# One-Dimensional Fluid-Structure Interaction Models in Presurized Fluid-Filled Pipes: A Review
## 2 Definitions and Basic Concepts
### Degrees-of-freedom in Fluid-Filled Pipes  
Degrees-of-freedom (DOF) are the number of independent coordinates or parameters that describe the position or configuration of a mechanical system at any time [26]. Systems with a finite number of degrees-of-freedom are called discrete systems, and those with infinite degrees-of-freedom are calledcontinuous systems. Pipe systems are continuous systems; however, these can be treated as discrete systems for numerical modelling purposes, with many DOFs depending on the number of nodes.  
Pipes are slender elements; therefore, a 1D approach assuming that the fluid pressure propagates axially during hydraulic transients is reasonable. However, transient pressures transmit forces over the pipe wall that make the pipe system move in a 3D space. The basic degrees-of-freedom for a rigid body in a 3D space are three for translation (i.e., heaving, swaying and surging) and three for rotation (i.e., pitching, yawing and rolling). An infinitesimal control volume of a pipe-segment (like in Figure 1) will have the referred six basic degrees-of-freedom. The pipe-wall control-volume is a hollow cylinder; therefore, axisymmetric vibration due to hoop (circumferential) strain must be considered as well, adding another degree-of-freedom. Also called pipe breathing, this degree-of-freedom has to do with the radial displacements of the pipe-wall. Additionally, the infinitesimal control volume of the 1D contained fluid accounts for another degree-of-freedom. In other words, the fluid is indeed an inner coaxial cylinder with only one degree-of-freedom in the axial direction. Henceforth, in the present 1D FSI analysis, eight degrees-of-freedom compose the infinitesimal control volume of a pipe.  
For each degree-of-freedom, momentum and mass conservation laws are applied, giving as a result a set of 16 partial differential equations (cf. Equations (1)-(16)), with time and space coordinates as independent variables, governing two basic dependent variables related with the loading and the movement in each degree-of-freedom (i.e., load and deformation relation). Depending on the pipe geometry, axial, shear, bending and torsional forces and displacements alternate throughout the pipe. A schematic of such displacements is shown in Figure 1.  
FSI models in 1D water-hammer analyses can be classified according to the pipe degrees-of-freedom as follows:  
* 1-DOF (fluid surging): only the axial fluid transient event is described.
* 2-DOF (breathing): radial inertia of the fluid and the pipe are taken into account.
* 3-DOF (solid surging): refers to the axial movement of the pipe.
* 4-DOF (swaying): includes the effect of horizontal displacement of the pipe.
* 5-DOF (heaving): includes the effect of vertical displacement of the pipe.
* 6-DOF (yawing): includes the rotation of the pipe in the \(\widehat{x}\)z plane.
* 7-DOF (pitching): includes the rotation of the pipe in the \(\widehat{y}\)z plane.
* 8-DOF (rolling): includes the rotation of the pipe on the \(\widehat{x}\)y plane.  
Figure 1: Spatial reference system and sign convention in a straight pipe element.