Spar Design Aid [Download]

Hey guys, I'd like to share a little project I've been working on lately. I was optimizing the mast design for my new racing dinghy, and had a deceivingly simple question, "how much wood can I remove to have the same stiffness if I reinforce that mast with fiberglass sleeve?". Boy oh boy, was that a rabbit hole I was getting into...

So I built a little Excel spreadsheet (no macros) with some pretty heavy math to answer this and other similar questions. I included the instructions inside, but I'll make a short summary / demo in this post so you can evaluate if this might be useful to you or not. There might be some bugs, and I might want to expand it later on, so the first version is V1.00, and I'll upload updated versions too. Also uploading some example spars made with it. Please let me know if you find any errors. I'll call it Spar Design Aid (SDA for short).



Intro

Purpose:

• SDA is meant to be a preliminary design tool to quickly make up a design of a spar (mast, boom, yard, whatever), and make some evaluations to see how much it will weight, how stiff will it be, what is the factor of safety, how optimized it is, what changes when you switch up materials, try out different diameters, wall thicknesses, tapers, reinforcement layups, etc.
• Unlike the traditional mast equations (like that of Norman Skene), SDA is capable of defining and calculating multi-material tapered spars, whether these materials are laminated on top of each other (like wood core mast with fiberglass skin), or have multi-material sections (like on ILCA - aluminum bottom, carbon fiber top).
• It is meant to be beginner-friendly, hiding most of the the complexity of spar calculations, but still flexible enough to allow advanced user to set up more complicated and non-standard designs. Almost every calculation can be overriden without breaking the rest if you know what you're doing.

SDA supports both metric and imperial units of measurement for both inputs and outputs.

Example spars that you can make with SDA:

• Round hollow aluminum/steel mast made from a piece of piping, single material, no taper. As basic as it gets.
• Square hollow wood mast, tapered at heel and tip, made from 4 planks, with plugs at each end plus at the partner;
• Round hollow wooden mast, tapered at both ends, made from two thick planks that were router-hollowed inside before gluing together;
• Round birdsmouth wooden mast, constant wall thickness, reinforced with 2 layers of carbon fiber biaxial sleeve;
• Round birdsmouth wooden mast, constant wall thickness, reinforced with 2 wraps of unidirectional carbon fiber and 1 wrap of woven carbon fiber;
• Round hollow carbon fiber mast, no taper, made from wraps of unidirectional fiber strands;
• Round hollow 2-piece mast with aluminum lower section and carbon fiber top section;
• Etc.

Features

Design:

• SDA allows you to input dimensions of your mast - length, outer diameter, inner diameter (or wall thickness) at heel, partner and tip. All dimensions (outer diameter, inner diameter, wall thickness) can be set individually. SDA automatically draws a tapering curve (straight or spline) to fill in the rest.
• Currently, SDA supports round solid, round hollow, round birdsmouth, square solid and square hollow spar profile shapes. You can also include plugs at heel, partner and tip if you want.
• You can set the materials for your mast - either a single material (wood/aluminum/steel/whatever), or multiple material laminate (for example, wood + fiberglass/carbon fiber reinforcement). You can define 1 core material and up to 5 reinforcement layers, each with different material, thickness, fiber angle, etc.
• You can go wild with the design - for example, have 1 reinforcement layer (for example fiberglass sleeve) for the entire spar length, but also add shorter extra reinforcement layers at critically loaded locations. Or, you can design a 2-piece mast with 2 different materials.

Design and material analysis:

• SDA will draw the side profile of your spar design, so you can visually verify if it looks correct.
• It will automatically calculate properties like maximum diameter, minimum wall thickness, flexural rigidity at different points on the spar, weight, weight component and stiffness contribution of each material, vertical center of gravity, etc.
• When making these calculations, SDA takes account of pretty much everything - taper type, wall thickness, material distribution, fiber angle, plugs, etc., so all your inputs matter.

Load case analysis:

• Right now this is very primitive, but you can input a bending load on the spar (either distributed load, or end point load), and SDA will calculate the maximum deflection, show a deflection graph. This is meant for free-standing masts and such.
• SDA calculates factors of safety for failure in tensile, compressive and shear modes, if appropriate values are defined in the material library. It will also plot FOS on a chart.

Goodies:

• SDA includes a material library with common materials (wood, fiberglass and carbon in twill, uni and biaxial sleeve configurations, aluminum, steel) with all the properties filled out, which you can select in drop-down menus when designing your spar.
• I included a wind pressure calculator meant for free-standing masts, so that if you know your sail area, CoE height and hull righting moment, it will calculate the force on the mast, which you can then use in the load case.
• Although the main goal of SDA is to be a design tool, I also included a chart that gives you a table of offsets for plotting and cutting fiberglass/carbon fiber strips for wrapping on the spar for reinforcements with 20 coordinate points.

Usage

Quick start:

1. Open the file, read the instructions in the first sheet, familiarize yourself with how it works, add any materials in the library if needed;
2. Make a copy of the Template sheet;
3. Fill the Design section with the dimensions and materials of your spar. Green fields are inputs, you can either write in them or use drop-down menus to select materials, units of measurement, etc. Yellow fields are "soft" equations, meaning they are there for convenience for typical scenarios, but you can override them if you need to. Blue fields are auto-calculated results, don't mess with these, but you can often change units of measurement in the green fields next to them.
4. If designing a mast, use the calculator in Tools sheet to estimate wind force, and input it in the Load Case section. If it is other kind of spar, just input something so that you can get some comparative data at least.
5. That should cover most common scenarios; but if you need something special, like some unusual taper profile, different materials for different sections of the mast or non-uniform thickness of reinforcement materials, you can override the yellow fields in the Spar Slices table.
6. You can continue making copies of the Template sheet or your own sheets in the same file to compare different designs, perhaps make some custom sheets to make proper evaluations, or use Excel's mathematical optimization tools to arrive at the perfect design. Be creative.

Misc.

Technical info:

• SDA uses numeric integration to split the spar along it's length into 1000 slices (by default), assigns properties to each slice (dimensions, materials, etc.), and then calculates stiffness, stresses, weight, and a whole bunch of other stuff for each slice. That info is then fed back to the top of the sheet.
• SDA is based on composite beam theory to calculate how even very dissimilar materials work together, to extract resultant stiffness, stresses, and other properties. Calculations are iterative and linear, so with large bending deflections the results might not be very accurate.
• For woven composite materials, SDA calculates each "layer" in the material separately, so for example if you lay a woven (twill) material at 0° angle, only half of it's fibers will contribute to spar stiffness; but if you lay it on, say, 45° angle, then fibers from both directions will contribute to the total stiffness, and this applies to any angle.
• For biaxial sleeves, SDA will account for change in sleeve's diameter and resulting fiber angle as the sleeve is stretched over a tapering spar.

Limitations:

• The load case definition is really primitive, just a single load of one of two types. Perhaps I will find the time to expand this bit. I tried including compressive loads and buckling failure mode, but after weeks of trying, I had to admit that it is way over my head.
• There is no calculation for interlaminar failure mode and it's associated FOS. I don't know how to do that, yet. It is an important failure mode though, so be careful with trusting SDA.
• By default, there are 1000 slices of the spar, which means that vertical "resolution" for a 5000mm spar is 5mm. This means that making very small changes (<5mm) to plug lengths and positions might not be reflected in the weight calculations. It could be fixed with more complex math, but I am too lazy.
• Birdsmouth masts with very small number of staves (like 4) are not handled well, because minimum and maximum wall thickness at any given slice is significantly different, and I haven't worked out the equations for that.

Disclaimer:
I am not a professional engineer nor boatbuilder, never had any formal education (my major is music), so there might be errors, mistakes or completely wrong assumptions in SDA. I did not test every possible combination of inputs. In fact, I only made a few tests so far, my baseline being "I guess that looks kinda right". Therefore I am not responsible for anything. Please do check the results, especially if they looks suspicious. Do not make maiden voyages out into the ocean with a mast made with this tool.

That being said... Have fun! Really looking forward to your comments, critique and suggestions.

 

Great work but I can't open the Excel file. Can anybody tell me how?

 

What does it say when you try to open it?

 

Got it. Somehow a small window pop asking me what app I will use to open it.

 

Oh I see. Perhaps file associations got lost or something. Glad you managed to open it. Please let me know if you have any comments. I see more than 20 people have downloaded this tool, yet no comments or anything. I hope I didn't make it too confusing. Just to clarify, the spar slices table is not important if you're doing a simple design where there is no change in materials in the vertical direction, it's just there to make the calculations automatically, you can do everything you need just in the top part of the sheet.

 

You are quite right. There is some heavy mathematics involved.

My first question is you have a material library. In it you have the fiber content but it seems you cannot change the material property even if you change the fiber content. I see you are using the Volume Fraction (VF) method and you have the matrix properties as well. The properties of the material should change if I change the Volume fraction, the properties of the fibers, or the properties of the matrix. I don't see any formula in the cell(s).

In the "tools" page there is a calculation of wind force and CE and there is a mention of lateen sail in the "instruction" page. Sails are normally triangular. trapezoidal, or ellipticals. Lateen are horizontal trapezoids.

Lastly, the title is "Spar design". Is this for a wing sail? Normally when it is a spar it is applicable to the wing as the pressure acts directly on the spar center area. Sails CE is always at a distance away from the MAST hence there is a twisting effort induced and mast shear becomes a critical load. To include the weight and the force of the sail, this becomes a "compound load" calculations.

There is a mention of the spar being subdivided into several points but no mention of lamination schedule (composite) or thickness (hollow wooden mast). I normally just divide it into 5 to 10 sections and define the diameter (or shape dimension) and the thickness of the laminations along the division.

Is this a free standing mast, supported mast, mast stayed, or mast stayed with deck support?

 

Thank you very much for taking a look into it and your feedback. Let me clarify.

It actually works the opposite way: you input the material properties of raw fiber and raw matrix, as well as the fiber content ratio in the green fields, and SDA calculates resulting Elastic Modulus, Tensile Strength and Compressive Strength using rule of mixtures. These values are in Material Library sheet, columns AB, AE, AH. Explanation of the column names: for example Ef means Elastic Modulus of Fibers, Em means Elastic Modulus of Matrix, and E means combined Elastic Modulus (as per rule of mixtures). So yes, when you change the fiber content percentage, you won't see changes in green fields as they are all inputs, only the output values in these blue columns change. This is intentional.

Yes, the wind force calculator calculates the load on the sail, without any assumptions of it's shape. It's up to the user to determine how to use that number for Load Case. The idea is that if the sail luff is attached to the mast, you can use that number directly with uniform load type, and if it's not (like on lug or lateen sails), you use point load at the tip. This is oversimplified, but the best I could do right now.

I meant the "spar" as in mast/boom/yard/gaff. The load case is aimed at free standing masts though. You are right about shear; I assume this is relevant if the mast cannot rotate freely. And yes, there are lots of other forces that I didn't include here, because I don't know how. For this first version I just wanted to add at least the simplest bending load to be able to make comparative calculations between different designs. If you can help me figure out how to build equations to support these additional forces, I'd be glad to include them in this tool.

I am not entirely sure what you meant here. In the Design->Materials section of the template sheet, you can input the lamination schedule. For example, if set up like in the pic below, it means: hollow wood mast core with 10mm wall thickness, followed by a 0.2mm wrap of unidirectional carbon fiber, followed by 0.2mm wrap of twill carbon fiber:



This assumes the schedule will be the same along the entire length of the mast. If you want to change that, you need to go into the Spar Slices table (below), and you can change the vertical distribution of the materials, for example have more layers of carbon fiber at the first 1/3rd of the mast and less above, etc.

Let me know if that clears things up, or if I misunderstood something myself.

 

Your material property sheet shows it is indeed using the Rule Of Mixtures (ROM) method. ROM is linear and is inaccurate as the curve falls off at higher glass content. It is further multiplied by a fiber directionality factor. I use the conservative formula based Gc formula by Lloyds or ISO or the VF based published data by the manufacturer. Use whatever you are comfortable with.

If the laminate (final layup) is thin, ROM is used but if thick, use the stress x distance squared or the tabulated method (LR or ISO).

As a refinement to your spreadsheet, I recommend to use the appropriate formulas to attain your goal. This all based on the first principle and the math involved is available from research materials or downloadable materials from the net.

Attached is my hierarchy of design which led to the design algorithm. It leans towards Composite Design but can be used in wood or metal construction. It can be made from simple to fairly complex by which my spreadsheet has become. I have inserted pictures to help understand it better.

Good luck

 

When you say "higher glass content", how high are we talking about? From what percentage does the ROM become inaccurate?

Same for thin/thick, what is the boundary point? Is it % of the diameter or something?

I will need to look up the formulas you mentioned.

I looked into your attached file, but it only seems to have pictures, there are no equations in any of the cells as far as I could find. Or is it unfinished?

 

It is finished, sort of. I have made dozens of spreadsheets each one being upgraded from the simplest to more complex but is now archived. I stopped at D section mast, CLT (matrix algebra) analysis of laminate, and Tsai Hill Failure theory. Tsai Hill and its variant is a better way of solving elastic and brittle failures instead of the Factor of Safety.

The spreadsheet does contain formulas. You only have to work it out and express yourself. Most walk throughs on solved problems are in YouTube or available in the net. Suggest you obtain a copy or visit the library for Martin Hollmann Aircraft Design Vol 1. It shows the step by step method on Spar design. Even has a program written in Basic. It is an old book, late 70's or early 80's.

 

If you are going to analyze a Spar or Wing mast, use M Hollmann's method.

If you are are going to analyze a sail mast, use this 07.2 Combined loading - Part A and 07.2 Combined loading - Part B whose picture appear in my spreadsheet.

You are confusing the two.