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

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


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)

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

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
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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

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)

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

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

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

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

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.