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6.3.10 RESPONSE SPECTRUM ANALYSIS
Products: Abaqus/Standard Abaqus/CAE
References
• “Dynamic analysis procedures: overview,” Section 6.3.1
• “Defining an analysis,” Section 6.1.2
• “General and linear perturbation procedures,” Section 6.1.3
• *RESPONSE SPECTRUM
• *SPECTRUM
• “Configuring a response spectrum procedure” in “Configuring linear perturbation analysis procedures,” Section 14.11.2 of the Abaqus/CAE User’s Guide, in the HTML version of this guide
• “Defining a spectrum,” Section 57.11 of the Abaqus/CAE User’s Guide, in the HTML version of this guide
Overview
A response spectrum analysis:
• provides an estimate of the peak linear response of a structure to dynamic motion of fixed points (“base motion”) or dynamic force;
• is typically used to analyze response to a seismic event;
• assumes that the system’s response is linear so that it can be analyzed in the frequency domain using its natural modes, which must be extracted in a previous eigenfrequency extraction step (“Natural frequency extraction,” Section 6.3.5);
• can use the high-performance SIM software architecture (see “Using the SIM architecture for modal superposition dynamic analyses” in “Dynamic analysis procedures: overview,” Section 6.3.1); and
• is a linear perturbation procedure and is, therefore, not appropriate if the excitation is so severe that nonlinear effects in the system are important.
Response spectrum analysis
Response spectrum analysis can be used to estimate the peak response (displacement, stress, etc.) of a structure to a particular base motion or force. The method is only approximate, but it is often a useful, inexpensive method for preliminary design studies.
The response spectrum procedure is based on using a subset of the modes of the system, which must first be extracted by using the eigenfrequency extraction procedure. The modes will include eigenmodes and, if activated in the eigenfrequency extraction step, residual modes. The number of modes extracted must be sufficient to model the dynamic response of the system adequately, which is a matter of judgment on your part.
In cases with repeated eigenvalues and eigenvectors, the modal summation results must be interpreted with care. You should add insignificant mass to the structure or perturb the symmetric geometry such that the eigenvalues become unique.
While the response in the response spectrum procedure is for linear vibrations, the prior response may be nonlinear. Initial stress effects (stress stiffening) will be included in the response spectrum analysis if nonlinear geometric effects (“General and linear perturbation procedures,” Section 6.1.3) were included in a general analysis step prior to the eigenfrequency extraction step.
The problem to be solved can be stated as follows: given a set of base motions, \ddot { u } _ { j } ^ { B } ( t ) ( j = 1 , 2 , 3 ) , specified in orthogonal directions defined by direction cosines t _ { k } ^ { j } \left( k = 1 , 2 , 3 \right) , estimate the peak value over all time of the response of any variable in a finite element model that is simultaneously subjected to these multiple base motions. The peak response is first computed independently for each direction of excitation for each natural mode of the system as a function of frequency and damping. These independent responses are then combined to create an estimate of the actual peak response of any variable chosen for output, as a function of frequency and damping.
The acceleration history (base motion) is not given directly in a response spectrum analysis; it must first be converted into a spectrum.
Specifying a spectrum
The response spectrum method is based on first finding the peak response to each base motion excitation of a one degree of freedom system that has a natural frequency equal to the frequency of interest. The single degree of freedom system is characterized by its undamped natural frequency, \omega _ { \alpha } , and the fraction of critical damping present in the system, \xi _ { \alpha } , at each mode . The equations of motion of the system are integrated through time to find peak values of relative displacement, relative velocity, and relative or absolute acceleration for the linear, one degree of freedom system. This process is repeated for all frequency and damping values in the range of interest. Plots of these responses are known as displacement, velocity, and acceleration spectra: S ^ { D } ( \omega _ { \alpha } , \xi _ { \alpha } ) , S ^ { V } ( \omega _ { \alpha } , \xi _ { \alpha } ) , and S ^ { A } ( \omega _ { \alpha } , \xi _ { \alpha } ) . The response spectrum can be obtained directly from measured data, as described in “Defining a spectrum using values of S as a function of frequency and damping,” below. You can also use a Fortran program to define a spectrum; an example of defining a spectrum from an acceleration record in this way is provided in “Analysis of a cantilever subject to earthquake motion,” Section 1.4.13 of the Abaqus Benchmarks Guide.
Alternatively, you can create the required spectrum by specifying an amplitude (time history record), the frequency range, and the damping values for which the spectrum will be built, as described in “Creating a spectrum from a given time history record,” below. The spectrum can be used in the subsequent response spectrum analysis, or it can be written to a file for future use.
For each damping value the magnitude of the response spectrum must be given over the entire range of frequencies needed, in ascending value of frequency. Abaqus/Standard interpolates linearly between the values given on a log-log scale. Outside the extremes of the frequency range given, the magnitude is assumed to be constant, corresponding to the end value given. (See “Material data definition,” Section 21.1.2, for an explanation of data interpolation.)
Any number of spectra can be defined, and each spectrum must be named. The response spectrum procedure allows up to three spectra to be applied simultaneously to the model in orthogonal physical directions defined by their direction cosines.
Defining a spectrum using values of S as a function of frequency and damping
You can define a spectrum by specifying values for the magnitude of the spectrum; frequency, in cycles per time, at which the magnitude is used; and associated damping, given as a ratio of critical damping.
Input File Usage: To define the spectrum on the data lines:
*SPECTRUM, NAME=spectrum name
Repeat this option to define multiple spectra for an analysis.
Abaqus/CAE Usage: To define a spectrum, do the following:
Step, Interaction, or Load module: Tools→Amplitude→Create;
Name: spectrum name, Type: Spectrum
To apply a spectrum to the model, do the following:
Step module: Create Step: Linear perturbation: Response
spectrum: Use response spectrum: select spectrum name for each
physical direction in which it should be applied
Specifying the type of spectrum
You can indicate whether a displacement, velocity, or acceleration spectrum is given. The default is an acceleration spectrum.
Alternatively, an acceleration spectrum can be given in g-units. In this case you must also specify the value of the acceleration of gravity.
Input File Usage: Use one of the following options to define a displacement, velocity, or acceleration spectrum:
*SPECTRUM, NAME=name, TYPE=DISPLACEMENT
*SPECTRUM, NAME=name, TYPE=VELOCITY
*SPECTRUM, NAME=name, TYPE=ACCELERATION
Use the following option to define an acceleration spectrum given in g-units:
*SPECTRUM, NAME=name, TYPE=G, G=g
Abaqus/CAE Usage: Use one of the following options to define a displacement, velocity, or acceleration spectrum:
Step, Interaction, or Load module: Tools→Amplitude→Create; Type:
Spectrum; Specification units: Displacement, Velocity, or Acceleration
Use the following option to define an acceleration spectrum given in g-units:
Step, Interaction, or Load module: Tools→Amplitude→Create; Type:
Spectrum; Specification units: Gravity, Gravity: g
Reading the data defining the spectrum from an alternate input file
The data for the spectrum can be specified in an alternate input file and read into the Abaqus/Standard input file.
Input File Usage: *SPECTRUM, NAME=name, INPUT=file name
Abaqus/CAE Usage: Step, Interaction, or Load module: Tools→Amplitude→Create; Type: Spectrum; click mouse button 3 while holding the cursor over the data table, and select Read from File
Creating a spectrum from a given time history record
If you have a time history of a dynamic event (e.g., acceleration, velocity, displacement), you can build your own spectrum by specifying the record type and the amplitude name that this record represents. If the amplitude record is given with an arbitrarily changing time increment, linear interpolation will be needed for the implicit integration scheme for the dynamic equation of motion for a single degree of freedom system subjected to this record. You can specify the frequency range for the integration scheme and the frequency increment. You can build a spectrum for every fraction of critical damping indicated in the list of damping values.
Input File Usage: *SPECTRUM, CREATE, AMPLITUDE=amplitude name, NAME=spectrum name, TIME INCREMENT=dt
Abaqus/CAE Usage: Creating a spectrum from a given time history record is not supported in Abaqus/CAE.
Specifying the type of spectrum to be created
You can indicate whether a displacement, velocity, or acceleration spectrum is to be created. The default is an acceleration spectrum.
Alternatively, an acceleration spectrum can be created in g-units. In this case you must also specify the value of the acceleration of gravity.
Input File Usage: Use one of the following options to create a displacement, velocity, or acceleration spectrum:
*SPECTRUM, CREATE, TYPE=DISPLACEMENT*SPECTRUM, CREATE, TYPE=VELOCITY*SPECTRUM, CREATE, TYPE=ACCELERATION
Use the following option to create an acceleration spectrum in g-units:
*SPECTRUM, CREATE, TYPE=G, G=g
Abaqus/CAE Usage: Creating a spectrum from a given time history record is not supported in Abaqus/CAE.
Specifying the record type that the time history represents
You can indicate whether a displacement, velocity, or acceleration amplitude is specified. The default is an acceleration amplitude.
Alternatively, an acceleration amplitude can be given in g-units. In this case you must also specify the value of the acceleration of gravity.
| Input File Usage: | Use one of the following options to indicate that the amplitude is defined in displacement, velocity, or acceleration units:*SPECTRUM, CREATE, EVENT TYPE=DISPLACEMENT*SPECTRUM, CREATE, EVENT TYPE=VELOCITY*SPECTRUM, CREATE, EVENT TYPE=ACCELERATIONUse the following option to indicate that an acceleration amplitude is given in g-units:*SPECTRUM, CREATE, EVENT TYPE=G, G=g |
| Abaqus/CAE Usage: | Creating a spectrum from a given time history record is not supported in Abaqus/CAE. |
Creating an absolute or relative acceleration spectrum
When you create an acceleration spectrum from a given time history record, you can create an absolute or relative response spectrum. The default is an absolute spectrum.
| Input File Usage: | *SPECTRUM, CREATE, TYPE=ACCELERATION, ABSOLUTE |
| *SPECTRUM, CREATE, TYPE=ACCELERATION, RELATIVE |
| Abaqus/CAE Usage: | Creating a spectrum from a given time history record is not supported in Abaqus/CAE. |
Generating the list of damping values for the fraction of critical damping
You must provide a list of damping values for the fraction of critical damping to create a spectrum. However, if the damping is evenly spaced between its lower and upper bound, you can automatically generate the list of damping values by providing the start value, end value, and increment for the fraction of critical damping.
| Input File Usage: | *SPECTRUM, CREATE, DAMPING GENERATE |
| Abaqus/CAE Usage: | Creating a spectrum from a given time history record is not supported in Abaqus/CAE. |
Writing the generated spectra to an independent file
You can write the generated spectra to an independent file. Otherwise, the generated spectra can be used only within the currently submitted job in subsequent response spectra procedures. You can inspect the generated spectra if you request that model definition data be printed to the data file (see “Model and history definition summaries” in “Output,” Section 4.1.1).
Input File Usage: *SPECTRUM, CREATE, FILE=file name
Abaqus/CAE Usage: Creating a spectrum from a given time history record is not supported in Abaqus/CAE.
Estimating the peak values of the modal responses
Since the response spectrum procedure uses modal methods to define a model’s response, the value of any physical variable is defined from the amplitudes of the modal responses (the “generalized coordinates”), q _ { \alpha } . The first stage in the response spectrum procedure is to estimate the peak values of these modal responses. For mode and spectrum k this is
(q _ {\alpha} ^ {m a x}) _ {k} = c _ {k} S _ {k} ^ {D} (\omega_ {\alpha}, \xi_ {\alpha}) \sum_ {j} t _ {j} ^ {k} \Gamma_ {\alpha j},
where
q _ { \alpha } is the modal amplitude for mode \alpha ;
c _ { k } is a scaling parameter introduced as part of the response spectrum procedure definition for spectrum S _ { k } ^ { D } ( \omega _ { \alpha } , \xi _ { \alpha } ) ;
S _ { k } ^ { D } ( \omega _ { \alpha } , \xi _ { \alpha } ) is the user-defined value of the spectrum (see “Specifying a spectrum”) in direction k interpolated, if necessary, at natural frequency \omega _ { \alpha } and the fraction of critical damping \xi _ { \alpha } in mode \alpha ;
t _ { j } ^ { k } is the jth direction cosine for the kth spectrum; and
\Gamma _ { \alpha j } is the participation factor for mode \alpha in direction j (see “Natural frequency extraction,” Section 6.3.5).
Similar expressions for ( \dot { q } _ { \alpha } ^ { m a x } ) _ { k } and ( \ddot { q } _ { \alpha } ^ { m a x } ) _ { k } can be obtained by substituting velocity or acceleration spectra in the above equation.
Combining the individual peak responses
The individual peak responses to the excitations in different directions will occur at different times and, therefore, must be combined into an overall peak response. Two combinations must be performed, and both introduce approximations into the results:
- The multidirectional excitations must be combined into one overall response. This combination is controlled by the directional summation method, as described below in “Directional summation methods.”
- The peak modal responses must be combined to estimate the peak physical response. This combination is controlled by the modal summation method, as described below in “Modal summation methods.”
Depending on the type of base excitation, either modal responses or directional responses are combined first.
Directional summation methods
You choose the method for combining the multidirectional excitations depending on the nature of the excitations.
The algebraic method
If the input spectra in the different directions are components of a base excitation that is approximately in a single direction in space, then for each mode the peak responses in the different spatial directions are summed algebraically by
q _ {\alpha} ^ {m a x} = \sum_ {k} (q _ {\alpha} ^ {m a x}) _ {k}.
After this summation is performed, the modal responses are summed. (Choosing the method used for modal summation is described below in “Modal summation methods.”) Since the directional components are summed first, the subscript k is not relevant and can be ignored in the modal summation equations that follow.
Input File Usage: *RESPONSE SPECTRUM, COMP=ALGEBRAIC, SUM=sum
Abaqus/CAE Usage: Step module: Create Step: Linear perturbation: Response spectrum: Excitations: Single direction or Multiple direction absolute sum
The square root of the sum of the squares directional summation method
If the spectra in different directions represent independent excitations, the modal summation is performed first, as explained below in “Modal summation methods.” Then, the responses in different excitation directions are combined by
(R ^ {i}) ^ {m a x} = \sqrt {\sum_ {k} \left((R ^ {i}) _ {k} ^ {m a x}\right) ^ {2}}.
Input File Usage: *RESPONSE SPECTRUM, COMP=SRSS, SUM=sum
Abaqus/CAE Usage: Step module: Create Step: Linear perturbation: Response spectrum: Excitations: Multiple direction square root of the sum of squares
The forty-percent method
If the spectra in different directions represent independent excitations, the modal summation is performed first, as explained below in “Modal summation methods.” Then, the responses in different excitation directions are combined by the 40% rule recommended by the ASCE 4–98 standard for Seismic Analysis of Safety-Related Nuclear Structures and Commentary, Section 3.2.7.1.2. This method combines the response for all possible combinations of the three components, including variations in sign (plus/minus), assuming that when the maximum response from one component occurs, the response from the other two components is 40% of their maximum value, using one of the following:
(R ^ {i}) ^ {m a x} = \pm [ (R ^ {i}) _ {1} ^ {m a x} \pm 0. 4 (R ^ {i}) _ {2} ^ {m a x} \pm 0. 4 (R ^ {i}) _ {3} ^ {m a x} ],
(R ^ {i}) ^ {m a x} = \pm [ (R ^ {i}) _ {2} ^ {m a x} \pm 0. 4 (R ^ {i}) _ {1} ^ {m a x} \pm 0. 4 (R ^ {i}) _ {3} ^ {m a x} ],
(R ^ {i}) ^ {m a x} = \pm [ (R ^ {i}) _ {3} ^ {m a x} \pm 0. 4 (R ^ {i}) _ {2} ^ {m a x} \pm 0. 4 (R ^ {i}) _ {1} ^ {m a x} ].
Input File Usage: *RESPONSE SPECTRUM, COMP=R40, SUM=sum
Abaqus/CAE Usage: Step module: Create Step: Linear perturbation: Response spectrum: Excitations: Multiple direction forty percent rule
The thirty-percent method
If the spectra in different directions represent independent excitations, the modal summation is performed first, as explained below in “Modal summation methods.” Then, the responses in different excitation directions are combined by the 30% rule recommended by the ASCE 4–98 standard for Seismic Analysis of Safety-Related Nuclear Structures and Commentary, Section 3.2.7.1.2. This method combines the response for all possible combinations of the three components, including variations in sign (plus/minus), assuming that when the maximum response from one component occurs, the response from the other two components is 30% of their maximum value, using one of the following:
(R ^ {i}) ^ {m a x} = \pm [ (R ^ {i}) _ {1} ^ {m a x} \pm 0. 3 (R ^ {i}) _ {2} ^ {m a x} \pm 0. 3 (R ^ {i}) _ {3} ^ {m a x} ],
(R ^ {i}) ^ {m a x} = \pm [ (R ^ {i}) _ {2} ^ {m a x} \pm 0. 3 (R ^ {i}) _ {1} ^ {m a x} \pm 0. 3 (R ^ {i}) _ {3} ^ {m a x} ],
(R ^ {i}) ^ {m a x} = \pm [ (R ^ {i}) _ {3} ^ {m a x} \pm 0. 3 (R ^ {i}) _ {2} ^ {m a x} \pm 0. 3 (R ^ {i}) _ {1} ^ {m a x} ].
Input File Usage: *RESPONSE SPECTRUM, COMP=R30, SUM=sum
Abaqus/CAE Usage: Step module: Create Step: Linear perturbation: Response spectrum: Excitations: Multiple direction thirty percent rule
Modal summation methods
The peak response of some physical variable (a component i of displacement, stress, section force, reaction force, etc.) caused by the motion in the th natural mode excited by the given response spectra in direction k at frequency \omega _ { \alpha } with damping \xi _ { \alpha } is given by
(R _ {\alpha} ^ {i}) _ {k} ^ {m a x} = \Phi_ {\alpha} ^ {i} (q _ {\alpha} ^ {m a x}) _ {k},
where \Phi _ { \alpha } ^ { i } is the ith component of mode , and there is no sum on . (In the case of algebraic summation the subscript k is not relevant and can be ignored in this equation and in those that follow.)
There are several methods for combining these peak physical responses in the individual modes, ( R _ { \alpha } ^ { i } ) _ { k } ^ { m a x } , into estimates of the total peak response, \left( R ^ { i } \right) _ { k } ^ { i n a \overline { { x } } } Most of the methods implemented in Abaqus/Standard follow the ASCE 4–98 standard for Seismic Analysis of Safety Related Nuclear Structures and Commentary. The updated documents, “Reevaluation of Regulatory Guidance on Modal Response Combination Methods for Seismic Response Spectrum Analysis” issued in 1999 (NUREG/CR-6645, BNL-NUREG-52276) and “Draft Regulatory Guide” (DG-1127) issued in 2005 contain new recommendations. You are advised to read the new recommendations before choosing a modal summation method from among those described below.
The absolute value method
The absolute value method is the most conservative method for combining the modal responses. It is obtained by summing the absolute values resulting from each mode:
(R ^ {i}) _ {k} ^ {m a x} = \sum_ {\alpha} | (R _ {\alpha} ^ {i}) _ {k} ^ {m a x} |.
This method implies that all of the responses peak simultaneously. It will overpredict the peak response of most systems; therefore, it may be too conservative to help in design.
Input File Usage: *RESPONSE SPECTRUM, COMP=comp, SUM=ABS
Abaqus/CAE Usage: Step module: Create Step: Linear perturbation: Response spectrum: Summations: Absolute values
The square root of the sum of the squares modal summation method
The square root of the sum of the squares method is less conservative than the absolute value method. It is also usually more accurate if the natural frequencies of the system are well separated. It uses the square root of the sum of the squares to combine the modal responses:
(R ^ {i}) _ {k} ^ {m a x} = \sqrt {\sum_ {\alpha} \left((R _ {\alpha} ^ {i}) _ {k} ^ {m a x}\right) ^ {2}}.
Input File Usage: *RESPONSE SPECTRUM, COMP=comp, SUM=SRSS
Abaqus/CAE Usage: Step module: Create Step: Linear perturbation: Response spectrum: Summations: Square root of the sum of squares
The Naval Research Laboratory method
The absolute value and square root of the sum of the squares methods can be combined to yield the Naval Research Laboratory method. It distinguishes the mode, \beta , in which the physical variable has its maximum response and adds the square root of the sum of squares of the peak responses in all other modes to the absolute value of the peak response of that mode. This method gives the estimate:
(R ^ {i}) _ {k} ^ {m a x} = | (R _ {\beta} ^ {i}) _ {k} ^ {m a x} | + \sqrt {\sum_ {\alpha \neq \beta} \left((R _ {\alpha} ^ {i}) _ {k} ^ {m a x}\right) ^ {2}}.
Input File Usage: *RESPONSE SPECTRUM, COMP=comp, SUM=NRL
Abaqus/CAE Usage: Step module: Create Step: Linear perturbation: Response spectrum: Summations: Naval Research Laboratory
The ten-percent method
The ten-percent method recommended by Regulatory Guide 1.92 (1976) is no longer recommended according to the “Reevaluation of Regulatory Guidance on Modal Response Combination Methods for Seismic Response Spectrum Analysis” document issued in 1999. It is retained here because of its extensive prior use. The ten-percent method modifies the square root of the sum of the squares method by adding a contribution from all pairs of modes and \beta whose frequencies are within 10% of each other, giving the estimate:
(R ^ {i}) _ {k} ^ {m a x} = \sqrt {\sum_ {\alpha} \left((R _ {\alpha} ^ {i}) _ {k} ^ {m a x}\right) ^ {2} + 2 \sum_ {\alpha < \beta} | (R _ {\alpha} ^ {i}) _ {k} ^ {m a x} (R _ {\beta} ^ {i}) _ {k} ^ {m a x} |}.
The frequencies of modes and \beta are considered to be within 10% of each other whenever
\frac {\omega_ {\beta} - \omega_ {\alpha}}{\omega_ {\beta}} \leq 0. 1, \quad \alpha < \beta .
The ten-percent method reduces to the square root of the sum of the squares method if the modes are well separated with no coupling between them.
Input File Usage: *RESPONSE SPECTRUM, COMP=comp, SUM=TENP
Abaqus/CAE Usage: Step module: Create Step: Linear perturbation: Response spectrum: Summations: Ten percent
The complete quadratic combination method
Like the ten-percent method, the complete quadratic combination method improves the estimation for structures with closely spaced eigenvalues. The complete quadratic combination method combines the modal response with the formula
(R ^ {i}) _ {k} ^ {m a x} = \sqrt {\sum_ {\alpha} \sum_ {\beta} (R _ {\alpha} ^ {i}) _ {k} ^ {m a x} \rho_ {\alpha \beta} (R _ {\beta} ^ {i}) _ {k} ^ {m a x}},
where \rho _ { \alpha \beta } are cross-correlation coefficients between modes and \beta _ { \ast } , which depend on the ratio of frequencies and modal damping between the two modes: