The analysis of structures involves determining the internal forces acting within the members of interconnected bodies. In statics, we primarily focus on structures that are in equilibrium, such as trusses, frames, and machines.

Consequence of truss assumptions: Because of these assumptions, each truss member acts as a two-force member. The internal force is directed along the axis of the member and can be either Tension (T) (pulling the joint) or Compression (C) (pushing the joint).

A simple planar truss can be analyzed to see if it is statically determinate internally based on its number of members, number of joints, and number of external reaction components. Since each joint provides two equations ($\Sigma F_x=0, \Sigma F_y=0$), the total available equations is $2j$.

• If $b + r = 2j$, the truss is exactly statically determinate (assuming proper constraint).
• If $b + r > 2j$, the truss has more members or supports than necessary and is statically indeterminate to the $(b+r-2j)^{\text{th}}$ degree.
• If $b + r < 2j$, the truss is unstable and will collapse as a mechanism.

There are two primary methods for analyzing planar (2D) trusses:

1. The Method of Joints
2. The Method of Sections

The Method of Joints is based on the principle that if a truss is in equilibrium, then every single joint in the truss must also be in equilibrium. Since all forces at a joint are concurrent, the moment equation $\Sigma M = 0$ is automatically satisfied. We only need to apply the force equilibrium equations:

The Method of Sections is used when you need to find the internal forces in only a few specific members of a truss, rather than the entire structure. For example, if you need to determine the force in a single member in the middle of a large bridge truss to check for potential buckling, you would use this method instead of solving every joint sequentially from the support. It involves "cutting" the truss into two distinct parts.

In some truss configurations, specific members carry no load. These are called zero-force members. They are often included to provide stability during construction, prevent buckling of long compressive members, or carry loads if the loading condition changes.

Identifying zero-force members simplifies the analysis significantly.

• Case 1: Two non-collinear members forming a joint with no external load or support reaction. If a joint connects only two members that are not in a straight line, and there is no external load or support reaction at that joint, both members are zero-force members. (Reason: $\Sigma F = 0$ along axes parallel and perpendicular to one member shows the other must be zero).
• Case 2: Three members forming a joint, two of which are collinear, with no external load or support reaction. If a joint connects three members, two of which lie in a straight line, and there is no external load or reaction at the joint, the third (non-collinear) member is a zero-force member. (Reason: Summing forces perpendicular to the collinear members directly shows the force in the third member must be zero).

A space truss is a three-dimensional framework of members joined at their ends. The basic element of a space truss is a tetrahedron formed by six members connecting four joints.

For a space truss, the conditions of equilibrium for each joint require that the resultant force acting on the joint be zero: $\Sigma \mathbf{F} = 0$. This resolves into three independent Cartesian scalar equations, which means we can solve for up to three unknown member forces at any given joint.

Unlike trusses, frames and machines are structures that contain at least one multi-force member (a member subjected to more than two forces, or members subjected to bending moments).

• Frames: Generally stationary structures designed to support loads.
• Machines: Structures containing moving parts designed to transmit or alter the effect of forces.