curvature ductility of rc sections based on eurocode
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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:
drsekaran@iitm.ac.in)
**Professor, Dept. of Structural Engineering, University of Naples Federico II, 21 via Claudio, 80125, Naples, Italy (E-mail: nunsci@unina.it)
***Professor, Dept. of Structural Engineering, University of Naples Federico II, 21 via Claudio, 80125, Naples, Italy (E-mail: serino@unina.it)
****Visiting Researcher, Dept. of Structural Engineering, University of Naples Federico II, 21 via Claudio, 80125, Naples, Italy (E-mail: fedcarran@libero.it)
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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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Srinivasan Chandrasekaran, Luciano Nunziante, Giorgio Serino, and Federico Carannante
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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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