According to Memory Alpha
The structural integrity field
) is a technology
developed by spacefaring cultures
to supplement the natural structural integrity
of the material their starship hulls
or other constructions are composed of. Engineers
use a structural integrity force field
to supplement the supports and bulkheads
that give a piece of architecture its shape. This technology was made necessary by the fantastic stresses and probability of a hull breach
presented by warp fields
and spatial phenomena
present in the galaxy
The field strength is usually measured in percentages. A structural integrity reading of 100% means that the field is accepting a maximum load of the stresses presented to the structure, and that all physical architecture is safe from that stress. As the percentage approaches zero, this means that more and more of the stress of weapons
and other phenomena is being experienced by the hull itself rather than the force field. As the hull becomes damaged or breaches, the SIF becomes less effective as the hull it was designed to hold together starts to disintegrate.
If we go by this information then it would seem that the SIF's primary function is to reinforce the ships hull durring the extreme presures put upon it by warp travel and various spaceborne phenomena. In all probability it is also active durring impulse flight but that doesn't necessarily mean that a ship without an active SIF would break apart going at full impulse.
Remember that 23'd and 24th century Starship hulls are made up of materials that we can't even conceive of. Perhaps the tensile strength of the alloys used in the Enterprise A allows the engine pylons and struts to be of a thinness which by our modern material standards would seem precarious but which is perfectly acceptable in the 23d century.
A perfect example of this can be seen in Star Trek IV when Scotty divulges the secret of Transparent Aluminum which can perform the job of a six inch thick piece of plexiglass but be only one inch thick.
Given that such advancements have probably made the construction of many things much lighter and yet sturdier (somewhat like the evolution of our own skyscrapers) The appearance of a frail engine strut may actually belie it's true hardiness.