How to identify natural and essential boundary conditions of this differential equation?

Question

I am trying to identify natural and essential boundary conditions from this differential equation.

Strong form is : $$\frac{d^2}{dx^2} \bigg[R(x)\frac{d^2w}{dx^2}\bigg] - q(x) =0, \hspace{6pt}\text{for } 0<x<L $$ (This is Euler-Bernoulli beam equation. ) Subjected to boundary conditions: $$w(0)=0, \hspace{10pt} \bigg(\frac{dw}{dx}\bigg)_{x=0} =0, \hspace{10pt} \bigg(b\frac{d^2w}{dx^2}\bigg)_{x=L}=M, \hspace{10pt} \bigg [ \frac{d}{dx} \bigg (b\frac{d^2w}{dx^2}\bigg)\bigg ]_{x=L}=F$$

To get the weak form, I multiply first equation by weight function $v$ and get following weak form after some algebra,

$$\int_0^L \bigg[b\frac{d^2v}{dx^2}\frac{d^2w}{dx^2}-vq \bigg]dx + \bigg[v\frac{d}{dx}\bigg(b\frac{d^2w}{dx^2} \bigg) - b\frac{dv}{dx}\frac{d^2w}{dx^2} \bigg]_0^L $$

My question is: in the boundary conditions listed, which one are natural and essential boundary conditions? And how to identify that? In the second order ODE, there are two boundary conditions. And the one having derivative term is essential boundary condition. But in higher order ODE, I am not sure how to identify this when there are multiple boundary conditions involving derivatives.

Answer 1

The third and second derivatives of the deflection ($w$) are known as the natural boundary conditions which relates the shear force and bending moments, respectively. The first and zero derivatives of $w$ are the essential boundary conditions, deflection and angle, respectively. See Na-Ho Kim. Introduction to Finite Element Methods, Wiley 2008.

The essential boundary conditions (BCs) are imposed directly in the solution imposed by the trial functions, which should satisfy them.

The natural BCs are imposed in the right-hand side of the resulting $KU=F$ system and are not necessarily satisfied by the trial functions. Indeed, they can be not prescribed and with the use of the essential BCs, these are not needed in the solution. Read the chapter 3 of the aforementioned book for further details and examples.

In general, mathematically speaking, I think that if your differential equation is of order $2n$, the conditions of order $0$ to $n-1$ are essential and the conditions of order $n$ to $2n-1$ are natural.

Answer 2

For the governing differential equation involving derivatives of order 2m, boundary conditions having derivatives of order at most (m-1) are essential and others are natural. Natural boundary conditions are embedded in the weak formulation of the problem. For a mechanical problem, these generally mean loads (shear force and bending moments) applied on the boundary of the body. The essential boundary conditions are imposed during the solution of the system of simultaneous algebraic equations. In a mechanical problem, the essential boundary conditions generally eliminate the rigid body motion. For the Euler-Bernoulli beam the governing equation is 4th order. Thus deflections and slopes prescribed at the end points constitute essential boundary conditions and the shear force and the bending moment as natural boundary conditions. One cannot uniquely find deflections unless it is prescribed at one end since the beam can then undergo rigid body motion.
Think of finding constants of integration if the 4th order ordinary differential equation is analytically integrated. To find the 4 constants of integration, one needs 4 conditions.