김경종
366 lines
23 KiB
Markdown
366 lines
23 KiB
Markdown
<!-- source-page: 291 -->
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$$
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\rho_ {\alpha \beta} = \frac {8 \sqrt {\xi_ {\alpha} \xi_ {\beta}} (\xi_ {\alpha} + r _ {\beta \alpha} \xi_ {\beta}) r _ {\beta \alpha} ^ {3 / 2}}{(1 - r _ {\beta \alpha} ^ {2}) ^ {2} + 4 \xi_ {\alpha} \xi_ {\beta} r _ {\beta \alpha} (1 + r _ {\beta \alpha} ^ {2}) + 4 (\xi_ {\alpha} ^ {2} + \xi_ {\beta} ^ {2}) r _ {\beta \alpha} ^ {2}},
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$$
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where $r _ { \beta \alpha } = \omega _ { \beta } / \omega _ { \alpha }$ .
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If the modes are well spaced, their cross-correlation coefficient will be small $( \rho _ { \alpha \beta } < < 1 )$ and the method will give the same results as the square root of the sum of the squares method.
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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.
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Input File Usage: \*RESPONSE SPECTRUM, COMP=comp, SUM=CQC
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Abaqus/CAE Usage: Step module: Create Step: Linear perturbation: Response spectrum:
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Summations: Complete quadratic combination
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# The grouping method
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This 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
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$$
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(R _ {g r} ^ {i}) _ {k} ^ {m a x} = | (R _ {1} ^ {i}) _ {k} ^ {m a x} | + | (R _ {2} ^ {i}) _ {k} ^ {m a x} | \dots + | (R _ {n} ^ {i}) _ {k} ^ {m a x} |,
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$$
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for $\ " { } \mathbf { n } \warrow $ frequencies within any “gr” group and then performing
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$$
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(R ^ {i}) _ {k} ^ {m a x} = \sqrt {\sum_ {g r = 1} ^ {g r = n} \left(\left(R _ {g r} ^ {i}\right) _ {k} ^ {\max}\right) ^ {2}}.
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$$
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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.
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The ten-percent method will always produce results higher in value than the grouping method.
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Input File Usage: \*RESPONSE SPECTRUM, COMP=comp, SUM=GRP
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Abaqus/CAE Usage: Step module: Create Step: Linear perturbation: Response spectrum: Summations: Grouping method
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# Double sum combination
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This 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:
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<!-- source-page: 292 -->
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$$
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(R ^ {i}) _ {k} ^ {m a x} = \sqrt {\sum_ {\alpha} \sum_ {\beta} (R _ {\alpha} ^ {i}) _ {k} ^ {m a x} c _ {\alpha \beta} (R _ {\beta} ^ {i}) _ {k} ^ {m a x}},
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$$
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where
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$$
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c _ {\alpha \beta} = \frac {1}{1 + \left(\frac {\omega_ {\alpha^ {\prime}} - \omega_ {\beta^ {\prime}}}{\xi_ {\alpha^ {\prime}} \omega_ {\alpha} + \xi_ {\beta^ {\prime}} \omega_ {\beta}}\right) ^ {2}},
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$$
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where
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$$
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\omega_ {\alpha^ {\prime}} = \omega_ {\alpha} \sqrt {1 - \xi_ {\alpha} ^ {2}},
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$$
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$$
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\xi_ {\alpha^ {\prime}} = \xi_ {\alpha} + \frac {2}{t _ {D} \omega_ {\alpha}}.
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$$
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Input File Usage: \*RESPONSE SPECTRUM, COMP=comp, SUM=DSC
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Abaqus/CAE Usage: Step module: Create Step: Linear perturbation: Response spectrum:
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Summations: Double sum combination
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# Selecting the modes and specifying damping
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You can select the modes to be used in modal superposition and specify damping values for all selected modes.
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# Selecting the modes
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You 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.
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Input File Usage: Use one of the following options to select the modes by specifying mode numbers:
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\*SELECT EIGENMODES, DEFINITION=MODE NUMBERS
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\*SELECT EIGENMODES, GENERATE, DEFINITION=MODE NUMBERS
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Use the following option to select the modes by specifying a frequency range:
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\*SELECT EIGENMODES, DEFINITION=FREQUENCY RANGE
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Abaqus/CAE Usage: You cannot select the modes in Abaqus/CAE; all modes extracted are used in the modal superposition.
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<!-- source-page: 293 -->
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# Specifying damping
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Damping is almost always specified for a mode-based procedure; see “Material damping,” Section 26.1.1. 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.
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<table><tr><td>Input File Usage:</td><td>Use the following option to define damping by specifying mode numbers:*MODAL DAMPING, DEFINITION=MODE NUMBERSUse the following option to define damping by specifying a frequency range:*MODAL DAMPING, DEFINITION=FREQUENCY RANGE</td></tr></table>
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<table><tr><td>Abaqus/CAE Usage:</td><td>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: ModesUse 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</td></tr></table>
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# Example of specifying damping
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Figure 6.3.10–1 illustrates how the damping coefficients at different eigenfrequencies are determined for the following input:
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<table><tr><td>*MODAL DAMPING, DEFINITION=FREQUENCY RANGE</td></tr><tr><td> $f_{1}, d_{1}$ </td></tr><tr><td> $f_{2}, d_{2}$ </td></tr><tr><td> $f_{2}, d_{3}$ </td></tr><tr><td> $f_{3}, d_{3}$ </td></tr><tr><td> $f_{4}, d_{4}$ </td></tr></table>
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# Rules for selecting modes and specifying damping coefficients
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The following rules apply for selecting modes and specifying modal damping coefficients:
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• No modal damping is included by default.
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• Mode selection and modal damping must be specified in the same way, using either mode numbers or a frequency range.
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• If you do not select any modes, all modes extracted in the prior frequency analysis, including residual modes if they were activated, will be used in the superposition.
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• If you do not specify damping coefficients for modes that you have selected, zero damping values will be used for these modes.
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<!-- source-page: 294 -->
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<details>
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<summary>line</summary>
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| frequency | damping values |
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| --------- | -------------- |
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| f₁ | d₁ |
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| λ₁ | d₂ |
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| f₂ | d₃ |
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| f₃ | d₄ |
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| f₄ | d₄ |
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| λ₃ | d₄ |
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</details>
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Figure 6.3.10–1 Damping values specified by frequency range.
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• Damping is applied only to the modes that are selected.
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• Damping coefficients for selected modes that are beyond the specified frequency range are constant and equal to the damping coefficient specified for the first or the last frequency (depending which one is closer). This is consistent with the way Abaqus interprets amplitude definitions.
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# Initial conditions
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It is not appropriate to specify initial conditions in a response spectrum analysis.
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# Boundary conditions
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All 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 two-dimensional model). You define the input spectra, $S ( \omega , \xi )$ , 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.
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# Loads
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The 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.
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# Predefined fields
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Predefined fields, including temperature, cannot be used in response spectrum analysis.
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<!-- source-page: 295 -->
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# Material options
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The density of the material must be defined (“Density,” Section 21.2.1). The following material properties are not active during a response spectrum analysis: plasticity and other inelastic effects, rate-dependent material properties, thermal properties, mass diffusion properties, electrical properties, and pore fluid flow properties—see “General and linear perturbation procedures,” Section 6.1.3.
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# Elements
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Other 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,” Section 27.1.3.
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# Output
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All the output variables in Abaqus/Standard are listed in “Abaqus/Standard output variable identifiers,” Section 4.2.1. The value of an output variable such as strain, E; stress, S; or displacement, U, is its peak magnitude.
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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,” Section 4.1.2):
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GU Generalized displacements for all modes.
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GV Generalized velocities for all modes.
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GA Generalized accelerations for all modes.
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SNE Elastic strain energy for the entire model per each mode.
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KE Kinetic energy for the entire model per each mode.
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T External work for the entire model per each mode.
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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 modal-based procedures as output to the data and/or results files (see “Output to the data and results files,” Section 4.1.2).
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Reaction force output is not supported for response spectrum analysis using eigenmodes extracted using a SIM-based 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 so-called, 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
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<!-- source-page: 296 -->
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superposition of the modal reaction forces can provide a reasonable approximation of a nodal reaction force in mode-based 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.
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Input file template
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```txt
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*HEADING
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...
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*BOUNDARY
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Data lines to define points to be excited by the base motion controlled by the input spectra
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*SPECTRUM, NAME=name1, TYPE=type
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Data lines to define spectrum “name1” as a function of frequency, ω, and
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fraction of critical damping, ξ
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*SPECTRUM, NAME=name2, TYPE=type
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Data lines to define spectrum “name2” as a function of frequency, ω, and
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fraction of critical damping, ξ
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**
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*STEP
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*FREQUENCY
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Data line to specify number of modes to be extracted
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*END STEP
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**
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*STEP
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*RESPONSE SPECTRUM, COMP=comp, SUM=sum
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Data lines referring to response spectra and defining direction cosines
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*SELECT EIGENMODES
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Data lines to define the applicable mode ranges
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*MODAL DAMPING
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Data lines to define modal damping
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*END STEP
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```
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# Additional reference
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• Rosenblueth, E., and J. Elorduy, “Response of Linear Systems to Certain Transient Disturbances,” Proceedings of the Fourth World Conference on Earthquake Engineering, Santiago, Chile, 1969.
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<!-- source-page: 297 -->
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# 6.3.11 RANDOM RESPONSE ANALYSIS
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Products: Abaqus/Standard Abaqus/CAE
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# References
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• “Defining an analysis,” Section 6.1.2
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• “General and linear perturbation procedures,” Section 6.1.3
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• “Dynamic analysis procedures: overview,” Section 6.3.1
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• \*RANDOM RESPONSE
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• \*PSD-DEFINITION
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• \*CORRELATION
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• “Configuring a random response 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
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# Overview
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A random response analysis:
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• is a linear perturbation procedure that gives the linearized dynamic response of a model to userdefined random excitation; and
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• uses the set of modes extracted in a previous eigenfrequency extraction step to calculate the power spectral densities of response variables (stresses, strains, displacements, etc.) and the corresponding root mean square (RMS) values of these same variables.
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# Random response analysis
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Random response analysis predicts the response of a system that is subjected to a nondeterministic continuous excitation that is expressed in a statistical sense by a cross-spectral density matrix. Since the loading is nondeterministic, it can be characterized only in a statistical sense; Abaqus/Standard assumes that the excitation is stationary and ergodic. These statistical measures are explained in detail in “Random response analysis,” Section 2.5.8 of the Abaqus Theory Guide. The random response procedure can, for example, be used to determine the response of an airplane to turbulence, the response of a car to road surface imperfections, the response of a structure to jet noise, or the response of a building to an earthquake.
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In the most general case the excitation is defined as a frequency-dependent cross-spectral density (CSD) matrix. Except in cases involving moving noise or user subroutine UCORR, it is assumed that for a given load case the CSD matrix can be separated into a product of a frequency-dependent, complexvalued scalar function and a frequency-independent, complex-valued, spatial correlation matrix. This assumption helps reduce both the computational time and the amount of required user input but implies that each element of the CSD matrix in a given load case has the same frequency dependence. You can define a different frequency dependence for each load case, but the loads in one load case will not be
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<!-- source-page: 298 -->
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correlated with loads in another. Consequently, the system CSD matrix is assembled by simply summing (superimposing) the CSD matrices of the individual load cases.
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The frequency-dependent scalar function can be composed of a weighted sum of user-defined, complex-valued, frequency functions. These user-defined frequency functions are specified in units of power spectral density. You assign weights to each frequency function as well as specify properties of the spatial correlation matrix that defines the correlation between excitations at different locations and in different directions for a particular load case. Frequency functions and correlations are discussed below; see “Defining the frequency functions,” and “Defining the correlation.”
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The loads can be defined as concentrated point loads, as distributed loads, as connector element loads, or as base motion excitations, as described below in “Boundary conditions,” and “Loads.” Multiple, uncorrelated load cases can be defined for concentrated point loads, connector loads, and base motions. Load case 1 is reserved for all distributed loads defined in a particular step. In these steps load case 1 cannot be used for any concentrated point load, connector load, or base motion. Thus, there cannot be any correlation between distributed loads and any other load. Moreover, base motion excitations are assumed to be statistically independent (no correlation) with any other load type even when the same load case number is used. The concentrated point and connector element loads are assumed to be correlated if the same load case number is used.
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The random response 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. The model can be preloaded prior to the eigenfrequency extraction. Initial stress effects are included in the stiffness used in the eigenfrequency extraction if geometric nonlinearities are included in the general analysis procedure used to apply the preloads (“General and linear perturbation procedures,” Section 6.1.3).
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The random response of the model is expressed as power spectral density values of nodal and element variables, as well as their root mean square values.
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# Defining the frequency range
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You specify the frequency range of interest for the random response procedure. The response is calculated at multiple points between the lowest frequency of interest and the first eigenfrequency in the range, between each eigenfrequency in the range, and between the last eigenfrequency in the range and the highest frequency in the range as illustrated in Figure 6.3.11–1. The default number of calculation points in each interval is 20; you can change this number when you define the step. Accurate RMS values can be obtained only if enough points are used so that Abaqus/Standard can integrate accurately over the frequency range. The bias function allows the points on the frequency scale to be spaced closer together at the eigenfrequencies, thus allowing detailed definition of the response close to resonant frequencies and more accurate integration.
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Input File Usage: \*RANDOM RESPONSE lower\_freq\_limit, upper\_freq\_limit, num\_calc\_pts, bias\_parameter, freq\_scale
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Abaqus/CAE Usage: Step module: Create Step: Linear perturbation: Random response
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<!-- source-page: 299 -->
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<details>
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<summary>flowchart</summary>
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```mermaid
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graph TD
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A["frequency points"] --> B["lower end of the range"]
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A --> C["mode n"]
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A --> D["mode n +1"]
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A --> E["mode n + 2"]
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A --> F["upper end of the range"]
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```
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</details>
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Figure 6.3.11–1 Division of range using modes and 5 calculation points.
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# The bias parameter
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The bias parameter can be used to provide closer spacing of the result points either toward the middle or toward the ends of each frequency interval. Figure 6.3.11–2 shows a few examples of the effect of the bias parameter on the frequency spacing.
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<details>
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<summary>flowchart</summary>
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```mermaid
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graph TD
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A["frequency points"] --> B["f̂₁"]
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A --> C["f̂₂"]
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B --> D["1"]
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C --> E["2"]
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D --> F["3"]
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E --> G["5"]
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style A fill:#f9f,stroke:#333
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style B fill:#ccf,stroke:#333
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style C fill:#ccf,stroke:#333
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style D fill:#cfc,stroke:#333
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style E fill:#cfc,stroke:#333
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style F fill:#fcc,stroke:#333
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style G fill:#fcc,stroke:#333
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```
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</details>
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Figure 6.3.11–2 Effect of the bias parameter on the frequency spacing for a number of points .
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The bias formula used to calculate the frequency at which results are presented is as follows:
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$$
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\hat {f} _ {k} = \frac {1}{2} (\hat {f} _ {1} + \hat {f} _ {2}) + \frac {1}{2} (\hat {f} _ {2} - \hat {f} _ {1}) | y | ^ {1 / p} \mathrm{sign} (y),
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$$
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where
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$$
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\boldsymbol {y} \quad = - 1 + 2 (k - 1) / (n - 1);
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$$
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<!-- source-page: 300 -->
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n is the number of frequency points at which results are to be given;
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k is one such frequency point $(k = 1, 2, \ldots, n)$ ; $\hat{f}_{1}$ is the lower limit of the frequency interval; $\hat{f}_{2}$ is the upper limit of the interval; $\hat{f}_{k}$ is the frequency at which the kth results are given;
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p is the bias parameter value; and $\hat{f}$ is the frequency or the logarithm of the frequency, depending on the chosen frequency scale.
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A bias parameter, p, that is greater than 1.0 provides closer spacing of the results points toward the ends of each frequency interval (as shown in the examples above), while values of p that are less than 1.0 provide closer spacing toward the middle of each frequency interval. The default value of the bias parameter for random response analysis is 3.0.
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# Defining the frequency functions
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To define the random loading, you specify a frequency function and a cross-correlation definition that refers to the frequency function. The frequency functions are defined as model data (i.e., they are step independent) and must be named. A log-log scale is used in interpolating between the given values.
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The type of units in the CSD matrix of the excitation are specified as part of the frequency function definition. The default type is power units. If the CSD matrix of the excitation is due to base motion, the units must be in g units and you should define the acceleration of gravity. Alternatively, decibel units can be specified; this type of units is explained below.
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Input File Usage: Use one of the following options to define the frequency function:
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*PSD-DEFINITION, NAME=name, TYPE=FORCE (default; power units)
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*PSD-DEFINITION, NAME=name, TYPE=BASE, G=g
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*PSD-DEFINITION, NAME=name, TYPE=DB, DB REFERENCE=P $_{ref}$
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Abaqus/CAE Usage: Load module: Create Amplitude; Type: PSD Definition; Specification units: Power, Decibel, or Gravity
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# Defining the cross-spectral density matrix in decibel units
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In Abaqus/Standard the decibel value $d b ( f )$ is related to the frequency function $P ( f )$ by the following full octave band conversion formula:
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$$
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d b (f) = 1 0 \log_ {1 0} \frac {P (f)}{\sqrt {2} P _ {\mathrm{ref}} / f _ {\mathrm{c}}},
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$$
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where $P _ { \mathrm { r e f } }$ is the user-specified reference power and $f _ { \mathrm { c } }$ is the midband frequency (see Table 6.3.11–1). Hence, the frequency function follows from the function defined in decibel units as
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$$
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P (f) = \frac {\sqrt {2}}{f _ {\mathrm{c}}} P _ {\mathrm{ref}} 1 0 ^ {d b (f) / 1 0}.
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$$
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