How Do You Calculate Forces in a Thermally Expanding Rod Attached to a Spring?

[Electrolight]

Piping stress analysis problem -- help please

well, hi all of you, well i am new to this forum but here is why i am here : )

the problem is:

a rod of certain length is fixed at one end, the other end of the rod is attached to a spring of stiffness k, then rod temperature is increased and the linear thermal expansion occurs in the rod, the spring on the end of the rod resists this expansion and gets compressed with a reaction force, the spring itself on the other end is attached and fixed.

please help me write the equation of forces and reactions generated in the system.

thanks,

 

[Pyrrhus]

First, what are you asked to find? do you understand the problem?

 

[Electrolight]

First, what are you asked to find? do you understand the problem?

well, i have posed the problem and all i want is to write the force equation in the system when delta T, co-efficient of linear expansion, modulus of elasticity, cross sectional area, length etc and all other things including the stiffness of spring are known.....

 

[Pyrrhus]

If you consider your system only the rod with its fixed support and disregard gravity (weight), you'll have 1 force acting on the rod becausse of the spring. This force's magnitude can be obtained by considering the thermal displacement is equal to the spring's displacement according to the geometry described.

 

[berkeman]

well, i have posed the problem and all i want is to write the force equation in the system when delta T, co-efficient of linear expansion, modulus of elasticity, cross sectional area, length etc and all other things including the stiffness of spring are known.....

Welcome to the PF. You posted in a different thread that you are an ME working professionally in the field. But this question of yours looks like basic lower division homework or coursework. Homework and coursework questions need to be posted in the correct Homework Help forum here on the PF, and you must show your own work so far in order for us to provide tutorial assistance. Should I move this question to the Homework Help forums?

 

[Electrolight]

Welcome to the PF. You posted in a different thread that you are an ME working professionally in the field. But this question of yours looks like basic lower division homework or coursework. Homework and coursework questions need to be posted in the correct Homework Help forum here on the PF, and you must show your own work so far in order for us to provide tutorial assistance. Should I move this question to the Homework Help forums?

OK, well sure, thanks for the comments, well yes i am ME working professionally as piping stress analyst in a JGC japan sub-company, well actually the problem i stated is not sooo much simple as i have put it...here in piping systems we have few lines known as critical lines and those need to be analysed by the use of computer FEA. i on the other hand am working just for my own pleasure to explore the design of the piping expansion loops - a piping configeration which is capable to absorb the linear thermal expansion by making use of the elbows and bends stiffnesses, u can can on net to have the idea of the shape of the loop. well in case of free expansion there would be no loads at all, but if a pipe of certain length is restrained then the force obtained easily by the use of ethod of super position since the problem is statically indeterminate, anyways in short the equation is obtained by equating the free expansion + the retrained expansion = 0, but here in the case of loop the problem is that the other end containing the bends may be considered as the one with flexible restraint.

so in this case what should be the equation:

thermal displacement + restrained displacement = spirng displacements ?

note that here i am trying to accumulate the elbow's flexibility characteristic into the spring contant factor...hope u understand now, other wise please tell and also tell me how to attach scan papers here on this forum ?

Regards.

 

[berkeman]

Fair enough. I moved this thread to the Mechanical Engineering forum, and re-named the title to something more descriptive of your question. You should get some good responses from the ME forum folks.

 

[Q_Goest]

Piping stress analysis in the US is governed by ASME code. ASME B31.3, Part 5, for example, covers piping flexibility analysis. As you've noted, there are plenty of computer programs that can also be used for analyzing pipe flexibility. I don't know what the applicable code is in Japan, but I have to believe they have very similar piping codes.

Regarding how to calculate stresses using basic principals, note that the 'spring' has some force which is dependant on the spring's displacement. Similarly, a pipe or any structure produces a force which must oppose the spring (equal and opposite) which is similarly dependant on the pipe's displacement. You can think of a pipe being nothing more than a spring with a spring constant which depends on pipe geometry and material properties. Simply equating these two forces should allow you to determine what the deflection and subsequent stress in the pipe is.

 

[Electrolight]

well, yes we also work on ASME B31.3, and use CAESAR II for analysis. code also gives u various formulas etc to evaluate the flexibility and reactions, but i was just reding at some other document that the problem of a length of pipe with elbow and then extending perpendicular to the original run may be taken as guided cantilever, but i didn't want to go using that equation as i intended to think of my own and came up with the simple model of the problem as i have poted it originally, so we may develop some equation of our own and this equation should work better too... well here is what i have currently.

the thermal expansion is the product of thermal co-efficient, length and delta Temperature, this is the free expansion, if this length is retrained then then restraints apply the redundant reaction force on the pipe not allowing it to expand, and a state of problem with stress and no strain at all is created, the problem then becomes the statically indeterminate. the easiest method to solve this is to first calculate the free expansion, then apply the restraint. and in case of that the deflection = (Force*Length)/(cross-sectional area*Modulus of Elasticity), now we have two deflections here, one free thermal expansion and one the restrained deflection for the same pipe, since the case is that total displacement = 0 as the restrains do not allw the pipe actually to expand, so using method of superposition we can obtain the force simply... but i am stuck here at a step further that in our case of loop there is no free expansion at all and there is no fixed restriant at all either, but partly the expansion of the length is somewhat opposed, so we here dealing the flexible restraints, in that case what should we do ?

should the equation be like that :

deflection (free thermal expansion) + deflection (restrained deflection of fixed restraints) = spring analogy deflection ?

help pleasezzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz

 

[Electrolight]

help pleasezzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz, i have to complete my work soonnnnnnnnnnnnnnnnnnnnnnnnn...

 

[Q_Goest]

all those zzzzzz's in the post make me sleepy ...

Am I getting paid for this? Please send the check to PhysicsForums

Not sure what you're asking, but I'll take a shot. You have something like a pipe with two fixed points and then you want to change the temperature which causes the pipe to want to expand but it can't because the pipe is fully restrained by two fixed points. Is that about right?

If so, then I think you're on the right track. You determine the free dimension of the pipe after thermal expansion/contraction and then determine what forces (ie: stresses) are needed to get the pipe to go back to the original dimension.

Take a simple example of a straight pipe between two fixed points. Let's say the pipe expands by some amount. For this expansion/contraction, assume there are no forces on the pipe. Now take the total length change and divide by the length to come up with how much the length changes per unit length. This is simply strain which is stress times modulus. Divide strain by modulus and you have stress in the pipe. Simple, right?

The problem with doing it this way is that this makes the assumption that there are fixed points. As I'm sure you're aware, pipe supports are almost never fixed, even when they're supposed to be. In reality, pipe supports almost always have some spring constant to them. Pipe systems are extremely difficult to analyze by hand because of all these interactions, which is why computers are so helpful in determining stresses, forces and deflections on piping systems.

 

[Q_Goest]

See also:
http://www.eng-tips.com/viewthread.cfm?qid=189629&page=1
for many more tips on piping stress analysis.

 

[Electrolight]

all those zzzzzz's in the post make me sleepy ...

Am I getting paid for this? Please send the check to PhysicsForums

Not sure what you're asking, but I'll take a shot. You have something like a pipe with two fixed points and then you want to change the temperature which causes the pipe to want to expand but it can't because the pipe is fully restrained by two fixed points. Is that about right?

If so, then I think you're on the right track. You determine the free dimension of the pipe after thermal expansion/contraction and then determine what forces (ie: stresses) are needed to get the pipe to go back to the original dimension.

Take a simple example of a straight pipe between two fixed points. Let's say the pipe expands by some amount. For this expansion/contraction, assume there are no forces on the pipe. Now take the total length change and divide by the length to come up with how much the length changes per unit length. This is simply strain which is stress times modulus. Divide strain by modulus and you have stress in the pipe. Simple, right?

The problem with doing it this way is that this makes the assumption that there are fixed points. As I'm sure you're aware, pipe supports are almost never fixed, even when they're supposed to be. In reality, pipe supports almost always have some spring constant to them. Pipe systems are extremely difficult to analyze by hand because of all these interactions, which is why computers are so helpful in determining stresses, forces and deflections on piping systems.

hmmmm, u got the point, but what u think abt the flexible retraint ?

anyways thanks for the link, nice link it is...

 

[MSS21413]

Hi,

I need to learn stress analysis using CEASAR II Ver 5.0 software. If anybody is out there who willing to teach online I will appreciate and compensate the time. I am unable to attend any seminars.
Appreciate for your response.

Thanks

 

[MikeLizzi]

Here’s my understanding of the problem

Start with:
You have a pipe fixed at one end. You have a spring also fixed at one end. The free end of the pipe touches the free end of the spring. Everything is in equilibrium and there are no forces.

Then:
You raise the temperature of the pipe and it gets longer. As it does it encounters a compressive force from the spring because it is compressing the spring.

Question:
What is the relationship between the change in length of the pipe and the change in temperature?

My Approach:
A) Recognize that the change in length of the pipe is the sum of two effects,
(change in length of pipe) = (thermal expansion) – (mechanical compression)

thermal expansion = l(dT)Alpha
where l = length of pipe
dT = change in temperature
Alpha = coefficient of thermal expansion

mechanical compression = Fl/AE
where F = longitudinal force on the pipe (also the force on the spring)
l = length of pipe
A = cross sectional area of pipe
E = modulus of elasticity of pipe

B) Recognize that the change in length of the spring is a result of only one effect, (I am assuming the spring is thermally insulated from the pipe)

(change in length of spring) = (mechanical compression of spring) = F/k

where F = force on the spring (also the force on the pipe)
k = spring constant.

C) Recognize that (change in length of pipe) = (change in length of spring)
So, l(dT)Alpha – Fl/AE = F/k

You can use the above equation to calculate the force on the spring (or pipe). Once you know that, you can calculate the change in length of the spring (or pipe).

Hope that helps.

 

[StressBom]

MSS21413,

Pipe stress analysis is not Caesar v5.0 or any software... It is just a software, a tool for pipe stress analyst/engineer.. having such software will just make you dangerous... you are capable of getting someone KILLED and LOST OF INVESTMENTS. Please work with your senior or at less more experience Pipe stress analyst..

Will happy to hear from you in http://www.coade.com/support_discussion.asp. I’m a member of this forum. FYI we have ASME committee members on board in that forum. I hope this will help you.

Regards,

bom