This module covers planar trusses, frames, and machines, detailing the theoretical assumptions required for manual analysis and the Methods of Joints and Sections. Trusses are a fundamental architectural element, allowing engineers to span massive distances—like stadium roofs and aircraft hangars—with minimal material weight.

By arranging members exclusively in triangles, the truss acts as a very deep, extremely lightweight beam. The triangle is the only 2D polygon that is inherently geometrically stable; it cannot change shape without physically altering the length of one of its sides. A rectangle, by contrast, will easily collapse into a parallelogram under lateral load unless cross-braced (which simply turns it into two triangles).

To mathematically calculate the internal forces inside a truss by hand, structural engineers must make three vital assumptions that simplify the complex physical reality into a solvable mathematical model:

Truss analysis often reveals that certain members carry exactly zero force under specific loading conditions. These are called Zero-Force Members. Why are they physically built into the real truss if they carry no load? They are required for overall geometric stability, to prevent long slender compression members from buckling out of plane, or to handle unexpected dynamic loads (like shifting wind directions or earthquakes).

Architectural trusses often contain structural members that mathematically carry zero internal force under a specific loading condition. Identifying them immediately simplifies the structural analysis equations. They provide critical stability to long, slender compression members to prevent sudden geometric buckling.

The Method of Joints is the required analytical method when an engineer needs to find the internal force in every single member of the entire truss framework.

The Concept: Since the entire global truss is in static equilibrium, every individual internal joint (node) must also be in perfect equilibrium. By isolating a single joint, we can treat it as a 2D Concurrent Force System. Because all forces pass through the pin, there are no moments to calculate ($\Sigma M = 0$ is naturally satisfied).

The Method of Sections is used when you only need to find the forces in a few specific members (often located near the middle of a long truss). It is drastically faster than the Method of Joints because it bypasses the need to calculate every single joint sequentially from the support.

The Concept: Instead of isolating a single concurrent joint, we draw an imaginary line slicing completely through the entire truss, cutting the specific members we want to analyze. This divides the truss into two separate rigid bodies. We discard one half and apply full rigid body equilibrium to the remaining half.

While trusses are incredibly efficient, many common architectural structures (like modern high-rise portal frames, A-frame cabins, and mechanical construction equipment) do not meet the strict assumptions of a pure truss. These are classified as Frames or Machines.

Method of Members: To analyze a frame or machine, you cannot simply isolate a point using the Method of Joints. Instead, you must carefully dismantle the entire physical structure into individual Free Body Diagrams (FBDs) for each separate member. Then, you apply the three rigid body equilibrium equations ($\Sigma F_x=0$, $\Sigma F_y=0$, $\Sigma M=0$) to each multi-force member independently.

Standard truss analysis fundamentally assumes that all joints are perfectly frictionless pins, meaning members only experience pure axial tension or compression.

In real architectural construction, steel trusses are bolted or welded together using large, thick metal plates called gusset plates. These rigid connections prevent the members from rotating freely. As the truss deflects slightly under a heavy roof load, the stiff gusset plates force the members to bend, inducing "secondary bending moments" alongside the primary axial forces. While often small enough to ignore in preliminary design, these secondary stresses must be verified in final engineering calculations.

While 2D planar trusses are common, modern stadiums and airports utilize 3D Space Trusses consisting of tetrahedral units. Analysis requires extending the Method of Joints into three dimensions ($X$, $Y$, and $Z$) simultaneously.

To simplify hand calculations, structural engineers treat real-world trusses as "ideal" models by enforcing three strict mathematical assumptions: