Analysis of Structures - Theory & Concepts

The analysis of structures involves determining the internal forces acting within the members of interconnected bodies. In statics, we primarily focus on structures that are in equilibrium, such as trusses, frames, and machines.

Analysis of Trusses

Truss

A structure composed of slender members joined together at their endpoints. They are commonly used in bridges and roof supports.

Important

Assumptions for Truss Analysis:
  1. All loadings are applied at the joints (nodes). The weight of the members is often neglected or applied half to each joint.
  2. The members are joined together by smooth pins.
Consequence: Because of these assumptions, each truss member acts as a two-force member. The internal force is directed along the axis of the member and can be either Tension (T) (pulling the joint) or Compression (C) (pushing the joint).

Truss Determinacy and Stability

A simple planar truss can be analyzed to see if it is statically determinate internally based on its number of members (bb), number of joints (jj), and number of external reaction components (rr). Since each joint provides two equations (ΣFx=0,ΣFy=0\Sigma F_x=0, \Sigma F_y=0), the total available equations is 2j2j.
The equation relating these variables is: b+r=2jb + r = 2j
  • If b+r=2jb + r = 2j, the truss is exactly statically determinate (assuming proper constraint).
  • If b+r>2jb + r > 2j, the truss has more members or supports than necessary and is statically indeterminate to the (b+r−2j)th(b+r-2j)^{th} degree.
  • If b+r<2jb + r < 2j, the truss is unstable and will collapse as a mechanism.
Note: Even if b+r≥2jb+r \ge 2j, a truss can still be unstable if the members form a collapsible shape (e.g., a square without a diagonal) or if the support reactions are concurrent or parallel.
There are two primary methods for analyzing planar (2D) trusses:

  1. The Method of Joints

The Method of Joints is based on the principle that if a truss is in equilibrium, then every single joint in the truss must also be in equilibrium. Since all forces at a joint are concurrent, the moment equation ΣM=0\Sigma M = 0 is automatically satisfied. We only need to apply the force equilibrium equations:

Joint Equilibrium Equations

Conditions for a truss joint to be in equilibrium.

Procedure for Method of Joints

  1. (Optional but recommended) Determine the external support reactions for the entire truss by treating it as a single rigid body.
  2. Select a joint with at least one known force and at most two unknown member forces.
  3. Draw the Free-Body Diagram (FBD) of the joint. By convention, assume unknown member forces are in Tension (pulling away from the joint).
  4. Apply ΣFx=0\Sigma F_x = 0 and ΣFy=0\Sigma F_y = 0 to solve for the unknowns. A positive answer means your assumed direction (Tension) was correct. A negative answer means the force is actually in Compression.
  5. Proceed to the next joint with at most two unknowns, using the newly found forces as knowns.

  1. The Method of Sections

The Method of Sections is used when you need to find the internal forces in only a few specific members of a truss, rather than the entire structure. It involves "cutting" the truss into two distinct parts.

Procedure for Method of Sections

  1. Determine the external support reactions for the entire truss.
  2. Pass an imaginary section (cut) through the truss, cutting through the members whose internal forces are desired. Try to cut through no more than three members with unknown forces.
  3. Choose the simpler of the two isolated halves of the truss and draw its FBD. The cut members will now expose their internal forces as external forces acting on the FBD. Assume they are in Tension.
  4. Apply the three rigid-body equilibrium equations (ΣFx=0\Sigma F_x = 0, ΣFy=0\Sigma F_y = 0, ΣM=0\Sigma M = 0) to solve for the unknowns.
  5. Tip: To solve directly for an unknown, sum moments about a point where the lines of action of the other two unknown forces intersect.

Zero-Force Members

In some truss configurations, specific members carry no load. These are called zero-force members. They are often included to provide stability during construction, prevent buckling of long compressive members, or carry loads if the loading condition changes.
Identifying zero-force members simplifies the analysis significantly.

Identifying Zero-Force Members

  • Case 1: Two non-collinear members forming a joint with no external load or support reaction.
    If a joint connects only two members that are not in a straight line, and there is no external load or support reaction at that joint, both members are zero-force members. (Reason: ΣF=0\Sigma F = 0 along axes parallel and perpendicular to one member shows the other must be zero).
  • Case 2: Three members forming a joint, two of which are collinear, with no external load or support reaction.
    If a joint connects three members, two of which lie in a straight line, and there is no external load or reaction at the joint, the third (non-collinear) member is a zero-force member. (Reason: Summing forces perpendicular to the collinear members directly shows the force in the third member must be zero).

Space Trusses (3D)

A space truss is a three-dimensional framework of members joined at their ends. The basic element of a space truss is a tetrahedron formed by six members connecting four joints.

Important

Assumptions for Space Truss Analysis:
  1. All loadings and support reactions are applied at the joints.
  2. The members are connected by smooth ball-and-socket joints.
Consequence: Just like planar trusses, these assumptions ensure that all members of a space truss act as two-force members carrying purely axial tension or compression.

Method of Joints (3D)

For a space truss, the conditions of equilibrium for each joint require that the resultant force acting on the joint be zero: ΣF=0\Sigma \mathbf{F} = 0
Which resolves into three independent Cartesian scalar equations:
  • ΣFx=0\Sigma F_x = 0
  • ΣFy=0\Sigma F_y = 0
  • ΣFz=0\Sigma F_z = 0
This means we can solve for up to three unknown member forces at any given joint.

Analysis of Frames and Machines

Unlike trusses, frames and machines are structures that contain at least one multi-force member (a member subjected to more than two forces, or members subjected to bending moments).

Classification

  • Frames: Generally stationary structures designed to support loads.
  • Machines: Structures containing moving parts designed to transmit or alter the effect of forces.

Important

Procedure for Analyzing Frames and Machines:
  1. Analyze the entire structure: If possible, draw the FBD of the entire frame/machine and determine the external support reactions.
  2. Dismember the structure: Take the structure apart and draw an FBD for every single member and every pin (if the pin connects more than two members).
  3. Newton's Third Law (Action-Reaction): This is the most critical step. When a multi-force member is separated from another member at a pin joint, the forces it exerts on the pin must be equal and opposite to the forces the pin exerts on it. Ensure that the assumed force components at shared joints (e.g., Ax,AyA_x, A_y) point in opposite directions on the interacting FBDs.
  4. Identify Two-Force Members: Look for members pinned at both ends with no intermediate loads. Their internal force lies along the axis connecting the pins.
  5. Apply Equilibrium Equations: Apply ΣFx=0\Sigma F_x = 0, ΣFy=0\Sigma F_y = 0, and ΣM=0\Sigma M = 0 to the FBDs of the individual members to solve for the unknown pin forces.

Interactive Truss Analysis Simulation

The following simulation demonstrates the Method of Joints on a simple triangular roof truss. Adjust the load position and magnitude to observe how internal forces transition between tension and compression to maintain equilibrium.

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
Key Takeaways
  • Trusses consist only of two-force members (tension or compression). Loads are applied only at the joints.
  • The Method of Joints analyzes equilibrium (ΣFx=0,ΣFy=0\Sigma F_x=0, \Sigma F_y=0) at each pin connection.
  • The Method of Sections cuts the truss to find forces in specific members using rigid body equations (ΣF=0,ΣM=0\Sigma F=0, \Sigma M=0).
  • Identifying Zero-Force Members by inspection simplifies truss analysis.
  • Frames and Machines contain multi-force members. Analysis requires dismembering the structure and applying Newton's Third Law (equal and opposite reaction forces) at the connecting pins.
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