1.46 Elements

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Element displacements are represented in general by polynomials. The

higher the degree of the polynomials, the more flexible an element is, the more

The type of an element, (see Fig. 1.161), is determined by the strains and

stresses which result if the element is deformed; that is, the type depends

foremost on the definition of the strain energy of the element.

1.47 Stiffness matrices 221

stress and strain states it can represent, and flexibility—think of the Green’s

functions—is very important in the FE method. There are three requirements

which any element must meet:

• Rigid-body motions: the element must be able to represent rigid-body motions

and constant strain states exactly, that is, it must be capable of

following the element through the first two terms in the Taylor series of

the displacement field, u(x) = u(0) + ∇u(0) x.

• Isotropy and rotational invariance: Theoretically a solution should not

depend on the orientation of the element, that is the elements should

not prefer particular directions. This is guaranteed if the polynomials are

complete.

• Continuity: At the interelement boundaries the displacements must be

continuous. Such elements are called C0-elements. If two neighboring

eral the displacements along the interelement boundary are the same. In

plate theory (Kirchhoff plates,KΔΔw) and beam theory (Euler–Bernoulli

beam, EI wIV ) the first-order derivatives must be C1 across interelement

boundaries.

The requirement that the polynomial shape functions be complete can be

relaxed: to have isotropy and rotational invariance it suffices that all terms

which are symmetric to the diagonal of Pascal’s triangle be included.

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