If restraints are not used to absorb the unbalanced pressure load, it must be handled by tie rods. These are tension-carrying rods attached to either end of the expansion joint, which prevent the ends from pulling apart. Since tie bars assure that there is no pressure reaction from the expansion joint, tie rods can be modeled in two different ways:
1 - implicitly, by omitting both the tie-rods and the pressure load from the model,
2 - explicitly, by including both the tie-rods and the pressure load in the model.
Assuming that the tie rods absorb 100% of the load, the net effect of both of these models on the piping system are the same.
Implicit Model of Tie Rods:
The first case noted above is obviously the simpler of the two. Omitting the tie rods is possible if there is no pressure load on the bellows; this can be omitted by defining an effective diameter equal to 0.0 for the expansion joint. Tie rods, besides absorbing the pressure load, also prevent extension and compression of the bellows under piping operating loads. Therefore, when the user leaves the tie bars and pressure thrust out of the model, it is also necessary to set the axial stiffness of the expansion joint to be essentially rigid (or actually to the total axial stiffness of the tie rods, which is AE/L). If the axial load on the expansion joint is tensile then the surrounding pipe is trying to stretch the tie bars even further. If the axial load on the expansion joint is compressive, then:
1 - If the compression is less than the pressure thrust load, there is not a problem. 2 - If the compression exceeds the pressure thrust load, then the tie rods will be in compression. The compression must be checked to ensure that it is not so great that it buckles the tie bars. If the tie rods are tension only (i.e., lock nuts are placed only on the outside of the expansion joint flanges) then some redesign is required, either:
a) put nuts on the tie bars on both sides of the expansion joint flanges, or
b) redesign the piping system so that the compressive load is not so great.
Explicit Model of Tie Rods:
When explicitly modeling the tie rods, the pressure load is included in the model by defining an effective diameter for the expansion joint. The tie rods can be modeled by using a structural element (of the same cross-sectional area as the tie rods) to connect the two ends of the expansion joint. The structural element used could be a pipe, a rigid element, or a user-defined structural steel element.
In the event that a single pipe element is used to represent the tie rods, the best way to provide the same axial stiffness as the tie rods is to leave the diameter equal to the diameter of the attached piping and set the wall thickness approximately equal to:
If a rigid element is used to model the tie rods, again the diameter should be set to that of the attached piping; the wall thickness should be set to:
The rigid element should be given a weight equal to the total weight of all of the tie rods, which, if made of steel, is approximately:
The tie rods are modeled in CAESAR II to resist only axial loads through the use of restraints with "CNODEs" (other nodes in the system to which a restraint is connected). Consider the expansion joint/tie rod assembly shown in Figure 3-43. The bellows element is modeled as running between the two node points 5 and 10. The tie rod element is then run from node point 5 to node point 20, using the same delta-coordinates as for the expansion joint. This puts node points 10 and 20 at a coincident location, without any actual attachment. The attachment is provided by placing a restraint at the far end of the tie rod (node point 20) in the direction of the expansion joint axis, in this case the Y direction. Placement of a restraint here in this manner restrains node point 20 (the end of the tie rod) against a rigid point in space; this can be adjusted by defining the restraint node point 10 as a CNODE. This means that node point 20 is not restrained against a point in space, but rather that it cannot move in the global Y-direction relative to node point 10 — the end of the expansion joint — an effective representation of a tied expansion joint.
The tie rods should be set to the ambient temperature, if they are outside of the piping insulation, or to a temperature closer to the operating temperature if they are inside of the insulation. Tie rods may also be modeled in a more complex fashion, using multiple rigid elements, as shown in Figure 3-44. In this model, loosening of the nuts on the rods due to rotation of the expansion joint flanges will be simulated.
As noted, tie rods must be checked for potential buckling loads after the analysis is complete. Or alternatively, they may be designed to take tension only. This is done by placing lock nuts only on the outside of the flange, as shown in Figure 3-45. In this case, the expansion joint is prevented from extending by the nut, while the flange can move freely during joint contraction. This configuration can be modeled in CAESAR II by using one-way restraints (or even gaps, if appropriate) between the end of the tie rod element and the CNODEs. For example, if the tie rod shown in Figure 3-42 was tension only, it would be modeled by placing a +Y restraint at node point 20, with 10 as the CNODE, indicating that the end of the tie rod cannot move down against the expansion joint (but can move up).
Complex expansion joint/tie rod models are cumbersome to build and check, but where hot, large diameter tight piping systems are being analyzed they yield the most accurate model. This is especially true where tie rods are long and not designed for compression. In these cases a slight rotation of the expansion joint can put one side of the tie rods in compression and the other side in a greater tension.
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