﻿ 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

Cantilever, End Load Deflection:  @ x = L

Slope:  @ x = L

Shear:

 V = +F

Moment:

 M = −F (L − x)
 Mmax = −FL @ x = 0
Cantilever, Intermediate Load 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

Fixed-Fixed, Center Load 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