Shear Force and Bending Moment Diagrams

Hello! Welcome to your third lesson in the module on Torsion and Bending in Beams.

In our last lesson, we explored how to analyze shafts under torsional loads. A key skill you practiced was creating an internal torque diagram by making imaginary "cuts" along the shaft and applying equilibrium to find the torque in each section. This method is fundamental, and today we will apply the exact same logic to beams under transverse loads (loads perpendicular to their length).

The goal of this lesson is to construct shear force and bending moment diagrams for beams under various loads. These diagrams are arguably the most important tools in beam analysis. They provide a complete picture of the internal forces and moments acting throughout the beam, allowing us to pinpoint locations of maximum stress.

1. What are Shear Force and Bending Moment?

When you apply loads to a beam, it doesn't just sit there; internal forces develop within the material to resist the external loads and keep the beam in equilibrium. We can represent these internal forces at any cut section with two resultants:

• Shear Force (V): The resultant of vertical internal forces. It's a measure of the tendency for one part of the beam to slide vertically with respect to an adjacent part.
• Bending Moment (M): The resultant of normal internal forces (tension and compression). It's a measure of the tendency for the beam to bend or rotate at the section.

The following video provides an excellent visualization of what these internal forces are and how they arise.

Sign Convention

To construct these diagrams consistently, we need a standard sign convention. While different conventions exist, the one based on how the beam deforms is the most common in mechanics of materials.

Here is a summary of the key points on sign convention:

• Bending Moment (M):

  • Positive (+M): Causes the beam to "sag" or hold water (concave up). This puts the bottom fibers in tension and the top fibers in compression.
  • Negative (-M): Causes the beam to "hog" or shed water (concave down). This puts the top fibers in tension and the bottom fibers in compression.

• Shear Force (V):

  • Positive (+V): The forces on the segment tend to cause a clockwise rotation.
  • Negative (-V): The forces on the segment tend to cause a counter-clockwise rotation.

2. Method 1: The Equation Method (Method of Sections)

This method is the most fundamental and is identical to the process you used to find internal torque. You will make a cut, isolate a segment of the beam, and use the equations of static equilibrium to derive functions for shear force, $V(x)$, and bending moment, $M(x)$.

The general procedure is:

1. Calculate Support Reactions: Draw a Free-Body Diagram (FBD) of the entire beam and use the equilibrium equations ($\sum F_x=0$, $\sum F_y=0$, $\sum M=0$) to find all unknown support reactions.
2. Make a Cut: For each region of the beam between loads, make an imaginary cut at a variable distance $x$ from the origin.
3. Apply Equilibrium: Draw an FBD of the segment to one side of the cut. Apply the equilibrium equations to this segment to solve for $V$ and $M$ as functions of $x$.
4. Plot the Equations: Plot the functions $V(x)$ and $M(x)$ for each region to create the final diagrams.

The following video demonstrates this "segment method" with a clear, step-by-step example.

This method is robust and always works, but it can be time-consuming for complex loading.

3. Method 2: The Graphical Method (Relationships)

A much faster way to construct these diagrams is by understanding the mathematical relationships between the load ($w$), shear force ($V$), and bending moment ($M$). These relationships allow us to sketch the diagrams by inspection.

The key relationships are:

Relationship (Differential)	Relationship (Integral)	Practical Meaning
$\frac{dV}{dx}=-w(x)$	$\Delta V=-\int w(x)dx$	The slope of the shear diagram at a point is the negative of the load intensity.
$\frac{dM}{dx}=V(x)$	$\Delta M=\int V(x)dx$	The slope of the moment diagram at a point is equal to the value of the shear force.

This means:

• The change in shear between two points is the negative area under the loading curve.
• The change in moment between two points is the area under the shear diagram.

The following video explains these powerful relationships.

Let's look at a practical example of applying this graphical method.

The image below shows diagrams for two common cases. On the left, point loads create constant shear sections and linear moment sections. On the right, a uniformly distributed load (UDL) creates a linear shear diagram and a parabolic (2nd-degree) moment diagram.

Test your understanding!

Look at the right-hand example in the image above (the cantilever beam).

1. At the fixed wall (left end), the shear force is at its maximum positive value. According to the relationship $dM/dx=V$, what should the slope of the bending moment diagram look like at that point?
2. Where the shear force diagram crosses the x-axis (V=0), what happens on the bending moment diagram?

Show answer

1. Since $V$ is large and positive at the wall, the slope of the bending moment diagram should be large and positive (steeply increasing). The diagram shows this.
2. Where $V=0$, the slope of the moment diagram $dM/dx$ must also be zero. This corresponds to a local maximum or minimum on the bending moment diagram. In this case, it's the point of maximum negative moment (maximum hogging).

4. A Comprehensive Example

Now let's see how these methods are applied to a beam with a mix of point loads and distributed loads. Calculating the reactions is always the first step. Then, you can use the graphical method to quickly sketch the shapes and find the values at key points.

The following text provides a few more standard examples.

Conclusion

In this lesson, you learned the two primary methods for constructing shear force and bending moment diagrams, which are indispensable tools for any engineer designing structures.

Key Takeaways:

• Shear force ($V$) and bending moment ($M$) diagrams are graphical representations of the internal forces along a beam's length.
• The Equation Method involves writing and plotting equations $V(x)$ and $M(x)$ derived from equilibrium. It is thorough but can be slow.
• The Graphical Method uses the relationships $\Delta V=-\int wdx$ and $\Delta M=\int Vdx$ to quickly sketch the diagrams. This is the preferred method for most engineers.
• A point load causes a "jump" in the shear diagram. A point moment causes a "jump" in the moment diagram.
• Maximum bending moment always occurs where the shear force is zero ($V=0$). This is the most critical point for design against bending failure.

Next Lesson Preview:
Constructing the bending moment diagram is not the end goal, but a means to an end. The entire purpose of this process is to find the maximum bending moment, $M_{max}$. In our next lesson, we will take the $M_{max}$ value from the BMD and plug it into the flexure formula to calculate the maximum bending stress in the beam, bringing us one step closer to a complete beam design.