# Cantilever Beams - Moments and Deflections

## Maximum reaction force, deflection and moment - 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 (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} = 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)*

### 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) *

#### Cantilever Beam - Single 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.

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)

### 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)

### 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})*