interpolating the beam nodal curvature values (Finite element question

[ThingsLearner]

I have the results of a beam FEM. I know that by using Hermition polynomials (shape functions) and having the nodal displacements, I can find the displacements anywhere along the beam by interpolation. I guess if I derive the cubic shape functions once (Hermit functions) and multiply them by the nodal displacements, I can find the slope at any arbitrary location along the beam but what about the curvature? I can not just compute the 2nd derivative of hermit functions and multiply it by nodal displacements to get curvature right? I think I do not get correct continuous results any more.

If I have the nodal responses (nodal displacements and even the nodal curvature values) what should I do to compute the curvature at any arbitrary location along the beam?

Should I use the moment values along the beam and the moment-curvature relationship instead?

Sorry if the question is elementary.

 

[prex]

You gave yourself the right answer: as bending moment is proportional to curvature and as, in the linear approximation that your elements likely adhere to, curvature is equal to the second derivative of deflections, then...
If you really want a good estimate (), you could correct it with:
1/r=(M/EI)/(1+y'²)^3/2

prex

http://www.xcalcs.com

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[ThingsLearner]

Prex, Thanks for your response. But
I know that the curvature is the 2nd derivative of deflection but when I derive the Hermit functions twice and multiply them with nodal displacements, I do not get a correct curvature response.
I guess Hermit shape functions satisfy the displacement and slope continuity but not curvature continuity.

So let me restate my question in this way:

Can we find the curvature in a beam directly from nodal displacements and not using the moment-curvature relationship?

 

[prex]

Don't understand your 2nd post: my suggestion is to use the bending moment distribution at nodes to obtain nodal curvature values.
But it all depends on what exactly you need and want to do, you should better explain.
If you want to interpolate the curvature over the elements, you can assume a linear variation (that of course is what you get from shape functions), and FEM cannot give you more that that as this is what it is based on, or you can use a polynomial or other smoothing curve over the nodal values: however this would be a kind of extrapolation.

prex

http://www.xcalcs.com

: Online tools for structural design

http://www.megamag.it

: Magnetic brakes for fun rides

http://www.levitans.com

: Air bearing pads

 

[prost]

Even for 'simple' beams, this computation is quite involved because of the mapping process in which the real coordinates (x,y) and real displacements (u,v) are mapped to parent elements whose coordinates are (chi,eta). First, what is the polynomial order of these Hermitian polynomials? I.e.,
y=mx+b---linear (1st order polynomial)
y=ax*x+b*x*c--quadratic (2nd order polynomial)...

Maybe you aren't computing the derivatives correctly, that is, you aren't including the mapping of the real coordinates that define the beam element to the basis or parent element. Because of this mapping, you have to compute the Jacobian derivatives that relate the (x,y) coordinates to the derivatives in the (chi,eta) coordinate system.

Also, if you are using theory of linear elasticity in your definition of the material behavior, and your beams are straight, that the curvature is computed with the displacement field, not the coordinates of the deflected shape.

 

[xerf]

The beam element formulation based on Euler-Bernoulli beam theory uses cubic Hermite polynomials.

 

[pja]

Your FEM based upon Hermites is only C1 continuous so your curvature will not be continuous across interelement boundaries. There is no way around that. If you want interelement continuous moments you will have to either use C2 continuos elements, which nobody does, or use a mixed functional where moments are included as field variables.

However within an element taking the second derivative of the Hermites should give the correct moment distribution within an element.

 

[ThingsLearner]

Thanks for all your responses.
pja, you are right. I do not observe inter-elemental continuity.

I have decided to use moment-curvature relationship instead of "differentiation of deflection" to find the curvature values.

Prex, you had asked about the problem I am solving:

I have analyzed a beam using a FEM code which deos everything correctly. we have measured curvatures along the beam in an experiment using sensors. Now I want to do parametric studies (material parameters in our model) to minimize the error between measured and computed curvature values. Thats why I need the exact values of curvature at the sensor locations (which are random) for each set of input parameters. So I am trying to write an automized MATLAB code to implement this Least Square process and give the optimum values of the input parameters.

 

[johnhors]

Most (if not all) commercial FE programs modify the beam element formulation to allow for shear deformation, which can be significant in short stubby beams.

 

[xerf]

The FE beam elements which allow shear deformation are based on Timoshenko beam theory.