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l9j\

= [1 - 3x2/L2 + 2x3/l\x- 2x2/L+x3/L2, 3 x2/L2 - 2x3/L\ -x2/L+x3jL2]

which has the same form as eqn. (9.32), in this case with shape functions

Nx = 1 - 3X2/L2 + 2jc3/L3 N2=x-lx2/L+xi/L2 Ni = 3x2/L2 - 2x3/L3 N4 = -x2/L +xi/L2 the variation of which is shown in Fig. 9.29.

Fig. 9.29. Shape functions for a simple beam element.

Element stiffness matrix in local coordinates

Longitudinal bending stress is given by simple bending theory as er = My/I

in which, from the differential equation of flexure, eqn. (9.48),

to give a — Ey d2v/dx2 Substituting eqn. (9.79) into eqn. (9.80) gives a = Ey[(\2x/L3 - 6/L2)(6x/L2 - 4/Q(-\2x/I? + 6/L2)(6x/L2 - 2/L)]

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