Generic Results of Solving for the Deflection of a Statically Determinate Beam

A Lesson in Mechanics of Materials

Introduction

Okay, in the above example (here) we solved for the deflection of a specific beam (specific material, support conditions, length, and load distribution). It was more interesting (in my opinion), but limiting. Of value (one might argue) would be a more generic equation giving us the deflection (especially the maximum deflection) for, say, any beam subject to the same kind of loading (triangular) and support conditions (free to rotate at both ends). Yeah, that would be cool, and (more) useful. In fact, there are pages in books with such `generic' solutions. And it is probably published somewhere, but let's pretend it (still) is not, and come up with the solution ourselves.

Procedure

There are really two main ways we can approach this. The first would be to go back to our starting place and make our beam `generic'. In other words, leave the length `L', the maximum load at the right end w_o, E as E, and I as I. And then solve, leaving all this stuff as variables. We would come up with `expressions' for the Shear, Moment, Slope, Curved Shape ... in terms of w<sub>o</sub>, L, E, I. It would be a good exercise. But there is another way ... kind of going `backward' from the result of a particular solution, like the one we have just finished. It is this latter approach I will employ here.

The Backward Generic Approach

From experience (or Dimensional Analysis) I propose that the answer to the maximum deflection equation will be of the form ...

... Δ = C w_o L⁴ / EI ... or ... C W L³ / E I

where

C is a dimensionless `constant' that involves (is determined by) the support conditions and how the load (either w_o or W) is distributed ... and will be different depending on whether we go the w_o or the W route;

... wo is the maximum value of distributed load in this case at the far (right) end;

(or ...) W is the `whole' load (in this case ½ w_o L);

L of course is the Length of the beam (strictly speaking the distance between the supports);

E is the Modulus of Elasticity, and

I is the Moment of Inertia of the beam.

If we did a Dimensional Analysis on the problem, we would probably come up with something like this ...