Challenges, Opportunities and Solutions in Structural Engineeringand Construction – Ghafoori (ed.)

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Nonlinear finite element analysis of unbonded post-tensionedconcrete beams

U. Kim, P.R. Chakrabarti & J.H. ChoiDepartment of Civil and Environmental Engineering, California State University, Fullerton, CA, USA

ABSTRACT: The main purposes of this study are to develop a sophisticated 3-D finite element model forsimulating the nonlinear flexural behavior of unbonded post-tensioned beams, to compare analysis results withexperimental results to verify the accuracy of the developed 3-D finite element model, and to investigate theeffects of various prestressing forces on the flexural behavior of post-tensioned beams. To investigate the non-linear flexural behavior of post-tensioned concrete beams, a 3-D finite element model was developed usingANSYS. ANSYS is a highly recognized and reliable commercial software that is used for finite element analysis.In order to validate the developed finite element model, four post-tensioned beams were tested at the structureslaboratory of California State University, Fullerton and the test results were compared with the analysis resultsusing ANSYS.

1 INTRODUCTION

The inelastic flexural behavior of unbounded post-tensioned concrete beams is inherently complicated,and reliable nonlinear behavior can usually be obtainedthrough physical tests on actual beams (Chakrabarti1995; Harajli 1991). However, tests are time consum-ing, expensive, and test results are generally limited tosurface measurements. Thus, this study was conductedto compare and analyze the results between a finite ele-ment method and experimental tests to develop a reli-able 3-D finite element model to simulate the flexuralbehavior of unbounded post-tensioned beams.

The experimental tests were performed at the struc-tures laboratory of CSUF (California State University,Fullerton). First, four post-tensioned test beams wereconstructed. Table 1 shows that two of the four exper-imental beams had an applied prestressing force of31.1 kN (7000 lb). The remaining two beams had anapplied prestressing force of 15.6 kN (3500 lb). Table 1also illustrates that beams 41 and 42 had two #3 rebarsin the upper top portion and two #3 rebars in thelower bottom portion of the beams, while beams 43and 44 had two #4 rebars in the lower bottom portion

Table 1. Post-tensioned concrete beam tests.

Beam Top Bottom Prestressingnumber reinforcement reinforcement force (kN)

41 2–#3 Rebars 2–#3 Rebars 31.142 2–#3 Rebars 2–#3 Rebars 15.643 2–#3 Rebars 2–#4 Rebars 31.144 2–#3 Rebars 2–#4 Rebars 15.6

of the beams. This study investigates the inelasticbehavior of unbonded post-tensioned beams using thefinite element method and experimental tests. Theobtained comparison and analysis will be discussed atthe end of this paper.

2 FLEXURAL TEST FOR POST-TENSIONEDCONCRETE BEAM

Figure 1 shows the dimensions of the experimentalpost-tensioned concrete beams.

The post-tensioned concrete beams were cons-tructed with double-harped strands which can be seen

Figure 1. Typical detail for post-tensioned concrete beams.

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Figure 2. Stirrup detail for post-tensioned concrete beams.

Figure 3. Test setup for post-tensioned concrete beams.

Table 2. Properties of strands, stirrups and concrete.

Area of Strands (mm2) 23.2Ultimate Strength of Prestressing Strand (MPa) 1862Area of Stirrups (mm2) 71Yield Strength of Stirrup, fy, (MPa) 414Compressive Strength of Concrete, f ′

c, (MPa) 32.4

in Figure 1. Figure 1 also illustrates the two point load-ings which were applied symmetrically on the tops ofthe beams. The beams were designed with a length of3.7 m, a width of 152 mm, and a depth of 254 mm.

Figure 2 illustrates the placement of the stirrupsin the beams. Each stirrup is spaced 114 mm apartand a total of 31 stirrups were used. The supports ofthe beams were located 51 mm from the edges of thebeams.

Figure 3 shows the test setup utilized for the flex-ural test of the post-tensioned concrete beams, whichwas performed at CSUF. Strain gages were installed tomeasure the strain values on the top and bottom rebarsand a LVDT was placed in the center of the test beam tomeasure the deflection. From this data, the stress levelof the rebars can be calculated and the load-deflectioncurve can be obtained.

Table 2 shows the material properties of the strands,stirrups, and concrete which were used in the construc-tion of the post-tensioned concrete beams.

3 FINITE ELEMENT MODEL

This chapter specifically describes the finite elementmodeling and analysis techniques used for simulat-ing the flexural behavior of post-tensioned concretebeams using ANSYS. ANSYS software is one of themost reliable and popular commercial finite elementmethod programs (Lawrence 2007).

3.1 Element types

Table 3 shows the details of the element types whichwere utilized to construct the finite element model.Steel plates were placed at both ends of the beam inorder to avoid unrealistic cracks due to stress concen-trations. If the steel plates were not added to the endsof the beams in the finite element model, the con-centrated prestressing forces would have been appliedat very small areas, which would ultimately inducecracks that would initiate at the ends of the beamsduring the analysis procedure. However, this type ofcracking mechanism would not occur during flexuraltests for post-tensioned concrete beams.

The concrete element type, Solid 65 was usedbecause both cracking in tension and crushing in com-pression can be considered.

3.2 Real constants

The real constants of the post-tensioned concretebeams are described in Table 4 and Table 5. Table 4

Table 3. Element types for post-tensioned concrete beams.

Material Steeltype Concrete plates Stirrups Strands

ANSYS Solid 65 Solid 45 Link 8 Link 8element type

Table 4. Real constants for PT beams (41 and 43).

Real Element Cross-sectional Initialconstant set type area (mm2) strain

1 Solid 65 Blank Blank2 Link 8 23.2 0.006823 Link 8 71.0 Blank

Table 5. Real constant for PT beams (42 and 44).

Real Element Cross-sectional Initialconstant set type area (mm2) strain

1 Solid 65 Blank Blank2 Link 8 23.2 0.003413 Link 8 71.0 Blank

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describes the real constants of beam 41 and beam 43.Table 5 demonstrates the real constants of beam42 and 44.

Tables 4 and 5 show the assigned initial strainvalues of the prestressing strands were 0.000682 forbeams 41 and 43, and 0.00341 for beams 42 and 44.These strain values were calculated from the appliedprestressing forces of the test beams. From Table 1, theapplied prestressing forces were 31.1 kN for beams 41and 43 and 15.6 kN for beams 42 and 44.

3.3 Material properties

This section explains the material properties of thepost-tensioned concrete beams. Figure 4 shows thestress-strain curve of the concrete.

The value used for the uniaxial tensile crackingstress of concrete was 3.6 MPa (520 psi). During theanalysis, if the tensile stress was over 3.6 MPa, crack-ing would begin to appear.

Material properties of the strands were input asmulti-linear isotropic material properties. Figure 5illustrates the stress-strain curve of the strands.

Figure 6 shows the stress-strain curve of the stirrupsand rebars as bilinear isotropic material properties.

3.4 Modeling

A steel plate was attached at the end of the concretebeam. The stirrups, steel plates, and strands were alsomodeled as shown in Figure 7. Figure 7 illustrates howthe double-harped shape of the strands was modeledas a finite element model. The strands are located

Figure 4. The stress-strain of the concrete.

Figure 5. The stress-strain curve of the strands.

Figure 6. The stress-strain for the stirrups and rebars.

Figure 7. Double-harped post-tensioned beam model.

Figure 8. The meshed post-tensioned concrete beam model.

76.2 mm (3.0 in.) from the bottom of the beam, inthe middle of the span. The reason why the strandswere not placed in the same places as the experimen-tal tests is due to limitations in the size of the mesh.

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If a mesh size less than 1.0 was used, the convergencewould have a high tendency to fail in the analysis.

3.5 Meshing

Figure 8 shows the mesh generation of the post-tensioned concrete beam model. A 1.0 mesh size wasused for this model. Therefore, the concrete beamwas meshed with cubes that have the dimensions of25.4 mm (1.0 in) × 25.4 mm × 25.4 mm.

3.6 Boundary conditions and loading

The loads were applied as two point loadings whichwere distributed on 3 nodes to avoid stressconcentration.

The boundary conditions were modeled as a simplysupported beam, which are the same as those of thetest setup.

4 SOLUTION CONTROL

This chapter describes the solution controls used toanalyze nonlinear materials. The values shown inTable 6 were used for simulating the post-tensionedconcrete beam model.

Table 6. Basics of the solution control.

Analysis options Small displacementCalculate prestress effects OffTime at end of loadstep 0Automatic time stepping OnNumber of substeps 5Max no. of substeps 30Min no. of substeps 2Write items to results file All solution itemsFrequency Write every Substep

Table 7. Nonlinear convergence for solution control.

Line search OffDOF solution predictor Program chosenMaximum number of iteration 20Cutback control Cutback according to

predicted number of iter.Equiv. plastic strain 0.15Explicit creep ration 0.1Implicit creep ration 0Incremental displacement 10,000,000Points per cycle 13Set convergence criteriaLabel UReference Value 1.6Tolerance 0.05Norm L2Min. Ref −1

Table 8. Load increment for analysis of finite elementmodel for beam 41.

Loading on TotalLoad step Sub step each node (N) loading (N)

1 4 Prestressing Force 02 4 Self Weight 33363 4 445 60054 4 890 86735 4 1334 113426 4 1779 140117 4 2002 153458 4 2224 166809 4 2446 1801410 6 2669 1934911 6 3114 2201712 4 3558 2468613 5 4003 2735514 4 4448 3002415 4 4893 3269316 5 5338 3536117 4 5560 3669618 5 5649 37363

The values in Table 7 were used for analyzing non-linear material properties. In this particular case, theconvergence criterion for force was discarded in orderto avoid convergence problems and the reference valuefor the displacement criteria was changed to 1.6. Oth-erwise, if this had not been done, the convergence forthe solution control would have had a high tendencyto fail in the analysis.

5 ANALYSIS PROCESS

In order for the nonlinear analysis to be done accu-rately, the loads are required to have a gradual applica-tion, and the nonlinear analysis also requires handlingof solution controls. Two point loadings were appliedin small incremental loads on beam 41 using the loadstep and sub step as shown in Table 8.

6 COMPARISON: TEST AND ANALYSIS

In this chapter, the comparison graphs of the load-displacement curves are illustrated in Figures 9through 12 for beams 41 through 44. The percent dif-ferences between the actual tests and results of ANSYSare summarized in Table 9, Table 10, Table 11, andTable 12 for beams 41, 42, 43, and 44, respectively.

These tables show the loads at specified deflectionsof 2.5 mm, 12.7 mm, and 22.9 mm for each beamspecimen. For beams 41, 42, and, 43, the percentagedifferences at a deflection of 2.5 mm range from 27%to 35%, while the percentage differences at deflectionsof 12.7 mm and 22.9 mm range from 1.3% to 7.3%.

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Figure 9. Beam 41 load-deflection graph.

Table 9. Percent difference of load-displacement betweenactual test and ANSYS for beam 41.

Displacement 2.5 mm 12.7 mm 22.9 mmLoad (ANSYS) 18,014 N 26,243 N 34,250 NLoad (TEST) 13,122 N 26,688 N 33,805 NDifference 27.16% 1.67% 1.3%

Table 10. Percent difference of load-displacement betweenactual test and ANSYS for beam 42.

Displacement 2.5 mm 12.7 mm 17.9 mmLoad (ANSYS) 15,568 N 23,130 N 28,956 NLoad (TEST) 10,008 N 22,018 N 27,527 NDifference 35.71% 4.81% 4.94%

Table 11. Percent difference of load-displacement betweenactual test and ANSYS for beam 43.

Displacement 2.5 mm 12.7 mm 22.9 mmLoad (ANSYS) 18,682 N 32,693 N 48,483 NLoad (TEST) 12,454 N 30,691 N 44,925 NDifference 33.33% 6.12% 7.34%

Compared to beams 41, 42 and 43, beam 44 had largerpercentage differences but still had similar structuralbehavior. These relative large discrepancies may beexplained by the idealized modeling related to materialproperties.

7 STRESS CONTOURS AND CRACKING

In this chapter, under the various levels of loadingsaccording to the different load steps, the contours ofthe Z-component of stress and change in crack pat-terns can be seen for beam 41. The number of cracksincreased and the region of cracking spread when theapplied loads were augmented as shown in Figure 13.

Figure 14 shows the crack pattern of the test beam.During the test, the sequence of crack development

Figure 10. Beam 42 load-deflection graph.

Figure 11. Beam 43 load-deflection graph.

Figure 12. Beam 44 load-deflection graph.

Table 12. Percent difference of load-displacement betweenactual test and ANSYS for beam 44.

Displacement 2.5 mm 12.7 mm 22.9 mmLoad (ANSYS) 16,458 N 30,246 N 44,925 NLoad (TEST) 8674 N 22,907 N 37,141 NDifference 47.30% 24.26% 17.33%

was marked with numbers and a total of 35 cracksdeveloped at 26,688 N.

Figure 15 shows the Z-component stress contour attotal loading of 34,027 N.

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Figure 13. Pattern of cracks at total loading of 14,678 N,21,350 N, and 34,027 N.

Figure 14. Crack pattern of the test beam at total loading of26,688 N.

Figure 15. The stress contour at total loading of 34,027 N.

8 CONCLUSIONS AND FUTURE WORK

From this research, the following conclusions can bereached.

1. The results of the fully prestressing case (31.1 kN)for post-tensioned concrete beams are closer to

actual beam test results than the partially prestress-ing case (15.6 kN).

2. The initial behavior shows more differences thanthe remaining behavior because the experimentalpost-tensioned concrete beams are not perfectlyelastic within the initial stage.

3. From the comparison results, a modification fac-tor of 0.75 is recommended to predict the load-deflection behavior of unbonded post-tensionedbeams using the proposed ANSYS model in thisstudy conservatively.

If this study proves to be applicable on more exper-imental beam tests through analysis, more accurateresults would be able to be investigated. More testresults should be further investigated to more preciselyevaluate the validity of the proposed FEM model. Fur-thermore, this FEM model can be used for simulatingthe nonlinear flexural behavior of a post-tensionedbeam repaired with FRP sheets by adding elementsof FRP to this model.

REFERENCES

Chakrabarti, P.R. 1995. Ultimate stress for unbonded post-tensioning tendons in partially prestressed beams. ACIStructural Journal 92(6): 689–697.

Harajli, M.H. & Kanj, M.Y. 1991. Ultimate flexural strengthof concrete members prestressed with unbounded tendons.ACI Structural Journal 88(6): 663–673.

Lawrence, K.L. 2007. ANSYS Tutorial. Mission: SDC.

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