Understanding the Principle of Continuity and Compatibility in Finite Element Modelling and Analyses

Finite element analyses are commonly adopted in undertaking analyses of complex structural and mechanical systems and components. There exists a preponderance of trainers in the public space, providing training on the use of several commercial software. Although most of the trainers may be fluent in the general use of the software; it is sometimes evident that there is a lack of understanding of the principle of the finite element method (FEM), to ensure continuity and compatibility of displacements. This is largely the case because most users of the finite element technique have learnt to use it on the job, with sometimes no formal education and training on the ‘Principles and Application of the Finite element (FE) technique’.

Using shell (2-D) and brick (3-D) elements with incompatible degrees of freedom creates a ‘gap’ phenomenon in the finite element model. The implication is that although the elements appear ‘graphically’ connected, they do not displace together as in real structures. If shell and brick elements are to share connectivity in an FE model, multi-point constraints (MPCs) or other numerical techniques must be used to ensure compatibility of strains and displacements (Figure 1). MPCs are also required for mesh refinement when transiting from coarse to fine mesh, to ensure compatibility and continuity of displacement (Figure 2).

Another common modelling error is the connectivity of plate and beam structures. For plates and beams (modelled as shell or beam elements) to experience the same displacements and strains at boundaries, the number and nodes of elements of the beam must be the same as the number of plate or shell elements at the points of connection. Without continuity and compatibility, the loads on the plate would not be transferred correctly to the beam. Hence the supporting beam may/would experience less loading, resulting in false member utilization and consequential failure of the unsupported plate structure. When constraining a beam stiffener to a shell, the general approach is to define the beam and shell elements with separate nodes. These nodes can then be constrained to each other using MPCs. An alternative and more economical approach is to use the same node for the beam and the shell nodes and then define the offset of the centre of the cross-section of the beam in the beam section data.

There is also a common mistake when creating X-connections in lattice structures, with no common joint at the point of intersection. As result, the members behave independently without the benefit of the bracing configuration intended.