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In-plane ultimate load-carrying capacity of tapered I columns
Advances in Steel Structures, Vol. I Chan, Teng and Chung (Eds.) © 2002 Elsevier Science Ltd. All rights reserved.
147
IN-PLANE ULTIMATE LOAD-CARRYING CAPACITY OF TAPERED I COLUMNS Yan-Lin Guo
Yong Pan
Department of Civil Engineering , Tsinghua University, Beijing 100084, P.R.China
ABSTRACT It is well known that the buckling of thin-walled steel columns is generally divided into three modes: local buckling, overall buckling and local-overall interactive buckling. Many previous research works have been carried out to investigate the local and overall interactive buckling behavior and ultimate load-carrying capacity of thin-walled columns. But all their works are focused on the steel prismatic columns with constant sections longitudinally. This paper is intended to present a study on the local-overall interactive buckling behavior and the ultimate load-carrying capacity of tapered I-section steel columns. Local buckling and post local buckling of plate components are considered by using large-deflection elasto-plastic shell elements. The behavior of local-overall interactive buckling is investigated in large deformation and elasto-plastic range by using shell element provided by ANSYS. Based on the nonlinear finite element structural analytic method, the effects of parameters on the ultimate load-carrying capacity, including width-thickness ratio of web and flange plates, tapering ratio and load eccentricity, are considered in the analysis. From the results obtained, it can be concluded that these parameters significantly affect both the buckling failure modes and the ultimate load-carrying capacity of tapered I-section columns. By comparing the results obtained with those of current Chinese code, some valuable conclusions are drawn and some advice is proposed for the design of tapered I-section steel columns. KEYWORDS Interactive Buckling, Local Plate Buckling, Ratio of component plates, Tapering Ratio, Tapered I-Columns, Ultimate Load Capacity
148
1 INTRODUCTION Lightweight steel portal-frame structure has been widely used in recent years. To reduce the consumption of the steel, cross-sections of members are always tapered longitudinally according to the variation of the bending moment. And the current Chinese code^'^ relaxes the restriction of the web plate width-thickness ratio, and thus the post-buckling strength of web plate can be fully used. It is well known that the buckling of thin-walled structures is generally divided into the following three buckling modes: (1) Local plate buckling (2) Overall member buckling (3) Interaction of local and overall buckling. The study of interactive buckling in thin-walled columns began in the 1950s(Bijlaard and Fisher^^^ 1953). Since then a large number of analytical studies and experiments have been carried out on the interactive behavior of local and overall buckling. Hancock^^^ studied the interactive behavior of I beams using finite strip method. Little^"*^ proposed a method to compute load deflection curves of locally buckled columns by integrating, along the member axis, moment-thrust-curvature relations which had been obtained from the results of large deflection elastic-plastic analysis of isolated plate elements. Usami and Fukumoto^^^ investigated the interaction of local and overall buckling of welded box columns. Yanlin Guo^^^ investigated the interactive behavior of short struts in elasto-plastic range by employing nonlinear finite strip method. And M-P-O curve obtained from short struts are applied to the analysis of long columns and beams. But most researches focused on the members with uniform cross-sections longitudinally, and researches on the tapered members are few. Therefore it is necessary to study the interactive behavior of tapered columns. Chinese code employs effective width method to consider the interaction of local and overall buckling. Its formula checking in-plane stability is: ^r n \A
where: NQ Design value of axial compressive force on the smaller cross-section; A/, Design value of bending moment on the larger cross-section; A^Q Effective area of the smaller cross-section; W^,^ Section modulus of effective area of the larger cross-section; (p^^ Coefficient of stability of columns subjected to axial compressive force (by slenderness ratio obtained according to the smaller cross-section); fi^^ equivalent coefficient of bending moment; N^, Euler load obtained according to the smaller cross-section.
2 COMPUTATIONAL MODEL 2,1 Boundary Conditions In general, if there is no crane load, side columns of lightweight steel portal frames are always simply supported at the bottom end. And the cross-sections of side columns are tapered according to the variation of bending moment. The bending moment diagram is shovm in Fig.l when portal frame is subjected to uniformly distributed load on roof The computational model of tapered columns is also shown in Fig.l. Both ends of tapered columns are simply supported. The larger cross-section of tapered columns are subjected to both axial compressive force and bending moment, while the smaller cross-section is only subjected to axial compressive force, as shown in Fig.l. All columns have end plates at both ends to simulate actual columns. Shell
149
element provided by ANSYS is employed in analysis to consider the local buckling of component plates. Since this paper only study in-plane ultimate load capacity of tapered I columns, enough lateral bracings are applied to all the examples to prevent them from lateral instability. For convenient manufacture and construction, only the width of web plate varies linearly, while the thickness of component plates and the width of flange plate are kept unchanged. This paper will just study this kind of members.
^M
/ P-Compressive Force b—Width of Flange Plate
M-Bending Moment
H—Length of Column
tf—Thickness of Flange Plate
h—Web plate height
t^—Thickness of Web Plate
Fig 1 Computational Model 2.2 Material and Geometric Non-linearity Generally speaking, the yielding of material will significantly reduce the stiffness of the tapered I columns and enlarge the buckling displacement. To consider the material non-linearity, the structural steel is assumed to be elastic-perfectly plastic isotropic hardening material in this paper, where Gy = 235 MPa. Geometric nonlinear analysis is required for the consideration of local buckling and post buckling strength of plate components. Arc length method is employed to solve the nonlinear equilibrium equation and trace a full load-displacement curve. 2.5 Initial Geometric Imperfection Initial geometric imperfection exists in all kinds of I columns. In this paper it is assumed that the columns to be studied have both plate initial geometric imperfection and axis initial geometric imperfection in its in-plane direction. The axis initial geometric imperfection is assumed as half sine-wave shape and its maximum value of imperfection is 0.00 IH, where H is the column length. Plate initial geometric imperfection is assumed as multiwave shape and its maximum value is 0.01 times web plate width-thickness ratio at the larger cross-section.
3 NUMERICAL STUDIES The interactive buckling behavior and in-plane ultimate load capacity of tapered I columns are affected by many factors, such as width-thickness ratios of component plate, tapering ratio, load eccentricity, column slenderness ratio, residual stress and initial geometric imperfection. This paper will focus on web plate width-thickness ratio, flange plate width-thickness ratio, tapering ratio and load eccentricity.
150
In this paper, load eccentricity is defined as Q = M / P, where M is the bending moment at the larger cross-section and P is the axial compressive force. 3.1 Web Plate Width-thickness Ratio Due to the linear variation of the web plate, the width-thickness ratio of web plate is not a constant longitudinally. In this paper, the width-thickness ratio refers to that of web plate at the larger cross-section. Dimensions of examples for web plate width-thickness ratio are shown in table 1. Table 1 Dimension of examples for web plate width-thickness ratio Length (m) 8 8 8 8
1 Where do, di
Flange plate (mm) Web plate (mm) Tapering ratio (di/do-1) 200X12 300-600X3.0 0.926 200X12 300-600X4.0 0.926 200X12 300-600X6.0 0.926 200X12 300-600X8.0 0.926 200X12 0.926 300-600X10.0 8 height of the smaller cross-section and the larger cross-section respectively. e=l .Om(finite element method) e= 1 .Om(chines code) e=2.0m(finite element method) e=2.0m(chines code) e=3.0m(finite element method) e=3.0m(chines code)
40
60
80
100
120
140
160
180 200
220
Web plate width-thickness ratio
Fig.2
Effects of Web Plate Width-thickness Ratio
The effects of web plate width-thickness ratio on the ultimate load capacity are shown in Fig.2. It can be observed that the larger the web plate width-thickness ratio is, the lower the ultimate load capacity of the columns is. And with the increase of load eccentricity, the effects of web plate width-thickness ratio become less significant. The reason for this is that when load eccentricity increases, the flange plate may buckle firstly, and thus cause the failure of the columns. By comparing the resuhs obtained by using finite element method with those of Chinese code, it can be concluded that Chinese code is conservative. 3.2 Flange Plate Width-thickness Ratio In this paper, flange plate width-thickness ratio is defined as (b-t^)/2/tj. for flange plate width-thickness ratio are shown in table 2.
Dimension of examples
151
Table 2 Dimension of examples for flange plate width-thickness ratio Length (m) 8 8 8 8 8
Flange plate (mm) 200X12.0 200X10.0 200X8.0 200X6.0 200X4.0
Web plate (mm) 300-^600X3.0 300-600X3.0 300-600X3.0 300-600X3.0 300-600X3.0
Tapering ratio (di/do-1) 0.926 0.926 0.926 0.926 0.926
-e=1.0m -e=1.0m -e=2.0m -e=2.0m -e=3.0m -e=3.0m
e=1.0m(finite element method) e=l .Om(chinese code) e=2.0m(finite element method) e=2.0m(chinese code) e=3.0m(finite element method) e=3.0m(chinese code)
(finite element method) (Chinese code) (finite element method) (Chinese code) (finite element method) (Chinese code)
2 1 H •
8
10
12
14
16
18
20
22
24
1
'
0.5
1
1.0
1
1
1.5
r-
2.0
2.5
3.C
Tapering ratio(d|/d^-l) Flange plate width-thickness ratio
Fig.3 Effects of Flange Plate Width-thickness Ratio
Fig.4 Effects of Tapering Ratio
3,3 Tapering Ratio Tapering ratio is a significant factor determining the ultimate load capacity of tapered I columns. In Chinese code it is defined as: / = (
152
3.4 Compressive Force and Bending Moment Interactive Curve Generally, the design formulae are expressed as compressive force and bending moment interactive curve, and the designers can use them conveniently. The interactive curve is an effective demonstration to study the behavior of columns subjected to both compressive force and bending moment. Dimension of examples is shown in table 4. Table 4 Dimension of columns for compressive force and bending moment interactive curve Length (m) 5 5 5 7 7 7
F^^~-r~A
• - < •
Flange plate (mm) 200X12 200X12 200X12 200X12 200X12 200X12
* A •
,
-mO -• O A A
Tapering Tapering Tapering Tapering Tapering Tapering
ratio 0.893(ANSYS) ratio 0.893(chinese code)" ratio 1 786(ANSYS) ratio 1 786(chinese code) ratio 2 678(A NSYS) ratio 2 678(chmese code)
Web plate (mm) 200-400X6.0 (8.0) 200--600X6.0 (8.0) 200-800X6.0 (8.0) 300-600X6.0 (8.0) 300-900X6.0 (8.0) 300-1200X6.0 (8.0) 1.0-
0.8A
A
'X i
•
A-
•
-m
Tapering Tapering Tapering Tapering Tapering Tapering
atio 0.893(ANSYS) atio 0.893(chinese code) atio 1.786(ANSYS) atio 1.786(chinese code) atio 2.678{ANSYS) atio 2.678(chinese code)
K
0.4-
•
• '' • O A ^
=^A
0.6-
Tapering ratio (di/do-1) 0.893 1.786 2.678 0.926 1.852 2.778
- T -
^ •
A
\
0.2-
A
% , 0.0-
-
,
^ — • * - — r
^
M/Mp
M/Mp
(b) web thickness=8mm, L=5m
(a) web thickness=6mm, L=5m
--»— D~ --•-O -A A
1— Tapering ra io 0.926(ANSYS) 1— Tapering ra io 0.926(chinese code) »— Tapering ra io 1.852(ANSYS) '— Tapering ra io 1.8S2(chine$ecode) k— Tapering ra io 2.778{ANSYS) ^^— Tapering ra io2.778(chinesecode)
0.0
0.2
0.4
Tapering Tapering Tapering Tapering Tapering Tapering
0.6
ratio 0.926(ANSYS) ratio 0.926(chinese code) ratio I 852(ANSYS) ratio 1 852(chinese code) ratio 2 778(ANSYS) ratio 2 778(chinese code)
0.8
1.0 ^/y^
1.2
(c) web thickness=6mm, L=7m (d) web thickness=8mm, L=7m Fig.5 Compressive Force and Bending Moment interactive curve Compressive force and bending moment interactive curve are shovm in Fig. 5, where Np is the ultimate load when the full smaller cross-section subjected to pure compressive force yields and M? is the ultimate bending moment when the full larger cross-section subjected to pure bending moment yields. It can be observed that when load eccentricity is very small, the ultimate load capacity of columns is
153
almost the same because the yielding of the smallest cross-section is the main cause of the failure of the tapered columns. But with the increase of the bending moment, the relative ultimate load capacity of columns with larger tapering ratio and larger web width-thickness ratio is lower. The reason lies in the fact that the greater the tapering ratio is, the more easily the web plate of the larger cross-section buckles. So local buckling is much easier to take place for larger tapering ratio columns. And its relative ultimate load capacity is smaller for larger load eccentricity. It can also be seen that interactive formulae employed in Chinese code to check in-plane stability are conservative and almost linear. 3,5 Failure mechanism The buckling deformation is shown in Fig.6, which shows the effects of web plate width-thickness ratio, load eccentricity and tapering ratio on ultimate load capacity of tapered columns. All the deformation is magnified 15 times. Dimensions and load eccentricities are shown in table 5. Table 5 Dimension of tapered columns Serial number A B C D
Length (m) 8 8 8 8
Flange plate (mm) 200X12 200X12 200X12 200X12
Web plate (mm) 300^-1200X6.0 300-600X3.0 300-600X6.0 300-600X6.0
e(m) 2.0 1.0 2.0 1.0
A B C Fig.6 Buckling deformation of tapered columns
Tapering ratio 2.778 0.926 0.926 0.926
D
From the buckling deformation diagram, plate local buckling can be obviously observed at larger compressive component plates near the larger cross-section when the columns reach their ultimate loads. And the overall buckling of the columns is also obvious. So local buckling and overall buckling are interactive. When load eccentricity is small, and flange plate is thick and web plate is thin, the local buckling of web plate is more obvious, as shown in Fig.6 (B) and Fig.6 (D). On the contrary, the local
154
buckling of flange plate is more obvious, as shown in Fig.6 (A) and Fig.6 (C). But local buckling of the smaller cross-section can hardly take place. Therefore, the local buckling of web plate and flange plate near the larger cross-section, as well as the interaction of local buckling and overall buckling, finally cause the failure of tapered I columns. Because the web plate width-thickness ratio of the larger cross-section is generally larger, local buckling is easier to take place at the larger cross-section. At the same time, due to the effects of the bending moment, the compressive stress of component plate near the larger cross-section is much larger. Therefore the component plates near the larger cross-section buckle more easily, which speeds the failure of overall stability of tapered I columns in the end.
4 CONCLUSIONS This paper investigated the effects of width-thickness ratio of web plate and flange plate, tapering ratio and load eccentricity on ultimate load capacity of tapered columns. The following conclusions can be reached: (1) Width-thickness ratios of web plate and flange plate both have significant influence on in-plane ultimate load capacity of tapered columns, but the latter effects are greater; (2) By increasing the height of the larger cross-section, the in-plane ultimate load capacity of tapered I columns can be greatly enhanced; (3) with the increase of load eccentricity, local buckling of the flange plate and web plate at the larger cross-section is more obvious. In this case the failure of tapered I columns is mainly caused by the local buckling of compressive component plates under large load eccentricity. (4) In-plane design formulae employed in Chinese code to check the in-plane stability of tapered columns are safe and reliable. By comparing the results obtained by using finite element method and those of Chinese code, it is suggested that it may reach a good economic efficiency for the tapered I column with larger web plate and smaller flange plate width-thickness ratio. Thicker flange plate can both enhance the overall stability of tapered columns and prevent local buckling of component plates. As a result, it can prevent the failure of tapered columns from the buckling of the flange plate. REFERENCES [1] Chinese code of lightweight steel portal frame, Beijing, 1999 [2] Bijlaard, RR & Fisher, G.R, Column strength of H-sections and square tubes in post-buckling range of component plates. Technical Note 2994, NACA, USA, 1963 [3] Hancock G J. "Local Distortional, and Lateral Buckling of I Beams," J. Struct. , Div ,ASCE 1978, 104(11), 1787-1798 [4] Little, G. H., "The Strength of Square Steel Box Columns—Design Curves and their Theoretical Basis," The Structural Engineer, Vol. 57, Feb., 1979, pp. 49-61 [5] Usami, T, and Fukumoto, Y., "Local and overall buckling of welded box columns," J. Struct. Div., ASCE, 1982, 108(3), 525-542 [6] Yanlin Guo., "Local and Overall Interactive Instability of Thin-Walled Box-Section Colums," J. Construct. Steel Research., 1992, 1-19
147
IN-PLANE ULTIMATE LOAD-CARRYING CAPACITY OF TAPERED I COLUMNS Yan-Lin Guo
Yong Pan
Department of Civil Engineering , Tsinghua University, Beijing 100084, P.R.China
ABSTRACT It is well known that the buckling of thin-walled steel columns is generally divided into three modes: local buckling, overall buckling and local-overall interactive buckling. Many previous research works have been carried out to investigate the local and overall interactive buckling behavior and ultimate load-carrying capacity of thin-walled columns. But all their works are focused on the steel prismatic columns with constant sections longitudinally. This paper is intended to present a study on the local-overall interactive buckling behavior and the ultimate load-carrying capacity of tapered I-section steel columns. Local buckling and post local buckling of plate components are considered by using large-deflection elasto-plastic shell elements. The behavior of local-overall interactive buckling is investigated in large deformation and elasto-plastic range by using shell element provided by ANSYS. Based on the nonlinear finite element structural analytic method, the effects of parameters on the ultimate load-carrying capacity, including width-thickness ratio of web and flange plates, tapering ratio and load eccentricity, are considered in the analysis. From the results obtained, it can be concluded that these parameters significantly affect both the buckling failure modes and the ultimate load-carrying capacity of tapered I-section columns. By comparing the results obtained with those of current Chinese code, some valuable conclusions are drawn and some advice is proposed for the design of tapered I-section steel columns. KEYWORDS Interactive Buckling, Local Plate Buckling, Ratio of component plates, Tapering Ratio, Tapered I-Columns, Ultimate Load Capacity
148
1 INTRODUCTION Lightweight steel portal-frame structure has been widely used in recent years. To reduce the consumption of the steel, cross-sections of members are always tapered longitudinally according to the variation of the bending moment. And the current Chinese code^'^ relaxes the restriction of the web plate width-thickness ratio, and thus the post-buckling strength of web plate can be fully used. It is well known that the buckling of thin-walled structures is generally divided into the following three buckling modes: (1) Local plate buckling (2) Overall member buckling (3) Interaction of local and overall buckling. The study of interactive buckling in thin-walled columns began in the 1950s(Bijlaard and Fisher^^^ 1953). Since then a large number of analytical studies and experiments have been carried out on the interactive behavior of local and overall buckling. Hancock^^^ studied the interactive behavior of I beams using finite strip method. Little^"*^ proposed a method to compute load deflection curves of locally buckled columns by integrating, along the member axis, moment-thrust-curvature relations which had been obtained from the results of large deflection elastic-plastic analysis of isolated plate elements. Usami and Fukumoto^^^ investigated the interaction of local and overall buckling of welded box columns. Yanlin Guo^^^ investigated the interactive behavior of short struts in elasto-plastic range by employing nonlinear finite strip method. And M-P-O curve obtained from short struts are applied to the analysis of long columns and beams. But most researches focused on the members with uniform cross-sections longitudinally, and researches on the tapered members are few. Therefore it is necessary to study the interactive behavior of tapered columns. Chinese code employs effective width method to consider the interaction of local and overall buckling. Its formula checking in-plane stability is: ^r n \A
where: NQ Design value of axial compressive force on the smaller cross-section; A/, Design value of bending moment on the larger cross-section; A^Q Effective area of the smaller cross-section; W^,^ Section modulus of effective area of the larger cross-section; (p^^ Coefficient of stability of columns subjected to axial compressive force (by slenderness ratio obtained according to the smaller cross-section); fi^^ equivalent coefficient of bending moment; N^, Euler load obtained according to the smaller cross-section.
2 COMPUTATIONAL MODEL 2,1 Boundary Conditions In general, if there is no crane load, side columns of lightweight steel portal frames are always simply supported at the bottom end. And the cross-sections of side columns are tapered according to the variation of bending moment. The bending moment diagram is shovm in Fig.l when portal frame is subjected to uniformly distributed load on roof The computational model of tapered columns is also shown in Fig.l. Both ends of tapered columns are simply supported. The larger cross-section of tapered columns are subjected to both axial compressive force and bending moment, while the smaller cross-section is only subjected to axial compressive force, as shown in Fig.l. All columns have end plates at both ends to simulate actual columns. Shell
149
element provided by ANSYS is employed in analysis to consider the local buckling of component plates. Since this paper only study in-plane ultimate load capacity of tapered I columns, enough lateral bracings are applied to all the examples to prevent them from lateral instability. For convenient manufacture and construction, only the width of web plate varies linearly, while the thickness of component plates and the width of flange plate are kept unchanged. This paper will just study this kind of members.
^M
/ P-Compressive Force b—Width of Flange Plate
M-Bending Moment
H—Length of Column
tf—Thickness of Flange Plate
h—Web plate height
t^—Thickness of Web Plate
Fig 1 Computational Model 2.2 Material and Geometric Non-linearity Generally speaking, the yielding of material will significantly reduce the stiffness of the tapered I columns and enlarge the buckling displacement. To consider the material non-linearity, the structural steel is assumed to be elastic-perfectly plastic isotropic hardening material in this paper, where Gy = 235 MPa. Geometric nonlinear analysis is required for the consideration of local buckling and post buckling strength of plate components. Arc length method is employed to solve the nonlinear equilibrium equation and trace a full load-displacement curve. 2.5 Initial Geometric Imperfection Initial geometric imperfection exists in all kinds of I columns. In this paper it is assumed that the columns to be studied have both plate initial geometric imperfection and axis initial geometric imperfection in its in-plane direction. The axis initial geometric imperfection is assumed as half sine-wave shape and its maximum value of imperfection is 0.00 IH, where H is the column length. Plate initial geometric imperfection is assumed as multiwave shape and its maximum value is 0.01 times web plate width-thickness ratio at the larger cross-section.
3 NUMERICAL STUDIES The interactive buckling behavior and in-plane ultimate load capacity of tapered I columns are affected by many factors, such as width-thickness ratios of component plate, tapering ratio, load eccentricity, column slenderness ratio, residual stress and initial geometric imperfection. This paper will focus on web plate width-thickness ratio, flange plate width-thickness ratio, tapering ratio and load eccentricity.
150
In this paper, load eccentricity is defined as Q = M / P, where M is the bending moment at the larger cross-section and P is the axial compressive force. 3.1 Web Plate Width-thickness Ratio Due to the linear variation of the web plate, the width-thickness ratio of web plate is not a constant longitudinally. In this paper, the width-thickness ratio refers to that of web plate at the larger cross-section. Dimensions of examples for web plate width-thickness ratio are shown in table 1. Table 1 Dimension of examples for web plate width-thickness ratio Length (m) 8 8 8 8
1 Where do, di
Flange plate (mm) Web plate (mm) Tapering ratio (di/do-1) 200X12 300-600X3.0 0.926 200X12 300-600X4.0 0.926 200X12 300-600X6.0 0.926 200X12 300-600X8.0 0.926 200X12 0.926 300-600X10.0 8 height of the smaller cross-section and the larger cross-section respectively. e=l .Om(finite element method) e= 1 .Om(chines code) e=2.0m(finite element method) e=2.0m(chines code) e=3.0m(finite element method) e=3.0m(chines code)
40
60
80
100
120
140
160
180 200
220
Web plate width-thickness ratio
Fig.2
Effects of Web Plate Width-thickness Ratio
The effects of web plate width-thickness ratio on the ultimate load capacity are shown in Fig.2. It can be observed that the larger the web plate width-thickness ratio is, the lower the ultimate load capacity of the columns is. And with the increase of load eccentricity, the effects of web plate width-thickness ratio become less significant. The reason for this is that when load eccentricity increases, the flange plate may buckle firstly, and thus cause the failure of the columns. By comparing the resuhs obtained by using finite element method with those of Chinese code, it can be concluded that Chinese code is conservative. 3.2 Flange Plate Width-thickness Ratio In this paper, flange plate width-thickness ratio is defined as (b-t^)/2/tj. for flange plate width-thickness ratio are shown in table 2.
Dimension of examples
151
Table 2 Dimension of examples for flange plate width-thickness ratio Length (m) 8 8 8 8 8
Flange plate (mm) 200X12.0 200X10.0 200X8.0 200X6.0 200X4.0
Web plate (mm) 300-^600X3.0 300-600X3.0 300-600X3.0 300-600X3.0 300-600X3.0
Tapering ratio (di/do-1) 0.926 0.926 0.926 0.926 0.926
-e=1.0m -e=1.0m -e=2.0m -e=2.0m -e=3.0m -e=3.0m
e=1.0m(finite element method) e=l .Om(chinese code) e=2.0m(finite element method) e=2.0m(chinese code) e=3.0m(finite element method) e=3.0m(chinese code)
(finite element method) (Chinese code) (finite element method) (Chinese code) (finite element method) (Chinese code)
2 1 H •
8
10
12
14
16
18
20
22
24
1
'
0.5
1
1.0
1
1
1.5
r-
2.0
2.5
3.C
Tapering ratio(d|/d^-l) Flange plate width-thickness ratio
Fig.3 Effects of Flange Plate Width-thickness Ratio
Fig.4 Effects of Tapering Ratio
3,3 Tapering Ratio Tapering ratio is a significant factor determining the ultimate load capacity of tapered I columns. In Chinese code it is defined as: / = (
152
3.4 Compressive Force and Bending Moment Interactive Curve Generally, the design formulae are expressed as compressive force and bending moment interactive curve, and the designers can use them conveniently. The interactive curve is an effective demonstration to study the behavior of columns subjected to both compressive force and bending moment. Dimension of examples is shown in table 4. Table 4 Dimension of columns for compressive force and bending moment interactive curve Length (m) 5 5 5 7 7 7
F^^~-r~A
• - < •
Flange plate (mm) 200X12 200X12 200X12 200X12 200X12 200X12
* A •
,
-mO -• O A A
Tapering Tapering Tapering Tapering Tapering Tapering
ratio 0.893(ANSYS) ratio 0.893(chinese code)" ratio 1 786(ANSYS) ratio 1 786(chinese code) ratio 2 678(A NSYS) ratio 2 678(chmese code)
Web plate (mm) 200-400X6.0 (8.0) 200--600X6.0 (8.0) 200-800X6.0 (8.0) 300-600X6.0 (8.0) 300-900X6.0 (8.0) 300-1200X6.0 (8.0) 1.0-
0.8A
A
'X i
•
A-
•
-m
Tapering Tapering Tapering Tapering Tapering Tapering
atio 0.893(ANSYS) atio 0.893(chinese code) atio 1.786(ANSYS) atio 1.786(chinese code) atio 2.678{ANSYS) atio 2.678(chinese code)
K
0.4-
•
• '' • O A ^
=^A
0.6-
Tapering ratio (di/do-1) 0.893 1.786 2.678 0.926 1.852 2.778
- T -
^ •
A
\
0.2-
A
% , 0.0-
-
,
^ — • * - — r
^
M/Mp
M/Mp
(b) web thickness=8mm, L=5m
(a) web thickness=6mm, L=5m
--»— D~ --•-O -A A
1— Tapering ra io 0.926(ANSYS) 1— Tapering ra io 0.926(chinese code) »— Tapering ra io 1.852(ANSYS) '— Tapering ra io 1.8S2(chine$ecode) k— Tapering ra io 2.778{ANSYS) ^^— Tapering ra io2.778(chinesecode)
0.0
0.2
0.4
Tapering Tapering Tapering Tapering Tapering Tapering
0.6
ratio 0.926(ANSYS) ratio 0.926(chinese code) ratio I 852(ANSYS) ratio 1 852(chinese code) ratio 2 778(ANSYS) ratio 2 778(chinese code)
0.8
1.0 ^/y^
1.2
(c) web thickness=6mm, L=7m (d) web thickness=8mm, L=7m Fig.5 Compressive Force and Bending Moment interactive curve Compressive force and bending moment interactive curve are shovm in Fig. 5, where Np is the ultimate load when the full smaller cross-section subjected to pure compressive force yields and M? is the ultimate bending moment when the full larger cross-section subjected to pure bending moment yields. It can be observed that when load eccentricity is very small, the ultimate load capacity of columns is
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almost the same because the yielding of the smallest cross-section is the main cause of the failure of the tapered columns. But with the increase of the bending moment, the relative ultimate load capacity of columns with larger tapering ratio and larger web width-thickness ratio is lower. The reason lies in the fact that the greater the tapering ratio is, the more easily the web plate of the larger cross-section buckles. So local buckling is much easier to take place for larger tapering ratio columns. And its relative ultimate load capacity is smaller for larger load eccentricity. It can also be seen that interactive formulae employed in Chinese code to check in-plane stability are conservative and almost linear. 3,5 Failure mechanism The buckling deformation is shown in Fig.6, which shows the effects of web plate width-thickness ratio, load eccentricity and tapering ratio on ultimate load capacity of tapered columns. All the deformation is magnified 15 times. Dimensions and load eccentricities are shown in table 5. Table 5 Dimension of tapered columns Serial number A B C D
Length (m) 8 8 8 8
Flange plate (mm) 200X12 200X12 200X12 200X12
Web plate (mm) 300^-1200X6.0 300-600X3.0 300-600X6.0 300-600X6.0
e(m) 2.0 1.0 2.0 1.0
A B C Fig.6 Buckling deformation of tapered columns
Tapering ratio 2.778 0.926 0.926 0.926
D
From the buckling deformation diagram, plate local buckling can be obviously observed at larger compressive component plates near the larger cross-section when the columns reach their ultimate loads. And the overall buckling of the columns is also obvious. So local buckling and overall buckling are interactive. When load eccentricity is small, and flange plate is thick and web plate is thin, the local buckling of web plate is more obvious, as shown in Fig.6 (B) and Fig.6 (D). On the contrary, the local
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buckling of flange plate is more obvious, as shown in Fig.6 (A) and Fig.6 (C). But local buckling of the smaller cross-section can hardly take place. Therefore, the local buckling of web plate and flange plate near the larger cross-section, as well as the interaction of local buckling and overall buckling, finally cause the failure of tapered I columns. Because the web plate width-thickness ratio of the larger cross-section is generally larger, local buckling is easier to take place at the larger cross-section. At the same time, due to the effects of the bending moment, the compressive stress of component plate near the larger cross-section is much larger. Therefore the component plates near the larger cross-section buckle more easily, which speeds the failure of overall stability of tapered I columns in the end.
4 CONCLUSIONS This paper investigated the effects of width-thickness ratio of web plate and flange plate, tapering ratio and load eccentricity on ultimate load capacity of tapered columns. The following conclusions can be reached: (1) Width-thickness ratios of web plate and flange plate both have significant influence on in-plane ultimate load capacity of tapered columns, but the latter effects are greater; (2) By increasing the height of the larger cross-section, the in-plane ultimate load capacity of tapered I columns can be greatly enhanced; (3) with the increase of load eccentricity, local buckling of the flange plate and web plate at the larger cross-section is more obvious. In this case the failure of tapered I columns is mainly caused by the local buckling of compressive component plates under large load eccentricity. (4) In-plane design formulae employed in Chinese code to check the in-plane stability of tapered columns are safe and reliable. By comparing the results obtained by using finite element method and those of Chinese code, it is suggested that it may reach a good economic efficiency for the tapered I column with larger web plate and smaller flange plate width-thickness ratio. Thicker flange plate can both enhance the overall stability of tapered columns and prevent local buckling of component plates. As a result, it can prevent the failure of tapered columns from the buckling of the flange plate. REFERENCES [1] Chinese code of lightweight steel portal frame, Beijing, 1999 [2] Bijlaard, RR & Fisher, G.R, Column strength of H-sections and square tubes in post-buckling range of component plates. Technical Note 2994, NACA, USA, 1963 [3] Hancock G J. "Local Distortional, and Lateral Buckling of I Beams," J. Struct. , Div ,ASCE 1978, 104(11), 1787-1798 [4] Little, G. H., "The Strength of Square Steel Box Columns—Design Curves and their Theoretical Basis," The Structural Engineer, Vol. 57, Feb., 1979, pp. 49-61 [5] Usami, T, and Fukumoto, Y., "Local and overall buckling of welded box columns," J. Struct. Div., ASCE, 1982, 108(3), 525-542 [6] Yanlin Guo., "Local and Overall Interactive Instability of Thin-Walled Box-Section Colums," J. Construct. Steel Research., 1992, 1-19