View source for File:Pushover & Dynamic Analyses of 2-Story Moment Frame with Panel Zones and RBS.doc

Example posted by:  <span style="color:blue"> Laura Eads, Stanford University</span>

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This example is an extension of the [[Pushover Analysis of 2-Story Moment Frame|Pushover Analysis of 2-Story Moment Frame]] and [[Dynamic Analysis of 2-Story Moment Frame| Dynamic Analysis of 2-Story Moment Frame]] examples which illustrates the explicit modeling of shear distortions in panel zones and uses reduced beam sections (RBS) which are offset from the panel zones.  Both pushover and dynamic analyses are performed in this example.  The structure is the same 2-story, 1-bay steel moment resisting frame used in the other examples where the nonlinear behavior is represented using the concentrated plasticity concept with rotational springs.  The rotational behavior of the plastic regions follows a bilinear hysteretic response based on the Modified Ibarra Krawinkler Deterioration Model (Ibarra et al. 2005, Lignos and Krawinkler 2009, 2010).  For this example, all modes of cyclic deterioration are neglected.  A leaning column carrying gravity loads is linked to the frame to simulate P-Delta effects.

The files needed to analyze this structure in OpenSees are included here:
* The main file:  [[MRF_2Story_Concentrated_PanelZone.tcl|MRF_2Story_Concentrated_PanelZone.tcl]] 
Supporting procedure files
* [[DisplayModel2D.tcl|DisplayModel2D.tcl]]  – displays a 2D perspective of the model
* [[DisplayPlane.tcl|DisplayPlane.tcl]]  – displays a plane in the model
* [[rotSpring2DModIKModel.tcl|rotSpring2DModIKModel.tcl]] – creates a bilinear rotational spring that follows the Modified Ibarra Krawinkler Deterioration Model (used in the concentrated model)
* [[rotSect2DModIKModel.tcl|rotSect2DModIKModel.tcl]] – creates a section with bilinear rotational response that follows the Modified Ibarra Krawinkler Deterioration Model and an elastic axial response (used in the distributed model)
* [[rotLeaningCol.tcl|rotLeaningCol.tcl]] – creates a low-stiffness rotational spring used in a leaning column
* [[rotPanelZone2D.tcl| rotPanelZone2D.tcl]] – creates a rotational spring to capture panel zone shear distortions
* [[elemPanelZone2D.tcl|elemPanelZone2D.tcl]] –creates eight elastic elements which form a rectangular panel zone
All files are available in a compressed format here:  [[Media:MRF_PanelZone_example.zip| MRF_PanelZone_example.zip]]

The rest of this example describes the model and presents the analysis results.  The OpenSees model is also compared to an equivalent model built and analyzed using the commercial program SAP2000 (http://www.csiberkeley.com/products_SAP.html). 

== Model Description ==

[[File: Eads_conc_PanelZone_sketch.PNG|frame|Figure 1.  Schematic representation of concentrated plasticity OpenSees model including explicit modeling of the panel zones.  Element number labels and [node number] labels are also shown.  A detail view of a typical panel zone is presented in Figure 2.  Note:  The springs are zeroLength elements, but their sizes are greatly exaggerated in this figure for clarity.]]

[[File: Eads_PanelZone_NumberingConventions.PNG|frame|Figure 2.  Schematic representation of a typical panel zone with element number labels and [node number] labels shown.  Note:  The spring is a zeroLength element, but its size is greatly exaggerated in this figure for clarity.]]

The 2-story, 1-bay steel moment resisting frame is modeled with [[elastic Beam Column Element|elastic beam-column 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.  The panel zones are explicitly modeled with eight [[elastic Beam Column Element|elastic beam-column elements]] and one [[zeroLength Element|zeroLength element]] which serves as rotational spring to represent shear distortions in the panel zone.  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.

A detailed description of this model is provided in [[Pushover Analysis of 2-Story Moment Frame|Pushover Analysis of 2-Story Moment Frame]].  This section merely highlights the important differences in this model, namely the inclusion of panel zones and reduced beam sections (RBS) which are offset from the panel zones.

The units of the model are kips, inches, and seconds.

=== Panel Zones ===
The panel zone is the joint region where beams and columns intersect.  In this model it consists of the rectangular area of the column web that lies between the flanges of the connecting beam(s).  The panel zone deforms primarily in shear due to the opposing moments in the beams and columns.  To capture these deformations, the panel zone is explicitly modeled using the approach of Gupta and Krawinkler (1999) as a rectangle composed of eight very stiff [[elastic Beam Column Element|elastic beam-column elements]] with one [[zeroLength Element|zeroLength element]] which serves as rotational spring to represent shear distortions in the panel zone (see Figure 2).  At the three corners of the panel zone without a spring, the elements are joined by a simple pin connection which is achieved by using the [[EqualDOF Command|equalDOF command]] to constrain both translational degrees of freedom.   The eight [[elastic Beam Column Element|elastic beam-column elements]] each have an area of 1,000.0 in<sup>2</sup> and a moment of inertia equal to 10,000.0 in<sup>4</sup> in order to give them high axial and flexural stiffness, respectively.  The elements are defined in [[elemPanelZone2D.tcl|elemPanelZone2D.tcl]].  The spring has a trilinear backbone which is created with the [[Hysteretic Material|Hysteretic material]] in [[rotPanelZone2D.tcl|rotPanelZone2D.tcl]].  This procedure also constrains the translational degrees of freedom at the corners of the panel zone.  The spring’s backbone curve is derived using the principle of virtual work applied to a deformed configuration of the panel zone (Gupta and Krawinkler 1999).

=== Reduced Beam Sections (RBS)  ===
Using an RBS which is offset from the beam-column joint ensures that the beam’s plastic hinge forms away from the column and thus protects the column’s integrity.  In this model, the decrease in moment of inertia at the RBS is neglected; however, the yield moment at the RBS is calculated based on the reduced section properties.  The plastic hinge is modeled by a rotational spring placed at the center of the RBS.  An [[elastic Beam Column Element|elastic beam-column element]] is used to connect the spring and the panel zone.  Since this element is not part of the spring-elastic element-beam subassembly described in the “Stiffness Modifications to Elastic Frame Elements” section of the [[Pushover Analysis of 2-Story Moment Frame|Pushover Analysis of 2-Story Moment Frame]] example, its moment of inertia and stiffness proportional damping coefficient are not modified by an “n” factor.

=== Application Points for Masses and Loading  ===
Since loads cannot be applied at the center of the beam-column joint, gravity loads are applied at the top node of the panel zone where it meets the column (node xy7 in Figure 2).  Both masses and lateral loads are applied at the centerline of the floor level along the right side of the panel zone (node xy05 in Figure 2).

=== Analysis ===
==== Pushover ====
The pushover analysis is identical to the analysis performed in the [[Pushover Analysis of 2-Story Moment Frame|Pushover Analysis of 2-Story Moment Frame]] example where the structure is pushed to 10% roof drift, or 32.4”.

==== Dynamic ====
The dynamic analysis is identical to the analysis performed in the [[Dynamic Analysis of 2-Story Moment Frame| Dynamic Analysis of 2-Story Moment Frame]] example where the structure is subjected to the 1994 Northridge Canoga Park record.

=== Results ===
[[File: Eads_Pushover_PZ_SAP.png|frame|Figure 3.  Pushover Curve: Comparison OpenSees & SAP2000 Models]]

The first and second mode periods of the structure obtained from an eigenvalue analysis are T<sub>1</sub> = 0.81 s and T<sub>2</sub> = 0.18 s, respectively.  These values agree with the SAP2000 model which had periods of T<sub>1</sub> = 0.81 s and T<sub>2</sub> = 0.20 s.

The periods of this OpenSees model are slightly smaller than the periods of the structure used in [[Pushover Analysis of 2-Story Moment Frame|Pushover Analysis of 2-Story Moment Frame]] which had periods of T<sub>1</sub> = 0.83 s and T<sub>2</sub> = 0.22 s.  This is expected because the including the panel zone regions makes the structure stiffer.

==== Pushover Results ====
A comparison of the pushover results from the OpenSees and SAP2000 models is shown in Figure 3.  As demonstrated by this figure, the results are nearly identical.

==== Dynamic Results ====
[[File: Eads_PZ_thist.png|frame|Figure 4.  Acceleration and Floor Displacement Histories]]

The floor displacement histories from the dynamic analysis are shown in Figure 4. 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 ==

# Gupta, A., and Krawinkler, H. (1999). "Seismic Demands for Performance Evaluation of Steel Moment Resisting Frame Structures," Technical Report 132, 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., 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).