Depending on the amount of damping inherent in the pipe then this determines how far away does the resonant frequency must be from the excitation frequency. An empty pipe has very low material (as opposed to viscous or friction) damping force available. Thus the first two or three natural mode frequencies should be at least 30% away from the excitation frequency. If the viscosity and pressure of the fluid the pipe is carrying is high then this "thick" high pressure fluid will yield some friction damping. However, if the viscosity and pressure is low then there will be little if any damping.
Having discussed the boundary conditions of the design it is now time to do some calculations to anticipate "ball park" expected natural frequencies. Depending upon the pipe and fluid I would approximate the system modulus of elasticity as a weighted mixture of pipe metal and flexible like modulus and by trial and error derive the stiffness of the system to calculate the first three modes: single sinusoid, double sinusoid and triple sinusoid. Tubular section stiffness calcs can be found in Blevin's "Formulas for Natural Frequencies and Modes" Once the stiffness is known then go back to Blevins to calculate natural frequencies. Going the whole route of an FEA simulation the same question will be encountered, what is the right Young modulus of elasticity for a pipe carrying a fluid under pressure? There are FEA modeling techniques that overcome the issue however, it may be an overkill if the first three physical modes are what is needed. Bracing should be placed at the antinodes of the modal responses. So far we discussed the simple approach.
Another factor here is that the above calculations do not anticipate pipe response to the excitation coming from "water hammer" that is reverse flow events nor the effects of acoustic wave excitation due to the fluid pulsations if any. "Water hammer effect" includes a wide spectrum of frequencies that will call for not only structural modal frequencies but also some type of elastomeric damper in the bracing. Fluid pulsation induced resonances call for the application of some type of inline type of preferably tuned fluidic damper.
Location of the pipe braces must also take into account that the pipe will grow/shrink due to either sun/cold-weather exposure or to exposure the fluid temperature. The pipe must be allowed to expand at one end freely otherwise if constrained at both ends, this will create a major thermal growth problem possible setting up the pipe in shape of one of its natural resonant modes. At refineries, they install loops in the pipes to dissipate thermal expansion and also this loop disarms some higher frequency acoustic pulsations. Tubas with their curved divergent piping amplify low frequencies . . . . however most fluid pulsation love straight runs to survive.
Therefore, depending upon how critical is the pipe application one must take into account all of the variables or some and then apply the proper analytical tool.
