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:
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
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
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
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)