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TABLE 9.1
Results for the Cross-Section of Fig. 9.14a
Torsional Constant
Maximum Stress*
Number of
Integration
Points/Nodes
BEM
BEM
Direct
Integration
Direct
Integration
Constant
Linear
Constant
Linear
12
0.1422
0.1366
0.1540
403.1
440.53
435.84
20
0.1411
0.1393
0.1456
448.8
467.1
462.57
28
0.14077
0.14008
0.1432
462.8
465.9
464.3
40
0.14074
0.1404
0.1418
474.6
473.75
472.74
56
0.14065
0.1405
0.1418
473.8
473.36
472.74
72
0.140627
0.14055
0.14096
475.89
474.95
474.7
FEM
0.14058
482.42
Results
Exact ∗∗
0.140577
482.16
Solution
The maximum stress is taken at the middle point of each side of the square.
∗∗ From Boresi and Chong (1987).
TABLE 9.2
Results for the Cross-Section of Fig. 9.14b
Integration Torsional Maximum
Points Constant Stress
18 0.1333 371.2
30 0.1344 412.9
42 0.13418 431.4
60 0.13414 401.3
84 0.13405 409.6
FEM Results 0.13400 408.97
The maximum stress is taken at the centroid of the
finite element with the sign
+
shown in Fig. 9.15.
The comparison shows that the results for direct integration of the torsional constant are
superior to those of the boundary element method for most of the schemes. For the stress
computations, the results from the direct integration are not as accurate as those from the
boundary element method for the first few boundary discretization schemes. However,
as the number of integration points increases, the stress results for direct integration are
about the same as those obtained from the boundary element method. Since no element
and hence no shape function is involved in the discretization of the boundary, the direct
integration method involves less computational effort in forming the system matrices
than the boundary element method and hence is a more efficient procedure for this two-
dimensional problem.
References
Boresi, A.P. and Chong, K.P., 1987, Elasticity in Engineering Mechanics , Elsevier, Amsterdam, Nether-
lands.
Brebbia, C.A. and Dominguez, J., 1992, Boundary Elements, An Introductory Course , 2nd ed., McGraw-
Hill, New York.
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