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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Expert-level abstract algebra knowledge. Use when working with groups, rings, fields, modules, Galois theory, representation theory, or algebraic structures. Also use when the user mentions 'group', 'ring', 'field', 'homomorphism', 'isomorphism', 'normal subgroup', 'quotient group', 'Sylow theorem', 'Galois theory', 'polynomial ring', 'ideal', 'module', 'vector space', or 'representation theory'.
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Expert-level acoustics knowledge. Use when working with sound waves, acoustic propagation, resonance, standing waves, Doppler effect, room acoustics, ultrasound, noise control, musical acoustics, or architectural acoustics. Also use when the user mentions 'sound wave', 'decibel', 'frequency', 'resonance', 'standing wave', 'Doppler effect', 'acoustic impedance', 'reverberation', 'ultrasound', 'noise control', 'harmonics', or 'sound intensity'.
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Expert-level aerodynamics covering subsonic and supersonic flow, lift and drag, airfoil theory, boundary layers, compressible flow, and CFD methods.
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Expert-level air quality engineering covering pollutant sources, atmospheric dispersion, air quality standards, emission controls, and air quality modeling.
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Expert-level aircraft structures covering structural design philosophy, wing and fuselage analysis, fatigue and damage tolerance, composite structures, and certification.
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Expert-level algorithms and complexity theory covering sorting, searching, graph algorithms, dynamic programming, greedy algorithms, NP-completeness, and algorithm design paradigms.
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