Stressed modal analysis requires finding the deformed position of piping (and structure, if modeled) along with the forces and stresses in the members prior to calculating the mode shapes with consideration of nonlinear boundary conditions (gaps, one-way supports, friction, multi-linear restraints) and P-delta effects if included in the model. Since modal analysis is itself a linear analysis, consideration of nonlinear boundary conditions and P-delta effects is accomplished through load sequencing in which the modal analysis is based on the end state of a nonlinear static load case.

CSiPlant enables users to define an unlimited number of modal cases, each of which can be based on the end state of a nonlinear static case. Since dynamic excitations typically occur when the piping is operational, it's more realistic to base modal cases on the end state of nonlinear static operating cases rather than on an unstressed state. This option can be used for both Eigen modal load cases and Ritz vector modal load cases.

The stressed Eigenvalue option often results in changes to the fundamental frequency/period of 50% to 200% in smaller models with even greater differences in large models. Load sequencing is accomplished by using the "Stiffness at End of" option shown below. In this example, the modal case is based on the end state of the GR→(T1+P1) operating case.

DNVGL-RP-D101 states "Modal analysis of a static model is usually not time consuming and should therefore be carried out to determine the lowest natural frequency of the system. A typical system supported in accordance with a good pipe support standard should result in a lowest natural frequency not less than 4 to 5 Hz."

In the example shown below, we'll compare modal analysis results between an unstressed modal case and a stressed modal case based on the end state of a nonlinear static operating case.

If you click the graphics below to enlarge, you will see that the Modal case on the left, which was analyzed as unstressed, reports its 1st mode fundamental frequency as 6.674 Hz, whereas the Modal-Stressed case reports its 1st mode as 3.842 Hz, which is below the DNV recommended threshold. The angle of the mode shape graphics makes it appear that pipe may have lifted off a one-way vertical stop support, but it didn't. When vertical pipe support reactions from the operating case are displayed below right, we can see that all pipe supports are carrying gravity-direction load, although there was load redistribution due to thermal displacement. CSiPlant model used in this example can be downloaded here.

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