Geometry Of Sections: 5. Thermal Stresses

So, in the previous video, we talked about Hooke's Law, or the relationship of stress to strain, or unit strain, and the concept of elasticity. So, up to a certain point, the elastic limit, or the yield point, a material is elastic, it goes back to its original shape, beyond which, beyond the elastic limit, the material is plastic and deformation is permanent. Elongation, shortening, bending, deflection, whatever it is, will be permanent. So again, if there is one equation you need to understand, not necessarily memorize, it's basically F equal P over A, and it's very clear in the units, you are taking pounds divided by square inch to get stress. Very good. Also, to calculate the elongation of a member or a shortening of a member along the axis, the equation is delta L equal PL original divided by AE, where E is the modulus of elasticity of the material. E depends on the material, A depends on the cross-section, and then PL0 is external loading and physical properties. Very good. So there's one more equation I need to go over, which is kind of important because it's the only time temperature comes in. So, thermal elongation, or delta L, along the axis is given by alpha, a coefficient of thermal expansion, times delta T, any change in temperature is going to cause an elongation or shortening, times the original length. So, alpha is the coefficient of thermal expansion, and it has certain units we're not going to talk about, inch per inch per degree Fahrenheit, very good. And the change in temperature in this equation is in degrees Fahrenheit. So this one is inch of elongation per inch of original length for every degree Fahrenheit. Excellent. So this must be given to us. So let's take a look at that alpha coefficient. Coefficient of thermal expansion and contraction is in inches per inch per degree Fahrenheit. you will notice it's a very small number. But you will also notice that aluminum is a big culprit. It has a coefficient of 0.40, 128. Fine, whatever that means. It indicates that aluminum is very susceptible to temperature change. The slightest change will make it expand or contract. That's why storefront, curtain wall, all that stuff, That's why you need to leave a gap and fill it with something soft, right? So that's your mullion and your, sorry, your gasket and whatnot, the rubber around the window, is to absorb movement. So this is the coefficient for aluminum. You will notice next in line is steel. It has five zeros, 65, which is almost half as much as aluminum. So for the same change in temperature and the same original length, aluminum will expand twice as much compared to steel. Very good. Next in line is concrete. And you will notice that this is almost very close in value. Concrete's coefficient of thermal expansion is very similar to that of steel, which allows us to do this. Basically, that's why we can put this material inside of that other material. So you have steel rebars inside of a concrete beam. They are thermally compatible. They expand together. They shrink together. Imagine if those rebars were aluminum. They would expand twice as much as the concrete and crack the concrete. So very good. Next is glass. And then finally, wood. Please don't mistake the coefficient of thermal expansion of wood with moisture content. That's a totally different story. Okay, so I would like to take an example of this formula here, which is the change in length due to temperature is equal to alpha delta T times L sub zero. Let's just look at one thing here. I would like the change in length to be in inches, which means if this one is feet, it's got to go. It's got to become inches because this one is inch per inch, not per foot, per degree Fahrenheit. Very good. So let's take an example of a parking deck, and we're going to put precast double Ts in there. The double Ts usually sit on an L beam, precast L beam of concrete. It looks like that. And the double T comes in and sits on there. Or else, if it's in the middle, it sits on a concrete, sorry, precast concrete upside-down tee. And this one can receive double tees from two sides. This one, if it's an end condition, receives from one side. These guys typically can span around 40 feet, let's say. 20 to 40 feet. Push it, maybe 50. The double T's that look like this usually are somewhere between 40 to maybe 75 feet. You have to watch out because it's precast and therefore there's transportation regulations on the highway. Can't be wider than 12 feet, can't be longer than 75 feet. Otherwise, you need the escort and it starts to get expensive, etc., etc. So parking decks, let's get some dimensions going. Typically, let's call this 20 feet. Could be 18. I don't want 18. I want a nice round number. This one is usually a minimum of 24 feet. I'm going to make it 25 feet just so I can deal with a nice even number. So let's say this bay is 65 feet wide. Typically, I don't know, maybe 9 feet by 18 feet is your typical stall. I don't know. So if these guys can go 40 feet, let's put a column here. Let's put another column. 1, 2, 3, 4 times 9, 36. Let's put one here. And let's put one over here. And let's go across and put another column in that middle between the two parking bays. And we're going to continue like this. Very good. So now this is approximately 36 to 40 feet, somewhere in there. Excellent. So we're going to put one of these L beams over here. We're going to put a T beam here because it's receiving double Ts from two sides and an L beam over here. And we'll do the same on the next one. Oops, we need a column here. There it is. And a T in the middle. And one more here. Very good. So these double Ts typically are dimensioned. Typically. It's the dimension of the bed at the precasting yard. So usually this is x, 2x, x. So 1 foot, 2 foot, 1 foot is a 4 foot double T. Maybe 1 and a half, 3, 1 and a half. That's a 6 foot. And an 8 foot is more typical. So 2, 4, 2. That gives me an 8 foot double T. Very good. So let's take an 8 foot or even a 10 foot double T, which would be two and a half, five, two and a half. And let's lay them out here. Let's put the first double T here. And if that's nine feet, that's maybe 10, a little bit more than nine. So that's my first double T. Here's the rib. Here's the other rib. And we'll put another one and another one. Sorry, not to scale. I don't know if I'm still at the 10 foot module. But basically, that's what we're doing. And we continue. Very good. So this is what it looks like in section. There's an L beam, there's a T beam, there's an L beam, and then there's a double T, double T, and another double T. Very good. And this span, we said, is 65 feet. Very good. So Let's assume that this is concrete and the span is 65 feet. So let's assume that the change in temperature is, just to keep a nice round number, 100 degrees Fahrenheit. That's summer to winter over the 25-year, 30-year span of the member. What is the worst case cold? What is the worst case hot? And then we have to account for movement for these critical conditions. So using this equation, I have, where is some room to write? Over here. Delta L, the change in length, is equal to the coefficient of thermal expansion of the material times the change in temperature times the original length. which means this is 0.12345 and then 55 times change in temperature, has to be in degrees Fahrenheit, times the original length. We have 65 feet. This equation necessitates that we change this into inches. Cannot work with feet. And the length, the span is always going to be in feet, and you always have to change it to inches in this equation. So doing the math I've calculated before I started this recording, Turns out to be 1, 2, 3, and 65 inches. Sorry, 0.5. I'm sorry, I messed up here. Let me erase this one. The math says it is 0.507 inches, which means this double T is going to want to expand 0.51 inches or contract 0.51 inches. That's half an inch. So you have to give it room to move. And the mistake I made was I gave the answer for the unit strain, which is delta L over L. You are going to change length of 0.507, and you are changing that much inches in that much length. Again, unit strain has no units, so this needs to become inches. So the change of length per original length is the unit strain, and it turns out to be 0.123 and 65 inches of elongation per every inch of the original 65 feet. Excellent. So that's the unit strain. It helps me understand because the concrete is going to freak out if the unit strain is 0.003 inch per inch. That's it. The concrete gives up intention and says, I can't do this. And you start getting cracks. So this must be avoided. I'm not saying we're close, but we need to keep an eye on it. Very good. So let's go in here and let's detail this double T a little bit. The point I'm trying to make here is you need to allow this double T to move on one end. So the other end is pinned. This is 65 feet. On this end, it's a roller to allow the 0.507 inches to move on one end. You cannot have two rollers because then it's totally unstable. You cannot have two pins because the 0.5 inches has no room to go, so it's going to bump up and crack the concrete. So this is a condition where you need a simply supported with a roller. Long span structures require a roller on one end, a pin on the other, in order to accommodate thermal movement, delta L equal alpha delta T times L sub zero.