Modal Decomposition and the Constrained Finite Strip Method

General

The application of the above definitions implies that the appropriate deformation constraints, which are formulated in the three criteria, must be introduced into FSM. As a consequence, the original number of degrees of freedom (DOF) is necessarily reduced since the member can only deform in accordance with the strain conditions. Thus, our goal is to work out how the DOFs are reduced due to the various strain assumptions, and how this DOF reduction can practically be handled. Although the derivations are not extremely complicated, they are much longer than the limits provided by this paper. For this reason, here only a small example will be presented to demonstrate the method, which, at the same time, highlights all the important features of the more general derivations. Complete derivations can be found in (Adany and Schafer 2005a).

Constraint Matrix Derivation

Consider the membrane deformation of a single finite strip, as shown in Figure 1. If the longitudinal distributions are assumed to be sinusoidal, as in the classical implementations of FSM, e. g., in CUFSM, the displacements can be expressed as a product of the assumed shape functions and the nodal displacements.

where U, u2, vb v2 are the transverse and longitudinal nodal displacements, m is the number of half­sine waves in the longitudinal direction, and a and b are the length and width of the strip, respectively.

Let us now introduce the criteria for zero transverse membrane strains:

du

ex = — = 0
dx

Substituting Eq. (2) into Eq. (4):

du – u, + u2 . rnrcy

є x = — =—– 1—– 2sin—- — = 0

dx b a

and since the sine function is generally not equal to zero, u, to satisfy the equality.

This implies that the transverse displacements of the strip’s two nodal lines must be identical, which is a natural consequence of the zero transverse strain assumption. In practice, the identical u

displacements prevent those deformations where the two longitudinal edges of the strip are not parallel, as illustrated in Figure 4.

The above derivation demonstrates that the introduction of a strain constraint reduces the number of DOFs, in this particular case from 4 to 3. Thus, we can define the new, reduced DOFs by u, v1 and v2, while the relationship of the original and reduced displacement vectors can be expressed as follows:

(6)

d = Rdr

where R is the constraint matrix, which is a representation of the introduced strain constraints.