# Problem of Model Verification for a Curved Beam Model

Dear all,

I have a problem in model verification for a curved beam model. I have formulated an Euler-Bernoulli beam model similar to the following paper (refered as Pai's model), but employing Euler angle description.

Pai,
P.F., Palazotto, A.N., 1996, “Large-deformation analysis of flexible beams,” International Journal of Solids Structures,
33(9), pp. 1335-1353.

My simulation results are always consistant with the cases provided in the paper. Of course, they are all matched with finite element analysis.

However, I find that for non-constant curvature, i.e. elliptic shaped beams, my model and Pai's model do not match with FEA. Since my formulation is equivalent with Pai's model, our results still match with each other, but not FEA. For example of an elliptic shaped cantilever beam subjected to pure moment at the tip, the nodal displacements calculated from my (or Pai's) model are almost as twice as those in FEA. But nodal rotation theta and moment M are the same.  For this case, the equations can be simplified as:

dtheta/ds = M / EI

du/ds = -1 + (1+e)*cos(theta) - k*w

dw/ds = -(1+e)*sin(theta) + k*u

where u and w are nodal displacements along the beam aixs and normal to the beam axis in the undeformed configuration, k is the initial curvature depending on the path length s, e is the axial strain (=0 in this case). I describe k as a polynomial of s, and I have verified that the model gives correct initial shapes under no loadings.

I also suspect if it is the problem of shear, so I try elements B33 and B32 in abaqus to test both Euler and Timoshenko beam models and both elements give the same results. I may say that Euler beam theory is valid in this case. By the way, the cross section is circular (with radius = 1mm), and the beam is a quarter ellipse (half major axis = 25mm,  half minor axis = 10mm).

I have spent a long time on this and have not found out what is wrong. Would anyone provide an idea / approach to detect the errors?

Thanks,

Jiajie

JiaJie,

Are your FEA results calculated using 3D brick elements? And also when you use B32 and B33, you mentioned that both elements give the same results. Are these same results also the same as what you will calculate from FEA using 3D brick elements?

Wenbin