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Or, more concisely,

which relates the nodal forces {Pj to the element forces {5} for the whole structure.

Internal element equilibrium

Internal equilibrium can be explained most easily by considering an axial force element. For static equilibrium, the axial forces at each end will be equal in magnitude and opposite in direction. If the element is pin-ended and has a uniform cross-sectional area, A, then for equilibrium within the element

in which the axial stress ax is taken to be constant over the cross-section.

9.5.2. Compatibility

External nodal compatibility

The physical interpretation of external compatibility is that any displacement pattern is not accompanied with voids or overlaps occurring between previously continuous members. In fea. this requirement is usually only satisfied at the nodes. Often it is only the displacement field which is continuous at the nodes, and not an element's first or higher order displacement derivatives. Figure 9.20 shows quadratically varying displacement fields for two adjoining quadrilateral elements and serves to illustrate these limitations.

Fig. 9.20. Quadrilateral elements with quadratically varying displacement fields.

External, nodal, displacement compatibility will be shown to be automatically satisfied by a system of nodal displacements. For the simple frame shown in Fig. 9.19, the nodal displacement column matrix is

{/>} = {P\P2Pl} = [U\V\,U2V2, u3v3) and the element displacement column matrix is

{5} = {{i(a)}{i<'')}{'S<C)}} = {U1V,U2V2, U2V2U3V3, U\V\U3V3]

It follows from the above that external, nodal, compatibility is satisfied by forming the relationship between the element and nodal displacements as

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