1. Introduction

When a beam is subjected to external loads, it bends. Due to this bending:

• The upper fibers of the beam are compressed
• The lower fibers are stretched (tension)

This internal resistance developed is called bending stress.

2. Definition of Bending Stress

🔹 Definition:

Bending stress is the internal stress induced in a material due to bending moment acting on it.

It varies linearly across the cross-section:

• Maximum at outermost fibers
• Zero at the neutral axis

3. Bending Stress Formula (Flexure Formula)

$\frac{M}{I} = \frac{\sigma}{y} = \frac{E}{R}$

🔹 Where:

• $M$M = Bending moment
• $I$I = Moment of inertia of cross-section
• $\sigma$σ = Bending stress
• $y$y = Distance from neutral axis
• $E$E = Modulus of elasticity
• $R$R = Radius of curvature

🔹 Important Form:

$$\sigma = \frac{M \cdot y}{I}$$

4. Assumptions of Simple Bending Theory

• Material is homogeneous and isotropic
• Beam is initially straight
• Plane sections remain plane after bending
• Stress is within elastic limit (Hooke’s law valid)
• Young’s modulus is same in tension and compression

5. Neutral Axis and Neutral Surface

🔹 Neutral Axis (NA):

• Line where bending stress is zero
• Passes through centroid of cross-section

🔹 Neutral Surface:

• Surface formed by neutral axis along beam length
• No change in length during bending

6. Stress Distribution in Bending

🔹 Key Points:

• Stress varies linearly from NA
• Maximum stress at outermost fiber
• Zero stress at neutral axis

🔹 Types of Bending:

• Sagging → Top in compression, bottom in tension
• Hogging → Top in tension, bottom in compression

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🔷 7. Section Modulus (Z)

🔹 Definition:

Section modulus is a measure of the strength of a section in bending.

🔹 Formula:

$$Z = \frac{I}{y_{max}}$$Z=ymax​I​

🔹 Bending Stress in Terms of Z:

$$\sigma = \frac{M}{Z}$$σ=ZM​

🔹 Importance:

• Higher Z → stronger section
• Used in beam design

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🔷 8. Moment of Inertia (I)

• Represents resistance to bending
• Depends on shape and size of cross-section

🔹 Examples:

• Rectangular section: $$I = \frac{bd^3}{12}$$I=12bd3​
• Circular section: $$I = \frac{\pi d^4}{64}$$I=64πd4​

9. Bending Stress in Different Sections

🔹 Rectangular Section:

• Maximum stress at top and bottom
• NA at center

🔹 Circular Section:

• Uniform distribution radially

🔹 I-Section:

• High strength with less material
• Widely used in construction

10. Beams of Uniform Strength

🔹 Concept:

A beam is said to have uniform strength when stress is constant throughout its length.

🔹 Achieved by:

• Varying cross-section
• Example: tapered beams

11. Applications of Bending Stress

• Design of beams and girders
• Bridge construction
• Machine shafts
• Structural components

12. Important Points

✔ Maximum bending stress occurs at outermost fiber
✔ Bending stress is directly proportional to bending moment
✔ Inversely proportional to moment of inertia
✔ Neutral axis passes through centroid

13. Limitations of Bending Theory

❌ Not valid beyond elastic limit
❌ Not suitable for short beams (shear effects significant)
❌ Assumes ideal conditions

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