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Local buckling strength of closed polygon folded section columns
J. Construct. Steel Research 20 (1991) 259-270
Local Buckling Strength of Closed Polygon Folded Section Columns* Tetsuhiko Aoki Department of Civil Engineering, Aichi Institute of Technology, Toyota, Japan
Yasuhiro Migita Department of Civil Engineering, Tokai University, Fukuoka, Japan
& Yuhshi Fukumoto Department of Civil Engineering, Osaka University, Suita, Osaka, Japan
ABSTRACT The local buckling behavior of regular polygonal, short length steel columns, fabricated by welding two half sections made of foided steel plates, is described. The polygonal sections are composed offive different section profiles with four to eight sides and each profile having component plates with one to four varied width-to-thickness ratios. A total of 15 specimens are used in the compression test, sustaining uniform compression stress in the fixed end condition. Accurate measurements of welding and cold-forming residual stresses and geometric imperfections were taken prior to testing and are presented in this paper. The test strengths are compared with the current plate buckling code in Japan and with the ECCS recommendations for unstiffened circular cylinders. The empirical design formula based on the test data is also presented to predict the local buckling strength of the polygonal section columns.
1 INTRODUCTION The cross-section profiles of boxes or pipes have been commonly used in compression members such as piers of highway bridges, columns of rigid * Paper presented at
the International Colloquium on Stability of Steel Structures, Budapest, 1990.
Preliminary Report. 259
260
T. Aoki, Y. Mioita and Y. Fukumoto
frames, and monopole tower structures like floodlight towers and transmission towers. In recent years, the possibility of the use of polygonal thin-walled steel members has been considered for those structures from a point of view of aesthetics in urban areas and from the demand of such freedom of designs for the shape of steel structures as concrete structures have. There have been, however, few studies on the buckling strength of the polygonal section steel members so far, on which the design formulas would have been established. In this research field, Wittrick and Curzon 1 had developed a theoretical method on local buckling of long polygonal tubes in combined compression and torsion. Bulson 2 performed the compression tests and analysis in the elastic range for the polygonal section members consisting of very thin plates with four to 40 sides. The strengths of the polygon members were estimated, replacing them with the pipe sections having the same section areas. Avent and Robinson 3 conducted the elastic stability analysis for the regular polygonal folded plate columns. Kurt and Johnson 4 also studied analytically the elastic stability of imperfect closed polygonal columns concerned in the local buckling domain. These studies are performed in relation to elastic buckling analysis. A more recent study presented by Koseko et ai. 5 describes the experiments and finite strip analysis of the local buckling strength of the short length, thin-walled steel members with regular and irregular octagonal cross-sections in the inelastic domain. In this paper, the load carrying capacities due to local buckling of the component plates of regular polygonal section members are investigated experimentally. The number of sides of the polygon is changed from four to eight and the width-to-thickness ratios of the component plates of each specimen are varied from one to four. The residual stresses caused by the welding and folding process and by the geometric imperfections are measured prior to the compression test.
2 TEST
2.1 Test specimens Test specimens were fabricated by welding two half section pieces made of folded plates of which the nominal thickness is 4-5 mm and the steel grade of nominal tensile strength is 400 N/mm 2. The cross-sectional profiles of the specimen consist of rectangular, pentagonal, hexagonal, heptagonal and octagonal regular shapes as shown in Fig. l(a), in which the welded
Local buckling strength of closed polygon folded section columns
• : Weldin 9
261
(a)
-Stiffener
(h)
(Unit:ram)
(c)
Fig. 1. Cross-sectional profiles and length of test specimens: (a) cross-sections; (b) compression test specimen; (c) for measurement of residual stress. locations in the cross-section are indicated by the solid triangular marks. The cross-sectional dimensions and properties are listed in Table 1. Each specimen is named with the short form of the polygon name followed by the plate width in centimeters, as given in Table 1. A total of 15 specimens were prepared for the compression test. For the compression test specimens, 1500ram was cut from the 2300mm length fabricated members and the rest was used for the measurement of residual stresses or tensile coupon test specimens, as shown Fig. l(b) and (c). The compression test specimens have diaphragms near both ends. The strains of local buckling of the component plates and longitudinal displacement were measured in the 1200 mm length between both diaphragms, which corresponds to four times the largest plate width (b = 300 ram) of the test specimens. One octagonal specimen was fabricated by welding eight flat plates at all their junctions to compare with the folded and welded specimens. From the tensile coupon tests the average yield strength ay=289 N/mm 2, and Young's modulus E = 2 1 5 kN/mm 2 were obtained. 2.2 Residual stresses
The 12 strain gages were attached to each couple on the outside and inside surfaces of the plate elements, around the peripheral of the specimens at the midlength. The method of sectioning was employed to release the residual stresses. An example of measured residual stress distribution for OCT-15 is shown in Fig. 2. For all measured specimens, the residual stresses on the
specimen
REC-20 REC-25 REC-30
sides
4
4-5 4"5 4"5 4"5
20 25 30
4"5
4-5 4-5 4"5
4.5 4.5
4.5 4.5 4.5
Thickhess, t (mm)
15
17
20 25 30
24 24
20 25 30
Width of a plate, b (cm)
Tensile yield strength: try = 289 N/ram 2. OCT-15w: all comers are welded.
8
HEP-17
7
OCT-15 ) OCT-15b OCT- 15w |. J OCT-20 OCT-25 OCT-30
HEX-20 HEX-25 HEX-30
6
PEN-24 PEN-24b
of
of
5
Name
Number
44"4 55"6 66"7
33"3
38" 1
44"4 55"6 66"7
53-3 53-3
44.4 55.6 66.7
Widththickness ratio (b/t)
0.86 1"07 1"29
0.64
0.74
0"86 1"07 1"29
1-03 1.03
0.86 1-07 1.29
Widththickness parameter (R)
72-0 90"0 108"0
54"0
54-0
54"0 67"5 81-0
54-0 54.0
36.0 45.0 54.0
Sectional area, A (cm 2)
TABLE 1 Nominal Cross-Sectional Properties and Maximum Stresses due to Test O'mx
278 274 275 244 206 173
271
251 206 177
213 207
261 201 210
(Nlmm ~)
0"96 0"95 0.95 0.84 0.71 0-60
0.94
0.87 0"71 0-61
0.74 0-71
0.90 0-70 ff72
O'rnax/O" Y
t-,,
,..,,,.
bO O~ bO
Local buckling strength of closed polygon folded section columns 1.0
Tension
0.8
0.~
I
263
• Outside o Inside • Welding R Corner
O
b>- 0 ~ b
-O
-O.
--O'v
Fig. 2.
Residual stress distribution in OCT-15.
outside and inside surfaces are almost the same value except in the corner parts, where the cold form effect of the folding plate causes the significant difference in the measured values of both sides. The distributions of the membrane residual stresses after averaging the outside and inside values, taking the half section between two welding lines from the section, are shown in Fig. 3. The residual stress distributions are categorized into three patterns by the shapes of their compression distributions and the number of corners between the welding locations rather than by the distance between the weldings, as shown in Fig. 3(a), (b) and (c). Figure 3(a) shows the results of the rectangular sections having two comers between the welding points and they seem to be of approximately trapezoidal shape for compression residual stress distributions. The distributions in Fig. 3(b) contain two triangles near the tensile residual stresses at the welding sites, and a small triangular tensile stress at the center, where three corners exist. The distributions of Fig. 3(c), being almost the same as those of Fig. 3(b), have
T. Aoki, Y. Migita and Y. Fukumoto
264
OCT 15
HEP 17
PEN 24
0"2~ 0"1 b
ol|
R /I
R
o
/
_
R
~
o
R
-0.2 -0.3 -0.4
~
-0.2 -0"3 -0.4
-0.4 REC25
b
o.,N e, l l
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Jl
R
o
RJ, I
-0.1
-0'4
R
-0.1
t-
0.2
i
0.2 -
/R~
-
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R
--
.
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0.2
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0.2 ~
0 CT 3 0
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o
R
-0-1
-
~
R
-0.2
:0:,3-
(a)
-0"4 t...
(b)
I
(c)
Fig. 3. Compressiveresidual stress distributions. four corners and a little trapezoidal tensile residual stress at the center, by observation. The changes of distribution along with the distance between the weldings are observed in each classification. The maximum compressive residual stresses varied from 0.25 ay to 0-3 ay for the rectangular sections and from 0.22 ay to 0.5 ay for other sections. 2.3 Geometric imperfections The geometric imperfections of the component plates forming the thinwalled polygonal test specimens were measured by dial gages to an accuracy of 0.01 mm per reading. Thirteen dial gages were fixed on a firm steel bar and the straightness was measured prior to the measurement of the geometric imperfections of the plates. All component plates of the test specimens were divided into grids consisting of six to eight equally spaced lines on the fiat part of the plates in the direction of the plate width and 13 equally spaced measuring points were taken along each line.
Local buckling strenoth of closed polyoonfolded section columns
, I
(a)
265
I
U (b)
Fig. 4. Examples of geometric imperfections (OCT-20): (a) welded panel; (b) nonwelded panel.
F ocrls-~
%
\
: Welding
Fig. 5. Geometric imperfections on the cross-sections at midlength of the specimens.
266
T. Aoki, Y. Migita and Y. Fukumoto
The test data for all specimens were analyzed by the least squares method, using Fourier series expansion consisting of seven terms, and the components of the half wavelengths were obtained. Examples of measurement results for the specimen OCT-20 are shown in Fig. 4(a) and (b). It was observed in almost all of the specimens that the shape of the geometrical imperfections of the nonwelded plates had one half wavelength mode predominantly, whereas random ripple modes appeared on the welded plates. The maximum values of imperfections were also found in the welded plates in all specimens except PEN-24 and OCT-15. The value of the maximum imperfection of all specimens was 0.014b (4.24mm) in the specimen OCT-30, where the plate width b=300mm. The mean value of all those measured was 1.21 mm, which corresponded to 1/1000 of the plate length. Figure 5 shows the geometric imperfections on the cross-section at midlength of the specimens, in which the deformations are scaled up to the cross-sectional dimensions for the illustration. 2.4 Compression test 2.4.1 Test method
The compression tests were conducted in the fixed end condition, setting the specimens between two rigid end plates with a couple of rotational beveled bearing circular plates to adjust the uniform compression stress conditions during loading. The 13 strain gages were attached longitudinally on the center line of the component plate, which was located adjacent to the welding plate, where the maximum compressive residual stresses were supposed to be contained. The axial displacements were measured with 1/100 mm accuracy dial gages attached at the four corners of each test specimen. 2.4.2 Test results
The longitudinal average stress-strain curves obtained from the dial gages as a whole column for octagonal specimens with different width-tothickness ratios are shown in Fig. 6. Each curve coincides well with others just before the maximum strength and after that the sudden unloading of the bearing load was observed. Figure 7(a) shows the change of strain distributions along the center line of the component plates of the specimen OCT-20. The uniform increase of the strains along the longitudinal lines is observed until the maximum stress am=x, in spite of the geometric imperfections of the plates. The local buckling mode is also observed clearly from the strain distributions after buckling. Figure 7(b) shows the stress and strain graph at the several
Local buckling strength of closed polygon folded section columns
267
1.0,
\
0-9
0CT15 (bit = 33.3) 0.6
Io
~.
0.5
" - ~-
E b
OCT 20 (bit
o,l
= 44-4)
ocT25 (bit =55.0) OCT 3 0 ( b i t = 66"7)
0.3 ¸
0-2,
0.1
0
lobo 2c;00 s&~o 46oo sc;oo 6c;00 7~bo 8doo exlO -6
Fig. 6.
Average stress-strain curves due to dial gages.
points on the center line shown in Fig. 7(a). It can be found from Fig. 7(b) that divergence of the strains starts from around 0-8 amax.
2.4.3 Maximum strength All the test specimens caused local buckling in the component plates. The maximum strength points due to the compression test are given in Table 1 and in Fig. 8, in which the current plate buckling formula of the Japan Highway Association's Specification,6 and the ECCS Recommendation's curve, 7 which shows the lower bound to the unstiffened circular cylinders, are also given. An empirical formula (eqn (1)) based on the present test data is also shown: trm,~/try= 1.35-0.55R
(R>0.636)
(1)
268
T. Aoki, Y. Migita and Y. Fukumoto
Upper edge Tension
Compression
O"m
ffrn
0"9
O-ma x
1"0
35
,o.9jy#~
35 p/.
1
37 A f t e r Emax :38 /
o.o.o., / o.o.o /
Shape of/d! initial deflection 42
/\
143 ~ 7 i
ffrnax
',,45 ~
m
4_5 0 ' 8 Crma x - -
Inwar_d
10.3~
Gage no.
, t
--'--
o. t!
47 48 49 5O
_ ~
____~ Outward [ Lower edge
(a)
-500
0
500 1000
40
2000 xlO -O
3000
(b)
Fig. 7. Examples of the change of strains on the specimen OCT-20: (a) the change of strain distributions along the center line; (b) stress-strain relations of the points on the center line.
R=x/ay/a~, a¢,=k 12(1_v2)
(2)
0)
(k=4)
(3)
Observing both the test results and eqn (1) in Fig. 8, it can be said that the maximum strength has a close relationship with the width-thickness ratios of the component plates of the polygonal section members. Since the profiles with the smaller width-thickness ratio may have the larger local buckling strength, polygonal sections having more sides among the thin-walled members with the same cross-sectional area, may become advantageous with regard to the local buckling strength and, therefore, the ultimate strength before to occur the shell type buckling mode for a large number of sides. Considering practical problems such as fabrication, the octagonal section profile may be desirable among polygonal members.
Local bucklino strenoth of closed polyoon folded section columns
269
1.2
1-0
"~...
\
E.~,.,"
0-8
b X
0-6
\J ~
E
b
0"4
0"2
m REC
• HEP • OCT
0.2
JH$
o'.4
o:6
' o:8 ' ~:o '
~:2
1:4
R or~,
Fig. $. Experimental maximum strengths.
3 C O N C L U D I N G REMARKS In this paper, an experimental study of load carrying capacity of the stub columns having thin-walled, regular polygonal section was presented. Test specimens were fabricated by folding plates, forming two halves of the cross-sections and welding them into one section. The ultimate strength of the uniformly compressed polygon stub columns under the fixed end condition showed stronger interaction with the width-thickness ratios of the component plates than with the crosssectional profiles. The polygonal, thin-walled short length members, therefore, seem to be better than box section members in respect to the ultimate strength of the members when they are affected directly by local buckling of the component plates under the condition of uniform compression. Among the polygonal sections, the octagonal section profile may have much possibility in practical use for the structures because of the convenience of fabrication and comparatively easy connection with other structural members, as well as the aesthetics considerations. The authors have been conducting the test program for medium length columns with the polygonal sections from a viewpoint of the interaction effect between local and overall buckling.
270
T. Aoki, Y. Migita and Y. Fukumoto
REFERENCES 1. Wittrick, W. H. & Curzon, A unified approach to the initial buckling of stiffened panels in compression. Aeronautical Quarterly, 19 (1968) 265-83. 2. Bulson, P. S., The strength of thin-wall tubes formed from flat elements. Int. J. Mech. Sci., 11 (1969) 613-20. 3. Avent, R. R. & Robinson, J. H., Elastic stability of polygon folded plate columns. Proc. ASCE, 102, (ST5) (1976) 1015-29. 4. Kurt, C. E. & Johnson, R. C., Cross sectional imperfections and columns stability. Proc. ASCE, 104 (ST12) (1978) 1869-83. 5. Koseko, N., Aoki, T. & Fukumoto, Y., The local buckling strength of the octagonal section steel columns. JSCE, 2 (330) (1983) 27-36. 6. Japan Highway Association, 1980: Specification for Highway Bridges (Steel Bridge). 7. European Recommendations for Steel Construction, 1983: Buckling of Shells ECCS-CECM-EKS, p. 7.
Local Buckling Strength of Closed Polygon Folded Section Columns* Tetsuhiko Aoki Department of Civil Engineering, Aichi Institute of Technology, Toyota, Japan
Yasuhiro Migita Department of Civil Engineering, Tokai University, Fukuoka, Japan
& Yuhshi Fukumoto Department of Civil Engineering, Osaka University, Suita, Osaka, Japan
ABSTRACT The local buckling behavior of regular polygonal, short length steel columns, fabricated by welding two half sections made of foided steel plates, is described. The polygonal sections are composed offive different section profiles with four to eight sides and each profile having component plates with one to four varied width-to-thickness ratios. A total of 15 specimens are used in the compression test, sustaining uniform compression stress in the fixed end condition. Accurate measurements of welding and cold-forming residual stresses and geometric imperfections were taken prior to testing and are presented in this paper. The test strengths are compared with the current plate buckling code in Japan and with the ECCS recommendations for unstiffened circular cylinders. The empirical design formula based on the test data is also presented to predict the local buckling strength of the polygonal section columns.
1 INTRODUCTION The cross-section profiles of boxes or pipes have been commonly used in compression members such as piers of highway bridges, columns of rigid * Paper presented at
the International Colloquium on Stability of Steel Structures, Budapest, 1990.
Preliminary Report. 259
260
T. Aoki, Y. Mioita and Y. Fukumoto
frames, and monopole tower structures like floodlight towers and transmission towers. In recent years, the possibility of the use of polygonal thin-walled steel members has been considered for those structures from a point of view of aesthetics in urban areas and from the demand of such freedom of designs for the shape of steel structures as concrete structures have. There have been, however, few studies on the buckling strength of the polygonal section steel members so far, on which the design formulas would have been established. In this research field, Wittrick and Curzon 1 had developed a theoretical method on local buckling of long polygonal tubes in combined compression and torsion. Bulson 2 performed the compression tests and analysis in the elastic range for the polygonal section members consisting of very thin plates with four to 40 sides. The strengths of the polygon members were estimated, replacing them with the pipe sections having the same section areas. Avent and Robinson 3 conducted the elastic stability analysis for the regular polygonal folded plate columns. Kurt and Johnson 4 also studied analytically the elastic stability of imperfect closed polygonal columns concerned in the local buckling domain. These studies are performed in relation to elastic buckling analysis. A more recent study presented by Koseko et ai. 5 describes the experiments and finite strip analysis of the local buckling strength of the short length, thin-walled steel members with regular and irregular octagonal cross-sections in the inelastic domain. In this paper, the load carrying capacities due to local buckling of the component plates of regular polygonal section members are investigated experimentally. The number of sides of the polygon is changed from four to eight and the width-to-thickness ratios of the component plates of each specimen are varied from one to four. The residual stresses caused by the welding and folding process and by the geometric imperfections are measured prior to the compression test.
2 TEST
2.1 Test specimens Test specimens were fabricated by welding two half section pieces made of folded plates of which the nominal thickness is 4-5 mm and the steel grade of nominal tensile strength is 400 N/mm 2. The cross-sectional profiles of the specimen consist of rectangular, pentagonal, hexagonal, heptagonal and octagonal regular shapes as shown in Fig. l(a), in which the welded
Local buckling strength of closed polygon folded section columns
• : Weldin 9
261
(a)
-Stiffener
(h)
(Unit:ram)
(c)
Fig. 1. Cross-sectional profiles and length of test specimens: (a) cross-sections; (b) compression test specimen; (c) for measurement of residual stress. locations in the cross-section are indicated by the solid triangular marks. The cross-sectional dimensions and properties are listed in Table 1. Each specimen is named with the short form of the polygon name followed by the plate width in centimeters, as given in Table 1. A total of 15 specimens were prepared for the compression test. For the compression test specimens, 1500ram was cut from the 2300mm length fabricated members and the rest was used for the measurement of residual stresses or tensile coupon test specimens, as shown Fig. l(b) and (c). The compression test specimens have diaphragms near both ends. The strains of local buckling of the component plates and longitudinal displacement were measured in the 1200 mm length between both diaphragms, which corresponds to four times the largest plate width (b = 300 ram) of the test specimens. One octagonal specimen was fabricated by welding eight flat plates at all their junctions to compare with the folded and welded specimens. From the tensile coupon tests the average yield strength ay=289 N/mm 2, and Young's modulus E = 2 1 5 kN/mm 2 were obtained. 2.2 Residual stresses
The 12 strain gages were attached to each couple on the outside and inside surfaces of the plate elements, around the peripheral of the specimens at the midlength. The method of sectioning was employed to release the residual stresses. An example of measured residual stress distribution for OCT-15 is shown in Fig. 2. For all measured specimens, the residual stresses on the
specimen
REC-20 REC-25 REC-30
sides
4
4-5 4"5 4"5 4"5
20 25 30
4"5
4-5 4-5 4"5
4.5 4.5
4.5 4.5 4.5
Thickhess, t (mm)
15
17
20 25 30
24 24
20 25 30
Width of a plate, b (cm)
Tensile yield strength: try = 289 N/ram 2. OCT-15w: all comers are welded.
8
HEP-17
7
OCT-15 ) OCT-15b OCT- 15w |. J OCT-20 OCT-25 OCT-30
HEX-20 HEX-25 HEX-30
6
PEN-24 PEN-24b
of
of
5
Name
Number
44"4 55"6 66"7
33"3
38" 1
44"4 55"6 66"7
53-3 53-3
44.4 55.6 66.7
Widththickness ratio (b/t)
0.86 1"07 1"29
0.64
0.74
0"86 1"07 1"29
1-03 1.03
0.86 1-07 1.29
Widththickness parameter (R)
72-0 90"0 108"0
54"0
54-0
54"0 67"5 81-0
54-0 54.0
36.0 45.0 54.0
Sectional area, A (cm 2)
TABLE 1 Nominal Cross-Sectional Properties and Maximum Stresses due to Test O'mx
278 274 275 244 206 173
271
251 206 177
213 207
261 201 210
(Nlmm ~)
0"96 0"95 0.95 0.84 0.71 0-60
0.94
0.87 0"71 0-61
0.74 0-71
0.90 0-70 ff72
O'rnax/O" Y
t-,,
,..,,,.
bO O~ bO
Local buckling strength of closed polygon folded section columns 1.0
Tension
0.8
0.~
I
263
• Outside o Inside • Welding R Corner
O
b>- 0 ~ b
-O
-O.
--O'v
Fig. 2.
Residual stress distribution in OCT-15.
outside and inside surfaces are almost the same value except in the corner parts, where the cold form effect of the folding plate causes the significant difference in the measured values of both sides. The distributions of the membrane residual stresses after averaging the outside and inside values, taking the half section between two welding lines from the section, are shown in Fig. 3. The residual stress distributions are categorized into three patterns by the shapes of their compression distributions and the number of corners between the welding locations rather than by the distance between the weldings, as shown in Fig. 3(a), (b) and (c). Figure 3(a) shows the results of the rectangular sections having two comers between the welding points and they seem to be of approximately trapezoidal shape for compression residual stress distributions. The distributions in Fig. 3(b) contain two triangles near the tensile residual stresses at the welding sites, and a small triangular tensile stress at the center, where three corners exist. The distributions of Fig. 3(c), being almost the same as those of Fig. 3(b), have
T. Aoki, Y. Migita and Y. Fukumoto
264
OCT 15
HEP 17
PEN 24
0"2~ 0"1 b
ol|
R /I
R
o
/
_
R
~
o
R
-0.2 -0.3 -0.4
~
-0.2 -0"3 -0.4
-0.4 REC25
b
o.,N e, l l
PEN 24
Jl
R
o
RJ, I
-0.1
-0'4
R
-0.1
t-
0.2
i
0.2 -
/R~
-
HEP 17
R
--
.
REC 30
0.2
HEX 30
0.2 ~
0 CT 3 0
0.1 ~ 011
R
R ,I,I
,o,1\ -F,.. vlII -0"3 - 0.4 L
o
R
-0-1
-
~
R
-0.2
:0:,3-
(a)
-0"4 t...
(b)
I
(c)
Fig. 3. Compressiveresidual stress distributions. four corners and a little trapezoidal tensile residual stress at the center, by observation. The changes of distribution along with the distance between the weldings are observed in each classification. The maximum compressive residual stresses varied from 0.25 ay to 0-3 ay for the rectangular sections and from 0.22 ay to 0.5 ay for other sections. 2.3 Geometric imperfections The geometric imperfections of the component plates forming the thinwalled polygonal test specimens were measured by dial gages to an accuracy of 0.01 mm per reading. Thirteen dial gages were fixed on a firm steel bar and the straightness was measured prior to the measurement of the geometric imperfections of the plates. All component plates of the test specimens were divided into grids consisting of six to eight equally spaced lines on the fiat part of the plates in the direction of the plate width and 13 equally spaced measuring points were taken along each line.
Local buckling strenoth of closed polyoonfolded section columns
, I
(a)
265
I
U (b)
Fig. 4. Examples of geometric imperfections (OCT-20): (a) welded panel; (b) nonwelded panel.
F ocrls-~
%
\
: Welding
Fig. 5. Geometric imperfections on the cross-sections at midlength of the specimens.
266
T. Aoki, Y. Migita and Y. Fukumoto
The test data for all specimens were analyzed by the least squares method, using Fourier series expansion consisting of seven terms, and the components of the half wavelengths were obtained. Examples of measurement results for the specimen OCT-20 are shown in Fig. 4(a) and (b). It was observed in almost all of the specimens that the shape of the geometrical imperfections of the nonwelded plates had one half wavelength mode predominantly, whereas random ripple modes appeared on the welded plates. The maximum values of imperfections were also found in the welded plates in all specimens except PEN-24 and OCT-15. The value of the maximum imperfection of all specimens was 0.014b (4.24mm) in the specimen OCT-30, where the plate width b=300mm. The mean value of all those measured was 1.21 mm, which corresponded to 1/1000 of the plate length. Figure 5 shows the geometric imperfections on the cross-section at midlength of the specimens, in which the deformations are scaled up to the cross-sectional dimensions for the illustration. 2.4 Compression test 2.4.1 Test method
The compression tests were conducted in the fixed end condition, setting the specimens between two rigid end plates with a couple of rotational beveled bearing circular plates to adjust the uniform compression stress conditions during loading. The 13 strain gages were attached longitudinally on the center line of the component plate, which was located adjacent to the welding plate, where the maximum compressive residual stresses were supposed to be contained. The axial displacements were measured with 1/100 mm accuracy dial gages attached at the four corners of each test specimen. 2.4.2 Test results
The longitudinal average stress-strain curves obtained from the dial gages as a whole column for octagonal specimens with different width-tothickness ratios are shown in Fig. 6. Each curve coincides well with others just before the maximum strength and after that the sudden unloading of the bearing load was observed. Figure 7(a) shows the change of strain distributions along the center line of the component plates of the specimen OCT-20. The uniform increase of the strains along the longitudinal lines is observed until the maximum stress am=x, in spite of the geometric imperfections of the plates. The local buckling mode is also observed clearly from the strain distributions after buckling. Figure 7(b) shows the stress and strain graph at the several
Local buckling strength of closed polygon folded section columns
267
1.0,
\
0-9
0CT15 (bit = 33.3) 0.6
Io
~.
0.5
" - ~-
E b
OCT 20 (bit
o,l
= 44-4)
ocT25 (bit =55.0) OCT 3 0 ( b i t = 66"7)
0.3 ¸
0-2,
0.1
0
lobo 2c;00 s&~o 46oo sc;oo 6c;00 7~bo 8doo exlO -6
Fig. 6.
Average stress-strain curves due to dial gages.
points on the center line shown in Fig. 7(a). It can be found from Fig. 7(b) that divergence of the strains starts from around 0-8 amax.
2.4.3 Maximum strength All the test specimens caused local buckling in the component plates. The maximum strength points due to the compression test are given in Table 1 and in Fig. 8, in which the current plate buckling formula of the Japan Highway Association's Specification,6 and the ECCS Recommendation's curve, 7 which shows the lower bound to the unstiffened circular cylinders, are also given. An empirical formula (eqn (1)) based on the present test data is also shown: trm,~/try= 1.35-0.55R
(R>0.636)
(1)
268
T. Aoki, Y. Migita and Y. Fukumoto
Upper edge Tension
Compression
O"m
ffrn
0"9
O-ma x
1"0
35
,o.9jy#~
35 p/.
1
37 A f t e r Emax :38 /
o.o.o., / o.o.o /
Shape of/d! initial deflection 42
/\
143 ~ 7 i
ffrnax
',,45 ~
m
4_5 0 ' 8 Crma x - -
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, t
--'--
o. t!
47 48 49 5O
_ ~
____~ Outward [ Lower edge
(a)
-500
0
500 1000
40
2000 xlO -O
3000
(b)
Fig. 7. Examples of the change of strains on the specimen OCT-20: (a) the change of strain distributions along the center line; (b) stress-strain relations of the points on the center line.
R=x/ay/a~, a¢,=k 12(1_v2)
(2)
0)
(k=4)
(3)
Observing both the test results and eqn (1) in Fig. 8, it can be said that the maximum strength has a close relationship with the width-thickness ratios of the component plates of the polygonal section members. Since the profiles with the smaller width-thickness ratio may have the larger local buckling strength, polygonal sections having more sides among the thin-walled members with the same cross-sectional area, may become advantageous with regard to the local buckling strength and, therefore, the ultimate strength before to occur the shell type buckling mode for a large number of sides. Considering practical problems such as fabrication, the octagonal section profile may be desirable among polygonal members.
Local bucklino strenoth of closed polyoon folded section columns
269
1.2
1-0
"~...
\
E.~,.,"
0-8
b X
0-6
\J ~
E
b
0"4
0"2
m REC
• HEP • OCT
0.2
JH$
o'.4
o:6
' o:8 ' ~:o '
~:2
1:4
R or~,
Fig. $. Experimental maximum strengths.
3 C O N C L U D I N G REMARKS In this paper, an experimental study of load carrying capacity of the stub columns having thin-walled, regular polygonal section was presented. Test specimens were fabricated by folding plates, forming two halves of the cross-sections and welding them into one section. The ultimate strength of the uniformly compressed polygon stub columns under the fixed end condition showed stronger interaction with the width-thickness ratios of the component plates than with the crosssectional profiles. The polygonal, thin-walled short length members, therefore, seem to be better than box section members in respect to the ultimate strength of the members when they are affected directly by local buckling of the component plates under the condition of uniform compression. Among the polygonal sections, the octagonal section profile may have much possibility in practical use for the structures because of the convenience of fabrication and comparatively easy connection with other structural members, as well as the aesthetics considerations. The authors have been conducting the test program for medium length columns with the polygonal sections from a viewpoint of the interaction effect between local and overall buckling.
270
T. Aoki, Y. Migita and Y. Fukumoto
REFERENCES 1. Wittrick, W. H. & Curzon, A unified approach to the initial buckling of stiffened panels in compression. Aeronautical Quarterly, 19 (1968) 265-83. 2. Bulson, P. S., The strength of thin-wall tubes formed from flat elements. Int. J. Mech. Sci., 11 (1969) 613-20. 3. Avent, R. R. & Robinson, J. H., Elastic stability of polygon folded plate columns. Proc. ASCE, 102, (ST5) (1976) 1015-29. 4. Kurt, C. E. & Johnson, R. C., Cross sectional imperfections and columns stability. Proc. ASCE, 104 (ST12) (1978) 1869-83. 5. Koseko, N., Aoki, T. & Fukumoto, Y., The local buckling strength of the octagonal section steel columns. JSCE, 2 (330) (1983) 27-36. 6. Japan Highway Association, 1980: Specification for Highway Bridges (Steel Bridge). 7. European Recommendations for Steel Construction, 1983: Buckling of Shells ECCS-CECM-EKS, p. 7.