Expert-level mechanics of materials covering stress and strain, axial loading, torsion, bending, shear, deflection, buckling, and failure theories.
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Normal stress: sigma = F over A, force per unit area perpendicular to surface. Shear stress: tau = V times Q over I times t, parallel to surface. Normal strain: epsilon = delta over L, deformation per unit length. Hooke law: sigma = E times epsilon for linear elastic material. Poisson ratio: lateral strain is negative nu times axial strain.
Deformation: delta = PL over AE. Statically indeterminate: compatibility equation provides additional equation. Thermal stress: sigma = E times alpha times delta T for constrained member. Stress concentration: geometric discontinuities increase local stress by Kt factor.
Shear stress: tau = T times rho over J, varies linearly with radius. Angle of twist: phi = TL over JG. Polar moment of inertia: J = pi r4 over 2 for solid circle. Thin-walled sections: tau = T over 2 times A_enclosed times t.
Flexure formula: sigma = M times y over I, linear stress distribution. Neutral axis: zero normal stress, centroidal axis for symmetric sections. Shear formula: tau = VQ over It, parabolic distribution in rectangular section. Deflection: integrate moment-curvature relation, M = EI times d2y over dx2.
Von Mises criterion: equivalent stress for ductile materials under combined loading. Tresca criterion: maximum shear stress theory, conservative for ductile materials. Buckling: Euler formula Pcr = pi squared EI over L_effective squared. Fatigue: S-N curve, endurance limit, stress concentration in fatigue.
| Pitfall | Fix |
|---|---|
| Wrong sign for bending stress | Define positive M direction and y direction consistently |
| Using gross section in buckling | Use minimum moment of inertia for column buckling |
| Ignoring shear in beam deflection | Include shear deflection for short deep beams |
| Applying Euler buckling to short columns | Check slenderness ratio before applying Euler formula |
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