# Stress and Deflections in Beams

## Beams and shafts - deflection and stress calculator

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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 L^{2}/ (8 I) (1)

where

σ = maximum stress (Pa (N/m^{2}), N/mm^{2}, 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 (m^{4}, mm^{4}, in^{4})

*1 N/m*^{2}= 1x10^{-6}N/mm^{2}= 1 Pa = 1.4504x10^{-4}psi*1 psi (lb/in*^{2}) = 144 psf (lb_{f}/ft^{2}) = 6,894.8 Pa (N/m^{2}) = 6.895x10^{-3}N/mm^{2}

Maximum **deflection** can be expressed as

δ = 5 q L^{4}/ (E I 384) (2)

where

δ = maximum deflection (m, mm, in)

E = modulus of elasticity (Pa (N/m^{2}), N/mm^{2}, 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/mm ^{2}*. The moment of inertia for the beam is

*78125000 mm*and the modulus of elasticity for the steel used in the beam is

^{4}*200000 N/mm*. The height of the beam is

^{2}*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/mm ^{2}) (5000 mm)^{2} / (8 (78125000 mm^{4})) *

* = 36 N/mm ^{2}*

* = 36000000 N/m ^{2} (Pa)*

The maximum deflection in the beam can be calculated* *

*δ = 5 (6 N/mm^{2}) (5000 mm)^{4} / ((200000 N/mm^{2}) (78125000 mm^{4}) 384)*

* = 3.13 mm*

#### Uniform Load Beam Calculator - Metric Units

*1 mm*^{4}= 10^{-4}cm^{4}= 10^{-12 }m^{4}*1 cm*^{4}= 10^{-8}m = 10^{4}mm*1 in*^{4}= 4.16x10^{5}mm^{4}= 41.6 cm^{4}*1 N/mm*^{2}= 10^{6}N/m^{2}(Pa)

#### Uniform Load Beam Calculator - Imperial Units

#### 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 in ^{4}*, modulus of elasticity

*29000000 psi*, with uniform load

*100 lb/in*can be calculated as

σ = y q L^{2}/ (8 I)

= (6.25 in) (100 lb/in) (100 in)^{2}/ (8 (285 in^{4}))

= 2741 (lb/in^{2}, psi)

The maximum deflection can be calculated as

δ = 5 q L^{4}/ (E I 384)

= 5 (100 lb/in) (100 in)^{4}/ ((29000000 lb/in^{2}) (285 in^{4}) 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/m^{2}), N/mm^{2}, 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 (m^{4},mm^{4}, in^{4})

Maximum **deflection** can be expressed as

δ = F L^{3}/ (E I 48) (4)

where

δ = maximum deflection (m, mm, in)

E = modulus of elasticity (Pa (N/m^{2}), N/mm^{2}, psi)

#### Single Center Load Beam Calculator - Metric Units

#### Single Center Load Beam Calculator - Imperial Units

#### 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 in ^{4}*, 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 in^{4}))

= 5482 (lb/in^{2}, psi)

The maximum deflection can be calculated as

δ = F L^{3}/ E I 48

= (10000 lb/in) (100 in)^{3}/ ((29000000 lb/in^{2}) (285 in^{4}) 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
- crane girders : span/600

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