Module 4 · Lesson 4.3

Trusses as beams, tension coefficients and space frames

Reading a truss as a deep beam, a faster hand method, and stepping into three dimensions.

Why this matters

Joints and sections will solve any plane truss, eventually. But a truss is really a beam with the material moved to where it is useful, and seeing it that way lets you estimate chord forces in your head. Tension coefficients then make the arithmetic mechanical — and, unlike the method of joints, extend to three dimensions without any new ideas.

By the end of this lesson you should be able to

Estimate chord and diagonal forces by treating the truss as a beam
Use tension coefficients to solve a joint
Recognise a compound truss and know why joints alone can stall on one
Apply the determinacy count m + r = 3j to a pin-jointed space frame

What you should already know

Method of joints and method of sections (this module)
Determinacy of plane trusses (this module)
Shear force and bending moment (Module 3)

Cut a parallel-chord truss with a vertical section and take the left-hand part as a free body. Whatever the internal arrangement, that free body must supply the same shear force and bending moment as a solid beam of the same span under the same load.

The truss supplies them by division of labour:

The chords are horizontal, so they cannot help with vertical shear. They resist the bending moment, acting as a push–pull couple: one chord in compression, the other in tension, separated by the depth d. So chord force = M/d.
The diagonals and verticals are the only members with a vertical component, so they carry the shear force in their entirety. For a diagonal at angle θ to the horizontal, the vertical component must equal the shear: F sin θ = V.

That is the whole of §4.5 in two sentences, and it turns a truss into something you can size on the back of an envelope.

Truss forces from beam actions

F_{c h or d} = \frac{M}{d}, F_{d ia g o na l} = \frac{V}{sin θ}

What it calculates: chord and diagonal forces from the equivalent beam's M and V

M: bending moment of the equivalent beam at the section (kNm)
V: shear force of the equivalent beam at the section (kN)
d: depth between the chord centrelines (m)
θ: angle of the diagonal to the horizontal (degrees)

This assumes

Parallel chords, so the chord forces form a couple of constant lever arm
Pin joints and loads applied only at the joints
One diagonal or vertical crossing the section

In plain terms: Chord force is inversely proportional to depth: double the depth of a truss and you halve the chord forces. That single relationship explains why long-span trusses are deep, and why a shallow truss is expensive.

Predict first

A parallel-chord truss is redesigned twice as deep, with the same span and loading. What happens to the chord forces?

The method of tension coefficients is a bookkeeping change that makes joint equilibrium almost mechanical. For a member from joint A to joint B carrying tension T over a length L, define

t = T/L

The force's components are then simply t·Δx and t·Δy, where Δx and Δy are the coordinate differences. Joint equilibrium becomes:

Σ t·Δx + X = 0 and Σ t·Δy + Y = 0

No trigonometry appears anywhere. You never compute an angle, never take a sine or a cosine — just coordinate differences, which you read straight off the geometry. Recover the forces at the end with T = tL.

The real prize is that this extends to three dimensions with a third equation, Σ t·Δz + Z = 0, and nothing else changes. Angles in three dimensions are miserable to handle directly; coordinate differences are not.

Note

Compound trusses, and where the method of joints stalls

A compound truss is two or more simple trusses joined together — by three non-concurrent, non-parallel members, or by a pin and a member. The determinacy count m + r = 2j still applies, and it will still be satisfied.

What can fail is the method of joints: you may find that every remaining joint has three or more unknown members, so there is nowhere to start. The truss is perfectly determinate; the sequential technique has simply run out of joints with only two unknowns.

The cure is a section cut through the connecting members. Three equilibrium equations for the cut free body give you three member forces at once, and the method of joints can resume from there. Recognising this situation, rather than assuming an error, is the practical skill.

Space frames are pin-jointed structures in three dimensions. Every joint now has three equations of equilibrium instead of two, so the determinacy condition becomes:

m + r = 3j

with m + r − 3j > 0 indicating indeterminacy and < 0 a mechanism — exactly the same logic as the plane case, and with the same caveat: the count is necessary but not sufficient. A badly arranged space frame can satisfy it and still be unstable.

The simplest useful case is a tripod: three legs from a common apex down to three anchors. It has m = 3, j = 4 and r = 9 (three pinned supports), so m + r − 3j = 3 + 9 − 12 = 0. Determinate, and if the arrangement is symmetric the three legs share the vertical load equally.

Worked example

A truss read as a beam, and a symmetric tripod

Given

Parallel-chord truss: span 24.0 m, depth 3.00 m, simply supported
Uniform load equivalent to 40.0 kN/m along the span
Separately: a tripod with apex 3.00 m above three anchors on a circle of radius 2.00 m, carrying 60.0 kN vertically

Find

The bottom chord force at mid-span, and the force in each tripod leg.

Practice

A parallel-chord truss spans 24.0 m with a depth of 3.00 m and carries the equivalent of 40.0 kN/m. What is the force in the bottom chord at mid-span, in kN?

Practice

A member runs from (0, 0) to (4.00 m, 3.00 m) and carries a tensile force of 50.0 kN. What is its tension coefficient, in kN per metre?

Practice

A pin-jointed space frame has 12 members, 6 joints and 6 restraint components. What is m + r − 3j?

Worked example

A compound truss that stops the method of joints

Given

Two simple triangular trusses, each spanning 6.00 m, joined side by side
They are connected by three members that are neither concurrent nor parallel
Total: 21 members, 12 joints, 3 reaction components

Find

Whether the truss is determinate, why the method of joints stalls, and how to restart it.

Practice

A compound plane truss has 21 members, 12 joints and 3 reaction components. What is m + r − 2j?

Summary

A parallel-chord truss is a beam: chords carry the moment as a couple, diagonals carry the shear
Chord force = M/d, so doubling the depth halves the chord forces
Diagonal force = V/sin θ
Tension coefficient t = T/L turns joint equilibrium into Σt·Δx = 0 and Σt·Δy = 0, with no trigonometry
The same method extends to three dimensions by adding Σt·Δz = 0
Space frames: m + r = 3j for static determinacy
A compound truss may need a section before the method of joints can start

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This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint