How can I use Euler–Bernoulli beam theory for non-constant beam cross-section ?

The beam is modeled as a continuous cone of known inner and outer diameters (both a function of $x$ the distance from one end of the beam).

Is there a way to correctly express the second moment of area of such a beam ?

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    Commented Oct 17, 2022 at 13:30

1 Answer 1


Let's split the answer in few paragraph.

Euler-Bernoulli beam

Euler-Bernoulli and other beam models are fine for elongated elements, with continuous variation of section, so that the distribution of internal stress on each section is approximately similar to the distribution of a beam with constant section.

Starting from the equilibrium equations of an infinitesimal beam element, you can get the dynamical equations of the beam. As an example, for axial solicitations we can write that

$m (x,t) \ddot{u}(x,t) \Delta x = $
$ = N(x+\Delta x,t) - N(x,t) + f(x,t)\Delta x =$
$ = N'(x,t) \Delta x + f(x,t)\Delta x + o(\Delta x)$

and thus, $m \ddot{w} = N' + f$. Now you can use the constitutive equation of a beam, relating the axial force $N$ with the strain of the beam element, $N(x,t) = EA(x) u'(x,t)$ to get the dynamical equations in $u(x,t)$

$ m \ddot{u} = (EA u')' + f$,

where $EA(x)$ is a function of the position in the beam and not uniform if the section of the beam or its structural properties are not uniform.

If you follow the very same procedure for bending (and shear stress, related in Euler-Bernoulli beam model), you get

$I \ddot{w} + (EJw'')'' = m$, with $EJ(x)$.

Observation. These equations are implemented quite naturally with FEM, with no problem at all in treating the non-uniform terms $EA(x)$, $EJ(x)$ thanks to integratino by parts.

Moment of the annular section

At each section, the moment of the annular section reads

$J(x) = \dfrac{\pi}{4}\left( R_{ext}^4(x) - R_{int}^4(x) \right)$,

as you can easily compute or check in https://en.wikipedia.org/wiki/List_of_second_moments_of_area.


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