Determine The Force In Each Member Of The Truss

Determining the force in each member of a truss is a fundamental problem in structural engineering. It involves analyzing the internal forces—tension or compression—acting within the individual components (members) of a truss structure. This analysis is crucial for ensuring the structural integrity and safety of bridges, roofs, and other engineering constructions. Several methods exist for performing this analysis, each with its advantages and applications. This article provides a comprehensive guide on how to determine the force in each member of a truss, covering the fundamental principles, different methods, practical examples, and considerations for accurate analysis.

Understanding Truss Structures

Before delving into the methods of analysis, it is crucial to understand the basic characteristics of truss structures:

• Definition: A truss is a structural framework composed of members joined together at their ends to form a rigid structure. The members are typically arranged in triangular units because this configuration provides inherent stability.
• Assumptions: Several simplifying assumptions are typically made to facilitate the analysis of trusses:
  • Members are connected at their ends by frictionless pins (hinges).
  • Loads are applied only at the joints (nodes).
  • Members are straight and weightless (or their weight is negligible compared to the applied loads).
• Types of Trusses: Trusses can be classified into different types based on their geometry and support conditions, including:
  • Simple Trusses: Consist of basic triangular units.
  • Compound Trusses: Formed by connecting two or more simple trusses.
  • Complex Trusses: Cannot be classified as simple or compound and require more advanced analysis techniques.

Fundamental Principles

The analysis of truss members relies on basic principles of statics, including:

• Equilibrium: A structure is in equilibrium when the sum of all forces and moments acting on it is zero. This condition must be satisfied for the entire truss and for each individual joint.
• Newton's Laws of Motion: Specifically, Newton's First Law (an object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by a force) is applied.
• Method of Joints: Analyzing the equilibrium of forces at each joint in the truss.
• Method of Sections: Analyzing the equilibrium of forces acting on a section of the truss.

Methods for Determining Forces in Truss Members

Several methods are commonly used to determine the forces in truss members. The two primary methods are the Method of Joints and the Method of Sections.

1. Method of Joints

The Method of Joints involves analyzing the equilibrium of forces at each joint in the truss. This method is based on the principle that if the entire truss is in equilibrium, then each joint must also be in equilibrium. The steps for applying the Method of Joints are as follows:

1. Check Determinacy and Stability:

   • Ensure the truss is statically determinate, meaning that the support reactions and internal forces can be determined using the equations of static equilibrium. The condition for determinacy is:

     m + r = 2j
     

     where:

     • m is the number of members,
     • r is the number of support reactions,
     • j is the number of joints.

   • Also, ensure the truss is stable and will not collapse under load.

2. Determine Support Reactions:

   • Calculate the external support reactions by applying the equations of equilibrium to the entire truss.

     ΣFx = 0 (sum of horizontal forces equals zero)
     ΣFy = 0 (sum of vertical forces equals zero)
     ΣM = 0 (sum of moments equals zero)
     

3. Analyze Joints Individually:
   • Start with a joint that has at most two unknown member forces.
   • Draw a free body diagram (FBD) of the joint, showing all forces acting on it, including external loads and member forces. Assume initially that all unknown member forces are in tension (i.e., pulling away from the joint).
   • Apply the equations of equilibrium (ΣFx = 0 and ΣFy = 0) to the joint to solve for the unknown member forces.
   • If the calculated force is positive, the member is in tension (as initially assumed). If the force is negative, the member is in compression.
4. Proceed to Other Joints:
   • Continue analyzing joints, one at a time, until the forces in all members have been determined. At each joint, use the known forces from previously analyzed joints.

Example: Method of Joints

Consider a simple truss with the following characteristics:

• A truss with four joints (A, B, C, D) and five members (AB, BC, CD, AD, BD).
• Joint A has a vertical support reaction (Ay) and a horizontal support reaction (Ax).
• Joint C has a vertical support reaction (Cy).
• An external vertical load (P) is applied at joint B.

Step 1: Check Determinacy and Stability

• m = 5 (number of members)
• r = 3 (number of support reactions: Ay, Ax, Cy)
• j = 4 (number of joints)

m + r = 2j
5 + 3 = 2 * 4
8 = 8

The truss is statically determinate.

Step 2: Determine Support Reactions

• Apply equilibrium equations to the entire truss:

  • ΣFx = 0: Ax = 0
  • ΣFy = 0: Ay + Cy - P = 0
  • ΣMA (moment about point A) = 0: Cy * L - P * (L/2) = 0 => Cy = P/2
  • Therefore, Ay = P - Cy = P/2 Step 3: Analyze Joints Individually

• Joint A:

  • Draw FBD of joint A.

  • Apply equilibrium equations:

    • ΣFx = 0: Ax + Fab * cos(θ) + Fad = 0 => 0 + Fab * cos(θ) + Fad = 0
    • ΣFy = 0: Ay + Fab * sin(θ) = 0 => P/2 + Fab * sin(θ) = 0

  • Solve for Fab and Fad. Fab = -P / (2 * sin(θ)) (Compression), Fad = P / (2 * tan(θ)) (Tension)

• Joint C:

  • Draw FBD of joint C.

  • Apply equilibrium equations:

    • ΣFx = 0: -Fcd - Fbc * cos(θ) = 0
    • ΣFy = 0: Cy - Fbc * sin(θ) = 0 => P/2 - Fbc * sin(θ) = 0

  • Solve for Fbc and Fcd. Fbc = P / (2 * sin(θ)) (Tension), Fcd = -P / (2 * tan(θ)) (Compression)

• Joint B:

  • Draw FBD of joint B.

  • Apply equilibrium equations:

    • ΣFx = 0: -Fab * cos(θ) + Fbc * cos(θ) - Fbd = 0
    • ΣFy = 0: Fab * sin(θ) + Fbc * sin(θ) - P = 0

  • Solve for Fbd. Fbd = 0

• Joint D:

  • Draw FBD of joint D.

  • Apply equilibrium equations:

    • ΣFx = 0: -Fad + Fcd + Fbd = 0
    • ΣFy = 0: 0 = 0

  • The forces are consistent with previous calculations.

Results:

• Fab = -P / (2 * sin(θ)) (Compression)
• Fad = P / (2 * tan(θ)) (Tension)
• Fbc = P / (2 * sin(θ)) (Tension)
• Fcd = -P / (2 * tan(θ)) (Compression)
• Fbd = 0

2. Method of Sections

The Method of Sections involves cutting the truss into sections and analyzing the equilibrium of forces acting on one of the sections. This method is particularly useful when determining the forces in specific members without having to analyze the entire truss. The steps for applying the Method of Sections are as follows:

1. Check Determinacy and Stability:
   • As with the Method of Joints, ensure the truss is statically determinate and stable.
2. Determine Support Reactions:
   • Calculate the external support reactions by applying the equations of equilibrium to the entire truss.
3. Cut the Truss:
   • Select a section that cuts through the members for which the forces are to be determined. The cut should not pass through more than three members with unknown forces.
4. Select a Section:
   • Choose one of the two sections created by the cut. The section should be chosen to simplify the analysis (e.g., minimize the number of forces or moments to be considered).
5. Draw Free Body Diagram:
   • Draw a FBD of the selected section, showing all forces acting on it, including external loads, support reactions, and member forces. Assume initially that all unknown member forces are in tension.
6. Apply Equilibrium Equations:
   • Apply the equations of equilibrium (ΣFx = 0, ΣFy = 0, and ΣM = 0) to the section to solve for the unknown member forces.
   • Choose the point about which to sum moments to eliminate one or more unknown forces from the moment equation.
   • If the calculated force is positive, the member is in tension. If the force is negative, the member is in compression.

Example: Method of Sections

Consider the same simple truss as before. Suppose we want to determine the force in member BD using the Method of Sections.

Step 1: Check Determinacy and Stability

• The truss is statically determinate (as shown in the previous example).

Step 2: Determine Support Reactions

• The support reactions are the same as before: Ay = P/2, Ax = 0, Cy = P/2.

Step 3: Cut the Truss

• Cut the truss through members AB, BD, and CD.

Step 4: Select a Section

• Choose the left section (containing joint A).

Step 5: Draw Free Body Diagram

• Draw a FBD of the left section, showing forces Ay, Fab, Fbd, and Fcd.

Step 6: Apply Equilibrium Equations

• ΣFy = 0: Ay + Fab * sin(θ) = 0 => P/2 + Fab * sin(θ) = 0
• ΣMA = 0: Choosing point A eliminates Fab. Thus ΣMA is not useful here.
• Take moments about point C:
  • ΣMC = 0: -Ay * L + Fab * sin(θ) * L = 0
  • P/2 * L + Fab * sin(θ) * L = 0 => Fab * sin(θ) = -P/2
• ΣFx = 0: Fab * cos(θ) + Fbd + Fcd = 0
• Note: By observation, the vertical force in BD must be 0, because there are no other vertical forces to balance the forces. Summing forces in the vertical direction confirms that. If this is not the case, the cut chosen would not provide enough information to resolve the force of member BD.

Results:

• Fab = -P / (2 * sin(θ)) (Compression)
• Fbd = 0

Additional Considerations

When analyzing trusses, it is important to consider the following:

• Zero-Force Members: These are members that do not carry any load under certain loading conditions. Identifying zero-force members can simplify the analysis. A zero-force member typically occurs when two members are collinear and no external load or support reaction is applied at their joint.
• Symmetry: If the truss and loading are symmetrical, the analysis can be simplified by considering only one half of the truss.
• Complex Trusses: For complex trusses, more advanced methods, such as the stiffness method or finite element analysis, may be required.
• Software Tools: Several software tools are available for analyzing trusses, such as SAP2000, ANSYS, and RISA. These tools can handle complex truss geometries and loading conditions.
• Accuracy: The accuracy of the analysis depends on the accuracy of the input data, including the truss geometry, material properties, and applied loads.

Practical Applications

Understanding how to determine forces in truss members is crucial for various engineering applications:

• Bridge Design: Trusses are commonly used in bridge design due to their high strength-to-weight ratio. Analyzing the forces in truss members is essential for ensuring the bridge can support the intended loads.
• Roof Structures: Trusses are used in roof structures to span large distances without the need for intermediate supports. Analyzing the forces in truss members is critical for ensuring the roof can withstand wind, snow, and other environmental loads.
• Crane Structures: Trusses are used in crane structures to provide lifting capacity and stability. Understanding the forces in truss members is essential for safe operation.
• Transmission Towers: Trusses are used in transmission towers to support high-voltage power lines. Analyzing the forces in truss members is necessary for ensuring the tower can withstand wind and ice loads.

Advanced Topics

• Influence Lines: Influence lines are used to determine the effect of a moving load on the forces in truss members. This is particularly useful in bridge design, where loads are constantly changing as vehicles move across the structure.
• Plastic Analysis: Plastic analysis considers the behavior of truss members beyond the elastic limit. This can provide a more accurate assessment of the structure's capacity under extreme loading conditions.
• Buckling Analysis: Buckling analysis is used to determine the stability of truss members under compressive loads. This is important because compressive members can fail by buckling before reaching their yield strength.

Common Mistakes to Avoid

• Incorrectly Determining Support Reactions: Accurate support reactions are essential for correct truss analysis.
• Incorrectly Drawing Free Body Diagrams: Free body diagrams must accurately represent all forces acting on the joint or section.
• Incorrectly Applying Equilibrium Equations: Ensure that the equations of equilibrium are applied correctly, including the correct sign conventions.
• Ignoring Zero-Force Members: Identifying zero-force members can simplify the analysis.
• Not Checking for Determinacy and Stability: Ensure the truss is statically determinate and stable before proceeding with the analysis.

FAQ Section

1. What is the difference between tension and compression in truss members?

   • Tension is a pulling force that tends to elongate the member. Compression is a pushing force that tends to shorten the member. In analysis, tension is typically considered positive, and compression is negative.

2. When should I use the Method of Joints versus the Method of Sections?

   • Use the Method of Joints when you need to determine the forces in all members of the truss. Use the Method of Sections when you only need to determine the forces in specific members.

3. What are zero-force members, and how can I identify them?

   • Zero-force members are members that do not carry any load under certain loading conditions. They typically occur when two members are collinear and no external load or support reaction is applied at their joint, or when three members meet at a joint, two of which are collinear and no external load is applied at that joint in the direction perpendicular to the two collinear members.

4. How do I handle complex trusses that are not statically determinate?

   • Complex trusses may require more advanced methods, such as the stiffness method or finite element analysis.

5. What software tools can I use for truss analysis?

   • Several software tools are available for analyzing trusses, such as SAP2000, ANSYS, and RISA. These tools can handle complex truss geometries and loading conditions.

6. Why is it important to check for determinacy and stability before analyzing a truss?

   • Checking for determinacy ensures that the truss can be solved using the equations of static equilibrium. Checking for stability ensures that the truss will not collapse under load.

Conclusion

Determining the force in each member of a truss is a fundamental aspect of structural engineering. The Method of Joints and the Method of Sections are two primary methods for analyzing truss structures. Understanding the basic principles of statics, drawing accurate free body diagrams, and applying the equations of equilibrium are essential for accurate analysis. By considering additional factors such as zero-force members, symmetry, and the use of software tools, engineers can effectively analyze truss structures and ensure their safety and stability. This comprehensive guide provides a solid foundation for analyzing truss members and understanding their behavior under load, essential for anyone involved in structural design and analysis.