Flexure Formula: Normal Stress in Beams

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

In our last lesson, we mastered the skill of constructing shear force and bending moment diagrams. The main purpose of that exercise was to find the maximum bending moment, $M_{max}$, acting on a beam. You learned that this critical value typically occurs where the shear force is zero.

Today, we will answer the question: "So what do we do with this maximum moment?" The goal of this lesson is to use the bending moment to calculate the resulting stresses inside the beam. You will learn to apply the flexure formula to calculate normal stress in beams due to bending.

This formula is a cornerstone of structural engineering. For your interest in aerospace, it's fundamental for analyzing the strength of aircraft components like wing spars and fuselage stringers, ensuring they can withstand bending loads during flight without failing.

1. From Bending Moment to Bending Stress

When a beam bends under a load, its material deforms. Imagine a simple beam bending downwards. The top surface gets squeezed together (compression), and the bottom surface gets stretched apart (tension).

Somewhere in the middle of the beam's cross-section, there is a surface where the material is neither compressed nor stretched. This is called the neutral surface, and its intersection with the cross-section is the neutral axis. The neutral axis of a beam's cross-section always passes through its centroid.

To understand the origin of these stresses and their relationship to the beam's deformation, the following video provides an excellent conceptual overview.

2. The Flexure Formula

As explained in the video, the relationship between the internal bending moment ($M$) and the normal stress ($\sigma$) it produces is defined by the flexure formula.

The flexure formula is:

$$\sigma =-\frac{My}{I}$$

Or, more commonly when considering just the magnitude:

$$\sigma =\frac{My}{I}$$

Where:

• $\sigma$ (sigma) is the normal stress at the point of interest.
• $M$ is the internal bending moment at the beam's cross-section. You find this from the bending moment diagram.
• $y$ is the perpendicular distance from the neutral axis to the point where you are calculating the stress.
• $I$ is the Area Moment of Inertia of the cross-section about the neutral axis. This property represents the beam's resistance to bending due to its shape.

The negative sign in the first version of the formula is a convention that depends on the coordinate system. For our purposes, it's more intuitive to determine the type of stress (tension or compression) by visualizing how the beam bends.

• A positive moment ("sagging") causes compression on the top ($y>0$) and tension on the bottom ($y<0$).
• A negative moment ("hogging") causes tension on the top and compression on the bottom.

Maximum Bending Stress

Stress is highest at the points farthest from the neutral axis. We call this maximum distance $c$. Therefore, the maximum bending stress in a section is:

$${\sigma }_{max}=\frac{Mc}{I}$$

Here, $c$ would be the distance from the neutral axis to the top or bottom edge of the beam. If the cross-section is not symmetric about its neutral axis (like a T-beam), the maximum tensile stress and maximum compressive stress will have different magnitudes because the value of $c$ will be different for the top and bottom fibers.

3. A Step-by-Step Calculation

To apply the flexure formula, you need to follow a clear procedure. The following video provides an excellent, complete worked example that brings together concepts from previous lessons (centroids, moment of inertia) to solve for bending stress.

The problem in the video is to find the maximum tensile and compressive stresses in a T-shaped beam subjected to a given bending moment. This requires three main steps:

1. Find the Neutral Axis: Calculate the centroid of the cross-section.
2. Calculate the Moment of Inertia: Determine $I$ for the composite shape about the neutral axis using the parallel-axis theorem.
3. Apply the Flexure Formula: Use the given $M$ and the calculated $I$ and $c$ values to find the stresses.

Let's break it down.

Step 1 & 2: Finding the Neutral Axis and Moment of Inertia

These are concepts from the first lesson of this module. The next video segment provides a great refresher on how to perform these calculations for a composite shape.

As a reminder, the parallel axis theorem is:
$I=I_c+A{d}^2$
Where:

• $I$ is the moment of inertia of an area about some axis.
• $I_c$ is the moment of inertia of that area about its own centroidal axis.
• $A$ is the area.
• $d$ is the distance between the two parallel axes.

Step 3: Calculating the Stresses

Now that we have all the components—$M$, $I$, and the distances to the extreme fibers—we can calculate the maximum stresses.

Test your understanding!

In the video example, the neutral axis was found to be at 7.85 inches from the bottom of an 11-inch tall beam. The maximum compressive stress occurred at the top, and the maximum tensile stress occurred at the bottom. Why wasn't the maximum stress the same value for both tension and compression?

Show answer

The cross-section was not symmetric about the neutral axis. The distance from the neutral axis to the top fiber was $c_{top}=11-7.85=3.15$ inches, while the distance to the bottom fiber was $c_{bottom}=7.85$ inches. Since stress is directly proportional to this distance ($\sigma =My/I$), the larger distance to the bottom fiber resulted in a higher tensile stress compared to the compressive stress at the top.

4. Summary of the Process

Calculating the maximum bending stress in a beam is a multi-step process that synthesizes much of what we've learned so far.

The six steps summarized are:

1. Solve for reactions and draw the Shear and Moment diagrams.
2. Find the maximum moment, $M_{max}$, from the moment diagram.
3. Calculate the location of the cross-section's centroid (the neutral axis).
4. Calculate the moment of inertia, $I$, about the centroid.
5. Determine the distance from the centroid to the extreme fiber, $c$.
6. Calculate the maximum stress using ${\sigma }_{max}=\frac{M_{max}c}{I}$.

Conclusion

In this lesson, you've connected the concept of internal bending moment to the actual stress experienced by the material. This is the crucial link that allows engineers to design beams that are strong enough to carry their intended loads.

Key Takeaways:

• The flexure formula, $\sigma =\frac{My}{I}$, is used to calculate the normal stress caused by bending.
• Bending stress is zero at the neutral axis (centroid) and increases linearly to a maximum at the fibers farthest from the neutral axis (the "extreme fibers").
• To find the maximum bending stress, you must first determine the maximum bending moment ($M_{max}$), the centroid location, and the area moment of inertia ($I$).
• For non-symmetrical cross-sections, the maximum tensile and compressive stresses will be different.

Next Lesson Preview:
We now know how to calculate the maximum stress in a given beam. But in a real-world design scenario, you often start with the loads and need to choose a beam that is strong enough. In our next lesson, we will focus on determining maximum bending stress and selecting appropriate beam sections for strength. We will introduce the concept of section modulus and learn how to use material properties (like yield strength) to pick a safe and efficient beam from a catalog of standard shapes.