Finite Element Analysis

Within a MasterFrame model, it is possible, using the MasterFrame FEA module, to include finite element surfaces within a single MasterFrame model.

The finite element method (FEM) provides a method to model the behaviour or a continuum structure, such as floor plates or walls elements. The structural element is subdivided into a series of small elements. In general, these elements may be two or three-dimensional. This subdivided structure may then be analysed by matrix methods which are an extension of those used in MasterFrame for analysis regular structural models using 1-dimensional line elements.

Each finite element is a mathematical formulation, consisting of a number of node points joined by edges. Each finite element is a mathematical idealisation representing a small part of the real structural element. Loads applied to a finite element are idealised to act at the node points. Finite elements are assumed to connect at the node points only, so compatibility between elements is taken to occur at the nodes.

Finite elements are then connected together to form a mesh, where this mesh is the analytic representation of the structural element to be modelled. From this mesh, the stiffness matrix for this FE structure can be compiled. By then converting this local stiffness matrix into global coordinates, the transformed matrix can be compiled as part of the analytic stiffness matrix of the full structure, thereby incorporating the FE surface into the analysis of the full structure.

The analysis of the structure calculates the deformation of the structure at the node points in the model, which includes the nodes of the finite element mesh. Thus, the analysis is calculating the deformations and forces at each node in the FE surface. Thus, the forces are displacements of each finite element is calculated at the node points of the element. However, since the FE surface is to model a continuum, an estimate of the forces and displacements across an element also needs to be calculated. This is done by the use of a suitable shape function. A shape function is a function which describe the distribution of the element stress and strain between the node points. The shape function needs to be continuous across the element, as does its first derivative and also needs to ensure compatibility with the results at the element nodes. For ease of calculation, polynomials are often chosen for shape functions.

Since the FEM is used to model and analyse the behaviour of a continua, the size of the FE mesh can have a significant impact on the accuracy of the results. Generally, reduction of the mesh leads to increased accuracy, but the increased number of nodes and degrees of freedom leads to a significant increase in the size of the stiffness matrix and a subsequent increase in the analysis time.

Within MasterFrame FE, 8-noded quadrilateral shell elements are employed. These are 2-dimensional shell elements, with nodes at each corner of the element and additional nodes at the mid-point of each side. The finite elements used employ Mindlin plate theory which enables the shear deformations of the elements to be included in the analysis. Mindlin theory assumes that there is a linear variation of displacement through the thickness of the element, but no change in the element thickness. This is similar to the assumption that plane sections remain plane in Bernoulli beam bending.

In MasterFrame, the 2-dimensional finite element surface is used to represent the centreline of the 3-dimensional element to be modelled and analysed. An FE surface therefore is used to model a structural element of constant thickness, with the FE surface representing the centreline of the structural element.