Pushover N2 Method

Contents Introduction of N2 method

The N2 Pushover Method ofEurocode8-1:2004 and NTC 2008

Normalization of mode shape Determination of lateral load pattern Determination of time period, yield force and yield displacement Determination of target displacement Verification

Program Version Revision Datehttp://en.midasuser.com

Gen 2011 (v1.1) Oct 18th 2010Midas Information Technology Co., Ltd.

Step

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Introduction

The N2 Pushover Method of Eurocode8-1:2004 and NTC2008

In Gen 2011, N2 method has been completely implemented for the determination of the target displacement and capacity for the EC8/Masonry and NTC 2008.

Pushover analysis is a non linear analysis carried out under conditions of constant gravity loads and monotonically increasing horizontal loads. It may be applied to verify the structural performance of newly designed and the existing buildings for the following purposes: (a) to verify the over strength ratio values. (b) to estimate the expected plastic mechanism and the distribution of damage. (c) to assess the structural performance of existing or retrofitted buildings. (d) as an alternative to the design based on linear analysis which uses the behavior factor q. In that case the target displacement should be used as the basis for the design.

Prominent Features: The N2 method combines the advantages of the visual representation of the capacity spectrum method with the physical basis of inelastic demand spectra. The N2 method leads to a transparent transformation from a multi degree of freedom (MDOF) model to an equivalent single degree of freedom model (SDOF). The N2 method takes into consider the period range of the structure for the determination of the

target displacement.

Procedure of the N2 method: Normalization of mode shape Generation of lateral load pattern for pushover analysis MDOF system to SDOF system

Determination of idealized elasto perfectly plastic force displacement relationship. Determination of time period and yield force of the SDOF system Determination of target displacement of the MDOF system corresponding to the response spectrum.

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Step

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New Load Pattern


For the implementation of the N2 method a new load pattern Normalized Mode Shape * Mass has been added to the Pushover Load Cases.



Step

02

Normalization of Mode Shape


The mode shape for which the pushover analysis has to be done is obtained by the eigenvalue analysis of the structure. In midas Gen normalization is done on the basis of the master node ( user defined) and hence after normalization the mode shape becomes such that the at master node becomes 1. Pushover analysis is stopped when the displacement of the master node reaches the specified maximum displacement. The lateral loads are applied on the centre of mass and the lateral load pattern is obtained by the normalized values of centre of mass. Normalization example has been provided for the following 3D structure. The obtained values of for the mode shape 1 for the master node and the centre of mass of the stories are specified in the table along with the normalized values. Centre of Mass Roof 3F

Master node

2F 1F

Normalization of Mode ShapeStory Roof (Master Node) Roof (Centre of Mass) Mode Shape .0621 .0346 .0326 .0212 .0121 Normalized Mode Shape 1 .557

3F (Centre of Mass)2F (Centre of Mass) 1F (Centre of Mass)

.5249.3413 .1948

*It is recommended that the master node be the node at the centre of mass . Otherwise in some cases, the normalized of the roof may not be 1 as in the table.



Step

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Normalization of Mode Shape


Example of normalization for a 2D model:

Defined Master Node .87 .65 .54 .21 1 .74 .63 .24

Model

Mode Shape

Normalized Mode Shape

Normalization of Mode ShapeStory Roof (Master Node) Roof (Centre of Mass) 3F (Centre of Mass) 2F (Centre of Mass) 1F (Centre of Mass) Mode Shape .87 .87 .65 .54 .21 Normalized Mode Shape 1 1 .74 .63 .24



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Lateral Load Pattern


The Lateral Load Pattern is required to generate the Pushover Curve for the SDOF model. Lateral loads are applied at each story ( mass centre) and the roof displacement is obtained . The load is applied to till the point when the displacement of master node reaches the maximum Displacement. The lateral load pattern is obtained by multiplying the mass of the story with the respective normalized and then normalizing it with the force obtained for roof story.

1 m4 = 400 m3 = 300 .74 .63 .24 1 .55

m2 = 200 m1 = 100

.31.06

Model

Normalized Mode Shape

Load Pattern

Lateral Load PatternStory Roof (Centre of Mass) 3F (Centre of Mass) 2F (Centre of Mass) 1F (Centre of Mass) Story Mass 400 300 200 100 Normalized Mode Shape 1 .74 .63 .24 Calculation (1X400)/(1X400) (.74X300)/(1X400) (.63X200)/(1X400) (.24X100)/(1X400) Load Factor 1 .55 .31 .06



Step

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Time Period and Yield Force


The yield force and the yield displacement is obtained from the bilinearized pushover curve . The F* and the d* values are obtained from the Pushover Curve generated for the SDOF model. They yield force fy* ,which represents the ultimate strength of the idealized system, is equal to the base shear force at the formation of the plastic mechanism. The initial stiffness of the idealized system is determined in such a way that the areas under the actual and the idealized force deformation curves are equal. Based on this assumption, the yield displacement (dy*) of the idealized SDOF system is given by:

where Em* is the actual deformation energy up to the formation of the plastic mechanism

dy*

AnnexB of EN 1998-2004

midas Gen - Determination of dy*

After determining the values of Fy* and dy*, the time period is calculated as:



Step

04

Determination of Transformation factor - Gamma


Transformation factor Gamma is calculated based on the following two methods: o 2D Behavior (EC8-1:2004 Annex B) o 3D Behavior 2D Behavior is based on EC8 -1 :2004 Annex B and determines the value of gamma by only considering the direction in which pushover analysis is performed . Hence the value of gamma is :

3D Behavior determines the gamma by considering lateral deflection in all the possible directions :



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Target Displacement


The target displacement of the structure for the MDOF model is obtained by first determining the target displacement for the SDOF mode and then multiplying it with . where

The target displacement for the SDOF model is obtained by first determining det*

Where: Se(T*) is the elastic acceleration response spectrum at the period T*

The target displacement is determined by determining whether the structure is in the short period range or the medium and long period range. The range is determined by comparing the values of T* and Tc where T* is the time period and the Tc is the corner point between the short and the medium period range in the response spectrum.

Short Period Range

Medium and Long Period Range



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Target Displacement


(a). T* < Tc (Short Period Range) (i) If Fy*/m*> Se(T*) , the response is elastic and thus:

(ii) If Fy*/m* Tc (Medium or long Period Range)

Target displacement for the MDOF model is obtained by multiplying the value obtained for the SDOF model by



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Target Displacement calculation for NTC 2008

Demand and Capacity

Text file containing the values of m*, T*, Fy* and target displacement



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Safety Verification for NTC 2008


Safety Verification is divided primarily into two different groups: Global Verification Local Verification Capacity for Global Verification The capacity for SLD is obtained as the displacement of the master node when the interstory drift becomes .005h , where h is the story height.

The capacity for SLO is 2/3 the capacity for SLDGlobal verification is performed by comparing target displacements (as obtained in the previous pages) and interstory drift capacity for both SLD and SLO limit states. For example: The following table shows the story drift table for a structure for different stories.

Story Drift Table for 1st floor

Story Drift Table for 2nd floor

Story Drift Table for 3rd floor

Story Drift Table for 4th floor It can be realized that the lowest step is obtained as the step 43 for which the inter story drift in 1st and second floor reaches more that .5%. Hence the capacity for global verification will be obtained by determining the displacement of master node at 43rd step which is the capacity for SLD, capacity for SLO = 2/3 SLDhttp://en.midasuser.com Midas Information Technology Co., Ltd.

Step

01

Capacity Determination NTC 2008


Capacity Determination for Local Verification The capacity is determined as shown in the table.

The demand (force, moment or rotation) for a member is obtained for the pushover step which is nearest to the target displacement in the spectral demand curve.

Eqn A.1

The local verification is done automatically in the program by generating the safety verification table. When the demand exceeds capacity the table shows NG for the particular element.



Step

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Safety Verification Table


Capacity values for different limit states can now be seen along with the user defined step The limit state for capacity can be defined along with the user defined step.

Safety Verification Table



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Pushover N2 Method