So, as the second part of two on bending moments, I covered frames and their behavior under loads.
Frames work in bending; which is a structurally inefficient way to carry load (see shoulder analogy from my last post). But, the majority of structures are not the most structurally efficient (and nor should they be) as we have greater issues to deal with such as needing flat floors, vertical walls, and open passage between the walls. Hence, the dominance of frames in our lives.
Overall, this is the most technical of classes so bear with me as we go through some of this…
My approach, as before, is to think about the direct loadpath of any applied load to the supports. Once we have that, any deviation produces bending. The greater the distance from structure to loadpath, the greater the bending. This is why I started this whole course with cables (and a bit on arches) as it’s the clearest way to think about structural behavior.
Here are some examples from simple to a bit trickier. I provide some explanation and follow with diagrams. The diagrams have blue for the frame and load, red for the direct loadpath, and green for the bending moment.
1) So let’s take the simplest frame. Pin joints at the base and a pin joint in the middle of the beam with a single load. The direct loadpath consists of straight compression lines to the supports (i.e. what a cable would do in reverse). So there is no bending moment at the pins (by definition) and linearly increasing bending moment as we move to the frame corners. These are the critical bending locations. Now, we draw bending moments perpendicular to the steel members but the frame corner is the frame corner…i.e. it has one bending moment. So I like to draw it curved around the corner to remind everyone.
Note how the compressive thrust is countered by a reaction in the opposite direction. i.e. there is a vertical and horizontal component to the reaction (just as there is for a hanging cable).
2) Now, if we keep everything the same but have the beam without a central pin, we still have straight compressive lines but they start higher up and cross through the beam.
What does this mean? Well, a number of things:
- First it means there are locations of zero bending moment in the beam. We saw this with fixed end beams in the last post.
- More importantly, we can immediately see the horizontal reaction at the base supports has been reduced. The thrustline is steeper – it’s like having a cable with greater sag.
- This all results in lower bending moments at the corners which were the critical locations. And this is as it should be for an overall structural frame that is now stiffer than before.
3) Last step for a basic frame with central singular load, if we make the base supports fixed, then our thrustlines cross both the beam and the columns. This tells us a great deal about the deflected shape of the frame. Remember how I explained about “smile” and frown” bending moments and their relationship to curvature in the last post. Well, this lets us immediately draw the deflected shape as shown below.
Notice how I haven’t done any equations or used numbers. This has all been qualitative work based on fundamentally understanding how structures carry loads. This can be extended to other locations of the load or multiple loads.
As a last example, here’s the same type of diagram for a lateral load. Notice how our thrustlines have both tension and compression.