Structural analysis and the iPad

If you’re curious about the iPad, you probably read this iFixit article, documenting the disassembly of an iPad. I thought it was well done and quite fun to read, but I was brought up short by this bit of design interpretation in Step 29:

  • The glass seems quite thick (~1.18 mm), which is not a huge surprise considering the size of the iPad. Compare that to about 1.02 mm for the iPhone.
    • The iPad would require thicker glass due to the increased “lever arm” caused by pressing down at the center of the screen. This is analogous to the difficulty of bending a one inch section of a ruler compared to bending the entire twelve inch ruler.

What iFixit seems to be saying is that Apple increased the glass thickness from 1.02″ to 1.18″ to make up for iPad’s increased width and height. Specifically, they’re saying the thickness was determined by the flexibility of the glass under the load of a finger pressing on the center of the screen. This is certainly incorrect.

I’m not saying there isn’t a lot of truth in what iFixit wrote: a long ruler is more flexible than a short one, and making the glass thicker will decrease its flexibility. But their sense of scale—which is essential to good structural design—is way off.

The internet is loaded with articles about how programmers think, and it’s not hard to find articles about how electrical engineers think. This isn’t surprising; the computer and electronics crowd are disproportionately represented online because online is their thing. Less common are articles about how mechanical and structural engineers think, so I’m going to use this small mistake by iFixit to write one.

The short explanation

The short explanation for why iFixit was wrong is, unfortunately, not especially short, but here goes:

The flexibility of a flat plate is governed by three things:

  1. The modulus of elasticity. Also known as Young’s modulus in honor of the English scientist Thomas Young and typically given the symbol [E], the modulus of elasticity is a property of the material of which the plate is made. The flexibility of a plate is inversely proportional to [E]—doubling [E] cuts the flexibility in half.
  2. The width. The flexibility of a plate is proportional to the square of its width—doubling the width quadruples the flexibility.
  3. The thickness. The flexibility of a plate is inversely proportional to the cube of its thickness—doubling the thickness decreases the flexibility to one-eighth of its previous value.

To compare the relative flexibilities of the iPhone and iPad glass, we compare these three properties.

We’ll start by assuming that the glass used in the two devices is of the same type and has the same modulus of elasticity. This may not be exactly true, but is probably pretty close,1 so the modulus will not factor into the iPad/iPhone difference.

The width of an iPad is about 3 times that of an iPhone, which gives it a 9-fold increase in flexibility. If Apple wanted to match the flexibility of the iPhone, it would have to increase the thickness of the iPad’s glass by a factor of

[\sqrt[3]{9} = 2.1]

This is much more than the change from 1.02 to 1.18 mm that iFixit measured. So Apple was not trying to match the flexibility of the iPhones’ glass. In fact, the iPad’s glass is nearly six times more flexible than the iPhone’s:

[\frac{3^2}{(\frac{1.18}{1.02})^3} = 5.8]

What is the governing criterion for the thickness of the glass? I don’t know, but it’s easy to think of a couple of possibilities:

If I had to bet, I’d put my money on impact resistance as the controlling factor for glass thickness.

The longer explanation

The following is mostly for my own amusement, so if you’re thinking about bailing out, don’t worry. You won’t offend me.

Remember when I said the flexibility of a plate was governed by three things? I lied, but just a bit. Here, from Timoshenko’s2 Theory of Plates and Shells, is the equation for the maximum deflection of a simply-supported rectangular plate with a concentrated load at its center:

It looks nasty, but it’s not really that complicated. Let’s start with a description of all the variables:

The flexibility is, by definition, the deflection generated by a given load, so if we want to compare the flexibility of the iPad glass to the iPhone glass, we need to compare their [w_{max}] values for the same [P].

Timo simplifies the formula down to just

[w_{max} = \alpha \frac{P a^2}{D}]


[\alpha = \frac{1}{2 \pi^3} \sum_{m = 1, 3, 5, \ldots}^\infty \frac{1}{m^3}\left( \tanh \alpha_m - \frac{\alpha_m}{\cosh^2 \alpha_m}\right)]

Substituting in the definition of [\alpha_m], we get

[\alpha = \frac{1}{2 \pi^3} \sum_{m = 1, 3, 5, \ldots}^\infty \frac{1}{m^3}\left( \tanh \frac{m \pi b}{2 a} - \frac{\frac{m \pi b}{2 a}}{\cosh^2 \frac{m \pi b}{2 a}}\right)]

and we see that [\alpha] is a function of the aspect ratio, [b/a], only.

Back to the simplified equation, substituting in the definition of [D] and rearranging a bit, we get

[w_{max} = \frac{12 (1- \nu^2)}{E}\;\alpha\; \frac{P\:a^2}{t^3}]

From this formula, you can see that the effects of the modulus of elasticity, the width, and the thickness are just as I described them in the short explanation. You can also see the two things I left out.

First is Poisson’s ratio, [\nu]. This is a material constant, and because we’re assuming the iPad and iPhone have the same type of glass, it should be the same for both devices and not factor into the flexibility difference at all.

(In fact, Poisson’s ratio doesn’t vary much, even across widely disparate materials; values between 0.2 and 0.4 are the most common. Plug those values into [(1 - \nu^2)] and you’ll see why changes in Poisson’s ratio are seldom worth worrying about, even if you’re comparing different materials.)

Second is the aspect ratio, [b/a], which, as we’ve seen, determines the value of [\alpha]. The aspect ratios of the two devices are different, about 1.3 for the iPad and about 1.9 for the iPhone. That difference does have an effect, but it’s pretty small.

Timo helpfully gives us a table from which we can look up values of [\alpha] for various aspect ratios. This sort of table was really helpful to working engineers back in 1940, when the book was first published, and was still helpful in 1959 when the second edition, the edition I have, was published. Nowadays we have a computer to calculate the infinite series. I wrote this little Octave script to calculate [\alpha] for any aspect ratio.

1:  #!/usr/bin/env octave -q
3:  # Solution for maximum deflection of centrally-loaded simply-supported plate.
4:  # See Timo Plates and Shells, article 34, pp. 141-143.
6:  r = eval(argv(){1})           # aspect ratio from command line
7:  m = linspace(1, 39, 20);      # odd indices only
8:  beta = pi*r*m/2;              # what Timo calls alpha_m
9:  alpha = sum((tanh(beta) - beta./cosh(beta).^2)./m.^3)/(2*pi^3)

For the iPhone, [\alpha = 0.0164]; for the iPad, [\alpha = 0.0143]. That was a lot of messing around for piddly 10-15% effect.

Now you can see why I lied to you in the simple explanation. The scale of the difference in flexibility is governed almost entirely by the modulus, the width, and the thickness. And because we’re comparing glass to glass, it all comes down to the width and the thickness.

  1. The strength of glass can be varied over a wide range by thermal and chemical processing, but the modulus tends to be more consistent from one glass to the next. 

  2. Stephen Timoshenko was one of the giants of 20th century mechanics. Not only was his own research influential, but his many books constitute an encyclopedia of analytical techniques in engineering mechanics. The computer science analog to Timo’s work would be Knuth’s Art of Computer Programming series. 

  3. Yet another material constant, this one named after the French scientist Siméon Poisson

  4. In the excerpted scan, Timo doesn’t include the restriction that the m takes only odd values, but it’s implicit in his earlier derivation of the formula.