## Equivalent Uniformly Distributed Load Method for the Deflection of Beams

In a simply supported beam, the maximum deflection induced by the applied loading always approximates the mid-span value if it is not equal to it. A number of standard frequently used load cases for which the elastic deformation is required are given in In many cases beams support complex load arrangements which do not lend themselves either to an individual load case or to a combination of the load cases given in Appendix 2. Provided that deflection is not the governing design criterion, a...

## Method of Tension Coefficients

The method of tension coefficients is a tabular technique of carrying out joint resolution in either two or three dimensions. It is ideally suited to the analysis of pin-jointed space-frames. Consider an individual member from a pin-jointed plane-frame, e.g. member AB shown in Figure 3.8 with reference to a particular X-Y co-ordinate system. If AB is a member of length LAB having a tensile force in it of TAB, then the components of this force in the X and Y directions are TAB Cos0 and TAB Sin0...

## Method of Joint Resolution

Considering the same frame using joint resolution highlights the advantage of the method of sections when only a few member forces are required. In this technique (which can be considered as a special case of the method of sections), sections are taken which isolate each individual joint in turn in the frame, e.g. In Figure 3.6 four sections are shown, each of which isolates a joint in the structure as indicated in Figure 3.7. Since in each case the forces are coincident, the moment equation is...

## Free and Fixed Bending Moments

When a beam is free to rotate at both ends as shown in Figures 4.70 a and b such that no bending moment can develop at the supports, then the bending moment diagram resulting from the applied loads on the beam is known as the Free Bending Moment Figure 4.70 Free Bending Moment Diagrams Figure 4.70 Free Bending Moment Diagrams When a beam is fixed at the ends encastre such that it cannot rotate, i.e. zero slope at the supports, as shown in Figure 4.71, then bending moments are induced at the...

## Example 83 Propped Cantilever

A propped cantilever is L m long and supports a collapse load of w kN m as shown in Figure 8.7. Determine the position of the plastic hinges and the required plastic moment The number of hinges required to induce collapse (ID+1) 2 (see Figure 8.1) The possible hinge positions are at the support A and within the region of a distributed load since these are the positions where the maximum bending moments occur. In this case the maximum moment under the distributed load does not occur at mid-span...

## Example 32 Two Dimensional Plane Truss

Consider the pin-jointed, plane-frame ABC loaded as shown in Figure 3.9. Construct a table in terms of tension coefficients and an X Y co-ordinate system as shown in Table 3.1. The equilibrium equations are solved in terms of the 't' values and hence the member forces and support reactions are evaluated and entered in the table as shown in Table 3.1. Consider joint B There are only two unknowns and two equations, hence Adding both equations substitute for in tlic first equation rc -4.76 Force...