In case of the more general strain-displacement constraints, and more general cross-sections, the derivations are somewhat more complicated, but finally the associated constraint matrices (R) can be defined, as shown in Adany and Schafer (2005a, b) for G and D modes and Adany (2004) for L and O modes, and thus apply for all the criterion summarized in Table 1. Since a different R matrix may be constructed for each of the modal classes: G, D, L, and O, taken together they span the entire original nodal basis and represent a transformation of the solution from the original nodal basis to a basis where G, D, L, and O deformation fields are segregated.
The columns of the R constraint matrices are the deformation fields associated with the G, D, L, and O spaces. For the C-section of example (a), modelled only with nodal lines at the corners and the free edge, making for a 24 DOF model, the columns of the R matrix are provided graphically in Figure 4. Figure 4a and b provide the warping displacements and transverse displacements for the G and D modes. An important characteristic of these modes is that the transverse displacements are uniquely defined by the warping displacements. Figure 4c provides the transverse displacements for the L modes (note, no warping occurs in the L modes). As shown in the figure, these L modes appear to be in the nodal DOF basis, but they are not identical to the original FSM nodal basis, because they represent only the part of the nodal rotations that meet the L constraints – note nodal rotations occur in the G and D modes as well. Finally, Figure 4d and e provide the O modes, associated with shear and transverse extension. As discussed in detail in Adany and Schafer (2005a, b) additional transformation inside the G, D, L, O spaces are possible and desirable. One attractive option is to use unit-member axial modes as detailed in Adany and Schafer (2005b). In many applications we are interest in only one mode class, and in such a case the R matrix truly represents a constraint on the original DOF and can be applied to reduce the problem size.