Prinicipal of minimum potential energy in 1D truss element derivation

[peteygee]

Hi,

I am struggling to get to grips with the derivation of a 1D statically loaded linear elastic truss element. In particular the concept of minimum potential energy.

I'll just run you through the steps described in my book, then ask you about my problems with it...

The truss is defined as nodes i,j with corresponding displacements Xi,Xj and displacements Ui, Uj.

Matrix terms used in following:
Displacement vector:
= [Ui; Uj]
Element Stiffness matrix:
[K] = [1 -1; -1 1] (; = next row)
Forces vector:
[F] = [Fi; Fj]
I have denoted a transpose as ^T e.g. ^T

Steps to derive element equations:
1) Derive strain energy (SE):
SE = AE/2L * [Ui; Uj][1 -1; -1 1][Ui; Uj]
= 0.5 * ^T [K]

2) Define "potential energy" (PE) to be:
PE = SE - W
(where W = work done by external forces)

3) Derive "Work done by external forces" (W):
Quoted from book: "For a bar element, the only external forces that can be applied are nodal forces (Fi and Fj) acting at the ends of the bar, so that the work done by external forces":
W = Ui*Fi + Uj*Fj = ^T [F]

4) Therefore for the single bar, the total potential energy is:
PE = 0.5 * ^T [K] - ^T [F]

5)For minimum PE, the displacements must be such that:
d(PE)/d = 0
which gives you...
[K] - [F] = 0


I understand all of this apart from steps 2 and 3.
- What is "potential energy" in this context? The way I am thinking is that the potential energy should just the stored up strain energy. By the definition here, if you subtract the work done, it will give you a total of zero "potential energy" ?
- Assuming that the "potential energy" is just an unfortunate name for a concept rather than the normal use of potential energy, why is the work done equal to force*distance ? Surely in a linear elastic case the force is proportional to distance, such that W = (Integral of)F.dx
which would give you 0.5*force*distance.

I have seen this derivation in several course notes now, without shining any light on the problem for me. If you can help explain this concept to me I will be very grateful because I have spent many hours now trying to understand what is going on here.

Thanks,

Peter.

 

[idly123]

Well good to think and question

Well potential energy here (PI) is refered to "Work Potential".
just read between the following lines:
.ability of a system to do work( in terms of mechanics analogy..something like moment of resistance of a section)

or ability to get work done on a system.
(in mechancis analogy we call it as applied bending moment)

can u make out the difference. this is what is basis of force and displacement method or flexibility and stiffness method

Well force represents a state of equlibrium. deformation is associated with compatibilty and material property.

How much work is done on a system is dependent on deformation , at the end of whic again the system s in equlibrium.

It is the work done by the force in moving through a displacemnt = work done= force * displacemnt

The derivation 1/2*force * displacemnt is because u are doing work from a zero to a maximum displacemnt or strain. so its the area under a stress block.

this is the work done on the system.
IN doing this the force also hase donea work by moving from is equlibrium to a new equlibrium.. which is force * distance.

Hope i have helped u to an extent

regds
raj Raj
Structural Engr.