Mechanics of Materials Cheatsheet

Cheatsheet Content

Stress and Strain Normal Stress ($\sigma$): Force perpendicular to area. $\sigma = \frac{P}{A}$ (P = axial force, A = cross-sectional area) Units: Pascals (Pa) or psi Shear Stress ($\tau$): Force parallel to area. $\tau = \frac{V}{A}$ (V = shear force, A = area parallel to force) Units: Pascals (Pa) or psi Normal Strain ($\epsilon$): Deformation per unit length. $\epsilon = \frac{\delta}{L_0}$ ($\delta$ = total deformation, $L_0$ = original length) Dimensionless Shear Strain ($\gamma$): Angular deformation. $\gamma = \frac{\delta_s}{L_0} = \tan \phi \approx \phi$ (for small angles) Dimensionless Material Behavior Under Tensile Loading (Stress-Strain Curve) Elastic Region: Material returns to original shape after unloading. Proportional Limit: Stress is directly proportional to strain (Hooke's Law: $\sigma = E\epsilon$). Elastic Limit: Maximum stress before permanent deformation. Yield Point: Stress at which significant plastic deformation begins. Yield Strength ($\sigma_y$): Stress at which a specified offset plastic strain (e.g., 0.2%) occurs. Plastic Region: Permanent deformation occurs. Ultimate Tensile Strength ($\sigma_{UTS}$): Maximum stress a material can withstand before necking. Fracture Point: Stress at which the material breaks. Mechanical Material Properties Young's Modulus (Modulus of Elasticity, $E$): Stiffness of a material in tension/compression. $E = \frac{\sigma}{\epsilon}$ (in elastic region) Units: Pa or psi Shear Modulus (Modulus of Rigidity, $G$): Stiffness of a material in shear. $G = \frac{\tau}{\gamma}$ (in elastic region) Units: Pa or psi Poisson's Ratio ($\nu$): Ratio of lateral strain to axial strain. $\nu = -\frac{\epsilon_{lateral}}{\epsilon_{axial}}$ Typically $0 Ductility: Ability of a material to deform plastically before fracture (e.g., percent elongation, percent reduction in area). Brittleness: Tendency to fracture with little or no plastic deformation. Toughness: Energy absorbed before fracture (area under stress-strain curve). Hardness: Resistance to indentation or scratching. Stress in a Bar Axial Stress: $\sigma = \frac{P}{A}$ Thermal Stress: Stress induced by temperature change when deformation is restricted. $\delta_T = \alpha L \Delta T$ (free thermal expansion) If constrained: $\sigma_T = E \alpha \Delta T$ ($\alpha$ = coefficient of thermal expansion) Elongation in Bars Single Bar under Axial Load: $\delta = \frac{PL}{AE}$ Composite Bar (Series): Total elongation is sum of elongations of individual segments. $\delta_{total} = \sum \frac{P_i L_i}{A_i E_i}$ Composite Bar (Parallel): Load shared by components. Compatibility: $\delta_1 = \delta_2 = \dots = \delta_{total}$ Equilibrium: $P_{total} = P_1 + P_2 + \dots$ Beams Types of Beams: Cantilever: Fixed at one end, free at the other. Simply Supported: Pinned at one end, roller at the other. Overhanging: Simply supported with one or both ends extending beyond supports. Fixed-Fixed (Propped Cantilever): Fixed at both ends. Continuous: Extends over multiple supports. Supports: Roller: Resists vertical force (1 reaction). Pin (Hinge): Resists vertical and horizontal forces (2 reactions). Fixed (Built-in): Resists vertical, horizontal forces, and moment (3 reactions). Loading: Concentrated Load: Single force at a point. Distributed Load: Load spread over a length (uniform, triangular, parabolic). Moment Load: External couple applied to the beam. Bending Stress in Beams (Flexural Stress) Flexure Formula: $\sigma_b = -\frac{My}{I}$ $M$: Bending moment at the section $y$: Distance from the neutral axis to the point where stress is calculated $I$: Moment of inertia of the cross-section about the neutral axis Maximum Bending Stress: Occurs at the top or bottom fiber (maximum $|y|$). $\sigma_{max} = \frac{|M|_{max} c}{I} = \frac{|M|_{max}}{S}$ $c$: Distance from neutral axis to the outermost fiber $S = \frac{I}{c}$: Section modulus Neutral Axis: Axis where bending stress is zero. Coincides with the centroid of the cross-section for symmetric beams. Structural Components of Aircraft Fuselage: Main body of the aircraft. Houses cockpit, cabin, cargo, and connects major components. Primarily subjected to bending, shear, and internal pressure. Common structures: Monocoque, Semi-monocoque (most common, uses skin, stringers, frames/formers). Wing: Generates lift. Main structural elements: Spars (main load-carrying members along span), Ribs (maintain airfoil shape, transfer loads to spars), Skin (carries shear, contributes to bending). Subjected to bending, shear, and torsion due to aerodynamic loads and fuel weight. Often contains fuel tanks. Empennage (Tail): Provides stability and control. Vertical Stabilizer (fin): Controls yaw. Horizontal Stabilizer (tailplane): Controls pitch. Includes movable control surfaces (rudder, elevators). Landing Gear: Supports aircraft on ground, absorbs landing loads. Control Surfaces: Ailerons, elevators, rudder, flaps, slats.