Stresses and Deflections in Beams

Beam and shaft deflection and stress calculator

The calculator below can be used to calculate maximum stress and deflection of beams with one or uniform loads.

Beam Supported at Both Ends, Uniform Load

Maximum Stress

Maximum stress in a beam with uniform load supported at both ends can be calculated as

σ = y q L² / 8 I (1)

where

σ = maximum stress (Pa (N/m²), psi)

y = Perpendicular distance from to neutral axis (m, in)

q = uniform load (N/m, lb/in)

L = length of beam (m, in)

I = moment of Inertia (m⁴, in⁴)

• 1 N/m² = 1x10^-6 N/mm² = 1 Pa = 1.4504x10^-4 psi
• 1 psi (lb/in²) = 144 psf (lb_f/ft²) = 6,894.8 Pa (N/m²) = 6.895x10^-3 N/mm²

Maximum deflection can be expressed as

δ = 5 q L⁴ / E I 384 (2)

where

δ = maximum deflection (m, in)

E = modulus of elasticity (Pa (N/m²), psi)

Metric Units

q - Uniform load (N/mm)

L - Length of Beam (mm)

I - Moment of Inertia (mm⁴)

E - Modulus of Elasticity (N/mm²))

y - Perpendicular distance from to neutral axis (mm)

• 1 mm⁴ = 1x10^-4 cm⁴ = 1x10^-12 m⁴
• 1 cm⁴ = 1x10^-8 m = 1x10⁴ mm
• 1 in⁴ = 4.16x10⁵ mm⁴ = 41.6 cm⁴
• 1 N/mm² = 1x10⁶ N/m² (Pa)

Imperial Units

q - Load (lb/in)

L - Length of Beam (in)

I - Moment of Inertia (in⁴)

E - Modulus of Elasticity (psi)

y - Perpendicular distance from to neutral axis (in)

Example - Beam with Uniform Load, English Units

The maximum stress in a "W 12 x 35" Steel Wide Flange beam, 100 inches long, moment of inertia 285 in⁴, modulus of elasticity 29000000 psi, with uniform load 100 lb/in can be calculated like

σ = y q L² / 8 I

    = 6.25 (in) 100 (lb/in) 100² (in²) / 8 285 (in⁴)

    = 2741 (lb/in² (psi))

The maximum deflection can be calculated as

δ = 5 q L⁴ / E I 384

    = 5 100 (lb/in) 100⁴ (in⁴) / 29000000 (lb/in²) 285 (in⁴) 384

    = 0.016 in

Beam Supported at Both Ends, Load at Center

Maximum Stress

Maximum stress in a beam with uniform load supported at both ends can be calculated as

σ = y F L / 4 I (1)

where

σ = maximum stress (Pa (N/m²), psi)

y = Perpendicular distance from to neutral axis (m, in)

F = load (N, lb)

L = length of beam (m, in)

I = moment of Inertia (m⁴, in⁴)

Maximum deflection can be expressed as

δ = F L³ / E I 48 (2)

where

δ = maximum deflection (m, in)

E = modulus of elasticity (Pa (N/m²), psi)

Metric Units

F - Load (N)

L - Length of Beam (mm)

I - Moment of Inertia (mm⁴)

E - Modulus of Elasticity (N/mm²)

y - Perpendicular distance from to neutral axis (mm)

Imperial Units

F - Load (lb)

L - Length of Beam (in)

I - Moment of Inertia (in⁴)

E - Modulus of Elasticity (psi)

y - Perpendicular distance from to neutral axis (in)

Example - Beam with a Single Center Load

The maximum stress in a "W 12 x 35" Steel Wide Flange beam, 100 inches long, moment of inertia 285 in⁴, modulus of elasticity 29000000 psi, with a center load 10000 lb can be calculated like

σ = y F L / 4 I

    = 6.25 (in) 10000 (lb) 100 (in) / 4 285 (in⁴)

    = 5482 (lb/in² (psi))

The maximum deflection can be calculated as

δ = F L³ / E I 48

    = 10000 (lb/in) 100³ (in³) / 29000000 (lb/in²) 285 (in⁴) 48

    = 0.025 in

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