Analysis of Trusses

Introduction

A truss is a structural system composed of straight, slender members connected at their ends to form a stable framework. Trusses are widely used in roof systems, bridges, towers, gantry structures, temporary works, and industrial platforms because they can carry substantial loads with relatively little material. Their efficiency results from arranging members so that loads are transferred primarily through axial tension and compression rather than bending.

Truss analysis determines support reactions, member axial forces, and the nature of each member force. The results support decisions about member sizing, connection design, constructability, inspection, and safety. This chapter develops methods for plane and space trusses, explains zero-force members and stability, and introduces computational approaches while emphasizing the assumptions that make ideal truss analysis valid.

Learning Objectives

By the end of this chapter, a student should be able to:

1. Explain the geometry, load-transfer mechanism, and idealizations of plane and space trusses.
2. Determine support reactions and member forces using the method of joints.
3. Select and apply suitable sections of a truss.
4. Identify zero-force members using equilibrium rules.
5. Assess determinacy, stability, and the limitations of ideal truss models.
6. Formulate basic matrix and computational approaches to truss analysis.
7. Interpret results for construction and engineering practice.

1. Truss Fundamentals

1.1 Terminology

A joint or node is the point at which members meet. A member is a straight element between two joints. A panel is a portion of a truss between adjacent joints or panel points. A chord is a principal longitudinal member, commonly the top or bottom boundary of a truss. Web members connect the chords. A support reaction is the force or moment developed by a support. A tension member is elongated by its axial force; a compression member is shortened.

A plane truss lies in one plane and is loaded by forces in that plane. A space truss is a three-dimensional arrangement whose members and loads do not necessarily lie in one plane.

1.2 Ideal truss assumptions

Elementary truss analysis normally assumes that:

• members are straight and connected only at their ends;
• joints are frictionless pins, so they do not transmit bending moments;
• external loads and reactions act only at joints;
• member self-weight is either neglected or replaced by equivalent joint loads;
• each member carries only axial force;
• the geometry remains sufficiently unchanged for small-displacement analysis;
• material behavior is linear elastic when deformation or stiffness methods are used.

Under these assumptions, a member is a two-force member. Its end forces are equal, opposite, and collinear, so the internal force is axial. Real connections have gusset plates, bolts, welds, eccentricities, and finite stiffness. Consequently, secondary bending and local stresses may occur even when the ideal model is statically determinate.

1.3 Load transfer

A joint load is resolved into member forces. Members in tension transfer force by pulling adjacent joints together, while compression members push them apart. The support reactions balance the external loads, and the complete force path runs from applied load through joints and members to the foundations.

A triangular arrangement is fundamental because a triangle is geometrically stable under pin connections. A rectangular panel without a diagonal can deform into a parallelogram without changing member lengths; it is therefore a mechanism unless a rigid connection or other bracing is provided.

2. Determinacy and Stability

For a plane truss with \(m\) members, \(j\) joints, and \(r\) independent reaction components, the usual count for static determinacy is

Here, \(m\), \(j\), and \(r\) are dimensionless counts. For a conventional plane truss with a pin and roller support, \(r = 3\), so the condition becomes

For a space truss, equilibrium at each joint supplies three independent equations, and the corresponding count is

A structure satisfying the count may be statically determinate, but the count alone does not guarantee stability. Members may be arranged in a geometrically unstable way, or some equations may be dependent. Conversely, a stable truss with more members than required is statically indeterminate and requires compatibility and stiffness relationships.

The degree of static indeterminacy, when the structure is stable, is commonly estimated as

for a plane truss and

for a space truss. A positive value indicates that additional deformation or constitutive equations are required. A negative value generally indicates a mechanism or instability, although geometry must still be examined.

3. Support Reactions

Analysis begins with the entire truss treated as a free body. Draw all applied loads and support reactions, then apply global equilibrium. For a plane structure:

where \(F_x\) and \(F_y\) are horizontal and vertical force components and \(M_O\) is moment about any convenient point \(O\). Use a consistent sign convention and consistent units, such as kN and m, producing moments in kN·m.

For a space structure:

and

A pin support in a plane generally supplies two force components, while a roller supplies one force component normal to its supporting surface. The support model must reflect the actual restraint; an incorrect model produces incorrect reactions and member forces.

4. Method of Joints

The method of joints isolates one joint at a time and applies particle equilibrium. Since all member forces at an ideal joint are concurrent, moments are not needed. For each plane joint:

The standard procedure is:

1. Calculate the support reactions from the whole-truss free body.
2. Choose a joint with no more than two unknown member forces, if possible.
3. Assume every unknown member force is tensile, directed away from the joint.
4. Resolve inclined forces into orthogonal components.
5. Solve equilibrium equations.
6. Interpret a positive result as tension and a negative result as compression under the assumed convention.
7. Move to an adjacent joint and continue.

For a member making angle \(θ\) with the positive horizontal axis, its force components are

where \(F\) is the axial force. The trigonometric functions are dimensionless, so the force components have the same units as \(F\).

Worked Example 1: Method of joints

Consider a symmetric triangular plane truss with supports at \(A\) and \(B\), span \(6\,m\), apex \(C\) located at midspan and height \(4\,m\). A downward load of \(20\,kN\) acts at \(C\). Support \(A\) is a pin and \(B\) is a roller. Members are \(AC\), \(BC\), and \(AB\).

By symmetry,

There is no horizontal external load, so \(A_x = 0\). Each inclined member has horizontal projection \(3\,m\), vertical projection \(4\,m\), and length \(5\,m\). Thus

At joint \(C\), assume \(F_{AC}\) and \(F_{BC}\) are tensile. Horizontal equilibrium gives equal and opposite horizontal components, so the two member forces have equal magnitude, say \(F\). Vertical equilibrium gives

Therefore,

The assumed tensile directions at joint \(C\) point downward along the two members, but a tensile member connected to the apex pulls away from the joint toward the supports, which indeed has downward components. Hence \(AC = BC = 12.5\,kN\) tension.

At joint \(A\), the inclined member contributes a horizontal component of \(12.5(3/5)=7.5\,kN\) toward the left or right depending on the joint direction. Equilibrium requires the bottom chord force to oppose it. Thus \(AB=7.5\,kN\) compression. The result is consistent with the typical behavior of a simple triangular truss: the upper inclined members are in tension under the apex load and the bottom member is in compression under this particular geometry and load arrangement. The sign should always be checked against the adopted joint directions.

5. Zero-Force Members

A zero-force member carries zero axial force for the stated loading and idealized geometry. It may nevertheless be essential for stability, construction, resistance to reversed loading, or restraint against out-of-plane movement.

Two reliable inspection rules for an unloaded joint are:

1. If only two non-collinear members meet at a joint, both members are zero-force members.
2. If three members meet at a joint, two are collinear, and no external load or support reaction acts at the joint, the non-collinear member is a zero-force member.

These rules follow directly from joint equilibrium. For example, in the second case, the two collinear members can balance each other along one line, but no force is available to balance a component in the transverse direction; therefore the third member force must be zero.

The rules must not be applied blindly. A joint load, support reaction, or collinearity error changes the conclusion. Zero-force members can also become force-carrying when loads change, when wind reverses direction, or when construction stages are considered.

6. Method of Sections

The method of sections is efficient when forces in only a few members are required. Pass an imaginary cut through the truss, separating it into two parts. The cut should generally pass through no more than three unknown member forces in a plane truss. Select one side, draw its free-body diagram, and apply

Choose the moment center strategically. If two cut-member lines intersect at a point, taking moments about that point eliminates both corresponding unknowns and directly determines the third. If cut members are parallel, use force equilibrium or take moments about another point.

Worked Example 2: Strategic section

Suppose a loaded plane truss has already been analyzed for reactions, and a section cuts members \(AB\), \(BC\), and \(CD\). To determine the force in \(BC\), choose the left portion and take moments about the intersection of the lines of action of \(AB\) and \(CD\). Their moments are zero. If the external loads on the selected portion produce total moment \(M_L\), and the perpendicular distance from the intersection to member \(BC\) is \(d\), then

The sign determines tension or compression according to the assumed direction. This calculation is valid only if \(d\) is measured perpendicular to the member force line and all moments use consistent units. The section method does not eliminate the need to check that the selected free body is in equilibrium.

7. Space Trusses

A space truss carries loads in three dimensions. At a joint, each member is represented by a force directed along its member axis. If a member extends from point \(i(x_i,y_i,z_i)\) to point \(k(x_k,y_k,z_k)\), define

and its length

The unit vector from \(i\) to \(k\) is

If the member force is \(F_{ik}\), its vector contribution at joint \(i\) is \(F_{ik}e_{ik}\). Joint equilibrium is

A space joint can therefore provide three scalar equations. The space-truss count \(m+r=3j\) is a preliminary test, not a proof of stability. Members must be arranged so that the framework resists translations and rotations in all relevant directions. A visually strong planar arrangement may still be weak out of plane.

8. Matrix and Computational Analysis

For larger or indeterminate trusses, the stiffness method is generally more efficient than repeated hand calculations. Each node has two degrees of freedom in a plane truss and three in a space truss. For a plane member connecting nodes \(i\) and \(j\), let \(c=cosθ\), \(s=sinθ\), \(A\) be cross-sectional area, \(E\) Young’s modulus, and \(L\) member length. Its global stiffness matrix is

The matrix relates the element nodal displacement vector to the element nodal force vector. Units are force per length, because \(AE/L\) has units of force/length. Assembly places each element matrix into the global structural matrix according to node connectivity:

where \(K\) is the assembled stiffness matrix, \(d\) is the vector of unknown nodal displacements, and \(P\) is the applied nodal-load vector. Boundary conditions are imposed by restraining the appropriate displacement components. After solving for \(d\), member extension and axial force can be obtained. If \(u_i\) and \(u_j\) are the global displacement components at the ends of a plane member,

and

where \(δ\) is axial extension and \(N\) is axial force. Positive \(N\) conventionally denotes tension. These equations assume small displacements, linear elasticity, pin-jointed axial behavior, and correctly specified connectivity and restraints.

Computational good practice

A software result is not a substitute for engineering judgment. A dependable workflow includes:

• checking node coordinates, units, member connectivity, areas, and material properties;
• verifying support restraints and load directions;
• inspecting the deformed shape for mechanisms or unrealistic motion;
• comparing reactions with global equilibrium;
• checking selected members by hand using joints or sections;
• identifying unusually large forces caused by modeling errors;
• performing sensitivity checks for load combinations and stiffness assumptions.

A singular stiffness matrix often indicates a mechanism, missing restraint, disconnected node, duplicate or incorrect connectivity, or an unconstrained rigid-body motion. It may also result from numerical problems or inconsistent units.

9. Assumptions, Limitations, and Design Interpretation

Ideal truss forces are not automatically design forces. A complete engineering assessment may require:

• tension yielding, rupture, bolt or weld failure, and net-section loss;
• compression buckling, including effective length and end restraint;
• slenderness limits and local plate buckling;
• connection eccentricity and gusset-plate behavior;
• member self-weight and load distribution between joints;
• second-order effects when compression members displace significantly;
• fatigue under repeated or fluctuating loads;
• fire, corrosion, accidental impact, seismic, wind, and construction-stage effects;
• serviceability limits such as deflection and vibration.

A compression member’s stability is governed not only by its axial force but also by its effective length, cross-sectional properties, imperfections, and restraint. A member may have a modest calculated axial force yet require careful detailing because its slenderness is high. Similarly, a tensile member may be safe in gross-section yield but unsafe at a connection because of net-section fracture.

Structural engineers must distinguish analysis models from physical reality, document assumptions, use applicable building codes, and obtain independent checks for safety-critical work. Computer output should be traceable, reproducible, and reviewed by a competent engineer. Ethical practice requires reporting uncertainty, not concealing unfavorable results, and never treating a convenient model as proof of safety.

10. Applied Case: Illustrative Roof-Truss Investigation

Background

A single-storey construction project uses prefabricated steel roof trusses at regular spacing. The architectural design places roof sheeting loads near panel points, but a mechanical contractor proposes hanging a concentrated duct from the middle of a chord panel.

Problem

The original ideal model applies loads only at joints. The proposed duct load acts between joints, creating bending in the chord and potentially changing the intended axial-force system. In addition, temporary construction loading may be unsymmetrical.

Analysis and approach

The engineer first establishes the support conditions, truss spacing, dead load, imposed roof load, wind actions, and duct load. The whole-truss reactions are checked using global equilibrium. The original joint-loaded model is analyzed by joints and independently by a matrix program. Selected chord and web forces are compared.

A second model applies the duct load at its actual location. The affected chord panel is modeled as a beam-column or is subdivided with an additional node and appropriate member properties. Load combinations are evaluated for gravity, uplift, and construction stages. Compression members are checked for buckling, tension members for yielding and connection capacity, and the duct hanger connection for local forces.

Outcome

The joint-loaded model predicts axial forces only, while the revised model shows additional chord bending and a higher connection demand. The engineer relocates the hanger to a panel point and adds a local reinforcing detail. Temporary bracing is specified until roof diaphragms are installed.

Learning points

1. Load location is part of the structural model, not merely a drafting detail.
2. A zero-force member under one load case may be active under uplift or construction loading.
3. Independent equilibrium checks can reveal software-input errors.
4. Truss analysis must be integrated with connection, stability, and construction-stage assessment.

11. Best Practices and Contemporary Tools

Modern practice combines hand methods, spreadsheet checks, finite-element software, building-information models, and field data. Parametric modeling can rapidly test geometry and load arrangements, while automated scripts can detect disconnected nodes, unrestrained degrees of freedom, and inconsistent units. Machine-learning tools may assist anomaly detection or preliminary design exploration, but they do not replace equilibrium, code compliance, model validation, or professional responsibility. The governing structural model, assumptions, version history, and review decisions should remain documented.

Chapter Summary

Trusses are frameworks whose ideal members carry axial tension or compression through pin-connected joints. Plane trusses are analyzed with two equilibrium equations at each joint; space trusses require three. Whole-truss equilibrium determines reactions, the method of joints determines forces sequentially, and the method of sections efficiently obtains selected member forces. Zero-force rules provide useful shortcuts but apply only under their stated loading and geometry conditions.

Member and joint counts provide preliminary determinacy checks, but geometry and stability must also be examined. Matrix stiffness analysis extends truss analysis to large and indeterminate systems by relating nodal displacements to loads and member forces. All methods depend on assumptions concerning joints, load placement, small displacement, material behavior, and axial action. Practical engineering must supplement calculated forces with buckling, connection, fatigue, serviceability, construction-stage, and code checks.

Key Terms

Truss: A framework of straight members arranged to carry loads primarily through axial forces.

Plane truss: A truss whose members, loads, and reactions lie in one plane.

Space truss: A three-dimensional truss requiring equilibrium in three translational directions.

Joint: The connection point at which truss members meet.

Member: A structural element connecting two joints.

Two-force member: A member subjected to equal, opposite, and collinear end forces.

Tension: Axial force that tends to elongate a member.

Compression: Axial force that tends to shorten a member.

Method of joints: Analysis method applying force equilibrium to individual joints.

Method of sections: Analysis method applying equilibrium to a cut portion of a truss.

Zero-force member: A member carrying zero axial force for a specified load case.

Determinacy: The condition in which forces can be found from equilibrium alone.

Stability: The ability of a structure to resist motion or mechanism formation.

Reaction: Force or moment supplied by a support.

Axial force: Internal force acting along a member’s longitudinal axis.

Degree of freedom: An independent displacement component of a structural node.

Stiffness matrix: Matrix relating nodal displacements to nodal forces.

Buckling: Instability involving sudden lateral deformation of a compression member.

Load path: The route by which loads travel through members and supports to the ground.

Review Questions

Short Answer Questions

1. State the principal assumptions of an ideal pin-jointed truss.
2. What is the difference between a plane truss and a space truss?
3. State the two common zero-force-member rules.
4. Why is the method of sections useful?
5. What does a negative member force indicate when tension was initially assumed?

Long Answer Questions

1. Explain how loads are transferred through a triangular roof truss from application point to foundation.
2. Derive and discuss the plane-truss determinacy relation \(m+r=2j\), including its limitations.
3. Describe the complete method-of-joints procedure and explain how member-force signs are interpreted.
4. Compare the method of joints, method of sections, and matrix stiffness method.
5. Discuss why ideal truss analysis may not adequately represent real connections and compression members.

Analytical / Application Questions

1. A symmetric triangular truss has a 30 kN load at its apex. The inclined members have horizontal and vertical projections of 3 m and 4 m. Determine the inclined-member forces and bottom-member force, stating whether each is in tension or compression.
2. A proposed truss has \(j=8\), \(m=13\), and three independent reactions. Assess its preliminary determinacy using the plane-truss count and explain why geometry must still be inspected.
3. A software model produces a singular stiffness matrix and a large displacement at one node. Identify at least four possible modeling causes and propose a systematic verification procedure.

Multiple Choice Questions with Answers

Question 1

For a stable plane truss with a pin and roller support, which relation indicates static determinacy?

A. m + 3 = 2j

B. m + 2 = 3j

C. m + j = 3

D. m = j + 3

Correct Answer: A. m + 3 = 2j

Question 2

At an unloaded joint with three members, two of which are collinear, the third member is generally:

A. In maximum compression

B. A zero-force member

C. In maximum tension

D. A support reaction

Correct Answer: B. A zero-force member

Question 3

Which equations are applied at an ideal plane-truss joint?

A. Only ∑M = 0

B. ∑Fx = 0 and ∑Fy = 0

C. ∑Fx = ∑Fy = ∑Fz = 0 only

D. Only compatibility equations

Correct Answer: B. ∑Fx = 0 and ∑Fy = 0

Question 4

A 20 kN vertical load acts at the apex of a symmetric triangular truss. Each inclined member has vertical direction cosine 4/5. What is the magnitude of the force in each inclined member?

A. 8 kN

B. 10 kN

C. 12.5 kN

D. 16 kN

Correct Answer: C. 12.5 kN

Question 5

The most effective moment center in a method-of-sections calculation is often:

A. A point where two cut-member lines intersect

B. The centroid of the entire truss only

C. The roller support in every case

D. Any point outside the free body only

Correct Answer: A. A point where two cut-member lines intersect

Question 6

For a space-truss member, the member length between coordinate points is calculated using:

A. L = Δx + Δy + Δz

B. L = √(Δx² + Δy² + Δz²)

C. L = ΔxΔyΔz

D. L = 2(Δx + Δy + Δz)

Correct Answer: B. L = √(Δx² + Δy² + Δz²)

Question 7

In the stiffness equation Kd = P, the vector d represents:

A. Material densities

B. Member angles only

C. Nodal displacements

D. Support reactions only

Correct Answer: C. Nodal displacements

Question 8

A negative result for a member force assumed tensile means that the member is:

A. Actually in compression

B. A mechanism in every load case

C. Definitely a zero-force member

D. Not connected to the truss

Correct Answer: A. Actually in compression

Question 9

Which condition most directly threatens the validity of an ideal axial-only truss model?

A. Loads applied exactly at ideal joints

B. A load applied between panel points on a chord

C. A triangular geometry

D. A pin support at one end

Correct Answer: B. A load applied between panel points on a chord

Question 10

A singular global stiffness matrix commonly indicates:

A. A perfectly overdesigned structure only

B. A missing restraint or structural mechanism

C. That all members are in tension

D. That reactions automatically satisfy equilibrium

Correct Answer: B. A missing restraint or structural mechanism