Confusion in Chandrupatla Textbook 1

[ThetaJC]

Previous threads have recommended this text as an excellent intro to FEA theory for engineers interested in wanting to begin developing a skill set, so I am covering the details in the third edition of the Introduction to Finite Elements in Engineering textbook. I am have a great deal of confusion on page 9 over the discussion of total potential energy of a system. They text defines the total potential energy of a structure as the sum of the strain energy and the work potential. The strain energy seems to intuitively make sense, however, the work potential terms do not. The referenced equation (1.26) lists terms that basically define the work potential. Now turning to the next page to show how they apply it will help make my case for what is causing me doubts. An example spring problem is outlined and total potential energy is defined as:

1/2k1d1^2+1/2k2d2^2+1/2k3d3^2+1/2k4d4^2-F1q1-F3q3

My doubt is with the last two terms. These terms reflect the work potential portion of the discretely connected system. For the sake of arguement and hopes to resolve my logical error, and I am challenging the last two terms. I feel like the forces F1 and F3 cannot simply be multiplied by the q1 displacement and q3 displacement, respectively. As the springs stretch, the force varies proportional to the net stretch of the spring. The simple definition of force times distance would only seem to work for me if the force was CONSTANT over the range of 0 to qi. In other words, the force is linearly proportional to the displacement to some degree. Going back to the last term in eq. (1.26) as an example (the other terms I have the same issue with) it seems strange that the summation of the product of ui and Pi (not 3.1415...) leads to a correct particular component of the work potential. So if I stretch a structure at a certain location with a point load, I simply consider the point load force times the displaced distance? How could this be, since the required force to displace a point a certain distance is proportional TO THE DISPLACEMENT OF THE POINT ITSELF. Work using a simple Force times distance approach only seems to lead credence to less complex situations like lifting a weight a certain distance of the ground. If I move a 5 lbf weight 10 inches, then I have done 50 lb-in of work. However, if I stretch a spring such that I am in equilibrium with the spring with a pull force of 5lbf and I have stretched the spring a total of 10 inches to achieve equilibrium, then I have expended only 25 lb-in of work. This is true because the energy stored in a stretched spring is 1/2kq^2, where k in this case is 1/2 lbf/in and q is 10 inches, then .5*.5*100= 25 lb-in.

So where have I made illogical conclusions? Thanks to anyone who can clarify this doubt.

 

[jerzy]

If you have access to Cook et al 4th Edition "Finite Elements: Concepts and Applications", the discussion on p138 onward might help. It is important to carefully distinguish between total potential energy of a system, strain energy in a body, and potential of loads.

 

[ThetaJC]

The strain energy in a discretely connected system of springs is the sum of all the 1/2kd^2 where d is the spring stretch. I can grasp this concept.

The work potential is force times distance. I understand what the book is describing. I just don't see a way to agree with it. How in the world can I call the work potential (or potential of loads) as you say, simply Fu, where u is the displacement vector. I surely can't maintain the entire force F over the span of 0 to u. With an ideal spring it is impossible. The force ramps up linearly proportional to the spring's stretch. I can't register this as only Fu since I only achieve the force F at the point u. Look at a plot of a stretched spring (F vs. x). The force ramps upward, it is not a simple horizontal line, which the textbook seems to imply. I WOULD BELIEVE that the work potential was Fu IF I could generate a CONSTANT force over the entire displacement u. So I guess maybe a better question would be how I would generate a constant force over the entire span of the displacement u. The force is generated by virtue of the resistance of the deformation of the spring. This concept is obviously not making sense to me.

The total potential energy is the difference between the internal strain energy and the work done on the system. I can grasp this concept as well.

I have researched enough publications to understand that I am missing something. I am arguing for sake of understanding. I have baffled my fellow engineers with this one. Any help would be greatly appreciated.

 

[jerzy]

Hello ThetaJC,
I appreciate your perplexity over this issue. I have been looking for a succinct clarification. The key lies in the fact that you can't equate the "potential of the loads" to the work done in stretching the springs. In fact you can't derive or deduce the expression for load potential because it is DEFINED as load times displacement, with no 1/2 factor. It is defined as such in the principle of minimum potential energy because the principle is derived from virtual work. During the virtual displacements used in this derivation forces remain constant. An expression defined as the "total potential energy of a system" results from this derivation, and this expression is a minimum when the system is in equilibrium. Total potential energy has two parts: strain energy of the body and another part equal to loads times displacements which is DEFINED as potential of loads. If you accept it as a DEFINITION and disassociate it from the strain energy in the spring, it might become less puzzling. If you want to dig deeper into some texts for a longer explanation, let me know and I'll pass along some citations.

 

[ThetaJC]

Jerzy,

Thanks much for the most informative response. I believe I am just thinking too hard as usual. When you put it in terms of accepting a definition, then I can appreciate the problem. After doing some more contemplating about the problem, I came to a comfortable mental compromise. I simply imagine the unstretched spring "floating" in space as state 1. Now instantaneously I apply a force that causes a displacement q. With the force now applied and the spring at a stable condition, I will mark this as state 2. Now to appreciate the potential energy of state 2 w/ respect to state 1, I imagine going from state 2 BACK to state 1. To do this I may "recieve" energy from the spring in the form of positive work of 1/2kq^2, however, I must deduct Fq from the total, beings I am now going from q to 0 and am "fighting" the force to come back to state 1. This leaves me with 1/2kq^2-Fq, and is in agreement w/ the books approach. This is of course assuming a constant force which can be explained by accepting the definition of total potential energy as like you said a DEFINITION. Thanks again for your help.

 

[jerzy]

Hello ThetaJC,
I think another cause for confusion on this point is that many Finite Element texts (especially those aimed at structures and solid mechanics) start out with a "direct" approach where forces acting on a node are equated to zero. Then they generalize using variational methods in which "total potential energy" is used: more abstract and less intuitive than the "direct" concrete approach. I'll keep looking for some clear but brief way to distinguish between the methods and make the definition a little easier to accept, because it certainly is counter-intuitive.
I think your probing "thinking too much" method will lead you to real insight into FEA, rather than a superficial knowledge of buzz words and acronyms.
jerzy

 

[ThetaJC]

Got Cook's book and found it to be a very nice supplement to the Chandrupatla book. Thanks for the suggested reference Jerzy. At this point I have put any remaining doubts due to the total potential energy on the back burner to be resolved at a later time. Now I am jumping into shape functions; what novel concepts! I have used FEA programs to solve problems, but I really appreciate the complexity now, just looking at a one dimensional element problem. Cool stuff!!!

 

[jerzy]

Hello ThetaJC,
The edition of Cook I'm currently using is the fourth: I figure books that reach four editions are probably pretty good. Another thought on potential energy confusion:
I know I sometimes confused the PRINCIPLE OF MINIMUM POTENTIAL ENERGY with the LAW OF CONSERVATION OF ENERGY.
You can use the conservation of energy to derive the stiffness formulation, and Cook does so on the bottom of p. 139. Here the 1/2 factor shows up because we are equating work done to strain energy in the spring. We don't equate work in the Minimum Potential Energy formulation, so there is no 1/2 factor. Enjoy your Cook Book.
jerzy

 

[dollarbulldog]

Hi

There is some confusion in the terms of load potential. The load potential is caused only by 'conservative force' type; internal forces and followed loads like damper or fluid pressure are non-conservative and they create no potential but destabilizing stiffness-like term. If you are considering the discreted end forces, that forces must be external conservative forces to cause potential. Work done from internal spring force in your case must be interpreted as strain energy since they originated from internal stress. The factor 1/2 in TPE is not universal but it's valid only for elastic material;anyway, the internal strain energy is indeed integration of internal potential just in case of hyperelastic media (no dissipation). Linear material is a special case.

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