📋 Table of Contents
- 3.1 Types of Loading
- 3.2 Types of Supports (2D and 3D)
- 3.3 Types of Beams + Stability
- 3.6–3.7 Shear Force and Bending Moment + Sign Conventions
- 3.8 Important Points about SFD and BMD (8 rules)
- 3.9–3.10 Standard SFD-BMD Cases
- 3.10.1 SFD and BMD by Integration Method
- 3.10.2 Effect of Concentrated Moment on BMD
3.1 Types of Loading
Figure 3.1 — Five Types of Loading on Beams (from source)
Fig. 3.1 — SFD is one degree higher than loading; BMD is one degree higher than SFD. Point load → step in SFD, triangle in BMD. UDL → linear SFD, parabolic BMD. UVL → parabolic SFD, cubic BMD.
3.2 Types of Supports
3.2.1 Two-Dimensional (2D) Supports
| Support Type | Reactions | Description |
|---|---|---|
| (a) Fixed Support | 3 reactions: Rₓ, Rᵧ, M_z |
Prevents translation in x, y directions and rotation. Also called built-in support. |
| (b) Hinge Support | 2 reactions: Rₓ, Rᵧ |
Prevents translation but allows rotation. Rₓ = horizontal, Rᵧ = vertical. |
| (c) Roller Support | 1 reaction: R (perpendicular) |
Only one reaction perpendicular to the supporting plane. |
| (d) Double Roller | 2 reactions: Rᵧ, M_z |
Constrained against translation but moment reaction exists. |
3.2.2 Three-Dimensional (3D) Supports
(a) 3D Fixed Support
6 reactions: Rₓ, Rᵧ, R_z, Mₓ, Mᵧ, M_z
Also called built-in support in 3D
Also called built-in support in 3D
(b) 3D Hinged Support
3 reactions: Rₓ, Rᵧ, R_z
Allows rotations about all three axes
Allows rotations about all three axes
3.3 Types of Beams + 3.4 Stability
Figure 3.2 — Six Types of Beams with Support Conditions
3.6–3.7 Shear Force, Bending Moment and Sign Conventions
Shear Force Sign Convention
Rule: Shear force having upward direction to the left hand side of the section = positive (clockwise). Downward direction to left = negative (anticlockwise).
Bending Moment Sign Convention
Sagging (concavity upward, like a smile) = +ve. Hogging (concavity downward, like a frown) = −ve.
NOTE from source: Bending moment at a section must NOT be confused with moment at a point. BM = summation of moments due to transverse forces either to left or right of section. This sum should equal zero for equilibrium.
3.8 Important Points about SFD and BMD
Key Relation 1
dS/dx = −w
−ve slope of SFD = downward loading rate
Key Relation 2
dM/dx = S
Slope of BMD = shear force at that section
Key Relation 3
d²M/dx² = −w
Second derivative of BMD = −loading rate
All 8 Important Points from Source (§3.8):
1
SFD is one degree higher than loading diagram and BMD is one degree higher than SFD
2
At any point where a concentrated point load or reaction acts → ordinate of SFD changes by the magnitude of that load (step change)
3
At a point where a concentrated moment (couple) acts → ordinate of BMD changes by the magnitude of that couple (step jump)
4
If shear force changes sign at a section → BM is either maximum or minimum at that section (inverse is NOT always true)
5
If BM changes sign at a section → curvature also changes → called Point of Contraflexure (same point is also called point of inflection on elastic curve)
6
Distance between two adjacent points of contraflexure = Focal Length
7
The portion of beam where shear force is constant = Shear Span
8
Relation between shear force and loading rate: dS/dx = −w; means −ve slope of SFD represents downward loading rate
3.9–3.10 Standard SFD and BMD Cases
Figure 3.3 — SFD and BMD for Simply Supported Beam with Central Load P
Standard Maximum BM Values for Simply Supported Beam (span L)
| Loading Condition | M_max | Location | SFD Shape | BMD Shape |
|---|---|---|---|---|
| Central point load P | PL/4 | Centre (x = L/2) | Two rectangles (step at C) | Triangle (linear) |
| UDL (w per unit length) | wL²/8 | Centre (x = L/2) | Linear (1°) | Parabolic (2°) |
| UVL: 0 at A → w at B | wL²/(9√3) | x = L/√3 from A | Parabolic (2°) | Cubic (3°) |
| UVL: 0 at both ends → w at centre | wL²/12 | Centre (x = L/2) | Cubic (3°) | 4th degree curve |
| Eccentric load P at ‘a’ from A | Pa(L−a)/L | At load point | Two rectangles | Triangle (linear) |
3.10.1 SFD and BMD by Integration Method
SFD and BMD can be plotted by integration method. The following relations are used (from source):
Basic Relations
dS/dx = −w …(i)
dM_x/dx = S_x …(ii)
d²M_x/dx² = −w …(iii)
dM_x/dx = S_x …(ii)
d²M_x/dx² = −w …(iii)
Integration Procedure (from source)
- Find rate of loading at any section as w
- Use equation d²M_x/dx² = −w
- Integrate once → dM_x/dx = S_x + C₁
- Integrate again → M_x + C₂
- Apply boundary conditions to find C₁ and C₂
Note: Equation (iii) valid only when loading is continuous
3.10.2 Effect of Concentrated Moment on SFD and BMD
- A concentrated moment (couple) at any point does NOT affect the SFD at that point
- However, it causes a sudden jump (step) in the BMD at that point equal to the magnitude of the couple
- Jump is upward if the couple is clockwise; downward if anticlockwise (by standard sign convention)
This is a very common GATE question!
Couple → NO change in SFD | Step jump in BMD = magnitude of couple.
Couple → NO change in SFD | Step jump in BMD = magnitude of couple.
Chapter 3: Shear Force and Bending Moment — Strength of Materials · MADE EASY Postal Study Package 2019