Cantilever Beam Formulas

Last updated August 23, 2023
By Ian Story

Case 1: Uplift resisted by point load at end of backspan

Point Load at Free End

$\Delta_m_a_x$ (at free end) $= \frac{Pa^2}{3EI}(a+b)$

Uniformly Distributed Load on Cantilever

$\Delta_m_a_x$ (at free end) $= \frac{wa^3}{24EI}(3a+4b)$

Case 2: Uplift partially resisted by backspan dead load

Backspan Reaction

This case uses available dead load on the backspan to help clamp down the beam, resisting overturning and reducing the slope at the start of the cantilever. There is assumed to be a wall below the beam that prevents negative deflection on the backspan. Because of this, not all of the backspan dead load can be used (some will just be transferred through the beam into the wall below). This case assumes a length of backspan dead load, $c$ , is used to clamp down the cantilever beam – the rest goes into the wall below. $c$ is set at the maximum value that does not produce negative deflection. Solving for $c$ requires finding the roots of a 4th degree polynomial function, so a numeric approximation (Newton’s method) is used instead.

For use in Excel calculations, here is the function for $c$ and its derivative:

$f(c) = 0$

$f(c) = D(c^4-2b^2c^2)+4Pab^2+2wa^2b^2$

$f'(c) = D(4c^3-4b^2c)$

Notes:

for D = 0, set c = b

Use c = max(c, b)

Point Load at Free End

$c =$ length of backspan dead load used to help resist uplift. Solving for $c$ requires finding the roots of a 4th degree polynomial function, so a numeric approximation is used instead. See the Desmos calculator.

$\Delta_m_a_x$ (at free end) $= \frac{Pa^2}{3EI}(a+b)-\frac{Dac^2}{24EIb}(2b-c^2)$

Uniformly Distributed Load on Cantilever

$\Delta_m_a_x$ (at free end) $= \frac{wa^3}{24EI}(3a+4b)-\frac{Dac^2}{24EIb}(2b-c^2)$

Combined Point Load at Free End and Uniformly Distributed Load on Cantilever

The length calculated for $c$ needs to take into account all loads applied to the cantilever. Once so calculated, superposition can be used for the remaining work.

$\Delta_m_a_x$ (at free end) $= \frac{wa^3}{24EI}(3a+4b)+\frac{wa^3}{24EI}(3a+4b)-\frac{Dac^2}{24EIb}(2b-c^2)$

Interactive Graph: Desmos Calculator

Case 3: Uplift resisted along backspan length

Note: this case assumes a tapered reaction force distribution on the backspan: the maximum tapered reaction, $w$ , occurs at the start of the cantilever, and tapers to zero at the end of the backspan. A point reaction at the end of the backspan provides any remaining required hold-down force. The tapered portion of the reaction is calibrated to provide a slope of zero at the end of the backspan and to ensure that the backspan deflection never goes negative. Unlike Case 2, this case assumes an unlimited amount of dead load is available to resist uplift.