An Efficient Computer Analysis of Large Three-Dimensional Piping Systems by the Finite Element Method
Authors: Martinez J.E. , Cetiner A.E. , Jones H.L. , Weiner P.D. 
Published: October 2008

ABSTRACT
The analysis of a large piping system by computer methods often requires solving a large number of simultaneous equations. This can demand a large computer memory and may consume vast amounts of computer time if the program is inefficient. Further, if the nonlinear effects of skid friction are considered, computational efficiency becomes mandatory as several solutions may be required before a single analysis is performed. Most existing pipeline computer programs not only neglect the effects of skid friction, but are usually limited to planar analyses.
This paper describes a mathematical model that employs the current state-of-the-art in addition to a new procedure for applying prescribed boundary conditions to predict the static response of large three-dimensional piping systems. The results obtained for several field problems that illustrate the capabilities of the computer model are also included.
The model, which is based on the Matrix Displacement Method of structural analysis, idealizes a piping system as an assemblage of twelve degree of freedom beam elements, and includes the effects of weight, differential temperature, internal pressure, settlement, elastic supports and skid friction. The computer program makes use of banding and a "collapse" of the structural stiffness matrix in an effort to obtain accurate solutions in relatively short periods of computer time. For example, the displacements of a pipeline idealized by 40 elements having 246 degrees of freedom were obtained in less than one second of IBM 360/65 computer time.
From the studies presently performed with this model, it can be concluded that:A three-dimensional pipeline structural analysis which includes the effects of skid friction and is exact within the limitations of linear beam theory can be obtained in relatively short periods of computer time.Use of the reduced stiffness matrix results in a computer time saving of approximately 25 percent over unreduced banded solution schemes.Due to its low core storage requirement, the model can be adapted for use by most small memory computers.

INTRODUCTION
The introduction of the high-speed computer has made it feasible for matrix methods of analysis to become extremely popular; the reason being that the computer can rapidly solve the large systems of simultaneous equations that are usually required by these methods. In setting up the problem, the approach taken in this paper was to employ the Matrix Displacement Method of Structural Analysis. This procedure is well established in the literature and idealizes a structure by an assemblage of discrete elements connected at nodes. [1–3] For the three-dimensional problem in question, each node has six unknown displacements and the total number of simultaneous equations that describe the response of the system is six times the number of nodes. Large order systems usually require a considerable number of elements and this leads to the solution of a large number of equations. In addition, if the problem is nonlinear, as may be the case when the effects of skid friction are considered, more than one solution is necessary before a single analysis can be performed.