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Cantilever Beams - Moments and Deflections

Maximum reaction force, deflection and moment - single and uniform loads

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Cantilever Beam - Single Load at the End

Cantilever beam - single load - deflection

Maximum Reaction Force

at the fixed end can be expressed as:

RA = F                              (1a)

where

RA = reaction force in A (N, lb)

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

Maximum Moment

at the fixed end can be expressed as

Mmax = MA

          = - F L                               (1b)

where

MA = 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

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

where

δB = 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)

Cantilever Beam - Single Load

Cantilever beam - single load - deflection

Maximum Reaction Force

at the fixed end can be expressed as:

RA = F                              (2a)

where

RA = reaction force in A (N, lb)

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

Maximum Moment

at the fixed end can be expressed as

Mmax = MA

          = - F a                               (2b)

where

MA = 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

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

where

δC = 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

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

where

δB = 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 (m4, mm4, in4)

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

Cantilever Beam - Uniform Distributed Load

Cantilever beam - uniform load - deflection

Maximum Reaction

at the fixed end can be expressed as:

RA = q L                                (3a)

where

RA = 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

MA = - q L2 / 2                              (3b)

Maximum Deflection

at the end can be expressed as

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

where

δB = 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)

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

Cantilever beam - declining distributed load - deflection

Maximum Reaction Force

at the fixed end can be expressed as:

RA = q L / 2                              (4a)

where

RA = 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

Mmax = MA

          = - q L2 / 6                               (4b)

where

MA = 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

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

where

δB = 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)

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