Certain types of bridge are susceptible to mechanical resonance effects even when marching soldiers are not involved.
Mechanical structures usually have one or more frequencies at which some part of the structure oscillates. A tuning fork has a well-defined natural frequency of oscillation. More complex structures may have a dominant natural frequency of oscillation. If some mechanical inputs (such as the pressure of feet walking in unison) have a frequency that is close to a natural frequency of the structure, these inputs will tend to initiate and, over a short time, increase the oscillating movements of the structure. Like pushing a child's swing at the right time.
London Millenium Bridge
One example is London's Millennium Bridge which was closed shortly after opening because low-frequency vibrations in the bridge were causing large groups of pedestrians to simultaneously shift their weight and reinforcing the oscillation. Dampers were fitted.
Maths in the city
All 690 tons of its steel-and-aluminium deck began to sway left and right like a giant, executive desktop toy, so much so that pedestrians, suspended above the Thames on slender steel cables, began to clutch at handrails to steady themselves, and throw themselves against the sway, to stay upright. As they did, so the swings began to get increasingly violent.
Report in The Guardian
Soon after the crowd streamed on to London's Millennium Bridge on the day it opened, the bridge started to sway from side to side: many pedestrians fell spontaneously into step with the bridge's vibrations, inadvertently amplifying them.
Crowd synchrony on the Millennium Bridge - Nature 438, 43-44 (3 November 2005)
other bridges – with completely different structures to the Millennium Bridge – have also moved laterally under large crowds. An example of this is the Auckland Harbour Road Bridge which has a box-girder structure. This bridge was filmed vibrating laterally at 0.7 Hz when a large crowd of demonstrators walked over it in 1975.
This is particularly significant because the Auckland Harbour Bridge is a large road bridge with a conventional structure.
Stabilising the London Millennium Bridge
There are reports of small suspension bridges collapsing partly due to resonance effects
Broughton Suspension Bridge
Angers Bridge Catastrophe
There's at least one PhD Thesis on the subject
Instructions to Soldiers
bridges should be crosssed at a slow gait, and marching men should break step.
The Examination & Repair of Bridges
WITH PRINCIPLES RELATING TO THEIR DESIGN,
Captain C. O. SHERRILL, Corps of Engineers
Instructor, Department of Engineering, 1909
companies of soldiers must
break step when crossing bridges due to the risk of creating large motions at resonant frequencies.
EVALUATION OF CONDITION OF
LAKE SUPERIOR REGULATORY STRUCTURE
SAULT STE. MARIE, MICHIGAN. 1981