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where L = length of member (distance between i and j) E = modulus of elasticity A = cross-sectional area of member I = moment of inertia about neutral axis in bending

In matrix notation, Eq. (3.146) for the ith member of a structure can be written

FIGURE 3.81 Member of a continuous structure. (a) Forces at the ends of the member and deformations are given with respect to the member local coordinate system; (b) with respect to the structure global coordinate system.

FIGURE 3.81 Member of a continuous structure. (a) Forces at the ends of the member and deformations are given with respect to the member local coordinate system; (b) with respect to the structure global coordinate system.

where S- = vector forces and moments acting at the ends of member i k = stiffness matrix for member i 8- = vector of deformations at the ends of member i

The force-displacement relationships provided by Eqs. (3.146) and (3.147) are based on the member's xy local coordinate system (Fig. 3.81a). If this coordinate system is not aligned with the structure's XY global coordinate system, these equations must be modified or transformed. After transformation of Eq. (3.147) to the global coordinate system, it would be given by

where Si = rfS- = force vector for member i, referenced to global coordinates k = rfk-ri = member stiffness matrix

8i = rfS- = displacement vector for member i, referenced to global coordinates ri = transformation matrix for member i

For the member shown in Fig. 3.81b, which is defined in two-dimensional space, the transformation matrix is cos a sin a 0 0 -sin a cos a 0 0 r = 0 0 1 0 0 0 0 cos a

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