Structural engineering: loads, signs and linear elasticity

Elastic formulas use modulus E and G in Pa, lengths in m, areas in m^2, second/polar moments in m^4, force in N and moment/torque in N m. Strain is dimensionless. Isotropic Poisson ratio must satisfy -1 < nu < 0.5. Euler buckling assumes a slender ideal elastic column, with effective length K*L; it does not model yielding, imperfections or post-buckling.

Beam convention

Positive point loads and uniform loads act downward. Reactions act upward, sagging moment is positive, and downward deflection is negative. A load at a support has zero span deflection. Shear at a point load uses the right-side value; plotted endpoints do not include the support reaction jumps. The Euler-Bernoulli relation is E*I*y’’=M with this sign convention. Small slopes, constant E*I, and negligible shear deformation are assumed.

from engineeringkit.structural_engineering import SimplySupportedBeam
result = SimplySupportedBeam(6, 200e9, 1e-5).solve_point_load(12000, 2)
assert result.reactions == (8000, 4000)
fig, axes = result.plot()

The tests check force/moment equilibrium, zero support deflections, numerical curvature versus M/(EI), and the uniform-load midspan result -5*w*L^4/(384*E*I). Axial strain energy U=P^2*L/(2*A*E) is differentiated independently to recover axial displacement.

Beams on any supports

Beam(length, E, I, supports) takes supports as (position, kind) with kind "pin", "roller" (both restrain deflection only) or "fixed" (deflection and slope), anywhere on the beam, so cantilevers, overhangs, propped cantilevers and continuous beams are all one call. solve accepts point loads (x, P) downward positive, couples (x, C) clockwise positive, and distributed loads (start, end, w) or (start, end, w_start, w_end) varying linearly, downward positive. The moment is written with Macaulay brackets and integrated exactly; the reactions and the two constants of integration come from one linear system of equilibrium and support conditions, so indeterminate beams need no extra input. A support layout that allows rigid-body motion raises a ValueError. The result is a BeamResult with one reaction per support, the reaction moments of the fixed supports (clockwise positive) and the slope. Shear at a load takes the value just to its right, except at the right end. superpose_beam_results adds the responses of one beam to several load cases; it requires the same stations and supports.

from engineeringkit.structural_engineering import Beam
beam = Beam(6.0, 200e9, 8e-6, ((0.0, "fixed"), (6.0, "roller")))
result = beam.solve(distributed_loads=[(0.0, 6.0, 1e3)])
result.reactions        # (3750.0, 2250.0): 5wL/8 and 3wL/8
result.support_moments  # (-4500.0,): -wL^2/8

Sections, unsymmetrical bending and shear

Cross-sections use Hibbeler’s and Gere’s axes: y vertical (up), z horizontal, beam axis x. Iz is the second moment about the horizontal centroidal axis, Iy about the vertical one and Iyz the product of inertia. composite_section builds a SectionProperties from rectangles (b, h, z, y[, weight]), with (z, y) the rectangle’s centre; weight -1 removes a hole and a modular ratio n = E/E_ref gives a transformed section, whose properties are in the reference material. The shapes i_section, t_section, channel_section, angle_section and box_section use overall dimensions, ignore fillets and corner radii, and put the bottom of the section at y = 0. SectionProperties.first_moment(y) is Q of the part above a cut, and width_at(y) the true width there (holes subtracted, modular ratios ignored).

unsymmetrical_bending_stress is the generalized flexure formula. Positive Mz compresses the fibres at positive y, so for a horizontal beam with downward loads Mz is the sagging moment; My follows the right-hand rule about y. Loads are assumed to pass through the shear centre. neutral_axis_angle is measured from z, counterclockwise. transverse_shear_stress is Jourawski’s VQ/(It), averaged across the cut; it is accurate for narrow cuts and thin walls, not across wide flanges. channel_shear_center uses centreline dimensions of a thin-walled channel and returns the distance behind the web centreline.

Torsion of thin-walled tubes, pressure vessels and stress at a point

Bredt’s formulas apply to closed thin-walled sections free to warp: thin_walled_torsion_shear_stress takes the area enclosed by the wall’s centreline and the local thickness; thin_walled_torsion_angle takes the integral of ds/t around the centreline. Open sections are not covered. thin_wall_vessel_stresses uses the inner radius and neglects the radial stress (valid for r/t of about 10 or more); thick_wall_cylinder_stresses is Lamé’s elastic solution away from the ends, with the axial stress of a closed cylinder. principal_stresses_3d returns the eigenvalues of the stress tensor in decreasing order; the absolute maximum shear is (sigma_1 - sigma_3)/2. strain_rosette takes three gauge readings at angles counterclockwise from x (rectangular 0-45-90 by default) and returns the engineering shear strain; plane_stress_from_strains is Hooke’s law for plane stress, valid at a free surface.

Column design

effective_length_factor returns theoretical K values; codes recommend larger ones because no connection is perfectly rigid. The secant formula assumes a linear elastic, initially straight column with an eccentric load and returns the peak compressive stress as a positive number; it is undefined at and beyond the Euler load. secant_formula_load solves for first yield by Brent’s method. johnson_critical_stress is the parabola for KL/r below Cc = sqrt(2 pi^2 E / sigma_y) and Euler’s hyperbola above, with no safety factor.

Trusses

Nodes have shape (n,2); member indices start at zero. DOFs are x0,y0,x1,y1,… and fixed DOFs have zero prescribed displacement. Members are pin-jointed, axial-only, and use a shared area and modulus. Member force is positive in tension. truss_determinacy returns Maxwell’s count m + r - 2j; it is necessary but not sufficient for stability. method_of_joints requires a count of zero and solves joint by joint from statics, raising a ValueError if every remaining joint has more than two independent unknowns. PlanarTruss.solve refuses a negative count, raises a ValueError when the reduced stiffness K_ff is singular to working precision, and otherwise returns cond(K_ff) in the result and issues a RuntimeWarning above cond_limit (1e12 by default): a near-mechanism gives a matrix that is invertible in floating point but physically meaningless.

Continuous beams and frames

three_moment_equation assumes simply supported ends, one modulus and a uniform load per span; support moments are sagging positive. influence_line is for a simply supported span and a unit downward load. moment_distribution assumes no joint translation (no sway) and uses Hardy Cross’s sign convention: member-end moments are clockwise positive. PlanarFrame has three DOFs per node (u, v, theta), counterclockwise rotations and moments, Euler-Bernoulli members with axial stiffness, and uniform member loads along the local -y axis converted to fixed-end nodal loads. Its end moments are counterclockwise positive, the negative of Cross’s. slope_deflection takes the same member tuples as moment_distribution and solves the joint equations directly, with clockwise rotations and moments; known chord rotations (support settlements, psi = Delta/L clockwise) can be given per member, but sway is not solved for. The tests check the three-moment equation, moment distribution and the frame solver against one another, the frame against cantilever and fixed-beam formulas, and the truss solvers against each other.

Stress, energy, stability and plasticity

Mohr circle uses (center, radius, sigma_max, sigma_min), all Pa; rotations in stress_transformation are counterclockwise. castigliano_deflection either returns a derivative dU/dP computed by the caller or integrates M (dM/dP)/(EI) along a beam with Simpson’s rule; only bending energy is included. Maxwell-Betti compares two supplied displacements; it does not establish linearity of an arbitrary model. tangent_modulus_load accepts a tangent modulus Et(sigma) that is positive at zero stress and nonincreasing, and solves sigma = pi^2 Et(sigma)/(KL/r)^2 by Brent’s method. plastic_collapse_load is rigid-plastic: no strain hardening, instability, or reduction of Mp by shear or axial force. unit_load_truss_deflection sums n N L/(A E) over the members; in an indeterminate truss the real forces must come from a compatible solution. unit_load_beam_deflection is the keyword form of castigliano_deflection under its virtual-work name.

Cables and arches

Cables are perfectly flexible and inextensible, with supports at the same level and the sag measured at midspan. parabolic_cable takes a load per unit horizontal length (a suspended deck), catenary_cable a weight per unit length of cable; both return the horizontal tension, the tension at the supports and the cable length. three_hinged_arch is a parabolic arch with pinned springings at the same level and a crown hinge at midspan, under a load uniform along the span and point loads, downward positive. Its moment is sagging positive (intrados in tension), its axial force tension positive, and the thrust is reported as a positive number. The shape is not updated under load.