Solid-Mechanics

The SolidExtensions class provides solid mechanics calculations centred on the Euler–Bernoulli beam equation $EI u^{\prime\prime\prime\prime} = q$, plus Hooke's law, second moment of area, the flexure formula, and analytical beam deflections.

Namespace: CSharpNumerics.Physics.SolidMechanics

🔩 Stress & Strain (Hooke's Law)

Formula	Method	Description
$\sigma = F/A$	NormalStress	Axial stress
$\varepsilon = \sigma/E$	NormalStrain	Axial strain
$\sigma = E\varepsilon$	HookesLaw	Hooke's law
$\tau = V/A$	ShearStress	Average shear stress
$G = E / 2(1+\nu)$	ShearModulus	Shear modulus

double sigma = force.NormalStress(area);        // σ = F/A
double eps   = sigma.NormalStrain(200e9);        // ε = σ/E  (steel)
double G     = (200e9).ShearModulus(0.3);         // ≈ 76.9 GPa

📐 Second Moment of Area

Cross-Section	Formula	Method
Solid rectangle	$I = bh^3/12$	RectangularSecondMoment
Solid circle	$I = \pi r^4/4$	CircularSecondMoment
Hollow tube	$I = \pi(R^4 - r^4)/4$	TubularSecondMoment

double I_rect = (0.10).RectangularSecondMoment(0.10);  // 8.33e-6 m⁴
double I_circ = (0.05).CircularSecondMoment();          // 4.91e-7 m⁴
double I_tube = (0.05).TubularSecondMoment(0.04);       // hollow tube

📏 Euler–Bernoulli Beam Equation

$EI \frac{d^4 u}{dx^4} = q(x)$

Relation	Method	Description
$M = EI\kappa$	BendingMoment	Moment from curvature
$\sigma = My/I$	BendingStress	Flexure formula
$V = EI u'''$	BeamShearForce	Shear from 3rd derivative
$q = EI u''''$	BeamLoadIntensity	Load from 4th derivative
$r = EI d^4u/dx^4 - q$	EulerBernoulliResidual	FD residual via Biharmonic1D

double M     = E.BendingMoment(I, curvature);          // M = EIκ
double sigma = M.BendingStress(y: 0.05, I);            // σ = My/I
double q     = E.BeamLoadIntensity(I, d4u);             // q = EIu⁗

// Discrete residual (should ≈ 0 for correct deflection)
var residual = u.EulerBernoulliResidual(E * I, dx, qVector);

📉 Analytical Beam Deflections

Case	Max Deflection	Method
Cantilever + point load P	$\delta = PL^3 / 3EI$	CantileverPointLoadMaxDeflection
Cantilever + uniform q	$\delta = qL^4 / 8EI$	CantileverUniformLoadMaxDeflection
Simply supported + uniform q	$\delta = 5qL^4 / 384EI$	SimplySupportedUniformLoadMaxDeflection
Simply supported + midpoint P	$\delta = PL^3 / 48EI$	SimplySupportedPointLoadMaxDeflection

Full deflection curves are also available: CantileverPointLoadDeflection(P, L, EI, x), etc.

// Steel cantilever: 10×10 cm, 1 m, 1 kN tip load
double EI  = 200e9 * (0.10).RectangularSecondMoment(0.10);
double max = (1000.0).CantileverPointLoadMaxDeflection(1.0, EI);  // PL³/(3EI)

// Deflection curve
for (double x = 0; x <= 1.0; x += 0.1)
    Console.WriteLine($"x={x:F1}  u={1000.0.CantileverPointLoadDeflection(1.0, EI, x):E3} m");