Stress Strain Curve

[NaikD]

All,

We are using a 0.148" wide, 0.45" long and 0.008" thick SS 301 full hard strip to preload another component. The strip has two clearance holes at both ends. Two M1.6 socket head cap screws are used to preload the strip.

So basically, it is a fixed beam with a center load. The deflection measured at the end of the strip is 0.012". FEM analysis shows that the stress, 151 ksi in the strip is in the plastic range of the stress strain curve.

Unfortunately due to CTE mismatch, during cold cycles, although the strip contracts on the component level, overall effect on the strip is adding more deflection on the strip as the both fixed ends move down on the assembly level. On stress strain curve, the operating point moves further in the plastic range, stretching the strip and adding more plastic strain and increasing the length of the strip and increase in the preload.

During hot cycle, strip relaxes due to overall expansion of the assembly, in addition the strip also expands. How can I represent this on the stress strain curve? Additional strain (along + ve x axis) due to thermal expansion of the strip and simultaneous reduction in the plastic stress (along - ve y axis) would not put the operating point on the stress strain curve.

Are there any formulaes/ideas how can I interpret this on the stress strain curve?

Your help is appreciated.

Thanks,

-NaikD

 

[israelkk]

"So basically, it is a fixed beam with a center load"
This assumption is may not be accurate. The M1.6 socket head cap screws that are used to preload the strip are not equivalent to the theoretical "fixed support". A much heavier support is needed to be considered even close to fixed support conditions. Only welded ends to much thicker base will give a real fixed support.

This is a common error of many designers and that is the reason for the larger deflection than expected and calculated.

When you release the load from the strip is it returning back to its free shape or has a permanent bend/deflection?

 

[rb1957]

perhaps it's too obvious (so there's a reason why not) but why not add a thermal loading to your analysis ? Then if you want to include thermal deformation in the problem, you'll need a geometrical non-linear FEA.

also, 0.012" deflection on a section 0.008" thick, make sure you've got the shear deflection flags on ... this deflection is probably enough to warrant a geo-non-linear model anyways (particularly as the "beam" is only 0.45" long ... it'll be quite curved in it's deflected shape).

btw, i'm picturing that the strip is in the horizontal plane, with reaction bolts and the applied load applying loads out-of-plane ... with this geometry a fixed end assumption probably isn't too unreasonable.

what about fatigue considerations ? particularly with full hard 301 ??

 

[NaikD]

Israelkk/rb1957

Thanks for quick response. You both are correct. I should have given more details in my earlier post.

1. After preloading the beam with M1.6 SHCSs, The screw heads were staked using epoxy, hence the assumption that the beam is fixed at both the ends. That means the beam will not be able to use bolt hole clearances to certain extent to allow beam deformations due to thermal cycles since it it staked.

2. We did non linear analysis and the results show that the beam is in plastic zone on the stress strain curve.

3. The beam is in horizontal plate and the screws along vertical axis just like we would draw a static line diagram for beam loading.


4. We have not taken the staking off to see whether the beam returns to its original position yet. In addition, while taking the staking off there is a danger of damaging the beam further since it is only 0.008" thick. The only indication here from FEM analysis is that beam has seen some plastic deformation.

I am trying to find out how can I represent the thermal elongation (increase in strain) during hot cycle and corresponding reduction in stress (in -ve y direction) on stress strain curve. Any ideas?

I am not good at plastic hand calculations something that I was wondering how to go about to get some numbers on final stress and beam length after all the thermal cycles.

Thanks again,

- NaikD

 

[rb1957]

if the beam is deflected into curve (and gone plastic), when you induce an elongation (due to thermal strain) i picture the beam deflecting more, rather than increasing its internal stress, because i doubt there is sufficient axial restraint in the beam. so would the increase in length be apparent between the points of inflection of the beam ? possibly these points of inflection would move slightly towards the ends (probably by a very small amount). maybe if you drew the deflected beam, then increased the arc length between the points of inflection by the thermal elongation, maybe this is near enough ???

what if the thermal condition is imposed before the deformation ? then the beam would develop the requisite thermal stress ('cause the end restaint is high) then impose you transverse load ...

if you have a NL model, you should be able to apply loads in time sequence ...

again, i hope this is a static condition, and not a fatigue loading ...

 

[NaikD]

rb1957

I think it is a classic textbook example of low cycle fatigue induced by thermal cycles. But I do not know how to represent it on the stress strain curve and compute preload after the end of thermal cycles.

The stresses are in plastic zone. Since the thermal cycles are less (only 26) there is no rupture/failure but the preload decreases in each hot cycle and increases in each cold cycle while the total length of the beam keeps on increasing irrespective of hot or cold cycle as the stresses are in plastic zone.

Although the beam contracts in the cold cycle on component level, it expands due to end constraints and due to CTE mismatch on the assembly level increasing preload (because the span of the beam increases on assembly level while beam is trying to contract on component level) and hence moving up/right on stress strain curve in plastic zone.

In hot cycle it expands (because the beam expands between the span on component level) on component as well as assembly level (because the span of the beam decreases at assembly level) reducing the preload. One would think that the beam would move back down to the left on the stress strain curve but the strain in this case is positive. The difference here is stress reduces while strain due to thermal expansion increases permanently at all times.

The strains in beam in cold as well as hot cycle are permanent, positive and accumulative.

The thermal cycles are applied after the preload on the assembly is in place and the screws are staked.

Thanks,

NaikD

 

[GBor]

Most non-linear FE codes have a temperature-dependent material option into which you can place stress-strain data for various temperatures...sounds like you may need to look in to this, if I'm understanding correctly.

 

[NaikD]

GBore,

Thanks for responding. The temperature range is -25 deg C to +60 deg C. The temperature stress strain curve for this material ASTM A666 (SS 301 full hard) that I can use is at ambient temperature as all other temperatures are very high (400, 600, 800 deg F). In addition since the strains are additive, I am more inclined to do hand calcs and present the successive points on stress strain curve.

If I can do that for first hot cycle, I will be able to do it for all thermal cycles. Any ideas?

Thanks,

NaikD

 

[GBor]

I would have to find out a little more about the material property data for fully hardened 301 stainless, but you can usually do some type of interpolation between these data points (i.e. how does the stress vary with temperature for a given strain, draw this curve through the data points in the ASTM and then pick out your particular temperature of interest from the graph). You can get some idea of the temperature dependent stress-strain data from this interpolation, or use a single temperature interpolation in your hand calc.

The hand calc idea is usually a good one, but your situation seems to be a little more complicated than looking up a formula in "Roarks' Formulas...". IsraelKK already raised questions about your end connection simplification. It isn't as easy to match a test with a hand calc as most would think.

Garland

Garland E. Borowski, PE
Borowski Engineering & Analytical Services, Inc.

http://www.borowskiengineering.com

 

[NaikD]

GBor,

Actually I realized that today, while thinking how I am going to go about it. And my thinking is similar on your lines. What I realized was I should not be looking at stress strain curve only at ambient temperature. The operating point as the system goes through thermal cycles jumps from one curve at -13 deg F to 68 deg F (ambient) to 140 deg F.

Now that I have stress strain curve for 68 deg F (ambient temp), 400, 600 and 800 deg F and since survival temperature is (-13 def F and +140 deg F), I need get datapoints from hardcopy and put those in excel spreadsheet (as I do not have digitizing software) and try to create/interpolate curves for -13 deg F and +140 deg F and then plot the operating point at a particualr temperature based on the stresses and strains due to thermal cycles.

I think I can do this in spread sheet with 26 columns to compute resulting stresses, strains and total beam length at the end of all 26 cycles.

Regarding the beam end conditions, since the screws are staked it is as good as fixed beam condition. However, it will not hurt to check what numbers you get for simply supported beam.

I feel comfortable now. Thanks for having a good discussion.

One last question, would this phenomenon fall in the category of "creep"? Normally creep happens at temperatures above 0.3 times melting temp. of metal. For steel that would be 500 deg F. I am tempting to call this low cycle thermal fatigue. Although the cycles are neglible for failure to occur. In my case only preload decreases due to thermal cycles.

Regards,

NaikD

 

[GBor]

I wouldn't expect creep, but you may want to run this question through the metallurgy and materials forum. You would likely get some interesting responses from them, and I suspect this question would be pretty elementary for most of the seasoned metallurgists.

 

[NaikD]

GBor,

I already have. According to our metallurgist, for ASTM A 666, (301 full hard) creep is not in play here although creep data for such low temperature do not exist, maybe for reason. I did extensive survey on creep. Creep is unimportant below temperatures close to 0.3 times melting temp (in Kelvin) which is about 500 deg C for BCC, body centered crystallographc stucture of 301 full hard. And we are no where operating around that temperature.

Thank you all,

Regards,

NaikD

 

[rb1957]

surely creep is only a factor for sustained loads ?