Domain
Structural Mechanics
Analyze how structural members respond to loads through internal forces, stress, deformation and stability, with interactive tools for axial members, beams, torsion, buckling and trusses.
Structural Mechanics combines focused calculators with five interactive Labs built on the same typed computation layer, under one published sign convention: tension-positive axial force, tension-positive normal stress, and counter-clockwise positive rotation. Use a calculator for a single well-posed stress, deformation, beam, torsion, buckling or truss question; use a Lab to keep geometry, loads, internal actions, stresses and deformations together in one workspace. The V1 scope is linear, small-deformation structural mechanics for determinate members and planar pin-jointed trusses, with explicit assumptions and honest failure states.
Axial Loading
Axially loaded bars and stepped members: internal axial force, elongation of a prismatic member, and the full N(x), σ(x) and ε(x) fields of a segmented member with concentrated loads.
- Normal Stress CalculatorAverage normal stress σ = N/A on a cross-section from an internal axial force and an area, with the tension or compression sense preserved.
- Normal Strain CalculatorEngineering normal strain ε = ΔL/L from a change in length and an original length, in both plain and microstrain form.
- Hooke's Law CalculatorSolve σ = Eε for stress, strain or Young's modulus in the uniaxial linear-elastic range.
- Axial Deformation CalculatorElongation δ = NL/(AE) of a prismatic axially loaded member, signed so that tension lengthens and compression shortens.
Stress & Strain
Normal stress and strain, the uniaxial linear-elastic relation, factor of safety, plane-stress transformation, principal stresses, Mohr's circle and the von Mises equivalent stress.
- Factor of Safety CalculatorFactor of safety n = S/|σ| against a declared yield, ultimate or custom reference strength, with the matching utilisation.
- Plane Stress Transformation CalculatorTransform a plane-stress state σx, σy, τxy through a counter-clockwise rotation θ and read the rotated components.
- Principal Stresses CalculatorPrincipal stresses σ1 ≥ σ2, their orientations, and the maximum in-plane shear of a plane-stress state.
- Mohr's Circle CalculatorCentre, radius, principal stresses and maximum in-plane shear of a plane-stress state, drawn as a true Mohr circle with τ positive upward.
- Von Mises Stress CalculatorVon Mises equivalent stress for a plane-stress state, with an optional yield strength giving utilisation and a yield-based factor of safety.
Beams
Determinate beams end to end: support reactions, shear and bending-moment diagrams with their true discontinuities, bending and transverse shear stress, and slope and deflection by direct integration.
- Beam Reactions CalculatorSupport reactions of a statically determinate beam — cantilever, simply supported or overhanging — from point loads, couples and uniform distributed loads, with equilibrium residuals and a determinacy verdict.
- Shear Force CalculatorInternal shear force at any station of a determinate beam, with both one-sided limits and the exact jump wherever a concentrated force or a support reaction acts.
- Bending Moment CalculatorInternal bending moment at any station of a determinate beam, sagging positive, with the exact −Mz jump at an applied couple and the analytically located extremes.
- Bending Stress CalculatorExtreme-fibre flexural stress σ = −My/I taken from the bending moment of the beam itself, with tension and compression named.
- Transverse Shear Stress CalculatorTransverse shear stress τ = VQ/(It) in a beam section, using the shear force the beam engine computes for the chosen station.
- Beam Deflection CalculatorSlope and deflection of a determinate beam by direct integration of EI v'' = M, with continuity at every load event and exact support conditions.
Torsion
Circular shafts in Saint-Venant torsion: polar second moment of area for solid and hollow sections, the linear shear-stress distribution τ = Tρ/J, and the angle of twist φ = TL/(GJ) accumulated along a stepped shaft. Non-circular sections warp and are not covered.
- Torsional Shear Stress CalculatorShear stress in a solid or hollow circular shaft under torsion, τ = Tρ/J, at the outer surface and at any radius, with the polar second moment reported alongside.
- Angle of Twist CalculatorRelative rotation of one end of a circular shaft with respect to the other, φ = TL/(GJ), in radians and degrees, with the torsional stiffness GJ/L.
Columns & Buckling
Elastic stability of slender columns: effective length KL for the idealised end conditions, slenderness ratio λ = KL/r, and the Euler critical load and stress. These are bifurcation values for an ideal column, not design capacities.
- Slenderness Ratio CalculatorEffective length KL and slenderness ratio λ = KL/r of a column, from a radius of gyration entered directly or derived from I and A, with the theoretical end-condition factors.
- Euler Buckling Load CalculatorElastic critical load Pcr = π²EI/(KL)² of an ideal slender column, with critical stress and slenderness, reported explicitly as a bifurcation load and not a design capacity.
Trusses
Planar pin-jointed trusses: support reactions, signed member axial forces with tension and compression, optional member stresses, determinacy classification and joint-equilibrium residuals.