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Beam Deflection — Simply Supported & Cantilever

Maximum deflection, slope, end reactions, and maximum moment for elastic beams under common loadings. Simply supported and cantilever boundaries; point load (mid-span or at free end), uniform distributed load, and superposition of both. Bending stress at extreme fiber.

in
lb (mid-span SS, at end for cant.)
lb/in
psi (steel: 29×10⁶; concrete: 4×10⁶)
in⁴
in (= h/2 for symmetric section)
—in
—— (≥ 360 for live load typical)
—in·lb
—psi

Defaults: 10-ft (120 in) steel beam, 500 lb mid-span point load + 10 lb/in UDL, I = 100 in⁴ (≈ W6×15). Stress check is gross-section only — does not include shear, local buckling, or LTB.

Simply supported, point at mid-span:
$$ \delta_P = \frac{P L^3}{48 E I}, \quad M_{max} = \frac{P L}{4} $$
Simply supported, UDL:
$$ \delta_w = \frac{5 w L^4}{384 E I}, \quad M_{max} = \frac{w L^2}{8} $$
Cantilever, point at free end:
$$ \delta_P = \frac{P L^3}{3 E I}, \quad M_{max} = P L \text{ (at fixed end)} $$
Cantilever, UDL:
$$ \delta_w = \frac{w L^4}{8 E I}, \quad M_{max} = \frac{w L^2}{2} \text{ (at fixed end)} $$
Bending stress at extreme fiber:
$$ \sigma = \frac{M c}{I} $$
P point load · w uniform distributed load · L span · E modulus of elasticity · I moment of inertia · c distance from neutral axis to extreme fiber · M internal moment.

Deflection limits in design codes

Total deflection includes both elastic deflection from this calculator and creep (concrete) or long-term effects. For prestressed concrete, camber is part of the design — the precaster deflects the beam upward at fabrication so the finished structure is flat under dead load.

Superposition is your friend

This calculator superposes a single point load at mid-span (or end) with a single UDL. For more complex loading (multiple point loads, off-center point loads, partial UDL, varying distributed loads), use a finite-element solver or apply superposition manually with the AISC Steel Construction Manual Beam Diagram tables.

The principle: each loading produces an independent deflection curve in elastic regime; total deflection at any point = sum of individual deflections. Linear superposition fails when the load level approaches the section's ultimate capacity (yielding, plastic hinge, P-Δ effects).

Why c < h/2 sometimes

For symmetric sections (rectangles, circles, I-beams about the strong axis), c = h/2 — same distance from the centroid to top and bottom fiber. For asymmetric sections (channels, tee sections, composite construction), the centroid is not at mid-depth and c is larger on one side. The critical fiber is the one with the larger c, governing maximum stress. For composite steel-concrete, transformed section analysis is required.

This is gross-section, elastic only

The Mc/I stress is the gross-section elastic prediction. It does NOT account for:

For complete steel beam design, use AISC 360 §F (flexure) and §G (shear). For concrete, ACI 318 §22.

Reference: AISC Steel Construction Manual, 16th ed. (2023). Hibbeler, R.C. (2014). Mechanics of Materials, 9th ed., Pearson, ch. 8-10.

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