Cantilever Beams - Moments and Deflections

Cantilever Beam - Single Load at the End

Maximum Reaction Force

at the fixed end can be expressed as:

R_A = F                              (1a)

where

R_A = reaction force in A (N, lb)

F = single acting force in B (N, lb)

Maximum Moment

at the fixed end can be expressed as

M_max = M_A

          = - F L                               (1b)

where

M_A = maximum moment in A (Nm, Nmm, lb in)

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

Maximum Deflection

at the end of the cantilever beam can be expressed as

δ_B = F L³ / (3 E I)                                      (1c)

where

δ_B = maximum deflection in B (m, mm, in)

E = modulus of elasticity (N/m² (Pa), N/mm², lb/in² (psi))

I = moment of Inertia (m⁴, mm⁴, in⁴)

b = length between B and C (m, mm, in)

Stress

The stress in a bending beam can be expressed as

σ = y M / I                                     (1d)            

where

σ = stress (Pa (N/m²), N/mm², psi)

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

M = bending moment (Nm, lb in)

I = moment of Inertia (m⁴, mm⁴, in⁴)

The maximum moment in a cantilever beam is at the fixed point and the maximum stress can be calculated by combining 1b and 1d to

σ_max = y_max F L / I                (1e)

Example - Cantilever Beam with Single Load at the End, Metric Units

The maximum moment at the fixed end of a UB 305 x 127 x 42 beam steel flange cantilever beam 5000 mm long, with moment of inertia 8196 cm⁴ (81960000 mm⁴), modulus of elasticity 200 GPa (200000 N/mm²) and with a single load 3000 N at the end can be calculated as

M_max = (3000 N)  (5000 mm)

         = 1.5 10⁷ Nmm

         = 1.5 10⁴ Nm

The maximum deflection at the free end can be calculated as

δ_B = (3000 N) (5000 mm)³ / (3 (2 10⁵ N/mm²) (8.196 10⁷ mm⁴)) 

    = 7.6 mm

The height of the beam is 300 mm and the distance of the extreme point to the neutral axis is 150 mm. The maximum stress in the beam can be calculated as

σ_max = (150 mm) (3000 N) (5000 mm) / (8.196 10⁷ mm⁴)

    = 27.4 (N/mm²)

    = 27.4 10⁶ (N/m², Pa)

    = 27.4 MPa

Maximum stress is way below the ultimate tensile strength for most steel.

Cantilever Beam - Single Load

Maximum Reaction Force

at the fixed end can be expressed as:

R_A = F                              (2a)

where

R_A = reaction force in A (N, lb)

F = single acting force in B (N, lb)

Maximum Moment

at the fixed end can be expressed as

M_max = M_A

          = - F a                               (2b)

where

M_A = maximum moment in A (N.m, N.mm, lb.in)

a = length between A and B (m, mm, in)

Maximum Deflection

at the end of the cantilever beam can be expressed as

δ_C = (F a³ / (3 E I)) (1 + 3 b / 2 a)                                       (2c)

where

δ_C = maximum deflection in C (m, mm, in)

E = modulus of elasticity (N/m² (Pa), N/mm², lb/in² (psi))

I = moment of Inertia (m⁴, mm⁴, in⁴)

b = length between B and C (m, mm, in)

Maximum Deflection

at the action of the single force can be expressed as

δ_B = F a³ / (3 E I)                              (2d)

where

δ_B = maximum deflection in B (m, mm, in)

Maximum Stress

The maximum stress can be calculated by combining 1d and 2b to

σ_max = y_max F a / I               (2e)   

Cantilever Beam - Single Load Calculator

A generic calculator - be consistent and use metric values based on m or mm, or imperial values based on inches. Default typical values are in metric mm.

F - Load (N, lb)

a - Length of beam between A and B  (m, mm, in)

b - Length of beam between B and C (m, mm, in)

I - Moment of Inertia (m⁴, mm⁴, in⁴)

E - Modulus of Elasticity (N/m², N/mm², psi)

y - Distance from neutral axis (m, mm, in)

Load Calculator!

Cantilever Beam - Uniform Distributed Load

Maximum Reaction

at the fixed end can be expressed as:

R_A = q L                                (3a)

where

R_A = reaction force in A (N, lb)

q = uniform distributed load (N/m, N/mm, lb/in)

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

Maximum Moment

at the fixed end can be expressed as

M_A = - q L² / 2                              (3b)

Maximum Deflection

at the end can be expressed as

δ_B = q L⁴ / (8 E I)                               (3c)

where

δ_B = maximum deflection in B (m, mm, in)

Cantilever Beam - Uniform Load Calculator

A generic calculator - use metric values based on m or mm, or imperial values based on inches. Default typical values are in metric mm.

q - Uniform load (N/m, N/mm, lb/in)

L - Length of beam (m, mm, in)

I - Moment of Inertia (m⁴, mm⁴, in⁴)

E - Modulus of Elasticity (Pa, N/mm², psi)

y - Distance from neutral axis (m, mm, in)

Load Calculator!

More than One Point Load and/or Uniform Load acting on a Cantilever Beam

If more than one point load and/or uniform load are acting on a cantilever beam - the resulting maximum moment at the fixed end A and the resulting maximum deflection at end B can be calculated by summarizing the maximum moment in A and maximum deflection in B for each point and/or uniform load. 

Cantilever Beam - Declining Distributed Load

Maximum Reaction Force

at the fixed end can be expressed as:

R_A = q L / 2                              (4a)

where

R_A = reaction force in A (N, lb)

q = declining distributed load - max value at A - zero at B (N/m, lb/ft)

Maximum Moment

at the fixed end can be expressed as

M_max = M_A

          = - q L² / 6                               (4b)

where

M_A = maximum moment in A (N.m, N.mm, lb.in)

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

Maximum Deflection

at the end of the cantilever beam can be expressed as

δ_B = q L⁴ / (30 E I)                                      (4c)

where

δ_B = maximum deflection in B (m, mm, in)

E = modulus of elasticity (N/m² (Pa), N/mm², lb/in² (psi))

I = moment of Inertia (m⁴, mm⁴, in⁴)

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