes 1 and 5 are ℎ1   = 500 mm and ℎ5   = 250 mm.

In your solution, you must show your element connectivity table. Do not change the node and element numbers in the figure.

Figure 2. Finite element mesh for seepage flow problem.

Question 3 (8 marks)

Figure 3. An assembly of three 2-node bar element with arbitrary orientation supporting concentrated force F.

Figure 3 shows an assembly of three 2-nodebar elements with arbitrary orientation supporting concentrated force F. Node numbers are shown next to the filled circles. The element numbers are in brackets. Use the finite element method to compute the displacement at node 3. The value of the force F = 60 KN. The members are made of aluminium with Young’s modulus E = 80 Gpa and cross sectional area A = 0.02 m2 . The length of element 1 is L(1)   = 6.5 m, element 2 is L(2)   = 6.0299 m and element 3 is L(3)   =  10.324 m.