Challenges, Opportunities and Solutions in Structural Engineering and Construction – Ghafoori (ed.) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-56809-8
Nonlinear f inite element analysis of unbonded post-tensioned concrete beams U. Kim, P.R. Chakrabarti & J.H. Choi Department of Civil and Environmental Environmental Engineering, California State University, University, Fullerton, CA, USA
ABSTRA ABSTRACT: CT: The main purposes purposes of this study are to develop develop a sophisticated sophisticated 3-D finite element element model for simulating the nonlinear flexural behavior of unbonded post-tensioned beams, to compare analysis results with experimental results to verify the accuracy of the developed 3-D finite element model, and to investigate the effects of various prestressing forces on the flexural behavior of post-tensioned beams. To investigate the nonlinear flexural behavior of post-tensioned concrete beams, a 3-D finite element model was developed using ANSYS. ANSYS. ANSYS ANSYS is a highl highly y recog recogniz nized ed and reliab reliable le comme commerci rcial al softw software are that that is used used for finite finite elemen elementt analy analysis sis.. In order to validate the developed finite element model, four post-tensioned beams were tested at the structures laboratory of California State University, Fullerton and the test results were compared with the analysis results using ANSYS. 1
INTR INTRODUCT ODUCTION ION
of the beams. beams. This This study study inve investi stigat gates es the inelas inelastic tic behavior behavior of unbonded post-tensioned beams using the finite element element method and experim experimental ental tests. The obtained comparison and analysis will be discussed at the end of this paper.
The inelastic flexural behavior of unbounded posttensioned concrete beams is inherently complicated, and reliable reliablenonl nonlinea inearr behavior behaviorcan can usually usually be obtained obtained through physical tests on actual beams (Chakrabarti 1995; Harajli 1991). However, tests are time consuming, expensive, expensive, and test results are generally limited to surface surface measureme measurements. nts. Thus, Thus, this study study was conducte conducted d to compar comparee andanalyze andanalyze the result resultss betw between a finite finite eleelement method and experimental tests to develop a reliable 3-D finite element model to simulate the flexural behavior behavior of unbounded unbounded post-tensioned beams. The The expe experim riment ental al tests tests were were perfor performed med at the strucstructures tures labora laborator tory y of CSUF CSUF (Cali (Califor fornia nia State State Unive Universi rsity ty,, Fullerton). First, four post-tensioned test beams were constructed. Table 1 shows that two of the four experimental beams had an applied prestressing force of 31.1 kN (7000 lb). The remaining two beams had an applie applied d prestr prestress essingforceof ingforceof 15.6 15.6 kN (3500 (3500 lb). lb). Table able 1 also also illu illust stra rate tess that that beam beamss 41 and and 42 had had two two #3 reba rebars rs in the upper top portion and two #3 rebars in the lower bottom portion of the beams, while beams 43 and 44 had two #4 rebars in the lower bottom portion
Table 1.
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FLEXURAL FLEXURAL TEST FOR POST-TEN POST-TENSION SIONED ED CONCRETE BEAM
Figure 1 shows the dimensions of the experimental post-tensioned concrete beams. The The postpost-ten tensio sioned ned concre concrete te beams beams were were consconstructed with double-harped strands which can be seen
Post-tens Post-tensioned ioned concre concrete te beam beam tests. tests.
Beam numb number er
To Top rein reinfo forc rcem emen entt
Bottom rein reinfo forc rcem emen entt
Prestressing forc forcee (kN) (kN)
41 42 43 44
2–#3 Rebars 2–#3 Rebars 2–#3 Rebars 2–#3 Rebars
2–#3 Rebars 2–#3 Rebars 2–#4 Rebars 2–#4 Rebars
31.1 15.6 31.1 15.6
Figure 1.
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Typical detail for post-tensioned post-tensioned concrete beams.
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FINITE ELEMENT MODEL
This chapter specifically describes the finite element modeling and analysis techniques used for simulating the flexural behavior of post-tensioned concrete beams using ANSYS. ANSYS software is one of the most reliable and popular commercial finite element method programs (Lawrence 2007). 3.1 Figure 2.
Stirrup detail for post-tensioned concrete beams.
Figure 3.
Test setup for post-tensioned concrete beams.
Table 2.
Area of Strands (mm2 ) Ultimate Strength of Prestressing Strand (MPa) Area of Stirrups (mm2 ) Yield Strength of Stirrup, f y , (MPa) Compressive Strength of Concrete, f c , (MPa)
Table 3 shows the details of the element types which were utilized to construct the finite element model. Steel plates were placed at both ends of the beam in order to avoid unrealistic cracks due to stress concentrations. If the steel plates were not added to the ends of the beams in the finite element model, the concentrated prestressing forces would have been applied at very small areas, which would ultimately induce cracks that would initiate at the ends of the beams during the analysis procedure. However, this type of cracking mechanism would not occur during flexural tests for post-tensioned concrete beams. The concrete element type, Solid 65 was used because both cracking in tension and crushing in com pression can be considered. 3.2
Properties of strands, stirrups and concrete. 23.2 1862 71 414 32.4
in Figure 1. Figure 1 also illustrates the two point loadings which were applied symmetrically on the tops of the beams. The beams were designed with a length of 3.7 m, a width of 152 mm, and a depth of 254 mm. Figure 2 illustrates the placement of the stirrups in the beams. Each stirrup is spaced 114 mm apart and a total of 31 stirrups were used. The supports of the beams were located 51 mm from the edges of the beams. Figure 3 shows the test setup utilized for the flexural test of the post-tensioned concrete beams, which was performed at CSUF. Strain gages were installed to measure the strain values on the top and bottom rebars and a LVDT was placed in the center of the test beam to measure the deflection. From this data, the stress level of the rebars can be calculated and the load-deflection curve can be obtained. Table 2 shows the material properties of the strands, stirrups, andconcrete which were used in the construction of the post-tensioned concrete beams.
Element types
Real constants
The real constants of the post-tensioned concrete beams are described in Table 4 and Table 5. Table 4 Table 3.
Element types for post-tensioned concrete beams.
Material type ANSYS element type
Table 4.
Concrete
Steel plates
Stirrups
Strands
Solid 65
Solid 45
Link 8
Link 8
Real constants for PT beams (41 and 43).
Real constant set
Element type
Cross-sectional area (mm2 )
Initial strain
1 2 3
Solid 65 Link 8 Link 8
Blank 23.2 71.0
Blank 0.00682 Blank
Table 5.
Real constant for PT beams (42 and 44).
Real constant set
Element type
Cross-sectional area (mm2 )
Initial strain
1 2 3
Solid 65 Link 8 Link 8
Blank 23.2 71.0
Blank 0.00341 Blank
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describes the real constants of beam 41 and beam 43. Table 5 demonstrates the real constants of beam 42 and 44. Tables 4 and 5 show the assigned initial strain values of the prestressing strands were 0.000682 for beams 41 and 43, and 0.00341 for beams 42 and 44. These strain values were calculated from the applied prestressing forces of the test beams. From Table 1, the applied prestressing forces were 31.1 kN for beams 41 and 43 and 15.6 kN for beams 42 and 44. 3.3
Material properties
Figure 6.
The stress-strain for the stirrups and rebars.
Figure 7.
Double-harped post-tensioned beam model.
Figure8.
Themeshed post-tensioned concrete beam model.
This section explains the material properties of the post-tensioned concrete beams. Figure 4 shows the stress-strain curve of the concrete. The value used for the uniaxial tensile cracking stress of concrete was 3.6 MPa (520 psi). During the analysis, if the tensile stress was over 3.6 MPa, cracking would begin to appear. Material properties of the strands were input as multi-linear isotropic material properties. Figure 5 illustrates the stress-strain curve of the strands. Figure 6 shows the stress-strain curve of the stirrups and rebars as bilinear isotropic material properties. 3.4
Modeling
A steel plate was attached at the end of the concrete beam. The stirrups, steel plates, and strands were also modeled as shown in Figure 7. Figure 7 illustrates how the double-harped shape of the strands was modeled as a finite element model. The strands are located
Figure 4.
Figure 5.
The stress-strain of the concrete.
The stress-strain curve of the strands.
76.2 mm (3.0 in.) from the bottom of the beam, in the middle of the span. The reason why the strands were not placed in the same places as the experimental 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 convergence would have a high tendency to fail in the analysis. 3.5
Table 8. Load increment for analysis of finite element model for beam 41.
Meshing
Figure 8 shows the mesh generation of the posttensioned concrete beam model. A 1.0 mesh size was used for this model. Therefore, the concrete beam was meshed with cubes that have the dimensions of 25.4 mm (1.0 in) × 25.4 mm × 25.4 mm. 3.6
Boundary conditions and loading
The loads were applied as two point loadings which were distributed on 3 nodes to avoid stress concentration. The boundary conditions were modeled as a simply supported beam, which are the same as those of the test setup.
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SOLUTION CONTROL
This chapter describes the solution controls used to analyze nonlinear materials. The values shown in Table 6 were used for simulating the post-tensioned concrete beam model.
Table 6.
Basics of the solution control.
Analysis options Calculate prestress effects Time at end of loadstep Automatic time stepping Number of substeps Max no. of substeps Min no. of substeps Write items to results file Frequency
Table 7.
Small displacement Off 0 On 5 30 2 All solution items Write every Substep
Equiv. plastic strain Explicit creep ration Implicit creep ration Incremental displacement Points per cycle Set convergence criteria Label Reference Value Tolerance Norm Min. Ref
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
4 4 4 4 4 4 4 4 4 6 6 4 5 4 4 5 4 5
Prestressing Force Self Weight 445 890 1334 1779 2002 2224 2446 2669 3114 3558 4003 4448 4893 5338 5560 5649
Total loading (N) 0 3336 6005 8673 11342 14011 15345 16680 18014 19349 22017 24686 27355 30024 32693 35361 36696 37363
The values in Table 7 were used for analyzing nonlinear material properties. In this particular case, the convergence criterion for force was discarded in order to avoid convergence problems and the reference value for the displacement criteria was changed to 1.6. Otherwise, if this had not been done, the convergence for the solution control would have had a high tendency to fail in the analysis.
ANALYSIS PROCESS
In order for the nonlinear analysis to be done accurately, the loads are required to have a gradual application, and the nonlinear analysis also requires handling of solution controls. Two point loadings were applied in small incremental loads on beam 41 using the load step and sub step as shown in Table 8.
Off Program chosen 20 Cutback according to predicted number of iter. 0.15 0.1 0 10,000,000 13 U 1.6 0.05 L2 −1
Sub step
5
Nonlinear convergence for solution control.
Line search DOF solution predictor Maximum number of iteration Cutback control
Load step
Loading on each node (N)
6
COMPARISON: TEST AND ANALYSIS
In this chapter, the comparison graphs of the loaddisplacement curves are illustrated in Figures 9 through 12 for beams 41 through 44. The percent differences betweenthe actualtests andresults of ANSYS are summarized in Table 9, Table 10, Table 11, and Table 12 for beams 41, 42, 43, and 44, respectively. These tables show the loads at specified deflections of 2.5 mm, 12.7 mm, and 22.9 mm for each beam specimen. For beams 41, 42, and, 43, the percentage differences at a deflection of 2.5 mm range from 27% to 35%, while the percentage differences at deflections of 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.
Figure 10.
Beam 42 load-deflection graph.
Figure 11.
Beam 43 load-deflection graph.
Figure 12.
Beam 44 load-deflection graph.
Table 9. Percent difference of load-displacement between actual test and ANSYS for beam 41. Displacement Load (ANSYS) Load (TEST) Difference
2.5 mm 18,014 N 13,122 N 27.16%
12.7 mm 26,243 N 26,688 N 1.67%
22.9 mm 34,250 N 33,805 N 1.3%
Table 10. Percent difference of load-displacement between actual test and ANSYS for beam 42. Displacement Load (ANSYS) Load (TEST) Difference
2.5 mm 15,568 N 10,008 N 35.71%
12.7 mm 23,130 N 22,018 N 4.81%
17.9 mm 28,956 N 27,527 N 4.94%
Table 11. Percent difference of load-displacement between actual test and ANSYS for beam 43. Displacement Load (ANSYS) Load (TEST) Difference
2.5 mm 18,682 N 12,454 N 33.33%
12.7 mm 32,693 N 30,691 N 6.12%
22.9 mm 48,483 N 44,925 N 7.34%
Compared to beams 41, 42 and 43, beam 44 had larger percentage differences but still had similar structural behavior. These relative large discrepancies may be explained by the idealized modeling related to material properties.
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STRESS CONTOURS AND CRACKING
In this chapter, under the various levels of loadings according to the different load steps, the contours of the Z-component of stress and change in crack patterns can be seen for beam 41. The number of cracks increased and the region of cracking spread when the applied 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
Table 12. Percent difference of load-displacement between actual test and ANSYS for beam 44. Displacement Load (ANSYS) Load (TEST) Difference
2.5 mm 16,458 N 8674 N 47.30%
12.7 mm 30,246 N 22,907 N 24.26%
22.9 mm 44,925 N 37,141 N 17.33%
was marked with numbers and a total of 35 cracks developed at 26,688 N. Figure 15 shows the Z-component stress contour at total loading of 34,027 N.
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actual beam test results than the partially prestressing case (15.6 kN). 2. The initial behavior shows more differences than the remaining behavior because the experimental post-tensioned concrete beams are not perfectly elastic within the initial stage. 3. From the comparison results, a modification factor of 0.75 is recommended to predict the loaddeflection behavior of unbonded post-tensioned beams using the proposed ANSYS model in this study conservatively.
Figure 13. Pattern of cracks at total loading of 14,678 N, 21,350 N, and 34,027 N.
If this study proves to be applicable on more experimental beam tests through analysis, more accurate results would be able to be investigated. More test results should be further investigated to more precisely evaluate the validity of the proposed FEM model. Furthermore, this FEM model can be used for simulating the nonlinear flexural behavior of a post-tensioned beam repaired with FRP sheets by adding elements of FRP to this model.
REFERENCES
Figure 14. 26,688 N.
Crack pattern of the test beam at total loading of
Figure 15.
The stress contour at total loading of 34,027 N.
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Chakrabarti, P.R. 1995. Ultimate stress for unbonded posttensioning tendons in partially prestressed beams. ACI Structural Journal 92(6): 689–697. Harajli, M.H. & Kanj, M.Y. 1991. Ultimate flexural strength of concrete members prestressed with unboundedtendons. ACI Structural Journal 88(6): 663–673. Lawrence, K.L. 2007. ANSYS Tutorial . Mission: SDC.
CONCLUSIONS AND FUTURE WORK
From this research, the following conclusions can be reached. 1. The results of the fully prestressing case (31.1 kN) for post-tensioned concrete beams are closer to
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