# Two-dimensional systems of bars — trusses

Bars are usually connected in two or three dimensions to form more complex structures called trusses. Common examples of truss structures are roof trusses and bridges.

A model for a truss is usually idealized according to the figure below.

The bars are assumed to be connected using hinge joints which can only transfer forces along the directs of the bars, i.e. $\alpha=0$ in the figure below. This is a modeling assumption — a simplification of reality — which is not fully accurate.

To see that the forces can only be transmitted along the bars, consider moment equilibrium around any joint. Thus, it is only the magnitude of the sectional force that is unknown, the direction of the force must coincide with the direction of the bar.

## Statically determinate trusses

There are two classic methods for analyzing statically determinate trusses: the method of joints and the method of sections. These methods will now be described in short.

### Method of joints

The method consists of making a free body diagram around each joint and establishing equilibrium for the join. For each joint one can establish two equations for equilibrium (three equations for a three-dimensional truss). The moment equation does not provide additional information since each sectional force has its line of action through the joint giving a leverage are that is zero.

Example: Method of joints

Description

Determine the bar forces in the truss below using the method of joints.

Solution

Study each joint and establish equilibrium for the unknown bar forces. One can start with any joint but it is often preferable to start with a joint which has few bars connected to it. In this case, we start with the upper middle joint since the structure is symmetrical.

Study each joint and establish equilibrium for the unknown bar forces. One can start with any joint but it is often preferable to start with a joint which has few bars connected to it. In this case, we start with the upper middle joint since the structure is symmetrical.

### Method of sections

Example: Method of sections

Description

Using the method of sections, determine the normal forces $N_1, N_2$ and $N_3$ for the bars marked $1-3$ in the truss below.

Solution

We will consider the part to the right of the section in the figure below. Global equilibrium for the free body diagram gives the sought normal forces $N_1, N_2$ and $N_3$.

\begin{align} \eqdown& N_2+ P=0 \qgives N_2 = -P \\\ \eqccwmom{N_3}& N_1 \ L +N_2 \ L – P \ 2L =0 \qgives N_1=3P \\\ \eqleft& N_3+ N_1=0 \qgives N_3=-N_1=-3P \end{align}

Note, we could also have studied the part to the left of the section but then we would have had to start by determining the reaction forces.

This method is useful when you are only interested in a few sectional forces and do not want to through joint by joint. For both methods, (method of joints and method of sections), it is easy to make mistakes and if you make one mistake everything that follows will be wrong. It is, therefore, important to check to that the equations of equilibrium are satisfied.