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Element stiffness matrix in global coordinates

12//L2)cosasina,

-A cos2 a-(121 sin2 a)/L2, (—A + 12//L2)cosasina,

A cos2 a + (12/ sin2 a)/L2, (A — 12//L2)cosasina, (61 sin a)/L,

(6/cosa)/L, (—A + 12//X2)cosasina, -A sin2 a - (12/ cos2 a)//,2, (61 cos a)/L, symmetric

Element stress matrix in global coordinates

Substituting from eqns. (9.73) and (9.74) into eqn. (9.25) gives the element stress matrix in global coordinates as:

cos a + 61 sin(a)/L cos a — 6bsin(a)/L cos a — 6t sin(a)/L cos a -f 6bsin(a)/L

Formation of structural governing equation and assembled stiffness matrix

Whilst the beam element matrices include rotational dof. terms, not present in the rod element matrices, the procedures of Section 9.7.1 still apply, and lead to the structural governing equation

and the assembled stiffness matrix

9.8.2. Formulation of a simple beam element using the principle of virtual work equation

As Section 9.7.2 the principle of virtual work equation will be invoked, this time to formulate the equations for a simple beam element.

Consider the simple beam element shown in Fig. 9.28, for which the local and global axes have again been taken to coincide to avoid need for the prime and hence to simplify the appearance of the equations. The two nodes are each taken to have only normal and rotational dof. The terms associated with the omitted axial dof. have already been derived for the linear rod element in §9.7 and will be incorporated once the other terms have been derived. The total of four dof. for this beam element permits the displacement to be interpolated by a fourth order polynomial, namely v(x) = a, 4- otix/L + a^x2/L2 + o^x3/L3

Fig. 9.28. Simple beam element, aligned with global x-axis.

where a\ to a4 are generalised coefficients to be determined. Utilisation of eqns. (9.48) and (9.49) shows this polynomial will provide for a linearly varying moment and constant shear force and hence will enable an exact solution for beams subjected to concentrated loads. Writing in matrix form, v(x) = [1 ,x/L, x2/L2,x3/L3]

Substituting into eqn. (9.77) gives

-Vi -

0 0

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