Engineering Structures 33 (2011) 1918–1929

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Engineering Structures

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A methodological approach for finite element modeling of pretensioned concretemembers at the release of pretensioningAmir A. Arab a,∗, Sameh S. Badie b, Majid T. Manzari ba TY Lin International, Alexandria, VA, 22312, USAb Department of Civil and Environmental Engineering, The George Washington University, Washington, DC, 20052, USA

a r t i c l e i n f o

Article history:Received 15 April 2010Received in revised form23 February 2011Accepted 23 February 2011Available online 26 March 2011

Keywords:Bond-slippageConcreteElastoplasticEmbedmentExtrusionFinite elementStrandFrictionPretensionedTransfer length

a b s t r a c t

This paper presents a methodological approach for finite element simulation of pretensioned concretemembers. The three-dimensional analysis presented in this paper involves a rectangular [150 mm (6in.) tall × 150 mm (6 in.) wide × 2440 mm (96 in.) long] concrete member hosting one 15-mm (0.6in.) diameter 7-wire low-relaxation Grade 1860 MPa (270 ksi) prestressing strand. The finite elementmodels are divided into two general classifications: (i) concentrically pretensioned, and (ii) eccentricallypretensioned. The finite element models are analyzed based on elastoplastic material behaviors as wellas mesh sensitivity. Two approaches are examined for finite element modeling of the pretensionedconcrete specimens: (i) the extrusion technique utilizing friction-based contact simulations, and (ii) theembedment technique simulating equivalent responses while being a computationally less expensivesolution. A comparative analysis is presented to measure the validity as well as accuracy of the findingsby the finite element techniques against the commonly used closed-form solutions based on elastic beamtheory. The validity of the finite element approach is further verified by comparative analysis of theanalytical data against the experimental findings. The paper concludes that the embedment techniqueprovides an accurate and numerically efficient alternative in comparison with the extrusion method forthe simulation of the pretensioned concrete members. While the extrusion technique provides moredetailed information corresponding to the regions located immediately around the prestressing strands,including the interface overstresses and bond slippage, the embedment technique appears to have theability to simulate the overall response of the concrete members with comparable accuracy.

© 2011 Elsevier Ltd. All rights reserved.

1. Objective

The current state of analysis and design of pretensioned con-crete members indicates a serious lack of unified and practicalguidelines for analytical simulation of precast prestressed con-crete members. There is a need for development of a validatedmethodological approach to analytically simulate the behavior ofpretensioned concretemembers during various stages of construc-tion and service life. This paper includes modeling methodologieswhich primarily focus on the response of the members immedi-ately after the release of the pretensioning.

The main objective of this research is to propose modeling ap-proaches that can adequately simulate the pretensioning mecha-nism and predict the inelastic response of a pretensioned memberimmediately after the release of the pretensioning.

Once a reliable analytical method is developed and confirmedagainst the available closed form solutions, a series of comparative

∗ Corresponding author. Tel.: +1 703 245 5212.E-mail address: aarab71@gwu.edu (A.A. Arab).

0141-0296/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2011.02.028

analyses are conducted to verify the simulation of concentricallypretensioned concrete prisms against experimental data. Thecomparative analyses not only show conformance of the analyticalresults with the test data, they also indicate the sensitivity of thesimulated models to passive confinement in the form of closestirrups along the length of themember. It is important to note thatavailability of analytical procedures to simulate the pretensioningmechanism can have direct impact on future research in the areaof precast/prestressed concrete considering the costly laboratoryexperiments using full or scaled specimens.

2. Literature review

In the past 30 years, many attempts have been made to pro-pose finite element models which can capture the response ofpretensioned concrete members. Dating back to 1978, Mirza andTawfik [1] presented a one-dimensional mathematical modelingwhich included a stiffness analysis for the detensioning (release)process. This model required the system stiffness matrix and loadvectors to be updated upon the releasing of each individual strands,resulting in a time consuming process.

A.A. Arab et al. / Engineering Structures 33 (2011) 1918–1929 1919

Table 1Cohesive stresses and target friction values recommended by AASHTO LRFD for shear friction design.

Categorization of shear interface CohesionMPa (psi)

Friction coefficient

Normal weight concrete cast monolithically 2.76 (400) 1.4Light weight concrete cast monolithically 1.65 (240) 1.0Normal weight concrete cast against clean (free of laitance) concrete surface intentionally roughened to a minimum amplitudeof 1/4 in.

1.93 (280) 1.0

Light weight concrete cast against clean (free of laitance) concrete surface intentionally roughened to a minimum amplitude of1/4 in.

1.65 (240) 1.0

Normal\light weight concrete cast against clean (free of laitance) concrete surface not intentionally roughened 0.52 (75) 0.6Normal\light weight concrete cast against clean steel surface 0.17 (25) 0.7Normal\light weight concrete anchored to structural steel by means of headed studs or reinforcing bars

Kannel et al. [2] utilized a three-dimensional finite elementmodel to investigate the effect of the releasemethodologies on theend cracking in the pretensioned concrete. This model includedcontinuum elements to model the concrete girder and trusselements to model the strands. The transfer length was indirectlymodeled using twomethodologies: (1) cross-sectional areas of thestrandswere linearly varied fromzero (at the end face of the girder)to themaximum (at the end of the theoretical transfer length), and(2) truss elements were constrained to the continuum elementsusing rigid-plastic springs. The validation of the analytical resultswas done through field observations.

A nonlinear finite element model using a commercial code(ANSYS) was used by Kaewunruen and Remennikov [3] to analyzerailway prestressed concrete sleepers. This model included brickelements to represent the concrete matrix with embedded three-dimensional truss elements simulating the prestressing.

Stephen [4] used a comprehensive three-dimensional finiteelement model utilizing a commercial code (ABAQUS) to simulatethe long-term behavior of precast/prestressed concrete bridges.This model included elastoplastic material modeling capable ofcapturing the nonlinear behavior of various concrete members(e.g., deck slab, pretensioned concrete girders) due to long termeffects such as creep and shrinkage. Themodel required an externalsubroutine to facilitate specific operations such as prestressing andapplication of the long term effects.

Rabczuk and Eibl [5] proposed a coupled element free Galerkinmethod to analyze prestrsessed concrete beams under quasi-staticloading. The constitutive law governing the concrete medium wasbased on a coupled damage-plasticity model. The reinforcementwas modeled as discrete beam elements so that the interactionbetween concrete and reinforcement can be modeled. The bondmodel included two modes of failure: pullout failure, and splittingfailure. The formulation of the bondmodel was based on the radialstress–radial strain relation with three distinct domains: the non-linear material behavior including the initiation and propagationof cracks, linear softening, and residual strength.

More recently, Ayoub and Filippou proposed a nonlinear modelfor simulation of the pretensioned prestressed concrete girders [6].The modeling approach consisted of three main components: con-crete girder simulated as a beam–column, prestressing strandsmodeled as truss elements, and a bond element to model the pre-stress transfer between the concrete and strands. The constitu-tive laws governing the nonlinear response of concrete and strandswere based on discretization of the media into fibers with uniax-ial hysteric models. The bond model at the interface between theconcrete and strand was formulated using special bond stress–sliprelations. The pretensioning mechanism was divided into discretetime steps representing various stages of the operation.

This paper presents alternative methodologies that use three-dimensional finite element modeling of pretensioned members,which are suitable for predicting elastoplastic response of concreteimmediately after the release of pretensioning. Since concrete ismodeled as a 3D continuum hosting the pretensioned strands, theglobal and local effects of pretensioning can be assessed in fulldetails.

3. Overview of pretensioning mechanism

Pretensioning of concretemembers refers to the practicewhereprestressing strands are tensioned prior to casting concrete. Onceconcrete attains adequate strength (commonly referred to as ini-tial compressive strength), strands are releasedwhile bondedwiththe host concrete. The process of load transfer between the pre-tensioned strands and concrete host is based on the followingmechanisms: (1) Adhesion at the interface between strands andconcrete, (2) Hoyer (wedging) effect, and (3)Mechanical interlock-ing between strands and concrete. The analytical techniques pro-posed in this paper simulate the combined effects of these factors.

The adhesion between the prestressed strands and concretematrix is the result of the chemical and physical bonding devel-oped at the interface of the two media. The adhesive bond isprimarily controlled by the shear strength of the weaker material(cementitious matrix) and greatly affected by the surface condi-tion of the strands. Table 1 includes the current cohesion valuesrecommended by AASHTO LRFD [7] applicable to shear–friction in-terfaces.

Immediately after the release of the prestressing strands, theadhesion between strands and concretemedium is expected to failin the vicinity of themember end faces. Once the adhesive bondingbetween strands and concrete is overcome, strands experienceslippage inside the concrete host along the length of adhesionfailure. The contribution of adhesion to the overall bond strengthis generally negligible in magnitude and will be terminatedat the onset of slippage. Therefore, adhesion may be ignoredwhen analyzing the ultimate bond strength between strands andconcrete after initiation of slippage. Fig. 1(a) and (c) schematicallyshow the Hoyer’s effect as well as the mechanical interlockingalong the corresponding transfer length. Transfer length, which isthe distance required for the strands to transfer their full effectivestress fpe to the concrete, depends on multiple parameters such ascompressive strength of concrete, surface condition of the strandsprior to pretensioning, number and spacing of strands, level ofconfinement, andnumber ofwires per strand. As shown in Fig. 1(b),the current state of practice in the US assumes a linear progressionof pretensioning along the transfer length, equivalent to sixty timesthe strand diameter [7].

4. Modeling approaches

Numerical modeling of pretensioned concrete members gener-ally consists of the following analytical foundations: (1) materialconstitutive modeling, and (2) finite element techniques to simu-late the anticipated pretensioning mechanism.

Material constitutive models control the elastoplastic responseof the simulated specimen immediately after the release of thepretensioned strands. The constitutive model for concrete mustbe able to correctly estimate the response of the member tocompressive and tensile stresses. This is an essential element ofmodeling which enables the analysis to predict the extent of the

1920 A.A. Arab et al. / Engineering Structures 33 (2011) 1918–1929

Fig. 1. Pretensioning mechanism: (a) Hoyer’s effect [8], and (b) AASHTO LRFD linearized transfer length model [7], and (c) Mechanical interlocking [9].

damaged regions (crushed or fractured). In addition to initiation ofcracks, the constitutivemodel should be able to realistically relievethe overstressed regions by redistributing the stresses within thecontinua.

4.1. Material simulation: concrete damage plasticity constitutivemodel

The elastoplastic response of pretensioned members is pre-dicted in accordance with the Concrete Damage Plasticity consti-tutive model. This constitutive model was first developed by Leeand Fenves in 1998 [10], which offers reliable predictions of plas-tic damage of concrete under monotonic and cyclic loading.

The initiation of cracks in concrete is based on a continu-ous formation during which the micro-cracks are connected. Thisphenomenon leads to the softening of concrete, during whichredistribution of stress occurs from the localized damaged areasto the neighboring elements. In addition to fracture and micro-cracking, concrete will experience significant stiffness degrada-tion. The complexity of this phenomenon arises from the fact thatconcrete will be able to recover some of the degraded stiffness as itis reloaded from tensile to compressive range while the previouslyinitiated crackswill close up under compression [11]. The Concrete

Damage Plasticity model is based on two main concepts: fracture-energy-based damage, and stiffness degradation. The correspond-ing yield function introduces two damage variables to distinguishdamages under tensile and compressive loads, as follows [9]:

F̄ =1

1 − α

αI1 +

3J2 + β

σ̂max

− cc (κ) (1)

whereα and β are dimensionless constants defined below, I1 = σii(First Invariant of stress matrix), J2 =

12 sijsij (Second invariant of

deviatoric stress tensor), and σ̂max is themaximumprincipal stress:

α =fb0 − fc02fb0 − fc0

, [0 ≤ α ≤ 0.5] (2)

β(κ) =cc(κ)ct(κ)

(1 − α)+ (1 + α). (3)

In Eq. (2), fc0 and fb0 refer to the initial uniaxial and biaxialcompressive stresses of concrete, respectively. Additionally, thevariable ct (κ) represents the tensile cohesion and cc (κ) representsthe compressive cohesion, indicating the frictional characteristicsof concrete as a cohesive medium. The evolution of the yieldfunction for Concrete Damage Plasticity model as defined byEq. (1) is partially based on the relationship between cohesion and

A.A. Arab et al. / Engineering Structures 33 (2011) 1918–1929 1921

Fig. 2. Barcelona yield function in plane stress state.

damage variables at the states of tension and compression. This isparticularly important since the evolution of compressive strengthhardly affects tensile strength and vice versa. As shown in Fig. 2,the yield function consists of two Drucker–Prager type functionscorresponding to compression and tension where c0 = −fc0, andfby and fty correspond to the states of biaxial compression andtension, respectively. Additional details of the model are providedin [10].

4.2. Finite element techniques

The analytical models presented in this research are con-structed using a commercial finite element package, ABAQUS [12].The features and functions utilized in this research are also offeredin other commercially available finite element packages.

4.2.1. Extrusion techniqueThe extrusion technique is based on modeling the interface be-

tween strands and concrete using contact formulations. This isfacilitated by modeling the strands within an extruded concretematrix as solid elements (e.g., 8-node elements)with common sur-faces. An extrudedmodel will require various contact properties inorder to optimize the simulation of the composite interaction be-tween concrete and strands, including friction, slippage, pressuredependency and over-closure.

The contact surfaces of extruded models will require varioussub-modules in order to accurately simulate the interactionbetween strands and concrete matrix during the pretensioningmechanism. A friction-base model is utilized to simulate thecollective effects of the pretensioning mechanisms as previouslydiscussed. Based on thepreliminary examinations by the authors, itwas decided that the built-in features in ABAQUS could adequatelyfacilitate such simulations while requiring proper calibration.

Based on the fundamental definition of friction-dominated in-terfaces, it is anticipated that shear and normal stresses will bedeveloped at the interface between strands and concrete host.Consequently, the proposed friction contact model shall be com-prised of the following components: (1) Tangential Behavior and(2) Normal Behavior.

The mechanical properties of the interaction between strandsand concrete matrix is modeled using the friction-based contactsurface algorithms.

The surface interaction is further coupled with the contactalgorithms to enforce additional controls for the simulation oftangential and normal interactions at the interfaces. Fig. 3(a) showsa typical scheme for finite element modeling of pretensionedconcrete members using extrusion technique.

4.2.1.1. Tangential behavior. The tangential behavior of the pro-posed contact model is governed by coefficient of friction, µf .

The coefficient of friction,µf , is treated as a variable parameterfor the parametric studies of this research, while the target valuesare based on the AASHTO LRFD recommendations for shear frictiondesign as summarized in Table 1 [7].

Slip tolerance is further defined as ‘‘the ratio of allowable max-imum elastic slip to characteristic contact surface face dimen-sion’’ [12]:γi = Ff li (4)where li corresponds to the characteristic contact surface length.The elastic slip, γi, will be controlled by the slip tolerance, Ff , whichis set at the default value of 0.005 for an optimized balance be-tween computational accuracy and efficiency.

Moreover, a parametric investigation is performed in order toidentify efficient while accurate methods to simulate the slippageof the strands within the concrete continuum based on thefollowing techniques (see Fig. 4):

Technique 1: Tie prestressing strand to the concrete host(extruded to host the strand), and model the interfaces of theconcrete host to allow for slippage.

Technique 2: Model the interface of the strand and concretehost to allow for direct slippage of strands within the concretematrix.

4.2.1.2. Normal behavior. The normal behavior at the interface be-tween strands and concrete matrix is simulated using a ‘‘hard’’pressure–overclosure relationship, which minimizes the penetra-tion of slave nodes into the master surface at the onset of con-tact. This formulation also prevents the transfer of tensile stresseswithin the contact interface. The applicable contact constraints aremathematically enforced using a Lagrange multiplier technique,which is concerned with the magnitude of contact pressure (p) asa function of overclosure (h). The contribution of virtual work (Π )can be described in the following linearized form [12]:dδΠ = (δp)dh + dp(δh) (5)where p = 0 when h < 0 (No contact), and h = 0 when p > 0(Contact initiated). This contact formulation includes no physical

1922 A.A. Arab et al. / Engineering Structures 33 (2011) 1918–1929

b

c

Fig. 3. Typical schemes of finite element modeling of pretensioned concrete members: (a) extrusion technique, (b) embedment technique, and (c) analytical model withstrand eccentricity parameter (ep) relative to the center of gravity of the member.

Fig. 4. Schematic presentation of the techniques used for simulation of slippagesurface upon releasing the strands: (a) concrete-over-concrete slippage, and (b)steel-over-concrete slippage.

softening, preventing separation once contact is initiated betweenthe adjacent surfaces.

4.2.2. Embedment techniqueEmbedment is a powerful finite element technique, which en-

ables one or more elements to be embedded inside a host ele-ment. One of the most significant advantages of the embedment

technique is the fact it does not require modeling of contact sur-faces, and therefore, eliminates the numerically expensive itera-tions associated with surface formulations.

An embedded element needs to be geometrically confinedwithin the host element. The degrees of freedomat the nodes of theembedded element will be eliminated and the nodes will become‘‘embedded nodes’’ while constrained to the interpolated valuesof the corresponding degrees of freedom of the host element. Theconstraint rules governing the relationship between master andslave nodes can be theoretically described as follows. Given Xm

and X s as position of the master and slave nodes, respectively, thereference configuration position, N , is defined as [12]

N = Xm− X s. (6)

Consequently, the position of the fully constrained slave nodeat any instance is found as

x̂s = xm + c(φm)N (7)

where C(φm) corresponds to the rotation matrix of the masternode. Fig. 3(b) shows a typical scheme for the finite elementmodel-ing of pretensioned concretemembers using the embedment tech-nique.

4.3. Simulation of pretensioning

Pretensioning of strands and the releasing mechanism canbe simulated using various modeling techniques. One effectivetechnique is based on the strain-compatibility between strand

A.A. Arab et al. / Engineering Structures 33 (2011) 1918–1929 1923

Table 2Summary of constitutive parameters used in finite element analyses.

Parameter Value used in analysis

Concrete

Compressive strength at time of release (f ′

ci) 40.0 MPa (5.800 ksi)Tensile [rupture] strength at time of release (fti) 4.0 MPa (0.578 (ksi))Density (wc ) 2400 kg/m3 (150 lb/ft3)Young’s elastic modulus (Eci) 3.2E + 04 MPa (4617 ksi)Poisson’s ratio (νc ) 0.2Dilation angle(ψ) 52.9°Flow potential eccentricity (εc ) 0.1Ratio of the initial equibiaxial compressive yield stress to initial uniaxial compressive yield stress (σbo/σo0) 1.16Ratio of the second stress invariant to that of the compressive meridian at the initial yield (Kc ) 0.778Viscosity parameter (µc ) 0.0

Prestressing strands

Ultimate tensile strength (fpu) 1860 MPa (270 ksi)Density (wp) 7865 kg/m3 (490 lb/ft3)Young’s elastic modulus (Ep) 197 GPa (28,500 ksi)Poisson’s ratio (νp) 0.2

Other structural steel (casting bed)

Density (wp) 7865 kg/m3 (490 lb/ft3)Young’s elastic modulus (Ep) 200 GPa (29,000 ksi)Poisson’s ratio (νp) 0.2

and concrete (the default pretensioning methodology used inthis work). The strain-compatibility technique includes varioussteps: (1) Pretensioning the strands as initial conditions before anyrelationship is established between the strands and concrete host(during this step the strain-compatibility is deactivated in orderto decouple the strands and concrete host while pretensioningis applied; this ensures that concrete is not strained while thestrands are pretensioned), and (2) Applying strain-compatibilitywhich simulates the release of the strands while the constraintsto the concrete host are established.

4.4. Simulation of casting bed

In order to capture the effect of the self-weight of the specimenssimultaneously applied with prestressing, the simulated modelsare supported on a casting bed (steel) which provides rigid supportwithout restraining the longitudinal and transverse movements ofthe pretensionedmember. The interface between the pretensionedmember and the casting bed is modeled as a friction-based surface(coefficient of friction is set at zero in this work), using the ‘‘hard’’pressure–overclosure relationship as previously discussed withthe exception that complete separation is allowed at the interfaceupon the cambering of the pretensioned member.

Alternatively, the effect of the casting bed may be simulated bymodifying the boundary conditions as follows: (1) Provide rollersat the release ends, (2) Provide symmetry conditions at the mid-span (if a half-symmetric model is used), and (3) Apply the self-weight of the member simultaneously with activation of straincompatibility. The advantage of this methodology is the elimina-tion of contact surface formulations which generally require com-plex numerical iterations upon cambering of the member over thecasting bed.

5. Modeling of a rectangular concrete beam with one preten-sioned strand

5.1. Introduction

The following section includes the finite element simulation ofa pretensioned concrete member with rectangular cross-sectionand one prestressing strand. The geometric characteristics of themodels are: Concrete Beam [15 cm (6 in.) × 15 cm (6 in.) ×

240 cm (96 in.)] and Prestressing strands [1.5 cm (0.6 in.)diameter seven-wire low relaxation strands]. The main objectivesof this simulation are: (1) material calibration (concrete, steel andprestressing strands), (2) model calibration based on extrusiontechnique, (3) model calibration based on embedment technique,and (4) comparison of responses between the embedment andextruded models as well as the closed form solutions.

Two independent classes ofmodels will be discussed to addressthe effect of eccentricity of the strand on the behavior of theconcrete beam at the time of release. The simulation presented inthis example investigates the response of pretensioned concretebeams immediately after the release of pretensioning withoutconsideration of any time-dependent effects such as creep andshrinkage.

5.2. Calibration of constitutive models

5.2.1. Calibration of concrete constitutive modelTable 2 includes the summary of the elastoplastic parameters

used in the finite element analyses discussed in this section. Inaddition to the elastoplastic parameters, the uniaxial stress–strainbehavior under tension and compression need to be included inthe consitutive model. Fig. 5 show the compressive and tensilestress–strain models used for the analytical purposes of thisresearch based on the initial compressive strength of 40.0 MPa(5.8 ksi) at the time of release. The calibration of the concreteconstitutive model was performed by analyzing the response ofa 5 cm (2 in.) by 20 cm (8 in.) finite element prism to uniaxialcompressive and tensile straining. Fig. 5(a) and (b) show theconvergence of the response of the calibrated models towardsthe theoretical uniaxial compressive and tensile stress–straincurves, respectively, after several iterations. Table 2 includes theconstitutive parameters yielding the final calibrations.

5.2.2. Constitutive parameters for prestressing strands and otherstructural steel

Table 3 gives the summary of the constitutive parametersbased on the default values used in the current practice of bridgeengineering in the US [7]. The prestressing strands will be stressedto a maximum threshold of fp0 ≤ 0.75 fpu, commonly referredto as jacking stress, where fpu corresponds to the ultimate tensile

1924 A.A. Arab et al. / Engineering Structures 33 (2011) 1918–1929

Fig. 5. Comparison of the stress–strain relationships obtained by the constitutive model with the theoretical stress–strain curves assumed for a concrete with initialcompressive strength of f ′

c = 40 MPa (5.8 ksi): (a) subject due to uniaxial compression, and (b) subject to uniaxial tension.

Table 3Summary of stresses obtained by the concentric and eccentric models.

Model Compressive stress MPa (psi) Tensile stress MPa (psi)

Concentrically pretensioned specimens

Extruded models

µ = 0.70 7.19 (1043) 0 (0)µ = 1.40 7.27 (1054) 0 (0)

Embedded models

Linear transfer length simulation 8.09 (1173) 0 (0)Nonlinear transfer length simulation based on µ = 0.70 8.11 (1177) 0 (0)Nonlinear transfer length simulation based on µ = 1.40 8.16 (1183) 0 (0)

Closed form solution

Gross section 8.03 (1165) 0 (0)Net section 7.96 (1154) 0 (0)Transferred section 8.16 (1184) 0 (0)

Eccentrically pretensioned specimens

Extruded models

µ = 0.70 18.88 (2739) 3.33 (485)µ = 1.40 18.97 (2751) 3.39 (492)

Embedded models

Linear transfer length simulation 19.33 (2803) 3.53 (513)

Closed form solution

Gross section 19.79 (2871) 3.73 (541)Net section 19.68 (2854) 3.70 (536)Transferred section 19.63 (2848) 3.67 (533)

capacity of the prestressing. Since the yield strength of 1860-MPa (270-ksi) pretensioned strands is approximately 90% fpu, theprestressing strands are anticipated to remain elastic throughoutthe analysis.

5.3. Finite element models

The extruded models are divided into two categories: (1)concentrically pretensioned, and (2) eccentrically pretensioned.

A.A. Arab et al. / Engineering Structures 33 (2011) 1918–1929 1925

Fig. 6. Typical finite element models for simulation of rectangular beams with one prestressing strand supported on casting bed: (a) extruded model, and (b) embeddedmodel.

The rectangular beams are modeled using linear 8-node cubeelements. The linear brick elements possess only translationaldegrees of freedom. The simulation of the prestressing strandsdepends on the modeling technique as previously discussed: (i)extrusion, where prestressing strands are simulated using three-dimensional linear brick elements, and (ii) embedment, whereprestressing strands are simulated using one-dimensional trusselements. Fig. 6 shows the typical finite element models of theconcentrically pretensioned rectangular beams.

The concentrically pretensionedmodels include the interactionbetween the concrete medium and the casting bed. In theeccentrically pretensioned models, the effect of casting beds issimulated using the boundary conditions as previously discussedearlier in this paper.

Based on the earlier discussion, the slippage of the strands ismodeled using the following techniques: (1) Slippage of concrete(with tied strand) over concrete host, and (2) Direct slippage ofstrands over concrete host.

The concentric models are pretensioned using one 15-mm(0.6-in.) diameter low-relaxation seven-wire strand positionedat the center of gravity of the concrete beams. Therefore, theconcrete member will be subjected to axial compression stressupon releasing of the strand.

At the time of release, losses due to relaxation of strands andelastic shortening are only considered. The anticipated relaxationof strands up to the time of release is already considered in thethreshold of jacking stresses (input parameter). Elastic shorteningis automatically taken into account through strain compatibilitybetween the pretensioned strand and concrete host. Alternatively,the theoretical loss due to elastic shortening may be estimatedin accordance with the provision of AASHTO LRFD. Given the

characteristics of the concrete beams combined with the level ofpretensioning, the elastic shortening losses are anticipated to be51.7 MPa (7.5 ksi).

The eccentric models are also pretensioned using one 15-mm(0.6 in.) diameter low-relaxation seven-wire strand positioned40 mm (1.556 in.) below the center of gravity of the concretebeams. Therefore, the concrete member is subjected to axialcompression combined with bending upon releasing of the strand.

5.3.1. Discussion of results: rectangular beams with one concentricstrand

Based on the available closed form solutions (elastic beamanal-ysis), the maximum anticipated compressive stress in the beamslightly varies depending onwhether gross, net or transferred crosssection is considered. The estimated compressive stress are sum-marized in Table 3.

Fig. 7 shows the simulated response of the concentricallypretensioned concrete specimen in accordance with: (i) concrete-over-concrete slippage, and (ii) strand-over-concrete slippage, forboundary friction coefficients of 0.70 and 1.40.

The main objectives of these parametric studies are to identify:(i) the optimized simulation of the slippage based on concrete-over-concrete versus strand-over-concrete, and (ii) the optimizedcoefficients of friction providing lower and upper bounds forpretensioning stress path over the simulated slippage surface(transfer length). Figs. 8 and 9(a) show the axial stresses andstrains developed in the strands immediately after the release ofpretensioning for two extrudedmodelswith coefficients of frictionof 0.70 and 1.40, respectively. It can be observed that both modelsconverge towards the theoretical transfer length along a nonlinear

1926 A.A. Arab et al. / Engineering Structures 33 (2011) 1918–1929

Fig. 7. Response of extruded models to concentric pretensioning based on variouscoefficients of friction at the slippage surface: (a) Coefficient of friction ofµ = 0.70,and (b) Coefficient of friction of µ = 1.40.

Fig. 8. Axial pretensioning stress in the concentric strands immediately after therelease (including elastic shortening losses).

stress variation path as opposed to the linear stress path assumedin current practice.

As previously discussed, the embedded models are not capableof predicting the bond stresses at the interface between the con-crete and strand(s) since they simulate the pretensioning mecha-nism based on nodal constraints rather than contact formulation.However, the extruded models are capable of estimating the bondstresses based on contact formulation. Fig. 9 shows typical resultscorresponding to the friction-based contact formulations, includ-ing the pressure and shear stresses developed at the interface be-tween the concrete and strand and the corresponding strains.

Since the concentrically pretensioned specimens are globallyin compression, optimized accuracy can be achieved withinnumerical efficiency. The optimized characteristic length of the

solid elements simulating the concrete medium is reached at75 mm (3 in.), evaluated against the closed form solutions. Theoptimization of the concrete Host2 (Fig. 4) is studied in order toverify the effect of the mesh sensitivity on the contact formulationused in the extruded models. Fig. 10 shows the convergence of thelongitudinal stresses based on the size of the concrete elementsrepresenting the zone of interface with the pretensioned strand(applicable to the extruded models only).

As indicated in Table 3, the compressive stresses resulted bythe embedded models are also consistent with the closed formsolutions.

5.3.2. Discussion of results: rectangular beams with one eccentricstrand

Table 3 includes the summary of top and bottom fiber stressesat the mid-span of the eccentrically pretensioned specimens. It isobserved that the results of the simulation with embeddedmodelsagree with the closed form solutions within a 5% range. In thecase of the extruded models, the results are generally lower thanthose obtained by the closed form solutions and/or embedded.Due to the flexural demand at top fibers of the concrete beam,the acceptable convergence of the model requires denser meshingwith the optimized characteristic length of 25 mm (1 in.).

Fig. 11 shows the longitudinal stress profiles, measured at thetop fiber, along the span of the pretensioned extrudedmodels. Theresults correspond to two different cases based on the slippagecoefficients (µf ) of 0.70 and 1.40 for friction at the interfacebetween the strand and concrete. Additionally, identical resultscorresponding to the embeddedmodelswith linear approximationof transfer length as prescribed by AASHTO LRFD are includedin Fig. 11. It can be observed that the longitudinal stresses bythe extruded models generally hover over the idealized linearapproximation of the stress variation within the end regions ofthe beam towards the location of the theoretical transfer length.Beyond the transfer length, the results from the extruded modelsconverge towards a similar plateau obtained from the embeddedmodels.

Three-dimensional finite element models of the pretensionedconcrete members are able to capture stress redistributions oncethe specimens enter the inelastic regimes. As previously discussed,the extrusion technique is capable of direct simulation of thestress transfer mechanism at the interface between concreteand pretensioned strands by utilizing friction-based contactformulations. This facilitates a more detailed prediction of theelastoplastic response of concrete aswell as potentials for cracking,crushing, and stress redistributions in the zones near the releasedstrands. Fig. 12 shows the distribution of the vertical stressesimmediately after the release of the axial pretensioning. The areaswith high stress concentrations are identifiable near the extrusion.

Since the primary objectives of the extruded finite elementmodels are the evaluation of the flexural stresses and identifyingthe areas with high stress concentrations, the results are observedto be highly mesh sensitive. Consequently, the extruded modelsgenerally become very numerically expensive, affected by theconvergence of the friction-based contact formulations. On thecontrary, the embeddedmodels are generally numerically efficientsince the convergence of the densely meshed models is not nearlyas demanding as the complex interface solutions required in theextrusion technique. In addition, a linear elastic analysis is notcapable of estimating the potential for plastic response of thezones with high stress concentrations. One example of such areasincludes the interface between concrete and pretensioned strandsas previously discussed in this paper. Additionally, past experienceindicates that end zones of pretensioned members are subject tocracking due to effects such as bursting and splitting immediately

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Fig. 9. Sample results for concentrically pretensioned specimens with friction-based contact formulation: (a) axial strain the strand after the release, (b) contact pressureat the interface between concrete host and strand, and (c) surface shear stresses at the interface between concrete and strand.

Fig. 10. Concrete mesh sensitivity at interface with the pretensioned strand(extruded model with one concentric strand).

after the release of pretensioning. Therefore, inclusion of anelastoplastic constitutive model is deemed essential for accuratesimulation of the response of concrete members to pretensioningdepending on the objectives of the analysis.

6. Comparison of analytical results with experimental data

In order to further verify the modeling approaches presentedin this paper, a number of numerical simulations are presentedfor a family of pretensioned concrete members that have beentested in the laboratory and for which sufficient experimental

Fig. 11. Response of extruded and embedded models to eccentric pretensioning.

data are available. The selected experimental data are based ona series of parametric tests by Akhnoukh [13] to investigatethe magnitude of transfer length for 18 mm (0.7 in.) diameterstrands as well as the effect of passive confinement on the transferlength. Fig. 13(a) shows a typical cross-section of 178 mm (7 in.)prisms with #10 (#3) closed stirrups (confinement rebars) spacedat 75 mm (3 in.), 150 mm (6 in.) and 225 mm (9 in.). Thespecimens are all 2440 mm (8 ft) long. The surface strains weremeasured using DEMEC points mounted on opposite side faces ofthe specimens. For the comparative purposes of this study, theDEMC readings are normalizedwith respect to their correspondingmaximumvalues. This allows the comparisons to focus on the paceof stress variation over the transfer length. Fig. 13(b)–(d) showthe comparison of the analytical results versus the experimentalmeasurements for the specimens with the #confinement stirrupsspaced at 75 mm (3 in.), 150 mm (6 in.) and 225 mm (9 in.),respectively. It can be observed that the extruded models with

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Fig. 12. Vertical stress contours (concrete matrix) for embedded model with one eccentric strand along the span and at the following distances from the end face of therectangular beam: (a) 0 mm (0 in.), (b) 75 mm (3 in.), (c) 152 mm (6 in.), (d) 305 mm (12 in.), (e) 915 mm (36 in.) (Theoretical Transfer Length) & (f) 1219 mm (48 in.)(Mid-span).

the friction coefficients of 0.70 and 1.40 provide lower and upperboundaries to envelop the experimental results, respectively.Additionally, the embedded models were calibrated such thatthe initial pretensioning conditions (model input) result in stress(strain) variations that provide best-fit to the experimental data.

Also consistent with the experimental findings, the analyticalresults indicate that the finite element models are sensitive tothe effect of the confinement on the transfer length althoughsuch effects are not significant. The friction contact formulationsincorporated in the extruded models result in stress (strain)plateaus comparable with the experimental observations.

7. Concluding remarks

The analytical results presented in this paper indicate that boththe embedment and extrusion techniques are viable methodolo-gies for the finite element modeling of the pretensioned concretemembers. Each technique has certain advantages and disadvan-tages depending on the objectives of analysis. Extruded modelsgenerally provide more details at the interface of the concrete andreleased strands (i.e., slippage, transfer length, etc.) while they areusually complex in nature and computationally expensive. On theother hand, the embedded models are generally less complicatedand computationally much less expensive. While embedded mod-els result in equivalent results compared with the corresponding

extruded simulations, the interaction of the released strandswith the concrete matrix can only be simulated and controlledduring the initial conditions through stepwise stressing of thestrands.

Parametric studies also show that: (i) direct slippage of strand-over-concrete using the friction simulation provides an acceptableapproximation of the pretensioning mechanism, including thetransfer length phenomenon, (ii) friction coefficients of 0.70and 1.4, in combination with Poisson’s effect and other contactparameters, appear to adequately provide lower and upper boundinterface interactions between the strands and concrete hostimmediately after the release of pretensioning, and (iii) thevariation of axial stresses in the strands along the transfer lengthzone is found to be of a nonlinear (parabolic) nature rather than thelinear approximation recommended by the codes and guidelinesused in current practice.

In addition to the modeling techniques used for the simulationof pretensioning mechanism, material nonlinearity with propercalibration of the required parameters can significantly affect theresponse of the prestressed beam. Depending on the objectivesof the analysis, it is important to note that unless the materialresponse remains within the elastic regime, accuracy of thefinite element analysis is significantly affected and governed bymaterial nonlinearity and hence by the simulative capabilities ofthe selected constitutive model.

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Fig. 13. Comparison of experimental measurements for strains versus the corresponding finite element results for 178 mm (7 in.) prisms with #10 (#3) confinementstirrups: (a) schematic analytical model, (b) stirrups spaced at 75 mm (3 in.) on center, (c) stirrups spaced at 150 mm (6 in.) on center, and (d) stirrups spaced at 225 mm(9 in.) on center.

Acknowledgement

The authors would like to thank Dr. Amin Akhnoukh forsharing the experimental results used in the comparative analysespresented in this paper.

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