Simple Stresses and Strains

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1. Basic Concepts Stress ($\sigma$): Internal resistance per unit area. Normal Stress: $\sigma = \frac{P}{A}$ (Force perpendicular to area) Shear Stress: $\tau = \frac{V}{A_s}$ (Force parallel to area) Strain ($\epsilon$): Deformation per unit length. Normal Strain: $\epsilon = \frac{\delta L}{L_0}$ (Change in length / Original length) Shear Strain: $\gamma = \frac{\delta x}{h}$ or $\gamma = \tan \phi \approx \phi$ (Angular distortion) St. Venant's Principle: The effect of a localized load application becomes negligible at a distance approximately equal to the largest dimension of the loaded area. Elasticity: Ability of a material to return to its original shape after load removal. Plasticity: Ability of a material to undergo permanent deformation without fracture. 2. Stress-Strain Diagram (Mild Steel) A typical stress-strain curve for ductile materials like mild steel shows distinct regions: Proportional Limit (A): Stress is directly proportional to strain (Hooke's Law holds). Elastic Limit (B): Maximum stress material can withstand without permanent deformation. Upper Yield Point (C): Stress at which material begins to deform plastically. Lower Yield Point (D): Stress drops slightly, then remains constant as plastic deformation continues. Ultimate Tensile Strength (E): Maximum stress the material can sustain. Fracture Point (F): Stress at which the material breaks. Stress-Strain Curve Regions: Elastic Region: O to B Plastic Region: B to F Strain Hardening: D to E Necking: E to F 3. Hooke's Law & Elastic Moduli Hooke's Law: $\sigma = E \epsilon$ (within proportional limit) Young's Modulus (Modulus of Elasticity, $E$): Ratio of normal stress to normal strain. $E = \frac{\sigma}{\epsilon}$ Modulus of Rigidity (Shear Modulus, $G$): Ratio of shear stress to shear strain. $G = \frac{\tau}{\gamma}$ Bulk Modulus ($K$): Ratio of volumetric stress to volumetric strain. $K = \frac{\Delta P}{\Delta V / V}$ Relationships between Elastic Moduli $E = 2G(1 + \nu)$ $E = 3K(1 - 2\nu)$ $G = \frac{3KE}{9K - E}$ $\nu = \frac{3K - 2G}{6K + 2G}$ 4. Working Stress & Factor of Safety Working Stress ($\sigma_w$): The maximum stress a material is allowed to withstand in design. Factor of Safety (F.S.): Ratio of ultimate stress (or yield stress) to working stress. $$F.S. = \frac{\text{Ultimate Stress}}{\text{Working Stress}} \quad \text{or} \quad F.S. = \frac{\text{Yield Stress}}{\text{Working Stress}}$$ Purpose: To account for uncertainties in material properties, loads, and manufacturing. 5. Lateral & Volumetric Strain, Poisson's Ratio Lateral Strain ($\epsilon_{lat}$): Strain perpendicular to the applied load. $\epsilon_{lat} = \frac{\delta D}{D_0}$ Poisson's Ratio ($\nu$): Ratio of lateral strain to longitudinal (axial) strain. $$\nu = - \frac{\text{Lateral Strain}}{\text{Longitudinal Strain}} = - \frac{\epsilon_{lat}}{\epsilon_{long}}$$ (The negative sign indicates contraction in lateral direction for positive longitudinal strain). Volumetric Strain ($\epsilon_v$): Change in volume per unit original volume. For uniaxial stress: $\epsilon_v = \epsilon_x + \epsilon_y + \epsilon_z = \epsilon_x (1 - 2\nu)$ For triaxial stress ($\sigma_x, \sigma_y, \sigma_z$): $$\epsilon_v = \frac{1}{E}(\sigma_x + \sigma_y + \sigma_z)(1 - 2\nu)$$ 6. Bars of Varying Section & Composite Bars Bars of Varying Section: Total deformation is the sum of deformations of individual sections. $$\delta L_{total} = \sum_{i=1}^{n} \frac{P L_i}{A_i E_i}$$ Composite Bars: Made of two or more materials rigidly joined. Under axial load: Deformation is equal: $\delta L_1 = \delta L_2 \Rightarrow \epsilon_1 = \epsilon_2$ Total load is shared: $P = P_1 + P_2$ Stress ratio: $\frac{\sigma_1}{\sigma_2} = \frac{E_1}{E_2}$ Modular Ratio ($m$): $m = \frac{E_1}{E_2}$ 7. Temperature Stresses Occur when thermal expansion/contraction is restrained. Thermal Strain ($\epsilon_T$): $\epsilon_T = \alpha \Delta T$ (where $\alpha$ is coefficient of thermal expansion, $\Delta T$ is temperature change) Thermal Stress ($\sigma_T$): If fully restrained, $\sigma_T = E \epsilon_T = E \alpha \Delta T$ Change in Length due to temperature: $\delta L = L \alpha \Delta T$ 8. Strain Energy & Resilience Strain Energy ($U$): Energy stored in a body due to deformation. $$U = \frac{1}{2} P \delta = \frac{1}{2} \sigma \epsilon (\text{Volume}) = \frac{\sigma^2}{2E} (\text{Volume}) = \frac{P^2 L}{2AE}$$ Resilience: Maximum strain energy stored per unit volume at the elastic limit. $$U_R = \frac{\sigma_{elastic}^2}{2E}$$ Proof Resilience: Maximum strain energy stored in a body up to the elastic limit. Modulus of Resilience: Proof resilience per unit volume. Toughness: Total strain energy stored in a body up to fracture (area under stress-strain curve). Modulus of Toughness: Toughness per unit volume. 9. Loadings & Impact Gradual Loading: Load applied slowly, from zero to maximum. $$\sigma = \frac{P}{A} \quad ; \quad \delta L = \frac{PL}{AE}$$ Sudden Loading: Load applied instantaneously. $$\sigma_{sudden} = 2 \sigma_{gradual} = \frac{2P}{A} \quad ; \quad \delta L_{sudden} = 2 \delta L_{gradual}$$ Impact/Shock Loading: Load applied with an initial velocity (e.g., falling weight). $$\sigma_{impact} = \frac{P}{A} \left[ 1 + \sqrt{1 + \frac{2hAE}{PL}} \right]$$ where $h$ is the height of fall. If $h \gg \frac{PL}{2AE}$, then $\sigma_{impact} \approx \frac{P}{A} \sqrt{\frac{2hAE}{PL}}$