Beam Sandbox

About Beam Sandbox — Interactive Deflection & Stress Visualizer

Textbook beam tables tell you the answer; this sandbox shows you why. Drag a point load along a simply supported beam and the elastic deflection curve follows in real time — watch the maximum deflection drift away from the load as it moves off-center, see the bending-moment triangle peak right under the load, and add a uniform load to see how superposition reshapes the curve. Drag the roller support to stretch or shrink the span and feel the L³ and L⁴ terms bite.

The math is the classic pair of closed-form cases from the AISC Steel Construction Manual Table 3-23 — "concentrated load at any point" and "uniformly distributed load" — superposed per the linear-elastic superposition principle, with the same forms appearing in Roark's Formulas for Stress and Strain. Reactions and the shear and moment diagrams reuse the statics solver behind the beam reactions calculator, and bending stress is σ = M/S from your section modulus input.

This tool is deliberately an educational visualization, not a design tool. It shows service-level elastic response for one idealized load case. Real beam design requires factored load combinations, lateral-torsional buckling, shear, bearing, and serviceability checks under the governing code — use the beam deflection and steel member tools for code-based checks, and a licensed engineer for design.

How It Works

1. Drag the red load arrow to move the point load P to any position a along the span — or type a in its field. Every draggable has a mirroring numeric field for accessibility and precision.
2. Drag the right roller support to resize the span L (the load position re-clamps automatically), and set the optional uniform load w, modulus E, moment of inertia I, and section modulus S.
3. The deflection curve is sampled from the superposed Table 3-23 closed forms each frame; its amplitude is exaggerated for visibility, and the marker pins the true maximum-deflection location found from the closed form (single load) or a search on the superposed curve.
4. Read the results panel: reactions, maximum deflection with its location and span ratio, maximum moment with its location, and σ = M/S. There is deliberately no pass/fail verdict — adequacy depends on code checks this tool does not perform.

Worked Example

A 50 kN point load sits at a = 2.4 m on a 6 m simply supported span (b = 3.6 m), with E = 200 GPa, I = 8,000 cm⁴, S = 500 cm³, and no uniform load. Reactions: R₁ = Pb/L = 30 kN, R₂ = Pa/L = 20 kN. Maximum moment: M = Pab/L = 50 × 2.4 × 3.6 / 6 = 72 kN·m at the load. Bending stress: σ = M/S = 72×10³ / 500×10⁻⁶ m³ = 144 MPa. Maximum deflection (Table 3-23, load left of midspan, so the mirrored closed form applies): Δ = Pa(L² − a²)^{3/2} / (9√3·L·EI) = 13.33 mm at x ≈ 2.82 m — note it is NOT at the load point, and the span ratio is L/450.

Formulas

Concentrated load at any point (AISC Table 3-23)

y(x ≤ a) = P·b·x·(L² − b² − x²) / (6·L·E·I), Δ_max = P·b·(L² − b²)^{3/2} / (9√3·L·E·I) at x = √((L² − b²)/3) for a ≥ b

Uniformly distributed load (AISC Table 3-23)

y(x) = w·x·(L³ − 2·L·x² + x³) / (24·E·I), Δ_max = 5·w·L⁴ / (384·E·I), M_max = w·L²/8

Superposition and bending stress

y_total(x) = y_point(x) + y_UDL(x); M_max point = P·a·b/L; σ = M/S = M·c/I

Standards & References

• AISC Steel Construction Manual, Table 3-23 "Shears, Moments and Deflections" — simple beam: concentrated load at any point; uniformly distributed load
• Roark's Formulas for Stress and Strain — simply supported span, intermediate concentrated load and uniform load (identical closed forms; used as the cross-check source)
• Linear-elastic superposition principle (both references) — the combined curve is the sum of the two cases

Frequently Asked Questions