I hope through experimentation you're able to get a better feel for columns and maybe discover what factors affect how a column will behave. so,how do columns fail? It turns out there's two different ways a column can fail. They can buckle. Buckling just means it displaces laterally or horizontally. In this case, the middle is displacing. Long slender columns like this one will tend to buckle. You could also have a compression failure. A compression failure is actually the material itself crushing or yielding. The shorter, wider columns will fail in compression. How do we predict failure? I'm going to start with a compression failure, and I'm going to use a small blue foam beam. When I push on it. If I push hard enough, I can shorten the column. That downward force when I push on it, I call that a compressive force, and I often model that with a downward arrow. Again, if I push hard enough, I can cause failure, but I can't push hard enough on this one to cause failure. I did bring a piece of chalk, also, a fairly short wide column with a circular cross section. I am not strong enough to push it to cause failure. But I think if I use my hammer, I can apply a large enough load to cause it to fail and that will mean that my applied force and stress is higher than the allowable. When I apply a compressive force to a column, it causes a stress in the column. What is stress? The definition is a force per unit area but what does that mean? If we use this blue foam column as an example, when I push on the top that's applying a compressive force, I usually denote that with a capital P and a downward arrow. That's my force at the top and that force ends up getting distributed throughout the column and over the column area as it gets internal stresses. Since this is a column with a circular cross section, that cross sectional area, which I denote as a capital A, will be Pi times this radius squared. That's easily calculatable and then I can use those values to calculate my compressive stress. My compressive stress would be the applied load P, divided by the cross-sectional area. The units will be force over area or Newtons per meter squared as an example. Failure and compression, which is a yielding or a crushing of the material, is something we want to avoid. Engineers will look up allowable stresses for different materials. Different materials will have different values for allowable stress. Wood has a much lower allowable stress than say, steel. When I look up those values, I can calculate an applied stress that I expect to my column and compare it to the allowable stress for the material I expect to use. That'll help me design and analyze my column. Now let's consider buckling. What is buckling again? Buckling is when I push on a column and it displaces laterally or horizontally. But when will that happen? What are the factors that will affect when a column will buckle? Well, a longer column buckle at a higher or lower load than a shorter column and how much of a factor is the length in the behavior. Other things that we can consider are material type, cross-sectional shape, and cross sectional area. Which of these will have an effect on how the load at which a column will buckle? The load at which a column will buckle is referred to as the buckling load, is also referred to as the Euler critical buckling load. It's named after a mathematician, Leonhard Euler, who discovered or derived the formula in 1757. If we look at this equation, the equation has P critical equals Pi squared EI over L squared. E is a material property. Different materials will have different modulus for the elasticity values, which is what E stands for. The modulus of elasticity or E for steel, will be much higher than the modulus of elasticity for wood, as an example. Moment of inertia or I in that equation is a property of the section type. The section shape really is what the moment of inertia is a function of. We'll discuss moment of inertia more when we discuss bending. For now, it's fine to just understand that there's a relationship between shape and the moment of inertia. As an example, a hollow tube will have a much higher moment of inertia than a solid tube with the same cross sectional area. Thus, this hollow tube will be able to support much more load before it buckles. Then, L is the length of the height of the column. When we look at a column, there's two main things we look at. We look at compressive stress and buckling load, and we're going to go over the equations that go along with those. Let's draw a simple column here with a rectangular cross section. That's the cross-section, give it some height. This is my column. This would be my cross-sectional area and it'll have a certain height or length. I usually use length for the column and call that L. The two different things we'll look at again, are compressive stress and buckling load. We have two equations that govern that. When I'm looking at compressive stress, I usually use the symbol Sigma, and to calculate the compressive stress in a column would be force over area. That would be the applied force. There'll be a downward applied force P, say, and that's my applied force and then the area is a cross-sectional area. Our A, is the cross-sectional area. That would be in meters squared, inches squared, some type of length squared. We calculate this value. For a given column, we will typically know the cross-sectional area and we'll know the applied force, and we'll know the material or will be designing the material. I want to keep this stress less than the allowable stress of the material. If we're dealing with steel, it would have a higher allowable stress than say, wood, and we would play around with these numbers to make sure that it didn't fail in compression. The other thing we have to look at is buckling. We usually look at the buckling load, which I wil refer to as P critical. That value is Pi squared times E times I over L squared. I'm not going to derive that equation here, but it was derived in the 1700s. Pi is a constant. We have E, which is our modulus of elasticity. Just as a reminder, that's a function of the material. Different materials will have different modulus of elasticity values. Steel again will have a higher modulus than say wood. We have moment of inertia. Moment of inertia is a function not of the material but of the cross-section, so the shape. This rectangular cross-section will have a certain moment of inertia and that'll be different than if it were circular or hollow tube. Then L is just the length. Using these equations or using this critical buckling load, if I know the material, I can look up E. I find out the shape of the cross section, I can look up the moment of inertia, and I will typically know the height. I can calculate this critical buckling load and I want to make sure that the applied load is less than this critical buckling load. You want P applied to be less than that critical buckling load so that you make sure it doesn't buckle. Those would be the two things we would look at if we are designing a column. Go ahead and experiment with the column simulator in the next segment of the course to see if you can build a tall but strong column and check out the column case studies. In addition to more standard columns and buildings, trees, bones, lamps, and many other things can be modeled as columns.
