Euler-Bernoulli Beam Theory and Nonlinear Differential Equations

[cmmcnamara]

I've been reading through my mechanics of materials textbook recently, notably in regard to the section on the deflection of beams. The well regarded Euler-Bernoulli beam theory relates the radius of curvature for the beam to the internal bending moment and flexural rigidity. However the theory approximates curvature by defining the derivative of the deflection function as much less than one causing the curvature function to approximate to the second derivative of the deflection. This essentially means the derived function is only valid for small angular deflections of the beam which is great for real world application because large angular deflections are not desirable in structures obviously. My book mentions that there are a small number of problems that the full relation of curvature can be used to solve because it produces a second order non linear degree one ODE. Out of curiosity, does anyone know of an example showing this put into use? I have been playing around with the ODE for a few hours but I am unsure of how to deal with the squared derivative term in the curvature equation as I have never encountered differential equations in my studies where the derivative terms are ever raised to a power. Thanks in advance!

 

[haruspex]

Don't know of any applications, but if you're interested in solutions you could try posting the equation.

 

[cmmcnamara]

$$\frac{\frac{d^2 v}{dx^2}}{[1+(\frac{dv}{dx})^2]^\frac{3}{2}}=\frac{M(x)}{EI}$$

EI is a constant which is determined by the beam material and cross section. Normally $\frac{dv}{dx}$ is considered small in comparison to one therefore the denominator is approximately one which reduces it to an approximation for small angles to a second order linear differential equation.

 

[AlephZero]

One simple solution is for a constant bending moment, when the deformed shape of the beam is the arc of a circle.

See the references in http://en.wikipedia.org/wiki/Elastica_theory for more.

 

[cmmcnamara]

Oh thanks, I just ordered the book "Nonlinear Problems of Elasticity" that was listed in the sources. I appreciate the push in the right direction!

 

[Studiot]

My book mentions that there are a small number of problems that the full relation of curvature can be used to solve because it produces a second order non linear degree one ODE.

I don't know that book but hope it will help you.
However you should be aware that non linear elasticity usually refers to non hookean ie materials where the stress - strain relationship is nonlinear. That is different from using the more exact curvature differential relationship.

large angular deflections are not desirable in structures obviously.

However they are common in industrial forming of components so I would suggest looking around the mechanical or production engineering section of the library. Sorry I can't suggest anything here it is not my field.

edit:
You also presumably realize that the equation you present is one dimensional - that is there is only bending in one plane and one single radius of curvature.

For real bending the is an effect across the breadth of the beam as well, especially if the beam has significant side to side loading as in a suspension bridge.

You should Google 'anticlastic bending'

http://www.sciencedirect.com/science/article/pii/0020768370900247

 

[AlephZero]

However you should be aware that non linear elasticity usually refers to non hookean ie materials where the stress - strain relationship is nonlinear. That is different from using the more exact curvature differential relationship.

There are two common meanings for "nonlinear" here and either or both may apply at the same time.

As you said, one is is "material nonlinearity", i.e. a nonlinear relation between stress and strain - for example creep or plasticity.

The other is "geometric nonlinearity" where the strains are small but the displacements (and particularly rotations) can be large, and most of the "rotation" at any point is a rigid body rotation of the object, not a deformation. That's what the OP was referring to. For eaxmple in something like a mechanical clock spring made by coiling a flat metal strip into a spiral with many turns, the strains in the spring are small and the matieral behaviour is completely linear, otherwise the it wouldn't survive many cycles of winding and unwinding.

 

[Studiot]

There are two common meanings for "nonlinear" here and either or both may apply at the same time.

Thank you for the comment but,

Only 2?

I already offered a third, which is probably the most common considering the volume of sheet (and other) material whose forming processes involve very large strains and/ or very large rotations.

I am also simply warning not to be disappointed if a textbook entitled 'non linear elasticity' is concentrates on non-hookean behaviour and supplying some alternative reference(s) that do not and offering a way of finding others that I have not mentioned, including watch and other springs.

 

[lywcy68526]

Bernoulli equations can only apply to continuous fluid espacially in narrow slit.
If there are holes in the sidewall of pipe, the lateral pressure will force part of fluid move towards side directions (Y,-Y,Z,-Z),
thus this: the momentum of flowing beam moving towards X direction will gradually reduce.

 

[cmmcnamara]

There are two common meanings for "nonlinear" here and either or both may apply at the same time.

Sorry I guess I hadn't been specific enough. The nonlinear terminology here I described was in complete reference to the type of differential equation presented, not the material behavior. Material mechanics is one of my favorite branches of engineering and so I pursue a lot of independent study in it, however all of my studies still are into regard to elastic material behavior. In fact the equation I presented is only valid in the elastic region because the M/EI term is a direct result of the flexure formula which is only applicable in the elastic region. The formula derived originally for a small differentially deformed element of a beam is: $\frac{1}{ρ}=\frac{M}{EI}$. The left side in my differential equation is just the substitution for plane curvature. I have been interested in how to solve that differential equation primarily. I suppose my interest is in the effect that the assumption $\frac{dv}{dx}<<1$ had on the elastica equation considering that initial conditions can cause interesting effects on the function.

Bernoulli equations can only apply to continuous fluid espacially in narrow slit.
If there are holes in the sidewall of pipe, the lateral pressure will force part of fluid move towards side directions (Y,-Y,Z,-Z),
thus this: the momentum of flowing beam moving towards X direction will gradually reduce.

If I am not mistaken I think you maybe referencing the Bernoulli equation from fluid mechanics that defines energy conservation along a streamline but I am unsure since you mention beams.