Stiffness Method vs Flexibility Method for Pipe Stress Analysis
Widely used pipe stress analysis software PASS/START-PROF  uses the structural mechanics methods for beam structures, see ,  and similar literature. The piping system is considered as consisting of straight and curved beams. Straight beams are pipes, rigid elements and valves, while curved beams are the bends.
Stiffness Method K*X=F
Traditionally, the most of stress analysis software use the direct stiffness method K*X=F , 
K - Element stiffness matrix;
X - Displacement vector (unknowns);
F - Load vector.
For 4 degree of freedom system (DOF) it will be:
The main disadvantage is high memory usage and low calculation speed for the big models. Advantage is very easy and transparent algorithms for programming.
In the stiffness method, displacements X (rather than forces) are taken as the unknown quantities. For this reason, the method is also called the displacement method. The unknown displacements X are obtained by solving equations of equilibrium (rather than equations of compatibility) that contain coefficients in the form of stiffnesses K.
Flexibility Method A*F=D
The flexibility method A*R=D , .
A - Element flexibility matrix;
R - Load vector (unknowns);
D - Displacement vector.
For 4 degree of freedom system (DOF) it will be:
The flexibility method advantage it that this method for tree-like systems like piping gives a several times less unknowns than stiffness method. The matrix is several times smaller. And solution time several times faster and require less memory. The main disadvantage of flexibility method is a very complex algorithms for programming.
The flexibility method is based upon the solution of equilibrium equations and compatibility equations. There will always be as many compatibility equations as redundants. It is called the flexibility method because flexibilities A appear in the equations of compatibility. Another name for the method is the force method because forces F are the unknown quantities in equations of compatibility.
Combined Stiffness and Flexibility Method used in START-PROF
PASS/START-PROF uses the combined flexibility and stiffness method . It use the combined matrix and combined unknowns (displacements in expansion joints and reaction forces in supports). Method uses the equations:
A - element flexibility matrix (displacements caused by unit forces, applied along the direction of unknown forces r);
K - element stiffness matrix (reactions caused by unit displacements, applied along the direction of unknown displacements x);
B - matrix of displacements from unit forces, applied along the direction of unknown displacements x;
R - vector of unknown forces;
X - vector of unknown displacements in expansion joints and cut-points of piping system;
D - vector of applied displacements;
F - vector of applied forces.
For example we have a 4 node frame (a) with anchors at nodes 1 and 4, hinge in node 2 and applied force at node 3
a - frame model, b - primary structure for flexibility method, c - primary structure for stiffness method
The primary system for flexibility method (b) can be obtained by removing the anchor 4 and applying the unknown loads r1, r2, r2. It will be a console system from anchor 1. The node 2 assumed as rigid.
The primary system for stiffness method (c) will be a system with the hinge in node c and unknown displacement x4.
The system of equation to solve this problem will be:
The following equation states, that the moment in hinge should be zero (lowest row of the system of equations):
This method has been chosen in 1965 when PASS/START-PROF was created, because of it's lowest memory usage and fastest solution time for typical tree-like piping systems. On the modern machines it allows to achieve the results of the complex nonlinear piping systems very fast.
For example we have the following system:
In classical finite element analysis (FEA) software with stiffness method this system will have 78 unknowns. The matrix K will be 78x78. Consume a lot of memory and need a lot of time for solution:
In PASS/START-PROF combined flexibility-stiffness method this system will have just 24 unknowns. The matrix will be 24x24 with same analysis results. The special cut-points A and B will be automatically added that breaks the piping system into 3 parts - "consoles":
In classical stiffness method the boundary conditions are zero displacements x at the nodes, where we place the supports. If we solve the nonlinear piping system assuming some pre-defined tolerance, the displacements at the support points will be always zeroes.
In flexibility method after matrix solution we obtain the unknown forces r instead of displacements. If we solve the nonlinear piping system assuming some pre-defined tolerance, the support reaction forces r will be obtained with some tolerance. After solution, PASS/START-PROF calculates the displacements at all nodes of each console, using the reaction forces r. The displacements will be obtained with some tolerance also. This can cause that some rigid supports will have non zero displacements, for example 0.5 mm. It is not an issue, it is just a solution tolerance. Usually it happening in systems that has a very-very long consoles. It doesn't need any action from user.
But if you want to increase the accuracy of displacements, add the special marker (M) objects. It breaks the long console into several parts. Additional 6 unknown reaction forces x are added at this point and consoles become shorter.
Nowadays, on the modern machines the main benefits of the flexibility method is several times faster analysis speed of nonlinear calculations of complex systems and low memory usage:
- Magalif V., Yakobson L. Pipe stress analysis on computers (Original PASS/START-PROF authors). Moscow, 1969
- Weaver W., Gere J. Matrix Analysis of Framed Structures
- Nagarajan P. Matrix methods of structural analysis