Methods of Simple Truss Analysis

A truss structure is composed of slender members joined together at their end points

Members are commonly wooden struts or metal bars

Joint connections are formed by bolting or welding the ends of the members to a common plate (gusset plate) or by simply passing a large bolt or pin through each of the members

Planar trusses lie in a single plane (often seen supporting roofs and bridges), 2D analysis of forces appropriate
1. Truss Analysis Method of joints

If a truss is in equilibrium, then each of its joints must also be in equilibrium

The method of joints consists of satisfying the equilibrium conditions for the forces exerted “on the pin” at each joint of the truss

Truss members are all straight twoforce members lying in the same plane

The force system acting at each pin is coplanar and concurrent (intersecting)

Rotational or moment equilibrium is automatically satisfied at the joint, only need to satisfy
∑ Fx = 0, ∑ Fy = 0 
Draw the freebody diagram of a joint having at least one known force and at most two unknown forces (may need to first determine external reactions at the truss supports)

Establish the sense of the unknown forces

Always assume the unknown member forces acting on the joint’s freebody diagram to be in tension (pulling on the “pin”)

Assume what is believed to be the correct sense of an unknown member force

In both cases a negative value indicates that the sense chosen must be reversed

Orient the x and y axes such that the forces can be easily resolved into their x and y components

Apply ∑ Fx = 0 and ∑ Fy = 0 and solve for the unknown member forces and verify their correct sense

Continue to analyze each of the other joints, choosing ones having at most two unknowns and at least one known force

Members in compression “push” on the joint and members in tension “pull” on the joint

Mechanics of Materials and building codes are used to size the members once the forces are known
Zeroforce members

Truss analysis using the method of joints is greatly simplified if one is able to determine those members which support no loading (zeroforce members)

These zeroforce members are used to increase stability of the truss during construction and to provide support if the applied loading is changed

If only two members form a truss joint and no external load or support reaction is applied to the joint, the members must be zeroforce members

If three members form a truss for which two of the members are collinear, the third member is a zeroforce member provided no external force or support reaction is applied.
2. Method of sections for trusses

Based on the principle that if a body is in equilibrium, then any part of the body is also in equilibrium
Procedure for analysis

Section or “cut” the truss through the members where the forces are to be determined

Before isolating the appropriate section, it may be necessary to determine the truss’s external reactions (then 3 equations equations of equilibrium can be used to solve for unknown member forces in the section)

Draw the freebody diagram of that part of the sectioned truss that has the least number of forces acting on it

Establish the sense of the unknown member forces

Apply 3 equations of equilibrium trying to avoid equations that need to be solved simultaneously

Moments should be summed about a point that lies at the intersection of the lines of action of two unknown forces

If two unknown forces are parallel – sum forces perpendicular to the direction of these unknowns
Frames and Machines

Structures are often composed of pinconnected multi force members

Frames are generally stationary and are used to support loads

Machines contain moving parts and are designed to transmit and alter the effect of forces

Can apply the equations of equilibrium to each member of the frame or machine to determine the forces acting at the joints and supports (assuming the frame or machine is properly constrained and contains no more supports or members than are necessary to prevent collapse)

Construct applicable freebody diagrams

Draw an outline of the shape

Show all forces or couple moments that act on the part

Indicate dimensions needed for determining moments

Identify all two force members in the structure

All loadings are applied at the joint

Members are joined together by smooth pins

Members have two equal but opposite forces acting at their points of application

The line of action of the forces are along the axis of the members

Forces common to any two contacting members act with equal magnitudes but opposite sense on the respective members

Apply the equations of equilibrium