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Nonlinear finite element analysis of prestressed concrete members using ADINA
Computers and Structures 81 (2003) 727–734 www.elsevier.com/locate/compstruc
Nonlinear finite element analysis of prestressed concrete members using ADINA Makoto Kawakami
a,*
, Tadahiko Ito
b
a
b
Kozo Keikaku Engineering Inc., 4-5-3 Chuo, Nakano, Tokyo, Japan Nishimatsu Construction Co., Ltd., 2570 Shimotsuruma, Yamato, Kanagawa, Japan
Abstract A prestressed concrete (PC) column and a precast segmental PC beam are analyzed using ADINA. Two dimensional finite element model is employed with material nonlinearities (concrete cracking/crushing and rebar plasticity) and geometrical nonlinearities (large displacement and contact/separation between the segments). The load-displacement relationship, the concrete cracking/crushing process, the load-strain relationship, and the gap opening/closing are in good agreement with those in the tests. Ó 2003 Elsevier Science Ltd. All rights reserved.
1. Introduction In recent years, prestressed concrete (PC) columns and precast segmental PC beams have been studied and designed because of their excellent mechanical characteristics and their practical advantages in construction. In the design of structures using such PC members, it is very important to predict their maximum strength, deformation performance, and material failure process in order to evaluate the reliability and safety of the structures. The prediction must take into consideration various mechanical nonlinearities, namely, (1) material nonlinearities arising from cracking/crushing of concrete, plasticity of reinforcing steel bars (rebars), and bond slipping between rebars and concrete, and (2) geometrical nonlinearities arising from large displacement of concrete members and PC bars/cables, and gap opening/closing between concrete segments. From the point of view mentioned above, we performed the nonlinear finite element analysis of PC members based on two load test results: [1] for a PC column which included material nonlinearities, and [2] for a pecast segmental PC beam which included geometrical nonlinearities as well as material nonlinearities.
*
Corresponding author.
For the calculation, we used the general-purpose finite element code ADINA [3]. The analyses of the PC column and the PC beam are discussed in Sections 2 and 3, respectively, in each of which, the load test is summarized and the calculated results are compared with the tested results. Finally, the concluding remarks are presented in Section 4.
2. Prestressed concrete column 2.1. Load test [1] The load test conditions are summarized here. Fig. 1 shows the test specimen, the concrete column of which is fixed on a reinforced concrete base. The column section is 30 cm 30 cm. The axial reinforcement consists of 12 D10 rebars (section area ratio ¼ 0:95%). The shear reinforcement D6 rebars are placed at 10 cm intervals (area ratio ¼ 0:21%) in the upper part of the column and at 7.5 cm intervals (area ratio ¼ 0:28%) in the lower part. Each of the two PC bars (/17) is subjected to the pretension of 154 kN without bonding between the PC bar and the concrete. The PC pretension and the constant axial load applied at the top of the column result in
0045-7949/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0045-7949(02)00486-8
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Fig. 1. Test specimen (PC column).
Table 1 Material properties (MPa)
Concrete
Axial rebar Shear rebar PC bar
D10 D6 £ 17
Fig. 2. Finite element model (PC column). Compressive strength
Tensile strength
YoungÕs modulus
34.9
2.71
2.64 104
Yield strength
Tensile strength
YoungÕs modulus
401 347 1225
578 550 1284
1.84 105 1.88 105 1.96 105
axial compressive stresses of 3.4 and 1.0 MPa, respectively, for the column. The horizontal shear load, applied 100 cm above the base, was under displacement control and was statically applied in a cyclic manner. Table 1 shows the material properties determined in the material test. The load test results are presented in Section 2.3 with calculated results. 2.2. Analysis model 2.2.1. Finite element model We analyzed only the column part of the test specimen. Fig. 2 shows the two-dimensional finite element model employed. The concrete was modeled using ninenode isoparametric plane stress elements, the thickness of which is the same as the column width normal to the two-dimensional plane. The rebars were modeled using three-node truss elements, each section area of which equaled the total section area of the rebars placed in parallel in the direction of the column width. The truss elements shared
nodes with the plane elements, which means bond-slipping effects were not considered. All of the nodes on the column baseline were fixed. Each PC bar was modeled using a truss element, the top and bottom nodes of which were connected with the top nodes of the column and fixed at the base position, respectively. Other than the connected/fixed nodes, no part of the truss element shared nodes with the plane elements. Therefore, pretension effect applied to the truss element (as described in Section 2.2.3) was transmitted only through the top of the column and, consequently, compressive prestress was generated in the plane elements. 2.2.2. Material model [4] Figs. 3 and 4 show the uniaxial stress–strain relationship and the biaxial stress failure envelope for the concrete model. The compressive part of the stress– strain relationship was determined based on Specifications for Highway Bridges (in Japan) [5], which included the effect that column concrete was constrained by shear rebars. The compressive stress–strain relationship was modified to increase the strength during the calculation depending on the current biaxial stress state in the compressive failure envelope. The tensile part of the stress–strain relationship included the linear tension stiffening effect after cracking. The crack model can represent the following capabilities: (1) smeared cracking where tensile (and compressive) failure is evaluated at each integration point of a plane element, (2) orthotropic stress–strain relationship using the reduced normal stiffness (reduction
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was applied cyclically), and then the forced displacement was unloaded until it became zero. 2.2.4. Solution strategy Static analysis with the small displacement formulation was performed. The nonlinear equilibrium equations were solved using the ‘‘simple’’ incremental approach with a constant displacement increment of 0.01 mm per load step. This was because the incremental approach with the equilibrium iterations was difficult to converge with larger increments. Fig. 3. Stress–strain relationship for concrete model.
Fig. 4. Stress failure envelope for concrete model.
factor ¼ 0:0001) and the reduced shear stiffness (reduction factor ¼ 0:5) about the cracked direction, (3) fixed initial crack direction during the solution, and (4) crack closing depending on strain normal to the cracked direction. PoissonÕs ratio was 0.167 (constant). For the stress–strain relationship of the rebar, the bilinear elastic–plastic model under the von Mises yield condition was assumed. The PC bar was always linearly elastic.
2.3. Analysis results and comparison with test results 2.3.1. Load–displacement relationship Fig. 5 shows the load–displacement relationship at the loading point, where the line with circles represents the analysis result and the bold line represents the ‘‘modified’’ test result obtained as follows. In the analysis, the base of the column was assumed to be fixed (no displacement), whereas, in the test, column base rotation was observed. Therefore, the measured displacement at the loading point was modified so as to remove the additional displacement caused by the column base rotation. The calculated load–displacement relationship is a result of the following failure sequence: (1) at 1 mm displacement, bending cracks occur on the tension side of the column base; (2) at 3 mm, the tensile rebar yields and then the concrete begins to be crushed on the compression side of the column base; (3) at 6.5 mm, the compressive rebar almost reaches the yield stress. This failure sequence and the load/displacement level at each stage of failure in the analysis are in good agreement with the behavior observed in the test. In addition, the calculated unloading path can represent such characteristics of PC structures that the stiffness of the unloading path is smaller than the initial stiffness
2.2.3. Applied load The pretension applied to the PC bar was modeled using initial strain of truss element based on the following equations: ei ¼ Ni =ðEAÞ;
r ¼ Eðe þ ei Þ;
N ¼ rA;
where ei , Ni , E, A, r, e, and N are initial strain, designed pretension load (154 kN), YoungÕs modulus, section area, axial stress, total strain, and axial force after deformation, respectively, all of which were for the PC bar truss element. After the constant axial load was applied to the top of the column, the prescribed horizontal displacement was applied up to the column deformation angle of 1/ 150 monotonically (though, in the test, the displacement
Fig. 5. Load–displacement relationship.
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and it is oriented toward the origin of the load–displacement relationship. 2.3.2. Concrete cracking and crushing Fig. 6 shows the calculated cracking and crushing extents at various load levels, where the symbols ¼ (two parallel lines), # (orthogonally crossed two parallel lines), and (asterisk) denote a single crack in the parallel direction, double cracks in the orthogonal directions, and crushing in all directions, respectively. The calculated failure sequence is as follows: (1) at 1 mm displacement, the bending cracks occur horizontally on the tension side of the column base; (2) with increasing load, the cracking progresses toward the upper part of the column; (3) at 3 mm, when the cracking reaches 40% of the column height, the tensile rebar yields at the column base, and then the concrete begins to be crushed on the compression side of the column base; (4) at 6.5 mm, the cracking no longer progresses upward but propagates downward through the depth of the column in a slanted direction; (5) at the same time, vertical cracking occurs along the compression side of the column base and crushing at the compressive bottom of the column progresses upward. During unloading, when the loading force becomes zero, 1 mm residual displacement is observed and the cracks appear closed, which is indicated by the smaller distance between the two parallel lines of the symbol ¼ . The calculated failure sequence (Fig. 6) was in good agreement with the tested failure sequence shown in Fig. 7, where the failure pattern appears nearly symmetrical because loading was cyclic in the test. In Fig. 7, the gray zones at the column base indicate where cracked/crushed concrete fell away from the column. Fig. 8 shows the detailed comparison between Figs. 6 and 7 for the most damaged state of the column. The
Fig. 7. Crack and crush distribution (test) caused by cyclic loading.
Fig. 8. Crack and crush distribution in detail.
two failure states are very similar in terms of properties such as the crack extent, the crack directions, and the correspondence of the concrete fallen zones in the test to the vertical cracking/crushing zone in the calculation.
Fig. 6. Crack and crush distribution (analysis) ( ¼ : single crack, #: double cracks, : crush).
2.3.3. Steel strain In Figs. 9–11, the calculated steel strains are compared with the tested strains. Fig. 9 shows the load–strain relationships for the axial rebars at the column base. The tensile rebar undergoes large plastic strain, whereas, the compressive rebar almost reaches the yield strain ()2100l). Fig. 10 shows the load–strain relationships for the PC bars. With increasing load, the initial tensile strain (about 3450l) is increased or decreased at the PC bar on
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is changed into small compressive strain at the first load stage, and then it turns back to tensile strain afterward. According to Figs. 9–11, the calculated strain behavior is in good agreement with the tested strain behavior.
3. Precast segmental prestressed concrete beam 3.1. Load test [2]
Fig. 9. Load–strain relationship of axial rebars.
Fig. 10. Load–strain relationship of PC bars.
The load test conditions are summarized here. Fig. 12 shows the test specimen, which consists of six precast segments with a T-shaped cross section. The segments were assembled into a beam using the pretension of four external cables (/12.7). The contact surfaces of the segments were not processed using any shear key and/or resin layer, which means that the shear force of a segment was transmitted to the adjacent segment only through the friction between the contact surfaces. Each external cable was fixed at both ends of the beam and set through the holes of two concrete deviators placed at the loading positions. The pretension of the cable was 102.9 and 29.4 kN for the upper and the lower cable, respectively, which resulted in the compressive prestress of 14.7 MPa at the midspan bottom face. The axial rebars (D13) were placed at the bottom of the web and at the top of the flange (section area ratio ¼ 0:55%), none of which were connected to the adjacent segments. The shear rebars (D6) were placed at 20 cm intervals (section area ratio ¼ 0:53%) in each segment. The simply supported beam was statically subjected to the vertical load in a cyclic manner at two deviator positions. Table 2 shows the material properties determined in the material test. The load test results are presented in Section 3.3 with calculated results.
Fig. 11. Load–strain relationship of shear rebar.
tension side (left-hand side) or compression side (righthand side). Although the strain is always in the elastic range (<6200l), the unloading paths do not come back to the initial strain positions; this is because the PC bars are connected with the column which undergoes the plastic deformation of concrete and rebars. Fig. 11 shows the load–strain relationships for the shear rebar at the column base. The initial tensile strain
Fig. 12. Test specimen (precast segmental PC beam).
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Table 2 Material properties (MPa)
Concrete
PC cable Rebar
£ 12.7 D6 D13
Compressive strength
Tensile strength
YoungÕs modulus
71.3
4.39
3.23 104
Yield strength
Tensile strength
YoungÕs modulus
1735 347 380
1901 550 555
1.96 105 1.92 105 1.99 105
3.2. Analysis model 3.2.1. Finite element model Fig. 13 shows the two-dimensional finite element model employed. Only the right half (three segments) of the beam was modeled because of the symmetry condition at midspan. The concrete was modeled using ninenode isoparametric plane stress elements with 6 cm thickness at the web and 30 cm thickness at the flange and the deviator. The contact surface calculation [6] enabled the transmission of the pressure and friction forces at the point of contact and no force at the separated parts between two adjacent segments. The friction coefficient was 0.3, which was determined from the ratio of the shear load to the pretension load in the test when adjacent segments began to move relative to each other. For the symmetry boundary condition at midspan, a fixed contact surface (rigid wall) was placed, which was used to evaluate contact conditions for the contact surface of the segment at midspan. The four cables in the test were modeled using two truss elements, each of which had the total section area of two cables. The rightmost node of the truss element (marked ‘‘A’’ in Fig. 13) was connected to the nearest concrete node of the beam end so that the corresponding nodes had the same displacements. The node of the truss element at the deviator position (marked ‘‘B’’ in Fig. 13) was connected to the nearest concrete node of the deviator so that the corresponding nodes had the same displacements in the vertical direction but were inde-
Fig. 13. Finite element model (precast segmental PC beam).
pendent in the horizontal direction, which enabled the cable to move freely through the deviator hole without friction. The pretension effect applied to the cable truss element (as described in Section 3.2.3) pressed the rightmost segment at the position ‘‘A’’ in Fig. 13; the compression force was transmitted to the leftward segments through the contact forces between the adjacent segments; consequently, compressive prestress was generated in the beam. The axial and shear rebars were modeled using twonode truss elements, each section area of which had the total section area of the rebars placed in the direction of the beam width. The truss elements shared nodes with the plane elements, which means bond-slipping effects were not considered. 3.2.2. Material model [4] Fig. 14 shows the uniaxial stress–strain relationship for the concrete model, the compressive part of which was determined based on Specifications for Highway Bridges (in Japan) [5], and the tensile part of which included the linear tension stiffening effect after cracking. All of the other conditions for concrete and steel material models were the same as described in Section 2.2.2. 3.2.3. Applied load The pretension applied to the cable was modeled using initial strain of the truss element, as described in Section 2.2.3. After the pretension load was applied to the cable truss elements, the vertical shear load was applied monotonically, though, in the test, the load was applied cyclically. 3.2.4. Solution strategy Static analysis using the large displacement formulation was performed. This means that geometrical nonlinearity was taken into consideration in this analysis, where stresses were calculated based on the updated coordinates of elements; for example, the axial force of
Fig. 14. Stress–strain relationship for concrete model.
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the cable truss element was calculated in the displaced current (not initial) position of the cable. The nonlinear equilibrium equations were solved using the incremental approach with the equilibrium iterations, where the constant load increment was 1.0 kN per load step. 3.3. Analysis results and comparison with test results 3.3.1. Load–displacement relationship Fig. 15a and b shows the calculated and tested deformations, respectively, at the maximum load level. Gaps between the segments are observed at the bottom face of the beam. The gap at midspan is the largest (13 mm at maximum load) and the other gaps become smaller with distance from midspan. In Fig. 15a, the segment at midspan is in contact with the boundary wall (vertical line ––) only through the upper part of the flange; similarly, in Fig. 15b, the two segments at midspan are in contact with each other only through the upper part of the flanges. Fig. 16 shows the relationship between the full-span shear load and the vertical displacement at the midspan bottom face, where the line with circles corresponds to the calculated result and the bold line to the test result. Around the load of 80 kN, the axial compressive stresses at the bottom face of the beam caused by the cable pretension are released by the axial tensile stresses
Fig. 16. Load–displacement relationship.
caused by the beam bending moment, and then gap openings begin to develop. Above that load level, the stiffness is reduced and becomes nearly constant because the gap openings no longer develop. Around the displacement of 25 mm in the calculation, the equilibrium iterations did not converge and the solution could not be obtained; correspondingly, around the displacement of 27 mm in the test, the midspan top face was crushed and measurement became impossible because of rapid failure of the beam. The above mentioned load–displacement behavior obtained from the calculation is in good agreement with the behavior observed in the test. 3.3.2. Concrete cracking and crushing Figs. 17 and 18 show the cracking and crushing distribution at the final state in the calculation and the test, respectively. The symbols used in Fig. 17, ¼ (two parallel lines), # (orthogonally crossed two parallel lines), and (asterisk), denote a single crack in the parallel direction, double cracks in the orthogonal directions, and crushing in all directions, respectively. The calculated failure sequence is as follows: (1) cracks occur horizontally at the upper part of the web (below the flange) adjacent to the midspan boundary; (2) cracking progresses at the flange adjacent to the midspan boundary and the right web of the midspan segment; (3) at the maximum load level, crushing occurs at the top
Fig. 15. (a) Deformation (analysis) (displacement magnification factor ¼ 10) and (b) deformation at midspan (test).
Fig. 17. Crack and crush distribution (analysis) ( ¼ : single crack, #: double cracks, : crush).
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Fig. 18. Crack and crush distribution (test).
cables, where the lines with symbols represent the calculated results and the bold lines the test results. The lines are always in the elastic range (<8900l). The relationship is nearly bilinear, which corresponds to the bilinear load–displacement relationship shown in Fig. 16. The calculated results are similar to the tested results except for the initial strain difference for the upper cable (the difference arose from the 10% reduction of the initial strain in the test from the designed pretension value). 4. Conclusions
Fig. 19. Load–strain relationship of concrete.
face adjacent to the midspan boundary, which corresponds to crushing observed in Fig. 15b. Most of the cracks are concentrated at the midspan segment only, whereas very few (or no) cracks are observed at the other segments. The calculated failure sequence was very similar to the tested failure sequence. Fig. 19 shows the relationship between the full-span load and the axial strain at the top face of the midspan segment, where the calculated maximum strain proceeds beyond the crushing strain ()3100l in Fig. 14). 3.3.3. Cable strain The cable strain was uniform throughout each cable in the calculation and the test. Fig. 20 shows the relationship between the full-span load and the strain for the
Fig. 20. Load–strain relationship of cables.
We analyzed a PC column and a precast segmental PC beam using a finite element method including material nonlinearities (concrete cracking/crushing and rebar plasticity) and geometrical nonlinearities (large displacement and contact/separation between the segments). The calculated results were in good agreement with the tested results in terms of properties such as the load–displacement relationship, the concrete cracking/ crushing process, and the load–strain relationship for the rebars and the PC bars/cables. Therefore, the analysis models we employed here were found to be very effective for gaining detailed knowledge of the nonlinear behavior of PC members. Acknowledgements The authors gratefully acknowledge Professor Shoji Ikeda (Yokohama National University) for lead and advice on our study. This paper is a slightly modified version of a paper presented at the first fib Congress in Osaka, Japan on October 13–19, 2002. References [1] Ito T, Yamaguchi T, Ikeda S. Seismic performance of reinforced concrete piers prestressed in axial direction. In: Japan Concrete Institute Annual Conference Proceedings on Concrete Research and Technology. vol. 19(2), 1997, p. 1197–202. [2] Ito T, Yamaguchi T, Ikeda S. Experimental study on flexural shear properties of precast segmental beams. J Prestressed Concrete, Jpn 1997;39(1):83–96. [3] ADINA R&D, Inc. ADINA theory and modeling guide. Report ARD97–7, 1997. [4] Bathe KJ, Walczak J, Welch A, Mistry N. Nonlinear analysis of concrete structures. Comput Struct 1989;32(3/4): 563–90. [5] Japan Road Association. Specifications for Highway Bridges, 1996, Part V Seismic design. [6] Bathe KJ, Chaudhary A. A solution method for planar and axisymmetric contact problems. Int J Numer Meth Eng 1985; (21):65–88.
Nonlinear finite element analysis of prestressed concrete members using ADINA Makoto Kawakami
a,*
, Tadahiko Ito
b
a
b
Kozo Keikaku Engineering Inc., 4-5-3 Chuo, Nakano, Tokyo, Japan Nishimatsu Construction Co., Ltd., 2570 Shimotsuruma, Yamato, Kanagawa, Japan
Abstract A prestressed concrete (PC) column and a precast segmental PC beam are analyzed using ADINA. Two dimensional finite element model is employed with material nonlinearities (concrete cracking/crushing and rebar plasticity) and geometrical nonlinearities (large displacement and contact/separation between the segments). The load-displacement relationship, the concrete cracking/crushing process, the load-strain relationship, and the gap opening/closing are in good agreement with those in the tests. Ó 2003 Elsevier Science Ltd. All rights reserved.
1. Introduction In recent years, prestressed concrete (PC) columns and precast segmental PC beams have been studied and designed because of their excellent mechanical characteristics and their practical advantages in construction. In the design of structures using such PC members, it is very important to predict their maximum strength, deformation performance, and material failure process in order to evaluate the reliability and safety of the structures. The prediction must take into consideration various mechanical nonlinearities, namely, (1) material nonlinearities arising from cracking/crushing of concrete, plasticity of reinforcing steel bars (rebars), and bond slipping between rebars and concrete, and (2) geometrical nonlinearities arising from large displacement of concrete members and PC bars/cables, and gap opening/closing between concrete segments. From the point of view mentioned above, we performed the nonlinear finite element analysis of PC members based on two load test results: [1] for a PC column which included material nonlinearities, and [2] for a pecast segmental PC beam which included geometrical nonlinearities as well as material nonlinearities.
*
Corresponding author.
For the calculation, we used the general-purpose finite element code ADINA [3]. The analyses of the PC column and the PC beam are discussed in Sections 2 and 3, respectively, in each of which, the load test is summarized and the calculated results are compared with the tested results. Finally, the concluding remarks are presented in Section 4.
2. Prestressed concrete column 2.1. Load test [1] The load test conditions are summarized here. Fig. 1 shows the test specimen, the concrete column of which is fixed on a reinforced concrete base. The column section is 30 cm 30 cm. The axial reinforcement consists of 12 D10 rebars (section area ratio ¼ 0:95%). The shear reinforcement D6 rebars are placed at 10 cm intervals (area ratio ¼ 0:21%) in the upper part of the column and at 7.5 cm intervals (area ratio ¼ 0:28%) in the lower part. Each of the two PC bars (/17) is subjected to the pretension of 154 kN without bonding between the PC bar and the concrete. The PC pretension and the constant axial load applied at the top of the column result in
0045-7949/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0045-7949(02)00486-8
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M. Kawakami, T. Ito / Computers and Structures 81 (2003) 727–734
Fig. 1. Test specimen (PC column).
Table 1 Material properties (MPa)
Concrete
Axial rebar Shear rebar PC bar
D10 D6 £ 17
Fig. 2. Finite element model (PC column). Compressive strength
Tensile strength
YoungÕs modulus
34.9
2.71
2.64 104
Yield strength
Tensile strength
YoungÕs modulus
401 347 1225
578 550 1284
1.84 105 1.88 105 1.96 105
axial compressive stresses of 3.4 and 1.0 MPa, respectively, for the column. The horizontal shear load, applied 100 cm above the base, was under displacement control and was statically applied in a cyclic manner. Table 1 shows the material properties determined in the material test. The load test results are presented in Section 2.3 with calculated results. 2.2. Analysis model 2.2.1. Finite element model We analyzed only the column part of the test specimen. Fig. 2 shows the two-dimensional finite element model employed. The concrete was modeled using ninenode isoparametric plane stress elements, the thickness of which is the same as the column width normal to the two-dimensional plane. The rebars were modeled using three-node truss elements, each section area of which equaled the total section area of the rebars placed in parallel in the direction of the column width. The truss elements shared
nodes with the plane elements, which means bond-slipping effects were not considered. All of the nodes on the column baseline were fixed. Each PC bar was modeled using a truss element, the top and bottom nodes of which were connected with the top nodes of the column and fixed at the base position, respectively. Other than the connected/fixed nodes, no part of the truss element shared nodes with the plane elements. Therefore, pretension effect applied to the truss element (as described in Section 2.2.3) was transmitted only through the top of the column and, consequently, compressive prestress was generated in the plane elements. 2.2.2. Material model [4] Figs. 3 and 4 show the uniaxial stress–strain relationship and the biaxial stress failure envelope for the concrete model. The compressive part of the stress– strain relationship was determined based on Specifications for Highway Bridges (in Japan) [5], which included the effect that column concrete was constrained by shear rebars. The compressive stress–strain relationship was modified to increase the strength during the calculation depending on the current biaxial stress state in the compressive failure envelope. The tensile part of the stress–strain relationship included the linear tension stiffening effect after cracking. The crack model can represent the following capabilities: (1) smeared cracking where tensile (and compressive) failure is evaluated at each integration point of a plane element, (2) orthotropic stress–strain relationship using the reduced normal stiffness (reduction
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was applied cyclically), and then the forced displacement was unloaded until it became zero. 2.2.4. Solution strategy Static analysis with the small displacement formulation was performed. The nonlinear equilibrium equations were solved using the ‘‘simple’’ incremental approach with a constant displacement increment of 0.01 mm per load step. This was because the incremental approach with the equilibrium iterations was difficult to converge with larger increments. Fig. 3. Stress–strain relationship for concrete model.
Fig. 4. Stress failure envelope for concrete model.
factor ¼ 0:0001) and the reduced shear stiffness (reduction factor ¼ 0:5) about the cracked direction, (3) fixed initial crack direction during the solution, and (4) crack closing depending on strain normal to the cracked direction. PoissonÕs ratio was 0.167 (constant). For the stress–strain relationship of the rebar, the bilinear elastic–plastic model under the von Mises yield condition was assumed. The PC bar was always linearly elastic.
2.3. Analysis results and comparison with test results 2.3.1. Load–displacement relationship Fig. 5 shows the load–displacement relationship at the loading point, where the line with circles represents the analysis result and the bold line represents the ‘‘modified’’ test result obtained as follows. In the analysis, the base of the column was assumed to be fixed (no displacement), whereas, in the test, column base rotation was observed. Therefore, the measured displacement at the loading point was modified so as to remove the additional displacement caused by the column base rotation. The calculated load–displacement relationship is a result of the following failure sequence: (1) at 1 mm displacement, bending cracks occur on the tension side of the column base; (2) at 3 mm, the tensile rebar yields and then the concrete begins to be crushed on the compression side of the column base; (3) at 6.5 mm, the compressive rebar almost reaches the yield stress. This failure sequence and the load/displacement level at each stage of failure in the analysis are in good agreement with the behavior observed in the test. In addition, the calculated unloading path can represent such characteristics of PC structures that the stiffness of the unloading path is smaller than the initial stiffness
2.2.3. Applied load The pretension applied to the PC bar was modeled using initial strain of truss element based on the following equations: ei ¼ Ni =ðEAÞ;
r ¼ Eðe þ ei Þ;
N ¼ rA;
where ei , Ni , E, A, r, e, and N are initial strain, designed pretension load (154 kN), YoungÕs modulus, section area, axial stress, total strain, and axial force after deformation, respectively, all of which were for the PC bar truss element. After the constant axial load was applied to the top of the column, the prescribed horizontal displacement was applied up to the column deformation angle of 1/ 150 monotonically (though, in the test, the displacement
Fig. 5. Load–displacement relationship.
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and it is oriented toward the origin of the load–displacement relationship. 2.3.2. Concrete cracking and crushing Fig. 6 shows the calculated cracking and crushing extents at various load levels, where the symbols ¼ (two parallel lines), # (orthogonally crossed two parallel lines), and (asterisk) denote a single crack in the parallel direction, double cracks in the orthogonal directions, and crushing in all directions, respectively. The calculated failure sequence is as follows: (1) at 1 mm displacement, the bending cracks occur horizontally on the tension side of the column base; (2) with increasing load, the cracking progresses toward the upper part of the column; (3) at 3 mm, when the cracking reaches 40% of the column height, the tensile rebar yields at the column base, and then the concrete begins to be crushed on the compression side of the column base; (4) at 6.5 mm, the cracking no longer progresses upward but propagates downward through the depth of the column in a slanted direction; (5) at the same time, vertical cracking occurs along the compression side of the column base and crushing at the compressive bottom of the column progresses upward. During unloading, when the loading force becomes zero, 1 mm residual displacement is observed and the cracks appear closed, which is indicated by the smaller distance between the two parallel lines of the symbol ¼ . The calculated failure sequence (Fig. 6) was in good agreement with the tested failure sequence shown in Fig. 7, where the failure pattern appears nearly symmetrical because loading was cyclic in the test. In Fig. 7, the gray zones at the column base indicate where cracked/crushed concrete fell away from the column. Fig. 8 shows the detailed comparison between Figs. 6 and 7 for the most damaged state of the column. The
Fig. 7. Crack and crush distribution (test) caused by cyclic loading.
Fig. 8. Crack and crush distribution in detail.
two failure states are very similar in terms of properties such as the crack extent, the crack directions, and the correspondence of the concrete fallen zones in the test to the vertical cracking/crushing zone in the calculation.
Fig. 6. Crack and crush distribution (analysis) ( ¼ : single crack, #: double cracks, : crush).
2.3.3. Steel strain In Figs. 9–11, the calculated steel strains are compared with the tested strains. Fig. 9 shows the load–strain relationships for the axial rebars at the column base. The tensile rebar undergoes large plastic strain, whereas, the compressive rebar almost reaches the yield strain ()2100l). Fig. 10 shows the load–strain relationships for the PC bars. With increasing load, the initial tensile strain (about 3450l) is increased or decreased at the PC bar on
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is changed into small compressive strain at the first load stage, and then it turns back to tensile strain afterward. According to Figs. 9–11, the calculated strain behavior is in good agreement with the tested strain behavior.
3. Precast segmental prestressed concrete beam 3.1. Load test [2]
Fig. 9. Load–strain relationship of axial rebars.
Fig. 10. Load–strain relationship of PC bars.
The load test conditions are summarized here. Fig. 12 shows the test specimen, which consists of six precast segments with a T-shaped cross section. The segments were assembled into a beam using the pretension of four external cables (/12.7). The contact surfaces of the segments were not processed using any shear key and/or resin layer, which means that the shear force of a segment was transmitted to the adjacent segment only through the friction between the contact surfaces. Each external cable was fixed at both ends of the beam and set through the holes of two concrete deviators placed at the loading positions. The pretension of the cable was 102.9 and 29.4 kN for the upper and the lower cable, respectively, which resulted in the compressive prestress of 14.7 MPa at the midspan bottom face. The axial rebars (D13) were placed at the bottom of the web and at the top of the flange (section area ratio ¼ 0:55%), none of which were connected to the adjacent segments. The shear rebars (D6) were placed at 20 cm intervals (section area ratio ¼ 0:53%) in each segment. The simply supported beam was statically subjected to the vertical load in a cyclic manner at two deviator positions. Table 2 shows the material properties determined in the material test. The load test results are presented in Section 3.3 with calculated results.
Fig. 11. Load–strain relationship of shear rebar.
tension side (left-hand side) or compression side (righthand side). Although the strain is always in the elastic range (<6200l), the unloading paths do not come back to the initial strain positions; this is because the PC bars are connected with the column which undergoes the plastic deformation of concrete and rebars. Fig. 11 shows the load–strain relationships for the shear rebar at the column base. The initial tensile strain
Fig. 12. Test specimen (precast segmental PC beam).
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Table 2 Material properties (MPa)
Concrete
PC cable Rebar
£ 12.7 D6 D13
Compressive strength
Tensile strength
YoungÕs modulus
71.3
4.39
3.23 104
Yield strength
Tensile strength
YoungÕs modulus
1735 347 380
1901 550 555
1.96 105 1.92 105 1.99 105
3.2. Analysis model 3.2.1. Finite element model Fig. 13 shows the two-dimensional finite element model employed. Only the right half (three segments) of the beam was modeled because of the symmetry condition at midspan. The concrete was modeled using ninenode isoparametric plane stress elements with 6 cm thickness at the web and 30 cm thickness at the flange and the deviator. The contact surface calculation [6] enabled the transmission of the pressure and friction forces at the point of contact and no force at the separated parts between two adjacent segments. The friction coefficient was 0.3, which was determined from the ratio of the shear load to the pretension load in the test when adjacent segments began to move relative to each other. For the symmetry boundary condition at midspan, a fixed contact surface (rigid wall) was placed, which was used to evaluate contact conditions for the contact surface of the segment at midspan. The four cables in the test were modeled using two truss elements, each of which had the total section area of two cables. The rightmost node of the truss element (marked ‘‘A’’ in Fig. 13) was connected to the nearest concrete node of the beam end so that the corresponding nodes had the same displacements. The node of the truss element at the deviator position (marked ‘‘B’’ in Fig. 13) was connected to the nearest concrete node of the deviator so that the corresponding nodes had the same displacements in the vertical direction but were inde-
Fig. 13. Finite element model (precast segmental PC beam).
pendent in the horizontal direction, which enabled the cable to move freely through the deviator hole without friction. The pretension effect applied to the cable truss element (as described in Section 3.2.3) pressed the rightmost segment at the position ‘‘A’’ in Fig. 13; the compression force was transmitted to the leftward segments through the contact forces between the adjacent segments; consequently, compressive prestress was generated in the beam. The axial and shear rebars were modeled using twonode truss elements, each section area of which had the total section area of the rebars placed in the direction of the beam width. The truss elements shared nodes with the plane elements, which means bond-slipping effects were not considered. 3.2.2. Material model [4] Fig. 14 shows the uniaxial stress–strain relationship for the concrete model, the compressive part of which was determined based on Specifications for Highway Bridges (in Japan) [5], and the tensile part of which included the linear tension stiffening effect after cracking. All of the other conditions for concrete and steel material models were the same as described in Section 2.2.2. 3.2.3. Applied load The pretension applied to the cable was modeled using initial strain of the truss element, as described in Section 2.2.3. After the pretension load was applied to the cable truss elements, the vertical shear load was applied monotonically, though, in the test, the load was applied cyclically. 3.2.4. Solution strategy Static analysis using the large displacement formulation was performed. This means that geometrical nonlinearity was taken into consideration in this analysis, where stresses were calculated based on the updated coordinates of elements; for example, the axial force of
Fig. 14. Stress–strain relationship for concrete model.
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the cable truss element was calculated in the displaced current (not initial) position of the cable. The nonlinear equilibrium equations were solved using the incremental approach with the equilibrium iterations, where the constant load increment was 1.0 kN per load step. 3.3. Analysis results and comparison with test results 3.3.1. Load–displacement relationship Fig. 15a and b shows the calculated and tested deformations, respectively, at the maximum load level. Gaps between the segments are observed at the bottom face of the beam. The gap at midspan is the largest (13 mm at maximum load) and the other gaps become smaller with distance from midspan. In Fig. 15a, the segment at midspan is in contact with the boundary wall (vertical line ––) only through the upper part of the flange; similarly, in Fig. 15b, the two segments at midspan are in contact with each other only through the upper part of the flanges. Fig. 16 shows the relationship between the full-span shear load and the vertical displacement at the midspan bottom face, where the line with circles corresponds to the calculated result and the bold line to the test result. Around the load of 80 kN, the axial compressive stresses at the bottom face of the beam caused by the cable pretension are released by the axial tensile stresses
Fig. 16. Load–displacement relationship.
caused by the beam bending moment, and then gap openings begin to develop. Above that load level, the stiffness is reduced and becomes nearly constant because the gap openings no longer develop. Around the displacement of 25 mm in the calculation, the equilibrium iterations did not converge and the solution could not be obtained; correspondingly, around the displacement of 27 mm in the test, the midspan top face was crushed and measurement became impossible because of rapid failure of the beam. The above mentioned load–displacement behavior obtained from the calculation is in good agreement with the behavior observed in the test. 3.3.2. Concrete cracking and crushing Figs. 17 and 18 show the cracking and crushing distribution at the final state in the calculation and the test, respectively. The symbols used in Fig. 17, ¼ (two parallel lines), # (orthogonally crossed two parallel lines), and (asterisk), denote a single crack in the parallel direction, double cracks in the orthogonal directions, and crushing in all directions, respectively. The calculated failure sequence is as follows: (1) cracks occur horizontally at the upper part of the web (below the flange) adjacent to the midspan boundary; (2) cracking progresses at the flange adjacent to the midspan boundary and the right web of the midspan segment; (3) at the maximum load level, crushing occurs at the top
Fig. 15. (a) Deformation (analysis) (displacement magnification factor ¼ 10) and (b) deformation at midspan (test).
Fig. 17. Crack and crush distribution (analysis) ( ¼ : single crack, #: double cracks, : crush).
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Fig. 18. Crack and crush distribution (test).
cables, where the lines with symbols represent the calculated results and the bold lines the test results. The lines are always in the elastic range (<8900l). The relationship is nearly bilinear, which corresponds to the bilinear load–displacement relationship shown in Fig. 16. The calculated results are similar to the tested results except for the initial strain difference for the upper cable (the difference arose from the 10% reduction of the initial strain in the test from the designed pretension value). 4. Conclusions
Fig. 19. Load–strain relationship of concrete.
face adjacent to the midspan boundary, which corresponds to crushing observed in Fig. 15b. Most of the cracks are concentrated at the midspan segment only, whereas very few (or no) cracks are observed at the other segments. The calculated failure sequence was very similar to the tested failure sequence. Fig. 19 shows the relationship between the full-span load and the axial strain at the top face of the midspan segment, where the calculated maximum strain proceeds beyond the crushing strain ()3100l in Fig. 14). 3.3.3. Cable strain The cable strain was uniform throughout each cable in the calculation and the test. Fig. 20 shows the relationship between the full-span load and the strain for the
Fig. 20. Load–strain relationship of cables.
We analyzed a PC column and a precast segmental PC beam using a finite element method including material nonlinearities (concrete cracking/crushing and rebar plasticity) and geometrical nonlinearities (large displacement and contact/separation between the segments). The calculated results were in good agreement with the tested results in terms of properties such as the load–displacement relationship, the concrete cracking/ crushing process, and the load–strain relationship for the rebars and the PC bars/cables. Therefore, the analysis models we employed here were found to be very effective for gaining detailed knowledge of the nonlinear behavior of PC members. Acknowledgements The authors gratefully acknowledge Professor Shoji Ikeda (Yokohama National University) for lead and advice on our study. This paper is a slightly modified version of a paper presented at the first fib Congress in Osaka, Japan on October 13–19, 2002. References [1] Ito T, Yamaguchi T, Ikeda S. Seismic performance of reinforced concrete piers prestressed in axial direction. In: Japan Concrete Institute Annual Conference Proceedings on Concrete Research and Technology. vol. 19(2), 1997, p. 1197–202. [2] Ito T, Yamaguchi T, Ikeda S. Experimental study on flexural shear properties of precast segmental beams. J Prestressed Concrete, Jpn 1997;39(1):83–96. [3] ADINA R&D, Inc. ADINA theory and modeling guide. Report ARD97–7, 1997. [4] Bathe KJ, Walczak J, Welch A, Mistry N. Nonlinear analysis of concrete structures. Comput Struct 1989;32(3/4): 563–90. [5] Japan Road Association. Specifications for Highway Bridges, 1996, Part V Seismic design. [6] Bathe KJ, Chaudhary A. A solution method for planar and axisymmetric contact problems. Int J Numer Meth Eng 1985; (21):65–88.