Strength of Materials Cheat Sheet

This cheat sheet summarizes key concepts in the strength of materials, covering stress, strain, beam behavior, torsion, and pressure vessel deformation.

Core Principles

• Hooke's Law: Stress is proportional to strain within the elastic limit.
• Stress and Strain: Fundamental measures of internal forces and deformation.
• Beam Theory: Analyzes bending moments, shear forces, and resulting stresses.
• Torsion: Examines twisting effects on shafts and springs.
• Pressure Vessels: Studies stresses and deformations in cylinders and shells.

Action Steps

• Identify material properties (E, G, K, ν).
• Determine applied loads (tension, compression, shear, bending, torsion).
• Calculate stress and strain using relevant formulas.
• Draw Shear Force Diagrams (SFD) and Bending Moment Diagrams (BMD).
• Analyze stress and deformation in beams, shafts, and springs.
• Apply Euler's theory for column buckling analysis.
• Calculate stresses and deformations in thin and thick cylinders/spheres.

Formulas

• Hooke's Law: $\sigma = E \epsilon$
• Shear Stress: $\tau = G \gamma$
• Bulk Modulus: $K = \frac{-P}{\Delta V / V}$
• Poisson's Ratio: $\nu = -\frac{\epsilon_{lateral}}{\epsilon_{axial}}$
• Bending Stress: $\sigma_b = \frac{My}{I}$
• Shear Stress in Beams: $\tau = \frac{VQ}{Ib}$
• Torsional Shear Stress: $\tau = \frac{Tr}{J}$
• Euler's Critical Load: $P_{cr} = \frac{\pi^2 EI}{(KL)^2}$

Key Terms

• Stress: Internal force per unit area within a material.
• Strain: Measure of deformation representing the relative displacement of particles.
• Elastic Limit: The maximum stress a material can withstand without permanent deformation.
• Shear Force Diagram (SFD): Graph showing the variation of shear force along the length of a beam.
• Bending Moment Diagram (BMD): Graph showing the variation of bending moment along the length of a beam.
• Neutral Axis: The line in a beam's cross-section where bending stress is zero.
• Moment of Inertia: A measure of a beam's resistance to bending.
• Torsion: The twisting of an object due to an applied torque.
• Hoop Stress: Circumferential stress in a pressure vessel wall.