ProductsAbaqus/StandardAbaqus/CAE Response spectrum analysisResponse 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 use mode combination rules that are appropriate for closely spaced modes. 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 perturbation procedures) 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, ${\ddot{u}}_{j}^{B}\left(t\right)$ ($j=1,2,3$), specified in orthogonal directions defined by direction cosines ${t}_{k}^{j}$ ($k=1,2,3$), 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 spectrumThe 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 $\alpha $. 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: ${S}^{D}\left({\omega}_{\alpha},{\xi}_{\alpha}\right)$, ${S}^{V}\left({\omega}_{\alpha},{\xi}_{\alpha}\right)$, and ${S}^{A}\left({\omega}_{\alpha},{\xi}_{\alpha}\right)$. 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. 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 loglog 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 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 dampingYou 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: ToolsAmplitudeCreate; 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 spectrumYou 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 gunits. 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 gunits: 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: ToolsAmplitudeCreate; Type: Spectrum; Specification units: Displacement, Velocity, or Acceleration Use the following option to define an acceleration spectrum given in gunits: Step, Interaction, or Load module: ToolsAmplitudeCreate; Type: Spectrum; Specification units: Gravity, Gravity: g Reading the data defining the spectrum from an alternate input fileThe 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: ToolsAmplitudeCreate; 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 recordIf 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 createdYou 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 gunits. 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 gunits: 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 representsYou 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 gunits. 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=ACCELERATION Use the following option to indicate that an acceleration amplitude is given in gunits: 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 spectrumWhen 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 dampingYou 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 fileYou 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). 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 responsesSince 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 $\alpha $ and spectrum k this is $${\left({q}_{\alpha}^{max}\right)}_{k}={c}_{k}{S}_{k}^{D}\left({\omega}_{\alpha},{\xi}_{\alpha}\right)\sum _{j}{t}_{j}^{k}{\mathrm{\Gamma}}_{\alpha j},$$
where
Similar expressions for ${\left({\dot{q}}_{\alpha}^{max}\right)}_{k}$ and ${\left({\ddot{q}}_{\alpha}^{max}\right)}_{k}$ can be obtained by substituting velocity or acceleration spectra in the above equation. Combining the individual peak responsesThe 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:
Depending on the type of base excitation, either modal responses or directional responses are combined first. Directional summation methodsYou choose the method for combining the multidirectional excitations depending on the nature of the excitations. The algebraic methodIf 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 $${q}_{\alpha}^{max}=\sum _{k}{\left({q}_{\alpha}^{max}\right)}_{k}.$$
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 methodIf the spectra in different directions represent independent excitations, the modal summation is performed first, as explained below in Modal summation methods. Then, the estimates of the total peak response in each excitation direction ${\left({R}^{i}\right)}_{k}^{max}$ are combined by $${\left({R}^{i}\right)}^{max}=\sqrt{\sum _{k}{\left({\left({R}^{i}\right)}_{k}^{max}\right)}^{2}}.$$
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 fortypercent methodIf 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 SafetyRelated 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: $${\left({R}^{i}\right)}^{max}=\pm \left[{\left({R}^{i}\right)}_{1}^{max}\pm 0.4{\left({R}^{i}\right)}_{2}^{max}\pm 0.4{\left({R}^{i}\right)}_{3}^{max}\right],$$
$${\left({R}^{i}\right)}^{max}=\pm \left[{\left({R}^{i}\right)}_{2}^{max}\pm 0.4{\left({R}^{i}\right)}_{1}^{max}\pm 0.4{\left({R}^{i}\right)}_{3}^{max}\right],$$
$${\left({R}^{i}\right)}^{max}=\pm \left[{\left({R}^{i}\right)}_{3}^{max}\pm 0.4{\left({R}^{i}\right)}_{2}^{max}\pm 0.4{\left({R}^{i}\right)}_{1}^{max}\right].$$
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 thirtypercent methodIf 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 SafetyRelated 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: $${\left({R}^{i}\right)}^{max}=\pm \left[{\left({R}^{i}\right)}_{1}^{max}\pm 0.3{\left({R}^{i}\right)}_{2}^{max}\pm 0.3{\left({R}^{i}\right)}_{3}^{max}\right],$$
$${\left({R}^{i}\right)}^{max}=\pm \left[{\left({R}^{i}\right)}_{2}^{max}\pm 0.3{\left({R}^{i}\right)}_{1}^{max}\pm 0.3{\left({R}^{i}\right)}_{3}^{max}\right],$$
$${\left({R}^{i}\right)}^{max}=\pm \left[{\left({R}^{i}\right)}_{3}^{max}\pm 0.3{\left({R}^{i}\right)}_{2}^{max}\pm 0.3{\left({R}^{i}\right)}_{1}^{max}\right].$$
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 methodsThe peak response of some physical variable ${R}^{i}$ (a component i of displacement, stress, section force, reaction force, etc.) caused by the motion in the $\alpha $th natural mode excited by the given response spectra in direction k at frequency ${\omega}_{\alpha}$ with damping ${\xi}_{\alpha}$ is given by $${\left({R}_{\alpha}^{i}\right)}_{k}^{max}={\mathrm{\Phi}}_{\alpha}^{i}{\left({q}_{\alpha}^{max}\right)}_{k},$$
where ${\mathrm{\Phi}}_{\alpha}^{i}$ is the ith component of mode $\alpha $, and there is no sum on $\alpha $. (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, ${\left({R}_{\alpha}^{i}\right)}_{k}^{max}$, into estimates of the total peak response, ${\left({R}^{i}\right)}_{k}^{max}$. 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/CR6645, BNLNUREG52276) and “Draft Regulatory Guide” (DG1127) 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 methodThe 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: $${\left({R}^{i}\right)}_{k}^{max}=\sum _{\alpha}\left{\left({R}_{\alpha}^{i}\right)}_{k}^{max}\right.$$
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 methodThe 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: $${\left({R}^{i}\right)}_{k}^{max}=\sqrt{\sum _{\alpha}{\left({\left({R}_{\alpha}^{i}\right)}_{k}^{max}\right)}^{2}}.$$
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 methodThe 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: $${\left({R}^{i}\right)}_{k}^{max}=\left{\left({R}_{\beta}^{i}\right)}_{k}^{max}\right+\sqrt{\sum _{\alpha \ne \beta}{\left({\left({R}_{\alpha}^{i}\right)}_{k}^{max}\right)}^{2}}.$$
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 tenpercent methodThe tenpercent 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 tenpercent method modifies the square root of the sum of the squares method by adding a contribution from all pairs of modes $\alpha $ and $\beta $ whose frequencies are within 10% of each other, giving the estimate: $${\left({R}^{i}\right)}_{k}^{max}=\sqrt{\sum _{\alpha}{\left({\left({R}_{\alpha}^{i}\right)}_{k}^{max}\right)}^{2}+2\sum _{\alpha <\beta}\left{\left({R}_{\alpha}^{i}\right)}_{k}^{max}{\left({R}_{\beta}^{i}\right)}_{k}^{max}\right}.$$
The frequencies of modes $\alpha $ and $\beta $ are considered to be within 10% of each other whenever $$\frac{{\omega}_{\beta}{\omega}_{\alpha}}{{\omega}_{\beta}}\le 0.1,\alpha \beta .$$
The tenpercent 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 methodLike the tenpercent 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 $${\left({R}^{i}\right)}_{k}^{max}=\sqrt{\sum _{\alpha}\sum _{\beta}{\left({R}_{\alpha}^{i}\right)}_{k}^{max}{\rho}_{\alpha \beta}{\left({R}_{\beta}^{i}\right)}_{k}^{max}},$$
where ${\rho}_{\alpha \beta}$ are crosscorrelation coefficients between modes $\alpha $ and $\beta $, which depend on the ratio of frequencies and modal damping between the two modes: $${\rho}_{\alpha \beta}=\frac{8\sqrt{{\xi}_{\alpha}{\xi}_{\beta}}\left({\xi}_{\alpha}+{r}_{\beta \alpha}{\xi}_{\beta}\right){r}_{\beta \alpha}^{3/2}}{{\left(1{r}_{\beta \alpha}^{2}\right)}^{2}+4{\xi}_{\alpha}{\xi}_{\beta}{r}_{\beta \alpha}\left(1+{r}_{\beta \alpha}^{2}\right)+4\left({\xi}_{\alpha}^{2}+{\xi}_{\beta}^{2}\right){r}_{\beta \alpha}^{2}},$$
where ${r}_{\beta \alpha}={\omega}_{\beta}/{\omega}_{\alpha}$. If the modes are well spaced, their crosscorrelation coefficient will be small (${\rho}_{\alpha \beta}\ll 1$) and the method will give the same results as the square root of the sum of the squares method. This method is usually recommended for asymmetrical building systems since, in such cases, other methods can underestimate the response in the direction of motion and overestimate the response in the transverse direction. Input File Usage RESPONSE SPECTRUM, COMP=comp, SUM=CQC Abaqus/CAE Usage Step module: Create Step: Linear perturbation: Response spectrum: Summations: Complete quadratic combination The grouping methodThis method, also known as the NRC grouping method, improves the response estimation for structures with closely spaced eigenvalues. The modal responses are grouped such that the lowest and highest frequency modes in a group are within 10% and no mode is in more than one group. The modal responses are summed absolutely within groups before performing a SRSS combination of the groups. Within the group responses are summed as $${\left({R}_{gr}^{i}\right)}_{k}^{max}=\left{\left({R}_{1}^{i}\right)}_{k}^{max}\right+\left{\left({R}_{2}^{i}\right)}_{k}^{max}\right\mathrm{\dots}+\left{\left({R}_{n}^{i}\right)}_{k}^{max}\right,$$
for “n” frequencies within any “gr” group and then performing $${\left({R}^{i}\right)}_{k}^{max}=\sqrt{\stackrel{gr=n}{\sum _{gr=1}}{\left({\left({R}_{gr}^{i}\right)}_{k}^{max}\right)}^{2}}.$$
The above expression includes all the groups; in addition, the group can consist of just one frequency response if this frequency does not have another member that is within the 10% limit. The tenpercent method will always produce results higher in value than the grouping method. Input File Usage RESPONSE SPECTRUM, COMP=comp, SUM=GRP Abaqus/CAE Usage Step module: Create Step: Linear perturbation: Response spectrum: Summations: Grouping method Double sum combinationThis method, also known as Rosenblueth's double sum combination (Rosenblueth and Elorduy, 1969), is the first attempt to evaluate modal correlation based on random vibration theory. It utilizes the time duration ${t}_{D}$ of strong earthquake motion. The mode correlation coefficients ${c}_{\alpha \beta}$, which depend also on the frequencies and damping coefficient $\xi $, lead to the following mode combination: $${\left({R}^{i}\right)}_{k}^{max}=\sqrt{\sum _{\alpha}\sum _{\beta}{\left({R}_{\alpha}^{i}\right)}_{k}^{max}{c}_{\alpha \beta}{\left({R}_{\beta}^{i}\right)}_{k}^{max}},$$
where $${c}_{\alpha \beta}=\frac{1}{1+{\left(\frac{{\omega}_{{\alpha}^{\prime}}{\omega}_{{\beta}^{\prime}}}{{\xi}_{{\alpha}^{\prime}}{\omega}_{\alpha}+{\xi}_{{\beta}^{\prime}}{\omega}_{\beta}}\right)}^{2}},$$
where $${\omega}_{{\alpha}^{\prime}}={\omega}_{\alpha}\sqrt{1{\xi}_{\alpha}^{2}},$$
$${\xi}_{{\alpha}^{\prime}}={\xi}_{\alpha}+\frac{2}{{t}_{D}{\omega}_{\alpha}}.$$
Input File Usage RESPONSE SPECTRUM, COMP=comp, SUM=DSC Abaqus/CAE Usage Step module: Create Step: Linear perturbation: Response spectrum: Summations: Double sum combination Separation of modal responses into periodic and rigid responsesEach spectrum can be divided into low, medium, and highfrequency regions. In the lowfrequency region the modes are usually uncorrelated. In the midrange they are partially correlated, and in the highfrequency region they tend to be correlated with the input data and, hence, between themselves. Each of these regions may, therefore, require special consideration. These considerations are addressed by the Gupta and LindleyYow methods. We divide the modal response into two parts: the rigid part and the damped periodic part. It is assumed that the rigid part and the damped periodic part are statistically independent. Let us define the rigid response coefficient for each mode by $0\le {\alpha}_{\beta}\le 1$, the rigid response by ${\left({R}_{\beta}^{i}\right)}^{r}{=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\alpha}_{\beta}\left({R}_{\beta}^{i}\right)$, and the periodic response by ${{\left({R}_{\beta}^{i}\right)}^{p}={R}_{\beta}^{i}\sqrt{1{\alpha}_{\beta}^{2}}}^{\text{\hspace{0.17em}}}$. Therefore, the relationship between both regions becomes: $${({\left({R}_{\beta}^{i}\right)}^{p})}^{2}+{({\left({R}_{\beta}^{i}\right)}^{r})}^{2}=\text{\hspace{0.17em}}1\text{\hspace{0.17em},\hspace{0.17em}}$$
which is equivalent to summing both regions according to the SRSS rule. The value of the rigid response coefficient is chosen depending on the method used. For the Gupta method this coefficient is estimated based on two userspecified frequencies: ${F}_{1}$ defining the beginning of the periodic region, and ${F}_{2}$ defining the beginning of the rigid region (which is often equivalent to the frequency associated with zero period acceleration). For the current frequency ${F}_{\beta}$, this expression for the Gupta method is $${\alpha}_{\beta}\text{\hspace{0.17em}}=\frac{\mathrm{log}\left(\frac{{F}_{\beta}}{{F}_{1}}\right)}{\mathrm{log}\left(\frac{{F}_{2}}{{F}_{1}}\right)}\mathrm{.}$$
For the LindleyYow method this coefficient is equal to $${\alpha}_{\beta}=\text{\hspace{0.17em}}\frac{ZPA}{{S}_{a\beta}},$$
where ZPA denotes the userspecified zero period acceleration and ${S}_{a\beta}$ denotes the acceleration spectrum for mode $\beta $. The LindleyYow method does not work well for the region of frequencies below the zero period acceleration on the lowfrequency end of the spectrum. Therefore, we set the coefficient ${\alpha}_{\beta}=0$ for all modes below the frequency of the peak acceleration value for this spectrum. The rigid response method works with only two modal summation rules: CQC and DSC with their respective definitions for the mode correlation coefficients ${c}_{\alpha \beta}$. By default, no rigid response is included. To include a rigid response, you must specify the rigid response method. Input File Usage Use the following option to specify the Gupta method for the rigid response: RESPONSE SPECTRUM, COMP=comp, RIGID RESPONSE=GUPTA Use the following option to specify the LindleyYow method for the rigid response: RESPONSE SPECTRUM, COMP=comp, RIGID RESPONSE=LINDLEYYOW Abaqus/CAE Usage You cannot specify a rigid response in Abaqus/CAE. Selecting the modes and specifying dampingYou can select the modes to be used in modal superposition and specify damping values for all selected modes. Selecting the modesYou can select modes by specifying the mode numbers individually, by requesting that Abaqus/Standard generate the mode numbers automatically, or by requesting the modes that belong to specified frequency ranges. If you do not select the modes, all modes extracted in the prior eigenfrequency extraction step, including residual modes if they were activated, are used in the modal superposition. Input File Usage Use one of the following options to select the modes by specifying mode numbers: SELECT EIGENMODES, DEFINITION=MODE NUMBERS SELECT EIGENMODES, GENERATE, DEFINITION=MODE NUMBERS Use the following option to select the modes by specifying a frequency range: SELECT EIGENMODES, DEFINITION=FREQUENCY RANGE Abaqus/CAE Usage You cannot select the modes in Abaqus/CAE; all modes extracted are used in the modal superposition. Specifying dampingDamping is almost always specified for a modebased procedure; see Material damping. You can define a damping coefficient for all or some of the modes used in the response calculation. The damping coefficient can be given for a specified mode number or for a specified frequency range. When damping is defined by specifying a frequency range, the damping coefficient for an mode is interpolated linearly between the specified frequencies. The frequency range can be discontinuous; the average damping value will be applied for an eigenfrequency at a discontinuity. The damping coefficients are assumed to be constant outside the range of specified frequencies. Input File Usage Use the following option to define damping by specifying mode numbers: MODAL DAMPING, DEFINITION=MODE NUMBERS Use the following option to define damping by specifying a frequency range: MODAL DAMPING, DEFINITION=FREQUENCY RANGE Abaqus/CAE Usage Use the following input to define damping by specifying mode numbers: Step module: Create Step: Linear perturbation: Response spectrum: Damping: Specify damping over ranges of: Modes Use the following input to define damping by specifying a frequency range: Step module: Create Step: Linear perturbation: Response spectrum: Damping: Specify damping over ranges of: Frequencies Example of specifying dampingFigure 1 illustrates how the damping coefficients at different eigenfrequencies are determined for the following input: Figure 1. Damping values specified by frequency range.
MODAL DAMPING, DEFINITION=FREQUENCY RANGE ${f}_{1},{d}_{1}$ ${f}_{2},{d}_{2}$ ${f}_{2},{d}_{3}$ ${f}_{3},{d}_{3}$ ${f}_{4},{d}_{4}$ Rules for selecting modes and specifying damping coefficientsThe following rules apply for selecting modes and specifying modal damping coefficients:
Initial conditionsIt is not appropriate to specify initial conditions in a response spectrum analysis. Boundary conditionsAll points constrained by boundary conditions and the ground nodes of connector elements are assumed to move in phase in any one direction. This base motion can use a different input spectrum in each of three orthogonal directions (two directions in a twodimensional model). You define the input spectra, $S\left(\omega ,\xi \right)$, as functions of frequency, $\omega $, for different values of critical damping, $\xi $, as described earlier in Specifying a spectrum. Secondary bases cannot be used in a response spectrum analysis. LoadsThe only “loading” that can be defined in a response spectrum analysis is that defined by the input spectra, as described earlier. No other loads can be prescribed in a response spectrum analysis. Predefined fieldsPredefined fields, including temperature, cannot be used in response spectrum analysis. Material optionsThe density of the material must be defined (Density). The following material properties are not active during a response spectrum analysis: plasticity and other inelastic effects, ratedependent material properties, thermal properties, mass diffusion properties, electrical properties, and pore fluid flow properties—see General and perturbation procedures. ElementsOther than generalized axisymmetric elements with twist, any of the stress/displacement elements in Abaqus/Standard can be used in a response spectrum analysis—see Choosing the appropriate element for an analysis type. OutputAll the output variables in Abaqus/Standard are listed in Abaqus/Standard output variable identifiers. The value of an output variable such as strain, E; stress, S; or displacement, U, is its peak magnitude. In addition to the usual output variables available, the following modal variables are available for response spectrum analysis and can be output to the data and/or results files (see Output to the Data and Results Files):
Neither element energy densities (such as the elastic strain energy density, SENER) nor whole element energies (such as the total kinetic energy of an element, ELKE) are available for output in response spectrum analysis. However, whole model variables such as ALLIE (total strain energy) are available for modalbased procedures as output to the data and/or results files (see Output to the Data and Results Files). Reaction force output is not supported for response spectrum analysis using eigenmodes extracted using a SIMbased frequency extraction procedure with either the AMS or Lanczos eigensolver. Reaction force output in response spectrum analysis using eigenmodes extracted with the default Lanczos eigensolver provides directional combinations of socalled, modal reaction forces weighted with maximal absolute values of corresponding generalized displacements. Directional and modal combination rules used for the reaction force calculation are the same as for other nodal output variables. Modal reaction forces are calculated in the frequency extraction procedure. They represent static reaction forces calculated for the normal mode shapes. Generally, they cannot adequately represent reaction force in dynamic analysis. For models with diagonal mass and diagonal damping matrices the superposition of the modal reaction forces can provide a reasonable approximation of a nodal reaction force in modebased analyses other than response spectrum analysis. In response spectrum analysis the model response can be better represented by requesting section stresses and section forces in structural elements containing supported nodes. Input file templateHEADING … BOUNDARY Data lines to define points to be excited by the base motion controlled by the input spectra SPECTRUM, NAME=name1, TYPE=type Data lines to define spectrum “name1” as a function of frequency, $\omega $, and fraction of critical damping, $\xi $ SPECTRUM, NAME=name2, TYPE=type Data lines to define spectrum “name2” as a function of frequency, $\omega $, and fraction of critical damping, $\xi $ ** STEP FREQUENCY Data line to specify number of modes to be extracted END STEP ** STEP RESPONSE SPECTRUM, COMP=comp, SUM=sum Data lines referring to response spectra and defining direction cosines SELECT EIGENMODES Data lines to define the applicable mode ranges MODAL DAMPING Data lines to define modal damping END STEP References
