One more post and we will finish the statics subject of our post series…
The beam example we have just given in previous post, was a simple beam. In that problem, we were easily able to solve all reactions, just by applying three equations, which were,
Total Fx = 0
Total Fy = 0
Total M = 0
So we had three equations that we were able to use. And luckily, we had only three unknowns. We had 2 unknown reactions from the pin support at point A, which were Fx and Fy, and we had 1 unkown reaction fro the roller support atpoint C, which was Fy only as a roller support cannot produce reaction in the direction it rolls. And noe of these supports could resist moment either, and so could not produce moment reaction. So we had a total of 3 unknowns.
So, three equations, and three unknowns, we were able to solve things easily, with simple math equations. this situation is called a “determinate structure”.
But what if, we had more unknowns? Which means, what if we had more supports, or different types of supports?
The equations available to us, does not change, they never change. They are still and always three, which are Fx, Fy and M = 0 (for a 2 dimensional problem as in here of course – for a 3D problem we have 6 equations where we also have Fz and My and Mx as well, but it means we need more supports too, but often times we can do structural analysis on 2d).
So, considering our 2 dimensional structures, which give us three equilibrium equations, what if we had more than 3 unknowns?
That would require more advanced methods, which we will not cover yet, and that case is called an “indeterminate structure”.
In real life, all structures you see are indeterminate, with so much more unknowns, than only 3. This means, there are a lot of redundancies, but it is a good and safe thing, because if one thing fails, another support can take over and our structure will not fail (whereas in our simple beam example, if one of the supports was damaged, we would not achieve Fx or Fy or M = 0 so the structure would move or in other words, fail.
Take a look at the beams and frames below:
statics 4 –
In the figure above, on the left a structure is given, and on the right, an exact equivalent of it, with all the available (possible) reaction forces are shown, if one or more external effects apply on that structure. As you can see, a fixed support is able to resist everything, and therefore it can provide 3 reactions as shown, and so on…
So we simply add the total available number of reactions we have, and then, we subtract the available equations we have, which is 3, for a 2 dimensional problem. Therefore, if for example we have 5 available reactions, since we only have 3 equations, it means 5-3 = 2 of these reactions are redundant. So it is said that this structure is indeterminate with 2 redundant reactions. Take a look at each case and see the corresponding reactions and degree of redundancy.
The last one is somewhat different. There we put an internal hinge. That is to show you that an internal hinge can provide extra equation, so now instead of 3, we will have 4 equations available to us. The extra equation comes because now we know that at that point moment must be zero, as a hinge can not resist moment.
The indeterminate structures can not be solved by the simple math logic we have seen in previous pages, to solve reactions for a determinate structure. There are more advanced methods available to solve indeterminate structures, which is beyond the scope of an introductory level text as this one. But the overall goal is the same. No matter how complicated the structure is and to what degree it is indeterminate, no matter how many members in what arrangement it has, all we do is, after the external reactions are solved, we can obtain axial force, shear force and bending moment diagrams for all the members in that structure, just like we did for our single, simply supported beam in previous pages. Then, since we will know all stresses and moment anywhere on our structure, we can come up with a required cross section and materials to design it, so that it can withstand these effects. That is what the structural design programs basically do. You can enter all the member arrangements and overall geometry, and show all external forces on it, such as load of the structure itself, loads from people, vehicles etc.. as input… Then, just like the general steps here the program calculates the external reactions and then it calculates the force and moment diagrams for all members, each and every of them, and finally it produces required design, after which, the engineers can refine.
All structures you see are indeterminate structures. We just need these redundancies so that when something goes wrong, the load can take another path which was redundant before. In real life, determinate structures have limited use such as simply supported bridge beams, that carry a bridge deck. We want those beams to be simply supported, and therefore move back and forth as needed during an earthquake, to avoid stresses on them.
In the next post, we will introduce the term “Strength / Mechanics of materials”