Formula sheet

Mechanical Stress Analysis Formula Sheet

Mechanical stress formulas for axial stress, strain, bending, torsion, shear, von Mises stress, stress concentration, deflection, buckling, thermal stress, and fatigue.

Branch: Mechanical Engineering
Content: Formula sheet
Updated: Jun 20, 2026
Revision: v1.1.0 · reviewed

This formula sheet collects common first-pass formulas used in mechanical stress analysis. The formulas assume simplified geometry and material behaviour. They are most useful for checking load paths, estimating order of magnitude, reviewing finite element results, and finding obvious design risks before detailed analysis.

Use consistent units. State sign convention, coordinate system, load case, material data, and whether stresses are nominal, local, principal, or equivalent.

Design margins and load-case traceability

Static yield factor:

\displaystyle N_y=\frac{\sigma_y}{\sigma_e}

Stress margin:

M_\sigma=\sigma_{allow}-\sigma_{demand}

Utilization:

\displaystyle u=\frac{\sigma_{demand}}{\sigma_{allow}}

Buckling factor:

\displaystyle N_{buckling}=\frac{P_{cr}}{P_{applied}}

Deflection utilization:

\displaystyle u_\delta=\frac{\delta_{predicted}}{\delta_{allowable}}

Record the load case for every margin. A positive static margin does not prove fatigue life, stiffness, buckling stability, leakage control, or proof-test acceptance.

Axial stress and strain

Normal stress in a straight member under axial load:

\displaystyle \sigma=\frac{F}{A}

Axial strain:

\displaystyle \epsilon=\frac{\Delta L}{L}

Linear elastic relation:

\sigma=E\epsilon

Axial deformation of a uniform member:

\displaystyle \delta=\frac{FL}{AE}

For members in series with different areas or materials:

\displaystyle \delta_{total}=\sum_i \frac{F_iL_i}{A_iE_i}

These relations assume uniform axial stress, small deformation, and appropriate end conditions.

Shear stress and shear strain

Average direct shear stress:

\displaystyle \tau_{avg}=\frac{V}{A}

Linear elastic shear relation:

\tau=G\gamma

For isotropic linear elasticity:

\displaystyle G=\frac{E}{2(1+\nu)}

where $G$ is shear modulus, $E$ is Young’s modulus, and $\nu$ is Poisson’s ratio.

Average shear formulas are not enough near holes, fasteners, welds, keyways, adhesive edges, and contact interfaces, where stress distribution may be highly nonuniform.

Bending stress

Elastic bending stress in a beam:

\displaystyle \sigma=\frac{My}{I}

Maximum bending stress:

\displaystyle \sigma_{max}=\frac{Mc}{I}

where $M$ is bending moment, $y$ is distance from the neutral axis, $c$ is distance to the outer fiber, and $I$ is second moment of area.

Section modulus:

\displaystyle S=\frac{I}{c}

so:

\displaystyle \sigma_{max}=\frac{M}{S}

Bending formulas assume beam theory conditions. Short, deep, curved, composite, cracked, or locally loaded members may need more detailed analysis.

Common second moments of area

Rectangle about centroidal bending axis:

\displaystyle I=\frac{bh^3}{12}

Solid circular section:

\displaystyle I=\frac{\pi d^4}{64}

Hollow circular section:

\displaystyle I=\frac{\pi(D^4-d^4)}{64}

For bending stiffness, material far from the neutral axis is especially valuable because $I$ scales with distance squared.

Torsion of circular shafts

Shear stress in a circular shaft:

\displaystyle \tau=\frac{Tr}{J}

Maximum shear stress:

\displaystyle \tau_{max}=\frac{Tc}{J}

Angle of twist:

\displaystyle \theta=\frac{TL}{JG}

Polar second moment of area for a solid circular shaft:

\displaystyle J=\frac{\pi d^4}{32}

For a hollow circular shaft:

\displaystyle J=\frac{\pi(D^4-d^4)}{32}

These simple torsion formulas apply to circular shafts in elastic torsion. Noncircular sections can warp and require different treatment.

Combined normal and shear stress

For a plane-stress element with normal stresses $\sigma_x$ , $\sigma_y$ and shear stress $\tau_{xy}$ , the principal stresses are:

\displaystyle \sigma_{1,2}=\frac{\sigma_x+\sigma_y}{2}\pm\sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2+\tau_{xy}^2}

Maximum in-plane shear stress:

\displaystyle \tau_{max}=\sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2+\tau_{xy}^2}

For uniaxial stress plus shear, with $\sigma_y=0$ :

\displaystyle \sigma_{1,2}=\frac{\sigma_x}{2}\pm\sqrt{\left(\frac{\sigma_x}{2}\right)^2+\tau_{xy}^2}

Principal stresses help distinguish tensile, compressive, and shear-driven failure risks.

Von Mises equivalent stress

For plane stress:

\sigma_e=\sqrt{\sigma_x^2-\sigma_x\sigma_y+\sigma_y^2+3\tau_{xy}^2}

For uniaxial normal stress plus shear:

\sigma_e=\sqrt{\sigma^2+3\tau^2}

For principal stresses:

\displaystyle \sigma_e=\sqrt{\frac{1}{2}\left[(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2\right]}

Static ductile yield check:

\displaystyle \sigma_e \leq \frac{\sigma_y}{N}

where $\sigma_y$ is yield strength and $N$ is the selected safety factor or design factor. Von Mises stress is not a universal failure criterion; it is mainly appropriate for ductile isotropic metals under yielding checks.

Stress concentration

The theoretical stress concentration factor is:

\displaystyle K_t=\frac{\sigma_{max}}{\sigma_{nom}}

For fatigue, a fatigue stress concentration factor is often used:

K_f=1+q(K_t-1)

where $q$ is notch sensitivity. The corrected alternating stress may be estimated as:

S_{a,local}=K_fS_{a,nom}

Do not apply stress concentration factors blindly. They depend on geometry, load mode, material behaviour, notch radius, local yielding, surface condition, and whether the analysis is static, fatigue, or fracture-based.

Beam deflection

For a simply supported beam with central point load $P$ :

\displaystyle \delta_{max}=\frac{PL^3}{48EI}

For a cantilever beam with end load $P$ :

\displaystyle \delta_{max}=\frac{PL^3}{3EI}

For a simply supported beam with uniform load $w$ :

\displaystyle \delta_{max}=\frac{5wL^4}{384EI}

For a cantilever beam with uniform load $w$ :

\displaystyle \delta_{max}=\frac{wL^4}{8EI}

Deflection formulas are sensitive to boundary conditions. Real supports, bolted joints, bearing compliance, weld flexibility, and contact can change stiffness.

Buckling

Euler elastic buckling load:

\displaystyle P_{cr}=\frac{\pi^2EI}{(KL)^2}

Slenderness ratio:

\displaystyle \lambda=\frac{KL}{r}

Radius of gyration:

\displaystyle r=\sqrt{\frac{I}{A}}

where $K$ is effective length factor, $L$ is unsupported length, $I$ is second moment of area, and $A$ is cross-sectional area.

Euler buckling is an ideal elastic model. Real columns require checks for imperfections, eccentric load, inelastic behaviour, residual stress, end fixity, and local buckling.

Thermal stress

Free thermal strain:

\epsilon_{th}=\alpha\Delta T

Fully constrained thermal stress:

\sigma=E\alpha\Delta T

Thermal stress in a component with partial restraint, gradients, or multiple materials usually requires compatibility analysis. High temperature can also introduce creep, relaxation, oxidation, and material-property changes.

Fatigue screening

Stress amplitude:

\displaystyle S_a=\frac{S_{max}-S_{min}}{2}

Mean stress:

\displaystyle S_m=\frac{S_{max}+S_{min}}{2}

Stress ratio:

\displaystyle R=\frac{S_{min}}{S_{max}}

Modified Goodman screening relation:

\displaystyle \frac{S_a}{S_e}+\frac{S_m}{\sigma_{UTS}}\leq\frac{1}{N}

where $S_e$ is endurance limit or finite-life fatigue strength, $\sigma_{UTS}$ is ultimate tensile strength, and $N$ is fatigue design factor.

Fatigue calculations must specify cycle count, load spectrum, surface finish, size effect, stress concentration, residual stress, material data, environment, and inspection assumptions.

Thin-walled pressure stress

For a thin-walled cylinder under internal pressure:

\displaystyle \sigma_{hoop}=\frac{pr}{t}

\displaystyle \sigma_{longitudinal}=\frac{pr}{2t}

where $p$ is internal pressure, $r$ is mean radius, and $t$ is wall thickness. These formulas assume a thin wall, closed ends, uniform pressure, elastic behaviour, and no local discontinuities. Nozzles, welds, supports, corrosion allowance, cyclic pressure, external pressure, and code rules require additional checks.

Mini example: shaft under bending and torque

A circular shaft has diameter:

d=30\ \text{mm}

It carries bending moment:

M=180\ \text{N m}

and torque:

T=120\ \text{N m}

For a solid circular section:

\displaystyle I=\frac{\pi d^4}{64}

\displaystyle J=\frac{\pi d^4}{32}

Using $d=0.03\ \text{m}$ :

I=3.98\times10^{-8}\ \text{m}^4

J=7.95\times10^{-8}\ \text{m}^4

Outer radius:

c=0.015\ \text{m}

Bending stress:

\displaystyle \sigma=\frac{Mc}{I}=\frac{180(0.015)}{3.98\times10^{-8}}=67.8\ \text{MPa}

Torsional shear stress:

\displaystyle \tau=\frac{Tc}{J}=\frac{120(0.015)}{7.95\times10^{-8}}=22.6\ \text{MPa}

Von Mises equivalent stress:

\sigma_e=\sqrt{\sigma^2+3\tau^2}=78.3\ \text{MPa}

If the material yield strength is:

\sigma_y=250\ \text{MPa}

then the static yield factor is approximately:

\displaystyle N=\frac{250}{78.3}=3.2

This is only a static nominal check. A real shaft review would also check fatigue, keyways, shoulders, surface finish, size effects, bearings, critical speed, torque transients, stress concentration, and manufacturing tolerances.

Validation notes

For a calculation record, state:

geometry revision, units, coordinate system, and load case;
whether stress is nominal, local, principal, or equivalent;
material source, heat treatment, product form, temperature, and strength values;
stress concentration, surface finish, residual stress, and manufacturing assumptions;
boundary conditions, contact assumptions, preload, and support stiffness;
acceptance criteria for yielding, fatigue, buckling, deflection, leakage, or proof test;
measured strain, displacement, inspection, or test evidence used to confirm the model.

Finite element results should be reconciled with at least one hand calculation or free-body check before release.

Common cautions

Do not compare nominal stress to yield strength without checking stress concentration and failure mode. Do not use beam formulas when support conditions or geometry violate beam assumptions. Do not use von Mises stress as a fatigue result without converting load history into stress amplitude and mean stress. Do not use Euler buckling for stocky columns or thin plates without checking applicable stability rules. Do not trust finite element stress peaks near sharp corners or point loads without understanding singularities and mesh convergence.

REF

Disciplines