Workshop Note on Pushover Analysis

IS-CAAD International Seminar on Computer Aided Analysis and Design

Performance Based Design and Push-over Analysis of Buildings

Workshop Note

Pushover Analysis Using ETABS and SAP2000

Asian Center For Engineering Computations and Software 1

Pushover AnalysisUsing ETABS and SAP2000

ByNaveed Anwar

Asian Center for Engineering Computations and SoftwareAsian Institute of Technology

In Association withComputers and Structures Inc., Berkeley, California, USA

June 18-19, Manila, PhilippinesFor

Association of Structural Engineers Philippines

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Acknowledgements• Some of the material presented in these

notes is based on following sources:– Class notes by Prof. Worsak Kanok-Nukulchai– Seminar notes from Computers and Structures

Incorporated, USA– Notes from various workshops conducted by

Naveed Anwar– SAP2000 User and Technical Manuals– ETABS User and Technical Manuals– ATC40, Applied Technology Council, USA– FEMA-273, Federal Emergency Management

Agency, USA



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Objectives• Introduce the basic Modeling and Analysis

Concepts• To provide an understanding of Static

Nonlinear Pushover Analysis for Seismic Performance

• To demonstrate the application of Pushover Analysis for buildings using ETABS and SAP2000 and to provide a comparison

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Modeling and Analysis



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Structural Model

EXCITATIONLoads

VibrationsSettlements

Thermal Changes

RESPONSESDisplacements

StrainsStress

Stress Resultants

STRUCTURE

pv

Structural System – Analysis Model



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Analysis of Structures

pv

∂σ∂

∂σ∂

∂σ∂

xx yy zzvxx y z

p+ + + = 0

Real Structure is governed by “Partial Differential Equations” of various order

Direct solution is only possible for:• Simple geometry• Simple Boundary• Simple Loading.

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The Need for ModelingA - Real Structure cannot be Analyzed:

It can only be “Load Tested” to determine response

B - We can only analyze a “Model” of the Structure

C - We therefore need tools to Model the Structure and to Analyze the Model



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Finite Element Method: The Analysis Tool

• Finite Element Analysis (FEA)– “A discretized solution to a

continuum problem using FEM”

• Finite Element Method (FEM)– “A numerical procedure for solving

(partial) differential equations associated with field problems, with an accuracy acceptable to engineers”

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Continuum to Discrete Model

pv

(Governed by partialdifferential equations)

CONTINUOUS MODELOF STRUCTURE

(Governed by eitherpartial or total differential equations)

DISCRETE MODELOF STRUCTURE

(Governed by algebraicequations)

3D-CONTINUM MODEL



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From Classical to FEM Solution

∂σ∂

∂σ∂

∂σ∂

xx yy zzvxx y z

p+ + + =0

σ εtvt

st

v

dV p udV p uds_ _ _

= + ∫∫∫

Assumptions

Equilibrium

Compatibility

Stress-Strain Law

(Principle of Virtual Work)

“Partial Differential Equations”

Classical

Actual Structure

Kr R=

“Algebraic Equations”

K = Stiffnessr = Response

R = Loads

FEM

Structural Model

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Simplified Structural System

F = K D

F

KD

Loads (F) Deformations (D)

Fv



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The Analysis System

EXCITATION RESPONSES

STRUCTURE

pv

• Static• Dynamic

• Elastic• Inelastic

Eight types of equilibrium equations are possible!

• Linear• Nonlinear

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The Equilibrium Equations

1. Linear-Static Elastic

2. Linear-Dynamic Elastic

3. Nonlinear - Static Elastic OR Inelastic

4. Nonlinear-Dynamic Elastic OR Inelastic

FKu =

)()()()( tFtKutuCtuM =++ &&&

)()()()()( tFtFtKutuCtuM NL =+++ &&&

FFKu NL =+



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Nonlinear-Inelastic-Dynamic AnalysisNonlinearInelasticDynamic

Linear-Inelastic-Dynamic AnalysisLinearInelasticDynamic

Nonlinear-Elastic-Dynamic AnalysisNonlinearElasticDynamic

Linear-Elastic-Dynamic AnalysisLinearElasticDynamic

Nonlinear-Inelastic-Static Analysis NonlinearInelasticStatic

Linear-Inelastic-Static AnalysisLinearInelasticStatic

Nonlinear-Elastic-Static AnalysisNonlinearElasticStatic

Linear-Elastic-Static AnalysisLinearElasticStatic

Basic Analysis TypeResponseStructureExcitation

Basic Analysis TypesPu

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Analysis TypeThe type of Analysis to be carried out depends on the Structural System

– The Type of Excitation (Loads)– The Type Structure (Material and

Geometry)– The Type Response



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Some More Solution Types• Non-linear Analysis

– P-Delta Analysis– Buckling Analysis– Static Pushover Analysis– Fast Non-Linear Analysis (FNA)– Large Displacement Analysis

• Dynamic Analysis– Free Vibration and Modal Analysis– Response Spectrum Analysis– Steady State Dynamic Analysis

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Static Vs Dynamic• Static Excitation

– When the Excitation (Load) does not vary rapidly with Time

– When the Load can be assumed to be applied “Slowly”

• Dynamic Excitation– When the Excitation varies rapidly with Time– When the “Inertial Force” becomes significant

• Most Real Excitation are Dynamic but are considered“Quasi Static”

• Most Dynamic Excitation can be converted to“Equivalent Static Loads”



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Elastic Vs Inelastic• Elastic Material

– Follows the same path during loading and unloading and returns to initial state of deformation, stress, strain etc. after removal of load/ excitation

• Inelastic Material– Does not follow the same path during loading and

unloading and may not returns to initial state of deformation, stress, strain etc. after removal of load/ excitation

• Most materials exhibit both, elastic and inelastic behavior depending upon level of loading.

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Linear Vs Nonlinear• Linearity

– The response is directly proportional to excitation– (Deflection doubles if load is doubled)

• Non-Linearity– The response is not directly proportional to

excitation– (deflection may become 4 times if load is doubled)

• Non-linear response may be produced by:– Geometric Effects (Geometric non-linearity)– Material Effects (Material non-linearity)– Both



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Elastic Vs Inelastic• Elastic Material

– Follows the same path during loading and unloading and returns to initial state of deformation, stress, strain etc. after removal of load/ excitation

• Inelastic Material– Does not follow the same path during loading and

unloading and may not returns to initial state of deformation, stress, strain etc. after removal of load/ excitation

• Most materials exhibit both, elastic and inelastic behavior depending upon level of loading.

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Linear Vs Nonlinear• Linearity

– The response is directly proportional to excitation– (Deflection doubles if load is doubled)

• Non-Linearity– The response is not directly proportional to

excitation– (deflection may become 4 times if load is doubled)

• Non-linear response may be produced by:– Geometric Effects (Geometric non-linearity)– Material Effects (Material non-linearity)– Both



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Elasticity and LinearityAc

tion

Deformation

Actio

n

Deformation

Actio

n

Deformation

Actio

n

Deformation

Linear-Elastic Linear-Inelastic

Nonlinear-Elastic Nonlinear-Inelastic

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Linear and Nonlinear

u

F

Non Linear Equilibrium

Ku = F

Ku - FNL = FFNL

FKu =)()()()( tFtKutuCtuM =++ &&&

)()()()()( tFtFtKutuCtuM NL =+++ &&&

FFKu NL =+

Nonlinear, Static and Dynamic

Linear, Static and Dynamic



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The Seven Degrees of Freedom • The General Beam

Element may have 7 degrees of freedom

• The seventh degree is Warping

• Warping is out-of plane distortion of the beam cross-section

z

y

xxu

yu

zu

xr

yr

zr

zw

Each section on a beam member can have seven Degrees Of Freedom (DOF) with respect to its local axis.

z

y

xxu

yu

zu

xr

yr

zr

zw

Each section on a beam member can have seven Degrees Of Freedom (DOF) with respect to its local axis.

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The Complete DOF Pictureuz Axial deformation Axial strain Axial stress

ux Shear deformation Shear strain Shear stress

uy Shear deformation Shear strain Shear stress

rz Torsion Shear strain Shear stress

ry Curvature Axial strain Axial stress

rx Curvature Axial strain Axial stress

wz Warping Axial strain Axial stress



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What is Stiffness ?• In structural terms, stiffness

may be defined as “Resistance to Deformation”

• So for each type of deformation, there is a corresponding stiffness

• Stiffness can be considered or evaluated at various levels

• Stiffness is also the “constant” in the Action-Deformation Relationship

uFK

FKuFu

=

=∞

For Linear Response

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The Structure Stiffness

Section Stiffness

Member Stiffness

Structure Stiffness

Material Stiffness

Cross-section Geometry

Member Geometry

Structure Geometry



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The Matrices in FEM

Element Nodal Deformations

Deformation in Element Space

Strain In Element Space

Stress in Element Space

Global Nodal DeformationsT-Matrix

Global-Local Cords.

N-MatrixShape Functions

B-MatrixStrain-Deforrmation

D-MatrixStress-Strain

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Methods of Dynamic Analysis• For Both Linear and Non-Linear Systems

– Step-by-Step Integration – Use of Mode Superposition with Eigen or Load-

Dependent Ritz Vector for FNA• For Linear Systems Only

– Transformation of frequency domain and FFT Method

– Response Spectrum Method – CQC - SRSS



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Step by Step Solution Method• Form Effective Stiffness Matrix• Solve Set of Dynamic Equilibrium Equations

for Displacement at Each Time Step• For Non-Linear Problems Calculate Member

Forces for Each Time Step and Iterate for Equilibrium – Brute Force Method

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Mode Superposition Method• Generate Orthogonal Dependent Vectors and

Frequencies • Form Uncoupled Modal Equations and Solve

Using Exact Method for Each Time Increment • Recover Nodal Displacement as a Function

of Time• Calculate Member Forces as a Function of

Time



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Load Dependent Ritz Vector

• Approximately Three Times Faster than the Calculation of Exact Eigen Vectors

• Results in Improved Accuracy using a Smaller Number of LDR Vector

• Computer Storage Requirements are Reduced

• Can be Used for Non-Linear analysis to Capture Local Static Response

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Dynamic Response of Beam

100 Pounds

10 @ 12" = 120"

Time

Force



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Pushover Analysis• One Dimensional Static Loads• No Energy Dissipation• Inertia Forces Not Considered• Defined One Failure Mode• Higher Mode Effects Neglected

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Fast Non-Linear Analysis• Evaluate LDR Vectors with Non-Linear

Elements Removed and Dummy Elements Added for Stability

• Solve All Modal Equations with Non-Linear Forces on the Right Hand Side

• Use Exact Integration within Each Time Step• Force and Energy Equilibrium are Satisfied at

Each Time Step by Iteration



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Fast Non-Linear Analysis• The FNA Method is Designed for Static and

Dynamic Analysis of Non-Linear Structures with a Limited Number of Pre-Defined Non-Linear Elements

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Base Isolation



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Building ImpactPu

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Dampers



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Gaps and JointsPu

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Hinges



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DampersPu

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Linear Viscous Damping• Does not Exist in Normal Structures and

Foundations• 5 or 10 Percent modal Damping Values are

Often Used to Justify Energy Dissipation Due to Non-Linear Effects

• If Energy Dissipation Devices are Used Then 1 Percent Modal Damping should be Used for the Elastic Part of the Structure



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Non-Linear Equilibrium Equations

• WhereFn= The Global Node Loads due to the Forces in the Non-Linear Elements

FFKuCvMa N =+++

NFFKuCvMa −=++

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Non-Linear Equilibrium Equations

• WhereKe= The Effective Linear Stiffness of the Non-Linear Element are of Arbitrary Values for Zero Damping

[ ] ukFFukKCvMa ENE +−=+++



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UpliftPu

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Non Linear Static Analysis



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Structure Types• Cable Structures

• Cable Nets• Cable Stayed

• Bar Structures• 2D/3D Trusses• 2D/3D Frames, Grids

• Surface Structures• Plate, Shell• In-Plane, Plane Stress

• Solid Structures

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Global Modeling of Structural Geometry

(b) Solid Model (c) 3D Plate-Frame (d) 3D Frame

(a) Real Structure

(e) 2D Frame

Fig. 1 Various Ways to Model a Real Struture

(f) Grid-Plate



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Some Sample Finite Elements

Truss and Beam Elements (1D,2D,3D)

Plane Stress, Plane Strain, Axisymmetric, Plate and Shell Elements (2D,3D)

Brick Elements

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The Modal Analysis



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The Modal Analysis• The modal analysis determines the inherent natural

frequencies of vibration• Each natural frequency is related to a time period

and a mode shape• Time Period is the time it takes to complete one

cycle of vibration• The Mode Shape is normalized deformation pattern• The number of Modes is typically equal to the

number of Degrees of Freedom• The Time Period and Mode Shapes are inherent

properties of the structure and do not depend on the applied loads

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Free Vibration Analysis• Definition

– Natural vibration of a structure released from initial condition and subjected to no external load or damping

• Main governing equation -Eigenvalue Problem

• Solution gives – Natural Frequencies – Associated mode shapes– An insight into the dynamic behavior and response of the structure

[ ] [ ] [ ] { } { }tttt

PuKucuM =+⎭⎬⎫

⎩⎨⎧+

⎭⎬⎫

⎩⎨⎧ •••



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The Modal Analysis

• The Modal Analysis should be run before applying loads any other analysis to check the model and to understand the response of the structure

• Modal analysis is precursor to most types of analysis including Response Spectrum, Time History, Push-over analysis etc.

• Modal analysis is a useful tool even if full Dynamic Analysis is not performed

• Modal analysis easy to run and is a fun to watch the animations

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Application of Modal Analysis

• The Time Period and Mode Shapes, together with animation immediately exhibit the strengths and weaknesses of the structure

• Modal analysis can be used to check the accuracy of the structural model– The Time Period should be within reasonable

range, (Ex: 0.1 x number of stories seconds) – The disconnected members are identified– Local modes are identified that may need

suppression



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Application of Modal Analysis

• The symmetry of the structure can be determined– For doubly symmetrical buildings, generally the

first two modes are translational and third mode is rotational

– If first mode is rotational, the structural is un-symmetrical

• The resonance with the applied loads or excitation can be avoided– The natural frequency of the structure should not

be close to excitation frequency

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Model Creation Tools• Defining Individual Nodes and Elements• Using Graphical Modeling Tools• Using Numerical Generation• Using Mathematical Generation• Using Copy and Replication• Using Subdivision and Meshing • Using Geometric Extrusions• Using Parametric Structures•



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Graphic Object Modeling• Use basic Geometric Entities to create FE

Models• Simple Graphic Objects

– Point Object Represents Node– Line Object Represents 1D Elements– Area Object Represents 2D Elements– Brick Object Represents 3D Elements

• Graphic Objects can be used to represent geometry, boundary and loads

• SAP2000, ETABS and SAFE use the concept of Graphic Objects

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Modeling Objects and Finite Elements• Structural Members are representation of

actual structural components • Finite Elements are discretized

representation of Structural Members• The concept of Graphic Objects can be used

to represent both, the Structural Members as well as Finite Elements

• In ETABS, the Graphic Objects representing the Structural Members are automatically divided into Finite Elements for analysis and then back to structural members for result interpretation



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Design Methods and Concepts

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The Response and Design

Building Response

Member Response

Section Response

Material Response

Building Analysis

Member Actions

Cross-section Actions

Material Stress/Strain Load Capacity

Applied Loads

From

Loa

ds to

Stre

sses

From

Stra

ins t

o R

espo

nse



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From Serviceability to PerformancePu

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From Serviceability to Performance• Satisfying one design level does not ensure

that other design levels will be satisfied– Serviceability design only ensures that

deflections and vibrations etc. for service loads are within limits but says nothing about strength

– Strength design ensures that a certain factor of safety against overload is available within a member or a cross-section but says nothing about what happens if load exceeds design level

– Performance design ensures that structure as a whole reaches a specified demand level. Performance design can include, both service and strength design levels



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From Serviceability to Performance

• The entire response of structure or a member can be determined, in an integrated manner from the Action-Deformation Curve

A – Serviceability B – Cracking LimitC – Strength LimitD – Failure Limit

∆

P

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The Cross-section ResponseMaterial Stress-Strain Curves

Cross-section Dimensions

Capacity Interaction Surface

M-M Curve

Moment-Curvature Curves

P-M Curve

Given P value

Given Moment Direction

Given Moments Given Axial Load

•Moment for Given Curvature•Curvature for Given Moment•Yield Moment•Stiffness•Ductility

•Moment for Given Load•Load for Given Moment•Capacity Ratio

•Mx for Given My•My for Given Mx•Capacity Ratio

Perfo

rman

ce

Stre

ngth



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Original Cross-sections

Plain concrete shape Reinforced concrete section Compact Hot-rolled steel shape

Compact Built-up steel section

Reinforced concrete, composite section

Composite section

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Sections After Strengthening



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Capacity Interaction Surface

MxMy

P

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P-M and M-M Interaction Curves



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The Moment Curvature CurvePu

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Cross-section Stresses



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Strength and Performance

• In Strength Design, every member and every cross-section must satisfy strength equation

• Even if all members and sections are designed for strength, the structure may not perform well in case of overload

• In Performance Based Design, only a few members on the critical load path need to perform well for the structure to perform well

• Therefore for strengthening of structures, we may only need to strengthen members or section in the critical load path

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Members on Critical Load Path

• In Performance Based Design, only a few members on the critical load path need to perform well for the structure to perform well

• Therefore for strengthening of structures, we may only need to strengthen members or section in the critical load path



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What Effects Serviceability?

• Anything that reduces cracking– The presence of appropriate amount of

reinforcement at appropriate locations• Anything that increases stiffness

– Reasonable sizes and proportions of member cross-sections

• Anything that reduces Creep/ Shrinkage– Presence of compressive reinforcement

• Anything that improves Durability– High strength concrete– Proper cver and protection of rebars

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What Effects Strength?

• The basic Material Strength– Concrete crushing strength– Reinforcement yield strength

• The Cross-section Dimensions• The amount of Rebars• The framing conditions



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What Effects Performance?

• Performance is generally of concern for lateral loads such as earthquake and wind

• The main factor that effects performance is the Ductility of the members on the critical load path

• In frame structures, the design of the joints between columns and beams is critical

• The performance of shear walls if great importance for lateral load demands

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Ductility – Definition and Usage• Ductility can be

defined as the “ratio of deformation and a given stage to the maximum deformation capacity”

• Normally ductility is measured from the deformation at design strength to the maximum deformation at failure

Yield/ Design Strength

Load

Deformation

Dy Du

Ductility = Dy / Du



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What Effects Ductility!

• The most important factor effecting ductility of reinforced concrete cross-section is the confinement of concrete– Amount of confinement steel– Shape of confinement steel

• Other factors include:– Presence of Axial Load– Stress-strain curve of rebars– Amount of rebars in tension– Amount of rebars in compression– The shape of cross-section

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Action – Deformation Curves

• Relationship between action and corresponding deformation

• These relationships can be obtained at several levels– The Structural Level: Load - Deflection– The Member Level: Moment - Rotation– The Cross-section Level: Moment - Curvature– The Material Level : Stress-Strain

• The Action-Deformation curves show the entire response of the structure, member, cross-section or material



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How to Get Action-Deformation Curves• By actual measurements

– Apply load, measure deflection– Apply load, measure stress and strain

• By computations– Use material models, cross-section dimensions to

get Moment-Curvature Curves• By combination of measurement and

computations– Calibrate computation models with actual

measurements– Some parameters obtained by measurement and

some by computations

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The Moment Curvature Curve



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The Moment-Curvature Curve• Probably the most important action-deformation

curve for beams, columns, shear walls and consequently for building structures

• Significant information can be obtained from Moment Curvature Curve to compute:– Yield Point– Failure Point– Ductility– Stiffness– Crack Width– Rotation– Deflection– Strain

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What is Curvature

• In geometry, it is rate of change of rotation

• In structural behavior, Curvature is related to Moment

• For a cross-section undergoing flexural deformation, it can computed as the ratio of the strain to the depth of neutral axis

C

e

Curvature = e / C (radian / unit length)



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How to Read M-Phi CurvePu

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Outputs from M-Phi Curve

1 -Yield Point2 -Failure Point

u

yDuctilityϕϕ

=3 -



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φ

φ

MEI

EIM

=

=

4 - Stiffness of the Section at given M and Phi

5 - Slope of the section at given Moment

dxEIMb

a∫=θ

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dxxEIMb

a∫ ⎟

⎠⎞

⎜⎝⎛=∆

6 - Deflection of the section at given Moment

7 - Strain at given Moment

cφε =c = distance from the NA tothe point where strain is required



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yXWXW s

φε

==

8 - Crack Width at given crack spacing

9 - Crack Spacing at given crack width

yWX

WXs

φ

ε

=

=

sε

φ

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Plot M-Phi Curve

Determine curvatureat known moment

Determine Flexural Stiffness (EI)

Determine Slope

Determine Deflection

Determine Strain

Determine CrackSpacing/Width

φMEI =

dxEIMb

a∫=θ

dxxEIMb

a∫ ⎟

⎠⎞

⎜⎝⎛=∆

cφε =

XW sε=s

WXε

=

Outputs from M-Phi Curve - Summary



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Outputs from M-Phi Curve - Example

φMEI =

For M=600 Phi = 0.00006From M-Phi Diagram

EI=600x12/0.00006EI=1.2E8 k-in^2

dxEIMb

a∫=θ

=600x7.5x144/1.2E8=0.0054 rad

Slope at Mid Span

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dxxEIMb

a∫ ⎟

⎠⎞

⎜⎝⎛=∆

sε

φ

Deflection at Mid Span

=600x7.5x144x15x12/(6x1.2E8)=0.162 in

Strain in Steel

M = 600 k-ft, y=16

=0.00006x16=0.00096

cφε =



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XW sε=

s

WXε

=

Crack Width Assuming crack spacing of 18 in

=0.00096 x 18=0.01728 in

Crack SpacingAssuming crack width of 0.02 in

=0.02/0.00096=20.8 in

Specified Crack Spacing = X

y

sε

φRebar Centroid

NA

W

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M-Phi Curve and Ductility

• Effect of Axial Load• Effect of Compression Steel• Effect of Confinement Model• Effect of Confinement Shape



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Axial Load and DuctilityEffect of Axial Load on Ductility

-100

0

100

200

300

400

500

600

700

-0.0002 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 0.0014 0.0016 0.0018

C ur v a t ur e ( r a d/ i n)

Axial Load = 0

Axial Load = 0.2Pu

Axial Load = 0.4Pu

Axial Load = 0.6Pu

Axial Load = 0.8Pu

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Compression Steel and DuctilityEffects of Compression Rebars on Ductility

-100

0

100

200

300

400

500

600

700

-0.0005 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040

C ur v a t ur e ( r a d/ i n)

a: Duct il it y Rat io = 2.65

b: Duct il it y Rat io = 3.32

c: Duct il it y Rat io = 4.68

d: Duct il it y Rat io = 9.25



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Confinement Model and DuctilityEffect of Concrete Confinement Model on Ductility of Cross-Section

0

50

100

150

200

250

300

350

0 0.001 0.002 0.003 0.004 0.005 0.006

Curvature (rad/in)

Mom

ent

(kip

-ft)

Whitney Rectangle

Mander Circular Confined

Mander Pipe Filled

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Confinement Steel and DuctilityEffect of Confinement Steel Spacing on Ductility

-20

0

20

40

60

80

100

120

140

160

-0.0005 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025

Curvature (in/rad)

Mom

ent (

kip-

ft)

Spacing = 3in

Spacing = 6 in

Spacing = 12 in



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Confinement Shape and DuctilityEffect of Confinement Shape on Ductility

-50

0

50

100

150

200

250

300

350

(0.0010) 0.0000 0.0010 0.0020 0.0030 0.0040 0.0050 0.0060 0.0070

C u r v a t u r e ( r a d / i n )

M ander Rectangular Confined M ander Circular ConfinedWhitney Rectangle

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IntroducingPushover Analysis



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The Pushover Analysis

• An alternate method of analysis for carrying out the Performance Based Design

• Pushover analysis is carried out after the Linear Analysis has been done and Serviceability and Strength design has been completed

• Pushover analysis is most suitable for determining the performance, specially for lateral loads such as Earthquake or even wind

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Why Pushover Analysis• Buildings do not respond as linearly elastic

systems during strong ground shaking• Improve Understanding of Building Behavior

– - More accurate prediction of global displacement

– - More realistic prediction of earthquake demand on individual components and elements

– - More reliable identification of “bad actors”• Reduce Impact and Cost of Seismic Retrofit

– - Less conservative acceptance criteria– - Less extensive construction

• Advance the State of the Practice



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Performance Based Design - Basics• Design is based not on Ultimate

Strength but rather on Expected Performance– Basic Ultimate Strength does not tell us

what will be performance of the structure at Ultimate Capacity

• Performance Based Design Levels– Fully Operational– Operational– Life Safe– Near Collapse– Collapse

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Pushover Spectrum



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Pushover Demand CurvesPu

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Earthquake Push on Building



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The Pushover CurvePu

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Pushover Capacity Curves



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Non-linearity in Pushover• Material nonlinearity at discrete, user-defined hinges

in frame/line elements.1. Material nonlinearity in the link elements.

• Gap (compression only), hook (tension only), uniaxialplasticity base isolators (biaxial plasticity and biaxial friction/pendulum)..

2. Geometric nonlinearity in all elements.• Only P-delta effects• P-delta effects plus large displacements

3. Staged (sequential) construction.• Members can be added or removed in a sequence of stages

during each analysis case.



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Important Considerations• Nonlinear analysis takes time and patience• Each nonlinear problem is different• Start simple and build up gradually.• Run linear static loads and modal analysis

first• Add hinges gradually beginning with the

areas where you expect the most non-linearity.

• Perform initial analyses without geometric non-linearity. Add P-delta effects, and large deformations, much later.

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Important Considerations• Mathematically, static nonlinear analysis

does not always guarantee a unique solution. • Small changes in properties or loading can

cause large changes in nonlinear response.• It is Important to consider many different

loading cases, and sensitivity studies on the effect of varying the properties of the structure

• Nonlinear analysis takes time and patience. Don’t Rush it or Push to Hard



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Procedure for Pushover Analysis• Create a model just like for any other analysis.• Define the static load cases, if any, needed for use in

the static nonlinear analysis (Define > Static Load Cases).

• Define any other static and dynamic analysis cases that may be needed for steel or concrete design of frame elements.

• Define hinge properties, if any (Define > Frame Nonlinear Hinge Properties).

• Assign hinge properties, if any, to frame/line elements (Assign > Frame/Line > Frame Nonlinear Hinges).

• Define nonlinear link properties, if any (Define > Link Properties).

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Procedure for Pushover Analysis• Assign link properties, if any, to frame/line elements

(Assign > Frame/Line > Link Properties).• Run the basic linear and dynamic analyses (Analyze

> Run).• Perform concrete design/steel design so that

reinforcing steel/ section is determined for concrete/steel hinge if properties are based on default values to be computed by the program.

• For staged construction, define groups that represent the various completed stages of construction.

• Define the static nonlinear load cases (Define > Static Nonlinear/Pushover Cases).



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Procedure for Pushover Analysis• Run the static nonlinear analysis (Analyze >

Run Static Nonlinear Analysis).• Review the static nonlinear results (Display >

Show Static Pushover Curve), (Display > Show Deformed Shape), (Display > Show Member Forces/Stress Diagram), and (File > Print Tables > Analysis Output).

• Perform any design checks that utilize static nonlinear cases.

• Revise the model as necessary and repeat.

Performance Based Designand

Pushover AnalysisTechnical Background

By:Iqbal Suharwardy, PhD, S.E

Director Development

Computers and Structures Inc., Berkeley, USA



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Performance Check for Structures

• Purpose– How will a structure perform when subjected to a

give level of earthquake?

• Definition of Structural Performance• Definition of Earthquake Level• Determination of performance level

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Performance Check for Structures

• Process– Recently released guidelines for Seismic

Rehabilitation of Buildings:• ATC-40• ATC-33 (FEMA 273 and 274)

– SEAOC Vision 2000 Framework



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Types of Performance Checks• Linear Static Analysis• Linear Dynamic Analysis• Non Linear Static Analysis (Pushover

Analysis)• Non Linear Dynamic Analysis

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Performance Check Using Pushover



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Steps in Performance Check• Construct Pushover Curve• Select Earthquake Level to check• Select Performance Level to check• Select acceptance criteria for each

Performance Level• Verify Acceptance

– ATC-40 Method– ATC-33 Method

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Constructing Pushover Curve• Define Structural Model

– Elements – Strength-Deformation properties

• Define Loads– Gravity – Lateral Load Patterns

• Select Control Displacements or Drifts • Perform Pushover Analysis



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Pushover Modeling (Elements)• Types

– Truss – Yielding and Buckling– 3D Beam – Major direction Flexural and Shear Hinging– 3D Column – P-M-M Interaction and shear Hinging– Panel Zone – Shear Yielding– In-Fill Panel – Shear Failure– Shear Wall – P-M-Shear Interaction!– Spring – for foundation modeling

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Pushover Modeling (Properties)



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Pushover Modeling (Beam Element)Pu

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Pushover Modeling (Column Element)



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Pushover Modeling (Column Element)Pu

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Pushover Modeling (Loads)• Start with Gravity Loads

– Dead Load– Same Portion of Live Load

• Select Lateral Load Patterns– Uniform– Code Static Lateral Load Distribution– First Mode– Combination of Modes



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Pushover Analysis (Control)• Force Controlled Analysis• Deformation Controlled Analysis

– Roof Displacement– Generalized Displacement Definitions

• Story Drift

• Limit of Analysis – Instability – Loss of Gravity Load Carry Capacity – Excessive Distortions

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Pushover Analysis (Solution Schemes)

• Event by Event Strategies – Manual

• Newton-Raphson Type Strategies– Constant Stiffness iteration– Tangent Stiffness iteraton

• Problem of Degradation of Strength• Ritz Method (Reduced Space) Strategies



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Pushover Analysis (Results)Pu

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Use of Pushover Curve (ATC-40)

• Construct Capacity Spectrum• Estimation of Equivalent Damping• Determine Demand Spectrum• Determine Performance Point• Verify Acceptance



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Use of Pushover Curve (ATC-40)Pu

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CECO

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Use of Pushover Curve (FEMA-273)

(Displacement Coefficient Method)

• Estimate Target Displacement • Verify Acceptance

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• Estimation of Target Displacement– Estimate effective elastic stiffness , Ke– Estimate post yield stiffness, Ks– Estimate effective fundamental period, Te– Calculate target roof displacement



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• Estimation of Target Displacement– Co, Relates spectral to roof displacement– C1, Modifier for inelastic displacement– C2, Modifier for hysteresis loop shape– C3, Modifier for second order effects

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SAP2000/ETABS Pushover Options

• Full 3D implementation• Single Model for

– Linear Static Analysis– Linear Response Spectrum Analysis– Linear Time History Analysis– Non Linear Time History Analysis– Non Linear Static Pushover Analysis– Steel and Concrete Design



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SAP2000/ETABS Pushover Options• Generally Follows ATC-40 and FEMA-273• Available Pushover Element Types

– Truss – Yielding and Buckling– 3D Beam – Major direction Flexural and Shear Hinging– 3D Column – P-M-M Interaction and shear Hinging– Shell, Solids, etc (Considered Linear)– Panel Zone – (later)– Shear Wall – (Later)– Non-Linear Spring – (Later)

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SAP2000/ETABS Pushover OptionsPu

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SAP2000/ETABS Pushover Options• Strength – Deformation and P-M-M curves

can be calculated by program for:– Steel beams (FEMA-273)– Steel columns (FEMA-273)– Shear Hinges in EBF Links (FEMA-273)

– Concrete Beams (ATC-40)– Concrete Columns (ATC-40)– Shear hinge in Coupling Beams (ATC-40)



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• Gravity Load Analysis– Nodal Loads– Element Loads– Load Controlled Analysis

• Pushover Analysis– Starts from Gravity loads– Nodal Load Patterns (User, Modal, Mass)– Multi-Step Displacement or Drift Controlled

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• Available Results for each step of Loading– Base Shear– Element Forces– Section Forces– Joint Displacement– Drifts– Element hinge Deformations– Limit Points reached



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• Pushover Curve Post-Processing (ACT-40)– Conversion to Capacity Spectrum– Calculation of Effective Period (per step)– Calculation of Effective Damping (per step)– Calculation of Demand Spectrum (per step)– Location of Performance Point– Limit Points (acceptable criteria) reached

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SAP2000/ETABS Pushover Options• Visual Display for Each Step

– Deformed Shape– Member Force Diagrams– Hinge Locations and Stages

• Graphs– Base Shear VS Roof Displacement – Capacity Curves– Demand Curves– Demand Spectra at different Damping – Effective Period Lines



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Examples

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Example 1



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Base Shear Vs DisplacementPu

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Capacity Spectrum



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Example 2

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Example 2

Measured Axial Displacement at Joint 2(in)

1000

2100

1700

0.1 0.6 0.8

Desired Behavior



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Find Column E• Determine Column E to

give Appropriate Initial Stiffness:

Column

= (1700 *12*12)/(24*24*0.1)= 4250 Ksi

∆=

APLE

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Find Column Deflection• Determine Elastic Column

Lengthening when loading from 1700 to 2100 K:

Column

= [(2100-1700) *12*12)]/(24*24*4250)= 0.0235 in AE

PL=∆



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Find Column Deflection• Determine Elastic Column

Lengthening when loading from 2100 to 1000 K:

Column

= [(2100-1000) *12*12)]/(24*24*4250)= 0.0647 in AE

PL=∆

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Find Column Deflection• Determine Elastic

Column Lengthening when loading from 1000 to 0 K:

Column

= 1000 *12*12)/(24*24*4250)= 0.0588 in

AEPL

=∆



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Find Hinge PropertiesPu

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Hinge Properties



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Pushover CurvePu

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Example 3



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Example 3Pu

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Example 3



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With W12x190 BracePu

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With W8x10 Brace



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Conversion to ADRS SpectraATC-40

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Response Spectrum Conversion

• Acceleration-Displacement Response Spectra (ADRS)

• Every Point on a Response Spectrum curve has a unique– Spectral Acceleration, Sa

– Spectral Velocity, Sv

– Spectral Displacement, Sd

– Time, T



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Response Spectrum Conversion

• For Each value or Sai and Ti determine the value of Sdi using the equation

• Spectral Acceleration and Displacement at period Ti are given by

gSTS aii

di 2

2

4π=

vi

ai ST

gS π2=

vi

di STSπ2

=

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Capacity Spectrum Conversion• Capacity Spectrum from Capacity or

Pushover Curve• Point by Point conversion to first mode

spectral coordinates• on capacity curves are converted

to corresponding Sai and Sdi on capacity spectrum using:roofi andV ∆

1αW

VS iai = ( )roof

roofdi PF

S,11 φ×

∆=



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Moment Hinge PropertiesUsing M-Fi Curve

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Procedure• Plot M-Fi curve for cross-section• Estimate EI value from M-Fi Curve using the

following equation

• Calculate Rotations from Curvature using:φ

φ

MEI

EIM

=

=

dxEIMb

a∫=θ



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Example• Reinforced Concrete

Beam-Column Cross-Section

• 24”x24” • Reinforced with 12 #9

bars• Length is 12 ft

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Example

0.00028

370



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Example

• So EI = 370/0.00028 = 1321428.6

• So = 0.00336 rad• Find for other Moment Values and input in

Hinge Property

φMEI =

dxEIMb

a∫=θ

θθ

pIEIM

=θ

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Considerations • Keep moment Constant over hinge length

when integrating or integrate over the whole member length with actual moment diagram

• Only one value of EI at Yield is sufficient

• Ip = h/2



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Comparisons of SAP2000 and ETABS

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SAP2000 vs ETABS• SAP2000

– General Purpose FEA Software

– Classic Finite Element Software

– Steel, and Concrete Frame Element Design

– Shear Wall Design Not Supported

– Fewer Automated Meshing Options

– Does not Support Composite Design

• ETABS– Specialized FEA Software

for Building analysis and design

– Fully Object based Modeling and Design

– Steel, concrete, composite Frame Element design

– Supports Shear wall design

– Full and practical auto meshing options

– Supports Composite Design



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– General output related to nodes and elements is reported

– Professional Report

– Powerful load cases, combinations, envelopes, multiple case, etc.

– Cables, Dampers, and NL Links and Hinges

• ETABS– Floor wise representation of

results such as story drift, floor mass participation, story shear, etc.

– General Report (text files)

– Relatively less ability to handle load combinations

– Only Nonlinear links and Hinges

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– Supports Solid Elements

– Relatively low versatility for defining and editing grid systems

• ETABS– Does not support solid

elements

– Powerful grid system definition and editing



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ETABS PushoverPu

shov

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ETABS Pushover



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ETABS PushoverPu

shov

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SAP2000 Pushover



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SAP2000 PushoverPu

shov

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IT

SAP2000 Pushover



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SAP2000 PushoverPu

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IT

SAP2000 Pushover



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Documents

Workshop Note on Pushover Analysis