Analysis

Structural analysis uses the concept of resolving forces, which can be analytically or graphically.

It is now necessary to introduce the concept of resolving a single force into its components in the horizontal and vertical directions. The magnitude of these components is such that their combined effect is the same as that of the single, original force.

It is worth noting here that in order to define a force completely it is necessary to specify its magnitude and direction. This can be done analytically or graphically simply by drawing the force to scale in the appropriate direction. Both representations are equivalent.

The analysis of trusses considers the equilibrium of the force components acting on each joint in turn.

If a number of forces meet at a point then each can be resolved, into two non-parallel components. The net effect is determined simply by summing these components in the two directions separately. If such a force system is in equilibrium, the sum of the force components in each direction must be zero. This principle can be applied to the system of forces acting at an individual joint within a truss. Because individual members of a truss are subject to direct forces only, the direction of such forces must be the same as the physical direction of the member itself. The equilibrium of a joint can therefore be considered by treating a system of forces acting in directions corresponding to the member meeting at the joint and resolving the forces into vertical and horizontal components.

This process of resolution of forces can be applied repeatedly at each joint in the truss and can be carried out either analytically or graphically.

The first step is to calculate support reactions; these are independent of the truss configuration.

For a simple truss, the first step in the analysis is to calculate the reactions in the same way as for a simple beam, by taking moments about one end support. For the truss arrangement shown (where all the truss angles, q, are 45°, and the length of each truss member is 1), taking moments about Joint 4:

(Reaction at joint 1) x 1.5 - 10 x 1 - 20 x cos(q) = 0

Hence the reaction = 13.3kN

This calculation is independent of the truss shape. If the structure were a beam, the calculations would be identical. The truss form comes into play when determining the value of the forces in the individual members.

The force system acting on each joint is then considered in turn.

Referring to the figure, and considering the joint at the left hand support (joint 1), the forces which meet at joint 1 can be resolved into their components.

By applying the condition that the sum of all vertical and horizontal components must be equal to zero, values for each can be obtained. If the angle of member 1-2, the diagonal member (with respect to the horizontal member 1-3) is q:

Resolving vertically: Reaction – Force in diagonal (1-2) ´ sin(q) = 0

hence F _1-2 = 18.84 kN

Resolving horizontally: AB x cos(q) - AC = 0

hence AC = 13.3 kN

These calculations require a consistent sign convention.

Implicit in these calculations is a convention that vertical forces are positive upwards and horizontal forces are positive to the left. This is simply a convention, and any convention can be used as long as it is adhered to throughout.

Looking at the direction of forces, it is easy to establish that for the loading conditions shown, 1-2 is subject to compression and 1-3 is subject to tension.

It is most convenient to start at a joint where there are just two unknown forces.

As a general strategy it is easiest to start the analysis at a joint where only two forces are unknown. In this case, joint 1 is a suitable starting joint and the forces at that joint can be established. Knowing the forces in members 1-2 and 1-3, it is clear that either joint 2 or 3 can now be analysed. Progressing in this way through the truss, the forces in all members can be established. This is often a time-consuming operation, and thankfully alternative methods exist which may be quicker and simpler.