# Mathematical Modelling

The purpose of mathematical modelling is to predict structural behaviour in terms of loads, stresses and deformations under any specified, externally applied force system. Since actual structures are physical, three-dimensional entities it is necessary to create an idealized model which is representative of the materials used, the geometry of the structure and the physical constraints e.g. the support conditions and the externally applied force system.

The precise idealisation adopted in a particular case is dependent on the complexity of the structure and the level of the required accuracy of the final results. The idealization can range from simple 2-dimensional 'beam-type' and 'plate' elements for pin-jointed or rigid jointed plane frames and space frames to more sophisticated 3-dimensional elements such as those used in grillages or finite element analyses adopted when analysing for example bridge decks, floor-plates or shell roofs.

It is essential to recognise that irrespective of how advanced the analysis method is, it is always an approximate solution to the real behaviour of a structure.

In some cases the approximation reflects very closely the actual behaviour in terms of both stresses and deformations whilst in others, only one of these parameters may be accurately modelled or indeed the model may be inadequate in both respects resulting in the need for the physical testing of scaled models.

### 1.3.1 Line Diagrams

When modelling it is necessary to represent the form of an actual structure in terms of idealized structural members, e.g. in the case of plane frames as beam elements, in which the beams, columns, slabs etc. are indicated by line diagrams. The lines normally coincide with the centre-lines of the members. A number of such line diagrams for a variety of typical plane structures is shown in Figures 1.4 to 1.9. In some cases it is sufficient to consider a section of the structure and carry out an approximate analysis on a sub-frame as indicated in Figure 1.8.

Figure 1.6

Figure 1.6

Figure 1.7

frame

Figure 1.9

frame

Figure 1.9

The support reactions for structures relate to the restraint conditions against linear and rotational movement. Every structural element and structure must be supported in order to transfer the applied loading to the foundations where it is dissipated through the ground. For example beams and floor slabs may be supported by other beams, columns or walls which are supported on foundations which subsequently transfer the loads to the ground. It is important to trace the load path of any applied loading on a structure to ensure that there is no interruption in the flow as shown in Figure 1.10.

Load palh for a typical frame

The loads are transferred between structural members at the joints using either simple or rigid connections (i.e. moment connections). In the case of simple connections axial and/or shear forces are transmitted whilst in the case of rigid connections in addition to axial and shear effects, moments are also transferred.

The type of connections used will influence the degree-of-indeterminacy and the method of analysis required (e.g. determinate, indeterminate, pin-jointed frame, rigid-jointed frame). Connection design, reflecting the assumptions made in the analysis, is an essential element in achieving an effective load path.

### 1.3.3 Foundations

The primary function of all structural members/frames is to transfer the applied dead and imposed loading, from whichever source, to the foundations and subsequently to the ground. The type of foundation required in any particular circumstance is dependent on a number of factors such as the magnitude and type of applied loading, the pressure which the ground can safely support, the acceptable levels of settlement and the location and proximity of adjacent structures.

In addition to purpose made pinned and roller supports the most common types of foundation currently used are indicated Figure 1.11. The support reactions in a structure depend on the types of foundation provided and the resistance to lateral and rotational movement.