Linear analysis of the Indian Point reactor feedwater line

Geometrical and material properties are taken from EPRI NP–3108 (1983). The supports are assumed to be linear springs for the purpose of these linear analyses, although their actual response is probably nonlinear. The spring stiffness values are those recommended by Tang et al. (1985). The pipe is assumed to be completely restrained in the vertical direction at the wall penetration.

In the experimental snap-back test used for the comparison (test S138R1SZ), the pipe is full of water. The density of the general beam section is, therefore, adjusted to account for the additional mass of the water by computing a composite (steel plus water) mass per unit length of pipe.

The pipeline is modeled with element type B31. This is a shear flexible beam element that uses linear interpolation of displacement and rotation between two nodes, with transverse shear behavior modeled according to Timoshenko beam theory. The element uses a lumped mass matrix because this provides more accurate results in test cases.

The coarse finite element model uses at least two beam elements along each straight run, with a finer division around the curved segments of the pipe to describe the curvature of the pipe with reasonable accuracy. Separate nodes are assigned for all spring supports, external loading locations, and all the points where experimental data have been recorded. The model is shown in Figure 2. This mesh has 74 beam elements.

In typical piping systems the elbows play a dominant role in the response because of their flexibility. This could be incorporated in the model by using the ELBOW elements. However, ELBOW elements are intended for applications that involve nonlinear response within the elbows themselves and are an expensive option for linear response of the elbows, which is the case for this study. Therefore, instead of using elbow elements, we modify the geometrical properties of beam elements to model the elbows with correct flexibility. This is done by calculating the flexibility factor, k, for each elbow and modifying the moments of inertia of the beam cross-sections in these regions. The flexibility factor for an elbow is a function of two parameters. One is a geometric parameter, $\lambda$, defined as

where t is the wall thickness of the curved pipe, R is the bend radius of the centerline of the curved pipe, r is the mean cross-sectional radius of the curved pipe, and $\nu$ is Poisson's ratio. The other parameter is an internal pressure loading parameter, $\psi$. For thick sections (like the ones used in this pipe), $\psi$ has negligible effect unless the pressures are very high and the water in this case is not pressurized. Consequently, the flexibility factor is a function of $\lambda$ only.

For the elbows in this pipeline $\lambda =$ 0.786 for the 203 mm (8 in) section and $\lambda =$ 0.912 for the 152 mm (6 in) section. The corresponding flexibility factors obtained from Dodge and Moore (1972) are 2.09 and 1.85. These are implemented in the model by modifying the moments of inertia of the beam cross-sections in the curved regions of the pipeline.

Abaqus provides two different options for introducing geometrical properties of a beam cross-section. For general beam sections all geometric properties (area, moments of inertia) can be given without specifying the shape of the cross-section. The material data, including the density, are given on the same option. Alternatively, the geometrical properties of the cross-section can be given by using a beam section. With this option the cross-section dimensions are given, and Abaqus calculates the corresponding cross-sectional behavior by numerical integration, thus allowing for nonlinear material response in the section. When this option is used, the material properties—including density and damping coefficients—are introduced in the material definition associated with the section. This approach is more expensive for systems in which the cross-sectional behavior is linear, since numerical integration over the section is required each time the stress must be computed. Thus, in this case we use a general beam section.

To verify that the mesh will provide results of adequate accuracy, the natural frequencies predicted with this model are compared with those obtained with another mesh that has twice as many elements in each pipe segment. Table 1 shows that these two meshes provide results within 2% for the first six modes and generally quite similar frequencies up to about 30 Hz. Based on this comparison the smaller model, with 74 beam elements, is used for the remaining studies (although the larger model would add little to the cost of the linear analyses, which for either case would be based on the same number of eigenmodes: only in direct integration would the cost increase proportionally with the model size).

In Abaqus the dynamic response of a substructure is defined by a combination of Guyan reduction and the inclusion of some natural modes of the fully restrained substructure. Guyan reduction consists of choosing additional physical degrees of freedom to retain in the dynamic model that are not needed to connect the substructure to the rest of the mesh. In this example we use only Guyan reduction since the model is small and it is easy to identify suitable degrees of freedom to retain. A critical modeling issue with this method is the choice of retained degrees of freedom: enough degrees of freedom must be retained so that the dynamic response of the substructure is modeled with sufficient accuracy. The retained degrees of freedom should be such as to distribute the mass evenly in each substructure so that the lower frequency response of each substructure is modeled accurately. Only frequencies up to 33 Hz are generally considered important in the seismic response of piping systems such as the one studied in this example, so the retained degrees of freedom must be chosen to provide accurate modeling of the response up to that frequency.

In this case the pipeline naturally divides into three segments in terms of which kinematic directions participate in the dynamic response, because the response of a pipeline is generally dominated by transverse displacement. The lower part of the pipeline, between nodes 1 and 23, is, therefore, likely to respond predominantly in degrees of freedom 1 and 2; the middle part, between nodes 23 and 49, should respond in degrees of freedom 2 and 3; and the top part, above node 49, should respond in degrees of freedom 1 and 3. Comparative tests (not documented) have been run to verify these conjectures, and two substructure models have been retained for further analysis: one in which the entire pipeline is treated as a single substructure, and one in which it is split into three substructures. In the latter case all degrees of freedom must be retained at the interface nodes to join the substructures correctly. At other nodes only some translational degrees of freedom are retained, based on the arguments presented above.

The choice of which degrees of freedom to retain can be investigated inexpensively in a case such as this by numerical experiments—extracting the modes of the reduced system for the particular set of retained degrees of freedom and comparing these modes with those of the complete model. The choices made in the substructure models used here are based somewhat on such tests, although insufficient tests have been run to ensure that they are close to the optimal choice for accuracy with a given number of retained variables. For linear analysis of a model as small as this one, achieving an optimal selection of retained degrees of freedom is not critical because computer run times are short: it becomes more critical when the reduced model is used in a nonlinear analysis or where the underlying model is so large that comparative eigenvalue tests cannot be performed easily. In such cases the inclusion of natural modes of the substructure is desirable. The substructure models are shown in Figure 2.

“Damping” plays an essential role in any practical dynamic analysis. In nonlinear analysis the “damping” is often modeled by introducing dissipation directly into the constitutive definition as viscosity or plasticity. In linear analysis equivalent linear damping is used to approximate dissipation mechanisms that are not modeled explicitly.

Experimental estimates of equivalent linear damping, based on three different methods, are found in EPRI NP–3108 (see Table 7–6, Table 7–7, and Figure 7–15 of that report). For the load case and pipe configuration analyzed here, those results suggest that linear damping corresponding to 2.8% of critical damping in the lowest mode of the system matches the measured behavior of the structure, with the experimental results also showing that the percentage of critical damping changes from mode to mode. Despite this all the numerical analyses reported here assume the same damping ratio for all modes included in the model, this choice being made for simplicity only.

For linear dynamic analysis based on the eigenmodes, Abaqus allows damping to be defined as a percentage of critical damping in each mode, as structural damping (proportional to nodal forces), or as Rayleigh damping (proportional to the mass and stiffness of the structure). Only the last option is possible when using direct integration, although other forms of damping can be added as discrete dashpots or in the constitutive models. In this case results are obtained for linear dynamic analysis with modal and Rayleigh damping and for direct integration with Rayleigh damping. For linear dynamic analysis based on the eigenmodes, modal damping can be specified using the mode numbers or using the frequency ranges.