Posts Tagged ‘engineering’

Is there concrete all around or is it in my head?

July 22nd, 2010 at 7:41 am

With In Our Time on hiatus until September and The Bugle in reruns for a month, I’ve been casting about for other podcasts to listen to. I decided to give Grammar Girl another try—I was a subscriber a couple of years ago—and downloaded several episodes.

One of the first I listened to was “Concrete versus Cement,” a usage issue near and dear to my heart. Unfortunately, the episode stabbed that heart with error after error. The show was written1 by Sal Glynn, a book editor who probably should have consulted a civil engineer before releasing the final copy.2

In fairness, the show gets its main point right: concrete and cement are not the same thing. Cement is a constituent of concrete and is almost never used as a building material by itself. From pools to patios, sidewalks to stoops; everything that’s commonly called cement is really made of concrete.

OK, fine. But things go south after that. Here’s the most egregious passage:

In 1824 Joseph Aspdin patented “Portland cement,” a powder made of limestone and clay. He called it Portland cement because when it was mixed with sand, gravel, pebbles, bits of rock, and water, the resulting dried concrete resembled the limestone from the English Isle of Portland.

This is the division: cement is a powder that is mixed with other materials and water to create the solid mass known as concrete. There is no such thing as a cement overpass, a cement porch, or a cement pond. These are all concrete. If you stub your toe on concrete you’ll yell like a Cretin. Get it? Concrete. Cretin.

And in case you were wondering, pavement is also not concrete. Pavement is a solid material made of sand, gravel, or crushed stone much like concrete, but the cement binder is replaced with asphalt or tar.

The history lesson is basically right, although I wouldn’t say Portland cement concrete “resembled” Portland stone (no matter what the Wikipedia article says). The Portland name was mostly a marketing gimmick. Aspdin was a successful salesman, not above pretending his cement had secret ingredients that made it superior.

Also, I don’t understand the distinction Glynn is trying to make between gravel, pebbles, and bits of rock. Gravel is made up of pebbles, each of which is a bit of rock. In concrete, they’re all categorized as coarse aggregate.

But the real problem in that first paragraph is the word dried.

Concrete doesn’t harden by drying.

This is a common but mistaken belief (as common and as mistaken as thinking that concrete and cement are the same). The cement in concrete undergoes a chemical reaction with the water added to it. That reaction, called hydration, produces a new material that becomes harder as the reaction progresses. The new material binds the sand and gravel together and fills the spaces between them.

Portland cement is a hydraulic cement—it can harden under water. Underwater concrete placement is fairly common, and even though that concrete never dries, it hardens just the same.

So never, ever look at newly-placed concrete and say it’s drying. Say it’s setting or setting up or hardening or curing. You can even say it’s hydrating if you want to get funny looks. Just don’t say it’s drying.

The real problem with drying is that it’s a word tha has a very specific meaning in reference to concrete and that meaning has nothing to do with hardening—it has to do with shrinking and cracking. Here’s why:

The hydration reaction requires a certain proportion of water to cement. When concrete is mixed, it usually has more water than is necessary to hydrate the cement. The extra water is added to make the fresh (i.e., unhardened) concrete a bit mushier and easy to work with. After the concrete sets, that extra water slowly migrates out of the hardened concrete and evaporates, and the concrete shrinks as a result. This is called drying shrinkage.

Usually, the concrete is prevented from shrinking as much as it wants to because it’s being held in place by something. Sidewalks, for example, can’t shrink the way they’d like to because the friction between their underside and the ground below prevents it. When that happens, tensile stresses develop in the concrete and—being weak in tension—it cracks.

These shrinkage cracks are so common, and so difficult to prevent, that in many structures prevention isn’t even attempted. You can’t, for example, eliminate shrinkage cracks from a sidewalk, so instead you try to control them. The joints that run across sidewalks—mistakenly called expansion joints by many people—are control joints. They’re put there to make the sidewalk weaker along those lines and cause the shrinkage cracks to form out of sight down at the bottom of the grooves rather than haphazardly over the walking surface.

So that’s the first paragraph. I dislike the second paragraph, too, because it presents an overly narrow definition of cement. The word cement is used not just for the gray powder, but also for the paste that’s formed after we mix that powder with water and for the hardened material that results from the hydration reaction. If the meaning isn’t clear from the context, we may be more explicit and refer to these latter two materials as cement paste and hardened cement paste, respectively, but they’re cement nonetheless. Saying that cement refers only to the powder, which Glynn does elsewhere in the episode, too, is simply wrong.

The third paragraph is just ridiculous. Pavement can’t be made of concrete? That would be news to

Of course pavements can be made of concrete. Back in Pavement Design class, we learned that those were called rigid pavements, while the ones made of asphalt are called flexible pavements. It’s right there on Page 5 of Yoder & Witczak.

Honestly, where would anyone, even a book editor, get the idea that pavement is exclusively asphalt?

I’m sure Mr. Glynn is a very nice person and quite knowledgeable in his field. But concrete is not his field. You don’t see me writing about gerunds.

  1. Grammar Girl herself, Mignon Fogarty, still narrates the episodes, but she seems to be outsourcing more and more of the scripts. ↩

  2. You might well ask why I’m bothering to correct a podcast that’s over a year and a half old. First, I just heard it a few days ago—I’d have corrected it earlier if I’d known about it. Second, it’s still sitting there on the Grammar Girl website, misinforming people who come across it. And third, I’m kind of pissed that a book editor thinks he can look up a few things on Wikipedia and present himself as an expert on a topic that I and thousands of others spent years learning. ↩

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Post #1321 | 9 Comments

Trussed up

June 11th, 2010 at 9:29 pm

June is Physical Science and Mechanics Month at Make magazine, and about a week ago Gareth Branwyn of Make asked me to answer one of the Ask Make questions a reader sent in: How do roof trusses distribute the load? This topic is, in fact, right in my wheelhouse, so after some guidance on how long the answer should be, I wrote up a short piece and sent it in. Make posted it yesterday, and I decided to put it here, too, so I’d have my own copy.

Trusses, like all structures, are devices for transferring loads from where you don’t want them to where you do. A roof truss takes the weight of the roof—and the snow on the roof if you live in that kind of climate—and transfers it out into the load-bearing walls of your house. A bridge truss takes the weight of the cars and trucks passing over it and transfers it to the piers. What makes a truss different from other structures—rafters, say, for a roof; or arches for a bridge—is the clever and efficient way it carries the load. Trusses tend to be very lightweight because they take advantage of geometry and the laws of statics.

Geometry

Imagine you have a set of flat sticks, like popsicle sticks or tongue depressors, and you drill holes at the ends of every stick so you can connect them with little bolts. If you connect three sticks together in a triangle, you get a structure that stays rigid even if you don’t tighten the bolts much.

If, on the other hand, you make a square with four sticks, it will be almost impossible to keep the structure from turning into a rhombus when you push on it, no matter how much tightening you do.

Unlike the triangle, the rigidity of this structure depends on the rigidity of the connections. To really keep a square rigid, you need to add a diagonal brace to create two triangles within the square.

This inherent rigidity of triangles is a geometric property. The triangle is the only polygon whose internal angles—and, therefore, shape—are uniquely defined by the lengths of its sides. There’s nothing like the side-side-side congruence postulate or the law of cosines for higher order polygons.

So if you want to make a structure that’s rigid regardless of the rigidity of its connections, you start with a triangle and build onto it by adding sticks to make more triangles. That, in a nutshell, is a truss.

Laws of statics

An ideal truss is like our assembly of popsicle sticks: a set of straight members or elements, pinned together at their ends, with forces applied only at the joints. (We’ll get to the relation between real trusses and ideal ones in a bit.) Under these conditions, each member in a truss is loaded only at its ends. Some of the loads may be coming from the externally applied forces, and others will be coming from the members to which it’s connected, but whatever their source, the loads are acting exclusively at the ends.

Applying the equations of statics, we can show that if a body is loaded at two points only, the resultant forces at those points

  1. are equal in magnitude;
  2. are opposite in direction; and
  3. act along the line between the two points.

For our truss members, this means that forces on the members are axial (that is, they act along the axis of the member), putting them in either pure tension or pure compression.

In trusses loaded by downward forces, the members along the top (the “top chord”) are in compression and the members along the bottom (the “bottom chord”) are in tension. The members connecting the top and bottom chords (the “web members”) may be tension or compression, depending on their angles and the distribution of the loads.

The forces in the members can be calculated in several ways. The traditional “by hand” methods are the method of joints and the method of sections. For truss analysis via computer, the finite element method is the standard technique.

Efficiency

The fact that the forces on each truss member are axial is the key to a truss’s efficiency. In an axially-loaded member, the force is carried equally by every part of the member—no part is wasted.

Contrast this to a beam. When you load a beam at the center, the stresses are much higher there than anywhere else. The material away from the center just isn’t doing as much work, lowering the efficiency of the structure.

You have, by the way, an instinctive understanding of this. If someone hands you a pencil and asks you to break it, you put your thumbs against the center and bend it. You’d never consider grabbing the two ends and pulling or pushing. You couldn’t even break a toothpick that way.

By sizing the members of a truss just right, you can tune it to carry huge loads while using very little material. This is how people win balsa wood bridge competitions. Real-world trusses can’t be optimized the way a balsa wood bridge can, because real-world trusses have to carry many different combinations of load, and a truss optimized for one set of loads won’t be optimal for another set. Still, even when they can’t be fully optimized, trusses are usually much lighter than alternative structures.

Theory and reality

Speaking of real-world trusses, remember how we defined our ideal truss? Straight members, pinned connections at either end, and loaded only at the joints. Only the first of those conditions is met in a real roof truss. The top and bottom chord members are often continuous though the joints, and the web members are connected through connector plates, not pins.

Furthermore, the top chord is loaded by the roof sheathing along its entire length, not just at the joints.

These deviations from the ideal do, in fact, generate additional stresses by imposing bending loads on the truss members. Fortunately, these additional stresses—structural engineers call them “secondary stresses”—don’t alter the truss’s behavior much and can be ignored in most cases. In trusses, the difference between theory and practice is small.

Uses

So if trusses are strong and stiff and efficient, why aren’t they used for every roof? Three reasons come to mind immediately:

  1. They take up space in your attic. Although the total volume of lumber used by roof trusses is less than the total volume used by a set of rafters and ceiling joists, the web members of the truss cut the attic up and make it less usable.
  2. They’re harder to adapt to some roof plans. If your roof has lots of valleys and hips, it’s easier to frame with rafters than with trusses.
  3. They require extra equipment to put in place. A truss has to be pre-assembled and then lifted as a complete unit up onto the framing. This is no big deal if you’re building an entire subdivision and can hire a crane to do several houses a day. But if you’re just doing a single house, the cost of renting a crane can be prohibitive. Rafters can be put in place by a small crew of framing carpenters.

But trusses are used in lots of roofs, both in residential and commercial construction. Warehouses and warehouse-style stores are almost always use steel trusses because they’re the cheapest way to hold up big, open expanses of roof. Next time you’re in a Costco or Sam’s Club, take a look up.

Postscript: I made the drawings in OmniGraffle and exported them out as PNGs. OmniGraffle’s magnets, auto-alignment tools, and arrows made the job go quickly.

The photo is mine, taken several years ago at a building collapse I was hired to investigate (I’ve done a lot of that over the years). I couldn’t remember the name of the job, but I grep’d for “truss” in my directory of old job reports and soon found the job I was looking for. Flipping through the directory of photos from that job—most of which were of splintered truss pieces, which didn’t seem appropriate—I found what may be the only photo I’ve ever taken of an unbroken roof truss.

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Post #1274 | Comments Off

Structural analysis and the iPad

April 22nd, 2010 at 3:02 pm

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}

where4

\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
2:  
3:  # Solution for maximum deflection of centrally-loaded simply-supported plate.
4:  # See Timo Plates and Shells, article 34, pp. 141-143.
5:  
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. ↩

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Post #1233 | 1 Comment

Chile’s earthquake

February 27th, 2010 at 9:59 pm

I’m not surprised that—so far, at least—the death toll from Chile’s earthquake is so low compared to Haiti’s, despite the much stronger quake. I was trained as a structural engineer, and my department was loaded with graduate students from South America. Latin American countries tend to take earthquakes very seriously, and their engineers are highly educated, both at home and abroad.

News reports will, of course, focus on the devastated areas, but most of the buildings must have done an excellent job of protecting the people within. This is not to say that the buildings weren’t damaged; there’s too much energy in a big quake to expect most buildings to escape unscathed. But I would expect to see most engineered buildings1 to have absorbed the energy without large-scale collapse.

  1. Buildings that were designed by engineers, like office and apartment buildings. Older, smaller residences are typically not engineered and usually suffer the most damage. ↩

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Post #1184 | Comments Off

The Toyota pedal problem

February 3rd, 2010 at 9:59 pm

I’m not a big fan of the Chicago Tribune,1 but I have to say it did a pretty good job yesterday explaining the Toyota pedal recall that’s been all over the news lately. The article is only so-so, but the graphic that accompanies it, by Phil Geib and the improbably-named Max Rust, answered many of the questions I had.

[Click on the graphic to get a slightly larger version.]

Comparing the drawings to a photo of an actual pedal (on my 2007 Camry), we see that the parts in question are all inside a plastic housing near the pedal’s pivot.

I couldn’t shoot a photo with the same point of view as the drawing because its a little cramped down there, but I think you get a sense of where everything is. In addition to the gas pedal, you can see the brake pedal (lower left foreground) and part of the steering mechanism (the shaft with the yellow-orange stripe at the upper left corner).

The gas pedal is a “drive-by-wire” system. There is no mechanical connection between the pedal and the engine; the pedal sends electrical signals that describe its position to a controller which runs the fuel injectors. (In my photo you can see the wires coming up off the top of the pivot housing and leading into a black corrugated conduit.) This is very different from the way cars used to work, and to give the pedal the traditional “feel,” Toyota has incorporated two sets of plastic teeth that rub against one another and provide some of the resistance felt by the driver’s foot.

Additional resistance comes from the pedal’s return spring, which, for clarity, isn’t included in the drawing. The return spring is what, under normal circumstances, pushes the pedal back up when you take your foot off the gas.

The toothed portions of the blue and tan parts slide on one another, the blue part rotating with the pedal and the tan part remaining stationary. The friction between the teeth is what generates the resistive force. Apparently, moisture on these plastic parts can increase the friction between them, and if the friction gets large enough, the return spring can’t overcome it and the pedal won’t return to the up position when you remove your foot. You may have heard this called “sudden acceleration” in news reports, which gives the impression of the gas pedal moving down on its own. “Stuck throttle” or “stuck accelerator” would be a more accurate name for the problem.

[Aside: Friction is an interesting force because it always opposes motion. When you press down on the gas, the friction and the return spring are acting in concert, both pushing up to resist the downward motion. When you take your foot off, the return spring continues to push up, but the friction suddenly changes direction to push down, resisting the spring.]

Toyota’s page for the pedal recall mentions wear of the plastic as another factor in the sticking pedal problem, so it’s not clear whether the problem is caused by an increased coefficient of friction, surface damage from galling, or some combination of both. Regardless, Toyota’s solution is to change the relative positions of the plastic teeth and reduce the friction.

Dealers will be installing what Toyota is calling a “reinforcement bar” (but should really be called a “spacer” or “shim” because it isn’t changing the strength of the parts, it’s changing their position) behind the tan part. The Tribune drawing does not do a good job of explaining how that shim is going to reduce the engagement of the two sets of teeth, but it seems clear that that’s the intent.

One thing I still don’t understand, and something I’ve not heard Toyota address publicly: How will the new, reduced friction affect the feel of the gas pedal? Presumably, the pedal resistance I get now will be lessened after I take my car in for the recall fix. Will it feel too floppy? If not, why did Toyota have the higher resistance to begin with?

Frankly, I’m curious why a friction mechanism was chosen in the first place. Toyota obviously wanted to get a particular relationship between the resistive force and the pedal deflection, but I don’t see why that couldn’t be achieved with a spring package of some sort. Mechanical engineers have been designing spring mechanisms for a long time and can get all kinds of force-deflection relationships.2

Maybe I’m just out of date and friction mechanisms like this are common in brake systems nowadays. If you know what other manufacturers do, I’d like to hear about it in the comments.

Update 2/4/10
I’ve known for a few days that there are “good” pedals and “bad” pedals, but it wasn’t until this morning that I learned who their manufacturers are. The good pedals, not subject to the recall, are made by Denso, and the the bad pedals, which are going to have the shim fix described above, are made by CTS.

It turns out that my pedal is a Denso. The Denso name is molded into the pivot housing, something you can’t see in my photo. Also, the housings of the two designs look very different; the CTS doesn’t have that circular area with radial spines sticking out of the side. You can see photos of the two designs at this site. Note: I am not endorsing anything said on that site, as I have not read through it in detail. I’m just linking to it as a source of photos showing the difference between the two types of pedal. It also has some nice photos taken with the housings opened.

Update 2/24/10
I’d like to say something about this week’s Congressional hearings into Toyota’s problems, especially since much of the testimony covered other explanations—mainly electronic—for sudden acceleration. But I don’t have a copy of the primary engineering report referenced in the testimony, so I don’t feel comfortable commenting yet. I was hoping the House subcommittee would post a PDF of the report, but it hasn’t so far. If you know where I can download a copy, leave a URL in the comments or send me an email directly.

  1. What’s wrong with the Trib? To start: its editorial position, its choice of columnists on the op-ed page, its placement of John Kass in the old Royko spot, and its many years of owning the Cubs. ↩

  2. If your experience with springs is limited to what you learned in physics class, you may think that all springs are linear. Ut tensio sic vis, and all that. But if you expand your concept of “spring” beyond helical coils of uniform wire many more things are possible. ↩

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Post #1166 | 5 Comments