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



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

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




C E C O M S ,

A I T

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



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 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



P


C E C O M S ,

A I T

Modeling and Analysis



P


C E C O M S ,

A I T

Summary

• The Purpose of Analysis

• The Significance of Modeling

• Analysis Types

• Linearity and Non-Linearity• Static and Dynamic Analysis



P


C E C O M S ,

A I T

Structural

Model

EXCITATION

LoadsVibrations

Settlements

Thermal Changes

RESPONSES Displacements

Strains

Stress

Stress Resultants

STRUCTURE

pv

Structural System – Analysis Model



P


C E C O M S ,

A I T

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.



P


C E C O M S ,

A I T

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



P


C E C O M S ,

A I T

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”



P


C E C O M S ,

A I T

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



P


C E C O M S ,

A I T

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



P


C E C O M S ,

A I T

Simplified Structural System

F = K D

F

K D

Loads (F) Deformations (D)

Fv



P


C E C O M S ,

A I T

The Analysis System

EXCITATION RESPONSES

STRUCTURE

pv

• Static

• Dynamic

• Elastic

• Inelastic

Eight types of equilibrium equations are possible!

• Linear

• Nonlinear



P


C E C O M S ,

A I T

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



P


C E C O M S ,

A I T

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



P


C E C O M S ,

A I T

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



P


C E C O M S ,

A I T

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



P


C E C O M S ,

A I T

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”



P


C E C O M S ,

A I T

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.



P


C E C O M S ,

A I T

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



P


C E C O M S ,

A I T

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



P


C E C O M S ,

A I T

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



P


C E C O M S ,

A I T

Basic Concepts for Analysis

• DOF (Degree of Freedom)

• Stiffness

• Static Analysis Process

• Dynamic Analysis Procedures



P


C E C O M S ,

A I T

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.



P


C E C O M S ,

A I T

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



P


C E C O M S ,

A I T

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



P


C E C O M S ,

A I T

The Structure Stiffness

Section Stiffness

Member Stiffness

Structure Stiffness

Material Stiffness

Cross-section Geometry

Member Geometry

Structure Geometry



P


C E C O M S ,

A I T

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



P


C E C O M S ,

A I T

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



P


C E C O M S ,

A I T

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



P


C E C O M S ,

A I T

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



P


C E C O M S ,

A I T

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



P


C E C O M S ,

A I T

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



P


C E C O M S ,

A I T

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



P


C E C O M S ,

A I T

Pushover Analysis

• One Dimensional Static Loads• No Energy Dissipation

• Inertia Forces Not Considered

• Defined One Failure Mode• Higher Mode Effects Neglected



P


C E C O M S ,

A I T

The Modal Analysis



P


C E C O M S ,

A I T

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



P


C E C O M S ,

A I T

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



P


C E C O M S ,

A I T

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



P


C E C O M S ,

A I T

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




C E C O M S ,

A I T

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




C E C O M S ,

A I T

Eccentric and Concentric Response

Mode-1 Mode-2 Mode-3

Symmetrical Mass and

Stiffness

Unsymmetrical Mass

and Stiffness




C E C O M S ,

A I T

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




C E C O M S ,

A I T

Special Analysis Problems




C E C O M S ,

A I T

Base Isolation

Isolators




C E C O M S ,

A I T

Building Impact

Building Impact

Analysis




C E C O M S ,

A I T

Dampers

Friction device

Concentrated damper

Nonlinear element




C E C O M S ,

A I T

Gaps and Joints

Bridge Deck ABUTMENT

Gap Element

Tension only element




C E C O M S ,

A I T

Hinges

2 Rotational DOF

Degrading Stiffness?

PLASTIC HINGES




C E C O M S ,

A I T

Dampers

Mathematical Model

F= ku

F= CvN

F= f(u,v,umax

)

Mechanical Damper




C E C O M S ,

A I T

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




C E C O M S ,

A I T

Uplift

Uplifting

Allowed

FRAME WITH UPLIFTING ALLOWED




C E C O M S ,

A I T

Structural Modeling




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




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




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




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

•




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




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




C E C O M S ,

A I T

Design Methods and Concepts



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




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




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




A C E C O M S ,

A I T


• 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



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


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




A I T

Cross-section Reponses

• Stresses – Tension

– Compression

– Shear > Tension-Compression

•Strains

– Normal strain

– Shear Strain

• Deformations

– Rotation

– Shortening

– Shearing

– Twisting




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




A I T

Capacity Interaction Surface

MxMy

P




A I T

P-M and M-M Interaction Curves




A I T

The Moment Curvature Curve






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




A I T

Sections After Strengthening




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




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




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




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




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




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




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




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




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




A C E C O M S ,

A I T

The Moment Curvature Curve




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




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)




A C E C O M S ,

A I T

How to Read M-Phi Curve




A C E C O M S ,

A I T

Outputs from M-Phi Curve

1 -Yield Point2 -Failure Point

u

y Ductility

3 -




A C E C O M S ,

A I T


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




A C E C O M S ,

A I T


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




A C E C O M S ,

A I T


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




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




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




A C E C O M S ,

A I T


dx x EI

M b

a

D


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




A C E C O M S ,

A I T


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


y

s

Rebar Centroid

NA

W

M Phi C d D ili




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





C fi t M d l d D tilit




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




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)






A C E C O M S ,

A I T

IntroducingPushover Analysis

Th P h A l i




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



Performance Based Design Basics




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




A C E C O M S ,

A I T

Pushover Spectrum

Pushover Demand Curves




A C E C O M S ,

A I T

Pushover Demand Curves

Earthquake Push on Building




A C E C O M S ,

A I T

Earthquake Push on Building



Pushover Capacity Curves




A C E C O M S ,

A I T

Pushover Capacity Curves

Demand Vs Capacity




A C E C O M S ,

A I T

Demand Vs Capacity

Non linearity in Pushover




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.



Important Considerations




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




A C E C O M S ,

A I T


• 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.





A C E C O M S ,

A I T


• 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).





A C E C O M S ,

A I T


•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.




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 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).





, A C E C O M S ,

A I T


•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




, 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



Performance Check for Structures




, 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



Types of Performance Checks




, 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



Steps in Performance Check




, 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




, 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)




, 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



Pushover Modeling (Beam Element)




, 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)




, A C E C O M S ,

A I T

g ( )

Plastic Hinge

Shear Hinge

Rigid Zone

Three Dimensional Column Element

Pushover Modeling




, 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







Use of Pushover Curve (ATC-40)




, A C E C O M S ,

A I T

( )

•Construct Capacity Spectrum

• Estimation of Equivalent Damping

• Determine Demand Spectrum

• Determine Performance Point






, 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







, 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)





, 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



Use of Pushover Curve (FEMA-273)




, A C E C O M S ,

A I T

•Displacement Coefficient Method

• Estimate Target Displacement


• Estimation of Target Displacement

– Estimate effective elastic stiffness , Ke

– Estimate post yield stiffness, Ks

– Estimate effective fundamental period, Te

– Calculate target roof displacement



SAP2000/ETABS Pushover Options




, 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







A C E C O M S ,

A I T

Force - Deformation Relationship

A

B

C

D E

Deformation

F o r c

e





A C E C O M S ,

A I T

Plastic Hinge

Flexible connection

Span Loads Shear Hinge

Rigid Zone

Three Dimensional Beam Element





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)





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





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





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





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





Base Shear Vs Displacement




A C E C O M S ,

A I T





Find Column E




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




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


(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





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


(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





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


(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




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




A C E C O M S ,

A I T

Pushover Curve



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




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



With W12x190 Brace




s ,

A C E C O M S ,

A I T

With W8x10 Brace




s ,

A C E C O M S ,

A I T

Conversion to




A C E C O M S ,

A I T

ADRS Spectra ATC-40

Response Spectrum Conversion




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




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




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




A C E C O M S ,

A I T

Using M-Fi Curve

Procedure




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




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




A C E C O M S ,

A I T

0.00028

370

Example

M




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




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




A C E C O M S ,

A I T

SAP2000 and ETABS

SAP2000 vs ETABS

SAP2000 ETABS




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




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




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




A C E C O M S ,

A I T

ETABS Pushover




A C E C O M S ,

A I T

ETABS Pushover




A C E C O M S ,

A I T

SAP2000 Pushover




A C E C O M S ,

A I T

SAP2000 Pushover



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




i s ,

A C E C O M S ,

A I T

SAP2000 Pushover




i s ,

A C E C O M S ,

A I T

SAP2000 Pushover




i s ,

A C E C O M S ,

A I T

ETABS Pushover1

Use Load Patterns

Steps to compute the




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



SAP/ETABS Pushover Output



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

Pushover Analysis using ETABS and SAP2000

Documents