interpolating the beam nodal curvature values (Finite element question
interpolating the beam nodal curvature values (Finite element question
(OP)
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.
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.





RE: interpolating the beam nodal curvature values (Finite element question
If you really want a good estimate (
1/r=(M/EI)/(1+y'2)3/2
prex
http://www.xcalcs.com : Online tools for structural design
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RE: interpolating the beam nodal curvature values (Finite element question
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?
RE: interpolating the beam nodal curvature values (Finite element question
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
RE: interpolating the beam nodal curvature values (Finite element question
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.
RE: interpolating the beam nodal curvature values (Finite element question
RE: interpolating the beam nodal curvature values (Finite element question
However within an element taking the second derivative of the Hermites should give the correct moment distribution within an element.
RE: interpolating the beam nodal curvature values (Finite element question
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.
RE: interpolating the beam nodal curvature values (Finite element question
RE: interpolating the beam nodal curvature values (Finite element question