Calibration of Maxwell Material: Difference between revisions

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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 is modeled with the [[nonlinearBeamColumn Element|nonlinearBeamColumn element]]. This element follow a [[Maxwell Material| Maxwell]] hysteretic response.  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.
 
The units of the model are mm, N, and seconds.


The units of the model are mm, kN, and seconds.


=== Basic Geometry ===
=== Basic Geometry ===
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=== Damper Section ===
=== Damper Section ===


A square section is used to define the area A of the damper (A<sub>damper</sub> = 10000.0mm<sup>2</sup>).
A square section is used to define the area A of the damper (A<sub>damper</sub> = 12000.0mm<sup>2</sup>).


=== Damper Links ===
=== Damper Links ===


[[nonlinearBeamColumn Element|nonlinearBeamColumn 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.
[[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 ===
=== Constraints ===
The frame columns are fixed at the base, and the leaning column is pinned at the base. To simulate a rigid diaphragm, the horizontal displacements of all nodes in a given floor are constrained to the leftmost beam-column joint node using the [[EqualDOF command|equalDOF]] command.
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 ===
=== Loading ===
Gravity loads are assigned to the beam-column joint nodes using the [[NodalLoad Command|nodal load command]].  Gravity loads tributary to the frame members are assigned to the frame nodes while the remaining gravity loads are applied to the leaning columns.  The gravity loads are applied as a [[Plain Pattern|plain load pattern]] with a [[Constant TimeSeries|constant time series]] since the gravity loads always act on the structure.
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.
 
In this example, lateral loads are distributed to the frame using the methodology of ASCE 7-10 (http://www.asce.org/Product.aspx?id=2147487569).  Lateral loads are applied to all the frame nodes in a given floor.  A [[Plain Pattern|plain load pattern]] with a [[Linear TimeSeries|linear time series]] is used for lateral load application so that loads increase with time.


=== 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 base shear reaction history
* 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
To record the moment and rotation histories in the springs, the [[Region Command|region command]] was used to assign all column springs to one group and all beam springs to a separate group, and the region was used as an input to the [[Element Recorder|element recorder]].
 
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 first analyzed under gravity loads before the pushover 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 pushover 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.
 
The pushover analysis is performed using a [[Displacement Control|displacement-controlled]] static analysis.  In this example, the structure was pushed to 10% roof drift, or 32.4”.  The roof node at Pier 1, node 13 in Figures 1 and 2, was chosen as the control node where the displacement was monitored.  Incremental displacement steps of 0.01” were used. This step size was used because it is small enough to capture the progression of hinge formation and generate a smooth backbone curve, but not too small that it makes the analysis time unreasonable.




== Results ==
== Results ==
=== Comparison of OpenSees Models ===
=== Comparison of OpenSees Model with MATLAB based Maxwell model ===
[[File:pushover_curve.png|frame|Figure 3.  Pushover Curve:  Comparison of OpenSees Models]]
[[File:MaxK1.png|300px|thumb|Right|Figure 2.  Viscous Damper Response and Comparison With MATLAB based Script]]
 
Theoretically, the results of the distributed plasticity model should approach those of the concentrated plasticity model as the length of the plastic hinge regions approaches zero.  Because of localized instability due to the stress-strain formulation of the [[Beam With Hinges Element|beam with hinges element]], the distributed plasticity model does not give reasonable results when the length of the plastic hinge is very small (i.e., 10e<sup>-5</sup>).  Therefore, the length of the plastic hinges was increased from 10e<sup>-5</sup> until the results of this model approached those of the concentrated plasticity model.  The plastic hinge length that led to agreement between the models was 0.4% of the frame member's total length.  The periods of the concentrated and distributed models are very close:  T<sub>1</sub> = 0.83 s (con) vs. 0.82 s (dist) and  T<sub>2</sub> = 0.22 s (con) vs. 0.21 s (dist).
 
The results of the pushover analyses from the OpenSees models are shown in Figure 3.  This figure shows the normalized base shear (weight of the structure divided by the base shear) versus the roof drift (roof displacement divided by the roof elevation).  The models are nearly identical until about 2.5% roof drift when their curves begin to diverge.  The descending branch of the the concentrated plasticity model is slightly steeper, but the two models agree reasonably well as there is less than 10% percent difference in the base shears at 10% roof drift.
 
 
== 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]
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.
# 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).

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:

Supporting procedure files

All files are available in a compressed format here: Calibration_Maxwell_example.zip

The rest of this example describes the model and shows the analysis results.

Model Description

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.

Recorders

The recorders used in this example include:

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.

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