Stress Propotionality 4

[a7x1984]

I checking some FEM output and came across an elementary question since being out of school for such a while. In a linear elastic model, is there any time when stress is NOT proportional to external loading due to geometry, loading distribution, etc?

The person running the program needed to change all of the external loads by the same factor, and he wanted to open it up, change them, and re-run them all over again. I told him just to multiply the critical internal forces, stresses and displacements (from the original run) by the same factor so as not to waste time. He didn't agree and thus I am here. I tried to explain that it would be no different than noticing (WL^2)/8 is proportional to normal stresses in simple beam theory.

"Structural engineering is the art of modelling materials we do not wholly understand into shapes we cannot..."...ah...screw it, we don't know what the heck we are doing.

 

[ZeroExperience]

It depends. If your FEA solver computed Von Mises stresses, you cannot simply multiply the Von Mises stress by some factor because the Von Mises stress equation is not linear at all. If you recall, the Von Mises stres equation is composed of square roots, squares, and differences between stresses.

If your solver computed purely bending moments, ie My/I, I think you can get away with it. Just be careful I guess.

The linear elastic model is reserved for behaviors that are modeled by linear differential equations. Such models have deflections and slopes proportional to applied loads. In such cases you can apply the Method of Superposition, but doesn't necessarily you can apply to stresses.

 

[a7x1984]

Zero - Yeah, that is pretty much what I was thinking. Thanks for the affirmation, sir.

"Structural engineering is the art of modelling materials we do not wholly understand into shapes we cannot..."...ah...screw it, we don't know what the heck we are doing.

 

[corus]

Of course the stress is proportional to the load in a linear elastic model and so is the Von Mises stress intensity. Simple maths will tell you that. The only time it won't be proportional is when you have non-linearity from large displacement or from contact.

 

[a7x1984]

Corus,

Would you like to explain better why those square roots and other operators/functions do not change the proportionality between external loading and maximum stresses?

"Structural engineering is the art of modelling materials we do not wholly understand into shapes we cannot..."...ah...screw it, we don't know what the heck we are doing.

 

[btrueblood]

Have to agree with Corus, and it is pretty simple math.

Given a body of linearly elastic, isotropic material, and a set of loads which can be characterized as proportional to some load factor, P.

A fundamental for linear elasticity (*) is that the components of the stress tensor vary linearly with the load vector, i.e. the principal stresses at any point in the body will vary linearly with P, i.e.

s₁ = k₁P
s₂ = k₂P
s₃ = k₃P

Von Mises' equation is

s_vm = sqrt[(k₁P - k₂P)^2 + (k₂P-k₃P)^2 + (k₃P - k₁P)²]

note we can factor out the P, giving

s_vm = P sqrt[(diff's of the k terms)]

Therefore von Mises stress magnitude is proportional to the load.

(*) I think it is, but couldn't find a reference. Others have made this statement above...if not a fundamental, then let's just make it an assumption for the discussion above, the math still holds.

 

[a7x1984]

I agree that that particular expression is correct and proportional. You must remember that not all engineers work with these combinations, but my original assumption was correct. Building codes, and typical steel, timber and concrete specs/codes do not speak much if at all about


"Structural engineering is the art of modelling materials we do not wholly understand into shapes we cannot..."...ah...screw it, we don't know what the heck we are doing.

 

[rb1957]

is your model truly linear ? no gap elements ?

assuming not, then the model is linear (which you can now verify looking at the rerun analysis). assuming not, i'd've done as your 1st thought, and factored the initial run.

 

[a7x1984]

rb1957, there are gap elements, in fact a lot of them. They are acting as shear studs connecting plates (concrete deck) to beam elements (Longitudinal bridge girders.

"Structural engineering is the art of modelling materials we do not wholly understand into shapes we cannot..."...ah...screw it, we don't know what the heck we are doing.

 

[rb1957]

then the gaps will non-linearise the results ... which you can now check

 

[a7x1984]

I did, and it wasn't.

"Structural engineering is the art of modelling materials we do not wholly understand into shapes we cannot..."...ah...screw it, we don't know what the heck we are doing.

 

[inline6]

gap elements in linear analysis become linear spring elements in some codes. What solver and solution was used?

Corus is correct if it is a true linear static analysis the ouput it proportional to input that is the whole premise of linear analysis.

there is such a thing as non linear elastic which includes geometric effects (gaps, stress stiffening, PD effects etc) but not linear material properties.

 

[crisb]

Am surprised that the composite bridge girder model you describe has any significant nonlinearity. I would stay with your initial instinct and check carefully there are no other problems with your model. I don't know what program you are using but are the "gap" elements intended to be rigid links or linear or nonlinear springs etc.

 

[rb1957]

i think gap elements/contact issues create the potential to non-linearise a linear model. the OP has the results in hand to see if there is a significant difference. and don't beat up corus 'cause he answered zero's post. zero is still in the wrong (as shown vM is linear if the model is truly linear), and so corus is right it critique zero's post.

 

[a7x1984]

Crisb, they are intended to perform as rigid links for composite action. The re-run vs. multiplying stresses by a factor was off by about 5%. I didn't create the model; I will have to check the geometry, etc..

rb, I will check it out some more to see where the difference originated. I agree zero is incorrect solely based on the square roots, etc. argument. However, my problem is that he attacked zero instead of kindly adjusting his mistake. The tone of this forum should covey respect, correct? I will leave it at that. Thanks for your help, gents.

"Structural engineering is the art of modelling materials we do not wholly understand into shapes we cannot..."...ah...screw it, we don't know what the heck we are doing.

 

[crisb]

If they are rigid links the model and results should be linear and proportional. The only explanation I can think of for the difference is if some of the loads have not been scaled up the same e.g. self weight loads or the composite action has been modelled as a staged construction with slab added later.

 

[btrueblood]

"However, my problem is that he attacked zero instead of kindly adjusting his mistake."

I repectfully disagree, the attack was prefaced with "no offence intended", and conveyed the tone that Zero should check his math, because he made a bonehead error. We all do it sometimes, no offense is intended.