The equations of Timoshenko’s beam theory are derived by integration of the equations of three-dimensional elasticity theory. A new formula for the shear coefficient comes out of the derivation. Numerical values of the shear coefficient are presented and compared with values obtained by other writers.
It usually makes little difference whether moments are extrapolated from values at the Gauss points or calculated directly at the corners; this result is in agreement with flat plate test cases. In the four-element mesh force NZB is considerably improved if extrapolation is avoided. However, the opposite trend is seen in certain of the one-element values of NxB. This effect may be explained by considering the variation of midsurface strain E,, which is directly proportional to N, because Poisson's ratio u = 0. In each element, E, varies quadratically with co-ordinate 4. Extrapolation imposes a linear variation of E, with 4. Apparently, the beneficial effect of calculating more accurate strain values at the Gauss points may or may not be overcome by the detrimental effect of discarding the quadratic variation.As was the case with flat plates,2 there seems no reason to use five internal freedoms per element rather than three. It is concluded that because internal freedoms are more often helpful than harmful in plate and shell problems, computer programs should permit the user to employ three internal freedoms in the eight-node element. As a general rule it seems that all strains at element corners should be calculated by extrapolation from the Gauss points rather than directly.
SUMMARYSeveral formulas are presented for the numerical integration of a function over a triangular area. The formulas are of the Gaussian type and are fully symmetric with respect to the three vertices of the triangle.
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