Principle of virtual work for continuous systems

[jkpennings]

I always thought that the principle of virtual work (PVW) is valid for all structures, including continuous structures (like bars, beams, plates, etc.). However, in his book 'Fundamentals of Structural Mechanics', Hjelmstad states that the PVW is only valid for discrete systems with N particles, and that for using the PVW for continuous systems, an approximation like Ritz' method should be used, see attachment. I actually do not understand what Hjelmstad means in this text. Can you please help? I'm really confused right now...

 

[PhanthomJay]

Are you in grad school or working on a complex project? Regardless ,you have every right to be confused, because Mr or Ms Hjelmstad's book on ' Fundamentals' is far beyond being fundamental, at least with thIs statement you have attached. Maybe he or she is just showing off. PVW is a valuable tool in analyzing indeterminate continuous beams and trusses. So maybe someone else can help you, but from my perspective, it is the basics of structural mechanics that are most important.
IMHO.

 

[jkpennings]

I am just a curious grad student in Applied Mechanics:) Do you understand why the PVW is not valid for continuous structures according to Hjelmstad?

 

[PhanthomJay]

I'm still trying to figure out what is a discrete system with N particles? Sounds like the author took a page out of Wikipedia. I took all sorts of advanced courses in grad school, none of which I remember, but all of which in some way was useful. If you are going on to a PhD level or Research, perhaps you need to investigate it more. I am wondering why you chose Applied Mechanics instead of Mechanical or Structural Engineering?

 

[jkpennings]

See attachment, this is what I mean by a discrete system with N particles :) The upper cantilever is a continuous structure, while the bottom cantiliver is a discrete system with N = 4 particles, in which the actual continuous stiffness is concentrated in those 4 nodes. If the number of N goes to infinity, the system becomes continuous.

 

[PhanthomJay]

See attachment, this is what I mean by a discrete system wit N particles :) The upper cantilever is a continuous structure, while the bottom cantiliver is a discrete system with N = 4 particles, in which the actual continuous stiffness is concentrated in those 4 nodes. If the number of N goes to infinity, the system becomes continuous.

Oh, it looks like your into some sort of dynamic vibration analysis, which is not my area of expertise. Any dynamic analyses I have done have often employed the use of dynamically 'equivalent' static loads, such as impact and earthquake loads where dynamic loads are adjusted to reflect for example static dead loads that are multiplied by a 'g' factor depending on natural frequency and with a lowered overload factor. For static analysis of determinate and indeterminate continuous systems ,PVW is a valuable means of analysis for determining deflections and reaction loads. Maybe it's no good for vibratory loading. Sorry I can't help further.