Static Pushover Analysis
Performance Based Design
Modeling for Pushover Analysis
Use of the Pushover Curve
M. Iqbal Suharwardy
Computers and Structures, Inc.
Static Pushover Analysis for Seismic Design
March 22, 1999
Performance Check of Structures

Purpose
How will a structure perform when subjected to
a given level of earthquake?
– Definition of structural performance
– Definition of earthquake level
– Determination of performance level
Performance Check of Structures

Process
Recently released guidelines for Seismic
Rehabilitation of Buildings:
– ATC-40
– FEMA 273 (ATC-33)
Types of Performance Checks




Linear Static Analysis
Linear Dynamic Analysis
Nonlinear Static Analysis
(Pushover Analysis)
Nonlinear Dynamic Analysis
Performance Check Using Pushover
Force Measure
Expected Performance Point
for given Earthquake
Performance Limits
(IO, LS, CP)
Deformation Measure
Goal is to predict peak response of building
and components for a given earthquake
Why Do Pushover Analysis?

Design Earthquakes cause nonlinear
behavior

Better understand building behavior
- Identify weak elements
- Realistic prediction of element demands

Less conservative acceptance criteria can be
used with consequences understood
Steps in Performance Check

Construct Pushover curve

Select earthquake level(s) to check

Select performance level(s) to check

Select acceptance criteria for each
performance level

Verify acceptance
Capacity Spectrum Method (ATC-40)
Displacement Coefficient Method (FEMA 273)
Constructing Pushover Curve

Define Structural Model
Elements (components)
Strength - deformation properties

Define Loads
Gravity
Lateral load pattern


Select Control Displacements or Drifts
Perform Pushover Analysis
Pushover Modeling
Definition of Structural Model
3D or 2D
Primary and Secondary Elements (components)
Non structural Elements
Foundation flexibility
P-Delta effects
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
Pushover Modeling (Properties)
Force-Deformation Relationship
Force
C
B
A
D
Deformation
E
Pushover Modeling (Properties)
Force
Force-Deformation (Back bone Curve)
Deformation
Pushover Modeling (Beam Element)
Three dimensional Beam Element
Flexible
Connection
Span Loads
Shear Hinge
Plastic Hinge
Rigid Zone
Pushover Modeling (Column Element)
Three dimensional Column Element
Shear Hinge
Plastic Hinge
Rigid Zone
Pushover Modeling (Column Element)
Axial Load - Moment Interaction (Concrete)
P
M
Pushover Modeling (Column Element)
Axial Load - Moment Interaction (Steel)
M CE  1.18 Z Fye 1  P / Pye 
Pushover Modeling (Loads)

Start with Gravity Loads
Dead Load
Some portion of Live Load

Select Lateral Load Pattern
Lateral Load Patterns (Vertical Distribution)
Lateral Load Horizontal Distribution
Torsional Effects
Orthogonal Effects
Pushover Modeling (Loads)
Lateral Load Patterns (Vertical Distribution)
Uniform
Code Lateral
Mode 1
Pushover Analysis (Control)

Force controlled analysis

Deformation controlled analysis
Roof Displacement
Generalized Displacement Definitions

Limit of analysis
Instability - loss of gravity load carrying capacity
Excessive distortions
Pushover Analysis (Solution Schemes)

Event by Event Strategies
Manual

Newton-Raphson Type Strategies
Constant stiffness iterations
Tangent stiffness iterations


Problem of degradation of strength
Ritz Modes (Reduced Space) Strategies
Pushover Analysis (Solution Schemes)
Base Shear
Event by Event Strategy
Roof Displacement
Pushover Analysis (Solution Schemes)
Base Shear
Problem of Degradation of Strength
Roof Displacement
Force Measure
Pushover Analysis (Results)
Deformation Measure
Pushover Analysis (Results)
Use of Pushover Curve


Capacity Spectrum Method
- detailed in ATC-40
- and as alternate method in FEMA-273
Displacement Coefficient Method
- detailed in FEMA-273
Use of Pushover Curve (ATC-40)





Construct Capacity Spectrum
Estimate Equivalent Damping
Determine Demand Spectrum
Determine Performance Point
Verify Acceptance
Use of Pushover Curve (ATC-40)
Base Shear
Spectral Acceleration
Constructing Capacity Spectrum
Roof Displacement
Spectral Displacement
Use of Pushover Curve (ATC-40)
Constructing Capacity Spectrum
The displaced shape at any point
on the pushover curve is used to
obtain an equivalent SDOF
system.
a is the mass participation and
relates the base shears
MDOF
PF is the participation factor and
relates the roof displacement to
Equivalent SDOF the SDOF displacement
Use of Pushover Curve (ATC-40)
Spectral
Acceleration
Constructing Capacity Spectrum
S a  V / W  / a1

S d   roof / PF1 * 1 ,roof
Spectral Displacement

Use of Pushover Curve (ATC-40)
Spectral
Acceleration
Estimation of Equivalent Viscous Damping
 eq   0  0.05
 0  (1 / 4 ) * ( ED / Eso )
 factor
Spectral Displacement
Use of Pushover Curve (ATC-40)
Spectral
Acceleration
Estimation of Equivalent Damping
Ed
Eso
Spectral Displacement
Use of Pushover Curve (ATC-40)
Spectral
Acceleration
Response Spectrum (5% damping)
2.5CA
CV/T
Time Period
Use of Pushover Curve (ATC-40)
Response Spectrum (5% damping)
CA and CV depend on:
- Seismic zone (0.075 to 0.4)
- Nearness to fault and source type (1 to 2)
- Soil Type (1 to 2.5)
- Level of Earthquake (0.5 to 1.5)
Use of Pushover Curve (ATC-40)
Spectral
Acceleration
Reduced Spectrum (Effective damping)
2.5CA/Bs
CV/(T BL)
Time Period
Use of Pushover Curve (ATC-40)
Acceleration-Displacement Response Spectrum
T0
T0
Time Period
Spectral
Acceleration
Spectral
Acceleration
Sd = SaT2/42
Spectral Displacement
Use of Pushover Curve (ATC-40)
Spectral
Acceleration
Performance Point
Demand Spectrum for effective
damping at performance point
Capacity Spectrum
Spectral Displacement
Use of Pushover Curve (ATC-40)
Spectral Acceleration
Performance Point
Spectral Displacement
Use of Pushover Curve (ATC-40)
Verification of Acceptance
Force Measure
Expected Performance Point
for given Earthquake
Performance Limits
(IO, LS, CP)
Deformation Measure
Use of Pushover Curve (ATC-40)
Use of Pushover Curve (FEMA-273)
(Displacement Coefficient Method)


Estimate Target Displacement
Verify Acceptance
Use of Pushover Curve (FEMA-273)
Estimation of Target Displacement
Estimate effective elastic stiffness, Ke
Estimate post yield stiffness, Ks
Estimate effective fundamental period, Te
Calculate target roof displacement as

2
 C0 C1 C2 C3 Sa Te
/(4 )
2
Use of Pushover Curve (FEMA-273)
Estimation of Target Displacement
C0 Relates spectral to roof displacement
C1 Modifier for inelastic displacement
C2 Modifier for hysteresis loop shape
C3 Modifier for second order effects
Use of Pushover Curve (ATC-40)
Estimation of Effective Elastic Period, Te
aKe = Ks
Base Shear
Vy
Estimate Te using Ke
.6Vy
Ke
Estimate Elastic Spectral Displacement

2
 S a Te
/(4 )
Roof Displacement
2
Use of Pushover Curve (FEMA-273)
Calculation of C0
Relates spectral to roof displacement
- use modal participation factor for control
node from first mode
- or use modal participation factor for
control node from deflected shape at the
target displacement
- or use tables based on number of
stories and varies from 1 to 1.5
Use of Pushover Curve (FEMA-273)
Calculation of C1
Modifier for inelastic displacement
Spectral
Acceleration
C1 = [1 +(R-1)T0/Te]/R
C1 = 1
T0
Time Period
R is elastic strength
demand to yield
strength
Use of Pushover Curve (FEMA-273)
Calculation of C2
Modifier for hysteresis loop shape
- from Tables
- depends on Framing Type
(degrading strength)
- depends on Performance Level
- depends on Effective Period
- varies from 1.0 to 1.5
Use of Pushover Curve (FEMA-273)
Calculation of C3
Modifier for dynamic second order effects
C3 = 1 if post yield slope, a is positive
else
C3 = 1 +[ |a|(R-1)3/2 ]/Te
Use of Pushover Curve (FEMA-273)
Force Measure
Verification of Acceptance
Target Displacement (or
corresponding deformation)
for given Earthquake
Performance Limits
(IO, LS, CP)
Deformation Measure
Use of Pushover Curve
Do these methods work?
Comparisons with:
- Nonlinear time history analysis
- Single degree of freedom systems
- Multi-degree of freedom systems
- Observed damage
How do they compare with each other?
SAP2000/ETABS Pushover Options



SAP2000 released September, 1998
Full 3D implementation
Single model for
- linear static analysis
- linear response spectrum analysis
- linear time history analysis
- nonlinear time history analysis
- nonlinear static pushover analysis
- steel and concrete design
SAP2000/ETABS Pushover Options


Generally follows ATC-40 & FEMA 273
Available Pushover Element Types
- 3D truss (axial hinge)
- 3D beam (moment and shear hinges)
- 3D column (P-M-M and shear hinges)
- Shells, Solids, etc. considered linear
- Panel zone (later)
- 3D column (Fiber hinge) (later)
- Shear wall (plasticity model) (later)
- Nonlinear springs (later)
SAP2000/ETABS Pushover Options
Force-Deformation Relationship
Force
C
D
B
A
E
Deformation
F
SAP2000/ETABS Pushover Options
Three dimensional Beam Element
Flexible
Connection
Span Loads
Shear Hinge
Plastic Hinge
Rigid Zone
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 hinges in coupling beams (ATC-40)
SAP2000/ETABS Pushover Options

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
SAP2000/ETABS Pushover Options

Available Results for each step of loading
- Base Shear
- Element Forces
- Section Forces
- Joint Displacements
- Drifts
- Element Hinge Deformations
- Limit Points (acceptance criteria) reached
SAP2000/ETABS Pushover Options

Pushover Curve Postprocessing (ATC-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 (acceptance criteria) reached
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 Curve
- Demand Curve
- Demand Spectra at different dampings
- Effective period lines