APPSC AEE Paper-2 Study Notes: Strength of Materials (Part A)

APPSC AEE PAPER-2: STRENGTH OF MATERIALS (PART - A)

This document contains quick revision notes tailored for the APPSC Assistant Executive Engineers (AEE) examination, covering the Strength of Materials syllabus.

1. Simple Stresses and Strains

Basic Concepts

• Stress ($\sigma$): Internal resistance offered by a body against deformation. $\sigma = \frac{P}{A}$
• Strain ($\epsilon$): Ratio of change in dimension to original dimension. $\epsilon = \frac{\delta L}{L}$
• Hooke's Law: Up to the proportional limit, stress is directly proportional to strain. $\sigma \propto \epsilon \implies \sigma = E \cdot \epsilon$

Stress-Strain Curve for Mild Steel (Ductile Material)

Key points on the curve: 1. Proportional Limit: Hooke's law holds true. 2. Elastic Limit: Maximum stress without permanent deformation. 3. Upper Yield Point: Yielding starts; sudden drop in stress. 4. Lower Yield Point: Actual yield stress used for design. 5. Ultimate Tensile Strength (UTS): Maximum stress the material can withstand. 6. Breaking/Rupture Point: Material fractures.

Important Properties & Ratios

• Factor of Safety (FOS):
  • For ductile materials: $FOS = \frac{\text{Yield Stress}}{\text{Working Stress}}$
  • For brittle materials: $FOS = \frac{\text{Ultimate Stress}}{\text{Working Stress}}$
• Poisson's Ratio ($\mu$ or $1/m$): Ratio of lateral strain to longitudinal strain.
  • $\mu = -\frac{\epsilon_{lateral}}{\epsilon_{longitudinal}}$
  • Range: $0 \le \mu \le 0.5$ (Cork = 0, Steel ≈ 0.3, Rubber ≈ 0.5)

Shear Stress & Elastic Constants

• State of Simple Shear: Equal and opposite shear stresses acting on parallel planes.
• Complementary Shear: A shear stress on a given plane is always accompanied by an equal shear stress of opposite sign on a mutually perpendicular plane.
• Elastic Constants:
  • Young's Modulus ($E$): $\frac{\text{Normal Stress}}{\text{Normal Strain}}$
  • Modulus of Rigidity / Shear Modulus ($G$ or $C$): $\frac{\text{Shear Stress}}{\text{Shear Strain}}$
  • Bulk Modulus ($K$): $\frac{\text{Normal Stress}}{\text{Volumetric Strain}}$
• Relations between Elastic Constants:
  1. $E = 2G(1 + \mu)$
  2. $E = 3K(1 - 2\mu)$
  3. $E = \frac{9KG}{3K + G}$
  4. $\mu = \frac{3K - 2G}{6K + 2G}$

Thermal Stresses

If a bar is subjected to a temperature change $\Delta T$ and its expansion/contraction is restricted, thermal stress develops. * Free Expansion (No stress): $\delta L = L \cdot \alpha \cdot \Delta T$ * Thermal Stress (Fully restricted): $\sigma_{th} = E \cdot \alpha \cdot \Delta T$ (Where $\alpha$ is the coefficient of thermal expansion).

2. Compound Stresses

When a body is subjected to multiple stresses, they can be resolved into principal stresses. * Principal Planes: Planes on which the shear stress is zero. * Principal Stresses: Normal stresses acting on the principal planes ($\sigma_1, \sigma_2$).

Equations for 2D Stress State ($\sigma_x, \sigma_y, \tau_{xy}$)

• Major Principal Stress: $\sigma_1 = \frac{\sigma_x + \sigma_y}{2} + \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$
• Minor Principal Stress: $\sigma_2 = \frac{\sigma_x + \sigma_y}{2} - \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$
• Maximum Shear Stress: $\tau_{max} = \frac{\sigma_1 - \sigma_2}{2} = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$

Mohr's Circle

A graphical method to determine stresses on any inclined plane. * Center of circle: $(C, 0) = \left( \frac{\sigma_x + \sigma_y}{2}, 0 \right)$ * Radius of circle: $R = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2} = \tau_{max}$

3. Shear Force and Bending Moment

• Shear Force (V): Algebraic sum of transverse forces acting on either side of a section.
• Bending Moment (M): Algebraic sum of moments of forces acting on either side of a section.

Differential Relations

• Load intensity ($w$) = rate of change of Shear Force: $\frac{dV}{dx} = -w$
• Shear Force ($V$) = rate of change of Bending Moment: $\frac{dM}{dx} = V$ (Max Bending Moment occurs where Shear Force is zero or changes sign).

Standard Cases (Max Values)

Beam Type	Loading	Max Shear Force ($V_{max}$)	Max Bending Moment ($M_{max}$)	Location of $M_{max}$
Cantilever (Length $L$)	Point load $W$ at free end	$W$	$W \cdot L$	Fixed End
Cantilever	UDL $w$ over entire length	$w \cdot L$	$\frac{w \cdot L^2}{2}$	Fixed End
Simply Supported	Point load $W$ at mid-span	$\frac{W}{2}$	$\frac{W \cdot L}{4}$	Mid-span
Simply Supported	UDL $w$ over entire length	$\frac{w \cdot L}{2}$	$\frac{w \cdot L^2}{8}$	Mid-span

4. Bending and Shear Stress in Beams

Theory of Simple Bending

Flexure Equation: $$ \frac{M}{I} = \frac{\sigma}{y} = \frac{E}{R} $$ * $M$ = Bending Moment * $I$ = Moment of Inertia * $\sigma$ = Bending Stress * $y$ = Distance from Neutral Axis (NA) * $E$ = Young's Modulus * $R$ = Radius of curvature

Note: Bending stress is maximum at the extreme fibers ($y = y_{max}$) and zero at the Neutral Axis.

Shear Stress Distribution

Shear stress equation: $\tau = \frac{V \cdot A \cdot \bar{y}}{I \cdot b}$

Standard Cross Sections: * Rectangular Section: Parabolic distribution. $\tau_{max} = 1.5 \cdot \tau_{avg}$ (at NA) * Circular Section: Parabolic distribution. $\tau_{max} = \frac{4}{3} \cdot \tau_{avg}$ (at NA) * I-Section: Max shear occurs at the NA of the web. The web carries almost all the shear force. * T-Section: Max shear is at the NA (which lies closer to the flange).

Torsion of Circular Shafts

Torsion Equation: $$ \frac{T}{J} = \frac{\tau}{r} = \frac{G \cdot \theta}{L} $$ * $T$ = Torque * $J$ = Polar Moment of Inertia ($\frac{\pi d^4}{32}$ for solid shaft) * $\tau$ = Shear stress (max at surface) * $r$ = Radius * $G$ = Shear Modulus * $\theta$ = Angle of twist * Power Transmitted: $P = \frac{2 \pi N T}{60}$ (where $N$ is RPM).

5. Columns and Struts

A member primarily subjected to axial compressive loads. * Strut: Compressive member in any direction (e.g., in a truss). * Column: A vertical strut fixed at both ends.

Euler's Theory (Long Columns)

Euler's critical buckling load: $$ P_e = \frac{\pi^2 E I}{L_e^2} $$ (Valid only for long columns where buckling stress is less than yield stress).

Effective Length ($L_e$): * Both ends hinged: $L_e = L$ * Both ends fixed: $L_e = L / 2$ * One fixed, one free: $L_e = 2L$ * One fixed, one hinged: $L_e = L / \sqrt{2}$

Rankine's Theory (Short & Long Columns)

$$ \frac{1}{P_R} = \frac{1}{P_c} + \frac{1}{P_e} $$ $$ P_R = \frac{\sigma_c \cdot A}{1 + a \cdot \lambda^2} $$ * $P_c$ = Crushing load ($\sigma_c \cdot A$) * $\lambda$ = Slenderness ratio ($L_e / r_{min}$) * $a$ = Rankine's constant

Eccentrically Loaded Columns

• Secant Formula: Used for calculating max compressive stress for eccentric loading. $\sigma_{max} = \frac{P}{A} \left[1 + \frac{e \cdot y_c}{r^2} \sec\left(\frac{L}{2r}\sqrt{\frac{P}{AE}}\right)\right]$
• Perry-Robertson Formula: Introduces an initial curvature eccentricity factor.

6. Deflections and Slopes

Methods to find deflection: Double Integration, Macaulay's, Moment Area, Conjugate Beam.

Standard Deflection and Slope Formulas

Where $E$ = Young's modulus, $I$ = Moment of Inertia, $L$ = Length.

Beam & Loading	Max Slope ($\theta$)	Max Deflection ($\delta$)	Location of Max Deflection
Cantilever (Point load $W$ at free end)	$\frac{WL^2}{2EI}$	$\frac{WL^3}{3EI}$	Free End
Cantilever (UDL $w$ over whole length)	$\frac{wL^3}{6EI}$	$\frac{wL^4}{8EI}$	Free End
Simply Supported (Point load $W$ at mid-span)	$\frac{WL^2}{16EI}$ (at ends)	$\frac{WL^3}{48EI}$	Mid-span
Simply Supported (UDL $w$ over whole length)	$\frac{wL^3}{24EI}$ (at ends)	$\frac{5wL^4}{384EI}$	Mid-span

(Fixed beams and propped cantilevers involve static indeterminacy and are solved using compatibility conditions).