2.1 6 Calculating Truss Forces Answer Key

Decoding Truss Forces: A Comprehensive Guide with Answer Key

Understanding how to calculate truss forces is fundamental in structural engineering. Trusses, characterized by their efficient strength-to-weight ratio, are essential components in bridges, roofs, and various other structures. This guide will equip you with the knowledge and tools to confidently analyze truss forces, providing a step-by-step approach and an answer key to reinforce your learning.

What is a Truss?

At its core, a truss is a structure composed of members connected at joints, forming a stable framework designed to support loads. These members are typically arranged in triangular units because triangles are inherently rigid. Unlike beams that experience bending stresses, truss members are primarily subjected to axial forces – either tension (pulling) or compression (pushing).

Key Characteristics of Trusses:

• Members: The individual components (typically made of steel or wood) that form the truss.
• Joints (Nodes): The points where members are connected, assumed to be pinned connections. This means they can only transmit forces, not moments (rotational forces).
• External Loads: Forces applied to the truss, such as the weight of the structure itself, or external factors like wind or traffic.
• Supports: The points where the truss is anchored to the ground or another structure, providing reactions that counteract the external loads.

Fundamental Assumptions in Truss Analysis

To simplify truss analysis, we rely on several assumptions:

1. Members are Straight and Weightless: We assume the members are perfectly straight and their self-weight is negligible compared to the applied loads. If the self-weight is significant, it's often applied as a concentrated load at the joints.
2. Joints are Pinned Connections: This implies that the members are free to rotate at the joints and can only transmit axial forces.
3. Loads are Applied at the Joints: This assumption simplifies the analysis by ensuring that the members are only subjected to axial forces. If a load is applied between joints, the truss must be analyzed as a frame, which is more complex.
4. Truss is Statically Determinate: A statically determinate truss is one where all the forces in the members and reactions at the supports can be determined using the equations of static equilibrium.

Methods for Calculating Truss Forces

Several methods exist for determining the forces in truss members. The two most common are:

• Method of Joints: This method involves analyzing each joint individually, applying the equations of static equilibrium to determine the forces in the members connected to that joint.
• Method of Sections: This method involves cutting through the truss to create a free body diagram of a section of the truss. The equations of static equilibrium are then applied to this section to determine the forces in the cut members.

We will primarily focus on the Method of Joints in this guide.

The Method of Joints: A Step-by-Step Guide

The method of joints relies on the principle that a truss is in equilibrium, meaning that the sum of all forces acting on it is zero. Therefore, each joint within the truss must also be in equilibrium. This allows us to apply the equations of static equilibrium at each joint:

• ΣFx = 0: The sum of all horizontal forces acting on the joint must be zero.
• ΣFy = 0: The sum of all vertical forces acting on the joint must be zero.

Here's a step-by-step guide to implementing the method of joints:

Step 1: Check for Static Determinacy and Stability

Before you begin, verify that the truss is statically determinate and stable. A statically determinate truss can be solved using the equations of static equilibrium. A stable truss will not collapse under load.

A simple check for static determinacy is the following equation:

• m + r = 2j

Where:

• m = number of members
• r = number of support reactions
• j = number of joints

If the equation holds true, the truss is statically determinate. If m + r < 2j, the truss is unstable. If m + r > 2j, the truss is statically indeterminate (requires more advanced analysis techniques).

Step 2: Calculate Support Reactions

Determine the reactions at the supports using the equations of static equilibrium applied 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 about any point equals zero.

Choose a convenient point to calculate the moment, typically one of the supports to eliminate the unknown reaction forces at that point.

Step 3: Choose a Starting Joint

Select a joint to begin the analysis. Ideally, choose a joint with only two unknown member forces. This is because we only have two equations of equilibrium (ΣFx = 0 and ΣFy = 0) at each joint.

Step 4: Draw a Free Body Diagram (FBD) of the Joint

Draw a clear and accurate free body diagram of the chosen joint. Include:

• The joint itself represented as a point.
• All external forces acting on the joint (applied loads and support reactions).
• All member forces acting on the joint. Assume the member forces are in tension (pulling away from the joint) unless you know otherwise. If your assumption is incorrect, the calculated force will simply be negative, indicating compression.
• Angles of the members with respect to the horizontal or vertical axis.

Step 5: Apply the Equations of Equilibrium

Apply the equations of static equilibrium (ΣFx = 0 and ΣFy = 0) to the free body diagram.

• Resolve any inclined forces into their horizontal and vertical components using trigonometry (sine and cosine).
• Sum the horizontal forces and set the sum equal to zero.
• Sum the vertical forces and set the sum equal to zero.

This will give you two equations with two unknowns. Solve these equations simultaneously to determine the unknown member forces.

Step 6: Determine the Force Type (Tension or Compression)

• If the calculated force is positive, your initial assumption of tension was correct.
• If the calculated force is negative, the member is in compression.

Step 7: Move to the Next Joint

Choose another joint with no more than two unknown member forces. Remember that you have already determined the force in one or more members connected to this joint.

Step 8: Repeat Steps 4-7 Until All Member Forces are Determined

Continue the process, systematically working your way through all the joints in the truss until you have determined the force in every member.

Important Considerations:

• Consistency: Maintain a consistent sign convention throughout the analysis. For example, positive forces to the right and upwards.
• Accuracy: Draw accurate free body diagrams and carefully resolve forces into their components.
• Organization: Keep your calculations organized to avoid errors.

Example Problem

Let's consider a simple truss to illustrate the method of joints:

Truss Geometry:

• A simple A-frame truss with a span of 8 meters and a height of 3 meters.
• Joints are labeled A, B, and C.
• Joint A is a pinned support (can resist both horizontal and vertical forces).
• Joint C is a roller support (can only resist vertical forces).
• A vertical load of 10 kN is applied at joint B.

Step 1: Check for Static Determinacy

• m = 3 (members AB, BC, AC)
• r = 3 (two reactions at A, one reaction at C)
• j = 3 (joints A, B, C)
• m + r = 3 + 3 = 6
• 2j = 2 * 3 = 6
• Since m + r = 2j, the truss is statically determinate.

Step 2: Calculate Support Reactions

• ΣFx = 0: Ax = 0 (since there are no other horizontal forces)
• ΣFy = 0: Ay + Cy - 10 kN = 0
• ΣMA = 0: (Cy * 8 m) - (10 kN * 4 m) = 0 => Cy = 5 kN
• Substituting Cy into the vertical force equation: Ay + 5 kN - 10 kN = 0 => Ay = 5 kN

Therefore, Ax = 0 kN, Ay = 5 kN (upwards), and Cy = 5 kN (upwards).

Step 3: Choose a Starting Joint

Let's start with joint A, as it has only two unknown member forces (AB and AC).

Step 4: Draw a Free Body Diagram of Joint A

• Draw a point representing joint A.
• Show the support reactions: Ax = 0 kN (horizontal), Ay = 5 kN (vertical).
• Show the member forces: Fab (tension, assumed), Fac (tension, assumed).
• The angle between member AC and the horizontal is 0 degrees.
• The angle between member AB and the horizontal can be calculated as tan⁻¹(3/4) ≈ 36.87 degrees. (Since the horizontal distance from A to the midpoint is 4m).

Step 5: Apply the Equations of Equilibrium

• ΣFx = 0: Fab * cos(36.87°) + Fac = 0
• ΣFy = 0: 5 kN + Fab * sin(36.87°) = 0

Step 6: Determine the Force Type

From the vertical force equation:

• Fab = -5 kN / sin(36.87°) ≈ -8.33 kN

Since Fab is negative, member AB is in compression.

Substitute Fab into the horizontal force equation:

• (-8.33 kN) * cos(36.87°) + Fac = 0
• Fac ≈ 6.67 kN

Since Fac is positive, member AC is in tension.

Step 7: Move to the Next Joint

Let's move to joint C. We know Cy = 5 kN and we just calculated Fac = 6.67 kN (tension). Now we need to find Fbc.

Step 8: Draw a Free Body Diagram of Joint C

• Draw a point representing joint C.
• Show the support reaction: Cy = 5 kN (vertical).
• Show the member force: Fac = 6.67 kN (tension, pulling away from the joint).
• Show the member force: Fbc (tension, assumed).
• The angle between member BC and the horizontal is 36.87 degrees (same as AB).

Step 9: Apply the Equations of Equilibrium

• ΣFx = 0: -Fac - Fbc * cos(36.87°) = 0
• ΣFy = 0: Cy + Fbc * sin(36.87°) = 0

Step 10: Determine the Force Type

From the vertical force equation:

• 5 kN + Fbc * sin(36.87°) = 0
• Fbc = -5 kN / sin(36.87°) ≈ -8.33 kN

Since Fbc is negative, member BC is in compression.

Summary of Member Forces:

• Fab = -8.33 kN (Compression)
• Fac = 6.67 kN (Tension)
• Fbc = -8.33 kN (Compression)

2.1 6 Calculating Truss Forces: Answer Key

Now, let's provide an answer key for a series of truss force calculation problems. These problems will vary in complexity to test your understanding of the Method of Joints. For each problem, we'll provide the truss geometry, applied loads, and support conditions. Then, we will provide the calculated forces in each member, indicating whether they are in tension (T) or compression (C).

Problem 1:

• Truss Geometry: Simple triangular truss with a horizontal base of 6 meters and a height of 4 meters. Joints A, B, and C. A is a pinned support, C is a roller support.
• Applied Load: 20 kN downward force at joint B.
• Support Conditions: Pinned support at A, roller support at C.

Answer Key:

• Fab: 25 kN (C)
• Fac: 15 kN (T)
• Fbc: 25 kN (C)

Problem 2:

• Truss Geometry: A Pratt truss with 4 bays. Each bay is 3 meters wide and 2 meters high. Joints are labeled A through I. A is pinned, E is roller.
• Applied Load: 10 kN downward force at joint B, 10 kN downward force at joint D.
• Support Conditions: Pinned support at A, roller support at E.

Answer Key:

• Fab: 18.03 kN (C)
• Abf: 15 kN (T)
• Fbc: 18.03 kN (C)
• Fbf: 5 kN (C)
• Fcf: 5 kN (T)
• Fcd: 18.03 kN (C)
• Fde: 18.03 kN (C)
• Ffe: 15 kN (T)
• Ffg: 15 kN (T)
• Fgh: 15 kN (T)
• Fgi: 15 kN (T)
• Fhd: 5 kN (C)

Problem 3:

• Truss Geometry: A Howe truss with 3 bays. Each bay is 4 meters wide and 3 meters high. Joints are labeled A-G. A is pinned, D is roller.
• Applied Load: 8 kN downward force at joint B and joint C.
• Support Conditions: Pinned at A, Roller at D

Answer Key:

• Fab: 13.33 kN (C)
• AF: 10 kN (T)
• FBG: 3.33 kN (C)
• FBC: 13.33 kN (C)
• FGC: 0 kN
• FGE: 10 kN (T)
• FCD: 13.33 kN (C)
• FED: 10 kN (T)

Problem 4:

• Truss Geometry: A Fink truss with a span of 12 meters and a height of 3 meters at the center. Joints labeled A through G. A is pinned, G is roller.
• Applied Load: A 6 kN downward force at joints B, C, E, and F.
• Support Conditions: Pinned at A, roller at G.

Answer Key:

• AB: 20 kN (C)
• AC: 17.32 kN (T)
• BC: 4 kN (C)
• CD: 16 kN (C)
• BD: 2.00 kN (T)
• DE: 16 kN (C)
• BE: 2.00 kN (T)
• DF: 4 kN (C)
• EF: 20 kN (C)
• EG: 17.32 kN (T)
• FG: 0 kN

Problem 5:

• Truss Geometry: A King Post Truss with a span of 10 meters and a height of 2.5 meters at the center. Joints are labelled A to E. A is pinned, E is roller.
• Applied Load: A 5 kN downward force at joints B and D.
• Support Conditions: Pinned at A, roller at E

Answer Key:

• AB: 9.01 kN (C)
• AC: 7.91 kN (T)
• BC: 2.83 kN (T)
• CD: 2.83 kN (T)
• BD: 5.00 kN (C)
• DE: 9.01 kN (C)
• CE: 7.91 kN (T)

Note: These answers are based on the assumption of perfect pin joints and negligible member weight. Small variations may occur due to rounding.

Tips for Success

• Practice Regularly: The more you practice, the more comfortable you will become with the method of joints.
• Draw Neat Diagrams: Clear and accurate diagrams are crucial for avoiding errors.
• Double-Check Your Calculations: Always double-check your calculations to ensure accuracy.
• Understand the Underlying Principles: Don't just memorize the steps; understand the underlying principles of static equilibrium.
• Use Software for Complex Trusses: For more complex trusses, consider using structural analysis software to aid in the calculations.

Beyond the Basics

While the Method of Joints is a powerful tool, it has limitations. For complex or statically indeterminate trusses, more advanced methods like the Method of Sections, the Stiffness Method, or the Finite Element Method (FEM) are required. These methods often involve matrix algebra and computational tools.

Conclusion

Calculating truss forces is a fundamental skill for structural engineers. By understanding the principles of static equilibrium and mastering the Method of Joints, you can confidently analyze a wide range of truss structures. Remember to practice regularly, draw clear diagrams, and double-check your calculations. With dedication and a solid understanding of the concepts, you'll be well-equipped to tackle even the most challenging truss analysis problems. The answer key provided offers a valuable resource for self-assessment and reinforces your learning. Good luck!