Understanding Ashby's Tables: Y-Intercepts

[_Bd_]

Hi, I am trying to understand how to use Ashby's tables, I have the book and the problems are just too ambiguous and I was wondering if anyone can help me understand this:

On whatever table, I understand you have to follow a certain sloped line that contains the features you want in your material selection, I also understand sometimes you have a minimum value for whichever axis, what I still don't understand is:

How do you know what the Y-intercept is?
I mean i see all his examples and stuff and they only plot the line, I understand the slope, but where does he get his Y-intercept from? its driving me nuts!

 

[greentlc]

Your slope and y-intercept come from equating your performance equations: p1 = p1. You re-arrange that equation to obtain: m1 = Cc x m2, where m1 and m2 are your material indices and Cc is called the coupling constant. The coupling constant is the numerical value you get from the geometric and functional indices. If you take the log of both sides of the equation you get log(m1) = log(m2) + log(Cc). This is an equation of the form y = mx + b. The slope is always 1 and log(Cc) is the y-intercept. This is the procedure to follow when you have multiple constraints in terms of a single objective function. Hope that helps.

GreenTLC

 

[greentlc]

I meant to say p1 = p2.

 

[_Bd_]

Thank you for yoru reply, that kind of helps,I've been re-reading the chapter, I am going to quote something from the book:

. . .Figure 5.9 shows as before, the modulus E plotted against density p on log scales. The material indices E/p, E^1/2 / p and E^1/3 / p can be plotted onto the figure. The condition
E/p = C or taking logs:

Log(E) = Log(p) + Log(C)

is that C the coupling constant? and anyways, how do you find it? who specifies it?
this is what confuses me the most, AFAIK E and p are supposed to be undefined, given the fact that we are looking for some material (with no defined E and p), but that C, is that the performance metric? (which I don't even know how to get, I only know its the multiplication of all 3 functions)
P < f(F) * f(G) * f(M)

3 functions of Functional requirements (F), Geometric Parameters (G) and Material properties (M)

This book just introduces the constant C in that phrase i quoted, and it doesn't explain anything about it!
after introducing it out of nowhere, it shows this:
http://img841.imageshack.us/img841/8196/picture1ra.jpg

I mean. . . where did he get that the intercepts are 10^-3, 10^-2 and 10^-1 respectively?
and then it skips to:

"It is now eas to read off the subset materials that optimally maximize performance"
and I am like. . .0.o============= EXAMPLE FROM ASHBY ==============

some panel that can withstand stuff, fixed Area but free width (h):

we have that mass (m) = ALp
the bending stiffness must be at least S*

S= CEI/L^3 > S*

the 2nd moemnt of area:

I= 1/12 * bh^3

eliminate h since its our free variable (L and b are constraints)

so you get that m = <(12S*/Cb)^1/3> * <bL^2> * <(p/E^1/3)>

where each bracket <> is the previous functions f(F),f (G) and f(M) respectively

so, what now? I can't solve for f(M) since the mass is not known, actually the mass is what we want to reduce as much as possible, so its a variable that we are targetting, which since f(G) and f(F) are fully defined (I can actually get numbers from those) you can say that that equationr educes to

m = C * f(M)
where m and f(M) are still variables, so I can't really solve for f(M) so I can't get a material index that would satisfy the equation M = E/p or whatever otehr index, that M would be the C that Ashby introduced, but still how do I get it? I am not sure if I am making any sense. . .writing this just confuses me more :P

 

[greentlc]

I was referring to something that is not introduced until chapter 7: multiple constraints, so it doesn't really apply here. What is shown in Figure 5.9 are simply design guidelines. If you refer to Figure 5.10 you see the guidelines plotted on a E vs p chart with values ranging from 0.2 to 5. These are the constants, and yes when the equation is in log form they are your intercepts. As the book states, "A material with M = 2 would give a panel that has one-tenth the weight of one with M = 0.2 that has the same stiffness.". Using them as seen in Figure 5.11 can be useful for down selecting the number of materials that you wish to look at in level 3 material universe for example. Have you done any of the case studies in Chapter 6? I definitely think that this is very confusing book because of its lack of worked through examples. As you noticed, concepts seem to appear out of no where. I hope this helps a little.

greentlc

 

[_Bd_]

I was referring to something that is not introduced until chapter 7: multiple constraints, so it doesn't really apply here. What is shown in Figure 5.9 are simply design guidelines. If you refer to Figure 5.10 you see the guidelines plotted on a E vs p chart with values ranging from 0.2 to 5. These are the constants, and yes when the equation is in log form they are your intercepts. As the book states, "A material with M = 2 would give a panel that has one-tenth the weight of one with M = 0.2 that has the same stiffness.". Using them as seen in Figure 5.11 can be useful for down selecting the number of materials that you wish to look at in level 3 material universe for example. Have you done any of the case studies in Chapter 6? I definitely think that this is very confusing book because of its lack of worked through examples. As you noticed, concepts seem to appear out of no where. I hope this helps a little.

greentlc

yes it helps a little, actually it helps a lot. ..Im looking at the case studies, and that table you mentioned, so those constants for M (.2, 2 etc.) are they just there . . .because Ashby's left *** wanted them to be there?

who decided those values? why not .5 and 50 and -100 ?
I mean, I am looking at some examples in other webpages and I found this:

Selection line for M index has slope =2
Positioned so that small group of materials situated above it.

So, do I just have to freely move the line so that there is a low amount of materials above it?
I mean TBH this sounds completely . . .not systematic. . .it just sounds like someone can just come and pick whatever. . .

 

[greentlc]

In a sense, yeah, you are kind of free to move that line around depending on what you are trying to do. At this stage in the game you aren't trying to select THE material, but rather a family or families of materials. Ill try to show you with an example from a project I was working on. I was in charge of selecting materials suitable for a high-voltage power line. I broke it up into 2 section: a tensile load section, and an electrically conductive section. For the conductive section I derived a material index M = ρ_e E, where ρ_e is the resistivity and E is Young's modulus. I wanted conductivity, however, its not available to plot in CES so I plotted the reciprocal of resistivity which is conductivity. Now I wanted to maximize conductivity(minimize resistivity) so I placed a line with a slope of negative 1, as per the log form of my material index suggests. I wanted to look at the top 8 materials that would meet my selection criteria, so I positioned the line so that only 8 materials passed. From there I went on to use other selection criteria because I had multiple constraints and objectives( you will get to that in later chapters) Here is a screen shot of my selection plot in the attached image. This is zoomed in quite a bit so you can read it. After this I looked at the relative costs associated with each material, etc, etc... Id like to post more however I have 2 finals tomorrow and one of them is for this class. Once again, I hope this helps. If not I am done on Friday and can help you more then.

greentlc

 

[_Bd_]

In a sense, yeah, you are kind of free to move that line around depending on what you are trying to do. At this stage in the game you aren't trying to select THE material, but rather a family or families of materials. Ill try to show you with an example from a project I was working on. I was in charge of selecting materials suitable for a high-voltage power line. I broke it up into 2 section: a tensile load section, and an electrically conductive section. For the conductive section I derived a material index M = ρ_e E, where ρ_e is the resistivity and E is Young's modulus. I wanted conductivity, however, its not available to plot in CES so I plotted the reciprocal of resistivity which is conductivity. Now I wanted to maximize conductivity(minimize resistivity) so I placed a line with a slope of negative 1, as per the log form of my material index suggests. I wanted to look at the top 8 materials that would meet my selection criteria, so I positioned the line so that only 8 materials passed. From there I went on to use other selection criteria because I had multiple constraints and objectives( you will get to that in later chapters) Here is a screen shot of my selection plot in the attached image. This is zoomed in quite a bit so you can read it. After this I looked at the relative costs associated with each material, etc, etc... Id like to post more however I have 2 finals tomorrow and one of them is for this class. Once again, I hope this helps. If not I am done on Friday and can help you more then.

greentlc

wow that actually answered everything. . .well I only had that one confussion as to the lack-of-systematic-M-value, but now that you say its just "choosing accordingly to a limit of materials" that coimpletely solves my doubts, thank you very very much :)

I still think its un-systematic, but w/e

thank you sir :)