6 Trusses, Frames, and Machines

Equilibrium and free-body diagrams

Trusses, frames, and machines are assemblies of connected members. Their unknown forces can be found by applying equilibrium to the whole assembly, selected members, or individual joints. The most efficient choice depends on how the members are connected and which forces are sought.

For a planar body in static equilibrium, the force and moment sums are zero:

A particle or joint with concurrent forces uses the two force equations. A general rigid body can use all three equilibrium equations. Choose moment centers strategically: forces whose lines of action pass through the chosen point create no moment about that point and can therefore be eliminated from the moment equation.

Ideal trusses and axial forces

A simple truss is modeled as straight, weightless two-force members connected by frictionless pins. Applied loads and support reactions act only at the joints. Each member carries an axial force: tension pulls away from a joint, while compression pushes toward it. These assumptions idealize real structures; connection details and member weights may require more detailed analysis.

Begin by drawing a free-body diagram of the entire truss and finding any required support reactions. Then use the method of joints or the method of sections. Rules for zero-force members can sometimes simplify the analysis before either method is applied.

Solving trusses joint by joint

Isolate one pin joint at a time. Show each connected member force along that member’s axis, together with any external loads or support reactions acting at the joint. Because the forces meet at the joint, apply $\sum F_x=0$ and $\sum F_y=0$.

Start at a joint with no more than two unknown member forces, then proceed to neighboring joints as forces become known. A convenient sign convention is to assume every unknown member force is in tension, with arrows directed away from the joint. A positive result confirms tension; a negative result means the member is in compression. When carrying a solved force to another joint, use an equal and opposite force there.

Example: A symmetric triangular truss has supports at $A$ and $B$, separated by $4\,\text{m}$, and a top joint $C$ centered $3\,\text{m}$ above the base. A downward load of $10\,\text{kN}$ acts at $C$. By symmetry and whole-truss equilibrium, the vertical reactions are $A_y=B_y=5\,\text{kN}$, and $A_x=0$. Each sloping member has length $\sqrt{13}\,\text{m}$, and its vertical direction ratio is $\frac{3}{\sqrt{13}}$. At joint $C$, the equal member forces must provide an upward resultant of $10\,\text{kN}$. Assuming tension gives

Both sloping members therefore carry about $6.01\,\text{kN}$ in compression. Equilibrium at either base joint gives a tensile force of $3.33\,\text{kN}$ in the bottom member.

Finding selected forces with sections

Use a section cut when only a few member forces are needed. Imagine cutting through the members of interest and draw a free-body diagram of one side. Each cut member contributes one unknown axial force along its axis. A useful planar cut generally exposes no more than three unknown member forces, because the section provides only three independent rigid-body equilibrium equations.

Choose the side with fewer loads and unknowns, then apply force and moment equilibrium to it. Taking moments about the intersection of the lines of action of two unknown cut forces can eliminate both and solve directly for the third. A cut need not be straight, but it must separate the truss into parts that can each be treated as a free body.

Use joints when many or all member forces are needed, and sections when selected forces are needed efficiently. The two methods can also be combined.

Analyzing frames and machines

A frame is a stationary assembly intended to support loads. A machine has parts that can move relative to one another and is often used to transmit or multiply force. Either may contain multi-force members, which can carry axial force, shear, and bending moment.

The truss joint and section methods rely on members being two-force bodies with known axial-force directions. They generally cannot be applied to a multi-force frame or machine member as if its force acted only along the member. Instead, isolate the whole structure, individual members, or useful groups of connected parts, and apply rigid-body equilibrium to each selected free-body diagram.

A practical analysis proceeds as follows:

1. Choose a free body. Start with the whole assembly if its support reactions can be found from equilibrium. Then isolate the members or groups needed to determine internal connection forces.

2. Draw each diagram completely. Include applied loads and couples, support reactions, relevant weights, and forces and moments at exposed connections. A pin connection between planar members typically transmits two force components but no couple; a fixed connection can transmit force components and a couple.

3. Use two-force-member information. A member loaded only at two points has equal, opposite, collinear end forces. Its force direction is known from the line joining those points, leaving only its magnitude unknown.

4. Keep interaction forces consistent. When two bodies are separated, the force exerted by one on the other and the force exerted back are equal in magnitude and opposite in direction. Show this action–reaction pair on the respective free-body diagrams.

5. Write equilibrium for each body. In two dimensions, a general member supplies up to three independent equations. Solve connected sets of equations together when one diagram alone has too many unknowns.

For example, a pinned link connected at only its two ends is a two-force member, so its end forces lie along the link. A lever attached to that link and also loaded by an external force is a multi-force body. Isolate the lever and write its force and moment equations, including the link force at the pin. The link force on the lever must be opposite the lever’s force on the link.

Checks and common pitfalls

Include only forces acting on the chosen free body. Forces between parts inside a whole-assembly diagram are internal and cancel in pairs; they appear when the parts are isolated separately.

Do not omit support reactions or a component’s weight when it is relevant to the model. Distinguish a pin’s force components from a moment-resisting connection: an ideal pin does not resist a couple.

Check that there are enough independent equations for the unknowns and that the structure is stable under the modeled supports. When possible, verify results using equilibrium of the whole structure or a different joint or member. A negative result usually indicates that an assumed force direction was reversed, not that the solution failed.

In summary, ideal trusses are analyzed through axial member forces, using joints for many unknowns or sections for selected members. Frames and machines require free-body diagrams of connected bodies because their multi-force members can carry forces and moments not restricted to the member axis. In every case, a reliable solution depends on choosing the free body carefully, modeling supports and connections correctly, keeping action–reaction pairs consistent, and applying equilibrium equations.

• multi-force member