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Many CAESAR II users question how to solve the friction problem. Unfortunately, things are not as simple as you would initially imagine.

There are two approaches to solving the friction problem. First, insert a force at the node which must be overcome for motion to occur, or second, insert a stiffness which applies an increasing force up to the value of Mu * Normal force. CAESAR II uses the restraint stiffness method. (An excellent paper on this subject is "Inclusion of a Support Friction Into a Computerized Solution of a Self-Compensating Pipeline" by J. Sobieszczanski, published in the Transactions of the ASME, Journal of Engineering for Industry, August 1972. A summary of the major points of this paper can be found below.)

Ideally, if there is motion at the node in question, the friction force is equal to Mu * Normal force. However, since we have a non-rigid stiffness at that location to resist the initial motion, **the node can experience displacements**. The force at the node will be the displacement * the stiffness. If this resultant force is less than the maximum friction force (Mu * Normal force), the node is assumed to be "not sliding," even though we see displacements in the output report.

The maximum value of the force at the node is the friction force, Mu * Normal force. Once this value is reached, the reaction at the node stops increasing. This constant force value is then applied to the global load vector during the next iteration to determine the nodal displacements.

Basically, here is what happens in a "friction" problem:

- The default friction stiffness is 50,000 lb/in in CAESAR II versions up to 3.24. Version 3.24 and later versions default to 1,000,000 lb/in. If necessary, this value should be decreased to improve convergence.

- Until the horizontal force at the node equals Mu * Normal force, the restraint load is the displacement times the friction stiffness.

- Once the maximum value of the friction force is reached, the friction force will stop increasing, since a constant effort force is inserted.

By increasing the friction stiffness in the setup file, the displacements at the node will decrease to some degree. This may cause a re-distribution of the loads throughout the system. However, this could have adverse affects on the solution convergence.

If problems arise during the solution of a job with friction at supports, reducing the friction stiffness will usually improve convergence. Several runs should be made with varying values of the friction stiffness to ensure the system behavior is consistent.

Summary of J. Sobieszczanski's ASME Paper

- For dry friction, the friction force magnitude is a step function of displacement. This discontinuity means the problem as intrinsically nonlinear and eliminates the possibility of using the superposition principle.
- The friction loading on the pipe can be represented by an ordinary differential equation of the fourth order with a variable coefficient that is a nonlinear function of both dependent and independent variables.
**No solution in closed form is known for an equation of this type**. The solution has to be sought by means of numerical integration to be carried out specifically for a particular pipeline configuration. - Dry friction can be idealized by a fictitious elastic foundation, discretized to a set of elastic (spring) supports.
- A well-known property of an elastic system with dry friction constraints is
**that it may attain several static equilibrium positions**within limits determined by the friction forces. - The whole problem, then, clearly does not have a deterministic character, but rather a stochastic one.

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