Derive Euler–Bernoulli equation from Navier-Cauchy equations

[jef445]

Hello

Is it possible to derive the Euler–Bernoulli equation:
[tex]\frac{d^2}{dx^2} \left(EI \frac{d^2w}{dx^2} \right) = q [/tex]
from Navier-Cauchy equations:
[tex]\left( \lambda + \mu \right)\nabla\left(\nabla \cdot \textbf{u} \right) + \mu \nabla^2\textbf{u} + \textbf{F} = 0[/tex]

I don't really know where to start because the Navier-Cauchy equations are 3 equations but the Euler–Bernoulli equation is just 1 equation.

 

[Achy47]

Hello,

Thank you for your question. Yes, it is possible to derive the Euler–Bernoulli equation from the Navier-Cauchy equations. The Euler–Bernoulli equation is a simplified form of the Navier-Cauchy equations that is specifically used for analyzing the bending of beams or other slender structures.

To derive the Euler–Bernoulli equation, we will need to make some assumptions and simplifications to the Navier-Cauchy equations. The first assumption we will make is that the deformation of the beam is small, which allows us to use the linear strain-displacement relationship. This means that we can neglect higher order terms in the strain-displacement equation, and the strain can be approximated as the first derivative of the displacement.

Next, we will assume that the beam is homogeneous and isotropic, meaning that the material properties (such as Young's modulus and Poisson's ratio) are constant throughout the beam and do not vary with direction. This allows us to simplify the Navier-Cauchy equations to just one equation, which is the Euler–Bernoulli equation.

Additionally, we will assume that the beam is subjected to a distributed load, which can be represented by the term "q" in the Euler–Bernoulli equation. This load is applied to the beam in the direction of the beam's length, and we will assume that it is constant along the length of the beam.

With these assumptions and simplifications, we can derive the Euler–Bernoulli equation by taking the second derivative of the displacement equation and substituting it into the Navier-Cauchy equations. This will result in the simplified form of the Navier-Cauchy equations, which is the Euler–Bernoulli equation.

I hope this helps to answer your question. If you have any further questions, please don't hesitate to ask.