Beam Deflection Calculator
The tip deflection of a cantilever with an end load is PL³/3EI: 10 kN on a 2 m steel cantilever with I = 8 × 10⁶ mm⁴ deflects 16.6667 mm.
1 cm⁴ = 10 000 mm⁴; 1 m⁴ = 10¹² mm⁴.
16.6666666666667 mm
Full answer, then cut to fewer decimals (not rounded):
- 16.6666666666667 mm
- 16.666666666666 mm
- 16.66666666666 mm
- 16.6666666666 mm
- 16.666666666 mm
- 16.66666666 mm
- 16.6666666 mm
- 16.666666 mm
- 16.66666 mm
- 16.6666 mm
- 16.666 mm
- 16.66 mm
- 16.6 mm
- Maximum slope θ
- 0.0125 rad
- Maximum slope θ
- 0.716197 °
- Maximum bending moment M
- 20000 N·m
- Maximum shear force V
- 10000 N
- Flexural rigidity EI
- 1600000 N·m²
Step-by-step solution
- δ = PL³ / 3EI
- EI = 200 × 10⁹ Pa × 8000000 × 10⁻¹² m⁴ = 1600000 N·m²
- δ = 0.3333 × 10000 × 2^3 ÷ 1600000 = 0.0166667 m = 16.6667 mm
Beam Deflection Formula
δ = PL³/3EI, wL⁴/8EI, PL³/48EI, 5wL⁴/384EI
P point load, w load per length, L span, E Young’s modulus, I second moment of area
Worked Example
A 2 m steel cantilever (E = 200 GPa, I = 8 × 10⁶ mm⁴) with 10 kN at the free end:
- δ = PL³ / 3EI
- EI = 200 × 10⁹ Pa × 8000000 × 10⁻¹² m⁴ = 1600000 N·m²
- δ = 0.3333 × 10000 × 2^3 ÷ 1600000 = 0.0166667 m = 16.6667 mm
Maximum slope θ: 0.716197 ° · Maximum bending moment M: 20000 N·m · Maximum shear force V: 10000 N · Flexural rigidity EI: 1600000 N·m²
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Example: A 100 × 200 mm rectangular beam carrying a 20 kN·m bending moment: Maximum bending stress σ = 30 MPa.
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Pcr = π²EI / Le²
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Example: A 3 m pinned steel column with I = 8 × 10⁶ mm⁴ and A = 5000 mm²: Critical buckling load Pcr = 1754.6 kN.
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Mohr’s Circle Calculator
σ₁,₂ = (σx + σy)/2 ± √[((σx − σy)/2)² + τxy²]
centre C = (σx + σy)/2, radius R = τmax
Example: σx = 80 MPa, σy = −40 MPa and τxy = 45 MPa: Major principal stress σ₁ = 95 MPa.
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Shaft Torsion Calculator
T/J = τ/r = Gθ/L
J = π(D⁴ − d⁴)/32, r = D/2, G modulus of rigidity
Example: A 50 mm solid steel shaft (G = 80 GPa), 1 m long, carrying 1 kN·m: Maximum shear stress τmax = 40.7437 MPa.
Frequently asked questions
What is the deflection of a simply supported beam with a central load?
δ = PL³/48EI at mid-span, 16 times less than a cantilever of the same span with the load at its tip (PL³/3EI).
How does deflection depend on span?
As L³ for a point load and L⁴ for a uniform load, so doubling the span multiplies the deflection by 8 or 16.
Sources
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
Cite this page
APA
Scientific Converters. (2026, October 15). Beam Deflection Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/beam-deflection-calculator/
IEEE
Scientific Converters, "Beam Deflection Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/beam-deflection-calculator/