# Cantilever Beams - Moments and Deflections

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

### 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 L^{3} / (3 E I) (1c)*

*where *

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

*E = modulus of elasticity (N/m ^{2} (Pa), N/mm^{2}, lb/in^{2} (psi))*

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

*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/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})*

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 cm ^{4}*

*(81960000 mm*, modulus of elasticity

^{4})*200 GPa (200000 N/mm*and with a single load

^{2})*3000 N*at the end can be calculated as

*M _{max} = (3000 N) (5000 mm)*

* = 1.5 10 ^{7} Nmm*

* = 1.5 10 ^{4} Nm*

The maximum deflection at the free end can be calculated as

*δ _{B} = (3000 N) (5000 mm)^{3} / (3 (2 10^{5} N/mm^{2}) (8.196 10^{7} mm^{4})) *

* = 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 10^{7} mm^{4})*

* = 27.4 (N/mm ^{2})*

* = 27.4 10^{6} (N/m^{2}, 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 a^{3} / (3 E I)) (1 + 3 b / 2 a) (2c)*

*where *

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

*E = modulus of elasticity (N/m ^{2} (Pa), N/mm^{2}, lb/in^{2} (psi))*

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

*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 a^{3} / (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 (m^{4}, mm^{4}, in^{4})

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

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

### 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 L^{2} / 2 (3b)*

#### Maximum Deflection

at the end can be expressed as

*δ _{B} = q L^{4} / (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 (m^{4}, mm^{4}, in^{4})

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

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

### 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 L ^{2} / 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 L^{4} / (30 E I) (4c)*

*where *

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

*E = modulus of elasticity (N/m ^{2} (Pa), N/mm^{2}, lb/in^{2} (psi))*

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