Draws the real deflected shape of a simply-supported or cantilever beam under a point or uniform load, computed from the actual beam deflection equation — not a straight-line schematic — with a live max-deflection readout.
This simulator computes and animates the true deflected shape of a beam under load, using the actual closed-form beam deflection equation for each of four standard cases: simply-supported or cantilever support, with either a point load or a uniformly distributed load. The curve you see is the real elastica shape from Euler-Bernoulli beam theory, sampled at 60 points along the span and scaled for visibility — not a simplified straight-line sketch.
Simply-supported beam, center point load P: y(x) = Px/(48EI)·(3L² − 4x²) for 0 ≤ x ≤ L/2 (symmetric beyond midspan), with δmax = PL³/(48EI) at midspan. Cantilever beam, end point load P: y(x) = Px²/(6EI)·(3L − x), with δmax = PL³/(3EI) at the free end. Simply-supported beam, uniform load w: y(x) = wx/(24EI)·(L³ − 2Lx² + x³), with δmax = 5wL⁴/(384EI) at midspan. Cantilever beam, uniform load w: y(x) = wx²/(24EI)·(x² − 4Lx + 6L²), with δmax = wL⁴/(8EI) at the free end. All four are derived by double-integrating the moment-curvature relationship EI·y″ = M(x) from Euler-Bernoulli beam theory.
For the same span, load, and section, a cantilever with an end point load deflects PL³/(3EI) — 16 times more than the same beam simply supported with the same point load at midspan (PL³/48EI). This is because a cantilever has no second support to share the bending moment; the full moment arm acts over the entire span from the single fixed end, and the deflection formula's cubic dependence on L makes this effect grow rapidly with span length. This is why cantilevered members (balconies, canopies, diving boards) require much stiffer sections (higher EI) than equivalent simply-supported spans.
Most building codes (IBC, referencing AISC and ACI) limit live-load deflection to L/360 for floor members supporting brittle finishes, and total load deflection to L/240 for roof members without such finishes — these limits control cracking of plaster/drywall and prevent a "bouncy" or visually sagging floor, and are independent of the strength (stress) check, which is calculated separately. A beam can easily pass a strength check yet fail a deflection check, especially for long spans in lighter materials like wood or aluminum, since deflection depends on L⁴ (or L³ for cantilevers) while bending stress depends only on L¹ to L².
For a uniformly loaded, simply-supported beam, δmax = 5wL⁴/(384EI). Doubling the span while holding w, E, and I constant increases deflection by a factor of 2⁴ = 16. This strong length-sensitivity is why deflection, not stress, usually governs the design of long-span floor and roof beams — a beam can be strong enough (adequate bending stress capacity) yet still deflect too much for serviceability.
EI is the flexural rigidity of the beam: E is the material's modulus of elasticity (stiffness) and I is the cross-section's moment of inertia (how efficiently the material is distributed to resist bending). Every standard deflection formula has EI in the denominator, so doubling either the material stiffness or the section's moment of inertia halves the deflection for the same load. Because I depends on the cube of a rectangular section's depth (I = bh³/12), increasing depth is a far more efficient way to reduce deflection than increasing width.
Comparing the two cantilever formulas at the same total load W: a concentrated end load P = W gives δmax = WL³/(3EI), while the same total load spread uniformly (w = W/L) gives δmax = wL⁴/(8EI) = WL³/(8EI) — the point-load case deflects 8/3 ≈ 2.67 times more than the same total load spread uniformly, because a concentrated end load produces the maximum possible moment arm, whereas distributing the same total load along the span reduces the average moment arm.
No — like all Euler-Bernoulli beam theory, these formulas assume small deflections (typically δmax < L/50) and linear-elastic material behavior. For very flexible members, large-deflection (geometrically nonlinear) analysis or elastica theory is required, and if the applied load exceeds the material's yield stress in bending, the section will yield and these purely elastic formulas no longer apply — see the companion Axial Loading Stress-Strain Simulator for material yielding behavior.
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