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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 L2 / 8 I (1)

where

σ = maximum stress (Pa (N/m2), N/mm2, psi)

y = Perpendicular distance from to neutral axis X (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 (lbf/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)

Metric Units

q - Uniform load (N/mm)

L - Length of Beam (mm)

I - Moment of Inertia (mm4)

► Wide Flange Beam ► Steel Beam Design ► Stress Analysis ► Steel Span

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E - Modulus of Elasticity (N/mm2))

y - Perpendicular distance from to 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)

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 X (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)

200000

150

Calculate!

100

100

285

29000000

6.25

Calculate!



E = modulus of elasticity (Pa (N/m2), N/mm2, 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) (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 � crane girders : span/600

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Related Topics

� Beams and Columns - Deflection and stress, moment of inertia, section modulus and technical information of beams and columns � Mechanics - Kinematics, forces, vectors, motion, momentum, energy and the dynamics of objects

Related Documents

� Aluminum I-Beams - Dimensions and static properties of aluminum I-beams - Imperial units � American Standard Beams - S Beam - American Standard Beams ASTM A6 - Imperial units � American Standard Steel Channels - Dimensions and static parameters of American Standard Steel Channels � American Wide Flange Beams - American Wide Flange Beams ASTM A6 in metric units � American Wide Flange Beams - W Beam - American Wide Flange Beams ASTM A6 in English units � Area Moment of Inertia - Second Moment of Inertia or Area Moment of Inertia � Area Moment of Inertia Converter - Conversion of Area Moment of Inertia � Beam Support Force Calculator - Calculate beams loads and support forces � British Universal Columns and Beams - Properties of British Universal Steel Columns and Beams � Cantilever Beams - Maximum reaction, deflection and moment - single and uniform loads � HE-A Steel Beams - Properties of HE-A profile steel beams � HE-B Steel Beams - Properties of HE-B profile steel beams � HE-M Steel Beams - Properties of HE-M profile steel beams

30000

5000

78125000

200000

150

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10000

100

285

29000000

6.25

Calculate!

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� Moment of Section Conversions - Convert Moment of Section (Inertia) units � Normal Flange I Beams - Properties of normal flange I profile steel beams � Section Modulus Conversions - Convert Section Modulus � Steel Angles - Dimensions and static parameters of steel angles with equal legs - imperial units � Steel Angles with Unequal Legs - Dimensions and static parameters of steel angles with unequal legs - imperial units � Steel Angles with Unequal Legs - Dimensions and static parameters of steel angles with unequal legs - metric units � W Steel Beams - Allowable Loads - Allowable uniform loads

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