Unit 4: Analysis of structures

1. Introduction to Trusses

In engineering mechanics, the analysis of structures involves determining the forces acting both externally on the structure and internally within its various components. A structure is fundamentally a system of connected parts used to support a load. Among the most common and efficient load-bearing structures used in engineering are trusses. Trusses are widely used in the construction of bridges, roof supports, cranes, and towers due to their high strength-to-weight ratio. They are designed to carry significant loads across long spans while utilizing a relatively small amount of material.

2. Definition of Trusses

A truss is a rigid engineering structure composed of slender structural members joined together at their end points.

• Members: The individual straight pieces that make up the truss (e.g., wooden struts, metal bars, or angle irons).
• Joints (or Nodes): The points where the members intersect and are connected. In ideal truss analysis, these are assumed to be frictionless pins.
• Planar vs. Space Trusses: A planar truss has all its members and applied forces lying in a single 2D plane (commonly used for roofs and bridges). A space truss is a 3D network of members.

Assumptions in Ideal Truss Analysis

To simplify the mathematical analysis of trusses, several fundamental assumptions are made:

1. All loadings are applied at the joints: Weight of the members is either neglected or assumed to be applied half at each end joint. External forces are never applied to the middle of a member.
2. Members are joined together by smooth, frictionless pins: This means the joints cannot resist moment (rotational) forces.
3. Members are straight: Every member connects two joints in a straight line.
4. Two-Force Members: Because loads are applied only at the joints and joints are frictionless pins, every member in a truss acts as a two-force member. The forces act along the longitudinal axis of the member.
   • Tension (T): Forces pull away from the joints, tending to elongate the member.
   • Compression (C): Forces push toward the joints, tending to shorten the member.

3. Simple Trusses

A rigid structure is one that does not collapse or drastically deform under a load. The simplest geometric figure that maintains its shape without requiring rigid joints is a triangle.

• Formation of a Simple Truss: A simple truss is constructed by starting with a basic triangular element composed of three members and three joints. From there, the truss is expanded by adding two new members and one new joint for each expansion.
• Because a triangle is inherently stable, a simple truss composed entirely of interconnected triangles will also be rigid.
• Determinacy and Stability Formula: For a planar simple truss, the relationship between the number of members () and the number of joints () is given by the equation:
  
  • If , the truss is statically determinate and internally stable.
  • If , the truss is deficient and unstable (mechanism).
  • If , the truss is statically indeterminate (has redundant members).

4. Analysis of Truss by Method of Joints

The Method of Joints is an analytical technique used to determine the internal forces in all members of a truss. It is based on the principle that if the entire truss is in equilibrium, then each individual joint must also be in equilibrium.

Step-by-Step Procedure:

1. Calculate External Reactions: Draw a Free Body Diagram (FBD) of the entire truss. Use the global equations of equilibrium (, , ) to find the support reactions. (Note: This step can sometimes be skipped if a joint is available with only two unknown member forces).
2. Select a Starting Joint: Choose a joint that has at least one known applied load (or support reaction) and no more than two unknown member forces.
3. Draw the Joint FBD: Isolate the joint and draw the forces exerted by the connected members.
   • Convention: It is common practice to initially assume all unknown member forces are in Tension (pulling away from the joint). If the mathematical result is positive, the member is in tension. If negative, it is in compression.
4. Apply Equilibrium Equations: Since the forces at a joint are concurrent (intersect at a single point), there is no moment equation. Apply the 2D concurrent force equilibrium equations:
5. Solve and Proceed: Solve for the two unknown forces. Move to an adjacent joint, carrying over the newly found forces, ensuring you again pick a joint with at most two unknowns. Repeat until all member forces are found.

Zero-Force Members

To expedite analysis, look for zero-force members. These members support no load but are essential for stability or for handling variable load conditions.

• Rule 1: If two non-collinear members form a joint with no external load or support reaction, both members are zero-force members.
• Rule 2: If three members form a joint for which two of the members are collinear, and there is no external load or support reaction at that joint, the third non-collinear member is a zero-force member.

5. Analysis of Truss by Method of Sections

The Method of Sections is highly efficient when you only need to find the forces in a few specific members of a truss, rather than all of them. This method relies on the principle that if a body is in equilibrium, any isolated portion (section) of that body must also be in equilibrium.

Step-by-Step Procedure:

1. Calculate External Reactions: Similar to the method of joints, determine the support reactions using the FBD of the entire truss.
2. Make a Virtual Cut: Draw an imaginary line (section) that cuts through the truss, dividing it into two completely separate parts.
   • Rule of Thumb: The cut should pass through the members whose forces you want to determine.
   • Constraint: Do not cut through more than three members with unknown forces, as you only have three equations of equilibrium available per section.
3. Draw the Section FBD: Choose to analyze either the left or the right side of the cut (usually, the side with fewer external loads is easier). Draw all external forces acting on that half, plus the internal member forces exposed by the cut.
   • Assume the exposed unknown internal forces act in tension (pulling away from the cut surface).
4. Apply Equilibrium Equations: Unlike a single joint, a section of a truss is subjected to a non-concurrent force system. Therefore, you can use all three equations of equilibrium:
5. Solve for Unknowns: A strategic application of the moment equation () is the key to this method. By taking the moment about a point where the lines of action of two unknown forces intersect, those two forces are eliminated from the equation, allowing you to solve directly for the third unknown force in a single step.