Beams

• long ($L >> B,D$)
• axis of the beam is straight
• constant cross-section throughout its length

Classified by supporting conditions

First 3 are the mandatory ones in s1.

u.d.l means uniformly distributed load.

Type	Image	Notes
Simply supported beam		At ends: Max SF and BM = 0
Cantilevered beam		At fixed end: Max SF and BM
Overhanging beam
Propped cantilevered beam
Continuous beam
Fixed beam		At ends: Max SF and BM

At a section

• $P$ - Normal force / Axial force
• $S_y, S_y$ - Shear forces along $y$ and $z$ axis
• $M_x$ - Twisting moment / Torque
• $M_y, M_z$ - Bending moments about $y$ and $z$ axis

Degress of freedom

A plane member have 3 degrees of freedom. Any of the 3 can be restrained.

• Displacement in $x$-direction
• Displacement in $y$-direction
• Rotation about $z$-direction

SFD & BMD

Sign convention

• Bending moment
  • Hogging (curves upwards in the middle) is (+) ve
  • Sagging (curves downwards in the middle) is (-) ve
• Shear force
  • Clockwise shear is (+) ve.
  • Counterclockwise shear is (-) ve.

Pure bending

A member is in pure bending when shear force is $0$ and bending moment is a constant. $$

Point of Contraflexure

The point about which bending moment is $0$, and changes its sign through the point. $$

Distributed load

Suppose a beam is under a distributed load of $w=f(x)$ per unit length. $$

$$\frac{\text{d}S}{\text{d}x}=-w$$ $$\frac{\text{d}M}{\text{d}x}=-S \;\; \land \;\; \frac{\text{d}^2M}{\text{d}x^2}=w$$

Deflection of a beam

Suppose a simply supported beam is applied a load of $W$ at mid-span. $$

$$S_{\text{max}} = \frac{WL}{4I} \;\; \land \;\; D_{\text{max}} = \frac{WL^3}{48EI}$$

Here:

• $S_\text{max}$ - Maximum stress
• $D_\text{max}$ - Deflection
• $W$ - Load
• $L$ - Span length
• $E$ - Young’s modulus
• $I$ - Second moment of cross-sectional area

Principle of Superposition

A beam with multiple loads can be split into multiple systems each with a single load. Reason for doing so is the ease of calculations.

Values will be the sum of each system’s corresponding value.