Strength of Material (ME-304) - Unit 3 Short Notes

UNIT 3: Comprehensive Study Notes (Based on RGPV Past Papers)

1. Fundamental Properties & Stress-Strain Relationships

Elastic Moduli:

• Young's Modulus (E): Ratio of normal stress to normal strain within elastic limit. $$\displaystyle E = \frac{\sigma}{\epsilon} $$.

• Shear Modulus (G) / Modulus of Rigidity: Ratio of shear stress to shear strain. $$\displaystyle G = \frac{\tau}{\gamma} $$.

• Bulk Modulus (K): Ratio of hydrostatic pressure to volumetric strain. $$\displaystyle K = -\frac{p}{\epsilon_v} $$.

• Relationship: $$\displaystyle G = \frac{E}{2(1+\nu)} $$ and $$\displaystyle K = \frac{E}{3(1-2\nu)} $$, where $\nu$ is Poisson's ratio.

Poisson's Ratio ($\nu$):

Definition: Ratio of lateral strain to longitudinal strain in the elastic region. For most metals, $$\displaystyle 0.25 < \nu < 0.35 $$.

Stress-Strain Curve Key Points:

• Proportional Limit: Stress is proportional to strain (Hooke's law valid).

• Elastic Limit: Maximum stress without permanent deformation.

• Yield Point: Stress at which material begins to deform plastically.

Strain Energy:

• Strain Energy (U): Work done by external load in straining a material.

• Strain Energy Density (u): Strain energy per unit volume.

• Gradual Loading: $$\displaystyle U = \frac{P^2 L}{2EA} $$ or $$\displaystyle U = \frac{1}{2} P \delta $$.

• Impact/Shock Loading: $$\displaystyle U = P(\delta_{st} + \delta_i) $$, where $$\displaystyle \delta_i = \delta_{st}\left(1 + \sqrt{1 + \frac{2h}{\delta_{st}}}\right) $$ for a falling weight.

Hooke's Law in 3D: For isotropic materials, generalized Hooke's law relates all six stress and strain components.

2. Axial Loading & Deformation

Uniform Bar:

$$\boxed{\delta = \frac{PL}{EA}}$$

Composite Bars (Different Materials in Series):

• Compatibility: Deformation of each segment is equal ($$\displaystyle \delta_1 = \delta_2 = ... $$).

• Equilibrium: Total load $$\displaystyle P = \sum P_i $$.

• Stress Calculation: Use $$\displaystyle \sigma_i = E_i \epsilon $$ and $$\displaystyle \sum \frac{P_i}{A_i E_i} = \frac{P}{E_{eq} A_{eq}} $$ approach.

Thermal Stresses in Composite Systems:

• Free Expansion: $$\displaystyle \delta_{th} = \alpha L \Delta T $$.

• Restrained Expansion: Stress develops. For a composite bar fixed at both ends:

$$\sum P_i = 0 \quad \text{and} \quad \frac{P_1 L}{A_1 E_1} + \alpha_1 \Delta T L = \frac{P_2 L}{A_2 E_2} + \alpha_2 \Delta T L$$

Solve for $$\displaystyle P_1, P_2 $$, then $$\displaystyle \sigma_1 = P_1/A_1 $$, $$\displaystyle \sigma_2 = P_2/A_2 $$.

Axially Varying Cross-Section (Tapered Bar - Circular):

For a bar tapering from diameter $D$ at $$\displaystyle x=0 $$ to $d$ at $$\displaystyle x=L $$:

$$d(x) = D - \left(\frac{D-d}{L}\right)x, \quad A(x) = \frac{\pi}{4} [d(x)]^2$$

$$\delta = \int_0^L \frac{P}{E A(x)} dx = \frac{4P}{\pi E (D-d)} \ln\left(\frac{D}{d}\right)$$

Statically Indeterminate Axial Problems:

• Use compatibility (deformation) conditions along with equilibrium.

• Example: Rigid plate with multiple bars → all bars have same deformation.

[!TIP] Common Pitfall: In composite bars, stresses are NOT equal; deformations are equal. Always write compatibility first.

3. Bending Stresses & Beam Deflection

Shear Force (V) & Bending Moment (M) Diagrams:

• Sign Convention: Sagging (+ve) BM, upward left of section (+ve) SF.

• Relationship: $$\displaystyle \frac{dM}{dx} = V $$, $$\displaystyle \frac{dV}{dx} = -w $$.

• Point of Contraflexure: Point where BM changes sign ($$\displaystyle M=0 $$).

Theory of Simple Bending:

• Assumptions:

  1. Material is homogeneous, isotropic.

  2. Beam is initially straight.

  3. Plane sections remain plane and perpendicular to neutral axis.

  4. Stress is proportional to strain (Hooke's law).

  5. Modulus of elasticity same in tension/compression.

• Flexure Formula:

$$\boxed{\sigma = \frac{My}{I}} \quad \text{or} \quad \boxed{\sigma_{max} = \frac{M}{Z}} \quad \text{where} \quad Z = \frac{I}{y_{max}}$$

Deflection of Beams - Methods:

Standard Deflection Formulas (EI constant):

• Cantilever (Point load W at free end): $$\displaystyle y_{max} = \frac{WL^3}{3EI} $$ at free end, $$\displaystyle \theta_{free} = \frac{WL^2}{2EI} $$.

• Cantilever (UDL w over entire span): $$\displaystyle y_{max} = \frac{wL^4}{8EI} $$, $$\displaystyle \theta_{free} = \frac{wL^3}{6EI} $$.

• Simply Supported (Central point load W): $$\displaystyle y_{max} = \frac{WL^3}{48EI} $$ at center.

• Simply Supported (UDL w over entire span): $$\displaystyle y_{max} = \frac{5wL^4}{384EI} $$ at center.

[!TIP] Scaling Laws: For similar beams (same shape, different size L, h, b), deflection scales as $$\displaystyle y \propto \frac{L^3}{h} $$ if load is force per unit length, or $$\displaystyle y \propto \frac{L^4}{h^3} $$ if load is total force.

4. Shear Stresses in Beams

Shear Formula:

$$\boxed{\tau = \frac{VQ}{It}}$$

• $V$ = Shear force at section.

• $Q$ = First moment of area above/below the point about NA: $$\displaystyle Q = \bar{y} A' $$.

• $I$ = Second moment of area about NA.

• $t$ = Thickness of material at the point.

Distribution:

• Rectangular Section: Parabolic. $$\displaystyle \tau_{max} = \frac{3}{2} \tau_{avg} $$ at NA ($$\displaystyle y=0 $$). $$\displaystyle \tau=0 $$ at top/bottom.

• Circular Section: $$\displaystyle \tau = \frac{4V}{3A} \sqrt{1 - \left(\frac{2y}{d}\right)^2} $$. Max at NA.

• I-Section: Shear carried almost entirely by web. $$\displaystyle \tau_{web} \approx \frac{V}{A_{web}} $$. $$\displaystyle \tau_{flange} $$ is small.

Maximum Shear Stress: Occurs at the neutral axis for symmetric sections (rectangular, circular, I-section).

[!TIP] Key Insight: For thin-walled sections (I-beam), assume uniform shear stress across the web thickness.

5. Torsion

Circular Shafts (Solid & Hollow):

• Shear Stress:

  • Solid: $$\displaystyle \tau = \frac{Tr}{J} $$, max at outer surface ($$\displaystyle r = R $$).

  • Hollow: $$\displaystyle \tau_{max} = \frac{T r_o}{J} $$, $$\displaystyle \tau_{min} = \frac{T r_i}{J} $$.

$$\boxed{J = \frac{\pi}{2} (r_o^4 - r_i^4)}$$

• Angle of Twist:

$$\boxed{\theta = \frac{TL}{GJ}} \quad (\text{in radians})$$

Power Transmission:

$$\boxed{P = \frac{2\pi N T}{60}} \quad \text{(Watts)}$$

where $N$ = rpm, $T$ = torque (N·m).

Solid vs. Hollow Shaft Comparison:

• Same Maximum Stress: Hollow shaft transmits more torque for same weight/material.

$$\frac{T_h}{T_s} = \frac{1 - \left(\frac{r_i}{r_o}\right)^4}{1 - \left(\frac{r_i}{r_o}\right)^2} \left(\frac{r_o}{r_o}\right)^2 \approx \left(1 - \left(\frac{r_i}{r_o}\right)^2\right)^{-1}$$

• Same Weight: $$\displaystyle T_h / T_s = (r_o/r_i)^2 $$ for thin-walled approximation.

Stepped Shafts:

• Compatibility: Total angle of twist $$\displaystyle \theta_{total} = \sum \frac{T_i L_i}{G_i J_i} $$.

• Given rotations at points (e.g., $$\displaystyle \phi_A, \phi_C, \phi_D $$), set up equations based on relative twists between sections to solve for internal torques $$\displaystyle T_i $$.

Combined Bending & Torsion:

• Principal Stresses:

$$\sigma_1, \sigma_2 = \frac{\sigma_b}{2} \pm \sqrt{\left(\frac{\sigma_b}{2}\right)^2 + \tau_t^2}$$

where $$\displaystyle \sigma_b $$ = bending stress (tensile +ve), $$\displaystyle \tau_t $$ = torsional shear.

• Equivalent Tensile Stress (for design):

  • Max Shear Theory: $$\displaystyle \sigma_{eq} = \sqrt{\sigma_b^2 + 4\tau_t^2} $$.

  • Von Mises: $$\displaystyle \sigma_{eq} = \sqrt{\sigma_b^2 + 3\tau_t^2} $$.

  Design for $$\displaystyle \sigma_{eq} \leq \frac{\sigma_y}{n} $$.

6. Stress Transformation & Mohr’s Circle

2D Stress Transformation (Plane at $\theta$):

$$\boxed{\sigma_n = \frac{\sigma_x + \sigma_y}{2} + \frac{\sigma_x - \sigma_y}{2} \cos 2\theta + \tau_{xy} \sin 2\theta}$$

$$\boxed{\tau_{nt} = -\frac{\sigma_x - \sigma_y}{2} \sin 2\theta + \tau_{xy} \cos 2\theta}$$

Principal Stresses & Planes:

• Principal Planes: Planes where $$\displaystyle \tau_{nt} = 0 $$.

• Principal Stresses:

$$\boxed{\sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}}$$

• Orientation: $$\displaystyle \tan 2\theta_p = \frac{2\tau_{xy}}{\sigma_x - \sigma_y} $$.

Maximum Shear Stress:

$$\boxed{\tau_{max} = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}}$$

Occurs on planes at $$\displaystyle 45^\circ $$ to principal planes.

Mohr’s Circle:

• Construction:

  1. Plot $$\displaystyle X(\sigma_x, \tau_{xy}) $$ and $$\displaystyle Y(\sigma_y, -\tau_{xy}) $$.

  2. Center $$\displaystyle C = \left(\frac{\sigma_x+\sigma_y}{2}, 0\right) $$.

  3. Radius $$\displaystyle R = \sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2 + \tau_{xy}^2} $$.

• Utility: Graphical determination of $$\displaystyle \sigma_n, \tau $$ on any plane, $$\displaystyle \sigma_{1,2} $$, $$\displaystyle \tau_{max} $$, and their orientations.

Obliquity of Resultant Stress ($\beta$):

• Angle between resultant stress $$\displaystyle \sigma_r $$ and normal to the plane.

• $$\displaystyle \tan \beta = \frac{\tau_{nt}}{\sigma_n} $$.

• Maximum Obliquity: Occurs when $$\displaystyle \frac{d}{d\theta}(\tan \beta) = 0 $$. $$\displaystyle \beta_{max} = \sin^{-1}\left(\frac{\tau_{max}}{\sigma_{avg} + \tau_{max}}\right) $$ or from Mohr's circle geometry.

Proof: Sum of normal stresses on two mutually perpendicular planes is constant.

$$\sigma_x + \sigma_y = \sigma_n + \sigma_{n-perp} = \text{constant}$$

7. Theories of Failure

Material Types:

• Ductile: Significant plastic deformation before failure (e.g., mild steel, Al, Cu). Fail by shear.

• Brittle: Little plastic deformation, fractures suddenly (e.g., cast iron, concrete). Fail by tensile on principal plane.

Failure Theories:

Graphical Representation ($$\displaystyle \sigma_1 $$-$$\displaystyle \sigma_3 $$ plane):

• Rankine: Vertical lines at $$\displaystyle \sigma_1 = \sigma_y $$, $$\displaystyle \sigma_3 = -\sigma_y $$.

• Tresca: Hexagon with corners at $$\displaystyle (\sigma_y, \sigma_y) $$, $$\displaystyle (\sigma_y, -\sigma_y) $$, etc.

• Von Mises: Ellipse inscribed in Tresca hexagon.

Application to Combined Loading:

• Axial + Torsion (Bolt): $$\displaystyle \sigma_1 = \frac{P}{A} + \sqrt{\left(\frac{P}{2A}\right)^2 + \tau^2} $$, $$\displaystyle \sigma_2 = \frac{P}{2A} - \sqrt{...} $$.

• Bending + Torsion (Shaft): As in Section 5.

• Biaxial (Thin-walled pressure vessel): $$\displaystyle \sigma_1 = \frac{pr}{t} $$ (hoop), $$\displaystyle \sigma_2 = \frac{pr}{2t} $$ (axial), $$\displaystyle \sigma_3 = 0 $$.

Factor of Safety (FOS):

$$n = \frac{\text{Yield Stress } (\sigma_y)}{\text{Allowable Stress from theory}}$$

[!TIP] Quick Recall: Ductile → Tresca or Von Mises. Brittle → Rankine. Von Mises is generally more accurate for ductile metals.

8. Columns & Buckling

Buckling vs. Crushing:

• Buckling (Elastic Instability): Long, slender columns fail by lateral deflection at load < crushing strength.

• Crushing (Material Failure): Short, stocky columns fail by direct compression at load > proportional limit.

• Slenderness Ratio: $$\displaystyle \lambda = \frac{L_e}{r} $$, where $$\displaystyle r = \sqrt{I/A} $$ (radius of gyration), $$\displaystyle L_e $$ = effective length.

Euler’s Formula (Derivation - Both Ends Pinned):

Assumptions: Perfectly straight, elastic, axially loaded, no initial crookedness.

1. Differential equation: $$\displaystyle EI \frac{d^2y}{dx^2} = -M = -P y $$.

2. $$\displaystyle \frac{d^2y}{dx^2} + k^2 y = 0 $$, where $$\displaystyle k^2 = P/(EI) $$.

3. General solution: $$\displaystyle y = A \sin kx + B \cos kx $$.

4. BCs: $$\displaystyle y(0)=0 \Rightarrow B=0 $$; $$\displaystyle y(L)=0 \Rightarrow \sin kL = 0 \Rightarrow kL = n\pi $$.

5. Critical load: $$\displaystyle P_{cr} = \frac{n^2 \pi^2 EI}{L^2} $$. Fundamental mode ($$\displaystyle n=1 $$) gives lowest $$\displaystyle P_{cr} $$.

$$\boxed{P_{cr} = \frac{\pi^2 EI}{L^2}}$$

Effective Length ($$\displaystyle L_e $$) for Different End Conditions:

Critical Stress & Slenderness:

$$\sigma_{cr} = \frac{P_{cr}}{A} = \frac{\pi^2 E}{(L_e/r)^2} = \frac{\pi^2 E}{\lambda^2}$$

• Limitation of Euler: Valid only if $$\displaystyle \sigma_{cr} < \sigma_{proportional limit} $$ (i.e., $$\displaystyle \lambda > \lambda_{cr} = \pi \sqrt{E/\sigma_{pl}} $$).

Rankine’s Formula (Empirical, for all lengths):

$$\boxed{P_{cr} = \frac{\sigma_c A}{1 + a \left(\frac{L_e}{r}\right)^2}} \quad \text{or} \quad \boxed{P_{cr} = \frac{\sigma_c A}{1 + \frac{\sigma_c}{\pi^2 E} \left(\frac{L_e}{r}\right)^2}}$$

• $$\displaystyle \sigma_c $$ = crushing strength (compressive yield stress).

• $a$ = Rankine constant ($$\displaystyle a = 1/(\pi^2 E/\sigma_c) $$ for Euler limit).

• For long columns ($\lambda$ large): $$\displaystyle P_{cr} \to \frac{\pi^2 EI}{L_e^2} $$ (Euler).

• For short columns ($\lambda$ small): $$\displaystyle P_{cr} \to \sigma_c A $$ (crushing).

Safe Load:

$$P_{safe} = \frac{P_{cr}}{n}$$

• Use Euler if $$\displaystyle \lambda > \lambda_{cr} $$ (long).

• Use Rankine for intermediate $\lambda$.

• Use Direct Crushing ($$\displaystyle P_{safe} = \sigma_c A / n $$) for short columns ($\lambda$ very small).

Crippling Load: Synonym for critical buckling load ($$\displaystyle P_{cr} $$).

[!TIP] Exam Strategy: For a given column, first compute $$\displaystyle \lambda = L_e/r $$. Compare with critical slenderness $$\displaystyle \lambda_{cr} = \pi\sqrt{E/\sigma_{pl}} $$ to decide formula. For built-up columns, use transformed section to find $I$ and $r$.

9. Specialized & Recurring Topics

Curved Beams (Circular Section):

• Neutral axis does not pass through centroid. It shifts towards the center of curvature.

• Stress Formula (at fiber at distance $y$ from NA, radius of curvature $\rho$):

$$\boxed{\sigma = \frac{Mc}{A r (e + y/\rho)}}$$

where $c$ = distance from NA to outer fiber, $r$ = radius to centroid, $$\displaystyle e = \frac{R}{A} \int \frac{dA}{y} $$ (eccentricity of NA from center).

• Finding NA: For pure bending, $$\displaystyle \int \frac{dA}{y} = 0 $$ (where $y$ measured from assumed NA).

Strain Energy:

• Bending: $$\displaystyle U = \int \frac{\sigma^2}{2E} dV = \int \frac{M^2}{2EI} dx $$.

• Torsion: $$\displaystyle U = \int \frac{\tau^2}{2G} dV = \int \frac{T^2}{2GJ} dx $$.

• Max Strain Energy in Shaft: $$\displaystyle U_{max} = \frac{\tau_{max}^2}{2G} \times \text{Volume} $$.

Composite Beams (Different Materials - Transformed Section):

1. Transform all materials to an equivalent section of a reference material using factor $$\displaystyle n_i = E_i / E_{ref} $$.

2. Find neutral axis of transformed section.

3. Bending stress in material $i$: $$\displaystyle \sigma_i = \frac{M y}{n_i I_{tr}} $$, where $$\displaystyle I_{tr} $$ = $I$ of transformed section about NA.

4. Ensure stresses do not exceed allowable for each material.

Impact Loading (Falling Weight on Vertical Bar):

• Instantaneous Deflection/Expansion:

$$\delta_i = \delta_{st} \left(1 + \sqrt{1 + \frac{2h}{\delta_{st}}}\right)$$

where $$\displaystyle \delta_{st} = \frac{PL}{EA} $$ is static deflection under load $P$.

• Derivation: Conserve energy: $$\displaystyle P h = \frac{1}{2} P \delta_i - \frac{1}{2} P \delta_{st} $$ (assuming no rebound).

Volumetric Strain ($$\displaystyle \epsilon_v $$):

$$\boxed{\epsilon_v = \epsilon_x + \epsilon_y + \epsilon_z}$$

For hydrostatic pressure $p$: $$\displaystyle \epsilon_v = -\frac{p}{K} $$.

Shear Modulus from E and ν:

$$G = \frac{E}{2(1+\nu)}$$

10. Short Notes & Definitions (Frequently Asked)

Principal Planes and Principal Stresses:

• Principal Planes: Planes on which shear stress is zero.

• Principal Stresses: Normal stresses on principal planes ($$\displaystyle \sigma_1 \geq \sigma_2 \geq \sigma_3 $$). They are the extreme values of normal stress.

Mohr’s Stress Circle:

• Construction: Plot $(\sigma, \tau)$ for two perpendicular planes. Circle center at $$\displaystyle (\frac{\sigma_x+\sigma_y}{2}, 0) $$, radius $$\displaystyle \sqrt{((\sigma_x-\sigma_y)/2)^2 + \tau_{xy}^2} $$.

• Utility: Graphical method to find normal/shear stress on any inclined plane, principal stresses, max shear stress, and their orientations without solving equations.

Theories of Failure (List):

1. Maximum Principal Stress (Rankine): Brittle materials.

2. Maximum Shear Stress (Tresca): Ductile materials.

3. Maximum Distortion Energy (Von Mises): Ductile materials (more accurate).

4. Maximum Normal Strain, Total Strain Energy (less common).

Equivalent Length of Columns:

• Definition: Length of a hypothetical pin-ended column with same $$\displaystyle P_{cr} $$ as the given column with actual end conditions.

• Values:

  • Both ends pinned: $$\displaystyle L_e = L $$

  • Both ends fixed: $$\displaystyle L_e = L/2 $$

  • One fixed, one free: $$\displaystyle L_e = 2L $$

  • One fixed, one pinned: $$\displaystyle L_e = 0.7L $$

Macaulay’s Method:

• Definition: Method for beam deflection using step functions (Macaulay brackets) to write bending moment equation as a single continuous expression for the entire beam.

• Advantages over Direct Integration:

  1. Handles multiple point loads easily.

  2. Single integration constant set for each boundary condition.

  3. No need to write separate moment equations for each segment.

Area-Moment Method:

• Theorems:

  1. First Theorem: Change in slope between two points = $$\displaystyle \frac{1}{EI} \times $$ (Area of M diagram between points).

  2. Second Theorem: Deflection of a point relative to tangent at another point = $$\displaystyle \frac{1}{EI} \times $$ (Moment of area of M diagram about that point).

• Advantages: Quick for standard cases, no integration.

• Limitations: Less convenient for varying $I$ or complex loading compared to other methods.

Crippling Load:

• Definition: The critical buckling load ($$\displaystyle P_{cr} $$) at which a column becomes unstable and deflects laterally.

• Euler’s Critical Load: $$\displaystyle P_{cr} = \pi^2 EI / L_e^2 $$ for pin-ended column.

Thermal Stresses:

• Definition: Stresses induced in a body due to restraint against free thermal expansion or contraction.

• Cause: When temperature change occurs but the body is prevented from expanding/contracting freely (e.g., fixed supports, composite bars with different $\alpha$).

Poisson’s Ratio ($\nu$):

• Definition: $$\displaystyle \nu = -\frac{\text{lateral strain}}{\text{longitudinal strain}} $$.

• Typical Values: 0.25-0.35 for metals, ~0.5 for rubber (incompressible), negative for some auxetics.

• Relationship: $$\displaystyle G = E/[2(1+\nu)] $$, $$\displaystyle K = E/[3(1-2\nu)] $$.

Composite Beams:

• Definition: Beams made of two or more different materials (e.g., steel-reinforced concrete, bimetallic strips).

• Analysis Method: Transformed Section Method (see Section 9). Transform all materials to equivalent section of one material using modular ratio $$\displaystyle n = E_{material}/E_{reference} $$.

Quarter-Fourth Rule:

• Brief: Approximate method to find moment of inertia ($I$) of built-up sections (like I-beams). Assume $$\displaystyle I \approx \frac{1}{12} b h^3 $$ for the flanges only, ignoring the web's contribution, or use $$\displaystyle \frac{1}{12}(b_f h_f^3 + b_w h_w^3) $$. Useful for quick estimates.