Strength of Material (ME-304) - Unit 5 Short Notes

UNIT 5: STRENGTH OF MATERIALS - EXAM-FOCUSED SHORT NOTES

I. AXIAL DEFORMATION & COMPOSITE BARS

Fundamentals

• Stress (σ): Force per unit area, $$\displaystyle \sigma = \frac{P}{A} $$.

• Strain (ε): Deformation per unit length, $$\displaystyle \epsilon = \frac{\delta}{L} $$.

• Hooke's Law: $$\displaystyle \sigma = E \epsilon $$, where E is Modulus of Elasticity (Young's Modulus).

• Strain Energy (U): Energy stored due to deformation.

  • Gradual loading: $$\displaystyle U = \frac{1}{2} P \delta $$.

  • In terms of stress/volume: $$\displaystyle U = \frac{\sigma^2}{2E} \times \text{Volume} $$.

[!TIP] Exam Alert: For impact/falling weight, use $$\displaystyle \delta_{max} = \delta_{static} \left(1 + \sqrt{1 + \frac{h}{\delta_{static}}}\right) $$, where h is drop height.

Composite/Segmented Bars

• Compatibility: Deformation of each segment is equal (if in series).

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

• Steps:

  1. Assume common deformation $ \delta $.

  2. Express stress in each material: $$\displaystyle \sigma_i = \frac{E_i \delta}{L_i} $$.

  3. Apply equilibrium: $$\displaystyle \sum \sigma_i A_i = P $$.

  4. Solve for $ \delta $ or unknown load.

  5. Check stress limits for each material.

Non-Uniform Cross-Section (Tapered Circular Bar)

• Diameter varies linearly: $$\displaystyle d(x) = d_1 + \frac{(d_2 - d_1)x}{L} $$.

• Area: $$\displaystyle A(x) = \frac{\pi}{4} [d(x)]^2 $$.

• Elongation:

$$ \delta = \int_0^L \frac{P}{E A(x)} dx = \frac{4P}{\pi E (d_2 - d_1)} \ln\left(\frac{d_2}{d_1}\right) \quad \text{(for } d_1 \neq d_2\text{)} $$

Thermal Effects

• Free expansion: $$\displaystyle \delta_{thermal} = \alpha L \Delta T $$.

• Thermal stress if expansion is restricted: $$\displaystyle \sigma = E \alpha \Delta T $$.

• For composite systems (e.g., rod & tube):

  • Compatibility: Total deformation = 0 (if rigidly connected).

  • Equilibrium: $$\displaystyle \sigma_s A_s + \sigma_t A_t = 0 $$.

  • Solve: $$\displaystyle E_s \alpha_s \Delta T + \sigma_s = 0 $$, $$\displaystyle E_t \alpha_t \Delta T + \sigma_t = 0 $$, with $$\displaystyle \sigma_s A_s + \sigma_t A_t = 0 $$.

II. SHEAR FORCE, BENDING MOMENT & BEAM DEFLECTION

SF & BM Diagrams

• Sign Convention:

  • S.F.: Positive if left section moves up on right.

  • B.M.: Positive if sagging (concave up).

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

• Point of Contraflexure: Where B.M. = 0 (changes sign).

Deflection Methods

Standard Cases (Simply Supported & Cantilever)

• Simply Supported, Central Point Load (P):

  • Max deflection: $$\displaystyle \delta_{max} = \frac{PL^3}{48EI} $$ at center.

  • Slope at ends: $$\displaystyle \theta_A = \theta_B = \frac{PL^2}{16EI} $$.

• Cantilever, UDL (w) over full span:

  • Free end deflection: $$\displaystyle \delta_B = \frac{wL^4}{8EI} $$.

  • Slope at free end: $$\displaystyle \theta_B = \frac{wL^3}{6EI} $$.

• Cantilever, UDL over part length (a from fixed end) – Area-Moment:

  • Deflection at free end: $$\displaystyle \delta_B = \frac{w a^2}{24EI} (4L^2 - a^2) $$.

[!TIP] Exam Tip: For overhanging beams, draw SF/BM carefully—negative BM possible over supports.

Section Properties

• Flexural Formula: $$\displaystyle \frac{M}{I} = \frac{\sigma}{y} = \frac{E}{R} $$.

• Section Modulus (Z): $$\displaystyle Z = \frac{I}{y_{max}} $$.

• Rectangular Section (b×d): $$\displaystyle I = \frac{bd^3}{12} $$, $$\displaystyle Z = \frac{bd^2}{6} $$.

• Proportioning: Given depth = 2×width ($$\displaystyle d = 2b $$), then $$\displaystyle Z = \frac{b(2b)^2}{6} = \frac{2b^3}{3} $$.

III. TORSION OF CIRCULAR SHAFTS

Fundamentals

• Torsion Equation:

$$ \frac{T}{J} = \frac{\tau}{r} = \frac{G \phi}{L} $$

• $ T $: Torque, $ J $: Polar moment of inertia, $ \tau $: Shear stress, $ r $: Radius, $ G $: Modulus of rigidity, $ \phi $: Angle of twist (radians), $ L $: Length.

• Solid Circular Shaft (diameter D):

  • $$\displaystyle J = \frac{\pi D^4}{32} $$

  • Max shear stress: $$\displaystyle \tau_{max} = \frac{16T}{\pi D^3} $$

  • Angle of twist: $$\displaystyle \phi = \frac{TL}{GJ} $$

• Hollow Circular Shaft (ID = d, OD = D):

  • $$\displaystyle J = \frac{\pi (D^4 - d^4)}{32} $$

  • $$\displaystyle \tau_{max} = \frac{16T}{\pi (D^4 - d^4)} \cdot \frac{D}{2} = \frac{TD}{2J} $$

Solid vs. Hollow Shaft Comparison (Same Material & Weight)

• Weight equality: $$\displaystyle \rho \cdot L \cdot A_{solid} = \rho \cdot L \cdot A_{hollow} \Rightarrow \frac{\pi D_s^2}{4} = \frac{\pi (D^2 - d^2)}{4} \Rightarrow D_s^2 = D^2 - d^2 $$.

• Torque capacity (same τ_max):

  • Solid: $$\displaystyle T_s = \frac{\pi \tau_{max} D_s^3}{16} $$

  • Hollow: $$\displaystyle T_h = \frac{\pi \tau_{max} D^3 (1 - k^4)}{16} $$, where $$\displaystyle k = d/D $$.

• Ratio:

$$ \frac{T_h}{T_s} = \frac{D^3 (1 - k^4)}{(D^2 - d^2)^{3/2}} = \frac{1}{\sqrt{1 - k^2}} > 1 \quad \text{for } k>0 $$

\boxed{\text{Hollow shaft transmits more torque for same weight and max stress.}}

Stepped Shafts & Angle of Twist

• Internal Torques: Cut sections, draw FBD, find T at each segment.

• Total Angle of Twist: $$\displaystyle \phi_{total} = \sum \frac{T_i L_i}{G_i J_i} $$.

• Given specified rotations at points (e.g., Dec 2024 Q3a), set up compatibility equations:

  $$\displaystyle \phi_{AC} = \phi_{AB} + \phi_{BC} $$, etc., and solve for unknown torques.

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, +ve tensile) and $$\displaystyle \tau_t $$ (torsional shear).

• Failure Theories:

  • Max Shear Stress (Tresca): $$\displaystyle \tau_{max} = \frac{\sigma_1 - \sigma_2}{2} \le \frac{\sigma_y}{2 \cdot FOS} $$ (ductile).

  • Max Distortion Energy (von Mises): $$\displaystyle \sigma_{eq} = \sqrt{\sigma_1^2 - \sigma_1\sigma_2 + \sigma_2^2} \le \frac{\sigma_y}{FOS} $$.

  • For pure shear (τ only): von Mises gives $$\displaystyle \sigma_{eq} = \sqrt{3} \tau $$.

IV. STRESS TRANSFORMATION & PRINCIPAL STRESSES

Transformation Equations (2D)

• Given $$\displaystyle \sigma_x, \sigma_y, \tau_{xy} $$, on plane at angle θ from x-axis:

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

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

• Proof of Sum Constant: $$\displaystyle \sigma_\theta + \sigma_{\theta+90^\circ} = \sigma_x + \sigma_y $$ (independent of θ).

Principal Stresses & Planes

• Principal planes: Where $$\displaystyle \tau_\theta = 0 $$.

• Principal stresses:

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

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

• Max Shear Stress: $$\displaystyle \tau_{max} = \frac{\sigma_1 - \sigma_2}{2} $$, occurs on planes at $$\displaystyle 45^\circ $$ to principal planes.

Mohr's Circle of Stress

• Construction:

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

  2. Plot point $$\displaystyle (\sigma_x, \tau_{xy}) $$ (sign: τ positive if causing clockwise rotation on x-face).

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

  4. Circle: $$\displaystyle (\sigma - \sigma_{avg})^2 + \tau^2 = R^2 $$.

• Read: Principal stresses (σ on σ-axis), max shear (τ at top/bottom), stresses on any plane (angle 2θ from x-axis).

Maximum Obliquity

• Obliquity (ψ): Angle between resultant stress $$\displaystyle \sigma_\theta $$ and normal to plane.

$$ \tan \psi = \frac{\tau_\theta}{\sigma_\theta} $$

• Max obliquity occurs when $$\displaystyle \frac{d}{d\theta}(\tan \psi) = 0 $$. Alternatively, from Mohr's circle, max ψ at point where line from origin is tangent to circle.

• Formula:

$$ \tan \psi_{max} = \frac{R}{\sqrt{(\sigma_{avg})^2 + R^2}} $$

, plane angle $$\displaystyle \theta = \frac{1}{2} \tan^{-1}\left(\frac{\tau_{xy}}{\sigma_{avg}}\right) $$.

V. THEORIES OF FAILURE

Material Classification

• Ductile: Significant plastic deformation before failure (steel, Al, Cu). Use shear-based theories.

• Brittle: Little plastic deformation (cast iron, concrete, ceramics). Use maximum normal stress theory.

Theories for Ductile Materials

1. Maximum Shear Stress Theory (Tresca):

   • Failure when $$\displaystyle \tau_{max} \ge \frac{\sigma_y}{2} $$ (simple tension yield).

   • Graphical: Hexagon inscribed in σ₁–σ₂ plane.

   • Conservative, safe for design.

2. Maximum Distortion Energy Theory (von Mises/Hencky):

   • Failure when $$\displaystyle \frac{1}{2}[(\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2] \ge \sigma_y^2 $$.

   • For 2D (σ₃=0): $$\displaystyle \sigma_{eq} = \sqrt{\sigma_1^2 - \sigma_1\sigma_2 + \sigma_2^2} \le \sigma_y $$.

   • Graphical: Ellipse in σ₁–σ₂ plane.

   • More accurate for ductile metals, matches experiments better.

Theory for Brittle Materials

• Maximum Principal Stress Theory (Rankine):

  • Failure when $$\displaystyle \max(|\sigma_1|, |\sigma_2|, |\sigma_3|) \ge \sigma_{ult} $$ (tensile or compressive).

  • Suitable for brittle materials in predominantly normal stress states.

Application to Thin Cylindrical Pressure Vessels

• Stresses: Hoop $$\displaystyle \sigma_h = \frac{p d}{2t} $$, Longitudinal $$\displaystyle \sigma_l = \frac{p d}{4t} $$, Radial $$\displaystyle \sigma_r \approx 0 $$.

• Thickness (t) from theories (with FOS = n, yield σ_y):

  • Max Principal Stress (Rankine): $$\displaystyle \frac{pd}{2t} \le \frac{\sigma_y}{n} \Rightarrow t = \frac{n p d}{2 \sigma_y} $$.

  • Max Shear Stress (Tresca): $$\displaystyle \tau_{max} = \frac{\sigma_h - \sigma_l}{2} = \frac{pd}{8t} \le \frac{\sigma_y}{2n} \Rightarrow t = \frac{n p d}{4 \sigma_y} $$.

  • Max Distortion Energy (von Mises): $$\displaystyle \sigma_{eq} = \sqrt{\sigma_h^2 - \sigma_h\sigma_l + \sigma_l^2} = \frac{pd}{4t}\sqrt{3} \le \frac{\sigma_y}{n} \Rightarrow t = \frac{n p d \sqrt{3}}{4 \sigma_y} $$.

[!TIP] Remember: For thin-walled cylinder, radial stress is negligible. Order of thickness: von Mises > Tresca > Rankine (most conservative).

VI. COLUMNS & BUCKLING (EULER'S & RANKINE'S FORMULAE)

Fundamentals

• Crippling Load (P_cr): Load at which buckling occurs.

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

• Equivalent Length (L_e): Depends on end conditions:

  | End Condition | L_e | |---------------|-----| | Both ends pinned (hinged) | L | | Both ends fixed | L/2 | | One fixed, one free | 2L | | One fixed, one pinned | L/√2 |

Euler's Formula

• For long columns ($$\displaystyle \lambda > \lambda_{critical} $$, material obeys Hooke's law at failure).

• Derivation (both ends pinned): From differential equation $$\displaystyle EI \frac{d^2y}{dx^2} = -M = -P y $$.

  Solution: $$\displaystyle y = A \sin\left(\frac{\sqrt{P}}{\sqrt{EI}} x\right) + B \cos\left(\frac{\sqrt{P}}{\sqrt{EI}} x\right) $$.

  Apply y(0)=0, y(L)=0 → $$\displaystyle \sin\left(\frac{\sqrt{P} L}{\sqrt{EI}}\right) = 0 $$ → $$\displaystyle \frac{\sqrt{P} L}{\sqrt{EI}} = n\pi $$.

  Fundamental mode (n=1):

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

• Assumptions:

  1. Column is perfectly straight, homogeneous, isotropic.

  2. Load is axial, centroidal.

  3. Material obeys Hooke's law.

  4. Buckling occurs in the plane of least I (for unsymmetrical sections).

• Valid for: $$\displaystyle \lambda > \lambda_{cr} $$, where $$\displaystyle \lambda_{cr} = \pi \sqrt{\frac{E}{\sigma_{proportional}}} $$.

Rankine's Formula

• For all lengths (intermediate & short columns).

$$ P_{cr} = \frac{f_c A}{1 + a \left(\frac{L_e}{r}\right)^2} $$

where $$\displaystyle f_c $$ = crushing strength, a = Rankine constant ($$\displaystyle a = \frac{\sigma_{proportional}}{f_c} \approx \frac{1}{n} $$, n = Euler's λ_cr²).

• Comparison with Euler:

  • For long columns ($$\displaystyle L_e/r \to \infty $$): $$\displaystyle P_{cr} \to \frac{\pi^2 E}{\left(\frac{L_e}{r}\right)^2} A $$ (Euler).

  • For short columns ($$\displaystyle L_e/r \to 0 $$): $$\displaystyle P_{cr} \to f_c A $$ (crushing).

• Length below which Euler not applicable: Set Euler's $$\displaystyle \sigma_{cr} = \frac{\pi^2 E}{(L_e/r)^2} = \sigma_{proportional} $$ → $$\displaystyle \left(\frac{L_e}{r}\right)_{cr} = \pi \sqrt{\frac{E}{\sigma_{proportional}}} $$.

Practical Column Problems

• Safe Load: $$\displaystyle P_{safe} = \frac{P_{cr}}{FOS} $$.

• Columns with different I about axes: Use minimum I (weakest axis) for $$\displaystyle L_e/r $$ calculation.

• Hollow tubular columns: $$\displaystyle r = \sqrt{\frac{D^2 + d^2}{4}} $$.

VII. ELASTIC CONSTANTS & VOLUMETRIC STRAIN

Elastic Constants

1. Young's Modulus (E): $$\displaystyle E = \frac{\text{Longitudinal stress}}{\text{Longitudinal strain}} $$.

2. Modulus of Rigidity (G): $$\displaystyle G = \frac{\text{Shear stress}}{\text{Shear strain}} $$.

3. Bulk Modulus (K): $$\displaystyle K = \frac{\text{Hydrostatic pressure}}{\text{Volumetric strain}} $$.

4. Poisson's Ratio (ν): $$\displaystyle \nu = -\frac{\text{Lateral strain}}{\text{Longitudinal strain}} $$.

Relationships

$$ E = 2G(1 + \nu) $$

$$ E = 3K(1 - 2\nu) $$

$$ E = \frac{9KG}{3K + G} $$

• From first two: $$\displaystyle G = \frac{3K(1 - 2\nu)}{2(1 + \nu)} $$.

[!TIP] Range of ν: For isotropic materials, $$\displaystyle -1 < \nu < 0.5 $$. Typical metals: 0.25–0.35.

Volumetric Strain

• Definition: $$\displaystyle \epsilon_v = \frac{\Delta V}{V} $$.

• Under Triaxial Stress (σ_x, σ_y, σ_z):

$$ \epsilon_v = \epsilon_x + \epsilon_y + \epsilon_z = \frac{1}{E} \left[ (\sigma_x + \sigma_y + \sigma_z) - 2\nu (\sigma_y + \sigma_z + \sigma_x) \right] = \frac{1-2\nu}{E} (\sigma_x + \sigma_y + \sigma_z) $$

• Hydrostatic Pressure (p): $$\displaystyle \sigma_x = \sigma_y = \sigma_z = -p $$ → $$\displaystyle \epsilon_v = -\frac{3p(1-2\nu)}{E} = -\frac{p}{K} $$.

• Proof of Sum: From 3D Hooke's law, sum of linear strains gives volumetric strain directly.

VIII. MISCELLANEOUS & THEORETICAL CONCEPTS

Curved Beams (Circular Section)

• Neutral Axis (NA) NOT at centroid. For pure bending:

$$ \frac{M}{A e \bar{r}_n} = \frac{\sigma}{r_n} $$

where:

• $ e $ = distance from NA to centroid,

• $$\displaystyle \bar{r}_n $$ = mean radius of NA,

• $$\displaystyle r_n $$ = radius of fiber where stress σ is calculated.

• Stress Distribution: $$\displaystyle \sigma = \frac{M y}{A e (r_n + y)} $$, where y measured from NA.

• NA Position: $$\displaystyle \int \frac{dA}{r_n + y} = 0 $$ (from equilibrium).

Shear Stress in Beams (Circular Section)

• Formula: $$\displaystyle \tau = \frac{V Q}{I b} $$.

• For solid circular shaft (diameter D):

  • $$\displaystyle I = \frac{\pi D^4}{64} $$, $$\displaystyle b = \sqrt{ (D/2)^2 - y^2 } $$.

  • $$\displaystyle Q = \int_{y}^{D/2} 2\sqrt{ (D/2)^2 - y'^2 } \cdot y' \, dy' $$.

• Distribution: Parabolic, max at neutral axis: $$\displaystyle \tau_{max} = \frac{4V}{3A} = \frac{4V}{3 \cdot \frac{\pi D^2}{4}} = \frac{16V}{3\pi D^2} $$.

• Plot:

  DiagramSEARCH: shear stress distribution circular beam

Short Notes (Frequently Asked)

1. Principal Planes & Principal Stresses

• Principal planes: Planes on which shear stress is zero and normal stresses are extremum (max/min).

• Principal stresses: Normal stresses on principal planes ($$\displaystyle \sigma_1 \ge \sigma_2 \ge \sigma_3 $$).

• Determination: Analytical (transformation equations) or graphical (Mohr's circle).

2. Mohr's Stress Circle

• Definition: Graphical representation of 2D/3D stress state.

• Construction: Plot (σ, τ) for two perpendicular planes; circle with center at $$\displaystyle (\sigma_{avg}, 0) $$, radius $$\displaystyle R = \sqrt{(\frac{\sigma_x-\sigma_y}{2})^2 + \tau_{xy}^2} $$.

• Utility: Easily find principal stresses, max shear, stresses on any inclined plane, and visualize stress transformations.

3. Composite Beams

• Beams made of two or more materials rigidly bonded.

• Assumption: Same strain at interface (perfect bond), different stresses.

• Analysis: Transform to equivalent section of one material using modular ratio $$\displaystyle m = \frac{E_1}{E_2} $$.

• Neutral axis: From transformed section centroid.

• Bending stress: $$\displaystyle \sigma = \frac{M y}{I_{transformed}} \cdot E_{material} $$.

4. Quarter-Fourth Rule (Rectangular Section)

• For shear stress distribution in rectangular beam:

  • Max shear at NA: $$\displaystyle \tau_{max} = 1.5 \tau_{avg} $$.

  • At $$\displaystyle y = \pm h/4 $$ from NA, $$\displaystyle \tau = 0.9375 \tau_{max} $$.

  • At $$\displaystyle y = \pm h/2 $$ (top/bottom), $$\displaystyle \tau = 0 $$.

• Used to approximate shear stress at critical sections.

5. Theories of Failure – Summary Table

6. Equivalent Length of Columns

• Definition: Length of an ideal pinned-pinned column with same buckling load as given column with actual end conditions.

• Values: See table in Section VI above.

• Importance: Euler's formula uses $$\displaystyle L_e $$: $$\displaystyle P_{cr} = \frac{\pi^2 EI}{L_e^2} $$.

END OF UNIT 5 NOTES
Focus on problem-solving using boxed formulas and understanding assumptions. Practice past paper problems for each topic.