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A STUDY ON DESIGN METHOD OF SHEAR BUCKLING AND BENDING MOMENT FOR PRESTRESSED CONCRETE BRIDGES WITH CORRUGATED STEEL WEBS Hiroyuki Ikeda Bridge and structural Engineering Division, Chubu Branch, Japan Highway Public Corporation Aichi, JAPAN Kenichirou Ashiduka Bridge and Suructural Engineering Division, Japan Highway Public Corporation Tokyo, JAPAN Toshimichi Ichinomiya Civil Engineering Department, Kajima Technical Research Institute Tokyo , JAPAN Yoshihide Okimi I.T. Solutions Department, Kajima Corporation Tokyo, JAPAN Toru Yamamoto Nagoya Branch, Kajima Corporation Aichi, JAPAN Masato Kano Research and Development Division, Bridge and Computer Engineering Co., Ltd. Osaka , JAPAN Keywords: shear buckling, bending moment, strain distribution, finite element analysis 1 INTRODUCTION Prestressed Concrete box girders with corrugated steel webs were put to practical use in France as alternatives with lighter weight to conventional prestressed concrete box girders. They have recently been used in Japan. Shear and flexural behavior of bridges with prestressed concrete girders with corrugated steel webs has been studied by basic tests and analyses and taken into consideration in actual design [1]. Shear buckling of corrugated steel webs has been checked using a formula for buckling strength that is based on Easley's formula and takes the inelastic range into consideration. Prestressed concrete girders with corrugated steel webs have been applied to medium spans around 80 m, so shear buckling has not been of predominant concern in design. When the depth of the girder affected by shear buckling of the web was large, the web exceeding 5 m in depth that is found on the existing bridges has been lined with concrete to prevent buckling. Effectively treating the cross section of girder that is expected to increase with the increase of bridge length demands greater accuracy of shear buckling analysis. For designing prestressed concrete bridges with corrugated steel webs considering bending moments and axial forces, only top and bottom concrete slabs of the girder are considered as the cross sections effective in resisting the forces while ignoring webs, and the Bernoulli's assumption that plane sections remain plane after bending is loaded. The Bernoulli's assumption is considered applicable because the strain distribution obtained in loading tests on specimens and actual bridges is close to 285 Composite structures Session 5 that obtained by calculation based on the assumption. Most of the tests conducted so far involved shear. It was therefore considered necessary to measure strain distribution using specimens with sufficiently long sections subject to pure bending to verify the applicability of the Bernoulli's assumption. In this study, therefore, the applicability of analysis considering both material nonlinearity and geometric was verified for accurately predicting shear buckling strength. Strain distribution on the top and bottom concrete slabs was measured using specimens with sections subject to pure bending, and the validity of the Bernoulli's assumption when bending was predominant was verified. 2 STUDY ON SHEAR BUCKLING 2.1 Specimen Efforts have been made to accurately predict shear buckling by analysis considering both material and geometric nonlinearity [2]. Studies have been made in the cases where shear yielding preceded shear buckling and where buckling occurred in the inelastic range. Specifications of the specimens used in the past test are shown in Table 1. Buckling strength considering the inelastic range is shown in Fig. 1. The specimen with a girder depth of 1.2 m and a wave height of 20 mm buckled in the inelastic range and the one with a wave height of 60 mm buckled after shear yielding occurred. In both cases, material nonlinearity had a great effect. For making composite nonlinear analysis to check the shear buckling of corrugated steel webs, verification of its validity in the range of high geometric nonlinearity is considered necessary. Then, the applicability of composite nonlinear analysis was examined in the case of buckling in the elastic range, which was greatly affected by geometric nonlinearity. Buckling in the elastic range could be caused by reducing the wave height or increasing the girder depth. The girder depth was increased because reduction of the wave height was likely to cause tests to be affected by fabrication accuracy. Table 1 Specifications of the specimens used in the past test Height Thickness Wave length Wave height (mm) (mm) (mm) (mm) No.1-1 [2] 20 No.1-2 [2) 30 No.1-3 [2) 60 1200 3.2 400 0.00 0.50 1.00 1.50 0.00 0.50 1.00 1.50 2.00 2.50 Shear buckling param eter λs Pu/Py Girder heig h t 1 .2 m W ave height 60m m ●：E xperiment △：A nalysis Grider height 1.2m W ave height 20m m Girder height 2.1m W ave height 20m m Pu/Py＝1／λs2 Yield fa ilu re Ineralsitc b u c k lin g E lastic buckling Girder heig h t 1 .2 m W ave height 30m m Fig.1 Shear buckling strength curve Proceedings of the 1st fib Congress 286 Analysis was made of the specimens used in the past tests and another specimen of greater girder depth using various analytical methods (Fig. 2). For the specimen with a girder depth of 1.2 m and a wave height of 60 mm, the results of composite nonlinear and material nonlinear analyses formed the same line. The specimen buckled at the range where the load became nearly constant after yielding occurred. This shows the predominance of material nonlinearity. The specimen with a girder depth of 2.1 m and a wave height of 20 mm, on the other hand, buckled before the effect of material nonlinearity became apparent, which indicates a predominant influence of geometric nonlinearity. The specimen with a girder depth of 1.2 m and a wave height of 20 mm buckled after the effect of material nonlinearity became apparent. Its behavior is at midway between the behavior of the above two specimens. As a result, in this study, a specimen with a girder depth of 2.1 m, a wave height of 20 mm, a wave length of 400 mm, a web thickness of 3.2 mm and a span of 4.2 m was tested. Fig. 3 shows the specimen. 2.2 Test The specimen was supported on bearings at both ends which allowing free rotation and longitudinal sliding, and load was applied at midspan by a hydraulic jack (Fig. 4). The load point and supports were reinforced by ribs with a thickness of 22 mm. In order to prevent the lateral buckling of the specimen, lateral displacement of the specimen was restrained at top and bottom ends of the supports and at the 0 1000 2000 3000 4000 5000 0102030 Vertical displacement (mm) Load (kN) 0 500 1000 1500 2000 2500 0 5 10 15 Vertical displacement (mm) Load (kN) (a) Girder height 2100mm, wave height 20mm 0 500 1000 1500 2000 2500 0 2.5 5 7.5 10 Vertical displacement (mm) Load (kN) Complicated nonlinearity Material nonlinearity Geometric nonlinearity Linear (b) Girder height 1200mm, wave height (c) Girder height 1200mm, wave height Fig.2 Parametric analysis 287 Composite structures Session 5 load point by roller bearings. The load and vertical displacement were measured by the load cell and displacement meter, respectively. Another displacement gauge was placed on the web of the specimen to measure out-of-plane deformation. The specimen was expected to be affected by its initial shape because it was designed to buckle in the elastic range, so its initial shape was measured. Initial shape of the specimen was measured at positions vertically placed at a spacing of 150 mm and at two points each 20 mm away from the center of the panel of the web that was parallel to the span of the specimen. 2.3 Analysis For analyzing the shear buckling of corrugated steel webs, "SLAP" [3], a complicated nonlinear analysis program, was used. A panel was divided into four elements horizontally. A nearly square mesh was formed. The measurements for the initial shape of the specimen described above were input as nodal coordinates of the analytical model. The results of a material test using test pieces obtained from the same production lot that provided the steel plates of the specimen, and the stress-strain curve for the analytical model are shown in Fig. 5. The load-displacement curve was found by prior studies to be greatly affected by the stress-strain relationship of the steel plate, so a multi-linear model was used to accurately reproduce material properties. 2.4 Test and analytical results Table 2 shows the values of buckling loads obtained by analysis and test. Fig. 6 shows the relationship between the load and vertical displacement. The analytical values deviated from the test values by 10% or less. Accurate prediction of shear buckling was verified. The relationship between the load and vertical displacement could be analyzed slightly more accurately when the initial shape was taken into consideration. The variance was, however, small. For specimen No. 2-3, loading was continued 20002000 2100 210019 19 4200 180 20 400 10110199 99 3,2 20 400 10110199 99 3,2 20 400 10110199 99 3,2 Fig.3 Dimensions of specimen F8-1 試験体 Fig.4 Loading method 0 50 100 150 200 250 300 350 0 1000 2000 3000 Strrain (×10-6) Stress (MPa) Material test Analytical model Fig.5 Stress-strain curve model Unit (mm) Proceedings of the 1st fib Congress 288 after buckling. Post-buckling strength was found to be about 80% of the maximum buckling strength. Fig. 7 shows the relationship between the load and out-of-plane displacement. Fig. 8 shows the distribution of out-of-plane displacement. According to these figures, it was verified that out-of-plane deformation that rules buckling can be accurately analyzed by faithfully considering the initial shape of the specimen. Fig. 9 is a diagram of out-of-plane deformation obtained by analysis under the maximum load. Photograph 1 shows buckling encountered during the test. It was known that specimens were subject to local buckling when they were made inaccurately because they had a small plate thickness. The specimen used in the test suffered total buckling on one side, which corresponded to the deformation identified by analysis. Table 2 Specifications of the specimens and buckling load With initial imperfection Without initial imperfection No.1-1 [2] 20 907 894 911 No.1-3 [2] 60 1153 1172 1229 No.2-1 998 1003 No.2-2 941 1020 No.2-3 995 1087 Wave heightHeight Thickness Wave length 20 1085 Dimension of web (mm) Analysis Buckling load (kN) 1200 2100 3.2 400 Experiment (a) Specimen No.2-1 0 200 400 600 800 1000 1200 0.0 2.0 4.0 6.0 Vertical displacement （ mm ） Load (kN) With initial imperfection Without initial imperfection Experiment (b) Specimen No.2-3 0 200 400 600 800 1000 1200 0.0 2.0 4.0 6.0 8.0 10.0 12.0 Vertical displacement （ mm ） Load (kN) With initial imperfection Without initial imperfection Experiment Fig.6 Load-vertical displacement relationship 0 200 400 600 800 1000 1200 -20 -10 0 10 20 30 Transverse dispacement (mm) Load (kN) With initial imperfection Without initial imperfection Experiment 0 0.35 0.7 1.05 1.4 1.75 2.1 -5 0 5 10 15 Transverse displacement (mm) Height of specimen (mm) With initial imperfection Without initial imperfection Experiment Fig.7 Load-lateral displacement relationship Fig.8 Lateral displacement distribution 289 Composite structures Session 5 3 STUDY ON BENDING MOMENTS AND AXIAL FORCES 3.1 Specimen The specifications of the specimen and its dimensions are shown Fig. 10. For the corrugated steel web, a 3.2-mm-thick steel plate was used. The height of the corrugated steel web was set at 1200 mm. Two variations of wave height, 30 mm and 60 mm, were applied. Concrete slabs with thickness of 250 mm thick and width of 800 mm were combined into the corrugated steel web by angled shear connectors. The total depth of the girder including the concrete slabs was 1.7 m. The shear span was set at 3.4 m, double the total girder depth. The length of the pure bending section was set at 5 m, about three times the girder depth. Fig.9 Deformation at maximum load (No.2-1) Photo. 1 Specimen after buckling (No.2-1) Shape of wave Angle shear connector Fig. 10 Dimensions of specimen Table 3 Strength of specimen Event Load (kN) Cracking of concrete slab 764 Yield of reinforcing bar 2,323 Yield of web plate 1,137 Unit (mm) Proceedings of the 1st fib Congress 290 The concrete had strength of 40 MPa. Eleven D13 reinforcing bars were placed in either the top or bottom slab as longitudinal reinforcement. In order to prevent cracking in the initial stages of loading, one and two prestressing bars of a diameter of 23 mm were placed in the top and bottom slabs, and prestresses of 152 kN and 583 kN were applied on the top and bottom slabs, respectively. The compressive stresses applied by prestressing were 0.5 MPa on the top edge of the top slab and 2.8 MPa on the bottom edge of the bottom slab. The shear and flexural strengths of the specimen are listed in Table 3. The specimen was designed to suffer the shear yielding of the corrugated steel web after cracking occurred and before the main reinforcement was subjected to flexural yielding. 3.2 Test method Loads were applied at two points of the specimen simply supported on either end using a 10000-kN loading machine. Loads were increased monotonously. Strains were measured at the positions shown in Fig. 11. Strains of the reinforcing bars, corrugated steel web and steel flanges were measured in three cross sections in the pure bending section including the midpoint in the specimen, and in another cross section in the shear span. The loads applied, vertical displacements, out-of-plane displacements and shear displacements in the concrete slab and corrugated steel web were also measured. 3.3 Analysis For analyzing the shear buckling of corrugated steel webs, "J-F-C-P" [4], a complicated nonlinear analysis program, was used. The analysis model is shown in Fig. 12. Only a half of the specimen was modeled for analysis because the specimen was symmetric. The stress-strain curves used for analysis are shown in Fig. 13. A bilinear model of the corrugated steel web was made based on material tests. The part of concrete in tension was represented by a bilinear model to facilitate convergence. The tensile strength of concrete was set at 0.5 MPa, about one-sixth the actual strength to prevent local cracking during prestressing. CL B F E D 1700 1700 1200 800 500 Fig. 11 Position of strain gauge Fig. 12 Analytical model Support Center of span Unit (mm) 291 Composite structures Session 5 3.4 Test and analytical results The distribution of strain immediately after prestressing is shown in Fig. 14. The strains in the figure almost agreed to the strains due to the axial force and eccentric loading that were calculated based on the Bernoulli’s assumption. Fig. 15 shows the relationships between the load and vertical displacement. Although the tensile strength of concrete was estimated low in the analysis, cracking load and rigidity after cracking were almost the same comparing the test results and analysis. 0 100 200 300 400 500 0 2000 4000 6000 8000 10000 Strain （ µ ） Stress （ MPa ） Test Analysis -10 0 10 20 30 40 50 -0.5 0.0 0.5 1.0 1.5 応力 (MPa) ひずみ (%) (a) Web plate (b) Concrete Fig. 13 Analytical model of stress-strain curves Strain (%) Stress ( MPa ) 0 250 500 750 1000 1250 1500 1750 -150 -100 -50 0 50 Strain (µ) Height (mm) Wave height 30mm Wave height 60mm Bernoulli's theorem 系 Fig. 14 Strain distribution for prestressing 0 200 400 600 800 1000 1200 1400 1600 0 20406080 Vertical displacement (mm) Load (kN) Test Analysis 0 200 400 600 800 1000 1200 1400 1600 1800 0 20406080100 Vertical displacement (mm) Load (kN) Test Analysis Fig. 15 Load-vertical displacement relationship Proceedings of the 1st fib Congress 292 In the specimen with a wave height of 30 mm, experiencing shear buckling and shear yielding almost simultaneously (Fig. 1), shear buckling occurred immediately following shear yielding and the load decreased rapidly. In the specimen with a wave height of 60 mm, suffering shear yielding before shear buckling (Fig. 1), the rate of increase in load decreased after shear yielding and the load started gradually decreasing. Then, shear buckling caused the load to decrease rapidly. Continued loading after shear buckling created a tension field in the specimen with a wave height of 30 mm and caused the load to increase gradually. The axial reinforcing bars yielded before the load was applied which induced shear buckling, decreasing the load, and deformation started to vary at the two load points. Then, loading was discontinued. In the specimen with a wave height of 60 mm, displacement increased before a tension field was created, and the concrete of the top slab suffered compressive failure between the load points. Fig. 16 shows the strain distribution for the specimen with a wave height of 30 mm at a load of 500 kN, the load before cracking occurred. The figure also shows the strain distribution obtained by three-dimensional finite element analysis and that obtained based on the Bernoulli’s assumption and assuming that the full cross section was effective in resisting forces. In cross sections E and F in the pure bending section, the strain distribution obtained based on the Bernoulli's assumption almost agreed to the test result for concrete slabs. The Bernoulli's assumption was evidently effective. In cross section D, although it is in the pure bending section, the Bernoulli's assumption was not true owing to the influence of concentrated loading. In cross section B in the shear span, the strain distribution identified by the analysis deviated slightly from the distribution obtained based on the Bernoulli's assumption because of the influence of shearing force, but the variance was small. In all cross sections, the results of the test and the three-dimensional finite element analysis almost agreed to each other. It Section B 0 250 500 750 1000 1250 1500 1750 -15 0 -10 0 -50 0 50 100 150 Strain （µ） Height （ mm ） Section D 0 250 500 750 1000 1250 1500 1750 -15 0 -10 0 -50 0 50 100 150 Strain （µ） Height （ mm ） Test ＦＥＭ Bernoulli's theorem Section E 0 250 500 750 1000 1250 1500 1750 -150 -100 -50 0 50 100 150 Strain （µ） Height （mm） Section F 0 250 500 750 1000 1250 1500 1750 -15 0 -10 0 -50 0 50 100 150 Strain （µ） Height （ mm ） Test ＦＥＭ Bernoulli's theorem Fig. 16 Strain distribution at load of 50kN 293 Composite structures Session 5 was therefore verified that the analysis could accurately predict the strains of corrugated steel plate and concrete not only in the case where the Bernoulli's assumption was true but also in the sections under the influence of concentrated loads. Similar results were also obtained for the specimen with a wave height of 60 mm. 4 CONCLUSIONS The test and analysis made in this study produced the following results. (1) It was verified that shear buckling strength and the relationship between load and deformation could be analyzed accurately even in the range under a great influence of geometric nonlinearity. Thus, the validity of the analysis method was verified. (2) It was revealed that the analysis considering the initial shape of the specimen could analyze the relationship between load and vertical displacement and the relationship between load and out-of-plane deformation more accurately than the analysis without such considerations. (3) As a result of a flexural load test on specimens with concrete slabs, it was found that the Bernoulli's assumption was true in the elastic range in the pure bending section. (4) In cross sections near the position where loads were concentrated, the Bernoulli's assumption did not hold true even in the pure bending section. The analysis, however, could predict the test results accurately. ACKNOWLEDGEMENTS The authors would like to thank the members in the technical committee of the long span composite bridges with corrugated steel webs (the chairman: Prof. E. Watanabe, Kyoto University) for their helpful suggestions and comments. They also would like to acknowledge the assistance of Mr. J. Okada (NKK Corporation) for accomplishment of the experimental program. REFERENCES [1] Research Group of Composite Structure with Corrugated Steel Web : Design Manual of PC Box girders with corrugated steel webs. Dec., 1998 (in Japanese) [2] Watanabe, E., Kadotani, T., Miyauchi, M., Tomimoto, M. and Kano, M. : Shear Buckling of Corrugated Steel Web. The First International Conference on Structural Stability and Dynamics Dec., 2000 [3] Okimi, Y. and Ukon, H. : A Frame Analysis System with Geometrical and Material Nonlinear Properties. Journal of the Japan Society of Civil Engineers, Vol. 80, Jan., 1995 (in Japanese) [4] Kano, M., T. Yamano, M. Nibu, and T. : A Computer Program, USSP, for Analyzing Ultimate Strength of Steel Plated Structures. Proceedings of the 5th International Colloquium on Stability and Ductility of Steel Structures, Jul., 1997 Proceedings of the 1st fib Congress 294 . 1.5 応力 (MPa) ひずみ (% ) (a) Web plate (b) Concrete Fig. 13 Analytical model of stress-strain curves Strain (% ) Stress ( MPa ) 0 250 500 750 1000 1250 1500 1750 -150 -100 -50 0 50 Strain ( ) Height. maximum load (No.2 -1) Photo. 1 Specimen after buckling (No.2 -1) Shape of wave Angle shear connector Fig. 10 Dimensions of specimen Table 3 Strength of specimen Event Load (kN) Cracking. 0 1000 2000 3000 4000 5000 0102030 Vertical displacement (mm) Load (kN) 0 500 1000 1500 2000 2500 0 5 10 15 Vertical displacement (mm) Load (kN) (a) Girder height 2100mm, wave height 20mm 0 500 1000 1500 2000 2500 0