2. Seismic Performance Evaluation of Reinforced Concrete Building by Displacement Principles

8/10/2019 2. Seismic Performance Evaluation of Reinforced Concrete Building by Displacement Principles

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The Institution of Engineers,

Malaysia

Universiti

Teknologi MARAUniversiti Malaya


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12th

International Conference on Concrete Engineering and Technology12 14 August 2014

Seismic Performance Evaluation of Reinforced Concrete

Buildings by Displacement Principles

Nelson LamReader in Civil Engineering

Infrastructure Engineering

The University of Melbourne

John WilsonExecutive Dean

Faculty of Engineering and Industrial Sciences

Swinburne University of Technology

Abstract

The post-yield displacement capacity of a structure subject to the transient actions of an

earthquake can be used to trade-off its ultimate strength requirements meaning that the

strength demand on the structure can be reduced should the capacity to displace be increased.

Thus, the ability of a structure to deform is as important as its ability to resist forces and

moments in ultimate conditions. However, contemporary design practices for modelling

deflection of a reinforced concrete structure are mostly aimed at ensuring fulfilment of

serviceability requirements. Force-displacement capacity models for lightly reinforced RC

columns and structural walls are introduced in this paper in a hand calculation format which

is suitable for use in design practices. The determination of the force-displacement capacityrelationships forms part of a performance based seismic evaluation of the structure. The

presented calculation techniques have been verified by comparisons with experimental results

as reported in the literature.

Keywords: Seismic performance, displacement-based design, reinforced concrete

1. Introduction

In designing reinforced concrete members for dead loads, live loads and wind loads bending

moment values of members at their critical cross sections are checked against the respectivedesign moment of resistance. Similar checks for shear and axial actions have to be

undertaken. This force based approach of design does not require the post-elastic deflection

of the member to be calculated. Contemporary design practices for modelling deflection of a

reinforced concrete structure are mostly aimed at ensuring fulfilment of serviceability

requirements in pre-yield conditions.

In reality, when a structure is subject to the transient actions of earthquakes, impact and blasts

its post-yield displacement capacity can be used to trade-off its ultimate strength

requirements meaning that the strength demand on the structure can be reduced should the

capacity to displace be increased. Thus, the ability of a structure to deform is as important as

its ability to resist forces and moments in ultimate conditions. However, in a conventionalforce-based design procedure, practising engineers need not be involved in modelling post-


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elastic displacement given that the trading-off phenomenon as described can be taken into

account by what is known as the behaviour factor (or structural response factor) which is

dependent on prescriptive design and detailing requirements (Eurocode 8, 2004).

The behaviour factor approach which relies on empirical information, and experience, to

justify its recommendations is simple to apply but has the shortcomings of lack oftransparencies. There are plenty of scope for making improvements to conventional design

practices which are mostly restricted to comparing applied loads with strength capacities.

Whilst empirical data on limited ductile structures (and their performance in a low seismicity

conditions) is scarce a more direct, and transparent, approach to design is warranted.

There has been ever growing references to the use of displacement principles for modelling

seismic actions in structures including RC building structures. Seismic design checks

incorporating displacement as the guiding principles can be made reliable and simple to

comprehend, and apply, provided that the displacement demand and displacement capacity

behaviour of the structure is known. Importantly, the trading-off phenomenon as described is

modelled effectively by this approach. The writing of the text book on displacement basedmethod of seismic design by Priestley Calvi & Kowalsky (2007) represents a milestone of

achievement in the development of this approach in the global context.

With new modeling techniques that have been developed by the authors and co-workers for

over a decade it is now possible to obtain predictions of how much horizontal drift a building

is expected to experience for given projected earthquake scenarios and subsoil conditions.

(eg. Lam & Wilson, 2004; Wilson & Lam, 2003 & 2006; Lumantarna et al., 2010, 2012 &

2013). Reliable, and unbiased, predictions of the seismically induced displacement demand of

the structure can now be made even for countries where little strong motion data has ever

been collected for conventional empirical modelling. Meanwhile, research has also been

undertaken to develop reliable correlations of the state of damage sustained by a reinforced

concrete structure with the amount of drift it is experiencing (eg. Wilson et al., in press;

Wibowo et al., 2013 & 2014a & b). The risks of collapse of the building in an earthquake can

also be ascertained should the amount of drift to cause the structure to lose its gravity load

carrying capacity be known. Refer also Wilson & Lam (2008) for a state-of-the-art review on

this topic from the perspectives of low and moderate seismicity regions.

This paper aims to provide a succinct summary of calculation techniques that are suitable for

use in a design office for estimating the displacement of a reinforced concrete column, or

structural wall, at the limit state of yielding, loss of horizontal load carrying capacity and

vertical load carrying capacity (i.e. limit state of collapse). These techniques were derivedfrom materials presented in international peer review journal articles of Wilson et al.(2014)

and Wibowo et al. (2013 & 2014a & b). Rigorous evaluation of the techniques based on

comparison with results from laboratory experimentations have been presented in the cited

academic articles and are not reproduced in the paper. The techniques presented have been

verified for the following range of parameters for structural walls:

- Aspect ratio (a) : 0.5 a4.0- Axial load ratio (n) : 0.02 n 0.50- Longitudinal reinforcement ratio (v): 0.2% v 2.0%- Transverse reinforcement ratio (h): 0.0% h 1.0%


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Schematic diagrams showing the monotonic force-displacement relationships for RC

columns and walls are shown in Figures 1a & 1b respectively. A force-displacement capacity

diagram representing structural walls and columns in support of a building can be

transformed and overlaid on an Acceleration-Displacement Response Spectrum Diagram

representing the projected seismic actions forming part of the seismic performance evaluation

process. Detailed descriptions of the methodology can be found in Wilson & Lam (2006).

Figure 1 Force-displacement models of RC columns and walls

2. Cracking (Fcr, cr)

The lateral moment of resistance to crackingMCR(and the associated lateral cracking strength

FCR) and displacement of a RC column, or a RC wall, can be calculated by employingclassical beam theory assuming linear elastic uncracked behaviour of the concrete in order

that gross-sectional properties may be assumed for all cross-sections. If provisions in the

Australian standard (AS3600) are to be adopted the design flexural tensile strength of

concrete may be taken as 0.6(f c) where fc is the characteristic cylinder strength of

concrete. Recommendations by another valid code of practice may be adopted as appropriate

given that the modelling principle is generic.

With lightly reinforced structural walls it is important to check the value ofMCR against the

ultimate moment of resistanceMuof the critical cross section. Should the value ofMCR>Mua single crack is likely to form at the base of the wall resulting in a potential concentration of

plastic deformation leading to non-ductile behaviour (Wibowo et al.,2013).

The curvature of the critical cross section at the state of cracking (cr) can be estimatedsimply by dividingMCR by the flexural rigidity,EIgross, of the cross section whereIgrossis the

gross second moment of area. The amount of displacement, cr, of a singly loaded cantilevermodel of length his accordingly equal to cr h

2/3 for a cantilever model and cr h

2/6 for a

column with both ends fixed. The corresponding drift ratio, cr/h , is typically in the order of0.10 % (Wilson et al., in press).

FCR

Faf

Fy

Flf

Fu

Lightly reinforced moderate and slender walls (a>1)

Lightly reinforced squat walls (a < 1)

FCR

Fy

Fu

Flf

CR y u lf af CR y peak m

(a) Columns (a) Walls


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3. First Yield (Fy, y)

For both RC columns and structural walls the moment of resistance of the critical cross

section at the state of initial yield,My(and the associated lateral initial yield strength Fy) may

be calculated using classical free body diagram analysis of the cross section which ensures

strain compatibility based on the assumption of plane sections remaining plane. The

calculations can be simplified further by assuming zero tensile strength of the concrete which

means ignoring contributions by tension stiffening which can be significant with lightly

reinforced and lightly axially loaded sections. This type of calculations can be implemented

by what is known as fibre element analysiswhich is operable in a user-friendly purposely

developed computational environment open to public access (eg. RESPONSE 2000;

CUMBIA 2007) or on EXCEL spreadsheets (Lam et al., 2011). For members subject to

significant axial pre-compression errors arising from the assumption of zero tensile strength

of concrete tends to be minor (Wibowo et al., 2014b). Furthermore, the effects of tension

stiffening is not so important in conditions of cyclic stress reversals which is expected in the

response of a structure to severe earthquake ground shaking (Wibowo et al., 2014b citingrecommendations by Park & Paulay, 1975).

An alternative simplified method of estimating the value of My circumventing the need of

fibre element analysis is to first undertake traditional (code based) calculations for

determining the ultimate flexural strength (Mu) of the cross section and then take My to be

equal to 0.8 timesMu(Wilson et al.,in press).

Section curvature at first yield, y, can be found simply by dividing the calculated value ofMyby the effective flexural rigidity, EIeff, of the cross section where Ieffis the effective second

moment of area and is not to be confused with the gross second moment of area, Igross. The

amount of displacement,y, is accordingly equal to y h2

/3 for a cantilever model and y h2

/6for column with both ends fixed. According to review of the literature by Wilson et al. (in

press) the following recommendations have been made for rectangular cross-sections to take

into account the important influence of the axial pre-compression of the cross section :

(a) FEMA356 (2000)

Ieff = 0.7Ig for axial load ratio n 0.5 (1a)Ieff = 0.5Ig for axial load ratio n 0.3 (1b)Ieff = 0.5Ig + (n 0.3)Ig for axial load ratio 0.3 < n < 0.5 (1c)

(b) Paulay & Priestley (1992)

gross

y

eff Inf

I

+=

100

(2)

The accuracies of results obtained from equations (1) and (2) have been evaluated by the

authors based on comparison with results from fibre element analyses. The comparative

analyses were based on a RC column with square cross section, longitudinal reinforcement

ratio, v, of 1% and axial pre-compression ratio ranging from 0 to 0.6 (Figure 2a). Thesensitivity of the moment-curvature relationships to changes in the value of axial pre-

compression is evident from results of the simulations (Figure 2b). A bi-linear model was

then calibrated to match with each of the curves that were simulated from the fibre elementanalyses. The effective stiffness value (EIeff) of the column was then taken as the slope of the


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calibrated bi-linear model. It is shown in the comparative analyses that estimates made by

equation (2) were highly consistent with those from fibre element analyses (Figure 2c).

Estimates of EIeff based on the use of equations (1a) (1c), hence estimates for yand y,have been found to be conservative meaning that estimates for the value of y could be

lower than the actual values. Estimates for yis expected to be particularly conservative for

shear dominated structural walls because shear deformation has been ignored in thecalculations. On the other hand, these same equations can underestimate the value ofEIeff for

lightly reinforced structural walls (v


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y, and plastic displacement,pl, which can be estimated using equation (3) the basis of which

can be illustrated using the schematic diagram of Figure 3. The value of u can be obtainedusing equation (4) in the final step of the calculation.

Figure 3 Curvature and Rotation of Plastic Hinge at Peak Conditions

( )

)4(

hingeplastictheofpointcentretheandloadingofpointthefromdistancetheis

entreinforcemallongitudinmaintheofdiametertheis

entreinforcemofstrengthultimateis(2007)Kowalsky&CalviPriestley,bytionsrecommendaperas

(3f)wallsstructuralfor0.0221.010.2

estimatecutfirstaas2

or2

astakenbecanchlength whihingeplastictheis

loadingofdirectionin thecolumntheofdimensiongrosstheis

steelMPa500for0.0025toequalisandyielding;ofonsetat thesteelofstraintheis

)3(2

or2

wallstructuraltheoflengththeisconcrete;ofdeptheffectivetheis

)3(6

1on takingbased024.0

or024.0

intosimplifiedbecan(3c)Eqn

zonencompressiotheofdepththeisor

0.004astakenbemayandspallingofonsetat theconcreteofstraintheis

)3(.

or.

)3(

(3a)where

w

plyu

b

u

byw

y

u

p

wp

y

w

yy

y

w

u

w

u

uu

cu

wu

cu

u

cu

upyup

ppl

Z

d

f

dfhf

fL

DL

D

eD

d

dk

d

KdK

cKdK

bL

Z

+=

++

=

=

==

==

=

5. Ultimate Conditions (Flf, lf or m)

RC columns and lightly reinforced squat walls (a 1) can be susceptible to rapiddeterioration in their lateral resistance when displaced beyond the peak limit because of

degradation in shear strength. The loss of lateral resistance due to shear degradation is

0.004

Ku.d

u

0.0025

D/2

y

Lp

p

u -y

(a) Curvature at onset of spalling

(b) Yield Curvature (c) Plastic Hinge Rotation


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6. Collapse Conditions (Faf,af)

The displacement limit of axial failure of RC columns has been found to be most sensitive to

the axial load ratio (n) as revealed by results of parametric studies (Wibowo et al., 2014b).

Transverse reinforcement ratio (h) is influential too as shown in Figure 5. The drift limitpredictions are based on 50% degradation in the lateral resistance of the member (ie. from Futo Faf= 0.5Fu). In regions of low and moderate seismicity RC columns and walls are typically

unconfined. Thus, the lowest curve in Figure 5 is to be adopted for determining the value of

af for this class of columns. Longitudinal reinforcement ratio (v) is also an importantparameter but influence of this parameter is minor in conditions of high pre-compression.

Conservative estimates of the value of af (in terms of percentage drift) are summarised inTable 1 for a range of axial load ratios based on recommendations made in Wibowo et al.

(2014b).

Figure 5 Axial load failure drift limitExcerpt from Wibowo et al. (2014b)

Table 1 Conservative estimates of axial load failure drift limit

Axial load ratio Axial load failure drift limit (%)

0.1 3.5

0.2 1.9

0.3 1.1

0.4 0.9

7. Worked Example

The methodology for constructing aforce-displacementcapacity model is illustrated with the

worked example of a column which is 4 m in length, has both ends fixed, and is subject to an

axial force of 1000 kN. A typical cross-section of the column is shown in Figure 2a. Concrete

grade is 40.


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Basic properties

( )

MPa5.7orkN/m567042.0

1000

MPa8.3406.0'

14.0104042.0

1000

'

kNm83200I

m0026.012

42.042.0I;m42.0

:propertiessectionalGross

GPa3240043.024

2

2

32

2g

43

g

22

5.1

==

==

=

==

=

=

==

==

g

t

cg

c

g

c

A

P

f

fA

Pn

E

A

E

At state of cracking

approx.mm4orm0037.06

40014.0

6

0014.083200117

kN5.582

117kNm117

42.0

10003800

6

42.0'

6

22

2

32

=

==

===

===

+=

+=

h

IEM

FA

Pf

BDM

cr

cr

gc

crcr

cr

g

tcr

At state of first yield


40088.0

60088.0

28290

250

kNm282908320034.034.014.0500

100100

kN1252

250

2

kNm2508.0Take

sectionsrrectangulaforchartsdesignfromkNm310

22

2

=

=====

===

+=

+=

===

=

=

h

IE

M

IEnfIE

IE

MF

MM

M

y

y

effc

y

y

effc

ygc

effc

y

y

uy

u

At peak

( )

( )

mm66orm066.0012.06.3023.0

rad012.02

42.00088.0067.0

067.036.0

024.0024.0

6.32

42.022

22

kN1552

310

2sectionsrrectangulaforchartsdesignfromkNm310

=+

==

====

=

=

+=

====

pl

p

upyup

ppppplplyu

u

uu

dL

Lh

MFM

At ultimate (allowing for degradation of shear strength)


12516.011

16.0

023.0

mm300ofspacingbarandmm12ofdia.barstirrupassumingkN300betoestimatedissectionofstrengthShear

16.04.8-9

e0.3k8.4

0.42

2and0.14n;

-9

e0.3kwhere

8.01

kN1252

250

2

8.08.0kNm2508.0

0.145.75.7n

=

+=

======

+=

====

lf

tot

tot

uy

lf

uuu

V

aa

V

Fk

k

MFM


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At Collapse (loss of axial resistance)

mm116orm116.04100

9.2

1Tableinlistedvaluesinginterpolatonbased9.204.01.0

9.15.35.31000.14For

kN772

155

2

5.05.0kNm1555.0

==

=

==

====

af

af

uuu

hn

MFM

Results of calculations for the force-displacement pairs of values are summarized in Table 2.

The capacity curve is accordingly shown in Figure 6.

Table 2 Force-Displacement values of limit states of column in worked example

Limit state Forces (kN) Displacement (mm)

Cracking 58.5 4

First Yield 125 23

Peak 155 66

Ultimate(loss of 20% of strength)

125 100

Collapse

(loss of 50% of strength)

77 116

Figure 6 Capacity model of column in worked example

8. Conclusions

Capacity models for lightly reinforced RC columns and walls have been presented, and

illustrated by example, in a hand calculation format which is suitable for use in design

practices. Estimates for the values of theforce-displacementpairs: Fcr- cr, Fy- y, Fu- uor

peak, Flf - lf and Faf - af required for the construction of the capacity models have beencovered. The determination of the force-displacement capacity relationships forms part of a

performance based seismic evaluation of the structure. The aim of this summary paper is to

illustrate calculation techniques which have been verified by comparisons with experimental

results as reported in the literature. A review of the academic literature in this topic and

details of the verifications of the presented relationships can be found in the cited referencesand are not reproduced herein.


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9. Acknowledgements

The authors acknowledge the financial assistance provided by the Australian Research

Council with grants DP1096753 and DP0772088. The proof reading of the script and

checking of the calculations in section 7 by Dr Elisa Lumantarna is also gratefully

acknowledged.

10. References

CUMBIA (2007) by Montejo, L.A. and Kowalsky, M.J., Set of Codes for the Analysis of Reinforced

Concrete Members, Report No. IS-07-01, Constructed Facilities Laboratory, North Carolina State

University, Raleigh, NC.

EN 1998-1, Eurocode 8: Design of structures for earthquake resistance Part 1: General rules, seismic

actions and rules for buildings, BSI, 2004.

FEMA 356 (2000) NEHRP guidelines for the seismic rehabilitation of buildings, Federal EmergencyManagement Agency, Washington DC, USA.

Lam, N.T.K., Wilson, J.L. (2004) Displacement Modelling of Intraplate Earthquakes Invited paper,

International Seismology and Earthquake Technology Journal (special issue on Performance Based

Seismic Design; Ed Nigel Priestley), 2004. Indian Institute of Technology, Vol.41(1), paper no. 439: pp.

15-52.

Lam, N.T.K., Wilson, J.L. and Lumantarna, E. (2011). Force-Deformation Behaviour Modelling of

Cracked Reinforced Concrete by EXCEL spreadsheets , Invited paper in Special Issue entitled

Computing Technologies and Concrete Structures inComputers and Concrete. 8(1): 43 57.

Lumantarna, E, Lam,.N.T.K., Wilson, J.L. and Griffith, M.C. (2010). Inelastic Displacement Demand of

Strength Degraded Structures,Journal of Earthquake Engineering.14: 487-511.

Lumantarna, E., Lam, N.T.K. & Wilson, J.L.(2013), Displacement Controlled Behaviour of

Asymmetrical Single-Storey Building ModelsJournal of Earthquake Engineering. 17: 902-917.

Lumantarna, E., Wilson, J.L. & Lam, N.T.K. (2012) Bi-linear displacement response spectrum model for

engineering applications in low and moderate seismicity regions Soil Dynamics and Earthquake

Engineering, 43: 85-96.

Park, R. and Paulay, T. (1975),Reinforced Concrete Structures, John Wiley & Sons Inc.

Paulay, T. and Priestley, M.J.N. (1992), Seismic design of reinforced concrete and masonry buildings,

John Wiley & Sons, Inc.

Priestley, M.J.N., Calvi, G.M. and Kowalsky, M.J.N. (2007), Displacement-Based Seismic Design of

Structures, IUSS Press, Pavia, Italy.

RESPONSE (2000) by Bentz, E.C., Sectional Analysis of Reinforced Concrete Members, PhD thesis,

University of Toronto.

Wibowo, A., Wilson, J.L., Lam, N.T.K. and Gad, E.F. (2014b), Drift Performance of Lightly Reinforced

Concrete Columns,Engineering Structures. 59: 522-535.

Wibowo, A., Wilson, J.L., Lam, N.T.K. and Gad, E.F. (2014a), Drift Capacity of Lightly Reinforced

Concrete Columns,Australian Journal of Structural Engineering. 15(2): 131-150.

Wibowo, A., Wilson, J.L., Lam, N.T.K. and Gad, E.F.(2013), Seismic performance of lightly reinforced

structural walls for design purposes,Magazine of Concrete Research. 65: 809-828.

Wilson, J.L., Wibowo, A., Lam, N.T.K. and Gad, E.F.(in press), Drift behaviour of lightly reinforcedconcrete columns and structural walls for seismic design applications, Manuscript no. S14-002R1

accepted for publication on 12 May 2014.


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Wilson, J.L., Lam, N.T.K. (2008), Recent Developments in the Research and Practice of Earthquake

Engineering in Australia, Special Issue of Invited papers on Earthquake Engineering, Australian Journal

of Structural Engineering. 8(1): 13-28.

Wilson, J.L., Lam, N.T.K.(2006), Earthquake Design of Buildings in Australia by Velocity and

Displacement Principles, Australian Journal of Structural Engineering Transactions, Institution of

Engineers, Australia. Vol. 6(2): 103-118. Awarded Warren Medal.

Wilson, J.L., Lam, N.T.K. (2003), A recommended earthquake response spectrum model for Australia,

Australian Journal of Structural Engineering.Institution of Engineers Australia, 5(1):17-27 (2003).

Documents

2. Seismic Performance Evaluation of Reinforced Concrete Building by Displacement Principles