For nonlinear beam models, there are some coupling stiffness being discovered, and such coupling stiffness often results in nonlinear phenomena even in isotropic beams, such as the trapeze effect in strip-like blades and the Poynting phenomena in rubber cylinders.
The question here is: "Is there any such nonlinear coupling behavior between bending and stretching even for isotropic beams? And is such behavior reported in the literature?"
Thanks a lot!
Hi Fang, Yes, the axial-bending coupling is present in beams that undergo finite deformation. Still, when kinematics is not large the axial-bending coupling is present, check this articles to see the influence of angular velocity (centrifucal force) in the bending frequencies of a beam.
Article Free vibration and dynamic stability of rotating thin-walled...
Rotor is a well-known case of coupling between modes when kinetic energy is increasing. Several types of rolling bearings are also sources of coupling that may be considered even for low rotating speed.
Dear Dr. Saravia and Professor Antoine,
Thanks a lot for your answers which are very helpful to me!
In addition, there is a observed phenomenon from our simulation:" the bending stiffness is reduced due to axial stretch of the isotropic beam". Any comments on this effect?
Again, thanks a lot for your info!
Best Regards,
Fang
Dear Fang, I can't think of an example where the flexural stiffness of a beam decreases with axial stretch. At first glance I would say that still in composites it is not possible that the bending stiffness is reduced due to axial stretch... Although the sectional stiffness terms could decrease due to Poisson effect (i.e. if you allow sectional deformation in its own plane), the geometrical non-linearity makes the flexural terms in the tangent stiffness matrix to increase (specially the geometrical stiffness ones).
As Dr. Saravia mentions, in the case of geometric non-linearities where bending causes in-plane stretching, the effect of the axial stretch is to "harden" or progressively stiffen the beam against further transverse deflection. This causes the beam to behave like a "hardening" spring. Such an effect becomes predominant when the deflection is around 1/10th of the beam thickness (as a rule of thumb). This is true for isotropic beams.
For finite deformations, the geometric coupling between bending and stretching comes about due to 2 major reasons: considering the Von Karman strains where the axial strain (Ex) has contributions from the slope of the deflection (w,x) terms and using a deformed state to define equilibrium, instead of the undefined configuration (as is required for finite deformations).
So, you should not be seeing an increase in deflections when using a non-linear beam model.
Dear Dr. Saravia and Pydah,
I am sorry but I got confused. Maybe I didn't clearly state what I was trying to ask.
Please let me clearly state the problem this time:
"Suppose we have a 3D slender structure (not 1D beam model)
One end of the 3D body is totally clamped
For the other end: (1) follower extensional force is applied, which means the extension force is always applied in the directional tangent to the beam reference line and normal to the end cross section (assuming Euler-Bernoulli type model); (2) at the same time, applied constant bending moment at the same tip. Please note here I do not assume constant cross-sectional bending stiffness and consider the cross-sectional in-plane and off-plane warping which is finite. Now I am trying to investigate the change of EI due to the increasing of axial force.
Question: Will the bending curvature in the deformed body increase when the bending moment on the tip is constant while the follower extensional force is increased? Please note here I focus on the CURVATURE of the DEFORMED 3D BODY, not on the deflections. And due to the extensional force, we have axial strain along the beam reference line.
Thanks a lot for your answers!
Best Regards,
Fang
Intuitively, an increase in the applied TENSILE axial load should reduce the curvature. The axial force has the effect of "stretching" out the curvature.
This is observed even in small deformations case.
Dear Pydah,
Thanks lot for your quick reply!
Could you please direct me to some literature showing what you have introduced to me?
Fang
The bending stretching coupling is intrinsic in the constitutive equations of non symmetric beams. Thus it is expected that it will take place in nonlinear beams. I mean, it will occur in any non symmetric beam, about the middle surface.
Dear Jiang,
I've attached an excerpt from the book "Plates- Theory and Applications" by Bhaskar and Varadan.
It pertains to small deformation analysis. But I believe you can expect the same sort of qualitative behaviour even in the case of finite deformations.
Hope this helps!
As Prof Neto points out, in the case of unsymmetric cross-sections, there would be an intrinsic bending stretch coupling because the centroidal axis is no longer the mid surface.
But in the case of combined loading, a tensile load would cause a decrease in the deflections, with a more significant effect in the case of the non-linear theory because of the hardening effect of the coupling.
Dear Fang, If you increase the axial force, the Poisson effect can make the bending stiffness of the deformed beam cross section to decrease (lower EI)... But this is a sectional property, the actual beam stiffness comes from the 3D elasticity, and this is why you must look at the tangent stiffness matrix and not the sectional properties to see what can occur to the beam curvature after increasing the axial load. As Dr. Pydah says, there are two contributions to the beam stiffening, look at the material and tangent stiffness matrices of a geometrical nonlinear formulation to clarify this.
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