Analysis of Trusses

Why it matters

Trusses are fundamental components in many structures, such as bridges, roofs, and towers. Understanding how to analyze trusses ensures that these structures can support the loads they encounter without failure, which is crucial for safety and reliability.

Key ideas

• Truss Definition: A truss is a structure composed of members connected at their ends to form a stable framework. Typically, trusses are assumed to be pin-jointed and loaded only at the joints.
• Types of Trusses: Common types include Pratt, Warren, and Howe trusses, each with distinct configurations and load distribution characteristics.
• Assumptions: Trusses are analyzed under the assumption that members are connected by frictionless pins and loads are applied only at the joints.
• Methods of Analysis:
  • Method of Joints: Involves isolating a joint and solving for the forces in the connected members using equilibrium equations.
  • Method of Sections: Involves cutting through the truss and solving for the forces in the members of interest using equilibrium equations.

Formulas

• ΣF_x = 0: Sum of horizontal forces must be zero.
• ΣF_y = 0: Sum of vertical forces must be zero.
• ΣM = 0: Sum of moments about any point must be zero.

Worked example

A triangular pin-jointed truss has A = (0,0), B = (6,0), C = (3,3) m and members AB, AC and BC. A is pinned; B is a roller on a horizontal surface. A 10 kN downward load acts at C. Neglect self-weight.

1. Global equilibrium gives A_x = 0 and A_y = B_y = 5 kN.
2. AC and BC each make 45° with the horizontal. Symmetry gives equal forces. At C, two compressive forces provide upward components: 2F sin45° = 10, so F_AC = F_BC = 7.071 kN compression.
3. At A, the horizontal component of AC is 7.071 cos45° = 5 kN to the left. Thus F_AB = 5 kN tension balances it.

Answer: AC and BC carry 7.071 kN compression each; AB carries 5 kN tension. Both inclined members contribute to the vertical equilibrium at the apex.

Common mistakes

• Incorrectly assuming all members are in tension or compression without proper analysis.
• Neglecting to consider the geometry of the truss when calculating angles.
• Forgetting to apply equilibrium equations correctly at each joint or section.

For GATE CE

Questions often involve analyzing a given truss to find forces in specific members using the method of joints or sections. Practice problems with different truss configurations and loading conditions to become proficient.

Quick check

1. What is the primary assumption in truss analysis?
2. Name two methods used to analyze trusses.
3. What is the angle θ in a right triangle with equal legs?

Answers: 1. Members are pin-jointed and loads are applied at joints. 2. Method of Joints, Method of Sections. 3. 45°.