# Beam with material change along beam axis

Hello there,

as the title says, I am trying to simulate a beam with a change in material (Young’s modulus) along the beam axis. However I found no real treatment of this problem in the literature.

My basic setup is a beam with a quadratic cross section, supported at both ends (displacement=0 in y direction). In the center there is a section consisting of a material with a different Young’s modulus. Tie constraint is used between the contacting surfaces. On the top surface of the central section a force is applied to a narrow area via surface traction.
To avoid rigid body motion, I applied zero displacement in the x and z direction for one point in the center of the beam (for reasons of symmetry).

To my understanding and according to Euler-Bernoulli theory, the change in Young’s modulus should change the maximum deflection of the beam but not the resulting stresses. However, in my model I see a drop in axial stresses as the Young’s modulus of the center piece decreases. (The theoretical value for the maximum stress would be 26.4MPa for this setup)

Result with centerpiece E=500MPa, outer pieces E=72000MPa:

Is there any reasonable explanation for this behavior? Any literature about it? Or is there something wrong with my FE model or analysis?

Markus

You’re asking if there is something wrong with your FE mdoel or analysis,
so my question have you done a very simple control calculation?
like your beam but with the same material and maybe with selfweight
you have to take care, that your beam will be a bit to stiff cause of your triangle elements!
wbr

If you are looking for agreement with beam theory, it would be better to use beam elements first. This way you wouldn’t have to use tie constraints (introducing potential spurious stress concentrations) either.

Problems like this (beams with variable stiffness) are quite common in literature but usually the stiffness changes are due to variable cross-section. Roark’s discusses that too - both gradual and sudden changes in cross-section.

the internal forces will not change as well as strains since this is an isostatic problem (if well posed), however as stress = Young * strain (read it as tensorial magnitudes) the stresses will certainly change.

It depends on the case. In simple linear static stress analyses of individual parts (constant stiffness), stresses don’t change when Young’s modulus is modified. From D. Madier’s book:

there is maybe always a better solution, but the questions is,
if these is helpful. there is a solution and these maybe works.
I’m not sure, if these is helpful to change to beam elements (with prepromax and netgen!?)
you can also elements and select the elements
in the middle to add them a spereate set, and give them different
material, so that there will be no use for tie contraints

No but the FEM workbench in FreeCAD supports CalculiX’s beam elements.

It looks far too stocky for Euler-Bernoulli beam theory to be accurate. You can already see the cross-section changing where the load is applied, and the discrete change in stiffness could make the middle section effectively an extremely short separate beam where shear stiffness is significant despite being neglected by Euler-Bernoulli.

I wouldn’t use beam elements because in CCX they’re not Euler-Bernoulli beams anyway so no reason to expect them to agree with the theory.

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Well, they aren’t beam elements at all, right ? They are expanded to solid elements and not using actual beam formulation. But they can often make modeling of slender structures easier and with no need for additional constraints. It’s true that the OP’s model doesn’t really fit the slender structure definition but if it was longer… Here at least 2D elements could be used.

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it seems something is missing at this, variable stiffness of beam will affect both of deflection and stress. As example given, ratio have nearly 150 times lower at center, thus make it act like a hinged.