Is dimensional analysis used outside fluid mechanics and transport phenomena? Most dimensionless numbers  (at least the ones easily found) used for dimensional analysis are about fluid dynamics, or transport phenomena, convection and heat transfer - arguably also sort of fluid mechanics.
My understanding of dimensional analysis is the following: Derive dimensionless numbers from the description of a system, find the ones physically meaningful, and use them to compare different situations or to scale experiments.
Is this possible in other fields, like classical mechanics, and their engineering applications?
Example: describe a horizontal beam by:
$$
X=\frac
{\text{forces acting on the beam}}
{\text{forces beam can withstand without plastic deformation}}
$$
Both parts of the ratio being functions of shape, density, gravity, material constants etc. 
My assumption is yes, it's possible, but most fields outside the sort-of fluid mechanics described above are easy enough to calculate without dimensional analysis. 
 A: Dimensional analysis is arguably the most important technique in any physicist's bag of tricks. It is used regularly throughout physics. Variations of the principle are also used in mathematics, biology and probably in other fields as well.
Sticking to physics, here is an example from classical mechanics: derive Kepler's third law.
The independent variables for a planet are its mass $m$, its distance $r$ from the sun, its period $t$, and the force $F$ acting on it. All other variables, like the velocity or acceleration, are functions of those. The only dimensionless number we can construct from the variables is
$$ \frac{F t^2}{m r} \,,$$
so this number must be the same for any planet in our solar system (assuming that there is some functional relation between the variables). We also have Newton's law of gravitation,
$$ F = \frac{G m m_s}{r^2} \,. $$
Together, we see that
$$ \frac{t^2}{r^3} = const \,. $$

most fields outside the sort-of fluid mechanics described above are easy enough to calculate without dimensional analysis.

I encourage you to pick up Polchinski's book on string theory and work through the first few chapters. Then review that statement.
A: I use dimensional analysis regularly, but not often to derive equations. I find it most useful when I've been doing some algebra and ended up with some equation - I plug in the dimensions of the variables in the equation to check that the left and right sides agree. This is a quick crosscheck for my working. Agreement doesn't prove I got it right, but disagreement certainly means I've made a mistake somewhere.
A: Particle physics uses dimensional analysis quite often, not only to derive and verify equations, but also to understand the physics behind many quantities that are not classical.
A: Dimensional analysis and nondimensionalizing expressions are very often incredibly enlightening and are therefore tools that any physicist will and should probably use often.
Especially when taking the limit of some quantity being "small" or "big" in relation to another quantity, you want an unambiguous way to express those vague notions numerically. Dimensionless variables allow you to do that. And often enough removing dimensions will help you solve integrals or differential equations with less or no assumptions about the values of certain quantities. Checking results from some calculation is also an important use of dimensional analysis, though that isn't really necessary if you instead work with dimensionless variables in the first place.
Additionally, dimensionless quantities can make it more obvious what is going on physically. For example, in solid state physics it is often helpful to express wave vectors in units of $k_F$, the Fermi wave vector, or some other typical wave vector scale. In an end result or on a plot, it is then often easy to spot what happens at which values of the wavevector in relation to that typical scale.
It's definitely not the case that the calculations in all fields other than fluid mechanics are too easy for dimensionless numbers to be of value. Definitely not. It's a technique used everywhere. Solid state physics, astrophysics, particle physics, you name it. They all frequently make use of it.
