peteygee
Mechanical
- Aug 1, 2002
- 1
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.
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.