Concrete Composite Cross-section

Response for Pushover Analysis

Asian Center for Engineering Computations and Software

Asian Institute of Technology, Thailand

Karachi, Pakistan

December ____, 2007

Presented By:

Naveed Anwar, D.Eng.

Objectives

• To present a unified approach in determining the

cross-sectional parameters needed for carrying-

out pushover analysis.

• To discuss the methodology that will handle

various factors such as the effects of pre-stress,

differential creep, shrinkage, confinement, staged

construction, strain hardening, tension stiffening,

relaxation and bond slippage etc., in a single

consistent computational procedure.

• To discuss cross-section capacity at specified

failure criteria.

Contents

• About Pushover Analysis

• Performance Based Design

• Action Deformation Curve

• Determining Cross Section Response

• Ductility and Stiffness

• Conclusions

About

Pushover Analysis

The Pushover Analysis

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

• Pushover analysis is a static, nonlinear procedure in which the magnitude of the structural loading is incrementally increased in accordance with a certain predefined pattern

• With the increase in the magnitude of the loading, weak links and failure modes of the structure are found (ATC-40, 1996)

• The procedure also helps to identify ductility requirements for various members

The Pushover Analysis

• The loading is monotonic with the effects of the cyclic

behavior and load reversals being estimated

• 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

• Static pushover analysis is an attempt by the structural

engineering profession to evaluate the real strength of

the structure and it promises to be a useful and

effective tool for performance based design.

Why Pushover Analysis

• Buildings do not respond as linearly elastic systems during strong ground shaking

• Improve Understanding of Building Behavior– More accurate prediction of global displacement

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

– More reliable identification of “bad actors”

• Reduce Impact and Cost of Seismic Retrofit– Less conservative acceptance criteria

– Less extensive construction

• Advance the State of the Practice

The Pushover Curve

Hinge Deformations

The roof displacement and pushover curve

is a result of rotations at hinges

Hinge Formation and Deformation

Pushover Modeling (Elements)

S.N.

Structural

Member

Type

Deformation Type of Hinge

1 Truss Yielding and Buckling Axial hinge

2 3D Beam Major direction

Flexural and Shear

M-M hinge, Shear (V)

hinge

3 3D Column P-M-M Interaction and

ShearP-M-M hinge

4 Panel Zone Shear Yielding Shear hinge

5In-Fill

Panel Shear Failure Shear hinge

6 Shear Wall P-M-Shear InteractionP-M hinge, Shear

hinge

7 Spring For foundation

modelingNon-linear spring

Hinge Properties

•Five points labeled A, B, C,

D, and E are used to define

the force deflection behavior

of the hinge

•Three points labeled IO, LS

and CP are used to define

the acceptance criteria for

the hinge

•IO- Immediate Occupancy

•LS- Life Safety

•CP-Collapse Prevention

Hinge Properties

• Point A is always the origin

• Point B represents yielding. No deformation occurs in the hinge up to point B, regardless of the deformation value specified for point B. The displacement (rotation) at point B will be subtracted from the deformations at points C, D, and E. Only the plastic deformation beyond point B will be exhibited by the hinge

• Point C represents the ultimate capacity for Pushover analysis

• Point D represents a residual strength for Pushover analysis

• Point E represents total failure. Beyond point E the hinge will drop load down to point F (not shown) directly below point E on the horizontal axis. To prevent this failure in the hinge, specify a large value for the deformation at point E

Hinge Properties

• Plastic Hinge/Zone Length

• The plastic zone length must be

chosen so that the actual M-ψ

relationship for the beam can be

used.

• This requires calibration against

experiment.

• A reasonable length seems to be 0.5

times the beam depth.

Elastic

MiMj

Inelastic

Plastic zone length

Inelastic

EI

Moment, M

Curvature, ψ

Hinge Properties

• Plastic Hinge/Zone Length – For a reinforced concrete cantilever column. Paulay and Priestley

(“Seismic Design of Reinforced Concrete and Masonry Buildings”, Wiley, 1992, p. 142) suggest the following plastic zone length:

• Lp = 0.08H + 0.15d fy

• where: – Lp = plastic zone length

– H = cantilever height

– d = reinforcing bar diameter

– fy = steel yield strength

– (all in kip-inch units)

Performance

Based Design

Spectr

al

accele

rati

on, S

a

Period, T

Spectr

al

accele

rati

on, S

a

Spectral displacement, Sd

T1

T2

T3

(a) (b)

Performance Based Design

• Performance is generally of concern for lateral

loads such as earthquake and wind.

• The main factor that affects 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 is also of great

importance for lateral load demands.

Performance Based Design

• The performance based design is basically a

comparison between the performance curve of

the section, member or entire structure against

the demand curve for the section, member or the

structure as a whole.

• Performance design ensures that structure as a

whole reaches a specified demand level.

• Performance design can include, both service

and strength design levels.

Seismic Performance

• The Seismic Performance evaluation or design

ensures that the structure can “Perform Well”

during a specified earthquake based on specified

criteria

• The Seismic Performance of a structure can be

evaluated by using a Static-Nonlinear Pushover

Analysis

• Pushover Analysis relies on properties and

performance of Plastic Hinges located in

structural model

Seismic Performance

• Stress Strain + Cross-

section

• Moment Curvature

Curve

• Hinge Rotation Curve

• Structure Load-

Displacement Curve

• Response Spectrum

Curve

• Local Factors

• ADRS Curve

• Expected

Performance

• Seismic

Performance

Point

Performance Design

Time Period

Spectr

al

Accele

ration

Spectral Displacement

ADRS

Spectr

al

Accele

ration

Spectral Displacement

Performance Point

Base

Shear,

V

Displacement,

Pushover Curve

Base

Shear,

V

Earthquake Response Spectra

F2

F1

Base Shear, V

Plastic

Hinge

dL A

A

B

D

Section A-A

Strain

Str

ess

Concrete

Strain

Str

ess

Steel

Rotation

Mom

ent

Curvature

Mom

ent

Seismic Performance Evaluation

Performance Curve

• The nonlinear response is described with the pushover curve, which plots the base shear versus the roof displacement.

• This pushover curve can be transformed into the “capacity spectrum” using the structure’s original elastic dynamic properties.

• This capacity spectrum is represented in the Acceleration Displacement Response Spectrum (ADRS) format, using spectral displacement and spectral acceleration.

• The response spectrum or so-called demand spectrum for considered ground motions could also be plotted in the ADRS format.

roof

F1

F2

F3

F4

V

Nonlinear Pushover

Analysis

Convert to

ADRS format

Equivalent SDOF

system

Spe

ctra

l ac

cele

rati

on, S

a

Spectral displacement, Sd

Capacity spectrum

Bas

e sh

ear,

V

Roof displacement, roof

Capacity curve

Structures Performance – Capacity

Spectrum

Capacity Spectrum Conversion

• Capacity Spectrum from Capacity or Pushover Curve

• Point by Point conversion to first mode spectral coordinates

• on capacity curves are converted to corresponding Sai and Sdi on capacity spectrum using:

roofi andV

1W

VS i

ai roof

roof

diPF

S,11

where, α1, PF1 are respectively modal mass coefficient and participation

factor for the first natural mode of the structure and Ø1, roof is the roof

level amplitude of the first mode.

Spectr

al

accele

rati

on, S

a

Period, T

Spectr

al

accele

rati

on, S

a

Spectral displacement, Sd

T1

T2

T3

(a) (b)

Seismic Demand Spectra

Standard Response Spectrum ADRS Format

Response Spectrum Conversion

• Acceleration-Displacement Response Spectra

(ADRS)

• Every Point on a Response Spectrum curve has

a unique

– Spectral Acceleration, Sa

– Spectral Velocity, Sv

– Spectral Displacement, Sd

– Time, T

Response Spectrum Conversion

• For Each value or Sai and Ti determine the value

of Sdi using the equation

• Spectral Acceleration and Displacement at period

Ti are given by

gST

S aii

di 2

2

4

v

i

ai ST

gS2

vi

di ST

S2

Performance Point

• The point where the capacity of structure

matches the demand for the specific earthquake

• The performance of the structure can be judged

based on the location of the performance point

• Performance Based Design Levels

– Fully Operational

– Operational

– Life Safe

– Near Collapse

– Collapse

Performance Point

Action

Deformation Curve

Load-Deformation Relationship

• A - The point up to which the relationshipbetween load and deformation can beconsidered nearly linear and the deformationsare relatively small

• B - The point at which the deformation startsto increase suddenly, at more or less constantload value or with relatively small increase inthe load

• C - The point at which the load value starts todrop with increasing deformations

• D - The point where load value become nearlyzero and member loses all capacity to carryany loads and collapses or fails completely

• Region OA corresponds to the serviceabilitydesign considerations and working strength orallowable strength design concepts related tolinear, small deformation state

DEFORMATION

LO

AD

PA

B C

D

O-A - Serviceability RangeA - Cracking LimitB - Strength LimitC-D - Failure Range

O

DEFORMATIONL

OA

D

PA

B C

D

O-A - Serviceability RangeA - Cracking LimitB - Strength LimitC-D - Failure Range

O

Load-Deformation Curves

1. Moment-curvature curve is probably the most important curve for flexural design of beams, columns, shear walls and consequently for building structures

2. It is also probably the least understood or utilized in normal design practice.

3. Many of the design codes and design procedures or design handbooks do not provide sufficient information for computation and use of these curves.

Axial Load and Axial

shortening

Shear force and shear

deformation

Moment and

curvature

Torsion and twist

What is Curvature

• In geometry, it is rate of change of rotation

• In structural behavior, Curvature is related to Moment through Stiffness

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

• It is simply the inverse of the radius of curvature that can be obtained from the slope of the strain profile

Curvature = e / C (radian / unit

length)

C

e

The Moment-Curvature Curve

• Significant information can be obtained from Moment Curvature Curve to compute:– Yield Point

– Failure Point

– Ductility

– Effective Stiffness

– Crack Width

– Rotation

– Deflection

– Strain

• The Moment-Curvature relationship is also the basis for

the capacity based, or Performance Based Design

methods, especially useful in the analysis of structures

using the Static Pushover Method, and in determining the

rotational capacity of plastic hinges formed during high

seismic activity

It is interesting to note that the Moment-Curvature curves obtained from the stress resultants are independent of the member geometry or the moment diagram, and for a given axial load is a property of the cross-section, just like the capacity interaction surface.

Determining

Cross Section Response

The Response and Design

Building Response

Member Response

Section Response

Material Response

Building Analysis

Member Actions

Cross-section Actions

Material Stress/StrainLoad Capacity

Applied Loads

Fro

m L

oad

s to

Str

esse

s

Fro

m S

tra

ins

to R

esp

on

se

General Cross-sections

• Based on Material– Un-reinforced, Reinforced, Prestressed Concrete

– Fiber Reinforced Concrete, Ferrocement

– Steel-Concrete Composite

– Hot Rolled Steel, Cold formed Steel

– Structural Timber

• Based on Geometry– Circular, Elliptical, Rectangular, Flanged, Polygonal

– Hollow, Solid, Voided

– Symmetrical, Unsymmetrical, Asymmetrical

• Based on Compressed Zone (for steel sections)– Slender Sections/Thin Walled Sections

– Compact Sections

– Plastic Sections

Generalized Cross-section

• Any attempt to integrate the design approaches must

begin at the cross-section level

x y

n

1i

iiz )y,x(fAdydxy,xfN

i

n

1i

ii

x y

x y)y,x(fAy.dydxy,xfM

i

n

1i

ii

x y

y x)y,x(fAx.dydxy,xfM

The three equations can be reduced to

two if the cross-section is oriented in

such a way that the bending axis

corresponds with Y-axis - Nz and My.

These 2 equations will generate the

capacity interaction surface.

Original Cross-sections

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

Compact Built-up steel

section

Reinforced concrete,

composite sectionComposite section

Retrofitted or Strengthened Sections

• Several Materials

• Different Material Properties

• Different Age

• Different Interface/ connection

MxMy

P

Capacity Interaction

Surface - PMM

Moment-Curvature

Curve

Determining Cross-section Response

Material Stress-Strain Curves

Cross-section Dimensions

Capacity

Interaction Surface

M-M Curve

Moment-Curvature Curves

P-M Curve

Given P value

Given Moment Direction

Given Moments Given Axial Load

•Moment for Given Curvature

•Curvature for Given Moment

•Yield Moment

•Stiffness

•Ductility

•Moment for Given Load

•Load for Given Moment

•Capacity Ratio

•Mx for Given My

•My for Given Mx

•Capacity Ratio

Per

form

ance

Str

eng

th

Generation of Moment-Curvature Curves

1. Fix the strain first and iterate on the depth of neutral axis .

2. compute moment and axial force using aforementioned equations, until equilibrium with the axial load is achieved.

3. The corresponding moment capacity at that depth of neutral axis and strain level is then used, along with the curvature at that point.

4. Once one moment-curvature set is obtained, the extreme fiber strain is changed and another solution is attempted to obtain yet another pair of moment and curvature.

5. Several points are computed, using a small strain increment to plot a smooth curve.

6. The generation of Moment-Curvature curve can be terminated based on any number of specific conditions such as, the maximum specified strain is reached, the first rebar reaches yield stress or any other strain level, the concrete reaches a certain strain level.

Note: During the generation of the Moment-Curvature curve the failure or key response points can be recorded and displayed on the curve

Ductility

and Stiffness

Lo

ad

Deformation

Dy Du

Ductility – Definition and Usage

• Ductility can be defined as the “ratio of deformation and a given stage to the maximum deformation capacity”

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

Yield/

Design

Strength

Lo

ad

Deformation

Dy Du

Ductility = Dy / Du

Stiffness

• To satisfy serviceability criteria, the following should be considered:

– Section and element geometry.

– Material property.

– Extent and influence of cracking in members.

– Contribution of concrete or masonry in tension.

• From principles of structural mechanics

– Relationship between geometric properties of members and the modulus of elasticity for the material.

• Deformations under the action of lateral forces are controlled.

Stiffness

Deformation

Fo

rce

Curvature

Mo

me

nt

Section Stiffness

Member Stiffness

Structure Stiffness

Material Stiffness

Structure Geometry

Member Geometry

Cross-section Geometry

Rotation

Mo

me

nt

Strain

Str

ess

Effective Stiffness

• Cross-section stiffness “EI” is only valid in the linear-elastic range.

• However, stiffness can indirectly (and more accurately) be determined from the slope of the M-φ curve, termed as “effective stiffness.”

• Effective stiffness includes effect of cracking, the stiffness contribution due to reinforcement and the change in the modulus of elasticity of concrete and steel due to certain level of strain.

Moment-Curvature

Relationship

Effective Stiffness

• If the moment curvature curve has been generated considering the short and long term effects, then the stiffness would automatically be adjusted for all those effects such as confinement, creep, shrinkage, temperature change, etc.

• It can be used directly in structural analysis such as stiffness matrix approach and the linear or non-linear finite element formulations.

• It can also be used to determine the deflections using the conventional deflection equations and formulae.

• The member and structure level non-linearity in stiffness can be incorporated by using P-∆ analysis and large displacement analysis while carrying the Pushover analysis as is done in SAP2000 and ETABS.

Strength Design

• The P-M interaction is in fact denoted by the

yield or capacity point on the Moment-

Curvature curve as shown in Figures.

• The factors that effect strength include the

basic material strength, the cross-section

dimensions, the amount of rebars and the

overall structural framing conditions.

• The strength design is a pre-requisite to the

performance design and infact required before

the Pushover Analysis can be carried out.

• This is because the cross-section dimension

and reinforcement should be known for

computation of the hinge properties that are

needed for Pushover analysis.

Serviceability Design

• The serviceability requirements are generally specified in terms of limit on stresses, limit on deflection and deformations and on crack width or crack spacing.

• The deflections can be directly computed by using the moment-Curvature curve and the moment diagram due to applied loads.

Plot M-Phi Curve

Determine curvature

at known moment

Determine curvature

at known moment

Determine Flexural

Stiffness (EI)

Determine Flexural

Stiffness (EI)

Determine SlopeDetermine Slope

Determine DeflectionDetermine Deflection

Determine StrainDetermine Strain

Determine Crack

Spacing/Width

Determine Crack

Spacing/Width

MEI

dxEI

Mb

a

dxxEI

Mb

a

c

XW s

s

WX

Computation of Deflection

1. Design the cross-section for moment and axial load at various locations along the beam-Column.

2. Generate the moment-curvature curves for the designed sections.

3. Plot the moment diagram and the axial load diagram. The axial load is generally constant along the length of the member so it can be used as the controlling variable while generating the Moment-Curvature curves.

4. For various locations along the member length, read the curvaturefor the applied moment using the appropriate moment-curvaturecurves. Use interpolation if the cross-section changes along thelength.

5. Plot the M/EI diagram along the length of the member.

6. To compute the deflection at a specific location, calculate the area ofthis M/EI diagram up to that point, starting one end of the member.

Conclusion

Conclusion

• The cross-sections and their properties are key

components in almost all aspects of theoretical

formulation, computer applications, practical

analysis and design.

• The cross section property of general section is

made up of several shapes and different

materials and requires good theoretical

knowledge and aid of computing tools.