Pin Jointed Frames Analysis: Method of Joints

Pin-jointed frames, also known as trusses, are structural systems made up of straight members connected at their ends by pin joints. These joints are assumed to be frictionless pins, allowing members to rotate freely relative to each other at the joint. The members are typically assumed to be subjected only to axial forces (tension or compression).

Method of Joints Explained

The method of joints is a technique used to determine the forces in the members of a pin-jointed frame. The method involves analyzing the equilibrium of forces at each individual joint of the structure. Since the joints are assumed to be pins, they can be treated as points where multiple forces (the member forces and any external loads or reactions applied at that joint) meet.

Equilibrium Equations for a Joint

For a system of concurrent forces acting at a point in a two-dimensional plane to be in static equilibrium, two conditions must be satisfied:

1. The sum of the horizontal components of all forces acting at the joint must be zero. This can be represented mathematically as $\sum F_x = 0$.
2. The sum of the vertical components of all forces acting at the joint must be zero. This can be represented mathematically as $\sum F_y = 0$.

The third equation of static equilibrium, which involves the summation of moments ($\sum M = 0$), is automatically satisfied for a point where all forces are concurrent. The moment of a force about the point through which it passes is zero. Therefore, considering the equilibrium of forces at a single joint, only the two force equilibrium equations ($\sum F_x = 0$ and $\sum F_y = 0$) provide independent information.

Number of Equations per Joint

When analyzing a pin-jointed frame using the method of joints, we consider each joint as a free body subjected to the forces from the connecting members and any external loads or reactions. For each such joint, we can write down the two equations of static equilibrium: one for the sum of forces in the horizontal direction and one for the sum of forces in the vertical direction.

Thus, for each pin joint in a 2D frame, there are exactly two independent equations of static equilibrium available to solve for the unknown member forces or reactions connected to that joint.

The number of equations of static equilibrium available for analyzing each joint in a pin jointed frame by the method of joints is 2.