Composite Calculations Worksheet

[tropostudio]

Spurred on by advice from from advice in the 'Mast loads for freestanding masts' thread, I'm doing legwork on composites design and engineering. Basically busting open and studying books I've owned since the 1990's, while accumulating even more!

Spreadsheets are a great tool for thinking through written design methods and for comparing values for variables that can be arrived at in several ways. For example, Transverse Tensile Modulus for a single-ply uni-d lamina, E22, can be formulated from Rule of Mixtures (basic mechanics of materials ), Halpin-Tsai, Chamis, Brintrup, DNV rule, et al.

I've started a spreadsheet to help me understand composite design principles and to compare values, with the hope that I can use it as design tool. A copy of the work-in-progress is attached in both XLXS and ODS formats. (I am using Excel; ODS is a 'save-as' version).

The Single Ply DNV worksheet was started to follow the design method in DNV-RU-Yacht; Part 3, Chapt. 5 - Composite Scantlings. DNV uses 'generic' values for transverse tensile modulus, EfT, for constituent fibers (sans resin). Looking for this value in manufacturers' datasheets came up empty. That led to worksheets for Fiber and Matrix data.

More research lead to the worksheet for Transverse Modulus calculations, in an attempt to justify the DNV formula for E22, which depends on a known value for EfT. None of the other methods in that worksheet require a constituent value for EfT. I still don't know where DNV gets their value, but now I can compare E22 between DNV and several other methods! BTW, the Halpin-Tsai model is really cool, as extremes for ℰ of ∞ and 0 converge on the upper and lower bounds for E obtained using the Rule of Mixtures (Voight-Reuss bounds).

As of 7/27/24, I've ended with the ability to graph the transformation of a single ply UD lamina into the global coordinate range (0-90 degrees).

I am looking for review of the workbook, a critique of any formulas or analyses, and suggestions for improvement. Thanks!

 

[C. Dog]

This is a great idea, and maybe could be converted to a small application for mobile when mature. On the question of ODS, I have been using Linux for 20 years and Open/Libre Office has matured into a wonderful suite of applications in that time. I find that the word processor software has some issues regarding interoperability with MS Office, mainly with rendering, but spreadsheets and whatever functions I have had in them work flawlessly between the two and I would suggest that you only need to provide the XLSX version, if there were issues then it is probable that the MS Office ODS version would carry them also.

I had a bad night and am not motivated to check formulas, but I am happy to contribute what I can to the effort. Is there a way of setting this up for collaboration where versioning will be taken care of etc., like Github for code? Then there is little possibility of losing data or upgrades making the document totally unusable.

 

[tropostudio]

Ah, I see the ODS version is problematic in Libre Open Office Calc. I think the Calc has all of the functionality to replicate the XLSX version, but it will require some manual 'tweaking.' Dropdown lists, named ranges, and a few other things work differently. I may keep up an ODS version alongside Excel, but that's a low priority.

I'll look into sharing it on GitHub - in the meantime, if I post updates in this thread I'll use the convention of 'Filename_6-digit date of issue_Initials.' At some point I will automate parts of the process in Excel using VBA to make it more 'usable.' I don't plan on hiding any formulas or macros, although protecting certain ranges would help . Right now it's easy to blow things up!

I'm working on a sheet using Martin Hollman's methods to compare results to DNV methods for single-plys in global coordinates. I have a copy of the inexpensive Windows program "The Laminator." Results of a runs of single ply laminates between the Single Ply DNV worksheet and "Laminator" show good agreement.

 

[tropostudio]

Not so much progress forward on the spreadsheet as hoped, but it's all good: the next move was to work through multiply analysis and comparison between methods. When I got into A. B and D matrices, I figured it was time to gain a through understanding of what each one matrix was considering. After going through all the PDF docs I've accrued, I picked up a used copy of Autar Kaw's Mechanics of Composite Materials. Definitely a good investment: It does a great job of backing up all the formulas with the physics and math behind them. You can skip to the harder stuff if you know, it, but you can also gain understanding from the bottom-up. You can find much of it online, but I prefer holding onto paper and flipping pages:


Here's a screenshot from the worksheet comparing transverse elastic moduli of a UD Ply using different methods using DNV values for Generic E-Glass and Epoxy Resin at 50% volume fiber fraction:

Rule of Mixtures for E22 would be a lower bound, and ROM for E11 would be an upper bound for lamina performance. DNV and Chamis don't coincide with E11 at fiber fraction =100%. Halpin-Tsai does, because the formulas are designed to do so. Regardless, realistic fiber fractions will be be on order of 30-70%, so you can see they aren't far apart for the useful regime. Not sure why the Brintrup formulas is mentioned so much in literature, other than it was part of early research in composites and it's in his PhD dissertation. From what I can find, his numbers were based off e-glass and polyester resin samples, and extrapolated from there.

Next are DNV values for a single UD ply rotated through global coordinates (o-90 degrees):

I haven't gotten into mapping a Tsai-Massard tensile and shear modulus yet, but looking at a 1985 copy of their 'Composite Design' report/book has me guessing it will be pretty close to DNV. I picked up several of Martin Hollman's books at an EAA Convention in the early 1990's. I finally got around to mapping out some of the formulas from his 'Composite Aircraft Design 'manual. Here are values for tensile and shear moduli for a typical UD E-glass and Epoxy lamina at 50% fiber volume fraction rotated through 0-90, compared with DNV values:

Martin Hollman was a smart guy with a lot of experience in aeronautical and composites engineering. From reading his self-published books, he did a lot testing on his own UTM. He notes his formulas for UD and BI-D tensile moduli were based on sample testing and comparison with other methods. Note the difference between his UD tensile modulus values vs DNV. Way different, way more conservative. Again, I haven't mapped any of Stephen Tsai's method onto this, but I am certain they are much closer to the DNV curve. Tsai's values are all based on curve fits to experimental values. I have found discrepancies between Hollman's formulas and graphed values (which would be ostensibly from his UTM?). I think Hollman needed a better proof-reader. Note the discrepancy between the columns "Hollmann Corrected' and 'Hollman Error?': The corrected values are a small change I made to his formula to match his graphs. The 'Error?' values graph his formula, but don't match his data graphs. Another concern: in his book he does not present the basis for deriving his formula for shear modulus. One has no way of knowing whether it is a fit to his own test values, a derivation of another formula, or something else. In any case, his published values are lower than DNV, so one might expect them to be extremely conservative. Onward...

 

[rxcomposite]

When I saw your modulus equation column k, row 45, I was looking for Youngs modulus equation (attached, highlighted in orange). This is an industry standard for finding the strength/modulus of a laminate on a given load angle. Image from BV, issue 1979. It is used by ISO, S. Mayers, BV, and others. It is called engineering constants and further expounded in the book Filament Winding by S. Mayers.

LR has a slightly different formula. It matters if you are building to Class Rules or just plain ISO regulations. There are also simplified ones like the Krenchel factors.

M. Hollmann is a pioneer in the composite industry when it was declassified in the early 80's. His discipline is in the aerospace industry and is not constrained by Class Rules or marine application. The curve he developed is representative of his test. I find his equation conforms to what he is describing in the book. You are right, there are many typo errors in the book, mainly the order of operations. Probably the proofreader is not so inclined. It occurs mostly in the long equations used.

 

[rxcomposite]

In your image of Longitudinal and transverse moduli you included the ROM method. This cannot be used based on elastic modulus. Modulus is a straight line (imaginary line, Hooke's Law) and thus will give you a linear progression result not quite consistent with the rest in your image.

A long time ago I posted here the method to use (ROM) using different fibers and different matrix materials. It is based on the Stress/strain curve of the matrix. Thus the curve will show the elastic, plastic, yield, and failure points. Not a straight line.

 

[tropostudio]

In your image of Longitudinal and transverse moduli you included the ROM method. This cannot be used based on elastic modulus. Modulus is a straight line (imaginary line, Hooke's Law) and thus will give you a linear progression result not quite consistent with the rest in your image.

Regarding the Transverse Moduli worksheet: E11 is the Young's modulus of the UD lamina in direction of fiber orientation. It is based on ROM, and one of the only formula's consistently agreed on in all the analytical methods I reviewed. E22 is the Young's modulus of the lamina perpendicular to fiber orientation (aka Transverse Tensile Modulus). E22 is calculated via the 5 different formulas below. The method using a rearranged ROM is the simplest 'mechanic's of materials' solution, and establishes the lower bound for any lamina. Brintrup is a close match. The E22 DNV formula is from their most current version (2022), and matches the highlighted formula for E2 in your 1979 BV report.

I found it interesting to compare how the different methods for calculating E22 varied depending on fiber type. First is a graph of 'generic' E-glass and epoxy resin at 50% fiber volume:



Next is a graph of 'generic' standard modulus carbon fiber and epoxy at 50% fiber volume:



The DNV and Chamis methods do not converge to E11 at 100% fiber volume. As mentioned earlier, fiber volumes in the range of 25-75% are realistic, so the extremes aren't that important.

The Halpin-Tsai equation is interesting. It is an extension of the ROM/mechanics of material formulas to allow for better fit with experimental results on the range of composite fibers, where the formulas for Poisson's ratio that work with isotropic materials don't hold up. By varying the value ℰ in cell M34, you cover the range of elastic moduli in a ply from longitudinal, to transverse, shear. A value of 2 is suggested by Tsai for most fibers as a best-fit of experimental results for E22. As ℰ approaches infinity, you converge on E11 (ROM). Here is e-glass/epoxy at 50% fv, with ℰ at 200 :



As ℰ approaches 0, you converge on E22 (ROM). Here is e-glass/epoxy at 50% fv, with ℰ at 0.10 :



I trust Stephen Tsai. His experiments and analyses for the Defense Dept. started in the early 1960's, and he is still publishing new methods with his long time associate Thierry Massard at age 95.

 

[tropostudio]

Still reading to get clear on basics of macromechanical analysis of laminates and stresses/strains in the 'ABD' matrix before setting up multi-ply laminate worksheets. A little bit to add to a single-ply UD through rotation from orientations 0-90 degrees using several methods (charts now list the selected fiber and resin inside a text box in the chart) :

Here is generic E-glass WEST systems 105:


Here is generic standard modulus (high-strength) carbon fiber with Gougeon Pro-Set:


The curves differ greatly in off-axis predictions. And mostly due constituent fiber properties (resin doesn't affect much - but that's a no-brainer). Hollman and Tsai or Tsai-Massard formulas are based on experimental fits. Because one can never have too many projects, it's time for my UTM machine:



Yes, the head is cocked...previous owner blew a ball-nut on it. 30k# rating and the 'Price was Right.' Repairs don't scare me...been there on many machines. Has a working PC and software, tensile grips, 2 each 30k load cells, and documentation. Psyched!

 

[tropostudio]

Sort of an update. Not composites calcs per-se, but it does accompany my interest in verifying anlayis with experiment. MTI 30 kilopound UTM pushed into the garage and under repair. Fixing this should be child's play compared to restoring the Howa 16-tool VMC behind it! Similar only in that both were purchased for scrap prices.

@rxcomposite - I know you suggested I just do the math and trust the experts; but as engineers - they were correlating formulas with experiment (some doing a better job than others). At a minimum, I will be able to correlate my own lay-ups to to claims of others and to theory.

 

[rxcomposite]

Is this collective, "you and I" or just assumptive (I know you)? Aren't the "experts" we are referring to engineers themselves?

Are we talking of predicting Material Properties or Failure theories?

 

[tropostudio]

Is this collective, "you and I" or just assumptive (I know you)? Aren't the "experts" we are referring to engineers themselves?

Missing a word or two, which I fixed : "I know you suggested I do the math..." I buy it, and I'm doing the math. The desire to verify samples myself just seems too cool, especially if I can get a 30 kilopound UTM running in my shop for under $2k USD! (It'll probably cost more than that to have the 2 load cells calibrated, if and when that's a concern).

So many tests I can see being run beside the typical tensile coupons. Panel deflection comparisons under cyclic loads through a shop-built hydramat, for example. I think of stuff along the lines of what Gougeon Bros has done over the years where they published a few comparative results for end users, but you don't get procedural details and full results.

Certainly the 'experts' being referred to are engineers. There also seems to be enough variation between predicted stress or strain values for lamina depending on formula's used and where the experimental data came from to warrant 'checking one's own work'. I can't think of a better way to see how lay-ups one produces correspond to prediction than either hiring a test agency or running your own tests

Are we talking of predicting Material Properties or Failure theories?

I'm not quite sure what this means. I would think the two are tied together. Are you referring to strain-limited vs stress-limited apporaches? Within a prescribed range of elastic deformation and fiber volume fraction, for instance, predictions may align pretty well. With a sufficient safety factor, you may be well within bounds using a number of different predictive methods.

 

[rxcomposite]

Your objective to predict/determine the laminate properties in the most accurate way is laudable but you have a long way to go. You are also jumping the gun by showing the behavior of the laminate at differing load angle. It is only as good as the data you put in.

There are three distinct phases in composite fabrication. 1. Predicting Laminate Properties 2. Determining Properties at differing load angle 3. Application of Failure criteria

You are still in no.1. Consult the archives of Composite Fabrication magazine, October 2001, Determining Laminate properties by Robert Schofield. Here you will find the data for determination of laminate properties. Schofield is a renowned Naval Architect specializing in composites. As a bonus there is a formula for converting Weight fraction to Volume fraction.

There is also Formula based method for determining properties by Lloyds and ISO 1221-5. Word of caution, these Rules based formula are the MINIMUM criteria required and are very conservative. You need to arrive at least 4 basic properties of Modulus, Strength, Poissons, Shear. You can plot it to see the trend in various Gc ratio.

For no. 2, Stick to industry standard. Use the formula I posted that is used by most Class and other specialized composite fabrication. Note that the formulas are interdependent. You need the modulus to arrive at the corresponding Poissons ratio and Shear modulus.

For multiple ply and differing fabric type use the Classical Lamination Theory (CLT) using the matrix algebra, ABD method.

No. 3 is a vast field of subject Ranging from Tsai Hill, Tsai Wu, Hashin-Rotem, Coulomb-Mohr (ductile and brittle Method). There is also the standard method of First ply failure (Stress and Strain method + shear interlaminar failure).

Tsai likes to stick his finger in every composite theory. I don’t think Halpin-Tsai works in predicting laminate properties. ξ is a variable and varies for every VF ratio and ranges from 0.5 to 2. I get the same result you have if I don’t vary it. The Curve is supposed to fit the number of test laminate. You need data. The ξ Is the fitted curve not the prediction.