Shear Force and Bending Moment free study note

1. Introduction to Shear Force and Bending Moment

When a beam is subjected to external loads, it experiences internal forces and moments that resist deformation. Two of the most important concepts in the analysis and design of beams are Shear Force and Bending Moment. Understanding these concepts is essential for designing safe and efficient structural and mechanical components such as bridges, machine frames, shafts, and building beams.

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

2. What is Shear Force (SF) ?

Shear Force is the internal force that acts parallel to the cross-section of a beam and tends to make one part of the beam slide relative to the adjacent part. 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. What is Bending Moment (BM) ?

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

Mathematical Expression:

$$M = \sum (F \times d)$$

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)

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)

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

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. Differences Between Shear Force and Bending Moment

Shear Force	Bending Moment
Internal force acting parallel to the cross-section	Internal moment causing bending
Measured in N or kN	Measured in N·m or kN·m
Causes sliding between adjacent sections	Causes curvature in the beam
Changes suddenly at point loads	Changes gradually along the beam
Represented by Shear Force Diagram (SFD)	Represented by Bending Moment Diagram (BMD)

---

External Reference