Shear Force And Bending Moment Examples

Shear force and bending moment are fundamental concepts in structural mechanics, crucial for analyzing the behavior of beams and other structural elements under load. Understanding these forces and moments is essential for engineers to design safe and efficient structures that can withstand external loads without failure.

Introduction to Shear Force and Bending Moment

Shear force is the internal force acting perpendicular to the cross-section of a beam, representing the sum of all vertical forces acting on either side of that section. It signifies the tendency of one part of the beam to slide or shear relative to the adjacent part. Bending moment, on the other hand, is the internal moment acting about the neutral axis of the cross-section. It represents the sum of the moments of all forces acting on either side of the section and indicates the internal resistance to bending deformation.

These internal forces and moments arise in response to external loads applied to the beam, such as concentrated loads, distributed loads, or moments. The distribution of shear force and bending moment along the length of the beam is typically represented graphically using shear force diagrams (SFD) and bending moment diagrams (BMD). These diagrams provide valuable insights into the internal stresses and deformations within the beam, allowing engineers to assess its structural integrity and predict its response to various loading scenarios.

Understanding Shear Force and Bending Moment Diagrams

Shear Force Diagram (SFD)

The SFD is a graphical representation of the shear force along the length of the beam. The vertical axis represents the magnitude of the shear force, while the horizontal axis represents the position along the beam. The SFD is constructed by considering the equilibrium of vertical forces at various sections along the beam.

Sign Convention: Shear force is typically considered positive if it causes a clockwise rotation to the part of the beam to the left of the section. Conversely, it is considered negative if it causes a counter-clockwise rotation.
Shape of SFD: The shape of the SFD depends on the type of loading applied to the beam. For example, a concentrated load will cause a sudden jump in the shear force, while a uniformly distributed load will result in a linear variation of shear force.
Key Points: The SFD can be used to identify the location of maximum shear force, which is critical for designing against shear failure.

Bending Moment Diagram (BMD)

The BMD is a graphical representation of the bending moment along the length of the beam. The vertical axis represents the magnitude of the bending moment, while the horizontal axis represents the position along the beam. The BMD is constructed by considering the equilibrium of moments at various sections along the beam.

Sign Convention: Bending moment is typically considered positive if it causes compression in the top fibers of the beam and tension in the bottom fibers (sagging). Conversely, it is considered negative if it causes tension in the top fibers and compression in the bottom fibers (hogging).
Shape of BMD: The shape of the BMD also depends on the type of loading applied to the beam. A concentrated load will cause a linear variation in the bending moment, while a uniformly distributed load will result in a parabolic variation.
Key Points: The BMD can be used to identify the location of maximum bending moment, which is crucial for designing against bending failure. The point where the bending moment is zero is called the point of contraflexure.

Steps to Draw Shear Force and Bending Moment Diagrams

Determine Support Reactions: First, calculate the reactions at the supports of the beam using static equilibrium equations (sum of vertical forces = 0, sum of moments = 0).
Establish Sign Conventions: Choose consistent sign conventions for shear force and bending moment (as explained above).
Cut Sections: Imagine cutting the beam at various locations (sections) along its length. It's often helpful to cut sections just before and just after a concentrated load or at the start and end of a distributed load.
Calculate Shear Force: For each section, calculate the shear force by summing all vertical forces acting on either side of the section (typically the left side). Remember to consider the sign conventions.
Calculate Bending Moment: For each section, calculate the bending moment by summing the moments of all forces acting on either side of the section about the section's location. Again, adhere to the sign conventions.
Plot the Diagrams: Plot the calculated shear forces and bending moments on their respective diagrams (SFD and BMD) with the position along the beam on the x-axis. Connect the points to create the diagrams.
Verify Results: Check the diagrams for consistency and accuracy. For example, the slope of the BMD at any point should be equal to the value of the shear force at that point.

Shear Force and Bending Moment Examples

Let's explore some examples to illustrate how to calculate shear force and bending moment and draw the corresponding diagrams.

Example 1: Simply Supported Beam with a Concentrated Load

Consider a simply supported beam of length L with a concentrated load P applied at its mid-span (L/2).

Step 1: Support Reactions:
- Due to symmetry, the vertical reactions at both supports are equal: Ra = Rb = P/2.
Step 2: Cut Sections:
- Section 1: 0 < x < L/2 (Left of the load)
- Section 2: L/2 < x < L (Right of the load)
Step 3: Calculate Shear Force:
- Section 1: V(x) = Ra = P/2 (Positive shear force)
- Section 2: V(x) = Ra - P = P/2 - P = -P/2 (Negative shear force)
Step 4: Calculate Bending Moment:
- Section 1: M(x) = Ra * x = (P/2) * x
- Section 2: M(x) = Ra * x - P * (x - L/2) = (P/2) * x - P * x + (P * L)/2 = (P * L)/2 - (P/2) * x
Step 5: Plot the Diagrams:
- SFD: The shear force is constant at P/2 from x = 0 to x = L/2, then it suddenly drops to -P/2 and remains constant until x = L.
- BMD: The bending moment increases linearly from 0 at x = 0 to (P * L)/4 at x = L/2. Then, it decreases linearly back to 0 at x = L.
Key Observations:
- Maximum shear force: P/2
- Maximum bending moment: (P * L)/4, occurring at the mid-span.

Example 2: Simply Supported Beam with a Uniformly Distributed Load (UDL)

Consider a simply supported beam of length L with a uniformly distributed load w (force per unit length) acting over the entire length.

Step 1: Support Reactions:
- Due to symmetry, the vertical reactions at both supports are equal: Ra = Rb = (w * L)/2
Step 2: Cut Section:
- Only one section is needed: 0 < x < L
Step 3: Calculate Shear Force:
- V(x) = Ra - w * x = (w * L)/2 - w * x
Step 4: Calculate Bending Moment:
- M(x) = Ra * x - (w * x) * (x/2) = (w * L * x)/2 - (w * x^2)/2
Step 5: Plot the Diagrams:
- SFD: The shear force varies linearly from (w * L)/2 at x = 0 to -(w * L)/2 at x = L. The shear force is zero at the mid-span (x = L/2).
- BMD: The bending moment varies parabolically from 0 at x = 0 to a maximum value at the mid-span (x = L/2), and then back to 0 at x = L.
Key Observations:
- Maximum shear force: (w * L)/2
- Maximum bending moment: (w * L^2)/8, occurring at the mid-span.

Example 3: Overhanging Beam with a Concentrated Load

Consider an overhanging beam with a length of L1 between the supports and an overhang of length L2 on one side. A concentrated load P is applied at the free end of the overhang.

Step 1: Support Reactions:
- Take moments about support A to find Rb, then use the sum of vertical forces to find Ra.
- Rb = (P * (L1 + L2)) / L1
- Ra = P + Rb = P + (P * (L1 + L2)) / L1 = (P * (2 * L1 + L2)) / L1
Step 2: Cut Sections:
- Section 1: 0 < x < L1 (Between the supports)
- Section 2: L1 < x < L1 + L2 (Overhang)
Step 3: Calculate Shear Force:
- Section 1: V(x) = Ra = (P * (2 * L1 + L2)) / L1
- Section 2: V(x) = -P
Step 4: Calculate Bending Moment:
- Section 1: M(x) = Ra * x = ((P * (2 * L1 + L2)) / L1) * x
- Section 2: M(x) = -P * (L1 + L2 - x) (Note: x is measured from the left end) This can also be expressed as M(x') = -P * x' where x' is the distance from the free end of the overhang.
Step 5: Plot the Diagrams:
- SFD: The shear force is constant at (P * (2 * L1 + L2)) / L1 between the supports and then drops to -P over the overhang.
- BMD: The bending moment increases linearly from 0 at x = 0 to a maximum positive value at support B (x = L1). Over the overhang, the bending moment is negative and varies linearly from 0 at the free end to -P*L2 at support B.
Key Observations:
- The bending moment changes sign, indicating points of contraflexure.
- The overhang introduces negative bending moment.

Example 4: Cantilever Beam with a Concentrated Load at the Free End

Consider a cantilever beam of length L with a concentrated load P applied at its free end.

Step 1: Support Reactions:
- The vertical reaction at the fixed end is equal to the load: Ra = P
- The moment reaction at the fixed end is equal to the load times the length: Ma = P * L
Step 2: Cut Section:
- Only one section is needed: 0 < x < L (x is measured from the free end)
Step 3: Calculate Shear Force:
- V(x) = -P (Constant shear force)
Step 4: Calculate Bending Moment:
- M(x) = -P * x (Negative bending moment, increasing linearly with distance from the free end)
Step 5: Plot the Diagrams:
- SFD: The shear force is constant at -P along the entire length of the beam.
- BMD: The bending moment varies linearly from 0 at the free end to -P * L at the fixed end.
Key Observations:
- Maximum shear force: P
- Maximum bending moment: P * L, occurring at the fixed end. The moment is negative, indicating hogging.

Example 5: Cantilever Beam with a Uniformly Distributed Load

Consider a cantilever beam of length L with a uniformly distributed load w (force per unit length) acting over the entire length.

Step 1: Support Reactions:
- The vertical reaction at the fixed end is equal to the total load: Ra = w * L
- The moment reaction at the fixed end is equal to the total load times half the length: Ma = (w * L^2) / 2
Step 2: Cut Section:
- Only one section is needed: 0 < x < L (x is measured from the free end)
Step 3: Calculate Shear Force:
- V(x) = -w * x
Step 4: Calculate Bending Moment:
- M(x) = -w * x^2 / 2
Step 5: Plot the Diagrams:
- SFD: The shear force varies linearly from 0 at the free end to -w * L at the fixed end.
- BMD: The bending moment varies quadratically (parabolically) from 0 at the free end to -(w * L^2) / 2 at the fixed end.
Key Observations:
- Maximum shear force: w * L
- Maximum bending moment: (w * L^2) / 2, occurring at the fixed end. The moment is negative, indicating hogging.

Practical Applications and Importance

The concepts of shear force and bending moment are not just theoretical exercises; they have immense practical importance in structural engineering. Understanding these concepts allows engineers to:

Design Safe Structures: By calculating the maximum shear force and bending moment, engineers can select appropriate materials and dimensions for structural members to ensure they can withstand the applied loads without failing.
Optimize Material Usage: Accurate analysis of shear force and bending moment distribution allows engineers to optimize material usage, leading to cost-effective designs. They can use more material where stresses are high and less where stresses are low.
Predict Deflections: Bending moment is directly related to the curvature of the beam, which in turn is related to deflection. Understanding the BMD allows engineers to predict the deflection of the beam under load.
Analyze Complex Structures: For more complex structures, numerical methods like the finite element method (FEM) are often used to determine shear force and bending moment distributions. However, understanding the fundamental principles remains crucial for interpreting the results of these analyses.
Ensure Structural Integrity: Regular inspections of structures often involve assessing the presence of cracks or deformations that may be indicative of excessive shear force or bending moment. Early detection of these issues can prevent catastrophic failures.

Common Mistakes and Pitfalls

While the principles of shear force and bending moment are relatively straightforward, there are some common mistakes that students and even experienced engineers can make:

Incorrect Sign Conventions: Consistent application of sign conventions is crucial. Confusing the sign conventions will lead to incorrect diagrams and inaccurate results.
Forgetting Support Reactions: Always start by calculating the support reactions accurately. Incorrect support reactions will propagate errors throughout the entire analysis.
Ignoring Distributed Loads: Remember to account for the distributed load's total force and its location when calculating shear force and bending moment. The equivalent point load is located at the centroid of the distributed load.
Incorrectly Calculating Moments: Ensure that the moment is calculated about the correct point and that the distances are measured accurately.
Misinterpreting Diagrams: Understand what the SFD and BMD represent. The SFD shows the internal shear force, and the BMD shows the internal bending moment. Don't confuse them with external loads.
Oversimplifying Assumptions: Be aware of the assumptions made in the analysis, such as the beam being perfectly straight, the material being linearly elastic, and the supports being perfectly rigid.

Conclusion

Shear force and bending moment are essential concepts in structural mechanics that provide valuable insights into the internal forces and moments within beams and other structural elements. Understanding how to calculate shear force and bending moment and how to draw the corresponding diagrams is crucial for designing safe, efficient, and durable structures. By carefully considering the applied loads, support conditions, and material properties, engineers can accurately predict the behavior of structures and ensure their structural integrity. The examples provided illustrate the application of these concepts in various scenarios, highlighting the importance of consistent sign conventions, accurate calculations, and a thorough understanding of the underlying principles. Mastering these concepts is a fundamental step towards becoming a proficient structural engineer.