Theory of Bending Stresses

TL;DR

When a beam bends, some parts are stretched (tension) and others are compressed, creating internal stresses. These bending stresses vary across the beam's cross-section, being zero at the neutral axis and maximum at the outer fibers. The flexure formula helps you calculate these stresses based on the applied bending moment and the beam's geometry.

1. The Mental Model

Imagine you're bending a rubber ruler. The top surface gets shorter (compressed), the bottom surface gets longer (stretched), and somewhere in the middle, there's a line that doesn't change length. That line is the neutral axis, and the internal forces trying to resist this bending are your bending stresses.

2. The Core Material

When a beam is subjected to a bending moment, internal stresses develop within its cross-section. These stresses are known as bending stresses (or normal stresses due to bending).

Assumptions for Pure Bending

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The theory of bending stresses relies on several key assumptions for "pure bending" (bending without shear force):
* The material is homogeneous and isotropic (properties are the same throughout and in all directions).
* The material obeys Hooke's Law (stress is proportional to strain).
* The beam is initially straight and has a constant cross-section.
* The beam has a longitudinal plane of symmetry, and the bending moment acts in this plane.
* Cross-sections perpendicular to the beam's axis remain plane and perpendicular to the deformed axis after bending. This is known as Bernoulli's assumption or the plane sections remain plane assumption.

Neutral Axis

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Due to bending, one side of the beam is in tension (stretched) and the other side is in compression (compressed). Between these two regions, there's a longitudinal surface where the material neither stretches nor compresses, meaning the normal stress is zero. This surface is called the neutral surface, and its intersection with the beam's cross-section is the neutral axis (NA). For symmetric cross-sections and elastic materials, the neutral axis passes through the centroid of the cross-section.

Bending Stress Distribution

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The strain, and consequently the stress, varies linearly from the neutral axis. It's zero at the neutral axis and reaches its maximum value at the outermost fibers of the beam.

graph TD
    A["Applied Bending Moment (M)"] --> B["Beam Bends"]
    B --> C["Top Fibers Compress"]
    B --> D["Bottom Fibers Stretch"]
    C --> E["Compressive Stress"]
    D --> F["Tensile Stress"]
    B --> G["Neutral Axis (No Stress/Strain)"]
    G --> H["Stress Varies Linearly from NA"]
    H --> E
    H --> F

The Flexure Formula

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The relationship between bending moment ($M$), bending stress ($\sigma_b$), and the geometry of the beam's cross-section is given by the flexure formula:

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

Where:
* $\sigma_b$ is the bending stress at a distance $y$ from the neutral axis.
* $M$ is the bending moment acting on the cross-section.
* $y$ is the perpendicular distance from the neutral axis to the point where the stress is being calculated.
* $I$ is the moment of inertia (or second moment of area) of the cross-section about the neutral axis. This value quantifies the cross-section's resistance to bending. A larger $I$ means a greater resistance to bending for a given material and moment.

The maximum bending stress ($\sigma_{max}$) occurs at the points farthest from the neutral axis (i.e., at the outermost fibers). If $c$ is the maximum distance from the neutral axis to the extreme fiber:

$\sigma_{max} = \frac{Mc}{I}$

The ratio $I/c$ is called the section modulus ($Z$), so the formula can also be written as:

$\sigma_{max} = \frac{M}{Z}$

A larger section modulus indicates a greater resistance to bending stress.

3. Worked Example

A rectangular timber beam is 150 mm wide and 250 mm deep. It is subjected to a maximum bending moment of 15 kNm. Calculate the maximum bending stress in the beam.

1. Identify dimensions and moment:

   • Width, $b = 150 \text{ mm}$
   • Depth, $h = 250 \text{ mm}$
   • Bending moment, $M = 15 \text{ kNm} = 15 \times 10^6 \text{ Nmm}$

2. Calculate the moment of inertia ($I$) for a rectangular section:
   For a rectangle about its centroidal axis (which is the neutral axis for a symmetric beam), $I = \frac{bh^3}{12}$.
   $I = \frac{150 \text{ mm} \times (250 \text{ mm})^3}{12} = \frac{150 \times 15,625,000}{12} = 195,312,500 \text{ mm}^4$

3. Determine the maximum distance ($c$) from the neutral axis:
   For a rectangle, the neutral axis is at the mid-depth, so $c = h/2$.
   $c = 250 \text{ mm} / 2 = 125 \text{ mm}$

4. Calculate the maximum bending stress ($\sigma_{max}$) using the flexure formula:
   $\sigma_{max} = \frac{Mc}{I}$
   $\sigma_{max} = \frac{(15 \times 10^6 \text{ Nmm}) \times (125 \text{ mm})}{195,312,500 \text{ mm}^4}$
   $\sigma_{max} = \frac{1,875,000,000}{195,312,500} \text{ N/mm}^2$
   $\sigma_{max} = 9.6 \text{ N/mm}^2 = 9.6 \text{ MPa}$

The maximum bending stress in the beam is 9.6 MPa. This stress will be tensile at the bottom fibers and compressive at the top fibers (assuming positive bending moment).

4. Key Takeaways

• Bending stresses are normal stresses caused by an applied bending moment.
• The neutral axis is the line within a cross-section where bending stress is zero.
• Bending stress is maximum at the outermost fibers of the beam and zero at the neutral axis.
• The flexure formula, $\sigma_b = My/I$, is fundamental for calculating bending stresses.
• The moment of inertia ($I$) describes a cross-section's resistance to bending.

Common Mistakes to Avoid:

• Incorrectly locating the neutral axis: For elastic bending, the neutral axis passes through the centroid of the cross-section.
• Using incorrect units: Ensure consistency, typically N, mm, and MPa (N/mm$^2$).
• Forgetting to convert moment units: Kilonewton-meters (kNm) need to be converted to Newton-millimeters (Nmm) for consistency with mm for dimensions.
• Confusing $y$ with $c$: $y$ is the distance to any point, while $c$ is the distance to the extreme fiber for maximum stress.
• Miscalculating moment of inertia: Use the correct formula for the specific cross-section geometry.

5. Now Try It

A solid circular shaft with a diameter of 80 mm is subjected to a bending moment of 2 kNm. Calculate the maximum bending stress in the shaft.

What to do:
1. Draw the cross-section and label dimensions.
2. Calculate the moment of inertia ($I$) for a circular section ($I = \frac{\pi d^4}{64}$).
3. Determine the maximum distance ($c$) from the neutral axis.
4. Apply the flexure formula to find the maximum bending stress.

What success looks like:
You should arrive at a maximum bending stress value in MPa. Ensure your units are consistent throughout the calculation.