Ultimate strength analysis of prestressed reinforced concrete sections under axial force and biaxial bending

Computers and Structures 89 (2011) 91–108

Contents lists available at ScienceDirect

Computers and Structures journal homepage: www.elsevier.com/locate/compstruc

Ultimate strength analysis of prestressed reinforced concrete sections under axial force and biaxial bending Francesco Marmo a, Roberto Serpieri b,⇑, Luciano Rosati a a b

Dipartimento di Ingegneria Strutturale, Università di Napoli Federico II, Napoli, Italy Dipartimento di Ingegneria, Facoltà di Ingegneria, Università del Sannio, Piazza Roma 21, 82100 Benevento, Italy

a r t i c l e

i n f o

Article history: Received 29 November 2009 Accepted 10 August 2010 Available online 16 September 2010 Keywords: Ultimate strength analysis Prestressed concrete Secant method Softening

a b s t r a c t A secant approach is illustrated for the ultimate limit state (ULS) analysis of prestressed reinforced concrete sections subjected to axial load and biaxial bending in presence of softening stress–strain laws. The stiffness matrix and the resultant loads are evaluated analytically by a novel methodology, termed fiberfree, which represents a computationally efficient alternative to fiber approaches. Extensive computations of the ULS domains of benchmark test cases show that the robustness of the proposed algorithmic strategy is substantially unaffected by the amount of reinforcement, prestressing and softening, though localized non-convex regions have been occasionally experienced in presence of softening. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction The ultimate limit state (ULS) analysis of RC sections under axial load and biaxial bending has been extensively investigated in the past, see e.g. [2,6,13,14,17,29,30] and overviews reported in [10,23], although only some of them have dealt with prestressed sections. More recently the topic of ULS analysis has found a renewed interest in the scientific community [3,4,9,18,23–28] mainly for three reasons. New methodologies are driven by the evolution of building codes since their prescriptions require more refined analyses. Second, ULS analysis of cross sections has to be performed for the majority of structural elements and for several load combinations so that the automation of such procedures is unavoidable in design practice. Finally, increased computational power allows for analyses that can solve denser discretizations of a section in fibers or finite elements equipped with advanced constitutive laws thus opening new research scenarios [11]. Some papers [13,23,24] have addressed sophisticated constitutive laws for concrete, including a linear softening branch as prescribed by ACI [1]. However they have dealt mainly with the automation of design procedures conceived for hand calculations rather than formulating the problem in a form more suitable to numerical computations. Nowadays the employment of numerical procedures that implement exactly the analysis for cross sections in fulfilment of building codes prescriptions proves to be compet⇑ Corresponding author. Tel.: +39 0824 305566; fax: +39 0824 325246. E-mail addresses: [email protected], [email protected] (R. Serpieri). 0045-7949/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2010.08.005

itive with simplified procedures [10,24]. In the past, in order to reduce the time spent in computations, the research in this subject was focused on providing simplified design aids as interaction diagrams or methodologies based on approximate integration procedures such as rectangular stress blocks [17]. The distinguishing features of the several numerical methods proposed so far are essentially the integration method (stress block, fiber methods, trapezoids, Gauss’ divergence theorem) [2,3,7,22] and the strategy exploited for seeking the equilibrium condition at the ultimate limit state. Regarding the former issue a refined integration of the concrete constitutive law has been performed in [13,23,24] being the stress–strain law piecewise polynomial. Equilibrium at the ultimate limit state can be found by exploiting approaches that follow the loading history path with strain driven or load driven strategies [10]. Within this second class of algorithms an original secant strategy has been developed by the authors [25] for the evaluation of the ULS load multiplier for an arbitrary affine load path. In particular, the ultimate load multiplier of arbitrary RC sections has been determined in [25] for the classical parabola–rectangle law prescribed by most building codes [8]. A twofold extension of the algorithms presented in [25] is developed in the present paper by introducing the effects of prestressing and by addressing softening stress–strain laws for concrete. The reason for introducing a softening branch is motivated by the need to develop algorithmic strategies for stress–strain laws which can better model the actual behaviour of concrete [12,15,16], e.g. in the unconfined part of the section, and which can also be employed in the nonlinear analysis of building structural models [11].

92

F. Marmo et al. / Computers and Structures 89 (2011) 91–108

In carrying out the benchmark tests, the constitutive laws, the structural assumptions and the general prescriptions of Eurocode 2 (EC2) [8] have been adopted. Note that the parabola–rectangle law commonly adopted for the ULS analysis has been replaced by the softening stress–strain law prescribed by EC2 for general nonlinear structural analyses. The ultimate strength domains N, Mx, My for a set of benchmarks of prestressed sections selected from the literature have been computed in order to carefully evaluate the effect of prestressing and softening on the computational performance of the proposed algorithm. In particular, an unsymmetric section with unsymmetric reinforcements layout has been considered in order to further validate the proposed methodology. A feature of the proposed algorithm is to achieve closed-form integration of the entries of the secant stiffness matrix and of the resultant loads. This is obtained by applying the integration formulas presented in [25] to the rational analytical expressions which typically characterize softening stress–strain laws. Remarkably, the proposed formulas provide both exact integration and considerable time savings in computations with respect to existing fiber formulations [19]. Numerical experiments have shown that the computational performances of the present algorithm are substantially similar to those reported in [25] in terms of robustness and convergence rate. As a matter of fact, the amount of reinforcement, the prestressing strain and the softening branch induce minor effects, the variation in the number of iterations being around 10%. Conversely, the shape of the domains occasionally modifies in the presence of softening and the ULS domains may exhibit a non-convex shape. In our numerical experiments this effect has been experienced only for sections having a non-convex geometry provided that the parameter governing the steepness of the softening branch exceeds a critical value. In any case no theoretical justification was found in order to put the existence of concavities in direct relation with the non-convexity of the section; actually, convex domains have been obtained also in presence of non-convex geometries of the section. The paper is organized as follows. Section 2 presents the kinematic and constitutive assumptions while the generalization of the secant strategy originally devised in [25] to account for prestressing is described in Section 3. The closed-form evaluation of the secant stiffness matrix of a section is reported in Section 4; the relevant analytical details can be found in the appendices. Finally, the results of the numerical analyses are presented and discussed in Section 5 in terms of correctness, robustness and convergence rate.

2. Kinematic and constitutive assumptions Let us consider a reinforced concrete section defined by a generic domain X which is assumed to be of polygonal shape. Accordingly, the section is defined by a discrete set of Nc vertices geometrically identified by position vectors ri = {xi, yi} where xi and yi are the cartesian coordinates of the ith vertex. The section is further equipped with Ns steel reinforcements and Np prestressing tendons. The former are located at rsj, j 2 {1, . . . ,Ns}, and have area Asj while the position and the area of the latter are respectively denoted by rpj and Apj, being j 2 {1, . . . ,Np}. As usual in the analysis of RC sections, reinforcements and tendons are treated as points so that the relevant area is not subtracted from the area of the concrete section. The ultimate strength analysis of prestressed RC sections subject to axial load and biaxial bending is developed on the basis of the usual assumptions of the classical Bernoulli–Navier hypothesis and of perfect bond between concrete and steel [25] for both ordinary and prestressing steel. This last hypothesis implies that the

strain field is the same for concrete, ordinary steel and prestressing tendons once concrete has hardened after precast. On account of the former assumptions the kinematic state of the section is defined by

uðtÞ ¼





e0 ðtÞ; g x ðtÞ; g y ðtÞ t ;

ð1Þ

where e0 is the longitudinal strain at the origin of the reference frame, while gx and gy are the components of the strain gradient g. Actually, introducing the vector q = {1, x, y}t = {1, r}t, the strain field of the section can be expressed as

eðq; tÞ ¼ uðtÞ  q ¼ e0 ðtÞ þ gðtÞ  r:

ð2Þ

The dependence on time t is introduced explicitly to account for time-dependent effects in concrete and tendons. In particular, time-dependent effects for concrete typically include shrinkage and creep. The former can be taken into account by assigning a uniform strain field ec,sh [5,23] to the concrete strain field ec according to the following expression

ec ðq; tÞ ¼ eðq; tÞ  ec;sh ðtÞ

ð3Þ

while the effects of creep are usually addressed by considering a time-dependent constitutive law. The strain esj related to the generic bar located at rsj is expressed as

esj ðtÞ ¼ es ðqsj ; tÞ ¼ eðqsj ; tÞ;

ð4Þ

t

where qsj = {1, rsj} . Similarly, at the generic prestressing tendon located at rpj, the strain epj is given by

epj ðtÞ ¼ ep ðqpj ; tÞ ¼ eðqpj ; tÞ þ ep0j ;

ð5Þ

t

where qpj = {1, rpj} and ep0j is the prestressing strain at time t = 0. As opposed to concrete, time-dependent effects in the tendons are more difficult to model, the only exception being the relaxation of the prestressed steel which is usually accounted for by considering a time-dependent modulus of elasticity [5] in the constitutive law. As a matter of fact the actual behaviour of tendons also depends upon additional inelastic effects such as cable friction and wedge draw-in of the anchorage devices. It is common practice in the design of prestressed concrete sections according to building codes to cumulatively take into account inelastic effects of both tendons and concrete by means of stress losses Drpj(t). The expression of such stress losses is usually assigned by building codes as function of the type and duration of loads acting on the structural model, of the initial prestressing strain and of the current strain in the section evaluated on the basis of a working stress analysis at the time of interest. Consequently, to perform the ULS analysis as prescribed by building codes, the generic stress losses Drpj(t) can be assigned once a nonlinear working stress analysis of the section has been carried out at the time of interest. To this end building codes require only a shortterm (t = 0) and a long-term (t = 1) working stress and ULS analyses. We shall actually adopt the approach prescribed by the Eurocode 2 and, being interested to illustrate an algorithm for performing the ULS analysis at a generic time of interest, we shall omit the explicit dependence upon time. 3. Ultimate limit state of prestressed concrete sections Let us denote by r the stress field acting on the section X and by

N¼

Z X

rðrÞ dX;



My Mx

 ¼

Z

rðrÞr dX;

ð6Þ

X

the resultant loads. More concisely the previous equations can be expressed as

93

F. Marmo et al. / Computers and Structures 89 (2011) 91–108 t f ¼ f N; M y ; M x g ¼

Z

rðrÞq dX:

ð7Þ

X

f ¼ KS ðuÞu;

By equilibrium, it turns out to be

f ¼ f c ðuÞ þ f s ðuÞ þ f p ðu; Drp Þ;

fc ¼

rc ðec Þq dX; f s ¼

Ns X

Xc

¼

ð8Þ

where the dependence of K upon the strain field is implicitly indicated by u. Specifically, the secant matrices of concrete and ordinary steel are given by:

KSc ¼

rs ðesj ÞAsj qsj ; f p

j¼1

Np X ðrp ðepj Þ þ Drpj ÞApj qpj

ð9Þ

j¼1

are the internal resultant loads. They are associated, respectively, with the concrete section Xc, the steel reinforcement and the prestressing tendons while rc, rs and rp denote the relevant uniaxial stress–strain laws. To give the treatment the utmost generality, we shall not refer to any particular analytical expression of the concrete constitutive law, though we specify that it can be an arbitrary nonlinear uniaxial law, possibly endowed with a softening branch. Similarly, arbitrary nonlinear constitutive laws can be considered for rs and rp. Usually one has to check that a given set of loads N, Mx, My, resulting from the analysis of the structural model, does fulfill the strain limits imposed by code regulations for each material. However, following the approach presented in [7,25], a more refined analysis can be carried out. Denoting by fd and fl the internal forces associated, respectively, with dead and live loads and setting

fðkÞ ¼ f d þ kf l

ð17Þ S

where Drp is the vector collecting the stress losses of all tendons and

Z

or, in matrix form [7],

ð10Þ

Z

ESc ðqÞq  qdX;

Xc

KSs ¼

Ns X

ESsj ðqsj Þqsj  qsj Asj ;

ð18Þ

j¼1

where ESc ðqÞ is the secant modulus of concrete at a generic point of the section and ESsj ðqsj Þ is the secant modulus of the jth reinforcement bar:

ESc ðqÞ ¼

rs ðes ðqsj ÞÞ rc ðec ðqÞÞ ; ESsj ðqsj Þ ¼ ; eðqÞ eðqsj Þ

ð19Þ

if e approaches zero the secant moduli defined above tend to the slope of the tangent at the origin of the given stress–strain laws. Extension to tendons of the previous definition of secant moduli yields

ESpj ðqpj Þ ¼

rp ðep ðqpj ÞÞ : eðqpj Þ

ð20Þ

Notice that the presence of the prestressing strain in the previous expression makes the numerator different from zero when the denominator, i.e. the strain in the concrete fibers surrounding the tendon, tends to zero. In order to overcome this problem, the equilibrium (14) are formulated in the following alternative form:

f d þ k f l  f p0 ¼ f c þ f s þ f p  f p0 ;

ð21Þ

we formulate the ULS problem of a prestressed section as follows: Find the minimum value k* of the positive load amplifier so as to fulfill at least one of the following conditions

where fp0 is the vector collecting the resultant loads due to prestress and stress losses:

minfg  r þ e0 g ¼ ecl ;

f p0 ¼

r2Xc



 0g

max fg  rsj þ e

ð11Þ ð12Þ

max fg  rpj þ e0 g ¼ epu ;

ð13Þ

j¼f1;...;Np g

where ecl, esu and epu are the strain limits for concrete, steel and tendons, respectively. The strain parameters g* and e0 are the components of the vector u* solution of the nonlinear problem:

f d þ k f l ¼ f c ðu Þ þ f s ðu Þ þ f p ðu ; Drp Þ:

ð14Þ

The vector u* is the ultimate limit state of the section, defined as a kinematic state such that the strain within each material belongs to the relevant admissible domain and, at least at one point of the section, either ecl, esu or epu have been attained. The ULS load amplifier k* provides a measure of how much the assigned load is distant from the boundary of the domain according to a given loading path within the space of internal forces. 3.1. Algorithmic strategy The nonlinear equilibrium equations (14) are solved by a secant approach, similar to the ones reported in [7,25], in which the generic constitutive law of the materials r(e) is replaced by a fictitious secant function ES defined pointwise within the section as follows:

rðeðqÞÞ ¼ ES ðeðqÞÞeðqÞ:

ð15Þ

Accordingly, the resultant loads can be written as

f¼

Z X

ES ðeðqÞÞeðqÞq dX;

ð22Þ

j¼1

¼ esu ;

j¼f1;...;Ns g

Np X ðrp ðep0j Þ þ Drpj ÞApj qpj :

ð16Þ

Accordingly, one can introduce the modified secant modulus for tendons

rp ðep ðqpj ÞÞ  rp ðep0j Þ b E Spj ðqpj Þ ¼ eðqpj Þ

ð23Þ

so that the secant stiffness

bS ¼ K p

Np X

b E Spj ðqpj Þqpj  qpj Apj

ð24Þ

j¼1

b S u ¼ f p  f p0 . fulfills the identity K p In conclusion the nonlinear equilibrium (21) are cast in the following equivalent form

  b S ðu Þ u : f d þ k f l  f p0 ¼ KSc ðu Þ þ KSs ðu Þ þ K p

ð25Þ

Set in this form the ULS problem (25) can be solved by a direct use of the secant strategy reported in [25], which the reader is referred to. In particular, among the criteria recommended therein for updating k, the strategy denominated SEI-PM-LA has been selected in the numerical examples reported herein. Remark. As mentioned in Section 2, most of the building codes lump the inelastic effects which characterize the behaviour of prestressed sections to a suitable stress loss for each tendon. Stress losses can be evaluated by means of a working stress analysis of the section at the time of interest. Although this issue is not of concern in this paper we specify that the working stress

94

F. Marmo et al. / Computers and Structures 89 (2011) 91–108

analysis of the section can be formally written as in (17) where the left-hand side is replaced by a known vector f, which contains the internal forces associated with the analysis of the structural model subject to the loads prescribed by the codes for evaluating stress losses, see e.g. [8]. A solution procedure for the working stress analysis of a section can be found in [21].

4.1. Secant stiffness matrix for a polygonal domain The integrals of interest can be expressed as finite sums of quantities depending upon the vertices of the section and the values assumed by suitable functions associated with the function ESc . Specifically, it turns out to be: S

Ac ¼

Nc X

4. Evaluation of the concrete secant stiffness matrix

sSc ¼

Nc X

ð1Þ ^  ni ÞUð0Þ li ðg i ðpES Þ;

^  ni Þ l i ðg



2

ASc Z 6 6 KSc ¼ ESc ðqÞq  q dX ¼ 6 sSc x 4 Xc ðsSc Þy

ðsSc Þx   J Sc

xx

ðJ Sc Þxy

3 ðsSc Þy   7 7 J Sc 7; xy 5 S ðJ c Þyy

ASc ¼

Xc

ESc ðqÞ dX;

JSc ¼

Z Xc

ESc ðqÞr dX;

ð28Þ

and the second secant elastic area moment

JSc

¼

Z Xc

Nc X

qÞr  r dX:

^  ni Þ li ðg



    ð1Þ ð0Þ ð2Þ ^ ; pES riþ1  Ui pES g c

c













ð1Þ ð1Þ ð1Þ ð2Þ ð1Þ Uð0Þ pES  2Ui pES þ Ui pES i c

c

c

 ðri  ri Þ

ð32Þ

where li is the length of the ith side of the polygon, ni is the unit vector normal to the ith side of the domain, directed outwards, and ^ ¼ g=ðg  gÞ. g Denoting by h a generic scalar function of the strain e, the quanð0Þ ð1Þ ð2Þ tities Ui ðhÞ; Ui ðhÞ and Ui ðhÞ associated with the ith side of the polygon are defined by

8 ð1Þ ð ei Þ < ph ðeiþ1 Þpð1Þ h

^  ðriþ1  ri Þ – 0; if g

:

^  ðriþ1  ri Þ ¼ 0; if g

ð0Þ i ðhÞ

¼

ð1Þ i ðhÞ

8 ð1Þ ð2Þ ð2Þ < ph ðeiþ1 Þ  ph ðeiþ1 Þph ðei Þ if 2 e  e e  e Þ ð iþ1 i iþ1 i ¼ :1 hðei Þ if 2

ð29Þ

Clearly, the evaluation of double integrals presents serious computational difficulties especially when the integration area has a complicated shape. This occurs quite often in the case of biaxial bending considered in this paper since, even in the case of regular sections, the neutral axis continuously rotates and migrates as N, Mx and My change. The integration problem is commonly addressed by dividing the section either in strips parallel to the current direction of the neutral axis [2,6] or in small rectangular elements (fibers) [26,28]. However, such approaches introduce unpredictable errors in the evaluation of the resultant loads, since the accuracy of the computation depends upon the size of the discretizing grid and the way in which non-rectangular areas occurring at the borders of the section are taken into account. In particular, reducing the size of the grid does not necessarily yield more accurate results for the resultant loads, especially in the presence of softening constitutive laws [18]. Thus it is more convenient, both in terms of accuracy and computational cost, to transform the double integrals into line integrals by means of the Gauss theorem [3,9,14,22,23,29]. All the above issues are completely circumvented in this paper since we adopt the integration formulas, proved in [18,25], which provide the exact value of the integrals associated with an arbitrary function; for the case at hand such function is represented by ESc appearing in (27)–(29) and defined according to (15). In particular, it has been proved in [18,25] that the integrals (16) as well as the secant elastic moments (27)–(29) can be computed in closed form either for polygonal and circular boundaries of the section domain. To make the paper self-contained we here detail the actual computation of the integrals (27)–(29) both for polygonal and circular domains with reference to an arbitrary continuous real-valued secant function ESc .

ð1Þ

ri þ Ui

c

U ESc ð



     ð1Þ ð1Þ ð2Þ ð1Þ þ Ui pES  Ui pES symðri  riþ1 Þ c c      ð2Þ ð1Þ ð0Þ ð2Þ þ Ui pES ðriþ1  riþ1 Þ  Ui pES c    c  ð1Þ ð2Þ ^Þ þ Uið1Þ pð2Þ ^Þ  Ui pES symðri  g symðriþ1  g ESc  c   ð0Þ ð3Þ ^g ^Þ ; þ2Ui pES ðg

ð26Þ

ð27Þ



c

i¼1

the first secant elastic area moment

sSc ¼



c

ð31Þ

by integrating over the concrete section Xc. The entries of the secant matrix are the secant elastic area

Z



ð1Þ ð1Þ ð1Þ Uð0Þ pES  Ui pES i

i¼1

The calculation of the resultant loads in (16) involves the evaluation of double integrals. In particular, according to the secant approach illustrated so far, we need to compute the concrete secant stiffness matrix

ð30Þ

c

i¼1

U

Uið2Þ ðhÞ ¼

8 > > > > <

eiþ1 ei

hðei Þ

ð1Þ

ph ðeiþ1 Þ

eiþ1 ei

ð3Þ

p

ð2Þ

 2 ðeh

ð33Þ

^  ðriþ1  ri Þ – 0; g

ð34Þ

^  ðriþ1  ri Þ ¼ 0; g

ðeiþ1 Þ

iþ1 ei Þ

2

ð3Þ

ph ðeiþ1 Þph ðei Þ

þ2 ðe e Þ3 > > iþ1 i > > :1 hðei Þ 3

^  ðriþ1  ri Þ – 0; if g

ð35Þ

^  ðriþ1  ri Þ ¼ 0; if g

ðkÞ where the expression ph denotes the kth primitive of the function ð1Þ ð1Þ h; thus, for instance, pES is the function such that d pES =d ¼ ESc . For c c

e

the reader’s convenience, the explicit expression of such primitives, related to the secant modulus associated with the stress–strain law of concrete adopted in the numerical examples are reported in Appendix A. The same formulas have been applied in [19,20] to evaluate the stiffness matrix of RC sections endowed with constitutive laws with history variables, see, e.g. [12,15]. 4.2. Secant stiffness matrix for a circular domain For circular domains, the integrals (27)–(29) can be specialized as follows. Let rC be the position vector of the center of a circular arc of radius R and let h be an angle assumed to be positive if measured in a counter-clockwise sense starting from an axis with origin at rC and oriented in the direction of g, see [25] for further details. We introduce the position vector r

~rðh; gÞ ¼ rC þ R cos hgu þ gg?u ;

ð36Þ g? u

where gu = g/jgj and g is the distance measured along the axis. We also introduce the function e E Sc which is the composition function of the concrete secant stiffness ESc with

F. Marmo et al. / Computers and Structures 89 (2011) 91–108

~eðhÞ ¼ eðrC Þ þ R cos hjgj:

ð37Þ

Accordingly, the entries of the secant stiffness matrix for a circular domain can be expressed as

ASc ¼

Z

Z

he

0

0

Z



R sin h

Z

hs

e E Sc ðhÞðR sin hÞdg dh

R sin h

0

0

e E Sc ðhÞðR sin hÞdg dh

¼ AESc ðhe Þ  AESc ðhs Þ; sSc

¼

Z

he

Z

Z

hs

0



R sin h

0

Z

0

ð38Þ

0

e E Sc ðhÞ~rðh; gÞðR sin hÞdg dh

¼ sESc ðhe Þ  sESc ðhs Þ; JSc

¼

Z

he

Z

Z

hs

0

0



R sin h

0

Z 0

ð39Þ

e E Sc ðhÞ~rðh; gÞ  ~rðh; gÞðR sin hÞdg dh

R sin h

slope of the softening branch in the sense that greater values of

ec0  ecu are associated with a more pronounced softening behaviour. The constitutive law considered for steel bars, see Fig. 2, has the following analytical expression:

8 > < fsy þ Esh ðes þ esy Þ if  esu 6 es 6 esy if  esy 6 es 6 esy rs ðes Þ ¼ Es es > : fsy þ Esh ðes  esy Þ if esy 6 es 6 esu

e E Sc ðhÞ~rðh; gÞ  ~rðh; gÞðR sin hÞdg dh

¼ JESc ðhe Þ  JESc ðhs Þ;

ð40Þ

8 > < fpy þ Eph ðep þ epy Þ if  epu 6 ep 6 epy if  epy 6 ep 6 epy rp ðep Þ ¼ Ep ep > : fpy þ Eph ðep  epy Þ if epy 6 ep 6 epu

5. Numerical examples The constitutive laws and the structural assumptions prescribed by Eurocode 2 for the ULS analysis have been considered in all tests presented in this section. As pointed out in the introduction, the presence of a softening stress–strain law has been taken into account to address a more realistic concrete behaviour. To this end the concrete stress–strain law prescribed by Eurocode 2 [8] for general nonlinear structural analyses has been considered:

kðec =ec0 Þ  ðec =ec0 Þ2 1 þ ðk  2Þðec =ec0 Þ

k ¼ 1:1Ec

ec0 fc

;

ecu 6 ec 6 0;

Fig. 2. Constitutive law of ordinary steel.

ð41Þ where fc denotes the concrete cylindrical compressive strength, ec0 the strain at peak stress, ecu the nominal ultimate strain and Ec the modulus of elasticity of concrete, see Fig. 1. Stresses and strains are assumed positive if tensile. The parameter ec0 in (41) governs the

Fig. 1. Concrete constitutive law.

ð43Þ

where Ep is the Young modulus, fpy the conventional yield stress of prestressing steel, epy the conventional yield strain, epu the ultimate strain and the slope Eph of the yielding branch is defined as in (42) in order to comply with EC2 specifications [8].

where hs and he are respectively, the initial and the final angles of the circular arc. The previous integrals are computed analytically with reference to the secant modulus function associated with the softening stress–strain law considered in the numerical examples. The resulting formulas are reported in Appendix B.

rc ðec Þ ¼ fc

ð42Þ

where Es is the Young modulus, fsy the yield stress, esy the design yield strain, esu the design ultimate strain and Esh represents the slope of the yielding branch. The following analytical expression of the constitutive law is assumed for the prestressing steel, see Fig. 3:

e E Sc ðhÞ~rðh; gÞðR sin hÞdg dh

R sin h

95

Fig. 3. Constitutive law of prestressing steel.

96

F. Marmo et al. / Computers and Structures 89 (2011) 91–108

As prescribed by EC2 [8], the complete set of inequalities that express the ULS conditions for concrete and steel is:

eðqÞ P ecl q 2 X; jeðqsi Þj 6 esu i 2 f1; . . . ; Ns g; jeðqpi Þj 6 epu i 2 f1; . . . ; Np g;

ð44Þ

where

(

ecl ¼ ecu if emax > 0; ec0 ecu emax ðecu ec0 Þ ecl ¼ if emax 6 0 ec0

ð45Þ

and emax is the maximum concrete strain over a section. The range of strain distribution allowed in the section by EC2 is shown in Fig. 4. Formula (45)2 expresses the prescription imposed by EC2 for predominantly compressive strain states emax 6 0, i.e. when the strain is negative in the entire section. Basically, in this circumstance, the strain field must attain the value ec0 at the points of the section whose distance from the most compressed fiber of the section is equal to ð1  eec0 Þh. Such distance is evaluated along cu a direction orthogonal to the neutral axis, see Fig. 5. This condition is expressed as a function of the maximum concrete strain over the section emax as follows:

emax  ecl h

¼

ec0  ecl : hð1  ec0 =ecu Þ

ð46Þ

The ULS condition (46) stems from the similarity of the triangles in Fig. 5 and, after some algebraic manipulations provides inequality (45)2. Several numerical tests have been selected in order to validate the proposed approach and the relevant results are illustrated

and discussed hereafter. Specifically, three groups of tests have been taken into account. A first group of two tests has been considered by performing a comparison between the results computed by the present approach and those available in the literature, either obtained experimentally or computed by means of different analytical methods. In a second group of tests the ultimate strength N, Mx, My domains have been computed for selected benchmark tests in order to evaluate the robustness and efficiency of the proposed algorithm. A representative set of the sections commonly encountered in design practice were studied. To investigate the computational performances of the proposed algorithmic strategy in the presence of prestressing and softening, a third group of tests has been carried out by applying them separately and jointly in order to evaluate their reciprocal influence on the overall ultimate behaviour of the section. Our numerical results have been obtained by adopting a numerical tolerance of 106 as the stopping criterion for convergence check. All the analyses refer to a generic time of interest and stress losses have been set to zero to allow for comparisons with existing results.

5.1. First group of tests: comparisons with results available in the literature Although not much literature exists on the subject, two papers [13,24] have been selected for validating our results by making a comparison with available experimental data. For greater clarity the numerical results obtained by the proposed approach are reported hereafter in two separate subsections.

Fig. 4. EC2 range of strain distribution allowed on a section.

Fig. 5. EC2 ULS for predominantly compressive strain states.

97

F. Marmo et al. / Computers and Structures 89 (2011) 91–108 Table 2 Exs. #2 and #6: box section with circular opening; material properties. Concrete

Reinforcement bars

Prestressing steel

Additional bars

fc = 40 MPa ec0 = 0.00224 ecu = 0.003 Ec = 33000 MPa

fsy = 350 MPa fsu = 350 MPa Es = 210000 MPa esu = 0.01 8£25 Concrete cover: 6 cm

fpy = 1650 MPa fpu = 1650 MPa Ep = 200000 MPa epu = 0.01 ep0 = 0.00381 2£50

9£25

Table 3 Exs. #2 and #6: box section – position of ordinary reinforcements. Fig. 6. Ex. #1: geometry of the rectangular section in Kawakami et al. [13].

Table 1 Ex. #1: rectangular section; material properties. Concrete

Reinforcement bars

Prestressing steel

fc = 37 MPa ec0 = 0.00224 ecu = 0.003 Ec = 34000 MPa

fsy = 350 MPa fsu = 350 MPa Es = 200000 MPa esu = 0.01 4£10 at the corners concrete cover: 3.1 cm

fpy = 1550 MPa fpu = 1550 MPa Ep = 200000 MPa epu = 0.01 ep0 = 0.0057 2£7 (filled circles)

#

x [cm]

y [cm]

#

x [cm]

y [cm]

1 2 3 4 5 6 7 8

44 44 31 31 44 44 31 31

26 39 39 26 26 39 39 26

9 () 10 () 11 () 12 () 13 () 14 () 15 () 16 () 17 ()

44 44 74 74 74 74 37 37 0

26.32 26.32 36.32 36.32 44 44 44 44 44

Fig. 7. Ex. #1: rectangular section by Kawakami et al. [13]; comparison of the results in terms of neutral axis with respect to [13,24].

Fig. 8. Exs. #2 and #6: geometry of the box section with a circular opening; additional reinforcements () used only in Ex. #6 and in the sensitivity analysis.

98

F. Marmo et al. / Computers and Structures 89 (2011) 91–108

5.1.1. Ex. #1: rectangular section by Kawakami et al. The first comparison regards the rectangular section analyzed both experimentally and numerically by Kawakami et al. in [13] and only numerically in [24]; geometry and material properties of the section are reported in Fig. 6 and Table 1. The constitutive parameters have been calibrated in order to achieve the best fit with the stress–strain laws considered in [13] although different analytical expressions are exploited. The computed ULS load and cross section strain field have been compared with the corresponding data experimentally measured by Kawakami et al. The components of the ultimate strain vector computed by the present method are gx = 9.62  105 cm1, gy = 6.50  104 cm1, e0 = 1.21  103 while the associated strain at the corner points is reported in Fig. 7. The computed 10 5 0

Table 4 Ex. #3: Y-shaped section; material properties. Concrete

Reinforcement bars

Prestressing steel

fc = 37 MPa ec0 = 0.00225 ecu = 0003 Ec = 34,000 MPa

fsy = 350 MPa fsu = 350 MPa Es = 200000 MPa esu = 0.01 26£16 Concrete cover: 3.1 cm

fpy = 1490 MPa fpu = 1790 MPa Ep = 200000 MPa epu = 0.01 ep0 = 0.006 43£8.1

Table 5 Ex. #3: Y-shaped section; position of ordinary reinforcements. #

x [cm]

y [cm]

#

x [cm]

y [cm]

#

x [cm]

y [cm]

1 2 3 4 5 6 7 8 9

16 16 5 5 8 8 10 10 0

4 4 4 4 37 37 67 67 88

10 11 12 13 14 15 16 17 18

0 13 13 27 27 27 27 54 54

102 100 100 98 107 98 107 107 112

19 20 21 22 23 24 25 26

54 54 78 78 82 82 82 82

107 112 122 122 118 122 118 122

N [MN]

−5 −10

Table 6 Ex. #3: Y-shaped section; position of prestressing reinforcements.

−15

−20 Mx − present method My − present method Mx − Rodriguez & Ochoa My − Rodriguez & Ochoa Nmax − Nmin axis

−25 −30

−35

0

1

2

3

4

5

6

7

Mx, My [MNm] Fig. 9. Boundary of the ULS domain for the box section (Ex. #2) for a ratio Mx/My = 1.732.

(a)

#

x [cm]

y [cm]

#

x [cm]

y [cm]

#

x [cm]

y [cm]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

10 0 10 10 0 10 5 0 5 5 0 5 5 0 5

5 5 5 10 10 10 15 15 15 20 20 20 25 25 25

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

5 0 5 5 0 5 5 0 5 5 0 5 5 0 5

30 30 30 35 35 35 40 40 40 45 45 45 50 50 50

31 32 33 34 35 36 37 38 39 40 41 42 43

5 0 5 5 5 5 5 20 20 35 35 70 70

60 60 60 75 75 90 90 100 100 105 105 115 115

(b)

Fig. 10. Ex. #3: geometry of Y-shaped section and distribution of reinforcements: (a) ordinary and (b) prestressing.

99

F. Marmo et al. / Computers and Structures 89 (2011) 91–108

3 My [MNm]

N [MN] -15,847 2

-14,043 -12,240 -10,436

1

-8,633 -6,829 0 -3

-2

-1

0

1

2

3

4

5 Mx [MNm]

-5,025 -3,222 -1,418

-1

0,385 2,189 -2

3,992

-3 Fig. 11. Ex. #3: Y-shaped section: interaction curves.

ULS bending moment is Mx = 7027 Nm while the corresponding ultimate load is 33068 N. This last value turns out to be closer to the ultimate load of 33200 N reported by Kawakami et al. than the value of 31630 N numerically obtained by [24]. Fig. 7 shows

also a comparison between the neutral axes at the ultimate state computed by the present method and the corresponding results reported in [13,24]. 5.1.2. Ex. #2: box section with a circular opening The second numerical test of the first group concerns the box section analyzed in [13,24], see Fig. 8. The same geometry, material properties, ordinary and prestressing reinforcements as in the original papers [13,24] have been used; they are reported in Table 2. Additional reinforcements with respect to the original ones considered in [13,24], denoted by crosses () in Fig. 8, have been considered only in Ex. #6 presented below. The position of the original and additional reinforcements is reported in Table 3. The computed ULS boundary curve associated with a ratio Mx/My = 1.732 is compared in Fig. 9 with the results reported in [24]. Fig. 9 shows also the projection of the axis of ULS domain, i.e. the segment connecting the tensile and compressive limit points. These points are characterized by maximum and minimum axial force over the whole domain and are associated with uniform positive and negative strain conditions. Such limit points can be computed by hand calculations. In particular, for the tensile limit point, stress is null in concrete due to the no-tension hypothesis; moreover the tendons work at the yield stress fpy = 1650.00 MPa while stress in ordinary bars is fsy = 350.00 MPa. Under this condition the axial force is Nt = 350  8  491  106 + 1650  2  1964  106 = 7.86 MN. Since eccentricity ey of the tendons with respect to the geometric centroid is 41.9 cm, the bending moments induced by the tendons are Mxt = 7.86  0.419 = 3.29 MNm and Myt = 0.00 MNm. 5.2. Second group of tests: assessment of the efficiency and robustness of the algorithm

Fig. 12. ULS domain of the Y-shaped section (Ex. #3).

In order to evaluate the performances of the proposed algorithm in terms of efficiency and robustness the ultimate strength

100

F. Marmo et al. / Computers and Structures 89 (2011) 91–108

Fig. 13. Ex. #4: unsymmetric roof slab.

3 My [MNm]

N [MN]

2

-8,825 -7,737 -6,649 1 -5,561 -4,473 -3,385 0 -2

-1

0

1

2 Mx [MNm]

-2,297 -1,209 -0,121 0,967

-1

2,055 3,143 -2

-3 Fig. 14. Ex. #4: unsymmetric roof slab: interaction curves.

N, Mx, My domains have been computed with reference to four benchmark tests. The results are presented in terms of {Mx, My}

interaction curves corresponding to twelve values of the axial force ranging between the admissible values defined by the Eurocode 2.

101

F. Marmo et al. / Computers and Structures 89 (2011) 91–108 Table 7 Ex. #4: unsymmetric roof slab; material properties. Concrete

Reinforcement bars

Prestressing steel

fc = 37 MPa ec0 = 0.00225 ecu = 0.003 Ec = 34000 MPa

fsy = 350 MPa fsu = 350 MPa Es = 200000 MPa esu = 0.01 16£20 Concrete cover: 3.1 cm

fpy = 1490 MPa fpu = 1790 MPa Ep = 200000 MPa epu = 0.01 ep0 = 0.003 15£8.1 (filled circles)

Table 8 Ex. #4: unsymmetric roof slab section; position of ordinary reinforcements. #

x [cm]

y [cm]

1 2 3 4 5 6 7 8

1 49 65 56 76 15 26 6

2 2 21 38 38 21 38 38

# 9 10 11 12 13 14 15 16

x [cm]

y [cm]

37 44 39 97 122 136 137 147

50 57 57 62 84 94 102 102

Table 9 Ex. #4: unsymmetric roof slab section; position of prestressing reinforcements.

Fig. 15. Ex. #4: ULS domain of the unsymmetric roof slab.

For completeness, we also report a three-dimensional overview of the whole ULS domain for each section. The first two examples are concerned with shapes commonly encountered in prestressed concrete design: a Y-shaped section with regular geometry and an unsymmetric roof slab with irregular geometry. The latter section has been intentionally considered in order to further validate the proposed methodology with a sufficiently general test case, even though it is worth remarking that all symmetric sections have been subjected to biaxial bending, that is unsymmetric loading conditions, to construct the relevant interaction domains. The third and fourth examples of this group, namely Exs. #5 and #6, are sections retrieved from the literature [24] and modified by including additional ordinary reinforcements with respect to the original configurations. This choice is motivated by the observation that, in the original sections, the reinforcements are exactly positioned along a horizontal line. This particular layout of the reinforcements causes the stiffness matrix to be singular in the region of the ULS domain for which the entire section is subject

#

x [cm]

y [cm]

#

x [cm]

y [cm]

1 2 3 4 5 6 7 8

10 5 0 5 10 15 20 25

15 10 5 5 5 5 5 5

9 10 11 12 13 14 15

30 35 40 45 50 55 60

5 5 5 5 5 10 15

to tensile strains. In this circumstance the section exhibits zero stiffness against bending along the vertical direction since not a single bar contributes to stiffness. Moreover, this reinforcements layout turns out to be an uncommon choice in real situations since, owing to code prescriptions, sections always have ordinary bars next to the boundary. 5.2.1. Ex. #3: Y-shaped section The geometry of the section, typically used in precast roof structures, is shown in Fig. 10 while material parameters as well as the positions of ordinary and prestressing steel reinforcements are reported in Tables 4–6, respectively. The interaction curves and the ULS domain are reported in Figs. 11 and 12. 5.2.2. Ex. #4: unsymmetric roof slab The geometry of the section is illustrated in Fig. 13 while Figs. 14 and 15 contain in turn the interaction curves and the ULS domain.

Fig. 16. Ex. #5: geometry of the bridge deck section with four circular openings.

102

F. Marmo et al. / Computers and Structures 89 (2011) 91–108 Table 10 Ex. #5: bridge deck section with four circular openings; material properties. Concrete

Reinforcement bars

Prestressing steel

fc = 30 MPa ec0 = 0.0022 ecu = 0.0035 Ec = 32000 MPa

fsy = 350 MPa fsu = 350 MPa Es = 210000 MPa esu = 0.01 70£30 disposed as follows: 8 bars at the nodes +33 bars at the top side eq. spaced +11 bars at the bottom side eq. spaced +18 bars at the lateral sides eq. spaced Concrete cover: 6 cm

fpy = 1530 MPa fpu = 1530 MPa Ep = 180000 MPa epu = 0.01 ep0 = 0.005 5£81 (filled circles)

Table 11 Ex. #5: bridge deck - position of corner reinforcements. Side reinforcements are equally spaced between the bars at corners. #

x [cm]

y [cm]

#

x [cm]

y [cm]

1 2 3 4

197.12 246.78 494 494

6 68.08 105.17 119

5 6 7 8

197.12 246.78 494 494

6 68.08 105.17 119

Material properties are reported in Table 7 while Tables 8 and 9 contain the positions of ordinary steel reinforcements and prestressing tendons, respectively.

5.2.3. Ex. #5: bridge deck with four circular openings and additional reinforcements This example refers to a prestressed bridge deck with four circular openings, see Fig. 16, first considered by Brondum–Nielsen

250 My [MNm]

200

N [MN] 150 -183,076 -163,091 100 -143,107 -123,122 50

-103,138 Mx [MNm]

0 -40

-20

-83,153 -63,169

0

20

40

60

80 -43,184

-50

-23,200 -3,215 16,769

-100 36,754

-150

-200

-250 Fig. 17. Ex. #5: bridge deck section with four circular openings – interaction curves.

103

F. Marmo et al. / Computers and Structures 89 (2011) 91–108

Fig. 20. ULS domain of the box section with circular opening and additional reinforcements (Ex. #6). Fig. 18. ULS domain of the bridge deck section with four circular openings (Ex. #5).

8 My [MNm]

N [MN]

6

-31,016 -27,947

4

-24,878 -21,810 2 -18,741 -15,672 0 -6

-4

-2

0

2

4

6

8 Mx [MNm]

-2

-12,603 -9,534 -6,465 -3,397

-4

-0,328 2,741

-6

-8 Fig. 19. Ex. #6: box section with a circular opening and additional reinforcements: interaction curves.

104

F. Marmo et al. / Computers and Structures 89 (2011) 91–108

Table 12 Box section with circular opening: average number of iterations as function of the diameter of rebars, of ec0 and ep0.

ec0

ep0

Average number of iterations TOL = 5.0  103

0.0019 0.0019 0.0019 0.0021 0.0021 0.0021 0.0030 0.0030 0.0030

0.00000 0.00381 0.00762 0.00000 0.00381 0.00762 0.00000 0.00381 0.00762

TOL = 1.0  106

£18

£25

£35

£18

£25

£35

26.95 22.18 23.48 21.53 27.46 23.10 20.45 21.16 21.67

24.81 26.36 27.26 24.22 25.53 26.09 24.44 25.67 26.25

20.46 21.79 22.07 20.31 21.01 21.62 20.31 20.85 21.52

44.17 46.18 50.27 47.57 49.29 56.93 45.81 46.67 47.94

44.49 48.26 48.84 48.72 51.03 54.93 45.03 47.93 47.67

45.68 49.05 47.97 47.85 57.90 49.51 45.27 50.44 47.03

[2] and later analyzed in [24]. The reader is referred to Table 10 for material parameters and is warned that the section has been modified with respect to the original one by adding ordinary reinforcements along its boundary. Table 11 reports the coordinates of corner reinforcements; the position of side bars, which are also shown in Fig. 16, are equally spaced on each side and can thus be inferred from the position of the relevant corner bars. Interaction curves in the Mx–My plane and the ULS domain are reported in Figs. 17 and 18, respectively. 5.2.4. Ex. #6: box section with a circular opening and additional reinforcements The geometry of the section is illustrated in Fig. 8 where now, different from Ex. #2, reinforcements indicated by a cross need to be included in the section. The position of such ordinary

Fig. 21. Ex. #6: comprehensive review of the ULS domains for increasing effects of prestressing and softening.

105

F. Marmo et al. / Computers and Structures 89 (2011) 91–108

have shown that the amount of reinforcement, prestressing and softening has no influence at all on the robustness of our procedure, since the algorithm converges in all cases. The data reported in Table 12 also show that the variability of these parameters has a minor influence on the number of iterations which exhibit a percentage variation of about 10% with respect to the average value. A comprehensive view of the relative influence of prestressing and softening on the shape of the ULS domains can be appreciated in Fig. 21 which reports an array of ULS domains for increasing values of ep0 and decreasing values of ec0. The dimension of the ULS

reinforcements is reported in Table 3. Interaction curves in the Mx– My plane and the ULS domain are reported in turn in Figs. 19 and 20. 5.3. Sensitivity analysis We report in Table 12 the results of a sensitivity analysis carried out by comparing the effects induced by ec0, ep0 and the amount of reinforcement on the average number of iterations necessary for determining the ULS boundary points for Ex. #6. Numerical results

N=−20 [MN]; ε =−0.003 c0

6

ε

=0

ε

= 0.0015

p0 p0

εp0 = 0.003

4

ε

= 0.0045

ε

= 0.006

p0 p0

My [MNm]

2

0

−2

−4

−6 −6

−4

−2

0 Mx [MNm]

2

4

6

Fig. 22. Ex. #6: interaction curves as function of the prestressing strain.

N=−20 [MN]; ε =0 p0

6

ε

= −0.003

ε

= −0.0023

c0 c0

εc0 = −0.0020

4

ε

= −0.0019

ε

= −0.00185

c0

My [MNm]

2

c0

0

−2

−4

−6 −6

−4

−2

0 Mx [MNm]

2

4

6

Fig. 23. Ex. #6: interaction curves as function of ec0.

106

F. Marmo et al. / Computers and Structures 89 (2011) 91–108

N=−20 [MN]; εp0=0 2 ε

= −0.003

ε

= −0.0023

ε

= −0.0020

c0

1.5

c0 c0

εc0 = −0.0019

1

ε

c0

My [MNm]

0.5

= −0.00185

0 −0.5 −1 −1.5 −2 −6

−5

−4 Mx [MNm]

−3

−2

Fig. 24. Ex. #6: interaction curves as function of ec0; detail of the interaction curves in proximity of the non-convexity point.

domains changes with ep0 and ec0; Fig. 22 shows the interaction curves corresponding to a representative value of the axial force, N = 20 MN, as function of increasing values of the prestressing strain. To evaluate the role of ec0 several interactions curves relevant to the value N = 20 MN are plotted in Fig. 23. It is worth noting first that the domain inflates with respect to the one in absence of softening, i.e. for ec0 = 0.003 in Fig. 22. Subsequently, for a value of ec0 ’ 0.0022, it starts reducing and modifying its shape by showing concave parts for values of ec0 greater than 0.0022. On the contrary no appreciable lack of convexity has been experienced for tensile axial forces. The cause for the presence of the small kinks detectable in the diagrams of Fig. 23 has been investigated in detail. Fig. 24 shows a zoom of the non-convex region of the interaction domain close to the kink located at zero My; each marker denotes a computed boundary point. It has been checked that such points are not associated with numerical instabilities of the algorithm but actually represent the solution for the ULS problem. This has been carefully checked by considering a dense cloud of points around kinks so as to compute, in their neighborhood, the relevant ULS.

6. Conclusions A general method has been presented, which is based on a secant approach, for performing the ULS analysis of prestressed reinforced concrete sections of arbitrary shape subjected to a given affine load path. The constitutive law with softening prescribed by the Eurocode 2 for nonlinear structural analysis has been adopted in the numerical examples. They have been concerned with the computation of the ULS domains of sections either reported in the literature or commonly adopted in the design practice. The adoption of a softening constitutive law allowed us, on one side, to assess the reliability of the proposed algorithm for its future use in nonlinear structural analyses and, on the other one, to illustrate the practical feasibility of the analytical integration

procedure presented in [25] for addressing a quite large family of constitutive laws. The results of the numerical simulations presented in the paper have proven that the general features of the secant method presented in [25] for ordinary RC sections are fully inherited by the proposed extension both in presence of prestressing tendons and of softening stress–strain laws. The sensitivity tests which have been carried out have shown that the robustness of the proposed secant strategy is practically unaffected by the amount of reinforcement, prestressing and softening. In a few cases, these effects produce only a slower convergence rate since the number of iterations slightly increases. Furthermore, non-convex ULS domains have been experienced in some cases related to an increase in softening. Acknowledgements The financial support of the Italian consortium ReLUIS (Rete dei Laboratori Universitari di Ingegneria Sismica) is gratefully acknowledged. Appendix A. Primitives of the secant modulus function of concrete In order to compute the integrals (27)–(29) by means of formuð1Þ ð2Þ ð3Þ ð4Þ las (30)–(32) the functions pES ðeÞ; pES ðeÞ; pES ðeÞ and pES ðeÞ need to c c c c be computed. ðiþ1Þ ðiÞ In this respect we remind that d pES =de ¼ pES and c c ð1Þ that d pES =de ¼ ESc , where the secant modulus associated with c (41) is:

ESc ðec Þ ¼





k  ec =ec0 ec0 1 þ ðk  2Þec =ec0 fc

ð47Þ

Setting

x ¼ ðk  2Þ

ec ; L ¼ logð1 þ xÞ; P ¼ 1 þ kðk  2Þ; ec0

ð48Þ

107

F. Marmo et al. / Computers and Structures 89 (2011) 91–108

!

the required primitives of ESc ðeÞ are

fc

ð1Þ

pES ðxÞ ¼ c

ð2Þ

pES ðxÞ ¼ c

ð3Þ

pES ðxÞ ¼ c

ð4Þ pES ðxÞ ¼ c

ðk  2Þ2 fc ec0

ðPL  xÞ;

ð49Þ þ

3

2ðk  2Þ fc e2c0

ð2PLð1 þ xÞ  Pð2x þ 2Þ  x2 Þ;

4

12ðk  2Þ fc e3c0

ð50Þ

ð6PLð1 þ xÞ2  Pð9x2 þ 18x þ 3Þ  2x3 Þ;

ð51Þ

and 2

72ðk  2Þ5

2

ð62Þ Finally the integrals in (40) are computed as   JESc ðhÞ ¼ R2 ½I1 ðhÞh0 ðrC  rC Þ  R3 ½I2 ðhÞh0 ðrC  gu þ gu  rC Þ

R3   ½I3 ðhÞh0 rC  g?u þ g?u  rC  R4 ½I4 ðhÞh0 ðgu  gu Þ 2 R4 R4    ½I5 ðhÞh0 gu  g?u þ g?u  gu  ½I6 ðhÞh0 g?u  g?u ; 2 3



d¼

k2

ec0

b¼

;

fc Rjgj

e2c0

;

c ¼1þ

k2

ec0

eðrC Þ;

Rjgj:

ð54Þ

Denoting by  h either the initial hs or final he values of the angle h, the two integrals in (38) are evaluated as  AESc ðhÞ ¼ R2 ½I1 ðhÞh0 ;

ð55Þ

 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðcdÞ tan2h 2 2 d  c2 k4 ðhÞ  192c2 ðbc  adÞðc2  d Þ arctanh pffiffiffiffiffiffiffiffiffi d2 c2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; I4 ðhÞ ¼ 5 2 2 96d d  c ð64Þ I5 ðhÞ ¼

k5 ðhÞ 5

96d

; I6 ðhÞ ¼

k6 ðhÞ 96d

5

ð65Þ

;

with 2

3

k4 ðhÞ ¼ 96bhc4 þ 96adhc3 þ 48bd hc2  48ad hc 2

3

4

 24d ðbc  adÞ sinð2hÞc þ 8bd sinð3hÞc þ 12bd h 2

4

2

4

þ 24dðad  bcÞðd  4c Þ sin h  8ad sinð3hÞ  3bd sinð4hÞ;

where

I1 ðhÞ ¼

ð63Þ where

ð53Þ

where

e2c0

3

ð12PLð1 þ xÞ  Pð22x þ 66x þ 30x þ 4Þ  3x Þ;

a þ b cos h e E Sc ðhÞ ¼ ; c þ d cos h fc eðrC Þ

2

4

The concrete secant stiffness e E Sc appearing in formulas (38)–(40) is the composition of the expression (47) with the function (37); the final result is



3

bðhÞ ¼ 12bhc3  12adhc2  6bd hc þ 6ad h þ 3dð4acd

Appendix B. Formulas for circular boundaries

fc k

ð61Þ

2

3

as they can be easily checked.

ec0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 d  c2 bðhÞ

þ bðd  4c2 ÞÞ sin h þ 3d ðbc  adÞ sinð2hÞ  bd sinð3hÞ: 3

ð52Þ

a¼

ðc  dÞ tan 2h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 d  c2

2

k2 ðhÞ ¼ 24cðbc  adÞðc2  d Þarctanh

ð66Þ

k1 ðhÞ

ð56Þ

3

4d

2

2

k5 ðhÞ ¼ 24dðad  bcÞð3d  4c2 Þ cos h  12d ð2acd  2bc

and

2

2

3

þ bd Þ cosð2hÞ  96ac3 d logðc þ d cos hÞ þ 96acd 2

k1 ðhÞ ¼ 4bhc2 þ 4adhc þ 2bd h ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc  dÞ tan 2h 2 2 þ 8ðbc  adÞ d  c arctanh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 d  c2

4

2 2

4

þ 3bd cosð4hÞ;

2



ð67Þ

ð57Þ

It can be checked that the values of fc, ec0 and Ec which typically characterize the constitutive law (41) make the term d2  c2 under square root in (57) always positive. Similarly, the two integrals in (39) become 

3

 cos hÞ  96bc d logðc þ d cos hÞ  8bcd cosð3hÞ

þ 4dðbc  adÞ sin h  bd sinð2hÞ:

sESc ðhÞ ¼ R2 ½I1 ðhÞh0 rC  R3 ½I2 ðhÞh0 gu 

4

 logðc þ d cos hÞ þ 8ad cosð3hÞ þ 96bc logðc þ d

4

3

k6 ðhÞ ¼ 3b sinð4hÞd þ 8ðad  bcÞ sinð3hÞd  24ðacd 2

2

4

3

R  ½I3 ðhÞh0 g?u ; 2

ð58Þ

2 2

3

4

þ 12ð8bc  8adc  12bd c þ 12ad c þ 3bd Þh 2

3

2

þ bðd  c2 ÞÞ sinð2hÞd  24ðad  bcÞð5d  4c2 Þd sin h

2 3=2

þ 192ðbc  adÞðd  c Þ

arctanh

! ðc  dÞ tan 2h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 d  c2

ð68Þ

where

I2 ðhÞ ¼

References

k2 ðhÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 4 2 12d d  c2

ð59Þ

2

I3 ðhÞ ¼  þ þ with

2

ð4bc þ 4adc þ 3bd Þ cos h 3

4d

þ

ðad  bcÞ cosð2hÞ 2

4d

b cosð3hÞ 12d   3 2 2 3 bc þ adc þ bd c  ad logðc þ d cos hÞ 4

d

;

ð60Þ

[1] American Concrete Institute. Building code requirements for structural concrete (ACI 318-05) and commentary (318R-05). Farmington Hills, Mich; 2005. [2] Brondum T. Ultimate flexural capacity of cracked polygonal concrete sections under axial loads and biaxial bending. ACI Concrete Int 1987;84(3):212–5. [3] Chen SF, Teng JG, Chan SL. Design of biaxially loaded short composite columns of arbitrary section. ASCE J Struct Eng 2001;127(6):678–86. [4] Chiorean CG. A fast incremental-iterative procedure for inelastic analysis of RC sections of arbitrary shape. Acta Tech Napocensis 2005;4:85–98. [5] Collins MP, Mitchell D. Prestressed concrete structures. Englewood Cliffs, NJ: Prentice Hall; 1991. [6] Davidster MD. Analysis of reinforced concrete columns of arbitrary geometry subjected to axial load and biaxial bending: a computer program for exact analysis. ACI Concrete Int Des Construct 1986;8:56–61.

108

F. Marmo et al. / Computers and Structures 89 (2011) 91–108

[7] De Vivo L, Rosati L. Ultimate strength analysis of reinforced concrete sections subject to axial force and biaxial bending. Comput Methods Appl Mech Eng 1998;166:261–87. [8] EN 1992-1-1. Eurocode 2: design of concrete structures – part 1-1: general rules and rules for buildings; 2004. [9] Fafitis A. Interaction surfaces of reinforced-concrete sections in biaxial bending. ASCE J Struct Eng 2001;127(7):840–6. [10] Furlong RW, Hsu TTC, Ali Mirza S. Analysis and design of concrete columns for biaxial bending – overview. ACI Struct J 2004;101(3):413–22. [11] Izzudin BA, Siyam AAFM, Lloyd Smith D. An efficient beam-column formulation for 3D reinforced concrete frames. Comput Struct 2002;80:659–76. [12] Karsan ID, Jirsa JO. Behavior of concrete under compressive loading. J Struct Div ASCE 1969;95:2543–63. [13] Kawakami M, Tokuda H, Kagaya M, Hirata M. Limit states of cracking and ultimate strength of arbitrary concrete sections under biaxial loading. ACI Struct J 1985;82(2):203–12. [14] Kwan KH, Liauw CT. Computerized ultimate strength analysis of reinforced concrete sections subjected to axial compression and biaxial bending. Comput Struct 1985;21(6):1119–27. [15] Mander JB, Priestley MJN, Park R. Theoretical stress–strain model for confined concrete. ASCE J Struct Eng 1988;114(8):1804–26. [16] Mander JB, Priestley MJN, Park R. Observed stress–strain behavior of confined concrete. ASCE J Struct Eng 1988;114(8):1827–49. [17] Marin J. Design aids for L-shaped reinforced concrete columns. ACI J 1979;76(11):1197–216. [18] Marmo F. A fiber-free approach to the inelastic analysis of reinforced concrete structures. PhD thesis, University of Naples, Federico II; 2007. [19] Marmo F, Rosati L. Incorporation of the Karsan–Jirsa model within the fiberfree approach. In: Proceedings of ReLUIS conference, Roma, Italy; 2008.

[20] Marmo F, Rosati L, Sessa S. Exact integration of uniaxial elasto-plastic laws for nonlinear structural analysis. In: Proceedings of 2008 seismic engineering conference commemorating the 1908 Messina and Reggio Calabria Earthquake, Reggio Calabria, Italy; 2008. [21] Marmo F, Rosati L, Serpieri R. Influence of prestressing and softening on the ultimate strength domains of RC sections subject to axial force and biaxial bending. In: Proceedings of symposium: advances in R/C and precast constructions – in honour of Giandomenico Toniolo, Milano, Italy; 2008. [22] Pisani MA. A numerical method to analyse compact cross-sections. Comput Struct 1996;59(6):1063–72. [23] Rodriguez-Gutierrez JA, Aristizábal-Ochoa JD. Biaxial interaction diagrams for short RC columns of any cross section. ASCE J Struct Eng 1999;125(6):672–83. [24] Rodriguez-Gutierrez JA, Aristizábal-Ochoa JD. Partially and fully prestressed concrete sections under biaxial bending and axial load. ACI Struct J 2000;97(4):553–63. [25] Rosati L, Marmo F, Serpieri R. Enhanced solution strategies for the ultimate strength analysis of composite steel – concrete sections subject to axial force and biaxial bending. Comput Methods Appl Mech Eng 2008;197:1033–55. [26] Sfakianakis MG. Biaxial bending with axial force of reinforced, composite and repaired concrete sections of arbitrary shape by fiber model and computer graphics. Adv Eng Software 2002;33:227–42. [27] Shuraim A, Naaman AE. A new methodology for the ultimate capacity of slender prestressed concrete columns. PCI J 2003;48(1):64–80. [28] Thanoon WA, Hamed AMM, Noorzaei J, Jaafar MS, Al-Silayvani BJ. Inelastic analysis of composite sections. Comput Struct 2004;82:1649–56. [29] Yan CY, Chan SL, So AKW. Biaxial bending design of arbitrarily shaped reinforced concrete columns. ACI Struct J 1993;90(3):269–78. [30] Yen JR. Quasi-Newton method for reinforced concrete column analysis and design. ASCE J Struct Eng 1991;117(3):657–66.