The only departure of eqn. (9.93) from the previous expressions is the replacement of the modulus of elasticity, E, by the elasticity matrix [£>], due to the change from a one- to a two-dimensional stress system.

Recalling, for the present case that the displacement fields are linearly varying, then matrix [5] is independent of the x and y coordinates. The assumption of isotropic homogeneous material means that matrix [D] is also independent of coordinates. It follows, assuming a constant thickness, /, throughout the element, of area, a, eqn. (9.93) can be integrated to give

Element stress matrix

The expression for the element direct and shear stresses is obtained by substituting from eqn. (9.90) into eqn. (9.92), to give

or, more fully,

Uj uk

Vj vi vk J

These stresses are with respect to the global coordinate axes and are taken to act at the element centroid.

Formation of structural governing equation and assembled stiffness matrix

As Sections 9.7.2 and 9.8.2, the structural governing equation is given by eqn. (9.28) and the assembled stiffness matrix by eqn. (9.47).

9.10. Formation of assembled stiffness matrix by use of a dof. correspondence table

Element stiffness matrices given, for example, by eqn. (9.23), are formed for each element in the structure being analysed, and are combined to form the assembled stiffness matrix [K], Where nodes are common to more than one element, the assembly process requires that appropriate stiffness contributions from all such elements are summed for each node. Execution of finite element programs will enable assembly of the element stiffness contributions by utilising, for example, eqn. (9.29) deriving matrix [a], and hence [a]T, from the connectivity information provided by the element mesh. Alternatively, eqn. (9.47) can be used, the matrix summation requiring that all element stiffness matrices, are of the same order as the assembled stiffness matrix [Jf], However, by efficient "housekeeping" only those rows and columns containing the non-zero terms need be stored.

For the purpose of performing hand calculations, the tedium of evaluating the triple matrix product of eqn. (9.29) can be avoided by summing the element stiffness contributions according to eqn. (9.47). The procedure to be adopted follows, and uses a so-called dof. correspondence table. Consider assembly of the element stiffness contributions for the simple pin-jointed plane frame idealised as three rod elements, shown in Fig. 9.19. The element stiffness matrices in global coordinates can be illustrated as:

C14" C24 C34 C44.

The procedure is as follows:

• Label a diagram of the frame with dof. numbers in node number sequence.

• Construct a dof. correspondence table, entering a set of dof. numbers for each node of every element. For the rod element there will be two dof. in each set, namely, u and v displacements, and two sets per element, one for each node. The sequence of the sets must correspond to progression along the local axis direction, i.e. along each positive x' direction. This is essential to maintain consistency with the element matrices, above, the terms of which have been shown in eqn. (9.23) to involve angle a, the value of which will correspond to the inclination of the element at the end chosen as the origin of its local axis. The sequence shown in Table 9.2 corresponds to ota = 330°, a/, = 180° and ac = 210°. The u and v dof. sequence within each set must be maintained.

• Choose an element for which the stiffness contributions are to be assembled.

• Assemble by either rows or columns according to the dof. correspondence table.

• Repeat for the remaining elements until all are assembled.

Table 9.2. Dof. correspondence table for assembly of structural stiffness matrix, [Jt].

Row and/or column in element stiffness matrix, [fc(<,)]

Row and/or column in assembled stiffness matrix, [K]

element a

element b

element c

0 0

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