Beam Deflection Calculator gives instant maximum deflection, shear, and bending moment for common beam cases using closed-form elastic solutions. Switch between Metric and Imperial units.
About span L. Span is the clear length between supports for simply supported beams, or the free length from the fixed face for cantilevers. Units: m (Metric) or ft (Imperial).
About elastic modulus E. E quantifies material stiffness (e.g., steel ≈ 200–210 GPa, concrete ≈ 25–35 GPa, aluminum ≈ 69–72 GPa). Units: GPa (Metric) or ksi (Imperial).
About second moment of area I. Also called the area moment of inertia; it captures cross-section stiffness against bending. Units: cm4 (Metric) or in4 (Imperial). Use a section tool if you don’t have I.
About loads. Uniform load w acts continuously along the span; point load P acts at a discrete location (midspan or free end in the provided cases). Units: kN/m and kN (Metric) or k/ft and k (Imperial).
Conversion / Calculation. The calculator evaluates classic Euler–Bernoulli beam solutions for prismatic members with small deflections. For each selected case it computes: (1) maximum deflection Δmax at the critical section, (2) maximum bending moment Mmax, and (3) maximum shear Vmax, using the formulas below with your L, E, I and load(s).
Formulas:
- Simply supported, uniform load w:
- Δmax = (5 w L4) / (384 E I) (deflection)
- Mmax = w L2 / 8 (maximum bending moment)
- Vmax = w L / 2 (maximum shear)
- Simply supported, midspan point load P:
- Δmax = (P L3) / (48 E I) (deflection)
- Mmax = P L / 4 (maximum bending moment)
- Vmax = P / 2 (maximum shear)
- Cantilever, uniform load w:
- Δtip = (w L4) / (8 E I) (tip deflection)
- Mmax = w L2 / 2 (maximum bending moment at the fixed end)
- Vmax = w L (maximum shear at the fixed end)
- Cantilever, end point load P:
- Δtip = (P L3) / (3 E I) (tip deflection)
- Mmax = P L (maximum bending moment at the fixed end)
- Vmax = P (maximum shear at the fixed end)
Examples:
- Simply supported, uniform load. L = 6 m, E = 200 GPa, I = 8 500 cm4, w = 10 kN/m → Δmax ≈ 9.93 mm; Mmax ≈ 45.00 kN·m; Vmax ≈ 30.00 kN.
- Simply supported, midspan point. L = 6 m, E = 200 GPa, I = 8 500 cm4, P = 20 kN → Δmax ≈ 5.29 mm; Mmax ≈ 30.00 kN·m; Vmax ≈ 10.00 kN.
- Cantilever, uniform load. L = 3 m, E = 200 GPa, I = 8 500 cm4, w = 8 kN/m → Δtip ≈ 4.77 mm; Mmax ≈ 36.00 kN·m; Vmax ≈ 24.00 kN.
Corresponding tools. Section Properties (to get I), Bending Stress Calculator (σ = M·c/I), Beam Reactions (support forces), Steel Weight Calculator (self-weight as uniform load), and Slope/Rotation tools for detailed beam line-shape analysis.
