﻿ Beam Deflection Tables | MechaniCalc

The tables below give equations for the deflection, slope, shear, and moment along straight beams for different end conditions and loadings. You can find comprehensive tables in references such as Gere, Lindeburg, and Shigley. However, the tables below cover most of the common cases.

For information on beam deflection, see our reference on stresses and deflections in beams.

### Cantilever Beams

Deflection:

 @ x = L

Slope:

 @ x = L

Shear:

 V = +F

Moment:

 M = −F (L − x)
 Mmax = −FL @ x = 0

Deflection:

 ( 0 ≤ x ≤ a ) ( a ≤ x ≤ L )
 @ x = L

Slope:

 ( 0 ≤ x ≤ a ) ( a ≤ x ≤ L )

Shear:

 V = +F ( 0 ≤ x ≤ a ) V = 0 ( a ≤ x ≤ L )

Moment:

 M = −F (a − x) ( 0 ≤ x ≤ a ) M = 0 ( a ≤ x ≤ L )
Cantilever, Uniform Distributed Load

Deflection:

 @ x = L

Slope:

 @ x = L

Shear:

 V = +w (L − x)
 Vmax = +wL @ x = 0

Moment:

 M = −w (L − x)2 / 2
 Mmax = −wL2 / 2 @ x = 0
Cantilever, Triangular Distributed Load

Deflection:

 @ x = L

Slope:

 @ x = L

Shear:

 Vmax = +w1L / 2 @ x = 0

Moment:

 Mmax = −w1L2 / 6 @ x = 0
Cantilever, End Moment

Deflection:

 @ x = L

Slope:

 @ x = L

Shear:

 V = 0

Moment:

 M = −M0

### Simply Supported Beams

Simply Supported, Intermediate Load

Deflection:

 ( 0 ≤ x ≤ a )

For a ≥ b:

 @

Slope:

 ( 0 ≤ x ≤ a ) @ x = 0 @ x = L

Shear:

 V1 = +Fb / L ( 0 ≤ x ≤ a ) V2 = −Fa / L ( a ≤ x ≤ L )

Moment:

 Mmax = +Fab / L @ x = a
Simply Supported, Center Load

Deflection:

 ( 0 ≤ x ≤ L/2 ) @ x = L/2

Slope:

 ( 0 ≤ x ≤ L/2 ) @ x = 0 @ x = L

Shear:

 V1 = +F / 2 ( 0 ≤ x ≤ L/2 ) V2 = −F / 2 ( L/2 ≤ x ≤ L )

Moment:

 Mmax = FL / 4 @ x = L/2
Simply Supported, 2 Loads at Equal Distances from Supports

Deflection:

 ( 0 ≤ x ≤ a ) ( a ≤ x ≤ L − a ) @ x = L/2

Slope:

 ( 0 ≤ x ≤ a ) ( a ≤ x ≤ L − a ) @ x = 0 @ x = L

Shear:

 V1 = +F ( 0 ≤ x ≤ a ) V2 = −F ( L − a ≤ x ≤ L )

Moment:

 Mmax = Fa ( a ≤ x ≤ L − a )
Simply Supported, Uniform Distributed Load

Deflection:

 @ x = L/2

Slope:

 @ x = 0 @ x = L

Shear:

 V = w (L/2 − x)
 V1 = +wL / 2 @ x = 0 V2 = −wL / 2 @ x = L

Moment:

 Mmax = wL2 / 8 @ x = L/2
Simply Supported, Moment at Each Support

Deflection:

 @ x = L/2

Slope:

 @ x = 0 @ x = L

Shear:

 V = 0

Moment:

 M = M0
Simply Supported, Moment at One Support

Deflection:

 @ x = L (1 − √3/3)

Slope:

 @ x = 0 @ x = L

Shear:

 V = −M0 / L

Moment:

 Mmax = M0 @ x = 0
Simply Supported, Center Moment

Deflection:

 ( 0 ≤ x ≤ L/2 )

Slope:

 ( 0 ≤ x ≤ L/2 )
 @ x = 0 @ x = L

Shear:

 V = +M0 / L

Moment:

 M = M0x / L ( 0 ≤ x ≤ L/2 ) Mmax = M0 / 2 @ x = L/2

### Fixed-Fixed Beams

Deflection:

 ( 0 ≤ x ≤ L/2 ) @ x = L/2

Shear:

 V1 = +F / 2 ( 0 ≤ x ≤ L/2 ) V2 = −F / 2 ( L/2 ≤ x ≤ L )

Moment:

 M = F (4x − L) / 8 ( 0 ≤ x ≤ L/2 )
 M1 = M3 = −FL / 8 @ x = 0 & x = L M2 = +FL / 8 @ x = L/2
Fixed-Fixed, Uniform Distributed Load

Deflection:

 @ x = L/2

Shear:

 V = w (L/2 − x)
 V1 = +wL / 2 @ x = 0 V2 = −wL / 2 @ x = L

Moment:

 M = w (6Lx − 6x2 − L2) / 12
 M1 = M3 = −wL2 / 12 @ x = 0 & x = L M2 = wL2 / 24 @ x = L/2

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Check out our beam calculator based on the methodology described here.

• Calculates stresses and deflections in straight beams
• Builds shear and moment diagrams
• Can specify any configuration of constraints, concentrated forces, and distributed forces