forked from 1b-agent/fieldcraft-upstream
Shells and beams: MITC4/MITC3 shells, Timoshenko beams, six-DOF nodes (M4) (#47)
* fem: MITC4/MITC3 flat shells and Timoshenko beams with profile properties (ElementDofs STRUCTURAL) Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm * solver: shells and beams on the DofLayout beside solids, 2D elements and trusses; analytic verification and tagged benchmarks Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm * engine: probe rotations, reaction moments and shell and beam element results; docs for shells and beams Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm --------- Co-authored-by: jyviko <2501648+jyviko@users.noreply.github.com> Co-authored-by: Claude Opus 5.5 <noreply@anthropic.com>
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//! The two-node Timoshenko beam (six degrees of freedom per node), with the section properties of
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//! the model's beam profiles. (Axial-only members are the trusses of [`crate::truss`].)
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//!
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//! **Local axes.** x runs from node 1 to node 2; y is the section's orientation vector (the
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//! profile's local 1 direction, its `width`) made normal to x; z = x × y (the profile's local 2
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//! direction, its `height`). Degrees of freedom per node: translations along x, y, z, then
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//! rotations about them; element matrices are built in local axes and rotated to the global ones
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//! ([`crate::frame`]).
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//!
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//! **Beam.** Axial and torsional stiffness are the exact two-node ones (EA/L, GJ/L). Each bending
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//! plane uses the interdependent interpolation of the Timoshenko beam: cubic deflection and
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//! quadratic rotation, coupled through Φ = 12 EI / (G A_s L²) so the shear strain is constant and
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//! the element is exact for end loads (any number of elements gives the nodal displacements of
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//! Timoshenko beam theory; J. S. Przemieniecki, *Theory of Matrix Structural Analysis*,
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//! McGraw-Hill 1968, §5.6; Z. Friedman, J. B. Kosmatka, *An improved two-node Timoshenko beam
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//! finite element*, Computers & Structures 47(3):473–481, 1993, whose shape functions are used
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//! here). With these functions w(ξ) and θ(ξ), ξ = x/L,
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//! K = ∫ (EI θ'ᵀ θ' + G A_s (w' − θ)ᵀ (w' − θ)) dx and the consistent mass
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//! M = ∫ (ρA N_wᵀ N_w + ρI N_θᵀ N_θ) dx (translational and rotary inertia) are integrated with
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//! four Gauss points, exactly; Φ → 0 recovers the Euler–Bernoulli (Hermite) element. The
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//! rotation about y is −dw/dx in the x–z plane, so that plane's rotations enter with a minus
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//! sign. Lumped mass: half the mass on each node's translations, half the polar inertia on the
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//! twist, and the consistent bending rotary diagonal scaled like the translations (Hinton, Rock &
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//! Zienkiewicz 1976). End forces are K u minus the consistent loads of the element's body force,
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//! reported as the section forces at each end (axial force N, shears V_y, V_z, torque T, moments
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//! M_y, M_z on the cut whose outward normal is +x).
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//!
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//! **Section properties.** Area, second moments I_y = ∫ z² dA and I_z = ∫ y² dA, the polar
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//! moment I_y + I_z (rotary inertia of torsion), the torsion constant J and the shear areas:
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//! - rectangle: J from Saint-Venant's series, J = a b³/3 (1 − 192 b/(π⁵ a) Σ_{n odd}
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//! tanh(nπa/2b)/n⁵) for sides a ≥ b (S. P. Timoshenko, J. N. Goodier, *Theory of Elasticity*,
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//! 3rd ed., McGraw-Hill 1970, §109); shear coefficient 10(1 + ν)/(12 + 11ν);
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//! - circle and pipe: J = I_y + I_z; shear coefficients 6(1 + ν)/(7 + 6ν) and, for a tube of
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//! radius ratio m, 6(1 + ν)(1 + m²)² / ((7 + 6ν)(1 + m²)² + (20 + 12ν) m²) (G. R. Cowper,
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//! *The shear coefficient in Timoshenko's beam theory*, J. Appl. Mech. 33(2):335–340, 1966);
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//! - I and box (thin-walled): J = Σ b t³/3 over the midline walls (open I) and Bredt's
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//! 4 A_m² t / p (closed box); shear areas the walls parallel to the shear, between the midlines
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//! of the walls across it (the web for shear along z, 5/6 of the flanges along y; the two
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//! webs or the two flanges of a box) (W. D. Pilkey, *Analysis and Design of Elastic Beams*,
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//! Wiley 2002, ch. 6).
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use modelcraft_model::{BeamProfile, Elastic};
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use crate::dense::Matrix;
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use crate::dof::{DofSet, ElementDofs};
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use crate::frame::{Axes, Vec3, cross, dot, norm, rotate_matrix, rotate_to_global, rotate_to_local, sub, to_local, translation, unit};
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use crate::quadrature::gauss_legendre;
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use crate::{Error, Result};
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/// Cross-section properties in the beam's local axes (y along the profile's width, z along its
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/// height).
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#[derive(Debug, Clone, Copy, PartialEq)]
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pub struct Properties {
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pub area: f64,
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/// ∫ z² dA: bending with deflection along z.
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pub iy: f64,
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/// ∫ y² dA: bending with deflection along y.
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pub iz: f64,
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/// Saint-Venant torsion constant.
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pub torsion: f64,
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/// Shear areas for shear along y and along z.
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pub shear_y: f64,
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pub shear_z: f64,
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}
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/// Saint-Venant's torsion constant of a solid rectangle.
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fn rectangle_torsion(width: f64, height: f64) -> f64 {
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let (a, b) = (width.max(height), width.min(height));
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let pi = std::f64::consts::PI;
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let series: f64 = (0..50).map(|k| f64::from(2 * k + 1)).map(|n| (n * pi * a / (2.0 * b)).tanh() / n.powi(5)).sum();
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a * b * b * b / 3.0 * (1.0 - 192.0 * b / (pi.powi(5) * a) * series)
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}
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/// The section properties of a profile (`poisson` enters the shear coefficients).
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pub fn properties(profile: &BeamProfile, poisson: f64) -> Properties {
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let pi = std::f64::consts::PI;
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let area = profile.area();
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match *profile {
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BeamProfile::Rectangle { width, height } => {
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let k = 10.0 * (1.0 + poisson) / (12.0 + 11.0 * poisson);
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Properties {
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area,
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iy: width * height.powi(3) / 12.0,
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iz: height * width.powi(3) / 12.0,
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torsion: rectangle_torsion(width, height),
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shear_y: k * area,
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shear_z: k * area,
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}
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}
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BeamProfile::Circle { radius } => {
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let i = pi * radius.powi(4) / 4.0;
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let k = 6.0 * (1.0 + poisson) / (7.0 + 6.0 * poisson);
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Properties { area, iy: i, iz: i, torsion: 2.0 * i, shear_y: k * area, shear_z: k * area }
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}
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BeamProfile::Pipe { radius, thickness } => {
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let inner = radius - thickness;
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let i = pi * (radius.powi(4) - inner.powi(4)) / 4.0;
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let m2 = (inner / radius).powi(2);
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let k = 6.0 * (1.0 + poisson) * (1.0 + m2).powi(2) / ((7.0 + 6.0 * poisson) * (1.0 + m2).powi(2) + (20.0 + 12.0 * poisson) * m2);
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Properties { area, iy: i, iz: i, torsion: 2.0 * i, shear_y: k * area, shear_z: k * area }
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}
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BeamProfile::I { height, width, web, flange } => {
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let inside = height - 2.0 * flange;
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Properties {
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area,
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iy: (width * height.powi(3) - (width - web) * inside.powi(3)) / 12.0,
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iz: (2.0 * flange * width.powi(3) + inside * web.powi(3)) / 12.0,
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torsion: (2.0 * width * flange.powi(3) + (height - flange) * web.powi(3)) / 3.0,
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shear_y: 5.0 / 6.0 * 2.0 * width * flange,
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shear_z: (height - flange) * web,
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}
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}
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BeamProfile::Box { width, height, thickness: t } => {
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let (wi, hi) = (width - 2.0 * t, height - 2.0 * t);
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let (wm, hm) = (width - t, height - t);
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Properties {
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area,
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iy: (width * height.powi(3) - wi * hi.powi(3)) / 12.0,
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iz: (height * width.powi(3) - hi * wi.powi(3)) / 12.0,
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torsion: 4.0 * (wm * hm).powi(2) * t / (2.0 * (wm + hm)),
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shear_y: 2.0 * wm * t,
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shear_z: 2.0 * hm * t,
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}
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}
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}
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}
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/// The Young's and shear moduli and Poisson's ratio of an isotropic material.
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/// (`what` names the section kind in errors.)
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fn moduli(elastic: &Elastic, what: &str) -> Result<(f64, f64, f64)> {
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let (young, poisson) = crate::material::isotropic_constants(elastic, what)?;
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Ok((young, young / (2.0 * (1.0 + poisson)), poisson))
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}
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/// A beam section: profile properties and an isotropic material.
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#[derive(Debug, Clone, Copy, PartialEq)]
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pub struct BeamSection {
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pub properties: Properties,
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pub young: f64,
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pub shear_modulus: f64,
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}
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impl BeamSection {
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pub fn new(profile: &BeamProfile, elastic: &Elastic) -> Result<BeamSection> {
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profile.validate().map_err(|e| Error::Invalid(e.to_string()))?;
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let (young, shear_modulus, poisson) = moduli(elastic, "beam")?;
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Ok(BeamSection { properties: properties(profile, poisson), young, shear_modulus })
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}
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}
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/// The section forces at the two ends of a beam: N, V_y, V_z, T, M_y, M_z in its local axes.
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#[derive(Debug, Clone, Copy, PartialEq)]
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pub struct BeamForces {
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pub axes: Axes,
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pub ends: [[f64; 6]; 2],
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}
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/// One bending plane's stiffness and mass (deflection, rotation at node 1, then node 2), with
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/// θ = dw/dx in the plane's own convention.
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fn bending_plane(ei: f64, gas: f64, rho_a: f64, rho_i: f64, l: f64) -> ([[f64; 4]; 4], [[f64; 4]; 4]) {
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let phi = 12.0 * ei / (gas * l * l);
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let mu = 1.0 / (1.0 + phi);
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let (mut k, mut m) = ([[0.0; 4]; 4], [[0.0; 4]; 4]);
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for &(x, w) in gauss_legendre(4) {
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let xi = 0.5 * (x + 1.0);
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let w = 0.5 * w * l;
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let x2 = xi * xi;
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let x3 = x2 * xi;
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let nw = [
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mu * (2.0 * x3 - 3.0 * x2 - phi * xi + 1.0 + phi),
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mu * l * (x3 - (2.0 + phi / 2.0) * x2 + (1.0 + phi / 2.0) * xi),
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-mu * (2.0 * x3 - 3.0 * x2 - phi * xi),
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mu * l * (x3 - (1.0 - phi / 2.0) * x2 - phi / 2.0 * xi),
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];
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let nt =
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[mu * 6.0 / l * (x2 - xi), mu * (3.0 * x2 - (4.0 + phi) * xi + 1.0 + phi), -mu * 6.0 / l * (x2 - xi), mu * (3.0 * x2 - (2.0 - phi) * xi)];
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// θ' and γ = w' − θ.
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let dt = [
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mu * 6.0 / (l * l) * (2.0 * xi - 1.0),
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mu / l * (6.0 * xi - 4.0 - phi),
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-mu * 6.0 / (l * l) * (2.0 * xi - 1.0),
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mu / l * (6.0 * xi - 2.0 + phi),
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];
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let dw = [
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mu / l * (6.0 * x2 - 6.0 * xi - phi),
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mu * (3.0 * x2 - (4.0 + phi) * xi + 1.0 + phi / 2.0),
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-mu / l * (6.0 * x2 - 6.0 * xi - phi),
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mu * (3.0 * x2 - (2.0 - phi) * xi - phi / 2.0),
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];
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let g: [f64; 4] = std::array::from_fn(|a| dw[a] - nt[a]);
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for a in 0..4 {
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for b in 0..4 {
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k[a][b] += w * (ei * dt[a] * dt[b] + gas * g[a] * g[b]);
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m[a][b] += w * (rho_a * nw[a] * nw[b] + rho_i * nt[a] * nt[b]);
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}
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}
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}
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(k, m)
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}
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/// A two-node beam element.
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#[derive(Debug, Clone, Copy)]
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pub struct Beam {
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axes: Axes,
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length: f64,
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section: BeamSection,
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}
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/// A line element's unit axis (node 1 to node 2) and length.
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fn line_axis(xyz: &[[f64; 3]]) -> Result<(Vec3, f64)> {
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let [a, b] = xyz else {
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return Err(Error::Invalid(format!("a two-node line element has 2 nodes, got {}", xyz.len())));
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};
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modelcraft_model::check_coordinates(xyz.as_flattened()).map_err(|e| Error::Invalid(e.to_string()))?;
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let d = sub(*b, *a);
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let length = norm(d);
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let ex = unit(d).ok_or_else(|| Error::Degenerate("a line element has zero length".into()))?;
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Ok((ex, length))
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}
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/// The beam's local axes from its end points and orientation vector.
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fn beam_axes(xyz: &[[f64; 3]], orientation: Vec3) -> Result<(Axes, f64)> {
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let (ex, length) = line_axis(xyz)?;
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let o = unit(orientation).ok_or_else(|| Error::Invalid("a beam's orientation vector must be nonzero and finite".into()))?;
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let proj = sub(o, ex.map(|c| c * dot(o, ex)));
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let ey = unit(proj)
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.filter(|_| norm(proj) > 1e-6)
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.ok_or_else(|| Error::Invalid("the beam section's orientation vector is parallel to the beam".into()))?;
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Ok(([ex, ey, cross(ex, ey)], length))
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}
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impl Beam {
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pub fn new(xyz: &[[f64; 3]], orientation: Vec3, section: BeamSection) -> Result<Beam> {
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let (axes, length) = beam_axes(xyz, orientation)?;
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Ok(Beam { axes, length, section })
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}
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pub fn axes(&self) -> Axes {
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self.axes
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}
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pub fn length(&self) -> f64 {
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self.length
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}
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/// The element's unknowns: three translations and three rotations per node.
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pub fn dofs(&self) -> ElementDofs {
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ElementDofs::new(DofSet::STRUCTURAL)
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}
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/// Local stiffness and consistent mass, 12 × 12.
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fn local(&self, density: f64) -> (Matrix, Matrix) {
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let (p, e, g, l) = (self.section.properties, self.section.young, self.section.shear_modulus, self.length);
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let (mut k, mut m) = (Matrix::zeros(12), Matrix::zeros(12));
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let pair = |k: &mut Matrix, m: &mut Matrix, dofs: [usize; 2], stiff: f64, mass: f64| {
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for (a, &i) in dofs.iter().enumerate() {
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for (b, &j) in dofs.iter().enumerate() {
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k.add(i, j, if a == b { stiff } else { -stiff });
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m.add(i, j, mass / 6.0 * if a == b { 2.0 } else { 1.0 });
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}
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}
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};
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pair(&mut k, &mut m, [0, 6], e * p.area / l, density * p.area * l);
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pair(&mut k, &mut m, [3, 9], g * p.torsion / l, density * (p.iy + p.iz) * l);
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// x–y plane: v and θ_z = dv/dx; x–z plane: w and θ_y = −dw/dx.
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let planes = [([1, 5, 7, 11], [1.0; 4], p.iz, p.shear_y), ([2, 4, 8, 10], [1.0, -1.0, 1.0, -1.0], p.iy, p.shear_z)];
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for (dofs, sign, i, shear) in planes {
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let (kp, mp) = bending_plane(e * i, g * shear, density * p.area, density * i, l);
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for a in 0..4 {
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for b in 0..4 {
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k.add(dofs[a], dofs[b], sign[a] * sign[b] * kp[a][b]);
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m.add(dofs[a], dofs[b], sign[a] * sign[b] * mp[a][b]);
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}
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}
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}
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(k, m)
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}
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pub fn stiffness(&self) -> Matrix {
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rotate_matrix(&self.local(0.0).0, &self.axes)
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}
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pub fn mass(&self, density: f64) -> Matrix {
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rotate_matrix(&self.local(density).1, &self.axes)
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}
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/// The lumped mass, block diagonal (3 × 3 rotation blocks) in global axes.
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pub fn lumped_mass(&self, density: f64) -> Matrix {
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let (_, m) = self.local(density);
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let (p, l) = (self.section.properties, self.length);
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let half = density * p.area * l / 2.0;
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let mut d = Matrix::zeros(12);
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// Bending rotations scaled like their plane's translations (equal in both planes).
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for (rotation, deflection) in [(5, 1), (4, 2)] {
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let scale = 2.0 * half / (m.get(deflection, deflection) + m.get(deflection + 6, deflection + 6));
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d.add(rotation, rotation, scale * m.get(rotation, rotation));
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d.add(rotation + 6, rotation + 6, scale * m.get(rotation + 6, rotation + 6));
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}
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for node in [0, 6] {
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for c in 0..3 {
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d.add(node + c, node + c, half);
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}
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d.add(node + 3, node + 3, density * (p.iy + p.iz) * l / 2.0);
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}
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rotate_matrix(&d, &self.axes)
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}
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/// Consistent nodal loads of a uniform force per volume b, in local axes: the consistent
|
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/// mass of unit density times the uniform nodal translation.
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fn local_body_force(&self, b: [f64; 3]) -> Vec<f64> {
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self.local(1.0).1.mul_vec(&translation(to_local(&self.axes, b), 2))
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}
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/// Consistent nodal forces and moments (global) of a uniform force per volume b.
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pub fn body_force(&self, b: [f64; 3]) -> Vec<f64> {
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rotate_to_global(&self.local_body_force(b), &self.axes)
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}
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/// Section forces at the ends for the nodal displacements and rotations `u` (global, 12)
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/// under a uniform force per volume `body` acting on the element.
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pub fn forces(&self, u: &[f64], body: [f64; 3]) -> Result<BeamForces> {
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if u.len() != 12 {
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return Err(Error::Invalid(format!("{} values for a beam element with 12 degrees of freedom", u.len())));
|
||||
}
|
||||
let ku = self.local(0.0).0.mul_vec(&rotate_to_local(u, &self.axes));
|
||||
let end: Vec<f64> = ku.iter().zip(self.local_body_force(body)).map(|(a, b)| a - b).collect();
|
||||
let at = |i: usize| end.get(i).copied().unwrap_or(0.0);
|
||||
Ok(BeamForces { axes: self.axes, ends: [std::array::from_fn(|c| -at(c)), std::array::from_fn(|c| at(6 + c))] })
|
||||
}
|
||||
}
|
||||
|
||||
#[cfg(test)]
|
||||
mod tests {
|
||||
use super::*;
|
||||
|
||||
const STEEL: Elastic = Elastic::Isotropic { young: 210e3, poisson: 0.3 };
|
||||
|
||||
/// Rigid translations and rotations about a point give no force.
|
||||
fn rigid_motions(xyz: &[[f64; 3]]) -> Vec<Vec<f64>> {
|
||||
(0..6)
|
||||
.map(|m| {
|
||||
xyz.iter()
|
||||
.flat_map(|x| {
|
||||
let mut d = [0.0; 6];
|
||||
if m < 3 {
|
||||
d[m] = 1.0;
|
||||
} else {
|
||||
let mut w = [0.0; 3];
|
||||
w[m - 3] = 1.0;
|
||||
let t = cross(w, *x);
|
||||
d[..3].copy_from_slice(&t);
|
||||
d[3..].copy_from_slice(&w);
|
||||
}
|
||||
d
|
||||
})
|
||||
.collect()
|
||||
})
|
||||
.collect()
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn one_beam_element_is_exact_for_a_tip_load() {
|
||||
// Cantilever along a skew axis; tip force along the local z axis: δ = PL³/3EI + PL/(G A_s).
|
||||
let profile = BeamProfile::Rectangle { width: 3.0, height: 5.0 };
|
||||
let section = BeamSection::new(&profile, &STEEL).unwrap();
|
||||
let xyz = [[1.0, 2.0, 3.0], [41.0, 32.0, 3.0]];
|
||||
let beam = Beam::new(&xyz, [0.0, 0.0, 1.0], section).unwrap();
|
||||
let l = beam.length();
|
||||
assert!((l - 50.0).abs() < 1e-12);
|
||||
let k = beam.stiffness();
|
||||
for r in rigid_motions(&xyz) {
|
||||
let f = k.mul_vec(&r);
|
||||
assert!(f.iter().all(|v| v.abs() < 1e-6), "{f:?}");
|
||||
}
|
||||
// Clamp node 1: solve the 6 × 6 block of node 2 for a unit force along local z.
|
||||
let axes = beam.axes();
|
||||
let mut k22 = Matrix::zeros(6);
|
||||
for i in 0..6 {
|
||||
for j in 0..6 {
|
||||
k22.add(i, j, k.get(6 + i, 6 + j));
|
||||
}
|
||||
}
|
||||
let mut f = vec![0.0; 6];
|
||||
f[..3].copy_from_slice(&axes[2]);
|
||||
crate::dense::Cholesky::new(&k22).unwrap().solve(&mut f);
|
||||
let p = section.properties;
|
||||
let want = l.powi(3) / (3.0 * section.young * p.iy) + l / (section.shear_modulus * p.shear_z);
|
||||
let got = dot([f[0], f[1], f[2]], axes[2]);
|
||||
assert!((got - want).abs() < 1e-12 * want, "{got} vs {want}");
|
||||
// Section forces at the clamp: V_z = 1, M_y = −L (the moment about y of a +z load at +x).
|
||||
let mut u = vec![0.0; 12];
|
||||
u[6..].copy_from_slice(&f);
|
||||
let forces = beam.forces(&u, [0.0; 3]).unwrap();
|
||||
assert!((forces.ends[0][2] - 1.0).abs() < 1e-9 && (forces.ends[0][4] + l).abs() < 1e-8, "{forces:?}");
|
||||
assert!((forces.ends[1][2] - 1.0).abs() < 1e-9 && forces.ends[1][4].abs() < 1e-8, "{forces:?}");
|
||||
// Mass: the translational entries sum to the mass along each direction.
|
||||
let m = beam.mass(7.8e-9);
|
||||
let total = 7.8e-9 * p.area * l;
|
||||
for d in 0..3 {
|
||||
let s: f64 = [d, 6 + d].iter().flat_map(|&i| [d, 6 + d].map(|j| m.get(i, j))).sum();
|
||||
assert!((s - total).abs() < 1e-12 * total, "{s} {total}");
|
||||
}
|
||||
let lumped = beam.lumped_mass(7.8e-9);
|
||||
assert!((lumped.get(0, 0) - total / 2.0).abs() < 1e-12 * total);
|
||||
assert!(Beam::new(&xyz, [40.0, 30.0, 0.0], section).is_err(), "orientation along the axis");
|
||||
assert!(Beam::new(&[xyz[0], xyz[0]], [0.0, 0.0, 1.0], section).is_err(), "zero length");
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn profile_properties() {
|
||||
// A square: J = 0.1406 a⁴ (Timoshenko & Goodier's table, 0.1406); thin strip J → b t³/3.
|
||||
let sq = properties(&BeamProfile::Rectangle { width: 2.0, height: 2.0 }, 0.3);
|
||||
assert!((sq.torsion / 16.0 - 0.1406).abs() < 1e-4, "{}", sq.torsion / 16.0);
|
||||
let strip = properties(&BeamProfile::Rectangle { width: 1000.0, height: 1.0 }, 0.3);
|
||||
assert!((strip.torsion / (1000.0 / 3.0) - 1.0).abs() < 1e-3, "{}", strip.torsion);
|
||||
// A thin pipe: the shear coefficient tends to 2(1 + ν)/(4 + 3ν), J = 2 π r³ t.
|
||||
let (r, t) = (100.0, 0.01);
|
||||
let pipe = properties(&BeamProfile::Pipe { radius: r, thickness: t }, 0.3);
|
||||
let k = pipe.shear_y / pipe.area;
|
||||
assert!((k - 2.6 / 4.9).abs() < 1e-3, "{k}");
|
||||
assert!((pipe.torsion / (2.0 * std::f64::consts::PI * r.powi(3) * t) - 1.0).abs() < 1e-3);
|
||||
// A box is a pair of I sections in bending: the same I_y with webs of twice the thickness.
|
||||
let b = properties(&BeamProfile::Box { width: 10.0, height: 20.0, thickness: 1.0 }, 0.3);
|
||||
let i = properties(&BeamProfile::I { height: 20.0, width: 10.0, web: 2.0, flange: 1.0 }, 0.3);
|
||||
assert!((b.iy - i.iy).abs() < 1e-9 && (b.area - i.area).abs() < 1e-12);
|
||||
let circle = properties(&BeamProfile::Circle { radius: 2.0 }, 0.3);
|
||||
assert!((circle.torsion - std::f64::consts::PI * 8.0).abs() < 1e-12);
|
||||
}
|
||||
}
|
||||
Reference in new issue
Block a user