How to Draw a Bending Moment Diagram for Frames

Drawing a bending moment diagram for frames involves several systematic steps that ensure accuracy and clarity in the structural analysis. This guide will walk you through the process, emphasizing key points and providing examples to help you understand and apply these principles effectively.

Steps to Draw a Bending Moment Diagram for Frames

1. Understand the Frame Structure

The first step in drawing a bending moment diagram is to understand the frame structure.

Identify the type of frame: Is it a rigid-braced frame, a truss, or a semi-rigid frame? Understanding the type of frame helps in determining its behavior under various loads. Note the supports and connections: Distinguish between pinned, fixed, and roller supports. Understanding the nature of these supports is crucial for applying the correct boundary conditions.

2. Determine the Loads

Accurately identifying and categorizing the external loads is essential.

Identify all external loads: This includes point loads, distributed loads, and moments. Additionally, consider the self-weight of the frame if necessary.

3. Static Equilibrium

Using static equilibrium equations to find support reactions is a fundamental step.

Sum of vertical forces: SigmaFy 0 Sum of horizontal forces: SigmaFx 0 Sum of moments about a point: SigmaMA 0

4. Create Shear Force Diagram (SFD)

The shear force diagram forms the basis for the bending moment diagram.

Start from one end: Calculate the shear force at various points along the length of the frame. Plot the shear force values: These values will be plotted against the length of the frame on the SFD.

5. Calculate Bending Moments

The bending moment at a section is the integral of the shear force diagram. Alternatively, moments can be calculated directly at key points and load applications.

M∫x1x2Vdx

6. Draw the Bending Moment Diagram (BMD)

Plot the calculated bending moments along the length of the frame. Indicating positive and negative moments is crucial.

Positive moments create a sagging effect (bending upwards). Negative moments create a hogging effect (bending downwards).

7. Tips for Bending Moment Diagrams

To ensure accuracy, consistency, and clarity:

Adopt a consistent sign convention: For example, counter-clockwise moments are positive. Pay attention to key points: Locations where loads are applied, supports, and points of zero shear are critical for the moment diagram. Use software for complex frames: For complex structures, structural analysis software can provide accurate results.

Example: A Simple Frame with a Vertical Load at the Center

Let's consider a simple example of a frame with a vertical load at the center.

1. Calculate Reactions at Supports:

Assume the frame is supported by two reactions, ( R_A ) and ( R_B ).

Using static equilibrium:

Sum of vertical forces: RA RB L1 L2 0 Sum of moments about point A: RB L1 0

2. Create the Shear Force Diagram (SFD)

The SFD will typically show a jump at the point of the load.

3. Calculate Bending Moments at Key Sections:

M∫x1x2Vdx

At the load application point: The moment will be maximum.

4. Plot the Bending Moment Diagram (BMD):

Show the moment values, indicating positive or negative moments.

Conclusion

By following these steps, you will be able to construct a bending moment diagram that accurately represents the internal moments within a frame structure under given loads. This process not only enhances your structural analysis skills but also ensures the safety and efficiency of your designs.