Stress and Deflections of Beams

The calculator below can be used to calculate maximum stress and deflection of beams with one single or uniform distributed 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), N/mm², psi)
y = Distance of extreme point off neutral axis (m, mm, in)
q = uniform load per length unit (N/m, N/mm, lb/in)
L = length of beam (m, mm, in)
I = moment of Inertia (m⁴, mm⁴, 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, mm, in)
E = modulus of elasticity (Pa (N/m²), N/mm², psi)

Note! - deflection is often the limit factor in beam design. For some applications beams must be stronger than required by maximum loads, to avoid unacceptable deflections.

Example - Beam with Uniform Load, Metric Units

A beam with length 5000 mm carries a uniform load of 6 N/mm². The moment of inertia for the beam is 78125000 mm⁴ and the modulus of elasticity for the steel used in the beam is 200000 N/mm². The height of the beam is 300 mm (the distance of the extreme point to the neutral axis is 150 mm).
The maximum stress in the beam can be calculated

σ = (150 mm) (6 N/mm²) (5000 mm)² / (8 (78125000 mm⁴))

  = 36 N/mm²

  = 36000000 N/m² (Pa)

The maximum deflection in the beam can be calculated

δ = 5 (6 N/mm²) (5000 mm)⁴ / ((200000 N/mm²) (78125000 mm⁴) 384)

   = 3.13 mm

Uniform Load Beam Calculator - Metric Units

q - Uniform load (N/mm)
L - Length of Beam (mm)
I - Moment of Inertia (mm⁴)
E - Modulus of Elasticity (N/mm²)
y - Distance of extreme point off neutral axis (mm)

• 1 mm⁴ = 10^-4 cm⁴ = 10^-12 m⁴
• 1 cm⁴ = 10^-8 m = 10⁴ mm
• 1 in⁴ = 4.16x10⁵ mm⁴ = 41.6 cm⁴
• 1 N/mm² = 10⁶ N/m² (Pa)

Uniform Load Beam Calculator - Imperial Units

q - Load (lb/in)
L - Length of Beam (in)
I - Moment of Inertia (in⁴)
E - Modulus of Elasticity (psi)
y - Distance of extreme point off 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 as

σ = y q L² / (8 I)
    = (6.25 in) (100 lb/in) (100 in)² / (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) (100 in)⁴ / ((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)         (3)
where
σ = maximum stress (Pa (N/m²), N/mm², psi)
y = Perpendicular distance from to neutral axis (m, mm, in)
F = load (N, lb)
L = length of beam (m, mm, in)
I = moment of Inertia (m⁴,mm⁴, in⁴)

Maximum deflection can be expressed as

δ = F L³ / (E I 48)         (4)
where
δ = maximum deflection (m, mm, in)
E = modulus of elasticity (Pa (N/m²), N/mm², psi)

Single Center Load Beam Calculator - Metric Units

F - Load (N)
L - Length of Beam (mm)
I - Moment of Inertia (mm⁴)
E - Modulus of Elasticity (N/mm²)
y - Distance of extreme point off neutral axis (mm)

Single Center Load Beam Calculator - Imperial Units

F - Load (lb)
L - Length of Beam (in)
I - Moment of Inertia (in⁴)
E - Modulus of Elasticity (psi)
y - Distance of extreme point off 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) (100 in)³ / ((29000000 lb/in²) (285 in⁴) 48)
    = 0.025 in

Some Typical Vertical Deflection Limits

• total deflection : span/250
• live load deflection : span/360
• cantilevers : span/180
• domestic timber floor joists : span/330 (max 14 mm)
• brittle elements : span/500
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