Short Answer:

The method of sections is a technique used in truss analysis to find the forces in specific members directly without analyzing the entire truss joint by joint. In this method, the truss is divided into two parts by passing an imaginary “cut” or “section” through it, and then equilibrium equations are applied to one of the parts.

This method is very useful when only a few member forces are required, as it saves time compared to the method of joints. The equilibrium conditions used are:

Detailed Explanation:

Method of Sections

The method of sections is one of the two main analytical techniques used to determine the internal forces in the members of a truss. Unlike the method of joints, which involves analyzing each joint separately, the method of sections allows engineers to find the force in selected members quickly and efficiently.

This method is based on the principle of static equilibrium, which states that a body is in equilibrium if the sum of all forces and moments acting on it is zero. By applying this principle to a portion of the truss, we can determine the unknown forces in the cut members directly.

It is particularly helpful when only a few specific member forces are required, such as those in the middle of a large truss, since it eliminates the need to solve for every joint.

Basic Principle

The method of sections works on the following simple concept:

A truss is divided into two parts by an imaginary section (cut) that passes through not more than three unknown members whose forces are to be found.
After cutting, one of the two parts is considered a free body and isolated for analysis.
The forces in the members cut by the section are then determined using the three equations of equilibrium:

By applying these equations, the magnitude and nature (tension or compression) of the forces in the cut members can be found.

Procedure of Method of Sections

To apply the method of sections correctly, the following steps should be followed:

Determine Support Reactions:
First, calculate the reaction forces at the supports using the overall equilibrium of the entire truss. This ensures that all external forces acting on the truss are known before making a cut.
Select the Section:
Draw an imaginary “section line” that cuts through the truss in such a way that:
- It passes through not more than three unknown members, since there are only three independent equilibrium equations available.
- The cut should expose the members whose forces are to be determined.
Isolate One Part of the Truss:
After making the cut, isolate either the left-hand or right-hand portion of the truss. Choose the side that has fewer external forces and members to simplify the calculations.
Indicate the Forces in Cut Members:
Assume the nature of the forces (tensile or compressive) in the cut members. If the calculated value is positive, the assumption is correct; if negative, the actual nature is opposite.
Apply the Equilibrium Equations:
For the selected part of the truss, apply the three static equilibrium conditions:
- → Sum of all horizontal forces equals zero.
- → Sum of all vertical forces equals zero.
- → Sum of moments about any point equals zero.

These equations allow you to solve for the unknown forces in the cut members.

Assumptions Used in Method of Sections

All members are connected by pin joints.
The loads act only at the joints, not along the members.
Each member carries only axial force (tension or compression).
The self-weight of members is neglected unless otherwise specified.
The truss is statically determinate, satisfying .

These assumptions simplify the calculations and make the analysis reliable for most practical applications.

Sign Convention for Member Forces

Tensile Force (T): The member is under tension if it pulls away from the joint or cut section.
Compressive Force (C): The member is under compression if it pushes toward the joint or cut section.

If the calculated value of a force is positive, the assumed direction is correct; if negative, the actual direction is opposite.

Advantages of Method of Sections

Time-Saving:
This method is much faster than the method of joints, especially when forces in only a few specific members are required.
Simplifies Large Trusses:
It is ideal for analyzing long or complex trusses without going through all the joints.
Applicable to Middle Members:
Useful for finding forces in interior members that are difficult to reach using the method of joints.
Practical and Accurate:
Provides accurate results using simple equilibrium equations.
Less Repetitive:
It eliminates the need to analyze every joint one by one.

Limitations of Method of Sections

Restricted to Three Unknowns:
The section must not pass through more than three unknown members since there are only three equilibrium equations available.
Requires Support Reactions:
Before making the section, support reactions must be known, which sometimes adds an extra step.
Geometric Complexity:
For trusses with inclined members, resolving forces can be more time-consuming.
Not Suitable for All Members:
If many forces are unknown in the cut section, this method cannot be applied directly.

Example (Conceptual)

Consider a simple truss where we need to find the forces in three interior members.

First, calculate the support reactions using the overall equilibrium.
Then, pass a section line cutting through the three required members.
Choose one part (left or right) of the truss for analysis.
Apply the equilibrium equations (ΣFx = 0, ΣFy = 0, ΣM = 0).
By solving these equations, the forces in the three cut members can be found directly without analyzing each joint separately.

This approach is extremely helpful in bridge or roof trusses, where the required members are located far from the supports.

Applications

Bridge Trusses: Used to determine forces in main and diagonal members.
Roof Trusses: Helps in finding forces in principal rafters or tie members.
Tower Structures: Used to analyze load-carrying members efficiently.
Cranes and Girders: Applied in analyzing internal load paths.
Mechanical Frames: Used in machines to check strength of supporting truss elements.

Conclusion

The method of sections is a powerful and efficient technique for analyzing trusses. It involves cutting the truss into two parts and applying the equations of static equilibrium to one part to determine the forces in selected members. This method is highly advantageous when only a few internal forces are needed, as it avoids the lengthy process of joint-by-joint analysis. Despite its limitations, it remains one of the most essential tools for engineers in designing safe and efficient truss structures.