The core mechanics of materials formulas on one page: axial stress and deformation, thermal stress, torsion, bending and transverse shear, beam deflection, stress transformation, column buckling and thin-walled pressure vessels. Each line lists its variables and SI units.
r = √(I/A) radius of gyration; valid while σcr < yield
Thin-walled cylinder
σhoop = p·r/t; σaxial = p·r/(2t)
p internal gauge pressure (Pa); r inner radius; t wall (m); r/t ≥ 10
Thin-walled sphere
σ = p·r/(2t)
Same in every direction in the wall
How to use this cheat sheet
Find the internal force first (axial force, torque, shear or moment at the section), using statics. Every formula here turns an internal force into a stress or a deformation; none of them finds the internal force for you.
Check which formula applies before you use it. The flexure formula assumes linear-elastic material and a prismatic beam; the torsion formulas need a circular cross-section; Euler buckling only holds below the proportional limit.
Write the units next to E, I and J. Mixing mm⁴ with m or GPa with N/mm² is the most common source of answers that are off by a factor of 1 000 or 10⁶.
Example: Stress and twist in a solid drive shaft
Given
Solid steel shaft, diameter d = 40 mm, length L = 1.2 m
Transmits P = 15 kW at n = 600 rpm
G = 79 GPa (typical value for steel)
Find
The maximum shear stress τ_max
The angle of twist φ over the length
Solution
Angular speed from rpm.ω = 2π·600/60 = 62.83 rad/s
Torque from power.T = P/ω = 15 000 W / 62.83 rad/s = 238.7 N·m
Polar moment of inertia of the solid section.J = π·(0.040 m)^4/32 = 2.513 × 10^-7 m^4
Maximum shear stress at the surface, c = 20 mm.τ_max = T·c/J = 238.7 × 0.020 / 2.513 × 10^-7 = 19.0 × 10^6 Pa
Angle of twist.φ = T·L/(J·G) = 238.7 × 1.2 / (2.513 × 10^-7 × 79 × 10^9) = 0.01443 rad
Convert to degrees.φ = 0.01443 × 180/π = 0.827°
Answer: τ_max ≈ 19.0 MPa; φ ≈ 0.0144 rad (0.83°).
Printable cheat sheet
The PDF is one US Letter page that also prints on A4 without cropping. The 2:3 PNG is the same sheet as an image, sized for a phone lock screen or a pin. Both are made from the formulas in the tables above, so they always match.
What is the difference between mechanics of materials and strength of materials?
They are two names for the same course. Both cover stress, strain and deformation in loaded members.
Why is the flexure formula negative?
With y measured upward from the neutral axis and a positive (sagging) moment, the top fibres are in compression. The minus sign makes compression negative. Many students use the magnitude M·c/I and decide tension or compression by inspection.
When is a pressure vessel thin-walled?
The usual rule is a radius-to-thickness ratio of 10 or more. Below that, the stress varies noticeably through the wall and thick-wall (Lamé) equations are needed.
