I sometimes hear engineers state ‘the beam is statically indeterminate’, which is often incorrect. Why is it indeterminate and what does the term really mean? I’ll try not to make this the same article that’s been written and re-written endlessly and will instead try to incorporate some new concepts and new drawings on a very old and sometimes confusing topic. Often the beam is deemed to be indeterminate because the boundary conditions are complicated and/or numerous, but this is a very poor criteria for defining what may or may not be indeterminate structure. Even attempting to solve the equilibrium equations and coming up short (meaning there are more unknowns than there are independent equations) can be a poor justification simply because the equilibrium equations may be written incorrectly from the start. Complexity just isn’t the driver when evaluating a beams ‘determinancy’ because a simple cantilever can cause significant problems for the analyst if not addressed properly.
Start with the example shown in Figure 1. Suppose two men are carrying a 10 foot long steel I-beam. The loading is from the weight of the beam only – there are no external applied loads. Because the beam’s weight of 20 lb/ft is evenly distributed along the length of the beam, the resultant load is half-way along the beams length and is 200 lb so each guy is supporting a 100 lb load.
Figure 1 – Statically determinate
Now take a look at Figure 2. A third guy decides he wants to help. Maybe. He positions himself at the halfway point of the beam, directly under the resultant and pushes up hard. Maybe. Because of his location he could now be supporting the entire load and the guys on the end are only keeping the beam in balance. Or he may be pushing up only slightly, grunting and groaning and making it look good but not really providing any benefit. Or… he may even be pulling down and making the situation worse. What’s he really doing? You can’t determine and this beam, despite very simple loading and boundary conditions is indeed statically indeterminate. Granted the example is somewhat extreme because a human has the ability to push up a lot or a little or even pull down if he wants to while a vertical column does not. But even if the three people are replaced with three columns the beam is still indeterminate because there are three unknowns and only two equations (one force and one moment).
Figure 2 – Statically indeterminate and pinned
So we make assumptions regarding the reactions, one being that the load is supported equally so each column carries 67 lb. Another would be based on supported length, so the center column carries 50% of the load and the two end columns carry 25% each as shown in Figure 2. What’s really happening can be solved analytically by energy methods or deflections, experimentally with strain gaging, and of course the finite element method but with basic statics, it’s indeterminate.
Now look at a cantilever as shown in Figure 3. The book says ‘you can’t sum moments about a cantilever’. Why not? The book never answers that question. It’s because you don’t know what’s happening inside the cantilever, meaning you don’t know how efficient it is at reacting a moment. In the example the intermediate support at ‘B’ may be reacting a significant portion of the 1000 lb applied load, it may be doing nothing at all or somewhere in between. It is entirely possible to remove the support at ‘B’ and have a simple diving board in static equilibrium. Or the cantilever at ‘A’ may fail as a result. Look at the equilibrium equations where moments about the cantilever are summed. Here there is one equation and one unknown and the force at ‘B’ is found to be 1667 lb. Incorrectly. It may be zero, or any value between zero and 1667 depending upon the effectivity of the cantilever. So again, this beam is statically indeterminate.
Figure 3 – Statically indeterminate and cantilevered
The simple examples quickly lend to the concept of end fixity in aerospace structure and why assuming half-way between fixed and pinned is often done when analyzing a fuselage frame, for example. It’s difficult to have a complex machining where ribs meet caps and idealize the ribs and axial members with simple, pinned end conditions. It’s equally difficult to rationalize them as fixed or cantilevered meaning no translation or rotation at the constraint. Pinned, fixed, cantilevered, rollers, sliders, etc.? It’s very important to understand the nature of the boundary conditions because they drive the determinancy, not the loading.