The Direct Integration Method

As a mathematician first and engineer second, I always focused on the math side of engineering. When taking a core course for my specialty, I recall the program glossing over a method called direct integration. In other words, a purely mathematical approach that focused on the one of the major concepts from single variable calculus. What’s more intriguing, the relationship between load, shear, moment, angle of rotation and deflection. Turns out they are related through derivatives and integrals. While the program at the university spent significant time on the moment area method and virtual work method, and almost no time on direct integration; I found that each had issues with more complex loading situations. The end goal was the same, calculate the deflection. The problem with direct integration, it was mathematically intense and often times, people would get stuck in solving for the unknowns. When I took finite elements in graduate school, we focused a lot on the matrix approach to solving problems. And that’s when it hit me. If I combine direct integration with linear algebra, and create the matrices; I could solve for all the unknowns at once. Then I’d have multiple equations completed that I could use to calculate anything I wanted in order to design the beam and verify it’s adequate. A purely mathematical approach to engineering, and an elegant approach it is. Rather than try to explain it in words, I thought a hand written example might explain it better. Also, rather than show a basic example, I thought a more complex one would help engineers decipher how to get past the point where they would usually get stuck. The attached PDF file shows an example of how to do that when your equations end up with 7 unknowns.