Curvature Ductility of RC Sections Based on Eurocode

8/10/2019 Curvature Ductility of RC Sections Based on Eurocode

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KSCE Journal of Civil Engineering (2011) 15(1):131-144

DOI 10.1007/s12205-011-0729-4

131

www.springer.com/12205

Structural Engineering

Curvature Ductility of RC Sections Based on Eurocode: Analytical Procedure

Srinivasan Chandrasekaran*, Luciano Nunziante**, Giorgio Serino***, and Federico Carannante****

Received October 12, 2008/Accepted March 16, 2010

Abstract

Correct estimate of curvature ductility of reinforced concrete members has always been an attractive subject of study as itengenders a reliable estimate of capacity of buildings under seismic loads. The majority of the building stock needs structuralassessment to certify their safety under revised seismic loads by new codes. Structural assessment of existing buildings, byemploying nonlinear analyses tools like pushover, needs an accurate input of moment-curvature relationship for reliable results. Inthe present study, nonlinear characteristics of constitutive materials are mathematically modelled according to Eurocode, currently inprevalence and analytical predictions of curvature ductility of reinforced concrete sections are presented. Relationships, in explicitform, to estimate the moment-curvature response are proposed, leading to closed form solutions after their verification with thoseobtained from numerical procedures. The purpose is to estimate curvature ductility under service loads in a simpler closed formmanner. The influence of longitudinal tensile and compression steel reinforcement ratios on curvature ductility is also examined anddiscussed. The spread sheet program used to estimate the moment-curvature relationship, after simplifying the complexities involvedin such estimate, predicts in good agreement with the proposed analytical expressions. Avoiding somewhat tedious hand calculationsand approximations required in conventional iterative design procedures, the proposed estimate of curvature ductility avoids errorsand potentially unsafe design.

Keywords: analytical solutions, concrete, curvature ductility, elasto plastic, reinforced concrete, seismic, structures, yield

1. Introduction

The focus of earthquake resistant design of Reinforced Concrete

(RC) framed structures is on the displacement ductility of the

buildings rather than on the materials like reinforcing steel.Critical points of interest are the strain levels in concrete and

steel, indicating whether the failure is tensile or compressive at

the instant of reaching plastic hinge formation (Pisanty and

Regan, 1998). Studies show that the estimate of ductility demand

is of particular interest to structural designers to ensure effective

redistribution of moments in ultra-elastic response, allowing for

the development of energy dissipative zones until collapse (see,

for example, Pisanty and Regan, 1993). In areas subjected to

earthquakes, a very important design consideration is the ductility

of the structure because modern seismic design philosophy is

based on energy absorption and dissipation by post-elastic defor-

mation for survival in major earthquakes (Paulay and Priestley,1992). Many old buildings show their structure unfit to support

seismic loads demanded by the structural assessment requests of

the revised international codes (see, for example, Chandrasekaran

and Roy, 2006; Chao Hsun Huang et al., 2006). Further, Sinan

and Metin (2007) showed that the deformation demand pre-

dictions by improved Demand Capacity Method are sensitive to

ductility as higher ductility results in conservative predictions.

Estimate of moment-curvature relationship of RC sections has

been a point of research interest since many years (Pfrang et al.,1964; Carrreira and Chu, 1986; Mo, 1992); historically, moment-

curvature relationships with softening branch were first intro-

duced by Wood (1968). Load-deformation characteristics of RC

structural members, bending in particular, are mainly dependent

on moment-curvature characteristics of the sections as most of

these deformations arise from strains associated with flexure

(Park and Paulay, 1975). As seen from the literature, in well-

designed and detailed RC structures, the gap between the actual

and design lateral forces narrows down by ensuring ductility in

the structure (see, for example, Luciano and Raffaele, 1988;

Pankaj and Manish, 2006). With regard to RC building frames

with side-sway, their response assessment is complicated notbecause of the influence of second order deformations, but also

due to the fact that considerable re-distribution of moments may

occur due to plastic behaviour of sections. Plastic curvature is

therefore a complex issue mainly because of interaction of various

*Associate Professor, Dept. of Ocean Engineering, Indian Institute of Technology Madras, Chennai 600036, India (Corresponding Aughor, E-mail:

[email protected])

**Professor, Dept. of Structural Engineering, University of Naples Federico II, 21 via Claudio, 80125, Naples, Italy (E-mail: [email protected])

***Professor, Dept. of Structural Engineering, University of Naples Federico II, 21 via Claudio, 80125, Naples, Italy (E-mail: [email protected])

****Visiting Researcher, Dept. of Structural Engineering, University of Naples Federico II, 21 via Claudio, 80125, Naples, Italy (E-mail: [email protected])


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Srinivasan Chandrasekaran, Luciano Nunziante, Giorgio Serino, and Federico Carannante

132 KSCE Journal of Civil Engineering

parameters namely: i) constitutive materials response; ii) mem-

ber geometry; as well as iii) loading conditions. Observations

made by Challamel and Hjiaj (2005) on plastic softening beams

show that the correct estimate of yield moment, a non-local

material parameter, is important to ensure proper continuity

between elastic and plastic regions during the loading process.

Experimental evidences on moment-curvature relationship of

RC sections already faced limited loading cases and support

conditions (see, for example, Ko et al., 2001). While Mo (1992)

suggested classical approach to reproduce moment-curvature

relationship with the softening branch carried out elastic-plastic

buckling analysis using finite element method, an alternative

approach proposed by Jirasek and Bazant (2002) uses a simpli-

fied model where this complex nonlinear geometric effect is

embedded in the nonlinear material behaviour of the cross

section. Experimental investigations also impose limitations in

estimating the plastic rotation capacity. For instance, studies

show that experimental results obtained from rotation-deflection

behaviour show good agreement with the analysis in elastic

regime; but for phase of yielding of reinforcing steel, theoretical

results do not agree with the experimental inferences (see, for

example, Lopes and Bernardo, 2003).

Studies reviewed above show that there exists no simplified

procedure to estimate curvature ductility of RC sections. While re-

sponse of RC building frames under ground shaking generally

results in nonlinear behaviour, increased implementation of displa-

cement-based design approach lead to the use of nonlinear static

procedures for estimating their seismic demands (ATC, 2005;

BSSC, 2003). An estimate of moment-curvature relationship be-

comes essential for performing non-linear analyses. Therefore, in

this study, an estimate of curvature ductility of RC sections, using

detailed analytical procedure is attempted. Calculations of moment-

curvature relationship are based on their nonlinear characteristics

in full depth of the cross section, for different ratios of longitudinal

tensile and compression reinforcements. They account for the vari-

ation on depth of neutral axis passing through different domains,

classified on the basis of strain levels reached in the constitutive

materials, namely concrete and steel. Obtained results, by employ-

ing the numerical procedure on example RC sections, are verified

with expressions derived from detailed analytical modelling.

2. Mathematical Development

Significant nonlinearity exhibited by concrete, under multi-

axial stress state, can be successively represented by nonlinear

characteristics of constitutive models capable of interpreting

inelastic deformations (see, for example, Chen 1994a, 1994b).

Studies conducted by researchers (Sankarasubramanian and

Rajasekaran, 1996; Fan and Wang, 2002; Nunziante et al., 2007)

describe different failure criteria in stress space by a number of

independent control parameters while the non-linear elastic

response of concrete is characterized by parabolic stress-strain

relationship in the current study, as shown in Fig. 1. Elastic limit

strain and strain at cracking are limited to 0.2% and 0.35%

respectively, as prescribed by the code, currently in prevalence

(DM 9, 1996; UNI ENV, 1991a, 1991b; Ordinanza, 2003, 2005;

Norme tecniche, 2005). Tensile stresses in concrete are ignored

in the study. Design ultimate stress in concrete in compression is

given by:

(1)

where, candRckare the partial safety factor and compressive cube

strength of concrete, respectively. The stress-strain relationship for

concrete under compressive stresses is given by:

(2)

where, parameters a, b and c in Eq. (2), are determined by

imposing the following conditions:

(3)

By solving, we get:

(4a)

c0 0.83( ) 0.85( )RcKc--------------------------------------=

c c( ) ac2

bc c+ +=

c c( ) c0=

c c( ) 0=

0 c c0

c0 c cu

c 0

c c 0=( ) 0=

c c c0=( ) c0=

dcdc--------

c c0=

0=

c 0=

ac02

bc0 c0=+

2ac0 b 0=+

a c0

c02

------- b 2c0

c0---------- c 0=,=,=

Fig. 1. Stress-strain Relationships: (a) Concrete, (b) Steel


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Curvature Ductility of RC Sections Based on Eurocode: Analytical Procedure

Vol. 15, No. 1 / January 2011 133

Stress-strain relationship for concrete is given by:

(4b)

Stress-strain relationship for steel, an isotropic and homogene-

ous material, is shown in Fig. 1. While the ultimate limit strain in

tension and that of compression are taken as 1% and 0.35%

respectively, elastic strain in steel in tension and compression are

considered the same in absolute values (see, for example, DM9,

1996). The design ultimate stress in steel is given by:

(5)

where s and y are partial safety factor and yield strength of

reinforcing steel, respectively. Stress-strain relationship for steel

is given by:

(6)

The fundamental Bernoullis hypothesis of linear strain over

the cross section, both for elastic and for elastic-plastic responses

of the beam under bending moment combined with axial force,

will be assumed. The interaction behaviour becomes critical when

one the following conditions apply namely: i) strain in reinforc-

ing steel in tension reaches ultimate limit; ii) strain in concrete in

extreme compression fibre reaches ultimate limit; as well as iii)

maximum strain in concrete in compression reaches elastic limit

under only axial compression. In the following section, only re-

ctangular RC sections under axial force, P and bending moment,

M will be considered.

2.1 Moment-curvature in Elastic Range

It is well known that the bending curvature is the derivative of

bending rotation, varying along the member length and at any

cross section, it is given by the slope of the strain profile. It depends

on the fluctuations of the neutral axis depth and continuously

varying strains. The moment-curvature relationship, in elastic

range, depends on both the magnitude and nature of the axial force

as well. Fig. 2 shows the variation of curvature with respect to

strain variation in constitutive materials. Magnitude of axial force

is assumed to vary in the range:

(Asc +Ast)s0


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limit elastic curvature and is derived in following section.

For further increase in curvature more than 0, concrete also

contributes to the compression resultant and the expressions for

axial force and bending moment take the form, as given below:

(15)

,

(16)

where, the coefficientsAi(for i = 0 to 3) andBi(fori = 0 to 4), as

a function of curvature are given by:

(17)

By solving Eq. (15) with respect to variable xc, three roots of

the variable are obtained as:

(18)

where,

(19)

Out of the above, only one root, namely xc3, closely matches

with the numerical solution obtained and hence by substituting

the rootxc3in Eq. (18), moment-curvature relationship in elastic

range is obtained as:

(20)

2.1.2 Axial Force Equal to Zero

The moment-curvature relationship is given by Eq. (20) for the

complete of [0, E].

2.1.3 Compressive Axial Force

Expressions for axial force and bending moment are given by:

(21)

(22)

where, the coefficientsEi= 0,1,2andFi=0,1are given by:

(23)

By solving the Eq. (21), position of neutral axis is determined

as:

(24)

By substituting the Eq. (24) in Eq. (22), we get:

(25)

where,

(26)

By imposing the condition (xc =D) in Eq. (24), limit curvature

0is determined as given above. Further increase in the curvature

changes the equilibrium conditions due to the contributions to

resultant compressive force by concrete. For curvature more than

0, moment-curvature relationship is given by Eq. (20).

Pe bc c y( )[ ]dy stAst scAsc+( )0

xc

=

A0 e( ) A1 e( )xc A2 e( )xc2

A3 e( )xc3

+ + +=

Me bc c y( )[ ]D

2---- y

dy stAst scAsc+( )D

2---- d

+0

xc

=

Me B0 e( ) B1 e( )xc B2 e( )xc2

B3 e( )xc3

B4 e( )xc4

+ + + +=

A0 e( ) b d D( )Dpt d pc pt( )+[ ]Ese ;=A1 e( ) b D d( )p c pt+( )Ese ;=

A2 e( ) bc0e

c0--------------- A3 e( )

bc0e2

3c02

--------------- ;=;=

B0 e( ) 12---b 2d2 3dD D2+( ) Dpt d pc pt+( )[ ]Ese ;=

B1 e( )1

2---b 2d

23dD D

2+( ) p c pt( )Ese B2 e( )

bDc0e2c0

-------------------- ;=;=

B3 e( ) bc0e 2c0 De+( )

6c02

-------------------------------------------- B4 e( ) bc0e

2

12c02

--------------- =;=

xc1 Pe e,( )1

6A3 e( )------------------ 2A2 e( )( )[=

2.5198A2

2e( ) 3A1 e( )A3 e( )( )

C1 ePe,( )--------------------------------------------------------------------------- 1.5874C

1

e

Pe

,( )+ +

xc2 Pe e,( )1

12A3 e( )---------------------=

4A2 e( )2.5198 4.3645i+( ) A2

2e( ) 3A1 e( )A3 e( )( )

C1 e Pe,( )-------------------------------------------------------------------------------------------------------

1.5874 2.7495i( )C1 e Pe,( )

xc3 Pe e,( )1

12A3 e( )---------------------=

4A2 e( )2.5198 4.3645 i( ) A2

2e( ) 3A1 e( )A3 e( )( )

C1 e Pe,( )-----------------------------------------------------------------------------------------------------

1.5874 2.7495i+( )C1 e Pe,( )

C1 ePe,( ) 4A 2

23A1A3( )

3 2A2

39A1A2A3 27A3

2A0 Pe( )+( )

2++

2A23

9A1A2A3 27A32A0 Pe( )+

1 3

=

Me B0 e( ) B1 e( )xc3 e Pe,( ) B2 e( )xc32

e Pe,( )+ += B3 e( )xc3

3e Pe,( ) B4 e( )xc3

rePe,( )+ + 0 E,[ ]

Pe bc c y( )[ ]dy stAst scAsc E0 E1xc E2xc2

+ +=+0

D

=

Me bc c y( )[ ]D

2---- y

dy stAst scAsc+( )D

2---- d

+0

D

=

F0 F1xe+=

E0 13---b 3d d D( )Espc 3d D( )2Espt D

2

c0 3c0 D+( )c0

2---------------------------------------- ,=

E1b dEs p c pt+( )c0

2 D Esp c pt+( )c0

2c0 2c0 D+( )+( )+[ ]

c02

------------------------------------------------------------------------------------------------------------------------------------- ,=

E2bDc0

2

c02

-------------------- ,=

F0b

12------[6d D 2d( ) D d( )Espc 6d D( )

22d D( )Espt+=

D3c0 2c0 D+( )

c02

---------------------------------------- ,

F1 b 3 D2

2d2

3dD+( ) p c pt( )Esc02

D3

c0[ ]6c0

2----------------------------------------------------------------------------------------------------------=

xcE1 E1

24E2E0 Pe( )+

2E0----------------------------------------------------------=

Me F0 Pe,( ) F1 Pe,( )xc+= 0 0,[ ]

0 3bc0 D d( )Es Dpc d pt pc( )+( ) D

2

c0+2bD

3c0

------------------------------------------------------------------------------------------------=

c0 3b 3b D d( )Esc0 Dpc d pt pc( )+( )( D2c0)

2+[ ] 4PeD

3c0

2bD3c0

----------------------------------------------------------------------------------------------------------------------------------------------------------


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By substituting Eq. (40) in Eq. (16), limit elastic bending mo-

ment can be obtained as follows:

(42)

where,

(43)

2.2.4 Case (iv): Strain in Extreme Compression Fibre in

Concrete Reaches Elastic Limit Value

Now, the depth of neutral axis is given by:

(44)

By substituting Eq. (44) in Eq. (15), expression for limit elastic

curvature is obtained as:

(45)

where the constantsRi(for i = 0 to 2) are given by:

(46)

By solving Eq. (45), the only real root (in this case, first root)

gives the limit elastic curvature as:

(47)

By substituting Eq. (47) in Eq. (16), limit elastic bending

moment,ME, can be obtained as follows:

(48)

where,

(49)

It may be easily seen that for percentage of tension steel

exceeding the maximum limit of 4%, as specified in many codes

(see for example Indian code (IS 456, 2000), case (iv) shall never

result in a practical situation. For the case (xc>D), the limits of

the integral in Eq. (15) will be from (0,D), which shall also result

in compression failure and hence not discussed. Expressions for

limit elastic moments are summarised as below:

(50)

wherept,el, for tow cases namely: i) axial force neglected; and ii)

axial force considered are given by the following equations:

(51)

(52)

2.3 Percentage of Steel for Balanced Section

Percentage of reinforcement in tension and compression for

balanced failure are obtained by considering both the conditionsnamely: i) maximum compressive strain in concrete reaches

ultimate limit strain; and ii) strain in tensile reinforcement

reaches ultimate limit. Balanced reinforcement for two cases is

considered namely: i) for beams where axial force vanishes; and

ii) for beam/columns where P-M interaction is predominantly

present. For sections with vanishing axial force, depth of neutral

axis is given by:

(53)

For vanishing axial force, governing equation to determine the

percentage of reinforcement is given by:

(54)

In explicit form, Eq. (53) becomes:

(55)

By solving, percentage of steel for balanced section is obtained

as:

(56)

For a known cross section with fixed percentage of compres-

sion reinforcement, Eq. (56) gives the percentage of steel for a

balanced section. It may be easily seen that for the assumed

condition of strain in compression steel greater than elastic limit,

Eq. (56) shall yield percentage of tension reinforcement for

balanced sections, whose overall depth exceeds 240 mm, which

is a practical case of cross section dimension of RC beams used

in multi-storey building frames. For sections where axial force is

predominantly present, percentage of balanced reinforcement

depends on the magnitude of axial force. By assuming the same

hypothesis presented above, depth of neutral axis is given by Eq.

(53); but Eq. (55) becomes as given below:

MEiii( ) b

2c02

---------M1

iii( )

Eiii( )2

----------- M2

iii( )

Eiii( )

----------- M3iii( )

M4iii( )

Eiii( )

M5iii( )

Eiii( )2

+ + + +=

M1iii( ) s0

3

s0 4c0( )c06

-------------------------------------- M2iii( ) D 2d( ) 3c0 s0( )s0

2

c03

---------------------------------------------------------- ,=,=

M3iii

d D( )s0 D 2d( )Es p t pc( )c02

d 2c0 s0( )c0[ ] ,=

M4iii( )

D d( ) D 2d( )2Esptc02 d

23D 2d( ) c0 s0( )c0

3--------------------------------------------------------- ,+=

M5iii( ) d

3d 2D( )c0

6--------------------------------=

xciv( ) c0

E------=

R0 R1E R2E2

0=+ +

R02bc0c0

3-------------------- R1 Pe b D d( )Esc0p c pt+( ) ,+=,=

R2 b D d( )Es Dpt d pt pe( )[ ]=

Eiv( ) R1 R1

24R0R2+

2R2---------------------------------------=

MEiv( ) M1

iv( )

Eiv( )2

---------- M2

iv( )

Eiv( )

---------- M3iv( )

M4iv( )

Eiv( )

+ + +=

M1iv( ) bDc0c0

3---------------------=

M2iv( ) 1

4---bc0

2c0=

M3iv( ) 1

2---b D

22d

23dD+( )Es p c pt( )c0=

M4iv( ) 1

2---b D

22d

23dD+( )Es Dpt d pc pt+( )[ ]=

MEME

ii( )if pt pt el, 0 if it is compression). For the

known cross section with fixed percentage of compression

reinforcement, Eq. (58) gives the percentage of steel for balanced

section. In the similar manner, percentage of compression rein-

forcement for a balanced section, by fixing pt, can be obtained by

inverting the relationship given in Eqs. (56) and (58) for respec-

tive axial force conditions.

2.4 Ultimate Bending Moment-curvature Relationship

Study in this section is limited to RC sections imposed with

tension failure as the compression and balanced failures do not

have any practical significance in the displacement-based design

approach, in particular. Let us consider two possible cases: i)

neutral axis position assumes negative values; and ii) neutral axis

position assumes positive values.

2.4.1 Neutral Axis Position Assuming Negative Values

By imposing the conditions: and solv-

ing Eq. (8) respect topt, for a specified range of tension steel per-

centage, , depth of neutral axis is

given by:

(59)

At collapse, the equilibrium equations become:

(60)

(61)

By solving Eq. (60) with respect to u, we obtain the ultimate

curvature, as reported below:

(62)

By substituting Eq. (62) in Eq. (61), ultimate bending moment

can be determined as:

(63)

It may be noted that the ultimate bending moment in this case

is similar to one given by Eq. (29) for elastic range.

2.4.2 Neutral Axis Position Assuming Positive Values

Under this condition at collapse, four different cases of tension

failure of RC sections are possible, namely:

(a)

(b)

(c)

(d) (64)

As the strain in tensile steel reaches its ultimate value (tensile

failure), in all the four cases mentioned above, equation for

computing the depth of neutral axis, as function of ultimatecurvature, will remain unchanged and is given by:

(65)

Axial force and bending moment in the cross section at

collapse, for case (a) are given by:

(66)

(67)

By substituting the Eq. (65) in Eq. (66) we get:

(68)

where the constantsJi(for i=0 to 3) are given by:

(69)

By solving Eq. (68), the real root (in this case, the third root)

gives the ultimate curvature as:

(70)

where,

(71)

By substituting Eq. (70) in Eq. (67), ultimate moment is given

by:

(72)

where the super-script (a) stands for the case (a); the constants of

Eq. (72) are given by:

,

b d D( ) c0c0 3cuc0 3 p e pt( ) cu su+( )s0[ ] P0=

pt bal, pc3cu c0( )c0

3 cu su( )s0--------------------------------

P0b D d( )s0---------------------------+=

xc 0 & su D d( )==

pt Pu bdEssu+( ) b d D( )s0( )

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Curvature Ductility of RC Sections Based on Eurocode