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:
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 (Nm, Nmm, 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)
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
σmax = ymax 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
Mmax = (3000 N) (5000 mm)
= 1.5 107 Nmm
= 1.5 104 Nm
The maximum deflection at the free end can be calculated as
δB = (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
σmax = (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:
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)
Maximum Stress
The maximum stress can be calculated by combining 1d and 2b to
σmax = ymax 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)
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)
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:
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)