Pushover Analysis using ETABS and SAP2000
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Pushover Analysis Using ETABS and SAP2000
June 18-19, Manila, Philippines For
Association of Structural Engineers Philippines
By
Naveed Anwar Asian Center for Engineering Computations and Software Asian Institute of Technology In Association with
Computers and Structures Inc., Berkeley, California, USA
Pushover Analysis Using ETABS (and SAP2000)
June 22-23, CEBU, Philippines
By
Naveed Anwar Asian Center for Engineering Computations and Software Asian Institute of Technology In Association with
Computers and Structures Inc., Berkeley, California, USA
Acknowledgements
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• 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
Pushover Analysis, ACECOMS, AIT
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
The Questions
Pushover Analysis, ACECOMS, AIT
• 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
Pushover Analysis, ACECOMS, AIT
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
Structural System – Analysis Model STRUCTURE
RESPONSES
Pushover Analysis, ACECOMS, AIT
EXCITATION Loads Vibrations Settlements Thermal Changes
Displacements Strains Stress Stress Resultants
pv
Structural Model
Analysis of Structures xx yy zz pvx 0 x y z Pushover Analysis, ACECOMS, AIT
pv
Real Structure is governed by “Partial Differential Equations” of various order Direct solution is only possible for: • Simple geometry • Simple Boundary • Simple Loading.
Pushover Analysis, ACECOMS, AIT
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 the Structure C - We therefore need tools to Model the Structure and to Analyze the Model
Finite Element Method: The Analysis Tool
• Finite Element Analysis (FEA) – “A discretized solution to a continuum problem using FEM”
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• 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 3D-CONTINUM MODEL
CONTINUOUS MODEL OF STRUCTURE
(Governed by partial (Governed by either differential equations) partial or total differential equations)
DISCRETE MODEL OF STRUCTURE
(Governed by algebraic equations)
From Classical to FEM Solution Equilibrium
Actual Structure
Pushover Analysis, ACECOMS, AIT
xx yy zz pvx 0 x y z “Partial Differential Equations”
FEM
Assumptions
Classical
Structural Model
Kr R
Stress-Strain Law Compatibility
t
_
_
“Algebraic Equations” _
dV p u dV p u ds t v
t s
v
(Principle of Virtual Work)
K = Stiffness r = Response R = Loads
Simplified Structural System Deformations (D)
Loads (F)
Pushover Analysis, ACECOMS, AIT
Fv
D
K
F
F=KD
The Analysis System STRUCTURE
RESPONSES
EXCITATION
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pv
• Static • Dynamic
• Elastic • Inelastic
• Linear • Nonlinear
Eight types of equilibrium equations are possible!
The Equilibrium Equations 1. Linear-Static
Elastic
Ku F
2. Linear-Dynamic
Elastic
Pushover Analysis, ACECOMS, AIT
Mu(t ) Cu(t ) Ku(t ) F (t )
3. Nonlinear - Static Elastic OR Inelastic Ku FNL F
4. Nonlinear-Dynamic Inelastic
Elastic OR
Mu(t ) Cu(t ) Ku(t ) F (t ) NL F (t )
Pushover Analysis, ACECOMS, AIT
Basic Analysis Types 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
Some More Solution Types
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• 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
Analysis Type
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The 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
Static Vs Dynamic • Static Excitation
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– 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”
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
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• 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.
Linear Vs Nonlinear • Linearity – The response is directly proportional to excitation – (Deflection doubles if load is doubled)
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• 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
Linear-Elastic
Action
Action
Elasticity and Linearity
Deformation
Action
Action
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Deformation
Linear-Inelastic
Nonlinear-Elastic Deformation
Nonlinear-Inelastic Deformation
Linear and Nonlinear Linear, Static and Dynamic
Ku F Pushover Analysis, ACECOMS, AIT
F
FNL
(t ) Cu (t ) Ku(t ) F (t ) Mu Ku = F Ku - FNL = F
Nonlinear, Static and Dynamic
Ku FNL F u
Non Linear Equilibrium
Mu(t ) Cu(t ) Ku(t ) F (t ) NL F (t )
Basic Concepts for Analysis
Pushover Analysis, ACECOMS, AIT
• • • •
DOF (Degree of Freedom) Stiffness Static Analysis Process Dynamic Analysis Procedures
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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 crosssection
ry uy y
u x rx x z uz rz wz
Each section on a beam member can have seven Degrees Of Freedom (DOF) with respect to its local axis.
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§ § § § § § §
DOF Picture uz The AxialComplete 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 r y Curvature Axial strain Axial stress rx Curvature Axial strain Axial stress wz Warping Axial strain Axial stress
Pushover Analysis, ACECOMS, AIT
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 ActionDeformation Relationship
For Linear Response
uF Ku F F K u
The Structure Stiffness Material Stiffness
Cross-section Geometry
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Section Stiffness
Member Geometry Member Stiffness
Structure Geometry Structure Stiffness
The Matrices in FEM Global Nodal Deformations T-Matrix Global-Local Cords.
Element Nodal Deformations Pushover Analysis, ACECOMS, AIT
N-Matrix Shape Functions
Deformation in Element Space B-Matrix Strain-Deforrmation
Strain In Element Space D-Matrix Stress-Strain
Stress in Element Space
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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
Methods of Dynamic Analysis • For Both Linear and Non-Linear Systems
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– Step-by-Step Integration – Use of Mode Superposition with Eigen or LoadDependent 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 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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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 • 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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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 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
Free Vibration Analysis • Definition – Natural vibration of a structure released from initial condition and subjected to no external load or damping
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• Main governing equation -Eigenvalue Problem M u c u K ut Pt t t
• Solution gives – Natural Frequencies – Associated mode shapes – An insight into the dynamic behavior and response of the structure
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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
Application of Modal Analysis
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• 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
Application of Modal Analysis
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• 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 unsymmetrical
• The resonance with the applied loads or excitation can be avoided – The natural frequency of the structure should not be close to excitation frequency
Eccentric and Concentric Response
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Unsymmetrical Mass and Stiffness
Symmetrical Mass and Stiffness
Mode-1
Mode-2
Mode-3
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Modes and Pushover • Generally the deformation pattern corresponding 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
Building Impact
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Building Impact Analysis
Dampers Friction device
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Concentrated damper Nonlinear element
Gaps and Joints Gap Element
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Bridge Deck ABUTMENT
Tension only element
Hinges
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PLASTIC HINGES
2 Rotational DOF
Degrading Stiffness?
Dampers
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Mechanical Damper F= f(u,v,umax)
F= ku
F= CvN Mathematical Model
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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
Uplift
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FRAME WITH UPLIFTING ALLOWED
Uplifting Allowed
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Structural Modeling
Structure Types • Cable Structures • Cable Nets • Cable Stayed
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• Bar Structures • 2D/3D Trusses • 2D/3D Frames, Grids
• Surface Structures • Plate, Shell • In-Plane, Plane Stress
• Solid Structures
Global Modeling of Structural Geometry
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(a) Real Structure
(b) Solid Model
(c) 3D Plate-Frame
(d) 3D Fram e
(f) Grid-Plate
(e) 2D Fram e Fig. 1 Various Ways to Model a Real Struture
Some Sample Finite Elements
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Truss and Beam Elements (1D,2D,3D)
Plane Stress, Plane Strain, Axisymmetric, Plate and Shell Elements (2D,3D)
Brick Elements
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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
Graphic Object Modeling
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• Use basic Geometric Entities to create FE Models • Simple Graphic Objects – – – –
Point Object Line Object Area Object Brick Object
Represents Node Represents 1D Elements Represents 2D Elements 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
From Loads to Stresses
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Applied Loads
Building Analysis
Member Actions
Cross-section Actions
Material Stress/Strain
From Strains to Response
The Response and Design Material Response
Section Response
Member Response
Building Response
Load Capacity
Three Design Approaches • Working Stress Design – Stress is primary concern and objective
• Ultimate Strength Design Pushover Analysis, ACECOMS, AIT
– Strain is primary concern
• Performance Based Design – Deformation is primary concern
From Serviceability to Performance
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Allowable material, control on deformation limits for design loads Material failure criteria, section capacity for factored loads Ductility considerations, deformation capacity, load capacity at large deformations. Extraordinary load considerations
Serviceability Design
Strength Design
Performance Design
From Serviceability to Performance
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• 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
From Serviceability to Performance
A – Serviceability B – Cracking Limit C – Strength Limit D – Failure Limit
P
D
C B
Load
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• The entire response of structure or a member can be determined, in an integrated manner from the ActionDeformation Curve
P
Δ A
Deformation
D
Cross-section Reponses • Stresses – Tension – Compression – Shear > Tension-Compression
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• Strains – Normal strain – Shear Strain
• Deformations – – – –
Rotation Shortening Shearing Twisting
Determining Cross-section Response Material Stress-Strain Curves Cross-section Dimensions
Performance
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Given P value
Given Moments
Given Axial Load
P-M Curve
M-M Curve
Moment-Curvature Curves
•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
Strength
Capacity Interaction Surface
Given Moment Direction
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Capacity Interaction Surface P
My
Mx
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P-M and M-M Interaction Curves
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The Moment Curvature Curve
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Cross-section Stresses
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Original Cross-sections
Plain concrete shape
Compact Built-up steel section
Reinforced concrete section
Composite section
Compact Hot-rolled steel shape
Reinforced concrete, composite section
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Sections After Strengthening
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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
Members on Critical Load Path
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• 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
What Effects Serviceability? • Anything that reduces cracking – The presence of appropriate amount of reinforcement at appropriate locations
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• 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
What Effects Strength? • The basic Material Strength
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– 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 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
Load
Ductility – Definition and Usage
Yield/ Design Strength
Dy
Du
Deformation Ductility = Dy / Du
What Effects Ductility! • The most important factor effecting ductility of reinforced concrete cross-section is the confinement of concrete
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– 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
Action – Deformation Curves
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• 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
How to Get Action-Deformation Curves • By actual measurements – Apply load, measure deflection – Apply load, measure stress and strain
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• 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
The Moment-Curvature Curve
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• 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
e C
Curvature = e / C (radian / unit length)
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How to Read M-Phi Curve
Outputs from M-Phi Curve
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2 -Failure Point
1 -Yield Point
y 3 - Ductility u
Outputs from M-Phi Curve
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4 - Stiffness of the Section at given M and Phi
M EI M EI 5 - Slope of the section at given Moment b
M dx EI a
Outputs from M-Phi Curve 6 - Deflection of the section at given Moment
M D x dx EI a Pushover Analysis, ACECOMS, AIT
b
7 - Strain at given Moment
c
c = distance from the NA to the point where strain is required
Outputs from M-Phi Curve 8 - Crack Width at given crack spacing
Specified Crack Spacing = X
W s X Pushover Analysis, ACECOMS, AIT
W yX
NA
y
Rebar Centroid
s
9 - Crack Spacing at given crack width W X
s
W X y
W
Outputs from M-Phi Curve - Summary Plot M-Phi Curve
EI
Determine curvature at known moment
Pushover Analysis, ACECOMS, AIT
Determine Flexural Stiffness (EI) b
a
M D EI a b
M
x dx
Determine Deflection
c
M dx EI Determine Slope
X
Determine Strain
W
W s X
s Determine Crack Spacing/Width
Outputs from M-Phi Curve - Example For M=600 Phi = 0.00006 From M-Phi Diagram
P=160 K
L/2 24 in
EI
M
36 in 15 ft
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EI=600x12/0.00006 EI=1.2E8 k-in^2
Slope at Mid Span M=600 k-ft b
M dx EI
a =600x7.5x144/1.2E8 =0.0054 rad
Outputs from M-Phi Curve - Example Deflection at Mid Span
M D x dx EI a Pushover Analysis, ACECOMS, AIT
b
=600x7.5x144x15x12/(6x1.2E8) =0.162 in
Specified Crack Spacing = X
Strain in Steel
c M = 600 k-ft, y=16
=0.00006x16 =0.00096
NA
y
Rebar Centroid
s
W
Outputs from M-Phi Curve - Example Crack Width Assuming crack spacing of 18 in
Specified Crack Spacing = X
W s X
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NA
=0.00096 x 18 =0.01728 in
Crack Spacing Assuming crack width of 0.02 in
X
W s
=0.02/0.00096 =20.8 in
y
Rebar Centroid
s
W
M-Phi Curve and Ductility
Pushover Analysis, ACECOMS, AIT
• • • •
Effect of Axial Load Effect of Compression Steel Effect of Confinement Model Effect of Confinement Shape
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Axial Load and Ductility
12#8 bars
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Compression Steel and Ductility
a)
b)
8#8 8#8bars bars
2#8 bars
8#8 bars
c)
4#8 bars
8#8 bars
d)
8#8 bars
8#8 bars
Confinement Model and Ductility Effect of Concrete Confinement Model on Ductility of Cross-Section 350
300
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Moment (kip-ft)
250
200
Whitney Rectangle Mander Circular Confined
150
Mander Pipe Filled 100
50
0 0
0.001
0.002
0.003
0.004
0.005
0.006
Curvature (rad/in)
a)
8#8 bars Whitney Rectangle (both)
b)
c)
8#8 bars 8#8 bars Whitney Rectangle (outside) Whitney Rectangle (outside) Mander Circular Confined (inside) Mander Pipe Filled (inside)
Confinement Steel and Ductility Effect of Confinement Steel Spacing on Ductility 160
140
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Moment (kip-ft)
120
100
Spacing = 3in 80
Spacing = 6 in 60
Spacing = 12 in
40
20 0 -0.0005
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
-20
Curvature (in/rad) a)
8#6 bars Mander’s Rectangular Confined
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Confinement Shape and Ductility
a)
8#6 bars Mander’s Rectangular Confined
b)
a)
8#6 bars Mander’s Circular Confined
8#6 bars Whitney Rectangle
Pushover Analysis, ACECOMS, AIT
Introducing
Pushover 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
Why Pushover Analysis
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• 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
Performance Based Design - Basics • Design is based not on Ultimate Strength but rather on Expected Performance
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– 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
Pushover Analysis, ACECOMS, AIT
Pushover Spectrum
Pushover Analysis, ACECOMS, AIT
Pushover Demand Curves
Pushover Analysis, ACECOMS, AIT
Earthquake Push on Building
Pushover Analysis, ACECOMS, AIT
The Pushover Curve
Pushover Analysis, ACECOMS, AIT
Pushover Capacity Curves
Pushover Analysis, ACECOMS, AIT
Demand Vs Capacity
Non-linearity in Pushover • Material nonlinearity at discrete, user-defined hinges in frame/line elements. 1. Material nonlinearity in the link elements.
Pushover Analysis, ACECOMS, AIT
• 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.
Important Considerations
Pushover Analysis, ACECOMS, AIT
• • • •
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 nonlinearity. • Perform initial analyses without geometric non-linearity. Add P-delta effects, and large deformations, much later.
Pushover Analysis, ACECOMS, AIT
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
Pushover Analysis, ACECOMS, AIT
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.
Pushover Analysis, ACECOMS, AIT
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).
Pushover Analysis, ACECOMS, AIT
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.
Pushover Analysis, ACECOMS, AIT
Procedure for Pushover Analysis • For staged construction, define groups that represent 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).
Pushover Analysis, ACECOMS, AIT
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.
Pushover Analysis, ACECOMS, AIT
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 displacements with the ability of the structure to reach those displacements without failing OR indicating that it will not reach those displacements
Performance Based Design and
Pushover Analysis Technical Background By:
Iqbal Suharwardy, PhD, S.E Director Development Computers and Structures Inc., Berkeley, USA
Performance Check for Structures • Purpose
Pushover Analysis, ACECOMS, AIT
– 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 for Structures • Process – Guidelines for Seismic Rehabilitation of Buildings:
Pushover Analysis, ACECOMS, AIT
• ATC-40 • ATC-33 (FEMA 273 and 274)
– SEAOC Vision 2000 Framework
Pushover Analysis, ACECOMS, AIT
Types of Performance Checks • Linear Static Analysis • Linear Dynamic Analysis • Non Linear Static Analysis (Pushover Analysis) • Non Linear Dynamic Analysis
Performance Check Using Pushover
Force Measure
Pushover Analysis, ACECOMS, AIT
Expected Performance Point for given Earthquake
Performance Limits (IO, LS, CP)
Deformation Measure
Steps in Performance Check
Pushover Analysis, ACECOMS, AIT
• • • •
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 • Define Structural Model – Elements – Strength-Deformation properties
Pushover Analysis, ACECOMS, AIT
• Define Loads – Gravity – Lateral Load Patterns
• Select Control Displacements or Drifts • Perform Pushover Analysis
Pushover Modeling (Elements)
Pushover Analysis, ACECOMS, AIT
• 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
Pushover Analysis, ACECOMS, AIT
C
B D A
Deformation
E
Pushover Modeling (Beam Element)
Three Dimensional Beam Element Pushover Analysis, ACECOMS, AIT
Span Loads
Shear Hinge
Flexible connection
Plastic Hinge
Rigid Zone
Pushover Modeling (Column Element)
Three Dimensional Column Element Pushover Analysis, ACECOMS, AIT
Shear Hinge
Plastic Hinge
Rigid Zone
Pushover Modeling
Pushover Analysis, ACECOMS, AIT
• Types of Deformation Properties – – – – –
Axial Moment only P-M : Uniaxial P-M Interaction P-M-M : Biaxial P-M Interaction Shear
Pushover Modeling (Loads) • Start with Gravity Loads – Dead Load – Some Portion of Live Load
Pushover Analysis, ACECOMS, AIT
• Select Lateral Load Patterns – – – –
Uniform Code Static Lateral Load Distribution First Mode Combination of Modes
Pushover Analysis (Control) • Force Controlled Analysis • Deformation Controlled Analysis – Roof Displacement – Generalized Displacement Definitions Pushover Analysis, ACECOMS, AIT
• Story Drift
• Limit of Analysis – Instability – Loss of Gravity Load Carry Capacity – Excessive Distortions
Pushover Analysis (Solution Schemes) • Event by Event Strategies – Manual
Pushover Analysis, ACECOMS, AIT
• Newton-Raphson Type Strategies – Constant Stiffness iteration – Tangent Stiffness iteration
• Problem of Degradation of Strength • Ritz Method (Reduced Space) Strategies
Pushover Analysis, ACECOMS, AIT
Use of Pushover Curve (ATC-40) • • • • •
Construct Capacity Spectrum Estimation of Equivalent Damping Determine Demand Spectrum Determine Performance Point Verify Acceptance
Use of Pushover Curve (ATC-40)
Spectral Acceleration
Pushover Analysis, ACECOMS, AIT
Capacity Spectrum
Spectral Displacement
Use of Pushover Curve (ATC-40) Response Spectrum (5% Damping)
Spectral Acceleration
Pushover Analysis, ACECOMS, AIT
2.5CA Cv/T
Time Period
Use of Pushover Curve (ATC-40) Reduced Spectrum (Equivalent Damping)
Spectral Acceleration
Pushover Analysis, ACECOMS, AIT
2.5CA/Bs
Cv /(T BL)
Time Period
Use of Pushover Curve (ATC-40)
Spectral Acceleration
Pushover Analysis, ACECOMS, AIT
Performance Point
Spectral Displacement
Use of Pushover Curve (ATC-40)
Force Measure
Pushover Analysis, ACECOMS, AIT
Expected Performance Point for given Earthquake
Performance Limits (IO, LS, CP)
Deformation Measure
Use of Pushover Curve (FEMA-273)
Pushover Analysis, ACECOMS, AIT
• • • •
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
Use of Pushover Curve (FEMA-273) • Estimation of Target Displacement
Pushover Analysis, ACECOMS, AIT
– – – –
Co, Relates spectral to roof displacement C1, Modifier for inelastic displacement C2, Modifier for hysteresis loop shape C3, Modifier for second order effects
SAP2000/ETABS Pushover Options
Pushover Analysis, ACECOMS, AIT
• 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
SAP2000/ETABS Pushover Options
Pushover Analysis, ACECOMS, AIT
• 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)
SAP2000/ETABS Pushover Options Force - Deformation Relationship
Force
Pushover Analysis, ACECOMS, AIT
C
B D A
Deformation
E
SAP2000/ETABS Pushover Options
Three Dimensional Beam Element Pushover Analysis, ACECOMS, AIT
Span Loads
Shear Hinge
Flexible connection
Plastic Hinge
Rigid Zone
SAP2000/ETABS Pushover Options • Strength – Deformation and P-M-M curves can be calculated by program for:
Pushover Analysis, ACECOMS, AIT
– 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 • Gravity Load Analysis
Pushover Analysis, ACECOMS, AIT
– 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
Pushover Analysis, ACECOMS, AIT
• 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
Pushover Analysis, ACECOMS, AIT
• 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 • Visual Display for Each Step – Deformed Shape – Member Force Diagrams – Hinge Locations and Stages
Pushover Analysis, ACECOMS, AIT
• Graphs – – – – –
Base Shear VS Roof Displacement Capacity Curves Demand Curves Demand Spectra at different Damping Effective Period Lines
Pushover Analysis, ACECOMS, AIT
Examples
Example 1 P=100 Kip Gravity Load m=3.6
W36x120
10 ft
Pushover Analysis, ACECOMS, AIT
Lateral Push to 0.5ft Disp
Default M3 Pushover Hinge
Pushover Analysis, ACECOMS, AIT
Base Shear Vs Displacement
Pushover Analysis, ACECOMS, AIT
Capacity Spectrum
Example 2 P=Unit Load
Axial Force, P (Kips)
24"x24" Conc Col
12 ft
Pushover Analysis, ACECOMS, AIT
Desired Behavior
User P Hinge
2100 1700 1000
0.1
0.6
0.8
Measured Axial Displacement at Joint 2 (in)
Find Column E • Determine Column E to give Appropriate Initial Stiffness:
PL E AD
= (1700 *12*12)/(24*24*0.1) = 4250 Ksi
Axial Force, P (Kips)
Pushover Analysis, ACECOMS, AIT
Column
Desired Behavior
2100 1700 1000
0.1
0.6
0.8
Measured Axial Displacement at Joint 2 (in)
Find Column Deflection
Pushover Analysis, ACECOMS, AIT
Column
PL D AE
= [(2100-1700) *12*12)]/(24*24*4250) = 0.0235 in
Axial Force, P (Kips)
• Determine Elastic Column Lengthening when loading from 1700 to 2100 K:
Desired Behavior
2100 1700 1000
0.1
0.6
0.8
Measured Axial Displacement at Joint 2 (in)
Find Column Deflection
Pushover Analysis, ACECOMS, AIT
Column D
PL AE
= [(2100-1000) *12*12)]/(24*24*4250) = 0.0647 in
Axial Force, P (Kips)
• Determine Elastic Column Lengthening when loading from 2100 to 1000 K:
Desired Behavior
2100 1700 1000
0.1
0.6
0.8
Measured Axial Displacement at Joint 2 (in)
Find Column Deflection
Column
PL D AE
= 1000 *12*12)/(24*24*4250) = 0.0588 in
Desired Behavior
Axial Force, P (Kips)
Pushover Analysis, ACECOMS, AIT
• Determine Elastic Column Lengthening when loading from 1000 to 0 K: 2100 1700 1000
0.1
0.6
0.8
Measured Axial Displacement at Joint 2 (in)
Find Hinge Properties Hinge Properties
B
D
E
2100 1700 1000
0.8
0.7412
0.4765 0.5412
A 0.0
Pushover Analysis, ACECOMS, AIT
1000
Axial Force, P (Kips)
C
2100 1700
Desired Behavior
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
0.1
0.6
0.8
Measured Axial Displacement at Joint 2 (in)
Pushover Analysis, ACECOMS, AIT
Hinge Properties
Pushover Analysis, ACECOMS, AIT
Pushover Curve
0.8 kip/ft
W14x90
Push 0 19 2x 0 W1 x1 W8
W1 2x 19 0 W8 x1 0
W14x90
Pushover Analysis, ACECOMS, AIT
Example 3 1.2 kip/ft 0.8 kip/ft
W24x55
Example 3 M3
M3
Pushover Analysis, ACECOMS, AIT
PMM
M3 V M3 M3 MR
MR
PMM
MR
P
PMM
M3
P Legend P = Axial Hinge MR = Moment Release M3 = Moment Hinge V2 = Shear Hinge PMM = PMM Hinge
PMM MR
Pushover Analysis, ACECOMS, AIT
With W12x190 Brace
Pushover Analysis, ACECOMS, AIT
With W8x10 Brace
Pushover Analysis, ACECOMS, AIT
Conversion to ADRS Spectra ATC-40
Response Spectrum Conversion
Pushover Analysis, ACECOMS, AIT
• 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 • For Each value or Sai and Ti determine the value of Sdi using the equation 2
Pushover Analysis, ACECOMS, AIT
Ti S di S ai g 2 4
• Spectral Acceleration and Displacement at period Ti are given by
2 S ai g Sv Ti
Ti S di Sv 2
Pushover Analysis, ACECOMS, AIT
Capacity Spectrum Conversion • Capacity Spectrum from Capacity or Pushover Curve • Point by Point conversion to first mode spectral coordinates • Vi and D roof on capacity curves are converted to corresponding Sai and Sdi on capacity spectrum using: Vi S ai W
1
S di
D roof
PF 1
1, roof
Pushover Analysis, ACECOMS, AIT
Moment Hinge Properties Using M-Fi Curve
Procedure
Pushover Analysis, ACECOMS, AIT
• Plot M-Fi curve for cross-section • Estimate EI value from M-Fi Curve using the following equation M EI M EI
• Calculate Rotations from Curvature using: b
M dx EI a
• Reinforced Concrete Beam-Column CrossSection • 24”x24” • Reinforced with 12 #9 bars • Length is 12 ft
24"
24"
Pushover Analysis, ACECOMS, AIT
Example
Pushover Analysis, ACECOMS, AIT
Example
370
0.00028
Example
Pushover Analysis, ACECOMS, AIT
M EI • So EI = 370/0.00028 = 1321428.6 b M M Ip dx EI EI a • So = 0.00336 rad • Find for other Moment Values and input in Hinge Property
Considerations
Pushover Analysis, ACECOMS, AIT
• 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
Pushover Analysis, ACECOMS, AIT
Comparisons of SAP2000 and ETABS
SAP2000 vs ETABS •
SAP2000 – General Purpose FEA Software
Pushover Analysis, ACECOMS, AIT
– 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
SAP2000 vs ETABS •
SAP2000 – General output related to nodes and elements is reported
•
ETABS – Floor wise representation of results such as story drift, floor mass participation, story shear, etc.
Pushover Analysis, ACECOMS, AIT
– General Report (text files)
– Professional Report – Powerful load cases, combinations, envelopes, multiple case, etc. – Cables, Dampers, and NL Links and Hinges
– Relatively less ability to handle load combinations – Only Nonlinear links and Hinges
SAP2000 vs ETABS • SAP2000 – Supports Solid Elements
Pushover Analysis, ACECOMS, AIT
– Relatively low versatility for defining and editing grid systems
• ETABS – Does not support solid elements – Powerful grid system definition and editing
Pushover Analysis, ACECOMS, AIT
ETABS Pushover
Pushover Analysis, ACECOMS, AIT
ETABS Pushover
Pushover Analysis, ACECOMS, AIT
ETABS Pushover
Pushover Analysis, ACECOMS, AIT
SAP2000 Pushover
Pushover Analysis, ACECOMS, AIT
SAP2000 Pushover
Pushover Analysis, ACECOMS, AIT
SAP2000 Pushover
Pushover Analysis, ACECOMS, AIT
SAP2000 Pushover
Pushover Analysis, ACECOMS, AIT
SAP2000 Pushover
Pushover Analysis, ACECOMS, AIT
1 Use Load Patterns Steps to compute the Displacement (Displacement not Monitored) Divide the Specified Displacement into Steps and apply loads to attain that displacement Monitor which DOF at what level/story Save Positive Results only 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
ETABS Pushover 4 Consider P-Delta effects and Large Displacements due to gravity loads caused by each step of lateral loading
1
2
4
3
5
5 For Construction Sequence analysis. Specify which Pushover case to be applied to which stage of construction or strengthening.
Pushover Analysis, ACECOMS, AIT
SAP Pushover
1 Weather to start from unstressed condition or if more than one Pushover cases are defined then may be start the later pushover case from the final state of the pervious case
2 When the load type in 3 is set to Loads this becomes irrelevant and if the Load Type in 3 is set to Acceleration then to find modal masses, select the analysis case from which the modal masses may be
3 Specify if Loads or Accelerations needs to be applied and what is the scale factor for each load case
4 Load Application Use full load application without monitoring the displacement or use the displacement control. Also specify the DOF to be Monitored and the Joint at which the DOF is to be monitored Results Saved Save Results at only final stage of Loading or after each step. Specify Max and Min number of steps Staged Construction For Construction Sequence analysis. Specify which Pushover case to be applied to which stage of construction or strengthening Nonlinear Parameters Those explained in 2 and 4 on previous slide
SAP/ETABS Pushover Output
2 3 4
Pushover Analysis, ACECOMS, AIT
1
1 V=Base Shear D=Displacement Sa=Spectral Acceleration Sd=Spectral Disp Teff=Effective Fundamental Period Beff=Effective Viscous Damping
5
2
3 Demand Curves plotted for these Damping Ratios 4 Grey Lines are the Constant Period Lines drawing for period specified here
5 If there is additional viscous damping provided in the structure, perhaps by viscous dampers that are not specifically included in the model The Structural Behavior Types A, B and C default to the values defined for those structural behavior types in Section 8.2.2.1.1 of ATC-40 . The User Defined Kappa
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