Beam Deflection Calculator
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Maximum deflection, δmax
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Location of maximum, x
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Flexural rigidity used, EI
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Step-by-step derivation
Reference formulas & sources
| Case | Formula for δmax | Where | Source |
|---|---|---|---|
| SS, midspan point P | δ = P L³ / (48 EI) | center | EngineeringToolbox |
| SS, point P at any a | wmax = (√3 P b (L²−b²)^{3/2}) / (27 L E I), b=L−a | x = √((L²−b²)/3) | Wikipedia table (asymmetric load) |
| SS, uniform w | δ = 5 w L⁴ / (384 EI) | center | EngineeringToolbox |
| SS, triangular (0→w₀ or w₀→0) | δ ≈ 0.00652 w₀ L⁴ / (EI) | x ≈ 0.519L (0→w₀), 0.481L (w₀→0) | Iowa State PDF |
| SS, end moment M (at support) | δ = M L² / (9√3 EI) | x = L/√3 | Iowa State PDF |
| Cantilever, tip point P | δ = P L³ / (3 EI) | tip | EngineeringToolbox |
| Cantilever, point P at a | δ = P a² (3L−a) / (6 EI) | tip | Iowa State PDF |
| Cantilever, uniform w | δ = w L⁴ / (8 EI) | tip | EngineeringToolbox |
| Cantilever, triangular 0→w₀ (max at free end) | δ = (11/120) w₀ L⁴ / (EI) | tip | Handbook of Civil Engineering Calculations |
| Cantilever, triangular w₀→0 (max at fixed) | δ = w₀ L⁴ / (30 EI) | tip | EngineeringToolbox |
Notes: formulas assume small deflection Euler–Bernoulli theory, prismatic members, linear elasticity.