Trusses, Frames, and Machines

Truss Assumptions

1. Members are connected by frictionless pins at joints
2. Loads are applied only at joints
3. Members are two-force members (forces act only along the member axis)
4. Members are either in tension (T) or compression (C)
5. Member weight is negligible (or split between joints)

Truss Determinacy

For a 2D truss with m members, j joints, and r reactions:

Method of Joints

Analyze equilibrium at each joint (2 equations: ΣFx = 0, ΣFy = 0).

Steps:

1. Find support reactions using overall equilibrium
2. Start at a joint with ≤ 2 unknowns
3. Assume all members are in tension (pulling away from joint)
4. Apply ΣFx = 0 and ΣFy = 0
5. Positive result = tension; negative result = compression
6. Move to the next joint with ≤ 2 unknowns

Method of Sections

Cut through the truss, isolating a portion, and apply equilibrium to that portion (3 equations: ΣFx = 0, ΣFy = 0, ΣM = 0).

Steps:

1. Find support reactions
2. Cut through at most 3 members whose forces you want
3. Draw FBD of one portion
4. Apply three equilibrium equations to solve for unknown member forces

When to use which method:

• Method of Joints: Finding forces in ALL members
• Method of Sections: Finding force in ONE specific member (faster for targeted analysis)

Identifying Zero-Force Members

A zero-force member carries no load under the given loading conditions. Two rules for quick identification:

Rule 1: If only two non-collinear members meet at an unloaded joint, both are zero-force members.

Rule 2: If three members meet at an unloaded joint and two are collinear, the third member is a zero-force member.

Important: Zero-force members are NOT useless — they prevent instability and may carry load under different loading conditions.

Frames and Machines

Unlike trusses, frames and machines contain multi-force members (forces at more than two points).

• Frames: Stationary structures that support loads
• Machines: Structures with moving parts that transmit/modify forces

Analysis Method

1. Find external reactions on the entire structure
2. Disassemble the structure at the joints
3. Draw FBD of each member
4. Apply equilibrium to each member
5. At connected joints, use Newton's Third Law (equal and opposite forces)

Key Difference from Trusses: Internal forces on frame members include axial forces, shear forces, AND bending moments — not just axial forces.