Calculate Bending Moment of Beam by Integration

What is Bending Moment?

The bending moment (M) is the internal moment within a structural element that resists the bending caused by an external load. It is typically measured in newton-meters (Nm) or pound-feet (lb-ft) and is a key parameter in beam design. The bending moment varies along the length of the beam and is crucial for determining the beam's strength and deflection.

Bending moments are caused by transverse loads applied to the beam. The maximum bending moment in a simply supported beam occurs at the point where the load is applied, while in a cantilever beam, it occurs at the fixed end.

Calculating Bending Moment

There are several methods to calculate the bending moment of a beam, including the moment-area method, the conjugate beam method, and the integration method. The integration method is particularly useful for beams with distributed loads or varying cross-sections.

The basic principle behind the integration method is that the bending moment at any point along the beam can be found by integrating the shear force diagram. This involves:

1. Determining the shear force diagram for the beam
2. Integrating the shear force diagram to obtain the bending moment diagram
3. Evaluating the bending moment at specific points of interest

Integration Method

The integration method for calculating bending moment involves the following steps:

1. Draw the free body diagram of the beam and identify all applied loads
2. Calculate the reactions at the supports
3. Determine the shear force at various points along the beam
4. Integrate the shear force to obtain the bending moment
5. Evaluate the bending moment at critical points

For beams with multiple loads, the shear force diagram will consist of piecewise linear segments, and the integration will involve summing the contributions from each segment.

Example Calculation

Consider a simply supported beam of length L with a uniformly distributed load of intensity w. We will calculate the bending moment at a distance x from the left support using the integration method.

1. First, calculate the reactions at the supports:
• Vertical reaction at the left support (A): R_A = wL/2
• Vertical reaction at the right support (B): R_B = wL/2
2. Determine the shear force diagram:
• From 0 ≤ x ≤ L: V(x) = R_A - wx = wL/2 - wx
3. Integrate the shear force to find the bending moment:
• M(x) = ∫ (wL/2 - wx) dx = (wL/2)x - (w/2)x² + C
4. Apply boundary conditions to find the constant of integration:
• At x = 0, M(0) = 0: 0 = 0 - 0 + C ⇒ C = 0
5. Final bending moment equation:
• M(x) = (wL/2)x - (w/2)x²

This equation gives the bending moment at any point along the beam. The maximum bending moment occurs at the center of the beam (x = L/2):

Interpretation

The bending moment diagram obtained from the integration method provides valuable information about the beam's behavior under load. Key points to interpret include:

• The maximum bending moment, which determines the required beam strength
• The points where the bending moment changes sign, indicating points of inflection
• The shape of the diagram, which can indicate the type of loading (uniform, triangular, etc.)

Engineers use this information to select appropriate beam sizes, check for deflection limits, and ensure the beam can withstand the applied loads without failure.