Analysis of Simple Structures

Learning Objectives

  • Distinguish ideal trusses from frames and machines based on member loading and connection behavior.
  • Explain the two-force-member idealization used for pin-jointed trusses.
  • Determine truss member forces conceptually using the method of joints.
  • Isolate a portion of a truss and apply the method of sections efficiently.
  • Identify common zero-force-member patterns without removing structurally necessary members from the actual design.
  • Interpret tension and compression in relation to architectural load paths.

Ideal Planar Truss

An assembly of straight, slender members connected at ideal pins, with external loads and reactions applied at joints so that each member is treated as a two-force member carrying axial tension or compression.

Trusses, Frames, and Machines

These structural idealizations use different member models:

  • An ideal truss consists of two-force members joined at pins; member force is axial.
  • A frame contains at least one member subjected to more than two forces or a force and couple, so shear and bending may occur in addition to axial force.
  • A machine is an assembly designed to transmit or modify forces and generally includes moving parts.

A real building connection should be treated as an ideal pin only when that idealization is justified for the intended analysis.

Tension and Compression

After solving a truss member force, its sign or assumed arrow sense indicates the axial state:

  • Tension pulls away from the joint and tends to elongate the member.
  • Compression pushes toward the joint and tends to shorten the member.

A common method is to initially assume every unknown member force is tensile. A negative solution then indicates compression.

Interactive Exploration

Change the truss loading and observe which members develop tension or compression. Use the visualization to connect joint equilibrium with the overall load path from the applied load to the supports.

Trusses, Frames, and Machines Suite

Concept and model scope

Select a joint on a physically proportioned truss and inspect its incident axial member forces and equilibrium residual.

Simulation purpose: Planar structural systems rendered from their actual member coordinates and dimensions, with solver residuals kept visible.

Model scope: Ideal pin-connected trusses use axial-only members; the compound-frame scenario shows the assembled system, frame disassembly separates interacting free bodies, and the linkage scenario isolates the two-force member and its collinear end actions. Machine examples use rigid ideal levers with explicit efficiency where shown.

Verification: All physical diagrams use one coordinate scale. Truss height changes must change member slopes; frame length and link run/drop must change actual geometry; linkage end-force arrows must remain collinear with the link axis; machine arm ratios must match the visible lever arms. Any dependent load-coordinate adjustment caused by shortening a frame is surfaced to the learner.

AB 66.7 CBC 66.7 TCD 66.7 TDE 66.7 CAC 53.3 TCE 53.3 TBD 106.7 CABC80.0 kNDEblue tension · red compression · amber selected/cut · one physical coordinate scale
Joint A residual
0.00e+0 kN
AB
66.667 kN C
AC
53.333 kN T
Joint load

Joint load

Vertical joint load used by the displayed truss analysis.

80 kN
Truss height

Truss height

Physical elevation of joints B and D. The same x-y scale is used, so changing height changes actual member slopes and force directions.

3.00 m
determinate
∑Fx=0,∑Fy=0,and for sections ∑MO=0\sum F_x=0,\qquad \sum F_y=0,\qquad \text{and for sections }\sum M_O=0

Equation concept

The solver and the diagram use the same node coordinates, member dimensions, load locations, and link directions. Physical x-y geometry is never independently stretched to fill the viewport.

Method of Joints

  1. Determine the external support reactions for the complete truss.
  2. Select a joint with no more than two unknown member forces whenever possible.
  3. Draw the isolated joint free-body diagram.
  4. Assume unknown member forces act in tension unless another consistent convention is preferred.
  5. Apply horizontal and vertical force equilibrium.
  6. Move to adjacent joints as newly known member forces reduce the number of unknowns.
  7. Interpret negative assumed-tension results as compression.

How to Use This Workflow

Follow the branch based on whether the task requires most member forces or only selected forces. If the equilibrium check fails, return to the free-body diagram, geometry, and modeling assumptions before continuing.
Choosing and Checking a Planar Truss Analysis Method
Choosing and Checking a Planar Truss Analysis MethodStart truss analysis → Solve external reactions; Solve external reactions → Need most member forces?; Need most member forces? — Yes → Use method of joints; Need most member forces? — No — selected forces → Cut through target member; Use method of joints → Solve a joint with ≤ 2 unknowns; Cut through target member → Use section equilibrium; Solve a joint with ≤ 2 unknowns → Equilibrium and signs consistent?; Use section equilibrium → Equilibrium and signs consistent?; Equilibrium and signs consistent? — Yes → Report tension / compression; Equilibrium and signs consistent? — No → Review FBD, geometry, and assumptions; Review FBD, geometry, and assumptions — Revise → Need most member forces?; Report tension / compression → End

Start truss analysis → Solve external reactions; Solve external reactions → Need most member forces?; Need most member forces? — Yes → Use method of joints; Need most member forces? — No — selected forces → Cut through target member; Use method of joints → Solve a joint with ≤ 2 unknowns; Cut through target member → Use section equilibrium; Solve a joint with ≤ 2 unknowns → Equilibrium and signs consistent?; Use section equilibrium → Equilibrium and signs consistent?; Equilibrium and signs consistent? — Yes → Report tension / compression; Equilibrium and signs consistent? — No → Review FBD, geometry, and assumptions; Review FBD, geometry, and assumptions — Revise → Need most member forces?; Report tension / compression → End

  • Start truss analysis: terminator
  • Solve external reactions: process
  • Need most member forces?: decision
  • Use method of joints: process
  • Cut through target member: process
  • Solve a joint with ≤ 2 unknowns: process
  • Use section equilibrium: process
  • Equilibrium and signs consistent?: decision
  • Review FBD, geometry, and assumptions: process
  • Report tension / compression: process
  • End: terminator

Method of Sections

The method of sections determines selected internal member forces without solving every joint. Pass an imaginary cut through the truss and isolate one side of the cut.

For a planar truss, choose a section that introduces no more than three unknown cut-member forces when possible. Then apply the three planar rigid-body equilibrium equations to the isolated portion.

Strategic moment centers can eliminate two unknown cut forces at once when their lines of action intersect at the selected point.

Method of Sections

  1. Determine the external reactions.
  2. Pass a section through the target member and as few additional unknown members as practical.
  3. Isolate the simpler side of the cut.
  4. Replace each cut member by an axial force along the member axis.
  5. Apply moment equilibrium first when it can isolate one unknown directly.
  6. Use force equilibrium for the remaining cut-member forces.
  7. State each result as tension or compression.

Zero-Force Members

Certain unloaded joints allow member forces to be recognized immediately:

  • If two non-collinear members meet at an unloaded joint with no support reaction, both are zero-force members.
  • If three members meet at an unloaded joint and two are collinear, the non-collinear member is a zero-force member.

These rules apply to the idealized loading case at that joint. A member that is zero-force for one load case may carry force for another and can still be necessary for stability, construction, buckling restraint, or load reversal.

Zero Force Does Not Mean Unnecessary

Never infer that a zero-force member can simply be deleted from a real structure. Its role may emerge under another load case or through stability, bracing, fabrication, or serviceability requirements.

Simple Truss Determinacy Screen

For a stable simple planar truss, the familiar relation m+r=2jm+r=2j is associated with static determinacy, where mm is the number of members, rr the number of external reaction components, and jj the number of joints.

This count is a screening relation, not a complete stability proof. Geometry matters: a truss satisfying the count can still be unstable if its members are arranged as a mechanism.

Planar Truss Count

A counting relation used as an initial determinacy screen for ideal planar trusses.

m+r=2jm+r=2j

Variables

SymbolDescriptionUnit
mmNumber of truss members-
rrNumber of external reaction components-
jjNumber of truss joints-

Keep Truss and Frame Models Distinct

If loads are applied between ideal truss joints, connections transfer moment, or members are not adequately modeled as two-force members, an ideal truss analysis can be inappropriate. The model must match the structural behavior being represented.

Key Takeaways
  • Ideal truss members carry axial force because the two-force-member assumptions remove member shear and bending from the model.
  • The method of joints uses particle equilibrium at individual truss joints.
  • The method of sections uses rigid-body equilibrium on a cut portion to solve selected member forces efficiently.
  • Zero-force-member rules depend on the joint loading and geometry of the idealized load case.
  • The relation m+r=2jm+r=2j is a useful count for simple planar trusses but does not replace a stability check.
  • Frames and machines generally contain multi-force members and require rigid-body free-body diagrams rather than pure two-force-member assumptions.

References