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^{4}), modulus of elasticity 200 GPa (200000 N/mm^{2}) and with a single load 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.
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.
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})
Insert beams to your Sketchup model with the Engineering ToolBox Sketchup Extension
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