Engineering Structures 40 (2012) 267–275

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Engineering Structures

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Nonlinear finite element modeling of cracking at ends of pretensioned bridge girders

Pinar Okumus a,⇑, Michael G. Oliva b, Scot Becker c

a Department of Civil and Environmental Engineering, University of Wisconsin – Madison, 1415 Engineering Drive, 1203 Engineering Hall, Madison, WI 53706, United Statesb Department of Civil and Environmental Engineering, University of Wisconsin – Madison, 1415 Engineering Drive, 1214 Engineering Hall, Madison, WI 53706, United Statesc Wisconsin Department of Transportation, 4802 Sheboygan Ave., Madison, WI 53707, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 23 September 2011Revised 22 December 2011Accepted 16 February 2012Available online 29 March 2012

Keywords:Nonlinear finite element analysisPrestressed girderEnd cracksConcrete plasticityPrestress transfer

0141-0296/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.engstruct.2012.02.033

⇑ Corresponding author. Tel.: +1 608 262 1262.E-mail addresses: okumus@wisc.edu (P. Okumus

Oliva), scot.becker@dot.wi.gov (S. Becker).

Recent bridge designs have created efficient prestressed concrete girder sections with thin webs, and highlevels of prestress. The transfer of the large stresses from strands to concrete causes these slender sec-tions to undergo cracking at the ends of the girders. Due to the large amount of cracking, a nonlinear anal-ysis is necessary to reveal and understand the behavior of the concrete and reinforcement bars atprestress release. Finite element modeling is an excellent tool to perform this task. The accuracy of theanalyses, however, depends on the input parameters, some of which are challenging to define for a non-linear problem. This paper identifies the input parameters and modeling features that have significantimpact on the results of nonlinear finite element analyses for pretressed concrete girder ends. The sen-sitivity of the results to the tensile properties of concrete, the bond properties between concrete and steelrebars and the form of prestress transfer to the concrete were evaluated. The impact of modeling relatedproperties such as the order of elements on the predicted results was also investigated. Available testdata was used to verify the modeling techniques. Once verified, the finite element modeling wasextended to girders where significant cracking is observed. The full field tensile strain patterns obtainedthrough the verified finite element models are used to explain observed cracking. The effectiveness of theend reinforcement bars, intended to control cracking, was examined.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The fundamental concept behind prestressed concrete design isthe transfer of compression force from steel prestressing strands tosurrounding concrete to achieve a crack free member under serviceloads. The properties of concrete and strand interface cause thistransfer of stress to occur within a relatively short distance fromthe girder end, with high stress conditions. As a result, an over-stressed girder end region may exhibit visible cracks in distinctivepatterns. This contradicts the crack free goal of prestressed con-crete. Heavily prestressed deep girders designed for long bridgespans appear to experience more severe end cracks than traditionalsmaller girders. Characteristic cracks observed in very similar pat-terns in all deep Wisconsin wide flanged I girders are shown inFig. 1 right after prestress release.

These cracks might lead to durability problems when they arenot enclosed in the end diaphragms and exposed to the environ-ment. If the salt water seeping through the cracks reaches strands,corrosion and loss of bond between strands and concrete may leadto structural capacity losses.

ll rights reserved.

), oliva@engr.wisc.edu (M.G.

Endeavors to explain the stress field at the girder ends, since theearly 1960s, utilized either empirical methods, simplified linearanalytical concepts, or strut and tie models. The cracks, to thisday, remain an issue despite the girder end design provisions ofcodes such as the AASHTO LRFD Bridge Design Specifications [1].Prompted by the lack of comprehensive nonlinear analyses of theprestressed girder end regions in the literature, this study em-ployed nonlinear finite element analysis (FEA) to assess the likeli-hood of cracking and the stress distribution at the girder endsduring and shortly after prestress release.

Among the most widely known girder end analytical studies isthe so called Gergely–Sozen model [2]. Assuming concrete behaveslinearly until cracking, the model formulates the area of web rein-forcement required to control crack size using free body analysis.Other analytical research [3,4] investigated the strut and tie meth-od for the design of the girder end region, some specifically forposttensioned girders [5]. The strut and tie method is most easilyapplicable after the material becomes inelastic, provides a lowerbound estimate of the strength capacity, and there is more thana single truss configuration for a given case.

The majority of the previous research work consists of empiricalor semi-empirical studies. The AASHTO LRFD Bridge Design Speci-fications, Article 5.10.10.1 [1] seems to have adopted the equationderived from the experimental study conducted by Marshall andMattock [6]. Tuan et al. [7] and Dunkman et al. [8] monitored

Fig. 1. Typical characteristic inclined, web, and Y cracks.

268 P. Okumus et al. / Engineering Structures 40 (2012) 267–275

strains on the vertical end zone stirrups. These strain values gaverise to practical recommendations on the end zone reinforcementefficiency. However, strain data collected by strain gages is boundto be discrete and provides limited information, particularly whencracking occurs away from a gage location. FEA can provide thestrain field for the entire continuum in any direction; can help ex-plain what triggers cracking and can also be used for parametricstudies with minimal cost.

The experimental study conducted by O’Callaghan [9] measurednumerous reinforcement bar strains at three different vertical lev-els along the girder length in the end region on Texas girders. Thevariation of the stresses the rebars carry along the girder lengthwas presented.

Kannel et al. [11] utilized FEA to simulate the restraint providedby the uncut strands, the effects of the order in which the strandsare cut, and debonding some strands at the girder end. The mainshortcoming of this model was the use of linear elastic materialproperties for concrete. Breen et al. [5] simulated a posttensionedgirder with two dimensional FEA, where the stress transfer mech-anism is concentrated at the girder end and much simpler com-pared to the gradual stress transfer in pretensioned girders.

2. The scope

The aim of this study was to achieve an accurate nonlinear finiteelement prediction of strain and stresses in pretensioned concretegirder ends. Due to nonlinearities in a girder end zone, particularlystiffness changes and redistribution of strains after cracking,explaining the strain distribution after the first cracks is beyondthe capabilities of linear analyses or free body diagrams. A nonlin-ear stress analysis of the crack prone pretensioned girder ends doesnot appear in the literature.

The accuracy of the FEA depends heavily on the input parame-ters. The tensile properties of concrete after cracking, the concrete–steel bond, and the concrete–strand bond are the importantparameters that define the cracking behavior. The sensitivity ofthe FEA results to varying values of these parameters was investi-gated. The role of the analysis type, and the element type on theefficiency and convergence of the models was studied. The resultsof each model were compared to available test results. The bestrepresentations of the input and modeling parameters were used

for the extended FEA study on Wisconsin girders with significantcracking.

The full field strain distribution at the girder end region and gir-der end zone reinforcement right after the prestress release is pre-sented. The stresses obtained through FEA are used to explain thecauses of characteristic cracks. The FEA was performed using Aba-qus/CAE.

3. Simulated girders

The base line for the verification of the FEA procedures was aTexas 1778 mm deep I girder tested by O’Callaghan [9]. This girderwas selected for the verification and sensitivity study due to thelarge amount of test data reported. O’Callaghan placed strain gageson selected rebars at 419 mm, 811 mm, 1575 mm from the bottomof the girder. These lines of gages will be referred in this paper asgage line I, II and III respectively. The stresses on rebars at the gagelocations were reported. In addition information on the distribu-tion of the bond stresses was also available through gages on se-lected strands in the end region. This girder had weldeddeformed rebar cage of 517 MPa average yield strength and addi-tional four rebars of 414 MPa average yield strength bundled at102 mm spacing closest to the end. The average concrete strengthwas 46 MPa. The girder had 46 low relaxation 1862 MPa ultimatestrength 15.2 mm diameter strands in the bottom flange at aneccentricity of 582 mm, in addition to four straight dummy strandson the top flange to facilitate construction.

The FEA study with confirmed modeling techniques and param-eters were then extended to resemble Wisconsin girders which ex-hibit the most significant cracking. The 1372, 1829 and 2083 mmdeep wide flange I girders represented heavily prestressed girderswith 40, 48, 46 strands, 8 of each were draped. Strain patterns aredemonstrated for a 1372 mm deep girder. The release strength ofgirder concrete studied was 47 MPa, the rebars had yield strengthof 414 MPa. The low relaxation strands were 15.2 mm diameterand had 1862 MPa ultimate strength. The cross section, strand pat-terns and end zone rebar details of these sections are given by theWisconsin Department of Transportation, Bureau of Structures[10].

Only a quarter segment of the full girder is modeled utilizingthe symmetric nature of the geometry and loading. Fig. 2 showsa typical meshed model with boundary conditions.

4. FEA sensitivity study

4.1. Material nonlinearity

Material nonlinearity, particularly for concrete in tension, was akey part of this cracking focused problem. Linear FEA studies existin the literature. In order to assess the level of accuracy loss withlinear models, a linear FEA was run and the results were comparedto the ones with nonlinear properties.

From a computational perspective, the FEA with the linear elas-tic concrete material model was much simpler and cheaper. Thetwo cases behave similarly until the concrete elements reachedtheir theoretical cracking strength. Once the cracking occurs inconcrete, the redistribution of stresses is expected to take placeand the rebars become engaged. Elastic material models whichdo not capture the stiffness loss of the concrete elements duringcracking are not capable of representing this and subsequent crackgrowth.

Fig. 3 presents the predicted principal tensile strains for a2083 mm deep Wisconsin girder after complete prestress releasewith and without concrete material nonlinearity. A qualitativecomparison between actual cracking in Fig. 1 and the two models

Fig. 2. The boundary conditions and the mesh.

Fig. 3. Principal tensile strain comparison of a linear and a nonlinear FEA model.

P. Okumus et al. / Engineering Structures 40 (2012) 267–275 269

in Fig. 3 shows the alignment of cracks with the high strain areas inthe nonlinear model.

The linear model causes the concrete elements maintain theirinitial stiffness, with maximum tensile strain and stresses up to314 microstrains and 1545 psi at the web. The cracking strainand strength for this concrete was calculated to be 124 micro-strains and 600 psi. The unrealistic excessive strain and stress val-ues prove that linear models are not suitable to simulateprestressed girder ends. For the nonlinear model, the magnitudesof maximum tensile stress did not exceed the cracking stress.Much higher tensile strains were obtained, magnitudes reaching0.38 mm when converted to crack opening. The highest strains

were in the inclined cracking region, matching the location wherethe largest cracks are observed in the field.

It was not possible to obtain rebar stresses within an acceptableerror range with the linear model. The maximum rebar stress witha linear model was predicted to be 51.7 MPa for the web bar closestto the end. On the other hand, the rebar stress for the same bar was193.1 MPa with a nonlinear model.

In order to decrease the computational cost, nonlinear materialproperties were only assigned in the end region. The length of theregion with nonlinear material properties was at least as long asthe girder depth from the girder end. Nonlinear regions shorterthan this resulted in significant errors in the results.

Fig. 4. Constitutive model for concrete in compression (left) and tension (right).

0

40

80

120

160

0 500 1000 1500

σrebar(MPa)

Distance from the girder end (mm)

Test by O'Callaghan [9]FEA, GF = 147 N/mFEA, GF = 115 N/mFEA, GF = 2x147 N/m

Fig. 5. The resultant rebar stresses along the girder length with varying levels of GF.

270 P. Okumus et al. / Engineering Structures 40 (2012) 267–275

4.2. Concrete material model

The material model used was the ‘‘concrete damaged plasticity’’model from the Abaqus/CAE material library. The constitutivemodel of concrete was based on FIB Model Code 2010 [12], andAASHTO LRFD Bridge Design Specifications [1] as shown in Fig. 4.

The initial modulus of elasticity is assumed to be the same fortension and compression and calculated per AASHTO LRFD BridgeDesign Specifications Section 5.2.4.2.The cracking strength of con-crete is calculated using the AASHTO LRFD Bridge Design Specifica-tions Section C5.4.2.7. This was judged to be a less conservative butmore accurate representation than the cracking strength given bythe FIB Model Code 2010, Volume 1, Section 5.1.5 based on theFEA results. For the nonlinear range, the constitutive model forconcrete in tension and compression were based on the mathemat-ical models given by FIB Model Code 2010, Volume 1, Section 5.1.8.

The concrete tension behavior is composed of two discreteparts: the pre cracking and post cracking stages. The post crackingstage can be defined in terms of strains, crack opening or the frac-ture energy. Mesh sensitivity is a potential issue due to narrowercrack widths with finer meshes [13]. A fracture energy based con-cept was used to define the stress–displacement (crack opening)behavior as opposed to a stress–strain relation in order to over-come this problem.

4.3. Rebar–concrete bond properties

Rebars are modeled as linear elastic elements, since end zonerebars are not expected to yield during strand detensioning. Theinteraction of the reinforcement bars and concrete was modeledthrough tension softening of the concrete implicitly. Tension stiff-ening, the term used by Abaqus, is introduced to concrete to repre-sent the added ductility that would be provided by the rebars aftercracking.

The rebar elements were ‘‘embedded’’ in concrete. In Abaqus,this means that the response of the concrete elements is used toconstrain the translational degrees of freedom of the rebar nodes.Once concrete elements reach their cracking limit, their stress car-rying capacity drops with increasing deformation, transferringtheir force to the steel rebars. Reinforcement bar elements distrib-ute the failure and softening to address any mesh sensitivity prob-lems with nonlinear analyses.

Tension softening is directly related to the fracture energy, GF asshown in Fig. 4. FIB Model Code 2010 Section 5.1.5.2 states thefracture energy should be determined by related tests. The fractureenergy depends on the water cement ratio, the maximum aggre-gate size, and the age of concrete and is affected by the curing

conditions and the size of the specimen [12]. In the absence oftests, the fracture energy in N/m is given to be estimated by Eq.(1), where fcm is the mean compressive strength of concrete in MPa.

GF ¼ 73 � f 0:18cm ð1Þ

This equation calculates the fracture energy for the base girdertested by O’Callaghan [9] as 145 N/m. The earlier version of theModel Code, CEB-FIB Model Code 1990 [14] estimates the fractureenergy for the same strength concrete to be between 70 N/m and115 N/m depending on the maximum aggregate size. Due to thenumber of parameters that affect the fracture energy measure-ments and the lack of uniaxial tension test data, the sensitivity ofthe rebar stresses to varying values of fracture energy needed tobe investigated. The results of FEA with varying levels of fractureenergy were compared. GF = 115 N/m given as the upper boundby the Model Code 1990 [14], GF = 145 N/m as given by the ModelCode 2010 [12], and GF = 2 � 145 N/m or double the FIB 2010 Mod-el Code value. The results as compared to the measured rebar stres-ses at gage line I of the base girder are shown in Fig. 5.

It is seen that the fracture energy of 145 N/m as prescribed bythe FIB Model Code 2010 is a good approximation that created only7% error in the peak rebar stresses. Using the material model pro-vided by the CEB-FIB Model Code 1990 introduced a 15% error inthe peak values. Using two times the fracture energy given bythe Model Code resulted in 13% error in the peak values, howeverconverged much faster.

It was observed that FEA with larger fracture energy levels con-verged significantly faster. Fracture energy values below 100 N/mdid not return converging results and therefore were judged notto be practical.

P. Okumus et al. / Engineering Structures 40 (2012) 267–275 271

4.4. Prestress transfer

The cracking in Wisconsin girders typically occurs during orright after the prestress release. Therefore the main loading consid-ered for the FEA was prestressing forces on the girder. Some minorwidening of cracks is observed weeks after the girders are cast;however the scope of this study only covered the immediate crack-ing upon release. Temperature loads were not included in the mod-els as the temperature changes during prestress release areminimal. Some amplification of the stresses due to the suddencut of the strands is expected. However, studying the dynamic ef-fects of the strand cutting process or the response of concrete un-der sudden loads was not within the scope of this project.

Modeling the exact condition between strand and concrete ascontrolled by the adhesion, friction due to the twisted strand con-figuration and Poisson radial expansion of the strand significantlyincreases the analysis cost. Modeling the entire girder end regionwould not have been possible with such an approach. Rather, theprestress load was applied directly on the concrete as surface loadsaround the diameter of the strands along the prestress transferlength as shown in Fig. 2. The loading from every two strandswas applied to the girder on a separate step to identify the stressesindividual strands created.

The research results on the strand transfer length and bondstress distribution in the literature are rather scattered as statedby Buckner [15], and Tabatabai and Dickson [16]. Therefore thethree different transfer length and bond stress distributions werestudied to pick the best representation for the problem.

The first model used the transfer length calculated by 60 timesthe strand diameter with bond stresses assumed uniform perAASHTO LRFD Bridge Design Specifications, Section C.5.11.4.2 [1].The transfer length for the 15.2 mm in strands was calculated as914 mm. The second model assumed a linear bond stress distribu-tion over 914 mm long transfer length. The third model used thebond stresses and transfer length as measured through the testsby O’Callaghan [9] via strain gages on strands. The results fromthree models are compared to the test results at gage line I in Fig. 6.

As expected, the FEA results aligned with the test results whenthe transfer length as measured through the tests was used. Thelinearly varying bond stresses over a 914 mm transfer length gaveresults similar to the test results with a total average error of 15%.The model with uniform strand–concrete bond stress over the914 mm as given in AASHTO LRFD [1] resulted in poor predictionof rebar stresses. The distribution of the bond stresses plays animportant role in the distribution of the girder end stresses.

0

40

80

120

160

0 500 1000 1500

σrebar (MPa)

Distance from the girder end (mm)

Test by O'Callaghan [9]

FEA, uniform bond

FEA, linearly varying bond

FEA, bond as measured [9]

Fig. 6. The resultant rebar stresses along the girder length with varying prestresstransfer rates.

4.5. Element type

The sensitivity of the results with respect to the element typeused in the nonlinear girder end region was investigated. The re-gion was meshed with first and second order elements of tetrahe-dral shapes. The results for the 1st order tetrahedral elements(C3D4) and 2nd order tetrahedral elements (C3D10) were com-pared to the test results provided by O’Callaghan [9] at gage lineI in Fig. 7. The numbers in parenthesis indicate the name of the ele-ments in ABAQUS. The hexahedral elements were not used as theyare prone to shear locking. It is also more difficult to mesh arbitraryvolumes with hexahedral elements.

The results of the analysis with 1st order tetrahedral elementshad the best correlation with the test results. The Abaqus [13] sug-gests that 2nd order elements generally give higher accuracy perdegree of freedom for the solution for elasticity problems. How-ever, for plasticity applications, it advises that the first order ele-ments are in general preferred for accuracy. The results of thisstudy, verify this recommendation.

The analysis time for the analysis with 2nd order elements wasin an order of magnitude longer compared to that with 1st orderelements. The use of 2nd order elements was found infeasible.

The steel rebars were modeled using one dimensional truss ele-ments, the order of which was compatible with the surroundingthree dimensional elements representing concrete (T3D2 for firstorder analysis and T3D3 for second order analysis). The steel pre-stressing strands were not included in the model as mentionedearlier in the previous section.

The smallest dimension of the elements used was 38.1 mm orless based on a balance between relative error and computationcost. The element size was gradually increased away from the gir-der end.

4.6. Additional verification

Based on the sensitivity studies described above, and usingwhat proved to be the best modeling parameters, the correlationof the analysis results for gage line II and gage line III with the testdata reported by O’Callaghan is as shown in Fig. 8. The single highrebar stress spike from O’Callaghan’s data at one gage location ongage line II is judged to be either an error in gage data or due toa gage location at a concrete crack. At other locations the FEAwas found to provide an accurate representation of the response.

5. Results extended to Wisconsin girders

Once the modeling techniques were verified using available testdata on a Texas girder, and the most suitable input parameters andmodeling techniques were selected; the FEA was extended to rep-resent the standard Wisconsin girders. The full field strains on con-crete and stresses in rebars are presented here for the Wisconsin1372 mm deep girder. Wisconsin girders of three different depthshave the same bottom and top flange, are often heavily prestressedand experience very similar end cracking patterns. Therefore, thestrain patterns shown here are representative of all Wisconsin Igirders.

5.1. Reactions in concrete

Of particular importance to this research were the principalcomponents of strains. The principal tensile strains and their direc-tions determine where and in which direction the concrete crack-ing would occur. After the prestressing force was transferred to theconcrete, the magnitudes of the principal tensile strains in the con-crete are as shown in Fig. 9. Positive strains indicate tension.

0

40

80

120

160

0 500 1000 1500

σrebar(MPa)

Distance from the girder end (mm)

Test by O'Callaghan [9]

FEA, 1st order tetra

FEA, 2nd order tetra

Fig. 7. The resultant rebar stresses along the girder length with varying element order.

-20

20

60

100

0 500 1000 1500

σrebar(MPa)

Distance from the girder end (mm)

Test by O'Callaghan [9]

FEA

-20

20

60

100

0 500 1000 1500

σrebar(MPa)

Distance from the girder end (mm)

Test by O'Callaghan [9]

FEA

Fig. 8. FEA results compared to the test results by O’Callaghan [9] for gage line II (left) and III (right).

Fig. 9. Principal tensile strain contours.

272 P. Okumus et al. / Engineering Structures 40 (2012) 267–275

Fig. 10. Principal strain directions in elevation on Section A–A.

P. Okumus et al. / Engineering Structures 40 (2012) 267–275 273

The principal strain direction plots as given in Fig. 10 for com-pression and tension can be used as a supplement to the contourplots to explain cracking. The direction of the principal tensilestrains, indicated by the short lines in the figures, determines thedirection of crack opening. The compression principal strains areconcentrated around the strands where compression struts spreadout into the girder as prestressing is transferred along the girderlength. The four regions of the girder where principal tensilestrains were significantly higher than average strains are markedin Figs. 9 and 10.

Region I: The highest concrete tension strains occur at region Ifor girders with draped strands. Region I is close to the drapedstrands where the inclined cracks are often visible as indicated inFig. 1.

The FEA shows that the principal tensile strain direction is per-pendicular to inclined cracking even without the draped strands.Comparison of FEA models with and without draped strands,however, showed that draped strands play a significant role in

Fig. 11. Principal tensile strain directions on

amplifying these strains. The radial tensile strains perpendicularto the compression strut around the draped strands align withthe principal tensile strains created by the straight strands. There-fore, the draped strands trigger the inclined crack formation.

Region II: The second largest strains are located in this web re-gion. The locations and directions of these high strains are repre-sentative of the horizontal web cracking shown in Fig. 1. Thesecracks develop across the tension tie that ties the two compressionstruts formed along the draped and straight strands.

These cracks are attributed to the eccentricity of the strands inthe direction of the girder depth. The draped and straight strandscreate moments in the web in opposite directions. These two reac-tion moments put the web into tension. The girder web then expe-riences cracking under heavy loads.

Region III: Region III is located in the bottom flange where Yshaped cracks are observed. These cracks form a separation line be-tween two sides of the bottom flange and web. They are close tothe strands and may form paths for corrosion agents to reach the

the cross section of the bottom flange.

274 P. Okumus et al. / Engineering Structures 40 (2012) 267–275

strands, raising the risk of structural capacity loss. The principaltensile strain directions as obtained through FEA, shown inFig. 11, agree with a Y shaped crack.

The vertical crack seems to be a result of the eccentricity of thestrands in the bottom flange across the width of the girder. Con-ceptually similar to web cracking, strands on each side of the bot-tom flange create moments in opposite directions on verticalsections. The Y shape forms when this vertical crack meets thelower horizontal web cracks in a transition zone between theweb and the bottom flange.

Region IV: Region IV marks high strains in the bottom flangearea peaking at a distance equal to the transfer length from the gir-der end. Fig. 11 shows the principal tensile strains on a cut fromthe bottom flange at the transfer length. These strains, sometimesreferred as the bursting stresses, form perpendicular to the com-pression force applied by the bottom flange strands, and suggestoutwards radial pressure. Although the FEA indicated that thesestrains exceed the elastic limit, no cracking in this region was vis-ible on girders inspected at plants. The lack of visible cracking maybe due to the fact that the predicted directions of the tensile strainsdo not align, making formation of a single crack unlikely. The bot-tom flange of the real girders might be experiencing numerousinterior micro cracks that are not visible during inspection.

5.2. Reactions in reinforcement bars

Determining the stresses in the rebars and in the surroundingconcrete is essential in evaluating the efficiency of the bars inresisting crack opening. The axial stresses in the web and bottomflange rebars in the end zone of the girder are shown by the linecolor in Fig. 12 on a quarter size model.

The vertical bars closest to the very end of the girder, crossingthe web cracks, exhibit the largest stresses. On the contrary, thestresses in the bottom flange confinement stirrup start lower atthe girder end and peak closer to the end of the transfer length.

The web reinforcement bars lay parallel to principal tensilestrains in the web cracking region as shown in Fig. 10, and there-fore they will be the most effective in restraining these cracks.The stresses, obtained through FEA, in vertical web reinforcementbars for heavily prestressed girders do not exceed 106.2 MPa,

Fig. 12. Longitudinal stresses (MP

182.7 MPa, and 199.3 MPa for 1372 mm, 1829 mm, 2083 mm deepheavily prestressed Wisconsin girders. The web reinforcement barsare only loaded up to 25–50% of their yielding capacities.

6. Conclusions

This paper describes the modeling procedures and input param-eters that lead to accurate results for nonlinear FEA results in pre-tensioned girder ends with cracks during prestress release. Thestrain and stress fields of concrete and rebars were presented toexplain the reasons behind cracking.

Linear models, even though they are computationally low costoptions, do not consider the stiffness loss of concrete upon crackingand largely under calculate the concrete strains. They fail to resem-ble the stress transfer from concrete and rebars due to cracking andtherefore underestimate the rebar stresses.

Underestimating the fracture energy as an input introducesconvergence problems. A 10–20% error in the rebar stresses is ex-pected when the fracture energy, as a concrete tensile property, ismispredicted. The fracture energy calculated using the equation inthe FIB Model Code 2010 lead to accurate results.

Representing the concrete–strand bond as uniform, linearly dis-tributed, or as measured by tests has a direct impact on the pre-dicted stress results for prestressed girder ends. The model withuniform concrete–strand bond stresses, per AASHTO LRFD BridgeDesign Specifications, predicted the peak rebar stresses lower thanthe test values. Using linearly varying bond stresses along thestrand transfer length is judged to be an acceptable substitutionfor the complex modeling of actual friction and bond effects forthe test girder where the bond distribution measured was similarto linear varying.

The computational cost for use of second order concrete ele-ments was very high and did not improve the predicted stresses.Therefore, these elements are not recommended for similar nonlin-ear plastic problems.

The locations and directions of the plastic strains predicted bythe FEA models coincide with the locations and crack openingdirections of the cracks observed on the girders. The inclined crackscloser to the top flange are attributed to the contribution of strainscreated by draped strands to the strains in this area caused by the

a) in the reinforcement bars.

P. Okumus et al. / Engineering Structures 40 (2012) 267–275 275

straight strands. The web cracks were attributed to the eccentricityof the bottom and draped strands over the girder depth. The Ycracks in the bottom flange are similarly a result of the transverseeccentricity of the bottom flange strands on each side of the flangeover the width of the girder. Finally, the FEA models captured highstrains growing along the bottom strands as the compression strutspreads out. The highest rebar stresses were observed in the web,and are less than half of the yielding strength.

Based on comparisons with data measured in girder tests inTexas, the FEA modeling techniques described here give a goodrepresentation of the complex stress and strain patterns at theends of girders where prestress transfer effects cause nonlinearbehavior with concrete cracking. These modeling procedures canbe effectively used in the future to investigate methods for elimi-nating or controlling the amount and size of concrete cracks in pre-stressed girders during the prestress transfer process.

Acknowledgements

This work was funded by the Wisconsin Highway Research Pro-gram of the Wisconsin Department of Transportation. The viewspresented are those of the authors and not the funding agency.

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[10] Standard Detail Drawings. Bureau of Structures, Wisconsin Department ofTransportation. <http://on.dot.wi.gov/dtid_bos/extranet/structures/LRFD/standards.htm> [updated 31.10.11; cited 31.10.11].

[11] Kannel J, French C, Stolarski H. Release methodology of strands to reduce endcracking in pretensioned concrete girders. PCI J 1997;42:42–54.

[12] Fédération Internationale du Béton. Fib Bulletins 55–56: Model Code 2010 –First complete draft. Fédération Internationale du Béton; 2010.

[13] Dassault Systèmes Simulia Corporation. Abaqus theory manual. In: Abaqus 6.9documentation. RI (USA): Dassault Systèmes; 2009.

[14] Comité Euro-International du Béton. CEB-FIP Model Code 1990: design code:London: T. Telford; 1993.

[15] Buckner CD. A review of strand development length for pretensioned concretemembers. PCI J 1995;40:84–99.

[16] Tabatabai H, Dickson TJ. The history of the prestressing strand developmentlength equation. PCI J 1993;38:64–75.