asymmetrically reinforced concrete piles in earth retaining systems

IABSE SYMPOSIUM MADRID 2014

Engineering for Progress, Nature and People

REPORT

International Association for Bridge and Structural Engineering IABSE

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ISBN: 978-3-85748-135-2Legal Deposit: B-17936-2014Printed in Spain

37th IABSE Symposium Madrid 2014iii

Martinez-Cutillas, Antonio – Chair, SpainGoicolea Ruigomez, José M. - Vice-Chair, SpainCamara Casado, Alfredo – Secretary, SpainAnderson, John, USABartlett, F. Michael, CanadaBeier, Marcos, BrasilBoegle, Annette, GermanyBriseghella, Bruno, ItalyBrühwiler, Eugen, SwitzerlandBucur, Carmen, RomaniaCampos e Matos, Jose, PortugalCerdeiriña, Roberto, SpainChrimes, Mike, UKde Boer, Ane, The NetherlandsDritsos, Stephanos, GreeceFuruta, Hitoshi, JapanG. Pulido, M. Dolores, SpainGiarlelis, Christos, GreeceGómez Hermoso, Jesús, SpainGuo, Tong, ChinaHuang, Dongzhou, USAHuang, Hongwei, ChinaJanjic, Dorian, AustriaKim, Ho-Kyung, KoreaKirk, Martin, UKLampropoulos, Andreas, UKLimsuwan, Ekasit, ThailandLozano Galant, José Antonio, SpainLu, Xinzheng, ChinaMalaga-chuquitaype, Christian, UKMcCall, Joanne, CanadaMcGormley, Jonathan, USMontens, Serge, FranceMorgenthal, Guido, GermanyNakamura, Shunichi, JapanOliveira-Santos, Luis, PortugalOrcesi, Andre, FrancePalmisano, Fabrizio, Italy

Payá Zaforteza, Ignacio, SpainPelke, Eberhard, GermanyRatay, Robert, USARevilla Angulo, Roberto, SpainRomo, José, SpainRuan, Xin, ChinaRuiz Teran, Ana Maria, SpainRus Jenni, Laurent, SpainSaad, Fathy, EgyptSantana, Guillermo, Costa RicaSchlaich, Mike, GermanySiebert, Geralt, GermanyStroetmann, Richard, GermanySubbarao, Harshavardhan, IndiaTeng, Jin-Guang, ChinaTraykova, Marina Doncheva, BulgariaTurmo Cordeque, José, SpainVejrum, Tina, DenmarkVergoossen, Rob, The Netherlands

Collaborators of the

Nguyen Gia, Khanh - Assistant Secretary, SpainDan, Sorin, RomaniaHernández Montes, Enrique, SpainLuna, Javier, SpainZanaica, Luca, ItalyPeters, Tom F., Switzerland

37th IABSE Symposium Madrid 2014iv

M. Dolores G. Pulido – ChairAntonio Martínez-CutillasRoberto CerdeiriñaJesús Gómez HermosoEnrique Hernández MontesAna Lorea ArnalJosé Antonio Lozano GalantPatricia Olazábal HerreroIgnacio Payá ZafortezaRoberto Revilla AnguloLaurent Rus Jenni

Volunteers

Clara Abella, University CEU San PabloMateo Álvarez Ríos, University CEU San PabloAlejandro Barberá Carpio, Technical University of ValenciaMaría del Reinio, University CEU San PabloMustapha El Hamdaoui, Technical University of MadridRafael Fernández Romero, University CEU San PabloCristina Fuente San Juan, University CEU San PabloAntonio Joaquín García Suárez, University of SevillaIrene Hernández Sánchez, University CEU San PabloAlberto Laorce, University Castilla–La ManchaFco de Asís Loustau, Technical University of MadridFrancisco Javier Luna, Eduardo Torroja Institute - CSICJesús Mazarro Serrano, University Castilla–La ManchaEnrique Pardo Goicoechea, Technical University of ValenciaDavid Pastor Moreno, Technical University of MadridLisbel Rueda García, Technical University of ValenciaAlicia Sánchez Turel, University CEU San PabloCarla María Sancho Pérez, Technical University of ValenciaAida Santos Santamaría, Technical University of MadridCarlos Sastre Mosquera, Technical University of MadridJosé Ramón Serra Santamaría, Technical University of ValenciaÁlvaro Soriano Ortiz, University Castilla–La Mancha

37th IABSE Symposium Madrid 2014xxx

Thomas Charles OSBORNE, Bartlomiej HALACZEK ................................................ 1671

Changing Neutral Axis for Effective Post-Tensioning of Long Span Concrete GirderTae Min KIM, Do Hak KIM, Yun Mook LIM, Moon Kyum KIM................................ 1678

A New Method for Reinforcement Design in Concrete StructuresDong XU, Yu ZHAO ..................................................................................................... 1685

Study on system extension and mechanical performance of outer convex spoke tensile structureYueqiang ZHANG, Jiemin DING .................................................................................. 1691

Temporary Multi-story Container House after Earthquake and Tsunami disaster on March 11, 2011Shigeru HIKONE, Masaki TOKUBUCHI ..................................................................... 1699

Seismic Behaviour Assessment of Long Viaducts: a combined FEM and SHM approachMariano AHIJADO, José M. SIMÓN-TALERO, Alejandro HERNÁNDEZ, Manuel SANTILLÁN, Alberto FRAILE, Lutz HERMANNS ...................................... 1707

Asymmetrically reinforced concrete piles in earth retaining systemsJuan F. CARBONELL MÁRQUEZ, Luisa Mª GIL-MARTÍN, M. Alejandro FERNÁNDEZ-RUÍZ, Enrique HERNÁNDEZ MONTES ............................................ 1714

Design of a High-Rise Steel Building to Resist Disproportionate CollapseKarl RUBENACKER, Ramon GILSANZ, Philip MURRAY, Eugene KIM ................. 1722

An alternative beam-to-column connection in building structuresAntonio AZNAR, José Ignacio HERNANDO, Jesús ORTIZ, Fernando MAGDALENA, Jaime CERVERA ............................................................... 1726

Fused seismic shock absorbers - an innovative solution for a high-speed rail viaduct of the AVE Granada Line, SpainBorja BAILLÉS, Gianni MOOR, Niculin MENG ......................................................... 1732

amhd_ugr

Resaltado

37th IABSE Symposium Madrid 20141714

Research Assistant Universidad Loyola AndalucíaSevilla, ESP [email protected]

JF Carbonell-Márquez received his civil engineering degree in 2010 and his PhD in 2014 both from the University of Granada, Spain. His main area of research is related to reinforced concretestructures

Associate Professor University of Granada Granada, ESP [email protected]

LM Gil-Martín, civil engineer, is the head of research group TEP-190 at University of Granada. She received her PhD in 1997 from University of Granada. She has published more than 40 research works relating steel and concrete structures in the main structural engineering journals

PhD CandidateUniversity of Granada Granada, ESP [email protected]

MA Fernández-Ruíz is civil engineer (University of Granada, 2012). He is working on his PhD on reinforced concrete shells and slabs since 2013. Other of his field of study are tension and compression only structures

MONTESProfessorUniversity of Granada Granada, ESP [email protected]

E Hernández-Montes received his civil engineering degree in 1992 and his PhD in 1995, both from the University of Granada, Spain. He has published more than 50 research works in steel and concrete structures andearthquake engineering in the top engineering journals

The asymmetric reinforcement in concrete pile members employed as retaining earth systems has recently been introduced. The use of non-symmetrical piles supposes savings up to 50% in weight of longitudinal reinforcing steel, compared with the traditional symmetrical piles. The behaviour of those new members under service both in short and long term loading is currently under study and new expressions regarding the contribution of concrete in tension have been introduced. Comparison between the behaviour of asymmetrically reinforced piles and its symmetrical counterpart is also presented.

reinforced concrete, service behaviour, asymmetric reinforcement, circular cross section, tension stiffening, effective area, deflection.

Introduction Concrete piles employed in earth retaining systems are traditionally reinforced by means of a symmetric layout of bars placed in the perimeter of the cage. Recent works of the TEP-190 research group at University of Granada have presented a new version of concrete piles reinforced with an asymmetric arrangement of longitudinal steel bars, based on previous studies regarding the optimal reinforcement solution of concrete members under flexure [1]–[5]. Although asymmetric reinforcement for circular concrete cross sections was first introduced by Weber and Ernst [6], the provided solution did not correspond with the minimum of reinforcement. The approach presented by Gil-Martín [7] allows bars of different diameter and spacing to be used and the absolute minimum reinforcement required in the ultimate strength design of circular sections can be computed.


Asymmetric reinforced pile cross sections have the steel bars placed where it is really needed – at tension zone of the section, Fig. 1 –. That’s why intuition suggests that an asymmetrically reinforced pile undergoes less deformation than a symmetric pile that provides the same ultimate flexural moment, under the same loads. Therefore, important savings in reinforcing steel – up to 50% – can be achieved and the behaviour of the system is even enhanced.Little attention has been paid to the behaviour in flexure of reinforced concrete members with circular cross – section in general literature or current codes of design as Eurocode 2 (EC2) [8] or CEB-fib Model Code 2010 (MC2010) [9]. In fact, when addressing serviceability, these codes of design provide expressions for the effective area of concrete in tension to take into account the so-called tension stiffening effect, but the only

consider rectangular cross –sections. As a consequence, Carbonell-Márquez [10] recently proposed a new expression for the effective concrete area in tension for circular concrete cross sections both symmetrically and asymmetrically reinforced, considering the tension stiffening model implicitly employed in EC2 and MC2010 and presented by Hernández-Montes [11]. An experimental campaign considering both short and long term conditions has been conducted in order to verify the analytically implemented models to capture the behaviour of the asymmetrically reinforced piles. Short term results of deflection are accurately predicted by the analytical model. Some analytical results have been obtained for long term loaded piles but further analysis is needed.

The numerical method employed to compute the M � relationship for a given RC section is a smeared crack approach and makes use of the Bernouilli’s hypothesis that plane sections remain plane after deformation, assuming that no slip of reinforcement occurs, so that the strain on any fiber of the section is given by:

( , , ) coscg cg R (1)

where cg is the strain at the center of gravity of the cross section and is the angle between the vertical principal axis of inertia of the section and the radio vector of the fiber where the strain is evaluated, Fig. 2. In the present work, compression strain, stresses and forces are considered as positive as well as, bending moments which cause compression at top fiber. In Fig. 2, ctm is the angle between the vertical principal axis of inertia and the radio vector of the fiber whose strain is the concrete limit strain of cracking, ctm = fctm/ Ecm with fctm the concrete tensile strength and Ecm is the concrete secant elastic modulus. Therefore, the value of the resultant axial force N in the section for a given pair of values

� cg is:

,

2, , Ø ,0

2 sinctm

c effc cg cTS cg j s j cgA

jN R d dA A y (2)

where AØj and yj are the cross sectional area and the vertical coordinate of each reinforcing steel bar, respectively, c, cTS and s are the stress-strain relationships for uncracked concrete, cracked concrete and steel, respectively, and Ac,eff is the effective area of concrete in tension; these relationships have been written directly as functions of the angle of the fiber, , the strain at the center of gravity of the section, cg, and the curvature of the section, :

A A

Section A – A

Fig. 1: Asymmetric piles in earth retaining systems

Center of gravity of the gross section

ctm R

ctm

CT( , cg, )

y=R cos

cg

Fig. 2: Bernouilli�s hypothesis and strain at the cross section and nomenclature


, , , ,cg cg (3)

Now, the M � relationship is calculated in an iterative way: given a value of = i, the strain on the center of gravity of the section, cgi, which causes axial equilibrium is found by employing the bisectional method. Once, cgi is known, the strain on each fiber of the section can be obtained with Eq.(1), and the corresponding bending moment, Mi, is computed as:

,

3 2, , Ø ,

0

sin cosctm

c eff

i c cg i cTS cg i j s j cg i jjA

M R d ydA A y y (4)

A new value of is imposed and the process is repeated.

2.2.1 Stress-strain model for steel The steel stress – strain model is the bilinear and symmetric model given in EC2. The hypothesis of symmetry is valid for the maximum spacing for the stirrups allowed by the majority of existing concrete design codes [12].

2.2.2 Stress-strain model for uncracked concrete The employed stress – strain model for uncracked concrete is the one proposed by EC2.

2.2.3 Stress-strain model for cracked concrete: tension stiffening effect The employed stress-strain model to account for the tension stiffening effect in cracked concrete is the one proposed by Hernández-Montes [11], an explicit expression deduced from the average strain in the reinforcement in a concrete prism subjected to pure tension, and given by CEB Design Manual on Cracking and Deformation [13]. Taking into account that tensile stresses and strains are

negative in this work, the expression that provides the tension stiffening contribution of cracked concrete is:

22 1

2 2eff eff

cTS s s ctm effE E f n (5)

being n = Es/Ec, eff = As/Ac,eff , Ac,eff is the effective area of concrete in tension and is a parameter that accounts for the duration of load and its repeatability ( = 1.0 in short term and = 0.5 in long term loading). This value of tension stress is

valid only until steel at any crack in the element reaches its tensile yield strain y = fy/ Es, with fy the yield limit of the steel and Es = 200 GPa. At that moment, the average strain of the concrete prism is ap and it can be found by force equilibrium between a generic section and a section at crack location. If the strain keeps growing, the axial load will be constant and the average tensile stress in concrete will be reduced linearly until the average strain of the prism reaches y. Therefore, the complete stress-strain expression for the cracked concrete is given in Eq. (9) and presented in Fig. 3:

,

22

0 if

if

1 if2 2

y

cTS apcTS y y

y ap

eff effs s ctm

ap

ap ctmeffE E f n

(6)

Fig. 3: tension stiffening model for cracked concrete

cracked

fctm

= l/l

ap - y ctm

cTS

c

cTS( ap)

N N

Ac,eff

As

l l/2 l/2


2.2.4 Effective area of concrete in tension Reinforced concrete design codes as EC2 or MC2010 provide the effective area of concrete in tension in case of rectangular cross sections. In the absence of treatment of circular cross-sections in the codes of design, Carbonell-Márquez [10] studied the evolution of the effective area in tension of circular cross sections so that, when employing the above given stress-strain models, the resulting moment-curvature, M- , relationship of the entire reinforced concrete cross section is the same than the one proposed by EC2 or MC2010 and given by the interpolation equation (10):

2 2

2 11 cr crM MM M

(7)

The effective area of concrete in tension, Ac,eff, is considered as a circular strip whose width is hc,effplaced on the tensile side of the cross-section below the horizontal fibre whose deformation is ctm, Fig. 4. The width hc,eff is distributed around the circle that links the center of gravity of the bars. hc,eff is divided into two portions: interior, hc,effint, and exterior, hc,effext. This division is made taking the rate TSztop/TSz; TSz – Tension Stiffening zone – is the distance that exists from the bottom fiber of the cross section and the fiber whose strain is ctm, and TSztop is TSz minus the distance from the center of

gravity of the lowest placed bar to the perimeter of the section, Fig 4. Therefore: 1

, , . , ,ØØ2

topc effint c eff s c effext c eff c effint

TSzh h R c h h h

TSz (8)

The evolution of hc,eff with the position of the neutral fibre in the cross-section was studied for more than 120 different cross-section dimensions and reinforcement configurations, including both symmetric and asymmetric layouts. The results can be adjusted by least square method and general expressions for the evolution of hc,eff with the position of the neutral fibre, x, can be proposed:

2 2 2 2, ,4/3 4/3

Symmetric case Asymmetric case1 11.765 11.343 9.375 1.117 8.657 7.132c eff c effh R R x x h R R x x

R R (9)

In Eqs. (9) the values of hc,eff, R and x are given in mm.

When considering long term loading, both concrete shrinkage and creep need to be taken into consideration. In order to do this, the concrete stress-strain model is affected consequently. Fig. 7 shows the modified stress-strain concrete model. First, the parameter in the tension stiffening model needs to be changed into 0.5 and the original stress-strain model for cracked concrete shown in Fig. 3 is modified adding a new linear portion (ab) after fctm, Fig. 7 (a), in order to be consistent with the evolution of the tensile force applied to the concrete prism considering long term affected tension stiffening. Then, the concrete modulus is affected by creep according to the effective modulus method [14], calculating the

=1.0a

b

c

As s( )

= l/l

N

Ncr

-Ny

ap - y ctm

d

(a) Force applied to the prism (Fig.3)

=0.5

d

fctm

cTS( )

cap - y ctm

c

b

a

=1.0

=0.5

(b) Concrete stress-strain w/ =0.5

fctm

c

ap,eff - y ctm,eff ,eff

Affected by

fctm

c

- y

(c) Concrete stress-strain affected by shrinkage

sh

(d) Concrete stress-strain affected by creep

Fig. 5: Long term affected concrete stress-strain model

Rhc,effext

Rhc,effintR

hc,effext

hc,effint

ctm

y ctm

1Ø Ø / 2sc

ctm

CT

hc,eff

hc,effint

hc,effextAc,eff

y

cg

1Ø

2Ø

Fig. 4: Definition and nomenclature for Ac,eff


creep coefficient as EC2 indicates. Finally, the shrinkage strain sh is computed is computed according to EC2 and the creep-affected model is displaced horizontally so that the strain at zero stress is equals to sh. Due to the applied level of loads in the long term experiments, only the linear portion of the compression model of concrete is considered. This model is employed to compute the M � relationship as part 2.1 indicates.

Four specimens of piles are tested, two in short term loading and two in long term loading. The members consist of 4 m long piles cross section diameter of 400 mm, subjected to 4–point bending so that there is a central segment of the specimen being subjected to a constant bending moment law of P KN·m, with P the applied load. In both experiments, short and long term, the deflection of the members at mid span is measured and strain gauges are located on the reinforcement bars and at the same section level in order to measure the longitudinal strain of those bars and, subsequently, deduce the curvature of the section assuming plane sections remain plane after deformation. Each of the two pair of tested elements consists of one symmetrically and one asymmetrically reinforced concrete piles. All the cross sections have a concrete cover to reinforcement c = 57 mm and, despite of the members are transversally reinforced with stirrups of Øs = 10 mm, no confinement effect is taken into account in the concrete model. For the steel, values of fy = 500 MPa and Es = 200 GPa are used in the computations. The symmetric cross section, shown in Fig. 9 (d), consists of 16 bars of Ø 16 mm (As = 3216.99 mm2) equally spaced. For the asymmetric cross section, shown in Fig. 9 (e), seven bars of Ø 16 mm are placed at the bottom of the section, with a separation between bars of 41 mm, and the rest of the cross section is reinforced by means of 3 bars of Ø 10 mm separated 193.21 mm (As = 1643.06 mm2).In the short term experiments, the piles were conducted to rupture without significant development of shear cracks. On the other hand, in the long term experiments a mechanism similar to a nutcracker is employed to apply two constant loads of 5 tons to the pile during three months.

Both piles were initially designed to be similar in bending resistance. In this way, concrete was intended to be a C30, that is fck = 30 MPa. With this concrete, the ultimate bending resistance of the piles would have been 142 KN·m for the symmetric cross-section and 133 KN·m for the asymmetric one. However, as the piles were not made at the same time, the differences in the concrete mixtures led to different values of concrete resistance. In this way, The mean compressive strength fcm of the concrete employed in the symmetrically reinforced member is 34.8 MPa after 28 days, whereas fcm = 31.2 MPa in the pile with asymmetric cross-section. Figure 7 represents the experimental and the analytically computed M � relationships for both, (a) symmetric and (b) asymmetric, cross-sections. The experimental relationships are obtained by computing the curvature that corresponds to the strain plane given by the gauges placed at the same vertical position in the unloaded state but at diametrically opposed steel bars. The gauges are

(b) Short term experiment

Shear

Bendingmoment

P P

3.6 m 1.0 m L (m)

(a) Test scheme

P

P

P

1.0 m

(c) Long term experiment

R = 400 mm c = 57 mm Øs = 10 mm Ø = 16 mm nØ = 16 s = 48.30 mm

s

R Ø

ØØ2sc

Fig. 6: Real scale tests in short and long term loading

R = 400 mm c = 57 mm Øs = 10 mm Ø1 = 16 mm nØ1 = 3 s1 = 193.21 mm Ø2= 16 mm nØ2 = 7 s2 = 41 mm

Ø1

s2

Rs1

Ø2

1ØØ2sc

(d) Symmetric cross-section (e) Asymmetric cross-section


placed at the central segment subjected to a constant bending moment law. The differences seen at the end of the graphics are due to the actual yield strain of the reinforcing steel, which is greater than the one employed in the simulations, at can been observed in Fig 8, which shows the stresses at most tensioned bar in both cross-sections, (a) symmetric and (b) asymmetric. There are slight discrepancies in the values of P in which concrete is assumed to be uncracked.

RegisteredComputed w/ Ac,eff according to Carbonell –Marquez et al. [10]

(a) Symmetric cross-section (b) Asymmetric cross-section

Fig. 7: Short term experimental Vs analytical M- relationship

(106 rad/mm)

M (KN·m)

50

100

150

5 10 15 20

200

0 5 10 15

(106 rad/mm)

50

100

150

0

M (KN·m)


50

100

150

P (KN)

5 10 15 20 25

(mm)

PP

(b) Symmetric cross-section 5 10 15 20 25

50

100

150

P (KN)

(mm)

P P

(c) Asymmetric cross-section

Fig. 9: Short term experimental deflection Vs analytical results

Fig. 8: Steel stress at most tensioned bar: experimental results Vs predictions


-300

-200

-100

-400

-500

-600

50 100 150

s (MPa)

Most tensioned bar

P (KN)

(a) Symmetric cross-section

-300

-200

-100

-400

-500

-600 s (MPa)

50 100 150 P (KN)

Most tensioned bar

(b) Asymmetric cross-section


The consequences of the good agreement between experimental and analytical M �relationships are observed in Figure 9 where the deflection results at mid-span of the member recorded during the experiments are compared against the analytically predicted deflections. In both cases, (a) symmetric and (b) asymmetric cross-sections, the analytical models accurately predict the behaviour of the member in bending. Fig. 10 compares the experimental deflection results in both short term tested members, symmetric and asymmetric cross-sections. It can be observed that deflection at member’s mid-span is slightly lower in the case of the asymmetrically reinforced cross-section even with a concrete of lower resistance than that of the symmetric cross-section. In this particular case, the savings in longitudinal reinforcing steel are 48.93%

As the long term tests are currently being conducted, no experimental data is still available. Figure 11 shows the predictions made for the symmetric and asymmetric cross-sections, employing the effective area of concrete in tension given by Carbonell-Márquez et al. [10] and the concrete stress-strain model affected by long term shrinkage and creep. The predictions show an important aspect: the expected displacement is greater in the case of the asymmetrically reinforced pile. As the reinforcement has an asymmetric layout, the concrete in around the thicker bars is more constrained against free shrinkage than the remaining of the section and, consequently, the shrinkage induced curvature is much greater than in the case of the symmetric cross-section.

ConclusionsTraditionally reinforced concrete piles with circular cross-sections have been compared against the newly introduced concrete pile cross-sections with an asymmetric longitudinal reinforcement layout. The employed material models and the model for the evolution of the effective concrete area in tension have been presented. The experimental campaign that is being conducted has been introduced and the main results of the short term loading tests have been presented. From these results, it can be deducted that an asymmetrically reinforced concrete pile undergoes less deformations when being loaded under the same level of forces than its symmetrical counterpart which has the same ultimate bending strength. Therefore, taking into account immediate response, important savings in longitudinal steel can be achieved as well as a better behaviour of the earth retaining system when using asymmetrically reinforced concrete piles. If measures against shrinkage are taken, as an adequate curing process, the long term deflection of an asymmetric can be low enough to be worthy of being adopted.

Fig. 10: Asymmetric cross-section Vs symmetric cross-section deflection results

5 10 15 20 25 (mm)

30

50

100

150

200 P (KN) Asymmetric cross-section Symmetric cross-section

P P

Fig. 11: Long term deflection predictions

Asymmetric cross-section Symmetric cross-section

50 KN 50 KN

11 (mm)

10

9

8

7

50 100 250 300150 200 t (days)


The present work was financed by the Ministry of Science and Innovation under the research project IPT-2011-1485-420000. This support is gratefully acknowledged.

References [1] HERNÁNDEZ-MONTES E., ASCHHEIM M., and GIL-MARTÍN L.M., “Impact of optimal

longitudinal reinforcement on the curvature ductility capacity of reinforced concrete column sections,” Magazine of Concrete Research, Vol. 56, Vo. 9, 2004, pp. 499–512.

[2] HERNÁNDEZ-MONTES E., GIL-MARTÍN L.M., and ASCHHEIM M., “Design of Concrete Members Subjected to Uniaxial Bending and Compression Using Reinforcement Sizing Diagrams,” ACI Structural Journal, Vol. 102, Vo. 1, 2005, pp. 150–158.

[3] ASCHHEIM M., HERNÁNDEZ-MONTES E., GIL-MARTÍN L.M., “Design of Optimally Reinforced RC Beam, Column and Wall Sections,” Journal of Structural Engineering - ASCE, Vol. 134, No. 2, 2008, pp. 231–239.

[4] HERNÁNDEZ-MONTES E., GIL-MARTÍN L.M., PASADAS-FERNÁNDEZ M., ASCHHEIM M., “Theorem of optimal reinforcement for reinforced concrete cross sections,” Structural and Multidisciplinary Optimization, Vol. 36, No. 5, 2007, pp. 509–521.

[5] LÓPEZ-MARTÍN D., CARBONELL-MÁRQUEZ J.F., GIL-MARTÍN L.M., and HERNÁNDEZ-MONTES, E., “Eccentricity-Based Optimization Procedure for Strength Design of RC Sections under Compression and In-Plane Bending Moment,” Journal of Structural Engineering - ASCE, Vol. 140, No. 1, 2014.

[6] WEBER K., and ERNST M., “Entwicklung von Interaktionsdiagrammen für asymmetrisch bewehrte Stahlbeton-Kreisquerschnitte.,” Beton- und Stahlbetonbau, Vol. 84, No. 7, 1989, pp. 176–180.

[7] GIL-MARTÍN L.M., HERNÁNDEZ-MONTES E., and ASCHHEIM M., “Optimization of piers for retaining walls,” Structural and Multidisciplinary Optimization, Vol. 41, No. 6, 2010, pp. 979–987.

[8] Comité Europeo de Normalizacion, Eurocódigo 2: Proyecto de estructuras de hormigon. Parte1-1: Reglas generales y reglas para edificación. UNE-EN 1992-1-1. Bruselas: Comité Europeo de Normalización, 2010, p. 244.

[9] fib-Special Activity Group 5, fib Bulletin 65: Model Code 2010 - Final Draft. Lausanne: International Federation for Structural Concrete (fib), 2012.

[10] CARBONELL-MÁRQUEZ J.F., GIL-MARTÍN L.M., FERNÁNDEZ-RUÍZ M.A., and HERNÁNDEZ-MONTES E., “Effective area in tension stiffening of reinforced concrete piles subjected to flexure according to Eurocode 2,” Engineering Structures (In press).

[11] HERNÁNDEZ-MONTES E., CESETTI A., and GIL-MARTÍN L.M., “Discussion of ‘An efficient tension-stiffening model for nonlinear analysis of reinforced concrete members’, by Renata S.B. Stramandinoli, Henriette L. La Rovere,” Engineering Structures, Vol. 48, No, 2013pp. 763–764.

[12] GIL-MARTÍN L.M., HERNÁNDEZ-MONTES E., ASCHHEIM M., and PANTAZOPOULOU S.J., “Aproximate expressions for the simulated response of slender longitudinal reinforcement in monotonic compression,” Magazine of Concrete Research,Vol. 60, No. 6, 2008, pp. 391–397.

[13] CEB, CEB Design Manual on Cracking and Deformations, Bulletin d’information 158. Paris: Comite Euro-International du Beton, 1985, p. 250.

[14] GILBERT R.I., WU H.Q., “Time-dependent stiffness of cracked reinforced concrete elements under sustained actions,” Australian Journal of Structural Engineering, Vol. 9, No. 2, 2009, pp. 151–158.

asymmetrically reinforced concrete piles in earth retaining systems

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