1. Introduction

When a beam is subjected to loads (point loads, distributed loads, etc.), internal forces develop at every cross-section:

• Shear Force (V) → tends to slide one part of the beam over the other
• Bending Moment (M) → tends to bend the beam

Understanding SF and BM helps in designing safe and strong structures.

2. Shear Force (SF)

Definition:

Shear force at a section is the algebraic sum of all vertical forces acting on either the left or right side of that section.

Mathematical Expression:

$$V = \sum F_y$$

Sign Convention:

• Upward force on left section → Positive SF
• Downward force on left section → Negative SF

Units:

• Newton (N) or kiloNewton (kN)

3. Bending Moment (BM)

🔹 Definition:

Bending moment at a section is the algebraic sum of moments of all forces about that section.

🔹 Mathematical Expression:

$$M = \sum (F \times d)$$M=∑(F×d)

🔹 Sign Convention:

• Sagging (concave upward) → Positive BM
• Hogging (concave downward) → Negative BM

🔹 Units:

• Newton-meter (N·m) or kN·m

4. Relationship Between Load, Shear Force, and Bending Moment

This is a very important concept:

$\frac{dV}{dx} = -w(x)$

$\frac{dM}{dx} = V(x)$

Interpretation:

• Rate of change of shear force = negative load intensity
• Rate of change of bending moment = shear force

Key Points:

• Maximum BM occurs where SF = 0
• Area under load curve → change in SF
• Area under SF curve → change in BM

5. Shear Force Diagram (SFD)

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🔹 Definition:

A graphical representation of variation of shear force along the length of the beam.

🔹 Characteristics:

• Sudden jump → point load
• Linear variation → uniformly distributed load (UDL)
• Parabolic → uniformly varying load (UVL)

6. Bending Moment Diagram (BMD)

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🔹 Definition:

Graph showing variation of bending moment along beam length.

🔹 Characteristics:

• Straight line → no load
• Parabolic curve → UDL
• Cubic curve → UVL

7. Types of Loads on Beams

• Point Load (Concentrated Load)
• Uniformly Distributed Load (UDL)
• Uniformly Varying Load (UVL)
• Moment Load (Couple)

8. Important Cases of Beams

🔹 (a) Simply Supported Beam

• Supports at both ends
• BM is zero at supports

🔹 (b) Cantilever Beam

• Fixed at one end, free at other
• Maximum BM at fixed end

🔹 (c) Overhanging Beam

• One or both ends extend beyond supports

9. Steps to Draw SFD and BMD

1. Calculate support reactions
2. Divide beam into sections
3. Apply equilibrium equations:
   • $\sum F_y = 0$∑Fy​=0
   • $\sum M = 0$∑M=0
4. Compute SF at key points
5. Plot SFD
6. Calculate BM using moments
7. Plot BMD

10. Points of Contraflexure

🔹 Definition:

Point where bending moment changes sign (from positive to negative or vice versa).

• Also called inflection point
• BM = 0 at this point

11. Applications

• Design of beams and girders
• Bridge construction
• Machine components (shafts, frames)
• Structural engineering analysis

12. Key Differences Between SF and BM

Aspect	Shear Force	Bending Moment
Definition	Sum of vertical forces	Sum of moments
Effect	Causes sliding	Causes bending
Unit	N or kN	N·m or kN·m
Diagram	SFD	BMD

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