NO, the physic of the problem is a bit more complicated.
This problem is easy and hard to explain, mainly because of the Classical mechanics involved, (small reference to Wikipedia).
Easy part, why yes
As RedGrittyBrick`s answers point out, there is "resonance effects" in bridges, in fact every object under Classical mechanics is affected for this.
Under Classical mechanics, the bridges can be oversimplify represented as a "mass spring system" (google that) like this image:
Reference: A physics forum
this "mass spring system" have a "natural frequency" in which they oscilate and in turn if excited with a force in such frequency the ressonance effect (see resonance in wikipedia) will appear, like in a swing, amplificating their displacements:
the formulae is increble simple:
Reference: Armonic oscilator, wikipeida
k: spring constant
and his effect can be displayed as follows (in this graphic "normalized frequency" is the quotient between "frequency of the applied force" and "natural frequency of the element"):
Reference:Wikipedia, or any physic book
As this the easy part many "Tesla followers" stop reading right here and start belive that the Earthquake machine exits or is posible
Easy part, why no
First, as I try to hint before is enormuosly complicated find the value of k, the spring part of the problem, without being a engineer / architect to make a computational model in software like Etabs, see examples Here
Also finding experimentally is hard to say at least because is not easily scalable, and a real scale experiment requieres a reaction wall (the gigantic structure in background weighting arounf 7500 tons) and isolate the bridge of external forces such winds, pedestrian, vehicules, surrondings, etc.
Refrence: Wikipedia again, the reaction wall is the concrete structure behind, the steel frame is the object being tested
Also I reserved to the end a last word damping (Reference in Wikipedia or a physics book). Damping (from friction with the air, heat dissipation, microcracking, element deformation to actual dampeners like this or this), besides add another hard to calculate variable to the analysis to complicate a bit our model, it damps the energy transmission.
Our oversimplistic model becomes:
Reference: More wikipedia, but the data and info are checked and up to date (2012.12.24)
And now our graphics looks like:
As you can see even small damping prevent the infinite energy acumulation/amplitude range
Hard part, why no
- Determinate k (spring coefficient) and C (damping coeficient) is hard, not to mention they will change as the structure degrades with local damage or the active movement of the troops/people, rendering imposible even to coordinated troops/people keep marching in the natural frequency who now varies in the time an is "almost" imposible to determinete in real time.
- As the problem adds degrees of freedom (displacement in many axis of many parts) the complexity scalates very quickly, the simplest physical object has 6 natural frequencies (feel free to check again the wikipedia in this regard)
- Depening of the force applied/mass of the bridge factor, along other factors, the time need to "make resonate" a bridge can vary enormously, our current technology allow the "artistic factor" to come in place, designing bridges too lightweight or too slender to being affect by the people using it in a short time frame.
- Depending of the desing, the structure can be enough damped to never resonate no matter how many soldiers march for how many time.
- Other things come in: the Tacoma Narrows Bridge, has sometimes been characterized in physics textbooks as a classical example of resonance. However, this description is misleading (see aeroelastic flutter).
Does a column of marching soldiers have to break their rhythm while crossing a bridge to prevent its collapse?
NO, unless the brigde where "artisticly shaped", as in this case of the Milennium Brigde even then, it would be only midly annoying/minor risk of fallin in or falling off the brigde, only long term stress would damage the bridge.
Humor note: I have been asking to military personel this very same question, but the only answers they gave was: How you get in here? / What are doing here?