pushover analysis using etabs and sap2000
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Pushover Analysis
Using ETABS (and SAP2000)
By
Naveed Anwar
Asian Center for Engineering Computations and Software
Asian Institute of Technology
In Association with
Computers and Structures Inc., Berkeley, California, USA
June 22-23, CEBU, Philippines
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Acknowledgements
• Some of the material presented in thesenotes 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 ManagementAgency, USA
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Objectives
• Introduce the basic Modeling and AnalysisConcepts
• To provide an understanding of Static
Nonlinear Pushover Analysis for SeismicPerformance
• To demonstrate the application of Pushover
Analysis for buildings using ETABS and
SAP2000 and to provide a comparison
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The Questions
• Why use Pushover Analysis
• What is Pushover Analysis
• How to carryout Pushover Analysis
• What to do before Pushover Analysis
• What to do after Pushover Analysis
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u s h o v e r A n a l y s i s , A
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Modeling and Analysis
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Summary
• The Purpose of Analysis
• The Significance of Modeling
• Analysis Types
• Linearity and Non-Linearity• Static and Dynamic Analysis
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Structural
Model
EXCITATION
LoadsVibrations
Settlements
Thermal Changes
RESPONSES Displacements
Strains
Stress
Stress Resultants
STRUCTURE
pv
Structural System – Analysis Model
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Analysis of Structures
pv
xx yy zz
vx x y z p 0
Real Structure is governed by
“Partial Differential Equations” of
var ious order
Direct solution is only possible for:
• Simple geometry
• Simple Boundary
• Simple Loading.
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The Need for Modeling
A - Real Structure cannot be Analyzed:
It can only be “Load Tested” to determine
response
B - We can only analyze a “Model” of theStructure
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 toengineers”
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Continuum to Discrete Model
pv
(Governed by partial
differential equations)
CONTINUOUS MODELOF STRUCTURE
(Governed by either
partial or total differential
equations)
DISCRETE MODELOF STRUCTURE
(Governed by algebraic
equations)
3D-CONTINUMMODEL
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From Classical to FEM Solution
xx yy zz
vx x y z
p 0
t vt
st
v
dV p u dV p u ds _ _ _
Assumptions
Equilibrium
Compatibility
Stress-Strain Law
(Principle of Virtual Work)
“Partial
Differential
Equations”
Classical
Actual Structure
Kr R“Algebraic
Equations”
K = Stiffness
r = Response
R = Loads
FEM
Structural Model
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Simplified Structural System
F = K D
F
K D
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
F Ku
)()()()( t F t Kut uC t u M
)()()()()( t F t F t Kut uC t u M NL
F F Ku NL
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Excitation Structure Response Basic Analysis Type
Static Elastic Linear Linear-Elastic-Static Analysis
Static Elastic Nonlinear Nonlinear-Elastic-Static Analysis
Static Inelastic Linear Linear-Inelastic-Static Analysis
Static Inelastic Nonlinear Nonlinear-Inelastic-Static Analysis
Dynamic Elastic Linear Linear-Elastic-Dynamic Analysis
Dynamic Elastic Nonlinear Nonlinear-Elastic-Dynamic Analysis
Dynamic Inelastic Linear Linear-Inelastic-Dynamic Analysis
Dynamic Inelastic Nonlinear Nonlinear-Inelastic-Dynamic Analysis
Basic Analysis Types
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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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Analysis Type
The type of Analysis to be carried out dependson the Structural System
– The Type of Excitation (Loads) – The Type Structure (Material and
Geometry)
– The Type Response
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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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Elasticity and Linearity
A c t i o n
Deformation
A c t i o n
Deformation
A
c t i o n
Deformation
A
c t i o 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 - F NL = FF NL
F Ku
)()()()( t F t Kut uC t u M
)()()()()( t F t F t Kut uC t u M NL
F F Ku NL
Nonlinear, Static and Dynamic
Linear, Static and Dynamic
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Basic Concepts for Analysis
• DOF (Degree of Freedom)
• Stiffness
• Static Analysis Process
• Dynamic Analysis Procedures
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The Seven Degrees of Freedom
• The General BeamElement may have
7 degrees of
freedom
• The seventh
degree is Warping
• Warping is out-of
plane distortion ofthe beam cross-
section
z
y
x
xu
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
x
xu
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 Picture§ u z
Axial deformation
Axial strain
Axial stress
§ u x
Shear deformation
Shear strain
Shear stress
§ u y
Shear deformation
Shear strain
Shear stress
§ r z
Torsion
Shear strain
Shear stress
§ r y
Curvature
Axial strain
Axial stress
§ r x Curvature Axial strain Axial stress
§ w z
Warping
Axial strain
Axial stress
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What is Stiffness ?
• In structural terms, stiffnessmay be defined as“Resistance to Deformation”
• So for each type of
deformation, there is acorresponding stiffness
• Stiffness can be consideredor evaluated at various levels
• Stiffness is also the“constant” in the Action-Deformation Relationship
u
F K
F Ku
F u
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 Deformations
T-Matrix
Global-Local Cords.
N-MatrixShape Functions
B-Matrix
Strain-Deforrmation
D-Matrix
Stress-Strain
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Linear-Static Analysis Process
• Generate Stiffness Matrix for each Element
• Form Global Stiffness Matrix
• Form Load Vector
• Modify for boundary conditions• Solve for unknown Displacements
• Compute element actions/ stresses from end
displacements
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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 Fast Nonlinear Analysis
(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 MemberForces for Each Time Step and Iterate for
Equilibrium – Brute Force Method
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Mode Superposition Method
• Generate Orthogonal Dependent Vectors andFrequencies
• Form Uncoupled Modal Equations and SolveUsing Exact Method for Each Time Increment
• Recover Nodal Displacement as a Functionof Time
• Calculate Member Forces as a Function ofTime
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Load Dependent Ritz Vector
• Approximately Three Times Faster than theCalculation 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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Fast Non-Linear Analysis
• Evaluate LDR Vectors with Non-Linear ElementsRemoved 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
• The FNA Method is Designed for Static and Dynamic
Analysis of Non-Linear Structures with a LimitedNumber of Pre-Defined Non-Linear Elements
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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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The Modal Analysis
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The Modal Analysis
• The modal analysis determines the inherent naturalfrequencies of vibration
• Each natural frequency is related to a time period
and a mode shape
• Time Period is the time it takes to complete onecycle 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
t t
t t
P u K ucu M
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The Modal Analysis
• The Modal Analysis should be run beforeapplying 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 fullDynamic 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, togetherwith 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 bedetermined
– 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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Eccentric and Concentric Response
Mode-1 Mode-2 Mode-3
Symmetrical Mass and
Stiffness
Unsymmetrical Mass
and Stiffness
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Modes and Pushover
• Generally the deformation patterncorresponding to the First Mode is used as
the basis for analysis
• This is acceptable for structures with time
period less than or equal to 1 second
• For more flexible structures, higher mode
contribution may become significant
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Special Analysis Problems
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Base Isolation
Isolators
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Building Impact
Building Impact
Analysis
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Dampers
Friction device
Concentrated damper
Nonlinear element
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Gaps and Joints
Bridge Deck ABUTMENT
Gap Element
Tension only element
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Hinges
2 Rotational DOF
Degrading Stiffness?
PLASTIC HINGES
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Dampers
Mathematical Model
F= ku
F= CvN
F= f(u,v,umax
)
Mechanical Damper
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Linear Viscous Damping
• Does not Exist in Normal Structures andFoundations
• 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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P u s h o v e r A n a l y s i s , A
C E C O M S ,
A I T
Uplift
Uplifting
Allowed
FRAME WITH UPLIFTING ALLOWED
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P u s h o v e r A n a l y s i s , A
C E C O M S ,
A I T
Structural Modeling
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P u s h o v e r A n a l y s i s , A
C E C O M S ,
A I T
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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P u s h o v e r A n a l y s i s , A
C E C O M S ,
A I T
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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P u s h o v e r A n a l y s i s , A
C E C O M S ,
A I T
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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P u s h o v e r A n a l y s i s , A
C E C O M S ,
A I T
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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P u s h o v e r A n a l y s i s , A
C E C O M S ,
A I T
Graphic Object Modeling
• Use basic Geometric Entities to create FEModels
• 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 representgeometry, boundary and loads
• SAP2000, ETABS and SAFE use the concept
of Graphic Objects
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P u s h o v e r A n a l y s i s , A
C E C O M S ,
A I T
Modeling Objects and Finite Elements
• Structural Members are representation ofactual structural components
• Finite Elements are discretizedrepresentation of Structural Members
• The concept of Graphic Objects can be usedto represent both, the Structural Members aswell as Finite Elements
• In ETABS, the Graphic Objects representing
the Structural Members are automaticallydivided into Finite Elements for analysis andthen back to structural members for resultinterpretation
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P u s h o v e r A n a l y s i s , A
C E C O M S ,
A I T
Design Methods and Concepts
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
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
F r o m L
o a d s t o S
t r e s s e s
F r o m S
t r a i n s t o
R e s p o n s e
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Three Design Approaches
• Working Stress Design – Stress is primary concern and objective
• Ultimate Strength Design
– Strain is primary concern
• Performance Based Design
– Deformation is primary concern
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
From Serviceability to Performance
Serviceability
Design
Strength Design
Performance
Design
Allowable material, control on
deformation limits for design loads
Material failure criteria, section capacity
for factored loads
Ductility considerations, deformationcapacity, load capacity at large
deformations. Extraordinary load
considerations
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
From Serviceability to Performance
• Satisfying one design level does not ensurethat other design levels will be satisfied – Serviceability design only ensures that
deflections and vibrations etc. for service loadsare within limits but says nothing about strength
– Strength design ensures that a certain factor ofsafety against overload is available within amember or a cross-section but says nothingabout what happens if load exceeds design level
– Performance design ensures that structure as awhole reaches a specified demand level.Performance design can include, both service andstrength design levels
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P u s h o v e r A n a l y s i s , A C E C O M S ,
A I T
From Serviceability to Performance
Deformation
L o a d
Δ
P
A
B
C D
• The entire response of structure or amember can be determined, in an
integrated manner from the Action-
Deformation Curve
A – Serviceability
B – Cracking Limit
C – Strength Limit
D – Failure Limit
P
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P u s h o v e r A n a l y s i s , A C E C O M S ,
A I T
Cross-section Reponses
• Stresses – Tension
– Compression
– Shear > Tension-Compression
•Strains
– Normal strain
– Shear Strain
• Deformations
– Rotation
– Shortening
– Shearing
– Twisting
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P u s h o v e r A n a l y s i s , A C E C O M S ,
A I T
Determining Cross-section Response
Material Stress-Strain Curves
Cross-section Dimensions
CapacityInteraction 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
P e r f o
r m a n c e
S t r e n
g t h
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P u s h o v e r A n a l y s i s , A C E C O M S ,
A I T
Capacity Interaction Surface
MxMy
P
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P u s h o v e r A n a l y s i s , A C E C O M S ,
A I T
P-M and M-M Interaction Curves
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P u s h o v e r A n a l y s i s , A C E C O M S ,
A I T
The Moment Curvature Curve
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P u s h o v e r A n a l y s i s , A C E C O M S ,
A I T
Original Cross-sections
Plain concrete shape Reinforced concrete section Compact Hot-rolled steel shape
Compact Built-up steel
section
Reinforced concrete,
composite sectionComposite section
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P u s h o v e r A n a l y s i s , A C E C O M S ,
A I T
Sections After Strengthening
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P u s h o v e r A n a l y s i s , A C E C O M S ,
A I T
Strength and Performance
• In Strength Design, every member and everycross-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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P u s h o v e r A n a l y s i s , A C E C O M S ,
A I T
Members on Critical Load Path
• In Performance Based Design, only a fewmembers 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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P u s h o v e r A n a l y s i s , A C E C O M S ,
A I T
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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P u s h o v e r A n a l y s i s , A C E C O M S ,
A I T
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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P u s h o v e r A n a l y s i s , A C E C O M S ,
A I T
What Effects Performance?
• Performance is generally of concern forlateral loads such as earthquake and wind
• The main factor that effects performance is
the Duct i l i ty 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 greatimportance for lateral load demands
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Ductility – Definition and Usage
• Ductility can bedefined as the “ratioof deformation and agiven stage to the
maximumdeformationcapacity”
• Normally ductility is
measured from thedeformation at designstrength to themaximumdeformation at failure
Yield/ Design
Strength
L o a d
Deformation
DyDu
Ductility = Dy / Du
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
What Effects Ductility!
• The most important factor effecting ductilityof 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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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Action – Deformation Curves
• Relationship between action andcorresponding 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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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
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 toget Moment-Curvature Curves
• By combination of measurement andcomputations
– Calibrate computation models with actualmeasurements
– Some parameters obtained by measurement andsome by computations
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
The Moment Curvature Curve
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
The Moment-Curvature Curve
• Probably the most important action-deformationcurve 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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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
What is Curvature
• In geometry, it is rateof change of rotation
• In structural behavior,
Curvature is related to
Moment
• For a cross-section
undergoing flexural
deformation, it cancomputed as the ratio
of the strain to the
depth of neutral axis
C
e
Curvature = e / C (radian / unit length)
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
How to Read M-Phi Curve
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Outputs from M-Phi Curve
1 -Yield Point2 -Failure Point
u
y Ductility
3 -
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Outputs from M-Phi Curve
M EI
EI
M
4 - Stiffness of the Section at given M and Phi
5 - Slope of the section at given Moment
dx EI
M b
a
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Outputs from M-Phi Curve
dx x
EI
M b
a
D
6 - Deflection of the section at given Moment
7 - Strain at given Moment
c c = distance from the NA tothe point where strain is
required
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Outputs from M-Phi Curve
yX W
X W s
8 - Crack Width at given crack spacing
9 - Crack Spacing at given crack width
y
W X
W X
s
Specified Crack Spacing = X
y
s
Rebar Centroid
NA
W
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Plot M-Phi Curve
Determine curvature
at known moment
Determine FlexuralStiffness (EI)
Determine Slope
Determine Deflection
Determine Strain
Determine Crack
Spacing/Width
M EI
dx EI
M b
a
dx x EI
M b
a
D
c
X W s
s
W X
Outputs from M-Phi Curve - Summary
O f C
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Outputs from M-Phi Curve - Example
15 ft
P=160 K
M=600 k-ft
L/2
36 in
24 in
M EI
For M=600 Phi = 0.000 0 6
From M-Phi Diagram
EI=600x12/0.00006 EI=1.2E8 k-in^2
dx EI M
b
a
=600x7.5x144/1.2E8
=0.0054 rad
Slope at Mid Span
O f M Phi C E l
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Outputs from M-Phi Curve - Example
dx x EI
M b
a
D
Specified Crack Spacing = X
y
s
Rebar Centroid
NA
W
Deflect io n at Mid Span
=600x7.5x144x15x12/(6x1.2E8)=0.162 in
Strain in Steel
M = 600 k-ft, y=16
=0.00006x16
=0.00096
c
O t t f M Phi C E l
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Outputs from M-Phi Curve - Example
X W s
s
W X
Crack Width Assuming crack spacing of 18 in
=0.00096 x 18 =0.01728 in
Crack Spacing
Assuming crack width of 0.02 in
=0.02/ 0.00096
=20.8 in
Specified Crack Spacing = X
y
s
Rebar Centroid
NA
W
M Phi C d D ili
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
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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C fi t M d l d D tilit
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Confinement Model and Ductility
Effect 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)
Moment(kip-ft)
Whitney Rectangle
Mander Circular ConfinedMander Pipe Filled
8#8 bars
Whitney Rectangle
(both)
a) b) c)
8#8 bars
Whitney Rectangle (outside)
Mander Circular Confined (inside)
8#8 bars
Whitney Rectangle (outside)
Mander Pipe Filled (inside)
C fi t St l d D tilit
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Confinement Steel and Ductility
Effect 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)
Moment(kip-ft)
Spacing = 3in
Spacing = 6 in
Spacing = 12 in
8#6 bars
Mander’ s Rectangular
Confined
a)
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
IntroducingPushover Analysis
Th P h A l i
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
The Pushover Analysis
• An alternate method of analysis for carryingout the Performance Based Design
• Pushover analysis is carried out after the
Linear Analysis has been done and
Serviceability and Strength design has beencompleted
• Pushover analysis is most suitable for
determining the performance, specially forlateral loads such as Earthquake or even
wind
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Performance Based Design Basics
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Performance Based Design - Basics
•Design is based not on Ultimate Strength butrather on Expected Performance – Basic Ultimate Strength does not tell us what will
be performance of the structure at UltimateCapacity
• Performance Based Design Levels – Fully Operational
– Operational
– Life Safe
– Near Collapse
– Collapse
Pushover Spectrum
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Pushover Spectrum
Pushover Demand Curves
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Pushover Demand Curves
Earthquake Push on Building
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Earthquake Push on Building
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Pushover Capacity Curves
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Pushover Capacity Curves
Demand Vs Capacity
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Demand Vs Capacity
Non linearity in Pushover
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Non-linearity in Pushover
•Material nonlinearity at discrete, user-defined hingesin frame/line elements.
1. Material nonlinearity in the link elements.
• Gap (compression only), hook (tension only), uniaxial
plasticity 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
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Important Considerations
• Mathematically, static nonlinear analysisdoes 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
Procedure for Pushover Analysis
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Procedure for Pushover Analysis
• Create a model just like for any otheranalysis.
• 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.
Procedure for Pushover Analysis
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Procedure for Pushover Analysis
• 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).
Procedure for Pushover Analysis
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Procedure for Pushover Analysis
•Assign link properties, if any, to frame/lineelements (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 basedon default values to be computed by the
program.
Procedure for Pushover Analysis
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P u s h o v e r A n a l y s i s
, A C E C O M S ,
A I T
Procedure for Pushover Analysis
•For staged construction, define groups thatrepresent the various completed stages of
construction.
• Define the static nonlinear load cases (Define
> Static Nonlinear/Pushover Cases).
• Run the static nonlinear analysis (Analyze >
Run Static Nonlinear Analysis).
Procedure for Pushover Analysis
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P u s h o v e r A n a l y s i s
, A C E C O M S ,
A I T
Procedure for Pushover 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.
Summary
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P u s h o v e r A n a l y s i s
, A C E C O M S ,
A I T
Summary
•We have to think in terms of“Displacements” and not in terms of loads,
stresses or strains
• The main idea is to compare expected
displacements or required displacementswith the ability of the structure to reach
those displacements without failing
OR
indicating that it will not reach those
displacements
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Performance Check for Structures
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P u s h o v e r A n a l y s i s
, A C E C O M S ,
A I T
Performance Check for Structures
•Purpose – How will a structure perform when subjected to a
given level of earthquake?
• Defini t ion o f Struc tural Performance • Defini t ion of Earthquake Level
• Determinat ion of perform ance level
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Types of Performance Checks
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P u s h o v e r A n a l y s i s
, A C E C O M S ,
A I T
Types of Performance Checks
•Linear Static Analysis
• Linear Dynamic Analysis
• Non Linear Static Analysis
(Pushover Analysis)
• Non Linear Dynamic Analysis
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Steps in Performance Check
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P u s h o v e r A n a l y s i s
, A C E C O M S ,
A I T
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
Constructing Pushover Curve
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P u s h o v e r A n a l y s i s
, A C E C O M S ,
A I T
Constructing Pushover Curve
•Define Structural Model – Elements
– Strength-Deformation properties
• Define Loads
– Gravity
– Lateral Load Patterns
• Select Control Displacements or Drifts
• Perform Pushover Analysis
Pushover Modeling (Elements)
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P u s h o v e r A n a l y s i s
, A C E C O M S ,
A I T
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 (Beam Element)
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P u s h o v e r A n a l y s i s
, A C E C O M S ,
A I T
g ( )
Plastic Hinge
Flexible connection Span Loads Shear Hinge
Rigid Zone
Three Dimensional Beam Element
Pushover Modeling (Column Element)
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P u s h o v e r A n a l y s i s
, A C E C O M S ,
A I T
g ( )
Plastic Hinge
Shear Hinge
Rigid Zone
Three Dimensional Column Element
Pushover Modeling
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P u s h o v e r A n a l y s i s
, A C E C O M S ,
A I T
g
•Types of Deformation Properties – Axial
– Moment only
– P-M : Uniaxial P-M Interaction
– P-M-M : Biaxial P-M Interaction – Shear
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Use of Pushover Curve (ATC-40)
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P u s h o v e r A n a l y s i s
, A C E C O M S ,
A I T
( )
•Construct Capacity Spectrum
• Estimation of Equivalent Damping
• Determine Demand Spectrum
• Determine Performance Point
• Verify Acceptance
Use of Pushover Curve (ATC-40)
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P u s h o v e r A n a l y s i s
, A C E C O M S ,
A I T
( )
S p e c t r a l
A c c e l e r
a t i o n
Spectral Displacement
Capacity Spectrum
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Use of Pushover Curve (ATC-40)
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P u s h o v e r A n a l y s i s
, A C E C O M S ,
A I T
( )
Cv/(T B
L)
2.5C A
/Bs
S p e c t r a l
A c c e l e r a t i o
n
Time Period
Reduced Spectrum (Equivalent Damping)
Use of Pushover Curve (ATC-40)
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P u s h o v e r A n a l y s i s
, A C E C O M S ,
A I T
S p e c t r a l
A c c e l e r a
t i o n
Spectral Displacement
Performance Point
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Use of Pushover Curve (FEMA-273)
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P u s h o v e r A n a l y s i s
, A C E C O M S ,
A I T
•Displacement Coefficient Method
• Estimate Target Displacement
• Verify Acceptance
• 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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SAP2000/ETABS Pushover Options
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P u s h o v e r A n a l y s i s
, A C E C O M S ,
A I T
•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
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Force - Deformation Relationship
A
B
C
D E
Deformation
F o r c
e
SAP2000/ETABS Pushover Options
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Plastic Hinge
Flexible connection
Span Loads Shear Hinge
Rigid Zone
Three Dimensional Beam Element
SAP2000/ETABS Pushover Options
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
• Strength – Deformation and P-M-M curvescan 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)
SAP2000/ETABS Pushover Options
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
•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
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
•
Available Results for each step of Loading – Base Shear
– Element Forces
– Section Forces
– Joint Displacement – Drifts
– Element hinge Deformations
– Limit Points reached
SAP2000/ETABS Pushover Options
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
•
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
SAP2000/ETABS Pushover Options
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
• 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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Base Shear Vs Displacement
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
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Find Column E
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
• Determine Column E to
give Appropriate Initial
Stiffness:
Column
= (1700 *12*12)/(24*24*0.1)
= 4250 Ksi
D
A
PL E
Measured Axi al Di splacement at Joint 2
(in)
A x i a l F o r c
e ,
P
( K i p s )
1000
2100
1700
0.1 0.6 0.8
Desired Behavior
Find Column Deflection
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
• Determine Elastic Column
Lengthening when loading
from 1700 to 2100 K:
Column
= [(2100-1700) *12*12)]/(24*24*4250)
= 0.0235 in
AE
PLD
Measured Axi al Di splacement at Joint 2
(in)
A x i a l F o r c
e ,
P
( K i p s )
1000
2100
1700
0.1 0.6 0.8
Desired Behavior
Find Column Deflection
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
• Determine Elastic Column
Lengthening when loading
from 2100 to 1000 K:
Column
= [(2100-1000) *12*12)]/(24*24*4250)
= 0.0647 in
AE
PLD
Measured Axi al Di splacement at Joint 2
(in)
A x i a l F o r c
e ,
P
( K i p s )
1000
2100
1700
0.1 0.6 0.8
Desired Behavior
Find Column Deflection
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
• Determine Elastic
Column Lengthening
when loading from 1000
to 0 K:
Column
= 1000 *12*12)/(24*24*4250)
= 0.0588 in
AE PLD
Measured Axi al Di splacement at Joint 2
(in)
A x i a l F o r c
e ,
P
( K i p s )
1000
2100
1700
0.1 0.6 0.8
Desired Behavior
Find Hinge Properties
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
1000
2100
1700
0 . 0
0 . 4
7 6 5
0 . 8
0 . 5
4 1 2
0 . 7
4 1 2
A
B
C
D E
B = 0.1 - 0.1 = 0
C = 0.6 - 0.1 - 0.0235 = 0.4765
D = 0.6 - 0.1 - 0.0235 + 0.0647 = 0.5412
E = 0.8 - 0.1 - 0.0235 + 0.0647 = 0.7412
Hinge Properties
Measured Axial Displacement at Joint 2
(in)
A x i a l
F o r c e ,
P
( K i p s )
1000
2100
1700
0.1 0.6 0.8
Desired Behavior
Hinge Properties
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Pushover Curve
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P u s h o v e r A n a l y s i
s ,
A C E C O M S ,
A I T
Example 3
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P u s h o v e r A n a l y s i
s ,
A C E C O M S ,
A I T
0.8 kip/ft0.8 kip/ft
1.2 kip/ft
Push
W
1 4 x 9 0
W
1 4 x 9 0
W24x55
W 1 2 x
1 9 0 W 1 2 x 1 9 0
W
8 x 1 0 W
8 x 1 0
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With W12x190 Brace
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P u s h o v e r A n a l y s i
s ,
A C E C O M S ,
A I T
With W8x10 Brace
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P u s h o v e r A n a l y s i
s ,
A C E C O M S ,
A I T
Conversion to
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
ADRS Spectra ATC-40
Response Spectrum Conversion
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
• 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
Response Spectrum Conversion
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
• For Each value or Sai
and Ti
determine the
value of Sdi using the equation
• Spectral Acceleration and Displacement atperiod Ti are given by
g S T
S aii
di 2
2
4
v
i
ai S T
g S 2 vi
di S T S 2
Capacity Spectrum Conversion
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
• Capacity Spectrum from Capacity or
Pushover Curve
• Point by Point conversion to first mode
spectral coordinates
• on capacity curves are convertedto corresponding Sai and Sdi on capacity
spectrum using:
roof i and V D
1 W
V S iai
roof
roof
di PF
S ,11
D
Moment Hinge Properties
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
Using M-Fi Curve
Procedure
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
• Plot M-Fi curve for cross-section
• Estimate EI value from M-Fi Curve using the
following equation
• Calculate Rotations from Curvature using:
M EI
EI
M
dx EI
M b
a
Example
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
• Reinforced Concrete
Beam-Column Cross-Section
• 24”x24”
• Reinforced with 12 #9
bars
• Length is 12 ft 2 4 "
24"
Example
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
0.00028
370
Example
M
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
• So EI = 370/0.00028 = 1321428.6
• So = 0.00336 rad
• Find for other Moment Values and input inHinge Property
M
EI
dx EI
M b
a
p
I EI
M
Considerations
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
• 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
Comparisons of
SAP2000 d ETABS
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
SAP2000 and ETABS
SAP2000 vs ETABS
SAP2000 ETABS
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
• SAP2000 –
General Purpose FEASoftware
– Classic Finite ElementSoftware
– Steel, and Concrete FrameElement Design
– Shear Wall Design NotSupported
– Fewer Automated MeshingOptions
– Does not Support
Composite Design
• ETABS –
Specialized FEA Softwarefor Building analysis anddesign
– Fully Object based Modelingand Design
– Steel, concrete, composite
Frame Element design – Supports Shear wall design
– Full and practical automeshing options
– Supports Composite Design
SAP2000 vs ETABS
SAP2000 ETABS
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
• SAP2000
– General output related tonodes and elements isreported
– Professional Report
– Powerful load cases,combinations, envelopes,multiple case, etc.
– Cables, Dampers, and NLLinks and Hinges
• ETABS
– Floor wise representation ofresults such as story drift,floor mass participation,story shear, etc.
– General Report (text files)
– Relatively less ability tohandle load combinations
– Only Nonlinear links andHinges
SAP2000 vs ETABS
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
• SAP2000
– Supports Solid Elements
– Relatively low versatility
for defining and editing
grid systems
• ETABS
– Does not support solid
elements
– Powerful grid system
definition and editing
ETABS Pushover
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
ETABS Pushover
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
ETABS Pushover
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
SAP2000 Pushover
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P u s h o v e r A n a l y s i s ,
A C E C O M S ,
A I T
SAP2000 Pushover
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P u s h o v e r A n a l y s
i s ,
A C E C O M S ,
A I T
SAP2000 Pushover
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P u s h o v e r A n a l y s
i s ,
A C E C O M S ,
A I T
SAP2000 Pushover
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P u s h o v e r A n a l y s
i s ,
A C E C O M S ,
A I T
SAP2000 Pushover
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P u s h o v e r A n a l y s
i s ,
A C E C O M S ,
A I T
ETABS Pushover1
Use Load Patterns
Steps to compute the
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P u s h o v e r A n a l y s
i s ,
A C E C O M S ,
A I T
p p
Displacement
(Displacement notMonitored)
Divide the Specified
Displacement into
Steps and apply loads
to attain that
displacement
Monitor which DOF at
what level/story
Save Positive Resultsonly
2
After a member fails
redistribute loads
locally around failed
members or reanalyze
structure using a new
stiffness matrix
3
Which Pattern Loads
to apply and what is
the scaling factor for
each loading case
included in the load
factor
1
2
3
4
Consider P-Deltaeffects and Large
Displacements due to
gravity loads caused
by each step of lateral
loading
45
For Construction
Sequence analysis.
Specify which
Pushover case to be
applied to which stage
of construction or
strengthening.
5
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SAP/ETABS Pushover Output
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i s ,
A C E C O M S ,
A I T
3
Demand Curves
5
If there is additional viscous
damping provided in the
structure, perhaps by viscous
1
2
3
4
5
21
V=Base Shear