2. SHEAR FORCE & BENDING MOMENT DIAGRAMS

For 4th Semester Polytechnic ME Students

Written by Garima Kanwar | Blog: Rajasthan Polytechnic

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Course Code ME 4002 (Same in MA/MP/MT 4002)
Course Title STRENGTH OF MATERIALS

2. Shear Force and Bending Moment Diagrams

This topic covers the analysis of beams under various loading conditions. Understanding the shear force (SF) and bending moment (BM) is essential in structural engineering to design safe and stable structures. Here's a breakdown of each sub-topic:

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2.1 Types of Beams

Beams are structural elements that support loads and resist bending. The different types of beams include:

• Simply Supported Beam: A beam supported at both ends. It can have a point load, uniform distributed load (UDL), or a combination of both. The beam can rotate at the supports but cannot displace vertically.

• Cantilever Beam: A beam fixed at one end and free at the other. It is subjected to loads at its free end or along its length.

• Overhanging Beam: A beam that extends beyond its supports. It can have one or both ends overhanging.

• Continuous Beam: A beam with more than two supports. It is supported at multiple points, which helps in distributing the load more efficiently.

• Fixed Beam: A beam that is fixed at both ends, offering resistance to both translation and rotation at the ends.

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2.2 Types of Loads

• Point Load (Concentrated Load): A load applied at a single point on the beam. It creates sharp changes in the shear force diagram and bending moment diagram.

• Uniformly Distributed Load (UDL): A load that is evenly spread out along the length of the beam. It results in a linear change in the shear force and a parabolic bending moment distribution.

• Uniformly Varying Load (UVL): A load that increases or decreases along the length of the beam, typically in a triangular shape. It results in a non-linear change in shear force and a cubic bending moment distribution.

• Couple or Moment Load: A moment applied to a point or along the length of the beam, creating rotation but no shear force.

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2.3 Shear Force and Bending Moment Diagrams for Various Types of Beams

Simply Supported Beam:

• The shear force diagram (SFD) is a plot of shear force along the length of the beam.
• The bending moment diagram (BMD) is a plot of bending moment along the length of the beam.

Cantilever Beam:

• In cantilever beams, the shear force and bending moment are both highest at the fixed end and decrease towards the free end.

Overhanging Beam:

• Overhanging beams have portions that extend beyond the supports, and their shear force and bending moment diagrams are more complex because of the additional reactions at the overhang.

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2.4 Analytical Method for Shear Force and Bending Moment of a Simply Supported Beam

For a simply supported beam with point loads and uniformly distributed loads, the following steps are used to draw the shear force and bending moment diagrams:

Steps for Drawing SFD and BMD:

1. Support Reactions:

   • First, calculate the reactions at the supports using static equilibrium equations.
   • Sum of forces in vertical direction ($\Sigma F_y = 0$).
   • Sum of moments about any point ($\Sigma M = 0$).

2. Shear Force (SF):

   • Starting from the leftmost support, move along the beam.
   • Add or subtract the loads to calculate the shear force at various points along the beam.
   • For point loads, the shear force changes abruptly.
   • For UDL, the shear force changes linearly along the length of the beam.

3. Bending Moment (BM):

   • Starting from the leftmost point, calculate the bending moment at different points by taking moments about a section of the beam.
   • For point loads, the bending moment changes in a linear fashion.
   • For UDL, the bending moment changes quadratically.

Example:

A simply supported beam has a length of 6 m and carries a point load of 10 kN at 2 m from the left support. The reactions at the supports can be calculated by equilibrium equations.

• Reaction at Left Support (R_A):

  $$R_A = \frac{10 \, \text{kN} \times (6 \, \text{m} - 2 \, \text{m})}{6 \, \text{m}} = 6.67 \, \text{kN}$$
  

• Reaction at Right Support (R_B):

  $$R_B = 10 \, \text{kN} - 6.67 \, \text{kN} = 3.33 \, \text{kN}$$
  

Now, plot the shear force diagram and bending moment diagram based on these reactions.

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2.5 Overhanging Beam with Point Loads

For an overhanging beam, the analysis is similar to the simply supported beam, with additional complexities due to the overhanging portion. The reaction forces are calculated in the same way as for simply supported beams, but additional reactions are considered for the overhanging portion.

Steps for Overhanging Beam:

1. Calculate Reactions: Use equilibrium equations to find reactions at the supports, including reactions due to loads on the overhanging section.
2. Shear Force (SF): The shear force will change at the support, especially at the point where the overhang begins.
3. Bending Moment (BM): The bending moment will be highest at the fixed support and decrease towards the free end, with the overhang contributing to a positive or negative moment depending on the direction of the load.

Example:

Consider an overhanging beam of length 6 m, with a point load of 8 kN at 3 m from the left support and the right overhang extending 2 m beyond the support. First, calculate the reactions at the supports and then determine the shear force and bending moment diagrams.

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2.6 Combination of Point Load and UDL for the Above; Related Numerical Problems

In most real-world cases, beams are subjected to a combination of point loads and uniformly distributed loads (UDLs). The analysis for such cases follows the same basic principles, but the reactions and the shear force and bending moment diagrams need to account for both types of loads simultaneously.

Steps for Combination Load:

1. Calculate Reactions: Use equilibrium equations to find reactions due to the combination of point loads and UDLs.
2. Shear Force (SF): The shear force will change at points where the load distribution changes (e.g., where a point load or UDL starts or ends).
3. Bending Moment (BM): The bending moment will be calculated by integrating the shear force diagram or by taking moments about specific points on the beam.

Example:

A simply supported beam of length 8 m has:

• A point load of 10 kN at 3 m from the left support.
• A UDL of 4 kN/m over the length from 4 m to 6 m.

1. Reactions at supports:

   • Calculate the reaction forces at the supports using equilibrium equations.

2. Shear Force Diagram (SFD):

   • The shear force will change at the location of the point load and at the start and end of the UDL.

3. Bending Moment Diagram (BMD):

   • Calculate the bending moment at various points along the beam using the shear force diagram or by directly summing moments.

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Important Formulas for Numerical Problems:

• Reaction at support (R):

  $$\Sigma F_y = 0 \quad \text{(for vertical equilibrium)}$$
  $$\Sigma M = 0 \quad \text{(for moment equilibrium)}$$
  

• Shear Force (V):

  $$V = \text{Reaction at support} - \text{Point loads or integral of UDLs}$$

• Bending Moment (M):

  $$M = \int \text{Shear Force} \, dx$$
  

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Sample Numerical Problems:

1. Problem 1: A simply supported beam of length 8 m carries a point load of 20 kN at the center. Find the shear force and bending moment at various points along the beam.

2. Problem 2: An overhanging beam of length 6 m has a 5 kN point load at 2 m from the left support and a 2 kN/m UDL over the last 2 m. Calculate the reactions at the supports and draw the shear force and bending moment diagrams.

3. Problem 3: A simply supported beam is subjected to a combination of a 10 kN point load at 3 m from the left end and a UDL of 4 kN/m between 4 m and 6 m from the left support. Calculate the shear force and bending moment at different points on the beam.

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Conclusion:

Understanding shear force and bending moment diagrams is crucial for analyzing the internal forces within structural elements under various load conditions. By calculating reactions and then using the principles of static equilibrium, you can determine the shear force and bending moment at key points along the beam. Practice with different types of load combinations and beam configurations to strengthen your understanding.

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