Thermal stress in thick walled cylinder

[NovaStark]

I am trying to approximate the stresses due to temperature and pressure in a thick walled cylinder. I attached the results of the calculation however, I suspect my formulas that I am using are incorrect.

I derived them by using the basic elastic theory equations. However as it can be seen, my axial stress is overly high so I am not sure if I am using the correct equations. Can anyone point me to the correct direction to get the proper equations or see what I am doing wrong in my calculation? I am trying to get API 530 to see what they use for stresses.

 

[prex]

Sure those equations must be wrong: there is no axial stress in a tube unless it is axially restrained or supported in some way. And where is the shear stress?
However without any clue as to the origin of those thermal equations and to what you intend to do, it is difficult to say more.


prex
[URL unfurl="true"]http://www.xcalcs.com[/url] : Online engineering calculations
[URL unfurl="true"]http://www.megamag.it[/url] : Magnetic brakes and launchers for fun rides
[URL unfurl="true"]http://www.levitans.com[/url] : Air bearing pads

 

[NovaStark]

Sure those equations must be wrong: there is no axial stress in a tube unless it is axially restrained or supported in some way. And where is the shear stress?
However without any clue as to the origin of those thermal equations and to what you intend to do, it is difficult to say more.

I am just trying to get the average stress that the tube experiences during service. It is a vertical tube inside of a furnace, so it is fixed to an outlet header at the bottom and to an inlet at the top. I used the thermal stress in cylinders and disk as given in attachment 1.

 

[davefitz]

prex is right- the only effective axial stress is that caused by fluid pressure and gravity load- the thermal stress axially is zero unless the cylinder is both constrained and compressive buckling is restrained. In a successfully designed unit, the cylinder is top supported and free to grow downwards.If axial growth was restricted, then compressive buckling would occur to releive the compressive stress.

There also is no thermal stress in the steady state if the temperature distribution is only due transient startup / shutdown conditions. If the temperatrue distribution is due to a steady heat transfer ( as from a flame in a furnace), then the thermal stress distribution shown may be correct if heated around the full circumference.

During a fast transient , the thermal stress in a thick walled cylinder can be found from the estimate provided in EN 12952-3 paragraph 13, and such a calculation can be used to compute fatigue damage.

"Nobody expects the Spanish Inquisition! "

 

[NovaStark]

prex is right- the only effective axial stress is that caused by fluid pressure and gravity load- the thermal stress axially is zero unless the cylinder is both constrained and compressive buckling is restrained. In a successfully designed unit, the cylinder is top supported and free to grow downwards.If axial growth was restricted, then compressive buckling would occur to releive the compressive stress.

There also is no thermal stress in the steady state if the temperature distribution is only due transient startup / shutdown conditions. If the temperatrue distribution is due to a steady heat transfer ( as from a flame in a furnace), then the thermal stress distribution shown may be correct if heated around the full circumference.

During a fast transient , the thermal stress in a thick walled cylinder can be found from the estimate provided in EN 12952-3 paragraph 13, and such a calculation can be used to compute fatigue damage.

Well the top of the tube is supported via a spring can connected to the trunions on the tube and the bottom of the tube is welded onto a header. The tube is also heated via a downfired burner from more or less all sides. But can I still use the equations for circumferential and radial stresses due to temp and pressure (excluding the axial due to temperature)? I'll have to look for that EN 12952-3 standard as I do not have that at the moment.

Also if the tube grows downwards, then it will in fact place extra load on the header, is there any way to estimate that? Or I use thermal strain = a*dT and then use stress = E*strain?

 

[IRstuff]

"Also if the tube grows downwards, then it will in fact place extra load on the header"

What's forcing the tube downward?

TTFN
faq731-376

Need help writing a question or understanding a reply? forum1529

 

[NovaStark]

When heated, should the tube not expand in both the circumferential and axial directions?

(I am not saying that is what definitely happens, just trying to sort out my confusions)

 

[IRstuff]

"should the tube not expand in both the circumferential and axial directions?"

Not if it's fixed at the header end and free at the other. The weight does not change, so the header's stress is unchanged.

TTFN
faq731-376

Need help writing a question or understanding a reply? forum1529

 

[NovaStark]

There is an inlet pipe connected at the top (about an inch OD) with the tube being about 5.2 inch OD, will that contribute to the top being fixed or is that still considered free at the top? Also, if the latter is the case, then all I need to consider is the stresses due to internal pressure right?

 

[IRstuff]

At right angles? if so, then the smaller pipe would take most of that stress, since it's less stiff.

A picture would certainly help.

TTFN
faq731-376

Need help writing a question or understanding a reply? forum1529

 

[NovaStark]

At right angles? if so, then the smaller pipe would take most of that stress, since it's less stiff.

A picture would certainly help.

I have attached the drawing of the top section of the tube (bottom fixed to an outlet header)

 

[corus]

I'm not sure of the design standard methods but if you're taking a 1D section then the equations you should be using are for generalised plane strain conditions, and not plane stress as those apply for thin discs where the axial stress is zero. For generalised plane strain conditions you will get an axial stress as the cooler inner temperature incurs a tensile stress due to the thermal expansion of the hotter outer temperature, and vice versa for the outer surface temperature.

I haven't got the analytical solution at hand but have the results for a spreadsheet I've adapted for a general double wall cylinder (splitting the wall into two equal parts of the same material). These are shown in the attached Word document. You can check your forumla against the results shown here. If you use (incorrectly) plane stress, then the stresses are lower with a maximum stress on the inner face of 89 MPa instead of 127 MPa for generalised plane strain. You need to add on the stresses from the pressure loading to these thermal stresses.

Note that it'd be better to use a FE analysis on the whole vessel and to use these results to check your model at some point away from the effect of features.

 

[desertfox]

Hi NovaStark

If the pipe is fixed top and bottom then axial compressive stresses will be present if the pipe isn't allowed to expand freely during heating.
If the pipe can expand freely without restraint from other components during the heating period then there will be no axial thermal stresses.
I've tried to follow your posts but its not clear to me whether the vertical pipe is restrained or not, although my gut feeling is that it is restrained.

 

[corus]

For a quick approximate solution refer to Roark. For a thin cylinder the hoop and axial stress is simply 1/2.E.alpha.Delta T/(1-v). This gives a stress of about 119 MPa.

 

[NovaStark]

Thanks corus, I will check your spreadsheet. My formulas are for plane strain conditions if I remember correctly. For the thick cylinder, I am supposed to get radial stresses correct? My formulas would give the radial thermal stress as zero at the inner radius.

Hi NovaStark
If the pipe is fixed top and bottom then axial compressive stresses will be present if the pipe isn't allowed to expand freely during heating.
If the pipe can expand freely without restraint from other components during the heating period then there will be no axial thermal stresses.
I've tried to follow your posts but its not clear to me whether the vertical pipe is restrained or not, although my gut feeling is that it is restrained.

It is a vertical tube (pipe), the top is the inlet and the bottom is the outlet.

At the top as shown in the drawing, there is a smaller pipe which enters it, this pipe is connected to a header at the other end. I don't know if to count this end as fixed.
The tube is supported via a spring support attached to the two trunions shown in the drawing as well.
At the bottom, the tube is connected to an outlet header so here is fixed.

 

[corus]

Sorry, I was referring to others who said that there should be no axial stress and as such implying plane stress conditions. As for the thermal stress at the inner radius, yes, the radial stress is zero, and also at the outer radius. If you include the pressure, however, the radial stress is equal to -Pressure at the inner radius, and the outer radial stress remains at zero.

 

[NovaStark]

Sorry, I was referring to others who said that there should be no axial stress and as such implying plane stress conditions. As for the thermal stress at the inner radius, yes, the radial stress is zero, and also at the outer radius. If you include the pressure, however, the radial stress is equal to -Pressure at the inner radius, and the outer radial stress remains at zero.

I understand this, but my equations are incorrect right (for thermal at least)?

If they are not, I'd need to get the stresses at a common point to sum them up otherwise, I won't get the effective stress.

 

[chicopee]

NovaStark, bear in mind that if the tubes expand, so will the shell to which the tube sheets are attached, thereby minimizing thermal stresses. Another comment is with tubes which normally they are not thick wall tubing and yet you have decided to use thick wall equations.

 

[corus]

Your calculation agrees with my results at the centre of the wall, but note that these are not maximum values, which occur at the inner and outer surfaces. The general expressions used for thermal stresses are attached in the Word document. The axial stress is just the sum of the radial and hoop stresses.

 

[NovaStark]

NovaStark, bear in mind that if the tubes expand, so will the shell to which the tube sheets are attached, thereby minimizing thermal stresses. Another comment is with tubes which normally they are not thick wall tubing and yet you have decided to use thick wall equations.

In general I'd just take D/t and see what ratio that gives me but I've seen that thick equations will be used if D/t >10 or in some cases 20. I decided to use the thick walled equations based on the pressure used (40 bars or 4 MPa). Is there a good rule of thumb to use?

Your calculation agrees with my results at the centre of the wall, but note that these are not maximum values, which occur at the inner and outer surfaces. The general expressions used for thermal stresses are attached in the Word document. The axial stress is just the sum of the radial and hoop stresses./quote]

Ah thank you then I will re-do my axial equation then as I have an extra term.