18.104.22.168 Bi-linear Stiffnesses
Restraints with bi-linear stiffnesses maintain a constant stiffness up until a specified loading is achieved, at which point the stiffness changes. These can be used to represent bottomed-outsprings (through the normal range of the spring the stiffness is the spring constant, but once the spring compresses completely, the support essentially becomes a rigid hanger); soil supports (clay normally has constant stiffness until its ultimate load bearing capacity is reached, at which point the clay liquefies and offers virtually no resistance); and plastic hinges (a model where pipe or restraint yields throughout the cross-section). An example of a pipe in a soil trench, along with a graphic representation of the restraint response is shown in Figure 3-105.
The iterative technique used for these restraints is explained below. The analysis is first done using a linear restraint with the first stiffness (K^). The restraint load (F) is then checked, if it is less than the transition load (Fy), then the analysis is complete. If not, some modification is done to the model. First, a CNODE is added at the end of the restraint. A displacement equal to Fy/Ki and a force equal to Fy are imposed on the CNODE, with a corresponding -Fy placed on the restrained point. The stiffness of the restraint is then changed to the second input stiffness K2, and the system is reanalyzed. The restraint load is then re-examined, and the process is repeated until the status of all restraints converges.
Bi-linear restraints are entered in CAESAR II through the use of the designation X2, Y2, Z2, RX2, RY2, or RZ2. The user is also required to enter the two stiffnesses, as well as the ultimate, or transition, load. This type of restraint is discussed further in Section 3.6.2.
Modeling And Analysis Of The Piping System
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