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Example posted by: <span style="color:blue"> Dimitrios G. Lignos, Ph.D., McGill University</span>
Example posted by: <span style="color:blue"> Dr. Dimitrios G. Lignos (McGill University)</span>
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The files needed to analyze this structure in OpenSees are included here:
The files needed to analyze this structure in OpenSees are included here:
* The main file: [[Maxwell_Calibrator.tcl|Maxwell_Calibrator.tcl]]
* The main file: [[File:Maxwell_Calibrator.tcl|Maxwell_Calibrator.tcl]]
Supporting procedure files
Supporting procedure files
* [[SquareSsection.tcl|SquareSsection.tcl]] – displays a square fiber section
* [[File:SquareSsection.tcl|SquareSsection.tcl]] – displays a square fiber section
* [[Damper.txt|Damper.txt]] – contains the displacement loading history of the damper in units of mm
* [[File:ViscousD.tcl|ViscousD.tcl]] – contains the displacement loading history of the damper in units of mm
All files are available in a compressed format here: [[Media:Calibration_Maxwell_example.zip|Calibration_Maxwell_example.zip]]
All files are available in a compressed format here: [[Media:Calibration_Maxwell_example.zip|Calibration_Maxwell_example.zip]]
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== Model Description ==
== Model Description ==
[[File:2s1b-Sketch.PNG|frame|Figure 1. Schematic representation of a viscous damper.]]
[[File:Maxwell-Fig1.png|300px|thumb|left|Figure 1. Schematic representation of a viscous damper.]]
The viscous damper shown in Figure 1 is modeled with [[nonlinear Beam Column Element|nonlinearBeamColumn elements]] connected by [[zeroLength Element|zeroLength elements]] which serve as rotational springs to represent the structure’s nonlinear behavior. The springs follow a [[Bilin Material|bilinear]] hysteretic response based on the Modified Ibarra Krawinkler Deterioration Model. A leaning column with gravity loads is linked to the frame by [[Truss Element|truss elements]] to simulate P-Delta effects. An idealized schematic of the model is presented in Figure 1.
The viscous damper is modeled with the [[Force-Based Beam-Column Element|Force-Based Beam-Column element]]. This element follow a [[Maxwell Material| Maxwell]] hysteretic response. An idealized schematic of the model is presented in Figure 1.
To simplify this model, panel zone contributions are neglected, plastic hinges form at the beam-column joints, and centerline dimensions are used. For an example that explicitly models the panel zone shear distortions and includes reduced beam sections (RBS), see [[Pushover and Dynamic Analyses of 2-Story Moment Frame with Panel Zones and RBS|Pushover and Dynamic Analyses of 2-Story Moment Frame with Panel Zones and RBS]].
The units of the model are mm, N, and seconds.
For a detailed description of this model, see [[Pushover Analysis of 2-Story Moment Frame|Pushover Analysis of 2-Story Moment Frame]].
The units of the model are kips, inches, and seconds.
=== Basic Geometry ===
The basic geometry of the viscous damper is defined by input variables for the length L=5000mm, and area A = 12000mm<sup>2</sup>. Two nodes are used for the geometry of the damper.
== Damping and the Rayleigh Command ==
=== Damper Section ===
This model uses Rayleigh damping which formulates the damping matrix as a linear combination of the mass matrix and stiffness matrix: '''c''' = a<sub>0</sub>*'''m''' + a<sub>1</sub>*'''k''', where a<sub>0</sub> is the mass proportional damping coefficient and a<sub>1</sub> is the stiffness proportional damping coefficient. A damping ratio of 2%, which is a typical value for steel buildings, is assigned to the first two modes of the structure. The [[Rayleigh Damping Command|rayleigh command]] allows the user to specify whether the initial, current, or last committed stiffness matrix is used in the damping matrix formulation. In this example, only the initial stiffness matrix is used, which is accomplished by assigning values of 0.0 to the other stiffness matrix coefficients.
To properly model the structure, stiffness proportional damping is applied only to the frame elements and not to the highly rigid truss elements that link the frame and leaning column, nor to the leaning column itself. OpenSees does not apply stiffness proportional damping to [[zeroLength Element|zeroLength elements]]. In order to apply damping to only certain elements, the [[Rayleigh Damping Command|rayleigh command]] is used in combination with the [[Region Command|region command]]. As noted in the [[Region Command|region command]] documentation, the region cannot be defined by BOTH elements and nodes. Because mass proportional damping assigns damping to nodes with mass, OpenSees will ignore any mass proportional damping that is assigned using the [[Rayleigh Damping Command|rayleigh command]] in combination with the [[Region Command|region command]] for a region of elements. Therefore, if using the region command to assign damping, the mass proportional damping and stiffness proportional damping must be assigned in separate steps.
A square section is used to define the area A of the damper (A<sub>damper</sub> = 12000.0mm<sup>2</sup>).
=== Modifications to the Stiffness Proportional Damping Coefficient ===
=== Damper Links ===
As described in the “Stiffness Modifications to Elastic Frame Elements” section of [[Pushover Analysis of 2-Story Moment Frame|Pushover Analysis of 2-Story Moment Frame]], the stiffness of the elastic frame elements has been modified. As explained in Ibarra and Krawinkler (2005) and Zareian and Medina (2010), the stiffness proportional damping coefficient that is used with these elements must also be modified. As the stiffness of the elastic elements was made “(n+1)/n” times greater than the stiffness of the actual frame member, the stiffness proportional damping coefficient of these elements must also be made “(n+1)/n” times greater than the traditional stiffness proportional damping coefficient.
== Dynamic Analysis ==
[[Force-Based Beam-Column Element|Force-Based Beam-Column elements]] are used to link the two nodes that define the geometry of the viscous damper with n=5 sections of integration and the damper section defined previously.
=== Constraints ===
Node 1 is pinned. A roller is assigned to node 2
=== Maxwell Material ===
To model the viscous damper the [[Maxwell Material| Maxwell]] is used. The input parameters that are selected for the damper example are as follows:
Axial Stiffness K = 500000.0 N/mm, Viscous Coefficient C=100000.0 N(s/mm)<sup>0.30</sup> and exponent a=0.30.
=== Loading ===
The viscous damper is loaded with a displacement history that is called Damper.txt in the axial loading direction. A linear pattern is selected and the node of application is Node 2.
=== Recorders ===
=== Recorders ===
The [[Recorder Command|recorders]] used in this example include:
The [[Recorder Command|recorders]] used in this example include:
* The [[Drift Recorder|drift recorder]] to track the story and roof drift histories
* The [[Recorder Command|localForce recorder]] to track the axial force of the viscous damper
* The [[Node Recorder|node recorder]] to track the floor displacement and base shear reaction histories
* The [[Recorder Command|stressStrain recorder]] to track the stress and strain history of the viscous damper in section 3.
* The [[Element Recorder|element recorder]] to track the element forces in the first story columns as well as the moment and rotation histories of the springs in the concentrated plasticity model
For the [[Element Recorder|element recorder]], the [[Region Command|region command]] was used to assign all column springs to one group and all beam springs to a separate group.
It is important to note that the recorders only record information for [[Analyze Command|analyze commands]] that are called after the [[Recorder Command|recorder commands]] are called. In this example, the recorders are placed after the gravity analysis so that the steps of the gravity analysis do not appear in the output files.
=== Analysis ===
=== Analysis ===
The structure is analyzed under gravity loads before the dynamic analysis is conducted. The gravity loads are applied using a [[Load Control|load-controlled]] static analysis with 10 steps. So that the gravity loads remain on the structure for all subsequent analyses, the [[LoadConst Command|loadConst command]] is used after the gravity analysis is completed. This command is also used to reset the time to zero so that the dynamic analysis starts from time zero.
In order to calibrate the viscous damper a multiple support excitation option is selected with application of displacement as the imposed motion. This is necessary since an integration step dt should be specified for integration of strain histories for each time step.
For the dynamic analysis, the structure is subjected to the Canoga Park record from the 1994 Northridge earthquake. To apply the ground motion to the structure, the [[Uniform Exciatation Pattern|uniform excitation pattern]] is used. The name of the file containing the acceleration record, timestep of the ground motion, scale factor applied to the ground motion, and the direction in which the motion is to be applied must all be specified as part of the [[Uniform Exciatation Pattern|uniform excitation pattern command]].
To execute the dynamic analysis, the [[Analyze Command|analyze command]] is used with the specified number of analysis steps and the timestep of the analysis. The timestep used in the analysis should be less than or equal to the timestep of the input ground motion.
== Results ==
== Results ==
=== Comparison of OpenSees Model with MATLAB based Maxwell model ===
[[File:MaxK1.png|300px|thumb|Right|Figure 2. Viscous Damper Response and Comparison With MATLAB based Script]]
The force - displacement relationship from the maxwell damper are shown in Figure 2. In order to obtain the displacement of the damper we multiply the strain in section 3 of the damper times the length L=5000mm. A comparison of the response with a MATLAB based script is also shown in the same figure. Results are identical.
The floor displacement histories from the dynamic analysis are shown in Figure 2. The top graph shows the ground acceleration history while the middle and bottom graphs show the displacement time histories of the 3rd floor (roof) and 2nd floor, respectively.
== References ==
# Ibarra, L. F., and Krawinkler, H. (2005). “Global collapse of frame structures under seismic excitations,” Technical Report 152, The John A. Blume Earthquake Engineering Research Center, Department of Civil Engineering, Stanford University, Stanford, CA. [electronic version: https://blume.stanford.edu/tech_reports]
# Ibarra, L. F., Medina, R. A., and Krawinkler, H. (2005). “Hysteretic models that incorporate strength and stiffness deterioration,” Earthquake Engineering and Structural Dynamics, Vol. 34, 12, pp. 1489-1511.
# Lignos, D. G., and Krawinkler, H. (2009). “Sidesway Collapse of Deteriorating Structural Systems under Seismic Excitations,” Technical Report 172, The John A. Blume Earthquake Engineering Research Center, Department of Civil Engineering, Stanford University, Stanford, CA.
# Lignos, D. G., and Krawinkler, H. (2010). “Deterioration Modeling of Steel Beams and Columns in Support to Collapse Prediction of Steel Moment Frames,” ASCE, Journal of Structural Engineering (under review).
# Zareian, F. and Medina, R. A. (2010). “A practical method for proper modeling of structural damping in inelastic plane structural systems,” Computers & Structures, Vol. 88, 1-2, pp. 45-53.
Latest revision as of 00:38, 28 February 2011
Example posted by: Dr. Dimitrios G. Lignos (McGill University)
This example demonstrates how to conduct a calibration of a viscous damper using the maxwell model.
The files needed to analyze this structure in OpenSees are included here:
Figure 1. Schematic representation of a viscous damper.
The viscous damper is modeled with the Force-Based Beam-Column element. This element follow a Maxwell hysteretic response. An idealized schematic of the model is presented in Figure 1.
The units of the model are mm, N, and seconds.
Basic Geometry
The basic geometry of the viscous damper is defined by input variables for the length L=5000mm, and area A = 12000mm2. Two nodes are used for the geometry of the damper.
Damper Section
A square section is used to define the area A of the damper (Adamper = 12000.0mm2).
Damper Links
Force-Based Beam-Column elements are used to link the two nodes that define the geometry of the viscous damper with n=5 sections of integration and the damper section defined previously.
Constraints
Node 1 is pinned. A roller is assigned to node 2
Maxwell Material
To model the viscous damper the Maxwell is used. The input parameters that are selected for the damper example are as follows:
Axial Stiffness K = 500000.0 N/mm, Viscous Coefficient C=100000.0 N(s/mm)0.30 and exponent a=0.30.
Loading
The viscous damper is loaded with a displacement history that is called Damper.txt in the axial loading direction. A linear pattern is selected and the node of application is Node 2.
The stressStrain recorder to track the stress and strain history of the viscous damper in section 3.
Analysis
In order to calibrate the viscous damper a multiple support excitation option is selected with application of displacement as the imposed motion. This is necessary since an integration step dt should be specified for integration of strain histories for each time step.
Results
Comparison of OpenSees Model with MATLAB based Maxwell model
Figure 2. Viscous Damper Response and Comparison With MATLAB based Script
The force - displacement relationship from the maxwell damper are shown in Figure 2. In order to obtain the displacement of the damper we multiply the strain in section 3 of the damper times the length L=5000mm. A comparison of the response with a MATLAB based script is also shown in the same figure. Results are identical.
