Beams, columns and trusses

The tool draws the beam with its loads, then the shear force, bending moment (sagging positive), slope and deflection along it, with the extreme of each marked. The cards give the largest moment, shear, deflection and bending stress σ = M·c/I, the safety factor on yield (or on the modulus of rupture for wood), and the deflection against the chosen span limit (L/180 … L/2000; a cantilever is judged as a span of twice its length). Shear stress is the peak VQ/(Ib) at the neutral axis.

The solver uses Euler–Bernoulli beam finite elements with nodes at every load and support (exact at the nodes for these loads), so indeterminate cases — fixed ends, propped cantilevers — need no special handling; shear and moment come from statics with the solved reactions. It reproduces the standard results tabulated in Roark’s Formulas for Stress and Strain (Young & Budynas, 8th ed., Table 8.1) and Gere & Goodno’s Mechanics of Materials (Appendix G): 5wL⁴/384EI and wL²/8 for a uniform load on a simple span, PL³/3EI for a loaded cantilever, wL⁴/384EI and wL²/12 at the ends of a fixed-fixed beam, 3wL/8 at the prop of a propped cantilever, wL²/(9√3) at L/√3 for a triangular load — the tests check each of these.

Area, centroid, second moments of area about both axes (and the principal values for angles), section modulus S = I/c top and bottom, radius of gyration, and the first moment Q and width at the neutral axis, for rectangles, round bar, round tube, box tube, I/H beams, channels, angles and tees. On its side turns the section 90°. The sections are idealised with sharp corners: rolled and extruded shapes have fillets and rounded corners, so catalogue values differ by a few percent — for a real part use the catalogue. Weight and cost of stock are in metal weight.

Euler’s critical load P_cr = π²EI/(KL)² for slender columns and Johnson’s parabola σ = f_y(1 − f_y(KL/r)²/(4π²E)) below the transition slenderness C_c = √(2π²E/f_y), drawn as a curve of critical stress against slenderness with your column marked. End conditions use the effective-length factor K — theoretical (1, 0.5, 0.7, 2) or the larger values recommended for design (AISC 360 Commentary, Table C-A-7.1: 1, 0.65, 0.8, 2.1), which allow for imperfect fixity. Buckling uses the section’s least moment of inertia. Real columns are crooked and loaded off-centre, so codes give lower strengths than Euler — keep a generous safety factor.

Draw a pin-jointed plane truss on the canvas, or load a preset (a 45° triangle, Pratt and Howe bridges, a Warren truss, a wall bracket, a Fink roof truss).

The truss is solved by the direct stiffness method — the matrix form of the method of joints, which also handles statically indeterminate trusses; the tool reports whether yours is determinate, and refuses a mechanism. Members are coloured blue in tension and red in compression (orange when the compression exceeds the member’s Euler load), with widths proportional to the force; the table gives each member’s length, force, stress, Euler load (K from the end condition you choose, on the member’s length and least I) and its use as a fraction of capacity times the safety factor. All members share one section and material.

Steels use the grade’s minimum yield (EN 10025-2, ASTM A36, A500, AISI bar data); aluminium 6061-T6 and 6063 from the Aluminum Association’s Aluminum Design Manual minimums. Woods are from the USDA Wood Handbook (FPL-GTR-282, 2021, Table 5-3a): E and the modulus of rupture of small clear specimens at 12% moisture. Those are averages, not design values — graded lumber with knots is far weaker (NDS or Eurocode 5 values are about a quarter to a half of MOR) — and wood creeps under sustained load. Printed plastics use moulded-bar datasheet values; a printed part is weaker, especially across its layers and at low infill (see the note shown with them). Every value can be overridden.

Related: shelf sag, metal weight, bolt torque, springs, welding, print tolerances, physics, levers and moment arms.

Links can be written by hand. Add the tab after #: beam, section, column or truss. Lengths in links are in metres (SI), whatever units are shown.

Presets: