# T312

## Propped Cantilever Beam with an Intermediate Load

This page provides formula for beam shear, moment and deflection
formula for a propped cantilever beam with an intermediate load.

Nomenclature:

F = Loading force

R = Reaction force

V = Shear force

M = Moment

l = beam length

a = Distance of load from left support

b = Distance of load from right support

x = location

E = Modulus of elasticity

I = Moment if Inertia

y = deflection

Relationships:
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R1 = F * b / (2 * l^3) * (3 * l^2 - b^2)
R2 = F * b / (2 * l^3) * (3 * l - a)
VAB = R1
VBC = -R2
M1 = F * b / (2 * l^2) * (b^2 - l^2)
MAB = F * b / (2 * l^3) * ((b^2 * l - l^3 + x * (3 * l^2 - b^2))
MBC = F * a^2 / (2 * l^3) * (3 * l^2 - 3 * l * x - a * l + a * x)
yAB = (F*b*x^2)/(12*E*I*l^3) * (3*l*(b^2 - l^2)) + x*(3*l^2 - b^2))
yBC = yAB - (F * (x - a)^3)/(6 * E * I)