[SOUND] This is module 31 of Mechanics of Materials III. We're continuing with the design of a beam in a real world engineering problem. Here is the worksheet of our model and the loading, and we said we're going to check three critical areas on the cross section. The first area we're going to check is the top or the extreme fiber of the cross section, the one that's furthest from the neutral axis whether it be at the top or the bottom. And that's where we see our maximum flexural stress. We're also going to check somewhere in between the outer surface and the neutral axis where we get a combination of flexural stress and sheer stress. And we're going to check part C, which is at the neutral axis where we see maximum transverse stress. To begin the design, we're actually going to assume that the extreme fiber, the cross section at point A is going to be critical. That's how we're going to select our beam cross section. And then we're going to go back though and check to make sure that also achieves and is good for points B and C as well. So we'll check all three locations. And so for our worksheet, let's begin by deciding what kind of failure theory are we going to use? We talked about a couple of different failure theories, for this problem are we going to use the maximum normal stress theory or the maximum shear stress theory? And what you should say is, we're going to use the maximum shear stress theory because this is a steel beam. And this theory is good for ductile materials, whereas the maximum normal stress theory is good for brittle materials. And so for this theory, failure occurs when the shear stress is greater than the shear stress that is defined for failure. And so we drew or draw our more circle from this simple tension test, and we found that tau max was equal to sigma max divided by two. And so for our point A, we've got our sigma max is our maximum flexural stress. So tau max is going to be that maximum flexural stress divided by two. And it's shown here. And so we can also show it in terms of the section modules. So we have tal max is equal to the maximum moment experience divided by two times the section modules. So we can rearrange this to put it in terms of the section modules. And for design we want that section modules that we pick. Or the beam that we pick, or the particular cross-section, to be greater than or equal to the largest moment we experience divided by two times the allowable shear stress that we'll be able to accept. And so, remember. Factor of safety is failure stress over actual stress. In this case, we're going to say the failure stress is the yield shear stress or we're going to say that yield is equal to the proportional limit. So that's the same as the yield, normal stress, divided by two or more circle. Or the normal stress actual divided by two by again, more circle. And so we can factor out the two, and we find out that here is our result. We're going to use sigma proportion limit, or yield, from a reference to be 250 mega pascals for our steel that we're using. And we'll go ahead and use a factor safety of 1.2 for the bridge, chord beam design. We'll go ahead and substitute in tau actual or allowed as being sigma actual allowed over two, the two's cancel and so now we want to design our section module list so that it's greater than or equal to the maximum moment of experience divided by this value that we came up with for the allowed normal stress over here. And so, what is the maximum moment experience? Do you remember what that was or how we would find it? And what you should say is okay, we gotta go back to our shear force and bending moment diagrams. The maximum moment that we experienced is actually right here. And it's 130 kilonewton meters. And so now that we have that, we know that s for our cross section has to be greater than or equal to 130.5 kilonewton meters divided by sigma actual allowed. And I do the conversion factors and I find that I need a section modulus for my wide flange I beam that's going to carry the load to be greater than or equal to 626.4 times 10 to the third millimeters cubed. What do I do now? Well, what you should do is find the table of wide flange I-beams and try to select an I-beam that has a section modules that is greater than this number. And so I've selected W356 by 45, it has a section modules at 688 times 10 to the third millimeters cube which is definitely greater than what's required or what's needed, and again you can find these values in a standard reference online. And so that's the beam we're going to to choose, and we'll design against that. We'll make sure now we've designed it based on the maximum flexural stress at the outer fiber. We'll now check to make sure it also works at point B where we have a combination of flexural stress, and shear stress. And at point C at the neutral axis where we experience maximum shear stress. [SOUND]
