## Ae

cos2 a

cos a sin a sin2 a

Element stress matrix in global coordinates

The element stress matrix found in local coordinates, eqn. (9.17), can be transformed to global coordinates by substituting from eqn. (9.18) into eqn. (9.16) to give

in which the element stress matrix in global coordinates is

Substituting from eqn. (9.17) and (9.19) into eqn. (9.25) gives the element stress matrix in global coordinates as

= £(e)[— cosa — sina cosa sina Ye)/L(e) (9.26)

Formation of structural governing equation and assembled stiffness matrix

With reference to § 9.5, external nodal equilibrium is satisfied by relating the nodal loads, {P}, to the element loads, {5}, via

Similarly, external, nodal, compatibility is satisfied by relating the element displacements, {s}, to the nodal displacements, {/>}, via

Substituting from eqn. (9.3) into eqn. (9.21) for all elements in the structure, gives:

in which [&] is the unassembled stiffness matrix. Premultiplying the above by [a]T and substituting from eqn. (9.1) gives:

which is the structural governing equation for static stress analysis, relating the nodal forces {P} to the nodal displacements {p} for all the nodes in the structure, in which the structural, or assembled stiffness matrix

9.7.2. Formulation of a rod element using the principle of virtual work equation

Here, the principle of virtual work approach, described in § 9.6, will be used to formulate the equations for an axial force rod element. As described, the approach permits the displacement field to be represented by approximating functions, known as interpolation or shape functions, a brief description of which follows.

### Shape functions

As the name suggests shape functions describe the way in which the displacements are interpolated throughout the element and often take the form of polynomials, which will be complete to some degree. The terms required to form complete linear, quadratic and cubic, etc., polynomials are given by Pascal's triangle and tetrahedron for two- and three dimensional elements, respectively. As well as completeness, there are other considerations to be made when choosing polynomial terms to ensure the element is well behaved, and the reader is urged to consult detailed texts.6 One consideration, which will become apparent, is that the total number of terms in an interpolation polynomial should be equal to the number of dof. of the element.

Consider the axial force rod element shown in Fig. 9.24, for which the local and global axes have been taken to coincide. The purpose is to simplify the appearance of the equations by avoiding the need for the prime in denoting local coordinate dependent quantities. This element has only two nodes and each is taken to have only an axial dof. The total of only two dof. for this element limits the displacement interpolation function to a linear polynomial, namely u{x) = a\ + a2x

Fig. 9.24. Axial force rod element aligned with global jr-axis.

where ot\ and a2, to be determined, are known as generalised coefficients, and are dependent on the nodal displacements and coordinates. Writing in matrix form

Or, more concisely, u(x) = [*]{a} At the nodal points, u(0) = w, and u(L) — uj. Substituting into eqn. (9.30) gives

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