Strength of Materials Cheatsheet

Cheatsheet Content

1. Stress & Strain Normal Stress ($\sigma$) Definition: Internal resistive force per unit area. Axial Stress: $\sigma = \frac{P}{A}$ Bending Stress: $\sigma_b = \frac{My}{I}$ Units: Pascal (Pa) or psi Shear Stress ($\tau$) Definition: Stress component parallel to the cross-section. Direct Shear: $\tau = \frac{V}{A}$ Torsional Shear: $\tau_t = \frac{Tr}{J}$ Units: Pascal (Pa) or psi Normal Strain ($\epsilon$) Definition: Deformation per unit length. Axial Strain: $\epsilon = \frac{\delta}{L}$ Units: Dimensionless (m/m, in/in) Shear Strain ($\gamma$) Definition: Angular deformation. $\gamma = \frac{\delta_s}{L} = \tan \phi \approx \phi$ (for small angles) Units: Dimensionless (radians) 2. Material Properties Hooke's Law Axial: $\sigma = E\epsilon$ Shear: $\tau = G\gamma$ Elastic Moduli Young's Modulus ($E$): Modulus of Elasticity (Normal Stress/Strain) Shear Modulus ($G$): Modulus of Rigidity (Shear Stress/Strain) Bulk Modulus ($K$): Volumetric Stress/Strain Relationship: $E = 2G(1+\nu)$ Poisson's Ratio ($\nu$) Definition: Ratio of transverse strain to axial strain. $\nu = -\frac{\epsilon_{trans}}{\epsilon_{axial}}$ Typical values: $0.25 - 0.35$ for metals Other Properties Yield Strength ($\sigma_y$): Stress at which plastic deformation begins. Ultimate Tensile Strength ($\sigma_{ult}$): Maximum stress a material can withstand. Ductility: Ability to deform plastically before fracture. Brittleness: Tendency to fracture with little plastic deformation. 3. Axial Loading Deformation $\delta = \frac{PL}{AE}$ For varying cross-section/load: $\delta = \sum \frac{P_i L_i}{A_i E_i}$ or $\int_0^L \frac{P(x)}{A(x)E(x)} dx$ Thermal Stress/Strain Thermal Strain: $\epsilon_T = \alpha \Delta T$ Thermal Deformation: $\delta_T = \alpha L \Delta T$ Thermal Stress (if restrained): $\sigma_T = E \alpha \Delta T$ Stress Concentration $\sigma_{max} = K \sigma_{nom}$ $K$: Stress concentration factor (depends on geometry) 4. Torsion Shear Stress in Circular Shafts $\tau = \frac{Tr}{J}$ $T$: Applied torque $r$: Radius from center $J$: Polar moment of inertia Solid Circular: $J = \frac{\pi}{2} c^4$ Hollow Circular: $J = \frac{\pi}{2} (c_o^4 - c_i^4)$ Angle of Twist ($\phi$) $\phi = \frac{TL}{JG}$ For varying torque/geometry: $\phi = \sum \frac{T_i L_i}{J_i G_i}$ or $\int_0^L \frac{T(x)}{J(x)G(x)} dx$ Power Transmission $P = T\omega = 2\pi f T$ $P$: Power (Watts or hp) $\omega$: Angular velocity (rad/s) $f$: Frequency (Hz) 5. Bending Flexural Stress (Normal Stress) $\sigma_b = -\frac{My}{I}$ $M$: Bending moment $y$: Distance from Neutral Axis (NA) $I$: Moment of inertia about NA Maximum stress: $\sigma_{max} = \frac{Mc}{I} = \frac{M}{S}$ $S = I/c$: Section Modulus Transverse Shear Stress $\tau = \frac{VQ}{It}$ $V$: Shear force $Q$: First moment of area above $y$ $t$: Width of cross-section at $y$ Rectangular beam max shear: $\tau_{max} = \frac{3}{2} \frac{V}{A}$ (occurs at NA) Circular beam max shear: $\tau_{max} = \frac{4}{3} \frac{V}{A}$ (occurs at NA) Moment of Inertia ($I$) Rectangle: $I = \frac{bh^3}{12}$ (about centroidal axis) Circle: $I = \frac{\pi d^4}{64}$ (about centroidal axis) Parallel Axis Theorem: $I_x = I_{xc} + Ad^2$ Beam Deflection Differential Equation: $EI \frac{d^2v}{dx^2} = M(x)$ Double Integration Method: Integrate $M(x)/EI$ twice to get $v(x)$. Superposition: Combine deflections from individual loads. Common Beam Deflection Formulas Beam Type Loading Max Deflection ($\delta_{max}$) Cantilever Point load $P$ at end $\frac{PL^3}{3EI}$ Cantilever Uniform load $w$ $\frac{wL^4}{8EI}$ Simple Support Point load $P$ at center $\frac{PL^3}{48EI}$ Simple Support Uniform load $w$ $\frac{5wL^4}{384EI}$ 6. Combined Stresses Plane Stress Transformation Normal Stress: $\sigma_x' = \frac{\sigma_x + \sigma_y}{2} + \frac{\sigma_x - \sigma_y}{2}\cos(2\theta) + \tau_{xy}\sin(2\theta)$ Shear Stress: $\tau_{x'y'} = -\frac{\sigma_x - \sigma_y}{2}\sin(2\theta) + \tau_{xy}\cos(2\theta)$ Principal Stresses $\sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$ Angle to principal planes: $\tan(2\theta_p) = \frac{2\tau_{xy}}{\sigma_x - \sigma_y}$ Maximum In-Plane Shear Stress $\tau_{max} = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$ Angle to max shear planes: $\tan(2\theta_s) = -\frac{\sigma_x - \sigma_y}{2\tau_{xy}}$ Mohr's Circle Graphical representation of stress transformation. Center: $C = \frac{\sigma_x + \sigma_y}{2}$ Radius: $R = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$ 7. Columns Euler Buckling Load (for long, slender columns) $P_{cr} = \frac{\pi^2 EI}{(KL)^2}$ $K$: Effective length factor (depends on end conditions) $KL$: Effective length Pin-Pin: $K=1.0$ Fixed-Free: $K=2.0$ Fixed-Pin: $K=0.7$ Fixed-Fixed: $K=0.5$ Critical Buckling Stress $\sigma_{cr} = \frac{P_{cr}}{A} = \frac{\pi^2 E}{(K L/r)^2}$ $r = \sqrt{I/A}$: Radius of gyration $L/r$: Slenderness ratio Short & Intermediate Columns Euler's formula is for long columns where buckling occurs elastically. For shorter columns, inelastic buckling or yielding may occur first. Use empirical formulas (e.g., J.B. Johnson, AISC) for intermediate columns.