Spectrum analysis attempts to estimate the maximum response developed in a system during a transient load. The results are a statistical summation of the maximum displacements, forces, reactions, and stresses. The individual responses do not represent an actual physical loading case because the maxima may all occur at different times. Spectrum analyses are especially useful when the loading profile is random, or not exactly known, such as with seismic loads. CAESAR II provides the ability to perform two types of spectrum analyses which may be combined: seismic and force spectra. Seismic loadings may be evaluated either uniformly over the entire system or applied through individual support groups with corresponding anchor movements. Force spectra analyses may be used to analyze impulse loadings, such as those due to relief valve, fluid hammer, or slug flow.
Seismic Spectrum Analysis
Seismic loads cannot be solved through time history analyses, because earthquakes cause random motion which may be different for each earthquake, even those occurring at the same site. To simplify the analytical definition of the earthquake, it is necessary to get the expected random waveform of acceleration (or velocity or displacement) versus time into a simple frequency-content plot. The most predominantly used frequency-content plot is the response spectrum. A response spectrum for an earthquake load can be developed by placing a series of single degree-of-freedom oscillators on a mechanical shake table and feeding a typical (for a specific site) earthquake time history through it, measuring the maximum response (displacement, velocity, or acceleration) of each oscillator.
The expectation is that even though all earthquakes are different, similar ones should produce the same maximum responses, even though the time at which they occur differs with each individual occurrence. Responses are based on the maximum ground displacement and acceleration, the dynamic load factors determined by the ratios of the predominant harmonic frequencies of the earthquake to the natural frequencies of the oscillators, and system damping. Response spectra for multiple damping values can be generated by plotting the maximum response for each oscillator. A plot of a set of typical response spectra is shown below:
Seismic response spectra resemble harmonic Dynamic Load Factor curves, because seismic loads indicate strong harmonic tendencies. As the damping value increases, the system response approaches ground motion. Seismic spectra also usually show strong evidence of flexible, resonant, and rigid areas. Spectra may have multiple peaks due to filtering by the building and/or piping system. Multiple peaks are usually enveloped in order to account for uncertainties in the analysis. Seismic response spectra peaks are typically spread to account for inaccuracies as well.
The idea behind the generation of the response spectra is that the modes of vibration of a system respond to the load in the exact same manner as a single degree-of-freedom oscillator. System response may be plotted in terms of displacement, velocity, or acceleration, because these terms of the spectra are all related by the frequency:
d = v / w = a / w2
Where:
d = displacement from response spectrum at frequency
v = velocity from response spectrum at frequency
w = angular frequency at which response spectrum parameters are taken
a = acceleration from response spectrum at frequency
Response Spectrum analysis proceeds according to the following steps:
Modes of vibration are extracted from the system using an Eigensolver algorithm. Each mode has a characteristic frequency and mode shape.
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The maximum response of each mode under the applied load is determined from the spectrum value corresponding to the natural frequency of the mode.
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The total system response is determined by summing the individual modal responses, using methods that reflect the time independence of the responses and the portion of system mass allocated to each mode.
There are four major sources of earthquake spectra available in CAESAR II:
El Centro
This predefined data is taken from J. Biggs’ Introduction to Structural Dynamics and is based on the north-south component of the May 18, 1940 El Centro California earthquake. The recorded maximum acceleration was 0.33 g. The spectrum provided here is intended to apply to elastic systems having 5 to 10 percent critical damping.
Nuclear Regulatory Guide 1.60
The predefined spectrum names are:
1.60H.5 1.60V.5 - Horizontal/vertical, 0.5% damping
1.60H2 1.60V2 - Horizontal/vertical, 2.0% damping
1.60H5 1.60V5 - Horizontal/vertical, 5.0% damping
1.60H7 1.60V7 - Horizontal/vertical ,7.0% damping
1.60H10 1.60V10 - Horizontal/vertical, 10.0% damping
These spectra are constructed according to the instructions given in Regulatory Guide 1.60 for seismic design of nuclear plants. They must also be scaled up or down by the maximum ground acceleration (ZPA—zero period acceleration), specified in the CAESAR II control parameter spreadsheet.
Uniform Building Code
The pre-defined spectrum names are:
UBCSOIL1 Spectrum for rock and stiff soils
UBCSOIL2 Spectrum for deep cohesionless or stiff clay soils
UBCSOIL3 Spectrum for soft to medium clays and sands
These spectra represent the normalized response spectra shapes for three soil types provided in Figure 23-3 of the Uniform Building Code (1991 Edition). When used, they must be scaled by the ZPA, which is the product of Z and I, where Z is the seismic zone coefficient and I is the earthquake importance factor, from UBC Tables 23-I and 23-L. The ZPA can be specific using the CAESAR II control parameter spreadsheet.
User defined spectra
User defined spectra may be entered with period or frequency as the range, and displacement, velocity, or acceleration as the ordinate. These spectra may be read in from a text file or entered directly into a spectrum table during dynamic input processing.
Independent Support Motion Applications
Earthquake ground motions are caused by the passing of acoustic shock waves through the soil. These waves are usually hundreds of feet long. If supports having foundations in the soil are grouped together within a several hundred-foot radius, they typically see the same excitation from the earthquake. If all supports for a piping system are attached directly to ground type supports, each support is excited by an essentially identical time waveform. This type of excitation is known as uniform support excitation. Often pipe is supported from rack, building, or vessel structures as well as from ground type supports. These intermediate structures sometimes filter or accentuate the effect of the earthquake. In this situation, the supports attached to the intermediate structure are not exposed to the same excitation as those that are attached directly to ground foundations. To accurately model these systems, different shocks must be applied to different parts of the piping system. This type of excitation is known as independent support motion (ISM) excitation. While the different support groups are exposed to different shocks, there are also relative movements between support groups that don’t exist for uniform support excitation. The movement of one support group relative to another is termed pseudostatic displacement, or seismic anchor movements. For uniform support excitation, there are spatial and modal response components available for combination. For independent support excitation, there are spatial and modal response components available for each different support group, plus pseudostatic components of the earthquake that must also be added into the dynamic response.
The major difference when running ISM type earthquake loads comes while building the shock load cases. In the uniform excitation case, the shock acts implicitly over all the supports in the system. In the ISM case, different shocks act on different groups of supports. The Spectrum Load Cases tab appears, with the following parameters:
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Spectrum (name)
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Factor
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Dir (direction)
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Start Node
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Stop Node
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Increment
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Anchor Movement
Name, Factor, and Dir are all that is required for uniform support excitations. For ISM type shocks, the group of nodes over which the shock acts must be specified as well, using Start Node, Stop Node, and Increment. Anchor Movement is used to explicitly define the seismic displacement of the restraint set. This displacement is used to calculate the pseudostatic load components. If omitted, the software defaults to the displacement derived from the response spectrum entry corresponding to the lowest frequency.
Force Spectrum Analysis
A similar method can be followed for non-random loads, such as an impulse load for which the force versus time profile is known. A look at the equation for the earthquake problem explains why the force spectrum solution is very similar to the earthquake solution:
The term on the right-hand side is a dynamic force acting on the piping system, such as F = Ma, so the analogous equation to be solved for the force spectrum problem is:
Where:
F = the dynamic load (water hammer or relief valve)
Instead of the displacement, velocity, or acceleration spectrum used for the seismic problem, a Dynamic Load Factor spectrum is used for a force spectrum problem. A DLF spectrum gives the ratio of the maximum dynamic displacement divided by the maximum static displacement. The earthquake response spectrum analysis method starts with the time history of an earthquake excitation. The force spectrum analysis method is done in the same way, except that the analysis starts with the force versus time profile. Just as for the earthquake, this time history loading is applied to a shake table of single degree-of-freedom bodies. A response spectrum (DLF versus natural frequency) is generated by dividing the maximum oscillator displacements by the static displacements expected under the same load. An alternate means of generating a response spectrum for an impulse load is to numerically integrate the dynamic equation of motion for oscillators of various frequencies under the applied load. Use Tools > DLF Spectrum Generator.
Process output from a spectrum analysis in two ways:
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Use the output processor to review the natural frequencies, mode shapes, participation factors, included mass/force, displacements, restraint loads, forces, or stresses in report form. Dynamic results also show the largest modal contributor, along with the mode and shock load responsible for that contribution.
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Animate the individual mode shapes extracted for the spectrum analysis.