Axial stiffness for individual pipe components or fittings is calculated using Equation 3.

k = aE/L Equation 3

Where:

a = cross sectional area of the pipe component or fitting (between outer diameter and inner diameter)

E = modulus of elasticity of the fitting material

L = length of the fitting or component

For axially flexible fittings using thrust rods, "a" is the total cross sectional area of all the rods, and "L" is the distance between the surfaces of the flanges where the nuts on the thrust rods resist the thrust. If an axially flexible pipe fitting (such as a sleeve type pipe coupling or rubber expansion joint) is used without thrust rods, the axial stiffness "k" of the fitting may be considered to be approximately zero for the range of axial deflection values of interest here. This also results in an effective axial pipe stiffness that approaches zero.

Consider a piping run that is restrained at the pump end only. The linear expansion is resisted by the piping as it stretches under the force of hydraulic pressure, but all of the expansion is away from the pump with little to no pressure reaction on the pump.

Typically, in such an installation, resistance from sliding friction in a pipe support or an orthogonal run of pipe created by an elbow or a manifold resists the axial deflection, and creates a reaction force.

For a piping run restrained at the pump end and the other end, the restraint opposite the pump can be in the form mentioned above or a pipe anchor. Other restraints include situations in which the pipe attaches to a manifold, passes through a wall or turns down into the floor. The ability of these restraints to resist the linear stretching is typically far greater than that of the pump.

For our purposes, the deflection at the restraint at the opposite end of the pipe is zero. Because the stretch in the pipe will be resisted by the pipe and the pump, this results in a reaction load of some magnitude on the pump that is also transmitted to the pump support structure, baseplate and foundation. This reaction load on the pump can be compared to the rated nozzle load.

Figure 2 can help explain this analysis approach. It shows the interaction of the pump and piping as a system of springs in parallel, with one spring representing the effective piping stiffness and one representing the pump and its support structure.

For a certain value of deflection to be achieved at the pump flange, the piping must stretch and the pump flange must deflect the same amount, with some component of F_{p} being restrained by the piping and some component being restrained by the pump. This calculated reaction load can be compared with the rated nozzle load for the pump to make sure the rated nozzle load is not exceeded.

In this model: d_{p} = F_{p}/k_{total} Equation 4

Where:

k_{total} = (k_{pipe} + k_{pump})

Equation 5

Recall that k_{pipe} is calculated for the various pipe components as discussed, and F_{p} = (P)(A).

Knowing F_{p}, k_{pipe} and k_{pump}, deflection d for the installation can be calculated. The reaction load on the pump is not F_{ p} = (P)(A) because this force is resisted by both the pipe and the pump. An exception would be a case where the pipe stiffness approaches zero.

This practice is strongly discouraged for the following:

- The hydraulic pressure force is easily constrained by thrust rods and easily transmitted to the piping that can handle such force.
- The hydraulic pressure reaction loads placed on the pump in such cases are typically relatively high compared with the pump capability.
- Any hydraulic pressure reaction force placed on the pump is transmitted to the support structure and foundation.
- A hydraulic pressure reaction force will load the pump, support structure and foundation each time the pump is started.

Because the reaction is proportional to the relative stiffness values of the piping and pump, the use of proper thrust rods across any axially flexible fitting provides for a piping system axial stiffness that reduces the pressure reaction force on the pump. The reaction force on the pump will be less than the F_{p} generated by the pressure in well-designed systems.

Another model (see Figure 3) is used to determine the reaction load on the pump to compare it to the rated nozzle load. Consider for any force F on the pump nozzle restrained only by the pump, the deflection is show in Equation 6.

d = F/k_{pump}

Equation 6

Using this model, the value of k_{pump} may be determined by applying a known force on the pump and measuring the resulting deflection at the flange.

k_{pump} = F/d

Equation 7

Historically, some pump nozzle load standards have been based on maximum allowable deflection when subjected to the nozzle load, an approach that lines up with this method of analysis.

Using Equations 4 and 5, Equation 8 for the reaction load R can be derived.

R = F_{p} – (d_{p})(k_{pipe})

Equation 8