Truss By Method Of Section

Analyzing Trusses Using the Method of Sections: A Comprehensive Guide

Determining the forces acting on individual members of a truss structure is crucial for design and safety. While the method of joints offers a systematic approach, it can become cumbersome for large trusses. The method of sections, however, provides a more efficient way to analyze specific members by strategically cutting through the truss and analyzing the resulting free-body diagram. This article provides a comprehensive guide to understanding and applying the method of sections for truss analysis.

Introduction to Trusses and the Method of Sections

A truss is a structural assembly composed of slender members interconnected at their ends to form a rigid framework. These members are typically subjected to axial tension or compression forces. Trusses are commonly used in bridges, roofs, and other structures where strength and lightness are important. Analyzing the internal forces in these members is essential to ensure structural integrity.

The method of joints analyzes each joint individually, solving for the forces in the members connected to that joint. This method is effective for smaller trusses, but becomes increasingly complex as the number of joints and members grows. The method of sections offers a more efficient alternative, especially when you need to determine the forces in only a few specific members. This method involves cutting through the truss with a section line, isolating a portion of the truss, and applying equilibrium equations to solve for the unknown forces.

Steps Involved in the Method of Sections

The method of sections involves a systematic process. Let's break it down into clear steps:

1. Identify the Target Members: Determine which members' forces you need to calculate. This will guide the placement of your section cut.

2. Draw the Section Line: Imagine a cutting plane slicing through the truss, passing through the members whose forces you want to find. This cut should ideally pass through no more than three members with unknown forces. If more than three unknowns are present in the section, you won't have enough equilibrium equations to solve for them.

3. Isolate a Section: After making the cut, select one of the two sections of the truss. Choosing the section with fewer external forces generally simplifies calculations.

4. Draw a Free Body Diagram (FBD): Create a free-body diagram of the isolated section. Include all external forces acting on this section (reactions at supports, applied loads). Also, represent the internal forces in the cut members as unknown forces acting along the member's axis. Remember to indicate the assumed direction of these forces (tension or compression). Incorrectly assuming the direction will simply result in a negative value for the force.

5. Apply Equilibrium Equations: Use the equations of static equilibrium to solve for the unknown member forces. These equations are:

   • ΣFx = 0: The sum of horizontal forces equals zero.
   • ΣFy = 0: The sum of vertical forces equals zero.
   • ΣM = 0: The sum of moments about any point equals zero.

   Choosing a convenient point for the moment equation can significantly simplify the calculations. Select a point where the lines of action of as many unknown forces as possible pass through. This eliminates those unknowns from the moment equation.

6. Solve for the Unknowns: Solve the system of equilibrium equations to find the magnitudes and directions of the unknown member forces. A positive value indicates tension, while a negative value indicates compression.

7. Repeat as Necessary: If more member forces need to be determined, repeat steps 2-6 for different section cuts.

Illustrative Example: Analyzing a Simple Truss

Let's analyze a simple truss to demonstrate the method of sections in action. Consider a truss with the following characteristics:

• A simply supported truss with a span of 12 meters.
• Two vertical loads of 10 kN each applied at joints B and D.
• Joint A is a pin support, and joint F is a roller support.

(Insert a clear diagram of the truss here showing dimensions, loads, and support reactions. This diagram should be properly labeled.)

Determining Support Reactions:

Before applying the method of sections, we need to determine the support reactions at A and F. Using the equations of equilibrium for the entire truss:

• ΣFy = 0: Ay + Fy - 10 kN - 10 kN = 0
• ΣMx = 0 (about A): (12m)(Fy) - (4m)(10 kN) - (8m)(10 kN) = 0

Solving these equations simultaneously yields Ay = 10 kN and Fy = 10 kN.

Analyzing Member Forces using Method of Sections:

Let's find the forces in members BC, CG, and FG. We'll cut through these members with a section line, isolating the left-hand portion of the truss.

(Insert a diagram showing the section cut and the isolated section.)

Free Body Diagram (FBD): The FBD of the isolated section includes the reaction force Ay (10kN), the two applied loads (10kN each), and the internal forces in members BC, CG, and FG. Let's assume that the forces in BC, CG, and FG are tensile (pulling away from the section).

(Insert the FBD clearly showing the forces and their assumed directions.)

Equilibrium Equations:

• ΣFx = 0: FBCcosθ + FCGcos(60°) + FGFG = 0 (θ is the angle of member BC with horizontal)
• ΣFy = 0: Ay + FBCsinθ - 10kN + FCGsin(60°) = 0
• ΣMA (moment about A): (4m)(10kN) + (8m)(10kN) – (3m)(FCGsin(60°)) - (6m)(FCGcos(60°)) = 0

Solving these three equations simultaneously, considering the geometric relationships of the truss, will give us the values for FBC, FCG, and FGFG. Remember that a negative value indicates compression.

Explanation of Equilibrium Equations and Solving Techniques

The equilibrium equations are based on Newton's laws of motion. Since the truss is in static equilibrium, the net force and net moment on any section must be zero. The choice of the point for calculating moments is crucial for simplifying the calculations.

Solving the system of equations can be achieved through various methods, including substitution, elimination, or matrix methods. In simpler trusses, substitution might be sufficient. For more complex scenarios, matrix methods offer a more systematic approach.

Advanced Considerations and Practical Applications

• Complex Trusses: For complex trusses with numerous members and loads, software tools are often employed to simplify the analysis. These tools can handle large systems of equations and provide detailed results.

• Influence Lines: The method of sections can be used in conjunction with influence lines to determine the maximum forces in truss members due to moving loads.

• Combined Methods: In some cases, a combination of the method of joints and the method of sections can be the most efficient approach. For example, you might use the method of joints to determine reactions and then the method of sections for specific members of interest.

• Influence of Material Properties: While the method of sections primarily focuses on the equilibrium of forces, it's crucial to consider the material properties (yield strength, modulus of elasticity) during the actual design process to ensure the selected members can withstand the calculated forces without failure.

Frequently Asked Questions (FAQ)

• Q: Can the method of sections be used for all types of trusses?

  • A: Yes, the method of sections is applicable to various truss types, including simple, compound, and complex trusses. However, its efficiency is most noticeable when analyzing specific members rather than the entire truss.

• Q: What happens if I assume the wrong direction for a member force?

  • A: If you assume the wrong direction, your solution will yield a negative value for the force. The magnitude of the force will be correct, but the sign indicates the actual direction is opposite to your initial assumption (tension instead of compression or vice versa).

• Q: How do I choose the best section cut?

  • A: Ideally, aim for a cut that passes through no more than three members with unknown forces. The cut should also be strategically placed to simplify the calculations (minimize the number of unknown forces in the FBD).

• Q: What are the limitations of the method of sections?

  • A: The method requires a good understanding of static equilibrium principles. For very complex trusses with many members, solving the simultaneous equations can be time-consuming, even with computational aids.

Conclusion

The method of sections provides a powerful and efficient tool for analyzing the internal forces in truss members. By strategically cutting through the truss and applying equilibrium equations, we can accurately determine the forces in specific members, which is crucial for design and safety assessment. While understanding the underlying principles of statics is essential, mastering this method allows for a more targeted and efficient analysis compared to the method of joints, especially when dealing with larger and more complex truss structures. Remember to always prioritize the clarity and accuracy of your free-body diagrams and calculations for reliable results. Careful attention to detail and a systematic approach will ensure success in applying this important structural analysis technique.