# Beams - Supported at Both Ends - Continuous and Point Loads

The stress in a bending beam can be expressed as

* σ = y M / I (1) *

* where *

* σ = stress (Pa (N/m ^{ 2 } ), N/mm ^{ 2 } , psi) *

* y = distance to point from neutral axis (m, mm, in) *

* M = bending moment (Nm, lb in) *

* I = moment of Inertia (m ^{ 4 } , mm ^{ 4 } , in ^{ 4 } ) *

- Beams - Supported at Both Ends - Continuous and Point Loads
- Beams - Fixed at One End and Supported at the Other - Continuous and Point Loads
- Beams - Fixed at Both Ends - Continuous and Point Loads

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 Continuous Distributed Load

The moment in a beam with uniform load supported at both ends in position x can be expressed as

* M _{ x } = q x (L - x) / 2 (2) *

* where *

* M _{ x } = moment in position x (Nm, lb in) *

* x = distance from end (m, mm, in) *

The maximum ** moment ** is at the center of the beam at distance * L/2 * and can be expressed as

* M _{ max } = q L ^{ 2 } / 8 (2a) *

* where *

* M _{ max } = maximum moment (Nm, lb in) *

* q = uniform load per length unit of beam (N/m, N/mm, lb/in) *

* L = length of beam (m, mm, in) *

#### Maximum Stress

Equation 1 and 2a can be combined to express maximum ** stress ** in a beam with uniform load supported at both ends at distance L/2 as

* σ _{ max } = y _{ max } q L ^{ 2 } / (8 I) (2b) *

* where *

* σ _{ max } = maximum stress (Pa (N/m ^{ 2 } ), N/mm ^{ 2 } , psi) *

* y _{ max } = distance to extreme point from neutral axis (m, mm, in) *

*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 ** :

* δ _{ max } = 5 q L ^{ 4 } / (384 E I) (2c) *

* where *

* δ _{ max } = maximum deflection (m, mm, in) *

* E = Modulus of Elasticity (Pa (N/m ^{ 2 } ), N/mm ^{ 2 } , psi) *

Deflection in position x:

* δ _{ x } = q x (L ^{ 3 } - 2 L x ^{ 2 } + x ^{ 3 } ) / (24 E I) (2d) *

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

Forces acting on the ends:

* R _{ 1 } = R _{ 2 } *

* = q L / 2 (2e) *

* where *

* R = reaction force (N, lb) *

#### Example - Beam with Uniform Load, Metric Units

A UB 305 x 127 x 42 beam with length * 5000 mm * carries a uniform load of * 6 N/mm * . The moment of inertia for the beam is * 8196 cm ^{ 4 } (81960000 mm ^{ 4 } ) * and the modulus of elasticity for the steel used in the beam is

*200 GPa (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 _{ max } = (150 mm) (6 N/mm) (5000 mm) ^{ 2 } / (8 (81960000 mm ^{ 4 } )) *

* = 34.3 N/mm ^{ 2 } *

* = 34.3 10 ^{ 6 } N/m ^{ 2 } (Pa) *

* = 34.3 MPa *

The maximum deflection in the beam can be calculated

* δ _{ max } = 5 (6 N/mm) (5000 mm) ^{ 4 } / ((200000 N/mm ^{ 2 } ) (81960000 mm ^{ 4 } ) 384) *

* = 2.98 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, Imperial 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 _{ max } = y _{ max } 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_{ max }= 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 ** moment ** in a beam with center load supported at both ends:

* M _{ max } = F L / 4 (3a) *

#### Maximum Stress

Maximum ** stress ** in a beam with single center load supported at both ends:

* σ _{ max } = y _{ max } F L / (4 I) (3b) *

* where *

* F = load (N, lb) *

Maximum ** deflection ** can be expressed as

* δ _{ max } = F L ^{ 3 } / (48 E I) (3c) *

Forces acting on the ends:

* R _{ 1 } = R _{ 2 } *

* = F / 2 (3d) *

#### 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 _{ max } = y _{ max } 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 _{ max } = F L ^{ 3 } / E I 48 *

* = (10000 lb) (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

### Beam Supported at Both Ends - Eccentric Load

Maximum ** moment ** in a beam with single eccentric load at point of load:

* M _{ max } = F a b / L (4a) *

#### Maximum Stress

Maximum ** stress ** in a beam with single center load supported at both ends:

* σ _{ max } = y _{ max } F a b / (L I) (4b) *

Maximum ** deflection ** at point of load can be expressed as

* δ _{ F } = F a ^{ 2 } b ^{ 2 } / (3 E I L) (4c) *

Forces acting on the ends:

* R _{ 1 } = F b / L (4d) *

* R _{ 2 } = F a / L (4e) *

### Beam Supported at Both Ends - Two Eccentric Loads

Maximum ** moment ** (between loads) in a beam with two eccentric loads:

* M _{ max } = F a (5a) *

#### Maximum Stress

Maximum ** stress ** in a beam with two eccentric loads supported at both ends:

* σ _{ max } = y _{ max } F a / I (5b) *

Maximum ** deflection ** at point of load can be expressed as

* δ _{ F } = F a (3L ^{ 2 } - 4 a ^{ 2 } ) / (24 E I) (5c) *

Forces acting on the ends:

* R _{ 1 } = R _{ 2 } *

* = F (5d) *

Insert beams to your Sketchup model with the Engineering ToolBox Sketchup Extension

### Beam Supported at Both Ends - Three Point Loads

Maximum ** moment ** (between loads) in a beam with three point loads:

* M _{ max } = F L / 2 (6a) *

#### Maximum Stress

Maximum ** stress ** in a beam with three point loads supported at both ends:

* σ _{ max } = y _{ max } F L / (2 I) (6b) *

Maximum ** deflection ** at the center of the beam can be expressed as

* δ _{ F } = F L ^{ 3 } / (20.22 E I) (6c) *

Forces acting on the ends:

* R _{ 1 } = R _{ 2 } *

* = 1.5 F (6d) *

## Related Topics

### • Beams and Columns

Deflection and stress, moment of inertia, section modulus and technical information of beams and columns.

### • Mechanics

Forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more.

### • Statics

Loads - forces and torque, beams and columns.

## 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 C Channels

Dimensions and static parameters of American Standard Steel C Channels

### American Wide Flange Beams

American Wide Flange Beams ASTM A6 in metric units.

### Area Moment of Inertia - Typical Cross Sections I

Typical cross sections and their Area Moment of Inertia.

### Area Moment of Inertia - Typical Cross Sections II

Area Moment of Inertia, Moment of Inertia for an Area or Second Moment of Area for typical cross section profiles.

### Area Moment of Inertia Converter

Convert between Area Moment of Inertia units.

### Beam Loads - Support Force Calculator

Calculate beam load and supporting forces.

### Beams - Fixed at Both Ends - Continuous and Point Loads

Stress, deflections and supporting loads.

### Beams - Fixed at One End and Supported at the Other - Continuous and Point Loads

Supporting loads, moments and deflections.

### Beams Natural Vibration Frequency

Estimate structures natural vibration frequency.

### British Universal Columns and Beams

Properties of British Universal Steel Columns and Beams.

### Cantilever Beams - Moments and Deflections

Maximum reaction forces, deflections and moments - single and uniform loads.

### Continuous Beams - Moment and Reaction Support Forces

Moments and reaction support forces with distributed or point loads.

### Drawbridge - Force and Moment vs. Elevation

Calculate the acting forces and moments when elevating drawbridges or beams.

### Floor Joist Capacities

Carrying capacities of domestic timber floor joists - Grade C - metric units.

### Floors - Live Loads

Floors and minimum uniformly distributed live loads.

### HE-A Steel Beams

Properties of HE-A profiled steel beams.

### HE-B Steel Beams

Properties of HE-B profiled steel beams.

### HE-M Steel Beams

Properties of HE-M profile steel beams.

### Normal Flange I-Beams

Properties of normal flange I profile steel beams.

### Section Modulus - Unit Converter

Convert between Elastic Section Modulus units.

### Square Hollow Structural Sections - HSS

Weight, cross sectional area, moments of inertia - Imperial units

### Steel Angles - Equal Legs

Dimensions and static parameters of steel angles with equal legs - imperial units.

### Steel Angles - Equal Legs

Dimensions and static parameters of steel angles with equal legs - metric units.

### Steel Angles - Unequal Legs

Dimensions and static parameters of steel angles with unequal legs - imperial units.

### Steel Angles - Unequal Legs

Dimensions and static parameters of steel angles with unequal legs - metric units.

### Stiffness

Stiffness is resistance to deflection.

### Stress

Stress is force applied on cross-sectional area.

### Three-Hinged Arches - Continuous and Point Loads

Support reactions and bending moments.

### Trusses

Common types of trusses.

### Typical Floor Loads

Uniformly and concentrated floor loads

### W Steel Beams - Allowable Uniform Loads

Allowable uniform loads.

### W-Beams - American Wide Flange Beams

Dimensions of American Wide Flange Beams ASTM A6 (or W-Beams) - Imperial units.

### Weight of Beams - Stress and Strain

Stress and deformation of vertical beams due to own weight.

### Wood Headers - Max. Supported Weight

Weight supported by a double or triple wood headers.