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Finite Element Analysis FINAL EXAMINATION (Closed Book) Answer all questions. All questions carry equal marks.
Maximum marks: 50 Time: 3 Hours
Question 1: Using the Rayleigh-Ritz method solve the following equation in a square region: 2 2 2 2 0 k T^ T g x y
^ ^
T 0 on sides x 1 and y 1 Tn (^) 0 (insulated) on sides x 0 and y 0
using a one-parameter approximation of the form T c 1 1 x^2^ 1 y^2 .
Question 2: Determine the forces and displacements of points B and C of the structure shown in the figure below.
Question 3: Derive Hermite interpolation functions for a two node beam element with three primary
variables at each node: w, , where dwdx , and
2 2 d w (^) dx.
Question 4: Evaluate the following integrals using the Newton—Cotes and Gauss quadratures when i
are the quadratic interpolation functions.
12 ^0 ^1 212 ^0 1 2
b (^) b
a a
x (^) x
x x
K x x d^ d dx G x x dx dx dx
^
where 1 1 2 1 , and 2 1 2 1 . Use the appropriate number of integration points.
Question 5: Determine the shape functions for the five-node rectangular element shown in the figure below.
Question 6: Determine the conditions on the location of node 3 of the quadrilateral element shown in the figure below. Show that the transformation equations are given by
x 1 4 1 2 1 a 1
y 1 4 1 2 1 b 1
- n =
- 0.00000
- 0.40584
- 0.74153
- 0.94910 - 0.41795 - 0.38183 - 0.27970 - 0.12948 - n =
- 0.52553
- 0.79666
- 0.96028 - 0.36268 - 0.31370 - 0.22238 - 0.10122 - n =
- 0.00000
- 0.32425
- 0.61337
- 0.83603
- 0.96816 - 0.33023 - 0.31234 - 0.26061 - 0.18064 - 0.08127 - n =
- 0.14887
- 0.43339
- 0.67940
- 0.86506
- 0.97390 - 0.29552 - 0.26926 - 0.21908 - 0.14945 - 0.06667
