Classical Structural Analysis Dec 2025

Intro

##
There is an elegance to the mathematical behavior of structures.

For me, elegance is beauty and efficiency somewhere between simple and complex.

My calculations do not meet that definition, but the derived formulas certainly do.

##
Recently I used the simple-beam-with-moment FEM — and I immediately recalled that the unit rotation on the left would generate a 1/2 unit rotation on the right.

As a formula, for a simple beam with a moment loading at the left end:

$\theta_{right} = \frac{1}{2} \theta_{left}$

From undergrad years and thinking back, I always thought that was intriguing.

I then thought I should prove the FEMs using the moment area method. Very satisfying for me but not quite enough. I proceeded to derive the equation for the elastic curve, and the magnitude and location of the maximum deflection. I found this to be quite enough.

##
And as a bonus, I thought I might do a refresher derivation of a couple centroid formulas.

the triangle ( $y=kx$ )
under the spandrel (powers of x)
over the spandrel)

(*) In calculus and engineering statics, a spandrel is the area bounded by a general curve function, typically defined as $y=kx^{n}\,$ and the straight lines of a rectangular frame or the x and y axes.

In architecture, a spandrel is the space between two arches or between an arch and a rectangular enclosure.

The Elastic Curve and FEMs

Centroids