Trusses

Look at a railway bridge, a roof over an arena, a transmission tower or the boom of a tower crane, and you will see the same pattern: slender straight members joined at their ends to form triangles. That is a truss. Trusses carry large loads over long spans with very little material, because every member is loaded only along its own axis, in pure tension or pure compression, which is the most efficient way to use steel or timber.

This chapter explains why trusses work, how to tell whether a truss can be analyzed by statics, and the two standard methods for finding member forces: the method of joints and the method of sections. It also shows how to spot members that carry no load, and finishes with frames and machines, structures whose members are not two-force members and must be taken apart piece by piece.

What makes a truss

A triangle is the only polygon that cannot change shape without changing the length of a side. Pin four bars into a square and it folds flat under a sideways push; add a diagonal and it becomes rigid. A truss is a structure built entirely from such triangles.

Different arrangements suit different jobs. In a Pratt truss under gravity load, the long diagonals are in tension and most of the shorter verticals are in compression, which suits steel, because long members in compression can buckle. A Howe truss reverses this, which suited the timber-and-iron bridges of the 1800s, when timber diagonals were good in compression and iron rods carried the tension. The Warren truss uses equilateral or isosceles triangles with no verticals. The Fink truss, with its W-shaped web, is what holds up most wood-frame house roofs in North America.

The top and bottom members are called chords, and the members between them make up the web. The points where members meet are joints (or panel points), and the space between adjacent verticals is a panel.

The modeling assumptions

Truss analysis rests on three assumptions:

• Members are connected at their ends by frictionless pins.
• Loads and reactions are applied only at the joints.
• The weight of each member is small compared with the loads, or is split equally between its two end joints.

Under these assumptions, every member has forces at only two points and no couples, so every member is a two-force member (Lesson 4). Its force acts along its own axis and is either tension (pulling on the joints, trying to stretch the member) or compression (pushing on the joints, trying to shorten it).

Real trusses are welded or bolted through gusset plates, not pinned, and their members do carry some bending. But because the members are slender and meet at a point, these secondary effects are usually small, and pin-jointed analysis gives member forces within a few percent. Floor and roof loads reach the joints through a deck and floor beams that sit on the panel points, which is why the second assumption holds in practice.

Is the truss determinate?

Each joint is a particle in equilibrium, giving two equations, so a planar truss with \(j\) joints gives \(2j\) equations. The unknowns are the \(m\) member forces plus the \(r\) support reactions. Comparing:

With the usual pin and roller (\(r = 3\)), a determinate simple truss has \(m = 2j - 3\). The count is necessary but not sufficient: the members must also be arranged in triangles so that no part of the truss can move as a mechanism. A truss built by starting with one triangle and adding two members and one joint at a time (a simple truss) always passes both tests.

The method of joints

The method of joints treats each joint as a particle and applies \(\sum F_x = 0\) and \(\sum F_y = 0\). Because each joint gives only two equations, you must start at a joint with no more than two unknown member forces, usually a support, after finding the reactions from the equilibrium of the whole truss.

A sign convention keeps the bookkeeping clean: assume every unknown member is in tension, which means drawing its force pulling away from the joint. A positive answer is tension (T); a negative answer is compression (C). Carry the sign forward: at the next joint, a member already found to be −37.5 kN is still drawn pulling away, with the value −37.5 substituted.

The pattern is typical of a truss carrying gravity load. The top chord is in compression and the bottom chord in tension, just like the top and bottom of a bent beam. In fact a truss is a beam with the unneeded material removed: the chords do the job of the flanges and the web members do the job of the web.

Zero-force members

Some members carry no force under a particular loading. Spotting them first saves work and avoids errors. Two rules, both from equilibrium of a single joint:

• If only two non-collinear members meet at a joint with no external load or support, both are zero-force members.
• If three members meet at an unloaded joint and two of them are collinear, the third member is a zero-force member.

In the Pratt truss, joint U2 is unloaded and has two collinear top chords plus the vertical U2L2. By the second rule, \(F_{U2L2} = 0\).

Zero-force members are not useless. They brace compression members against buckling by cutting their unsupported length in half, they keep the truss stable during construction, and they pick up load when the loading changes. Under a single moving vehicle, or a wind load, U2L2 would carry force. Engineers design every member for the worst of many load cases.

The method of sections

If you need the force in one member in the middle of a large truss, working joint by joint from the support is slow. The method of sections goes straight to it. Cut the truss into two pieces with an imaginary line that passes through the member you want, cutting no more than three members whose forces are unknown. Then apply the three equilibrium equations to either piece.

The trick is to choose moment points well. Take moments about the point where two of the cut members intersect; those two drop out, leaving one equation in the third.

The vertical-force equation in that example reveals something useful. The net vertical force on the left piece, \(30 - 20 = 10\) kN, is the shear force in that panel, and in a truss with horizontal chords the diagonal is the only member that can carry it. The moment about L2, 120 kN·m, is the bending moment at that point, carried by the couple between the chords: \(30 \text{ kN} \times 4 \text{ m} = 120\) kN·m. Deeper trusses therefore have smaller chord forces, which is why long-span trusses are deep.

Choosing a method

• All member forces wanted: method of joints, starting from a support.
• One or two members in the middle of a large truss: method of sections.
• Both are often combined: sections to reach the middle quickly, joints to finish nearby members.

In professional practice, trusses are analyzed by computer using the matrix stiffness method, which solves the same joint equilibrium equations all at once. Hand methods remain essential for checking computer output: an engineer who seals a design is responsible for the results, whatever software produced them.

Why compression members matter

A tension member fails when the material yields or breaks, so its strength depends only on its cross-sectional area. A compression member can fail at a much lower load by buckling: bowing sideways and collapsing. Long slender compression members are weak, which is why designers keep them short (by adding bracing members, often the "zero-force" ones), make them stockier, or choose a truss form, like the Pratt, that puts the long members in tension.

The first Quebec Bridge, which you met in ENGG-100, collapsed in 1907 when a built-up lower chord member in the south anchor arm, next to the main pier, buckled. The chord forces had been calculated, but the member's self-weight had been underestimated and the design of its lacing (the light bracing that ties the parts of a built-up member together) was inadequate for the compression it carried. Statics gives the member forces; the next courses in mechanics of materials and steel design teach how to make members that can carry them.

Frames and machines

Not every structure is a truss. In a frame (a structure that stays put) or a machine (one that transmits forces and moves), at least one member is a multi-force member: it has forces at three or more points, or carries loads along its length. Its forces are not along its axis, so the shortcuts of truss analysis do not apply.

The method is to take the structure apart. Draw a free-body diagram of each member separately, showing the pin forces as unknown components. Where two members share a pin, the forces they exert on each other are equal and opposite (Newton's third law), so use the same symbols with opposite directions on the two diagrams. Identify any two-force members first, since each adds only one unknown. Then write three equations for each multi-force member.