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Indeterminacy - A curse or A boon

A course in structural engineering starts from a particular type of structural analysis called determinate structural analysis. Here you study anything regarding determinate structures, whether it may be a simply supported beam, a 3 hinged arch, a truss. Everything is determinate and that means number of unknowns is equal to the number of equations. That is great, you see the unknowns, form the equations and figure out the value of the unknowns. As simple as that. But this isn't everything about structural engineering. When you design a building or a bridge, there is more to it than just a determinate structure. 

Let us take a simply supported beam and a fixed ended beam.In case of a simply supported beam, you apply the load and the amount of the load carried by the beam is equal to the strain energy stored in the beam. Let us assume a beam with a load capacity of 100 kN. When you load the beam, then it will be deflected by a certain amount (x). So the work done by the load will be equal to the strain energy stored in the beam in the form of bending moment. The strain energy is stored in the form of bending moment acting in the beam. Now, the point comes to what happens when we increase the load gradually beyond 100 kN? Does it fail immediately? No, it does not..!! Let me show it to you by drawing it on a paper...!!

 Figure 1(a): Load of 100kN applied on the beam

 Figure 1(b): Bending stress graph of the beam

Figure 1(c): Simplified strain energy graph for the beam

The above picture represents how the energy is absorbed in the beam as the load is applied. It shows the connectivity between the strain energy ad the internal stresses of the beam. If you wish to know more about the beam bending then click here.

Figure 2(a): Beam loaded beyond it's capacity and forming a hinge in the center

A plastic hinge means that the beam has been loaded beyond it's capacity and now the beam will no more resist any amount of loading, instead it will simply deflect.

Figure 2(b): Bending stresses in beam at plastic moment

Figure 2(c): Strain energy absorbed by the beam

Now, what does figure set 1 and set 2 represent? This shows that if a particular member is loaded beyond it's capacity, it still has energy to carry a higher amount of load. This energy comes from the in elastic capacity of the beam. Here I have used the analysis form of Elasto-plastic, because it is simpler than using a non linear stress strain graph and it gives a good understanding to reader. In figure set 1, the beam is loaded till the elastic point is reached, but in set 2, the beam is loaded such that it's entire cross section is yielded which means more absorption of energy. This is clearly reflected in figure 2(c) where the stress remains the same but the strain reaches till the breaking point where the plastic hinge is formed. Did you now see what just happened over here? But I directly showed you the jump from one state to another, what about a state in between when you gradually increase the load till the plastic stage? The figure below describes that. Look carefully at the bending stress graphs.

What has all this to do with the indeterminacy? I gave a title about the curse or a boon? How can indeterminacy can be a curse? Well for a structural engineer, to carry out the calculations of an indeterminate structure is much more difficult than to carry out the calculations of a determinate structure. To analyze an indeterminate structure, we need 100 more papers and more complex matrix formulations. We use approached like finite element methods, stiffness matrix approach, flexible matrix approach just to calculate an indeterminate structure. So this becomes a curse. But is it really so bad? Is it bad to have indeterminacy in a structure? No..!! Not at all..!! In fact, indeterminacy has always been an old friend of a structural engineer and here is why.

We know that when a member loses it's capacity, it turns into a mechanism. This means that it gets deformed without any resistance in the direction of the applied force. As seen in the case above, a simply supported beam turned into a mechanism when it lost it's capacity. 

But now let us consider a fixed ended beam. As shown in image below, the beam is loaded with a UDL of
w kN/m and the bending moments at the end and at the middle of the beam are shown with the strain energy diagram and the plastic hinges. This shows that the bending moment is less in case of a fixed ended beam as compared to a simply supported beam. Over and above there are three energy absorbing locations.

Now, it has been shown in the figure that the energy absorbed by the ends at this stage is more as compared to the center of the beam, this is because of the higher restrains at the end. The end has a higher resisting moment with a factor of 1/12 while the center has just 1/24. This shows that first the ends will fail, but still the beam has not turned into a mechanism, it still has a central location where the beam has still some amount of energy absorbing capacity. This will lead to higher load capacity. When that particular load is reached then the beam will turn into a mechanism. One more thing to note is, the maximum bending moment acting in a simply supported beam has a factor of 1/8 while this has 1/12. This reflects reduction in the bending moment in a fixed ended beam.

How is the above factor important? Well, we design a structure for a particular amount of predicted load. But what if that load increases? Will the member have the capacity to take the load beyond it's designed capacity? Will a particular building have the capacity to resist a higher earthquake force? All the answers lie in the indeterminacy of the structure which leads us to alternate load path method or redundancy or robustness of a building. 

Suppose I make a building which is completely determinate and I design it for one person every floor. But suppose if accidently there are two people on one floor. You know what will happen? First the local member will fail and then the structure will lose it's stability and then the structure will collapse. But if I have a highly indeterminate structure, then even if a local damage will occur, the load will find out alternate load paths by concentrating energy into other locations of the building and tada..!! The building is still safe. You can do this experiment using Popsicle sticks and glue and then you will say that indeterminacy is actually a boon..!!

I hope you got a insight to the matter. I hope you enjoyed it.

Have a good day ahead and keep carrying the madness.