Analysis of Statically Determinate Trusses

Techniques to determine the internal forces in members of a truss, including the Method of Joints and the Method of Sections.

Introduction to Trusses

Truss

A framework composed of members joined at their ends to form a rigid structure. In a planar truss, all members lie in a single plane. Loads are assumed to act only at the joints (nodes), and all members are two-force members, meaning they are in either pure tension or pure compression.

Truss Members and Forces

  • Tension Members (T): Members subjected to axial forces that tend to stretch them. The internal force pulls away from the joint.
  • Compression Members (C): Members subjected to axial forces that tend to crush or buckle them. The internal force pushes towards the joint.

Determinacy and Stability of Trusses

Verifying if a truss can be analyzed using static equilibrium alone.

Determinacy Formula

The static determinacy of a planar (2D) truss is evaluated by comparing the total number of unknown forces to the total number of available equilibrium equations.
Let:
  • bb = number of bar members
  • rr = number of external support reactions
  • jj = number of joints (nodes)
Since there are two equilibrium equations (∑Fx=0\sum F_x = 0 and ∑Fy=0\sum F_y = 0) available at each joint, the total number of equations is 2j2j.
  • Statically Determinate: b+r=2jb + r = 2j
  • Statically Indeterminate: b+r>2jb + r > 2j (The degree of indeterminacy is D=b+r−2jD = b + r - 2j)
  • Unstable: b+r<2jb + r < 2j

Classification of Trusses

  • Simple Trusses: Formed by starting with a basic triangular element and connecting two new members to a new joint to expand the framework. Simple trusses are inherently stable.
  • Compound Trusses: Formed by connecting two or more simple trusses together. Common methods of connection include a single joint and a single bar, or three non-concurrent bars.
  • Complex Trusses: A truss that cannot be classified as simple or compound. They often have overlapping members and cannot be easily analyzed by standard Method of Joints or Sections alone; they frequently require the Method of Substitute Members or matrix analysis.
  • Space Trusses: A three-dimensional framework of members joined at their ends. The basic element is a tetrahedron formed by six members. The determinacy condition for a space truss is b+r=3jb + r = 3j, because there are three equilibrium equations available at each 3D joint (∑Fx=0\sum F_x = 0, ∑Fy=0\sum F_y = 0, ∑Fz=0\sum F_z = 0).

Zero-Force Members

Identifying Zero-Force Members

Identifying zero-force members simplifies truss analysis by reducing the number of unknowns. These members carry no axial force under the current loading condition, although they provide stability and carry loads if the loading pattern changes.
  • Rule 1: If only two non-collinear members form a joint, and no external load or support reaction is applied to that joint, then both members are zero-force members.
  • Rule 2: If three members form a truss joint, for which two of the members are collinear, and there is no external load or support reaction at the joint, then the third (non-collinear) member is a zero-force member.

Method of Joints

The Method of Joints involves isolating each joint of the truss and applying the equations of equilibrium to the forces acting there. Since the forces all intersect at the joint (concurrent force system), the moment equation ∑M=0\sum M = 0 is automatically satisfied, leaving only two independent equations: ∑Fx=0\sum F_x = 0 and ∑Fy=0\sum F_y = 0.

Procedure

  1. Determine the global support reactions by treating the entire truss as a rigid body.
  2. Select a joint with no more than two unknown member forces.
  3. Assume a direction for the unknown forces (usually tension, pulling away from the joint).
  4. Apply the equations of equilibrium (∑Fx=0\sum F_x = 0 and ∑Fy=0\sum F_y = 0) to solve for the unknowns. A negative result indicates compression.
  5. Proceed to the next joint with at most two unknowns, using the newly found forces as knowns.

Method of Sections

Definition

The Method of Sections involves passing an imaginary cut (section) through the truss to divide it into two parts. Equilibrium is then applied to either part to determine the internal forces in the cut members. This is useful when the force in only a few specific members is required.

Procedure

  1. Determine the support reactions if necessary.
  2. Pass an imaginary section through the truss, cutting the members whose forces are to be determined. The section should not cut more than three members with unknown forces, as there are only three equations of equilibrium available (∑Fx=0\sum F_x = 0, ∑Fy=0\sum F_y = 0, ∑M=0\sum M = 0).
  3. Draw a free-body diagram of either the left or the right portion of the cut truss.
  4. Assume all cut members are in tension (pulling away from the section).
  5. Apply the equations of equilibrium to solve for the unknown forces. Taking moments about the intersection point of two unknown forces directly yields the third unknown force.

Interactive Tool: Truss Analysis Simulation

Visualize and calculate internal forces in members of basic truss configurations under point loads.

Truss Analysis Simulator

METHOD OF JOINTS

Member Forces Results:

  • AB:0 N
  • BC:0 N
  • AC:0 N

Support Reactions:

  • Ay = 0.0 N (Up)
  • Cy = 0.0 N (Up)
  • Ax = 0.0 N

Tension (T) members are shown in blue and pull away from joints. Compression (C) members are shown in red and push into joints. Zero-force members are gray.

ABC600 N

Space Trusses

Extension of planar truss analysis to three dimensions.

3D Truss Determinacy

A space truss is a three-dimensional framework where members are connected by ball-and-socket joints.
The determinacy equation for a space truss is b+r=3jb + r = 3j
Where:
  • bb = number of members (bars)
  • rr = number of reaction components
  • jj = number of joints
If b+r>3jb + r > 3j, the truss is statically indeterminate. If b+r<3jb + r < 3j, it is unstable.

Complex Trusses

Method of Substitute Members

Complex trusses are structures that cannot be classified as simple or compound and cannot be solved directly by the method of joints or sections alone. A classical approach to solving complex trusses is the Method of Substitute Members (Henneberg's Method), which involves temporarily removing a member to make the truss simple, applying an artificial force, and using superposition to solve for the true member forces.

Common Types of Trusses

Standard truss configurations used in bridges and roofs.

Roof and Bridge Trusses

Different truss patterns have been developed to optimize material use for specific spans and loads:
  • Pratt Truss: Characterized by vertical members in compression and diagonal members in tension (except near the center). Diagonals slope down towards the center.
  • Howe Truss: The opposite of the Pratt truss. Vertical members are in tension and diagonals are in compression. Diagonals slope up towards the center.
  • Warren Truss: Consists of equilateral or isosceles triangles. Diagonals alternate between tension and compression. Often used without verticals to simplify construction.
  • K-Truss: Used for very long spans to reduce the unbraced length of compression members, thereby reducing buckling.
Key Takeaways
  • Trusses consist of two-force members joined at nodes.
  • Zero-force members can be identified by inspection using two simple rules, which significantly speeds up analysis.
  • The Method of Joints solves for forces by applying equilibrium (∑Fx=0\sum F_x = 0, ∑Fy=0\sum F_y = 0) to individual nodes.
  • The Method of Sections determines forces by analyzing a cut portion of the truss using all three equilibrium equations (∑Fx=0\sum F_x = 0, ∑Fy=0\sum F_y = 0, ∑M=0\sum M = 0).
  • Tensile forces pull away from nodes, while compressive forces push towards nodes.
  • Trusses are composed of two-force members connected by frictionless pins.
  • Determinacy is checked using b+r=2jb + r = 2j (2D) or b+r=3jb + r = 3j (3D).
  • The Method of Joints isolates individual pins using ∑Fx=0\sum F_x = 0 and ∑Fy=0\sum F_y = 0.
  • The Method of Sections isolates a portion of the truss using moment equations to solve directly for internal member forces.
  • Identifying zero-force members simplifies truss analysis significantly.
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