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Ultimate strength of uniformly compression edge-stiffened thin-walled sections
,L Construct. Steel Res. Vol. 36, No. 1, pp. 31-51, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0143°974X(94)00038-7 0143-974X/96 $15.00 + 0.00
ELSEVIER
Ultimate Strength of Uniformly Compression Edge-Stiffened Thin-Walled Sections L. K. Seah School of Mechanical & Production Engineering, Nanyang Technological University, Singapore 2263 (Received 24 January 1994; revised version received 4 July 1994; accepted 15 August 1994)
ABSTRACT This paper reports an investigation" into the collapse behaviour of uniformly compressed edge-stiffened thin-walled sections. The term 'edge stiffener' refers to a stiffener in the form of a single or double right-angled lip formed at the free edge of the un;~tiffened element. The primary function of the edge stiffener is to economically enhance the local buckling of an open thin-walled section. An outline of a series of tests on edge-stiffened thin-walled sections of various geometries is given. Design procedures are proposed based on the modifications of the draft Eurocode to predict the ultimate strength of such edge-stiffened thin-walled section,s. The empirical results obtained are then compared with existing experimental res~,~lts, and close agreement is found. A further comparison with an independent source of experimental results confirms the rationality of the approach.
LIST O F S Y M B O L S
AR
A stiff b bl b2 bw b 'eR
effective area of the edge stiffeners be1, be2 and including the adjacent plane element, beR; AR=(beR+ b~l +b¢2)t, to be determined at o-c=0.5ay, beR and b~l are estimated as a stiffened element and bo2 as an unstiffened element. ~Lrea of stiffeners only, equal to t. (be~ +b~2) at 1/2 yield. width of plate element to be stiffened width of first stiffener width of second stiffener width of adjacent web. effective adjacent plane element at twice yield: 31
32
D E IR I Rmin IR n
P inad Pu Ps t O'y
L. K. Seah
plate flexural rigidity Young's modulus of elasticity moment of inertia of the stiffener with area A R about axis a-a, as shown in figure (7). Minimum required value of moment of inertia of the stiffener for adequacy, set by equation (1). The moment of inertia of the actual stiffener with area A R determined at yield stress, about the axis a-a shown in figure (7) variable index Pb non-dimensional axial l o a d , / ~ - g 2 D Ultimate strength of an inadequately stiffened element Ultimate strength of an unstiffened element Ultimate strength of an adequately stiffened element plate thickness yield stress
1 INTRODUCTION Local buckling is a prominent behaviour of thin-walled sections. It causes a reduction in the stiffness of a section and a lowering of the load-carrying capacity. In order to improve these buckling characteristics, edge stiffeners are widely used to stiffen the flat compressive elements of thin-walled structural members which can be found in various engineering fields. The primary function of the edge stiffener is to conomically enhance the local buckling strength of an open thin-walled section. The buckling behaviour of edge-stiffened thin-walled sections has attracted the attention of large number of researchers over many years ~- s One of the most common types of edge stiffener used in cold-rolled sections is formed by turning the free edge of the unstiffened flange inwards or outwards to form a lip. Past investigations of this type of edge stiffener have tended to concentrate on single-fold lipped plate elements of thin-walled sections. However, for sections with high width to thickness ratios, the width required for a simple lip would be such that the lip itself would begin to have stability problems. In this paper, the collapse behaviour of compound right-angled lipstiffened sections is examined. The investigations here are confined to the case of shot, asymmetrically edge-stiffened, thin-walled channel struts, subjected to uniform end compression. The edge stiffened sections under these circumstances can fail, due to elastic local buckling of plate and stiffener, torsional buckling of the edge stiffened flange or a
Strength of thin-walled sections
33
combination of these two buckling modes, depending on the stiffener's flexural rigidity. An outline of a series of tests on outwardly-and inwardly-turned lipped channel sections of various geometries is given. Design procedures are proposed based on the draft Eurocode 6 to predict the ultimate strength of such edge-stiffened thin-walled sections. Reasonably good agreement is obtained between the empirical and experimental results.
2 EXPERIMENTAL INVESTIGATION
2.1 Tes! specimens A total of 50 specimens were manufactured and tested under uniform end compression, of which 26 specimens were outwardly-turned lipped tophat secl:ions and 24 specimens were inwardly-turned lipped channel sections, The prime objectives of the investigation were to examine the effects of adjacent elements on the behaviour of compound edge-stiffened elements and to devise a simple design criterion for sections having compound edge stiffeners. In the following sections, the tests perfomed in this investigation are described. The cross-sectional geometry, dimensions and material yield stress of all the specimens are shown in Table 1. Tensile test specimens were cut and tested from all sheets of material used, so that each specimen could be analysed on the basis of the true yield stress. The three nominal thicknesses of steel sheet used were 0.8, 1-0 and 1-2 mm. These specimens were manufac,tured by cold-folding, using a 4 ft long Oliver Pan Folder, to normal engineering tolerances. The ends of the test specimens were carefully milled and filed to ensure a fiat and plane surface perpendicular to the specimen's longitudinal axis. The ends were then tested for flatness against a surface table and any gap exceedirLg the maximum tolerance of 0.05 mm was not allowed. To ensure that both ends of the specimen did not warp during the experiment, aluminium plates of 5 mm thickness were glued onto both the open ends of the specimen, using quick-setting Araldite 2002. These aluminium plates would also help to take up any slight irregularities on the specimen ends, which may still exist after careful machining and filing. The length of the test specimens was required to be sufficiently short so that the behaviour will not be affected by the overall buckling, but long enough to reflect the local buckling behaviour of the individual component plates. These requirements, therefore, set bounds on the length of the specimens.
11'88 9'19 7'19 8"38 12.38 9'23 7'38 7'88 12 9'18 7'94 8'13 8.6
C0-8/1/1 C0"8/1/2 C0-8/2/1 C0"8/2/2 C0"8/3/1 C0"8/3/2 C0"8/4/1 C0"8/4/2 CI'0/1/1 C1"0/1/2 C1.0/2/1 C1-0/2/2
267.36 248.82 248'82 291.85 337'59 265.12 337.59 337'59 263"61 293"13 293.13 226-77 226-77
26.38 19'5 14.5 9'63 26'5 19.34 15.38 9'75 26'5 18.23 14'16 15'9 9"88
0'89 0'89 0'89 0"88 0"99 1'09 0"98 0"99 1.15 1"26 1-26 1.15 1"15
TO'8/1/1 T0"8/2/1 T0"8/3/1 T0-8/4/1 TI'0/1/1 Tl'0/2/1 T1.0/3/1 T1.0/4/1 T1.2/1/1 T1.2/2/1 Tl'2/3/1A T1.2/3/1B T1'2/4/1
75'4 76-8 75'5 75'9 76'6 76.02 74'42 74'91 76'4 76'74 76'36 75'21 75.11
No.
80.6 77.4 77.4 79"4 75"8 77.4 79-4 78.2 78"2 77-3 77"1 79"8 79"8
Spec.
try
b2
(N/mm2)
b
b1
I
I b2
t
bw
bw
b
Top-hat section
b
[
bI
Spec. No.
b2l
b3
0-81 0-79 0"81 0.79 0-8 0"81 0"79 0"78 0"96 0-96 0"96 0"96
t
TABLE 1 Specimen Dimensions (All Dimensions in mm)
75"56 74.91 75-46 76"41 75-71 74.12 75-46 75-82 75.24 75.34 75-81 75.67
bw
bt
76'15 76'59 75"6 76"15 74.84 74"83 75-59 74.22 76.46 76-69 76"56 76"38
b
21"72 21"79 18"17 18.18 13.04 13"06 8-27 6"27 21.78 21"04 18'03 18'61
b1
Channel section
bI
5'81 6'6 8-84 9.45 6"56 6.15 7'25 6'45 6'1 6'73 9"23 9"26
b2
ff y
281"7 281.7 281'7 287 281.7 281'7 281.7 281"7 304"45 304'45 304'45 304"45
(N/mm2)
ta,a
T0-8/1/2 T0.8/2/2A T0.8/2/2B T0.8/3/2 T0-8/4/2 T1.0/1/2 T1.0/2/2 T1.0/3/2 TI.0/4/2 T1.2/1/2 T1.2/2/2 T1-2/3/2 T1.2/4/2
0'87 0'87 0'897 0-86 0"887 0"99 0"98 0'98 0'99 1"13 1-16 1"15 1'15
77"79 80-17 78'58 77"4 76"99 77"4 77-4 77'4 76"99 75"2 78-58 77'4 78'18
50"4 51 50"9 49"61 50"01 50"4 50-2 49"6 50"4 50"8 50"4 50'8 50"2 25"9 20 19"63 15 9'31 26"88 19"18 15"63 9"625 24"5 19"85 15"38 9-125 9"5 9"5 7'75 8'38 12'26 9-25 7"69 8-13 11"31 9"88 7"63 8-25
11"75
248'82 248'82 267'36 248"82 248"82 337'59 337"59 337'59 337"59 226"77 263'61 226"77 226"77 C1.2/2/1 C1.2/2/2 C1.2/3/1 C1.2/3/2 C1.2/4/1 C1-2/4/2
c1-2/1/2
C1.2/1/1
cFo/4/2
C1.0/3/1 C1-0/3/2 C1.0/4/1 0'98 0"98 0"96 0"96 1"15 1"14 1'14 1'13 1-13 1"14 1"13 1"15
75"9 75'82 75'01 75"94 75"52 75-46 75"06 75'77 75'7 75-56 76"59 77"68 76"01 76"23 76"34 75-71 76"57 76"31 74-39 76'37 76-22 76"43 76"78 77"54
14'52 14'33 8-61 8"53 21"22 21"2 18-69 18'71 14'8 14"8 7"81 6-76
7"0 6-73 7"88 7'45 6"2 7"03 9-03 9"36 6"83 7"36 7"81 7'56
304"45 304"45 304"45 304"45 235"45 296-8 235"45 296"8 235"45 296"8 235"45 296"8
,g
t~
36
L. K. Seah
2.2 Test Equipment A Tinius Olsen electro-mechanical testing machine was used in applying uniform end displacements during the experimental investigation. The compressive load magnitudes and applied end-displacements were plotted automatically on the machine drum plotter with the use of a deflectometer. The test rig consisted of a special loading head and ground plates. This special loading head was designed to adapt onto the Tinius Olsen machine's top platten and was able to provide uniform compression. A linear variable differential transformer (LVDT), used as a contact probe, and a position transducer were used in conjunction to record the out-of-plane deflection of the flange. As the contact probe was moved along the longitudinal direction of the stiffened flange, the recorder would record any small out-of-plane deviations and the corresponding distance in that direction. Hence, when the probe was run along the stiffened flange, the resulting trace represented the combined component of local and torsional deflections.
2.3 Test procedures A prepared test column was placed in the Tinius Olsen testing machine. The two ends of the column, with glued aluminium end plates, were set between the ground plate and loading head, which had been fixed onto the lower and top platten of the machine, respectively. The loading head was then carefully lowered into position. The screws on the loading head were tightened. The LVDT probe was then fixed into position and moved along the entire length of the stiffened flange to ensure the whole moving frame carrying the LVDT probe was lying parallel to the test specimen. Initial imperfections along the flange were then recorded at the zero loading condition. At this point, the specimen as then pre-stressed to one-third of the theoretical buckling load, to get rid of any slackness in the set-up. It was then unloaded to the zero loading condition again. Each specimen was loaded axially by uniformly lowering the top platten of the testing machine at a rate of 0.05 in./min. At suitable loading intervals, the loading machine was stopped momentarily to take recordings of the out-of-plane deflection of the stiffened flange. The process of loading was continued until any further increase in applied displacement caused a drop in the load-carrying capacity of the specimen. For observation purposes, further loading into the post-failure range was also carried out.
37
Strenoth of thin-walled sections
3 EXPERIMENTAL RESULTS AND OBSERVATIONS The values of the experimental failure load and the deflected form of the stiffened flange for all the 50 specimens were recorded. From the experiments, it was observed that generally there are three types of failure; the plate local buckling mode, the stiffener buckling mode and the interaction of the two modes. The two principal mode are shown in Fig. 1. The local buckles formed on the web and flanges are characteristic of the plate local buckling mode, and this is the governing failure mode for sections with adequately stiffened flanges. The stiffener buckling mode is characterized by the displacement of the edge stiffeners in the direction perpendicular to the flanges and is the governing failure mode for sections with inadequately stiffe11Ledflanges. Figures 2-4 show some of the typical flange deflection variations along the length of the struts. From these figures, it is observed that sections with lip size No.4 usually failed by the stiffener buckling mode. All other lip sizes tended to be adequately stiffened and the failure was initiated by the plate local buckling mode. For failure initiated by the stiffener buckling mode, in each case some local buckles were formed prior to collapse. This
1
|--
Fig. 1. Bucklingmode of edge-stiffenedsections.
Specimen No.
: C0.8/1/2
0
\".-.-Jl
Fig. 2. Out-of-plane deflectionof flange.
38
L. K. Seah
1
Fig. 3. Out-of-plane deflection of flange.
i
m, - -
o
r I~ ~f--~"='qE~"
Specimen No. : TI.0/3/2
Fig. 4. Out-of-plane deflection of flange.
type of failure is characterized by the rapid growth of torsional deflection of the flange, with the web remaining plane and straight. The presence of the lip does help to minimize or eliminate local buckling of the flange, right up to failure by sudden collapse. For failure initiated by the plate local buckling mode, in most of these cases there were initially some torsional deflections of the flange which were caused by initial imperfections. The local buckles formed and grew very rapidly and were symmetrical with stationary node points. There is another mode of failure which is characterized by the sudden collapse of the web. Figure 4 shows a sudden increase in the flange out-of-plane displacement during the failure of the web. This usually occurred in those top-hat specimens with flanges width of 50 mm. It can be seen from this figure that the flange remains straight or with a little waving, right up to the collapse of the specimens. Figures 5 and 6 show some collapsed test specimens. The specimens shown were compressed significantly into the post-failure range and therefore tend to show deformations additional to those which first occurred at failure.
Strength of thin-walled sections
Fig. 5. Collapsed top-hat sections.
Fig. 6. Collapsed channel sections.
39
L. K. Seah
40
4 ULTIMATE STRENGTH DESIGN CONSIDERATION Generally, the ultimate strength of a thin-walled section made up of edge-stiffened elements is determined by treating the section as an assembly of stiffened and unstiffened plate elements. Stiffened elements are elements which have both unloaded edges supported, and unstiffened elements have only one edge supported. The summation of the ultimate load-carrying capacity of each component plate element, determined using the effective width approach, gives the total ultimate load-carrying capacity of the whole section. Stiffened elements can be further classified into adequately and inadequately stiffened elements, depending on the flexural rigidity of the stiffener. Therefore, the design rules for use in predicting the ultimate strength of an edge-stiffened thin-walled section are basically equations for assessment of: (i) (ii) (iii) (iv)
The The The The
stiffener adequacy. effective width of an adequately stiffened element. effective width of an inadequately stiffened element. effective width of an unstiffened element.
Also used are limiting b/t ratios for particular types of stiffener. The effective width equations in the current revised version of Eurocode 6 with some modifications will be u/sed to predict the ultimate strength of a compound edge-stiffened member. In calculating the effective section properties, the effective width is assumed to be located next to the supported edges, equally disposed between these edges for the case of a stiffened plate element. For an unstiffened plate element, the effective width is assumed to be located next to the singly-supported edge. The calculation of effective width is based on the middle line dimensions of the element.
4.1 Edge Stiffener Rigidity Requirements The addition of longitudinal stiffeners, though representing a relatively small part of a structure, substantially enhances the local buckling strength. If, however, the flexural rigidity of the stiffener is not adequate, the stiffeners will buckle in addition to the local plate buckling, this is referred to as the stiffener buckling mode. The edge stiffener must therefore possess a certain specified minimum flexural rigidity, in order to constrain an otherwise unstiffened element to behave and fail as a stiffened element. It is also known that the stiffener rigidity requirements for such an element are substantially dependent on the restraint against rotation by the adjacent element. That is to say, the difference in geometries of sections
41
Strength of thin-walled sections
will influence the stiffener rigidity requirements, this influence is clearly shown by Lim 5 in his theoretical and experimental investigation of single-fold lipped channel sections, and by Rhodes's 7 finite strip investigation. Rlhodes 7 and Lim 5 show that when the rotational restraint 'R' increased, the stiffener rigidity requirements reduced proportionally. When the b/t ratio increases, the stiffener rigidity requirements increase for a particular section geometry. To the authors' knowledge, only the current proposed Eurocode, 6 and the design rules proposed by Ref. [8], have taken into account the influence of section geometry on stiffener rigidity requirements. In the proposed Eurocode, 6 the rigidity requirements for the edge stiffener were based on an effective moment of inertia of the edge stiffener area, including the adjacent part of the plane element. The requirements are as given below:
IR >0.121 (1 + b~) (__~)2(~) 3
(1)
where IR = m o m e n t of inertia of the stiffener with area AR about axis a-a, as shown iLn Fig. 7, AR =effective area of the edge stiffeners be1; and be2 and including the adjacent plane element heR; Aa=(b~R÷ bet +b~2)t, to be determined at ac=0"5ay; b~R and b~l are estimated as a stiffened element and be2 as an unstiffened element; bw = width of adjacent web; b = width of plate element to be stiffened.
The stiffener rigidity requirements, set out by eqn (1) are different from the present British Standard 9 and AISI specification, t° which do not consider b ben
b I-
belt
i
i
beA
=
I~
&it, lit, (s) Simple lip
(b) Compound lip
Fig. 7. Edge-stiffenedelement.
b~R
_
L. K. Seah
42
the influence of section geometry on the stiffener rigidity requirements. Apart from that, the AISI specification 1° is not quite applicable to predicting the ultimate load of a compound or a double-fold edge-stiffened element, since the determination of the elastic buckling coefficient for a compound edge-stiffened element is not clear. 4.2 Adequately edge-stiffened element An edge-stiffened element is referred to as an adequately stiffened element in the proposed Eurocode 6 when the stiffener rigidity satisfies the requirement given by eqn (1). The proposed Eurocode, 6 which is in draft form at present, and possibly subjected to alterations, is used with some modifications in predicting the ultimate strength of the adequately stiffened element. The effective width equations are rewritten here for convenience: be
b
P
(2)
The reduction factor p at yield is given by, when 2y ~<0.673, p=l when 2 r > 0.673,
p --
(1_°22]
(3)
,,~y
where p ~<1, K is the buckling coefficient equal to 4 and
=
trCR
Referring to stiffened, the area of the according to
0"CR= K
12(1 --V
2)
(4)
Figure 7, when the edge stiffened element is adequately above equations are used to determined beA. The effective edge stiffeners and the adjacent plane element at yield, the proposed Eurocode, 6 is then
ARef = 0"5XAR
(6)
Strength of thin-walled sections
43
That is to say, the stiffened member does not fully behave as adequate when it is adequately stiffened. This is to account for the reduction in load-carrying capacity due to slight bending of the stiffeners caused by the Shifting of the effective stiffener neutral axis, as shown in Fig. 8. The change in the effective neutral axis position is, in turn, due to the different rate of reduction in effectiveness of the plate and stiffeners. The effect on the load-carrying capacity will be more noticeable for compound edge-stiffened members than for simple edge-stiffened members. One interesting point is that this bending induced at the lips is probably the reason why the top-hat and channel section buckle in the opposite direction at failure; this is illustrated in Fig. 8 and is also clearly shown in Figs 5 and 6 of the collapsed test specimens. As mentioned previously, the proposed Eurocode 6 imposes a reduction in the effective area of stiffener and adjacent plane element, i.e. ARe f =0"5XAR, at yield. However, this has a disadwmtage when the b/t ratio is small, because the element would then be fully effective in reality. Hence the proposed Eurocode 6 will give over-conservative results for elements with a small value of the b/t ratio. This overt-conservatism is very noticeable when 2 is less than 0.673, that is when b/t is less than about 35 for a yield stress of 280 N/mm 2, since at this lower range of b/t ratios, the element should be considered as fully effective. To overcome this disadvantage, the following design rules are suggested.
_
,I ,I
.able
i I
I I
I
I I
~". I
I
% I ~%!
I
!,
I I I I
!
II I
II
(a) Top-hat section . . . .
Initial neutral axis
. . . .
Shifted neutral axis
(b) Channel section
Fig. 8. Buckled shape of edge-stiffened sections.
L. K. Seah
44
At yield, the effective area of the stiffeners and adjacent plane element are (referring to Fig. 7): A stiff ARe f = beRt + - -
(7)
2
where Astiff=area of stiffeners only, equal to t(bel +be2) at half yield, b'eR = the effective adjacent plane element at twice yield. Figure 9 shows a comparison of the ultimate load predicted by the proposed design rules with the proposed Eurocode 6 for a given adequately compound edge-stiffened channel section with h=0.2, bw/b=l and b2/bl =0-1. In this figure, the ultimate load predicted by the current British specification is taken as the reference. The main purpose of this figure is to show the over-conservatism of the proposed Eurocode at bit ratios less than 35. The proposed design rules are in better agreement with the British specification at this low range of b/t ratios and, as the bit ratio increases, the proposed design rules become very close to the proposed Eurocode. This is further confirmed by comparing with the experimental results of Desmond et al. 2 on single-fold lipped channel sections, shown in Table 2. In this table, specimens with adequate lips and b/t less than 35 are E-21.4-6.69, E-23-9-2.87 and E-23.9-5.26. For these three specimens the proposed Eurocode underestimates by about 21-25% and the proposed rules are in much better agreement with Desmond's experimental results (about a 11-15% underestimate). 0.93 0.92 0.91 0.90 0.89 0.88 0.87 0.86 0.85 0.84 0.83 0.82 0.81 0.80 0.79 0.78 -0.77
Proposed
,t
•, ,"
try = 280 N m m "2
I
t
Eurocode
Fig. 9. C o m p a r i s o n
p
] 20
i
I I t I
....
Pee / PBS
~
P~/Pss
t
I 4O
I
I
6O
8O
I00 b/t o f u l t i m a t e l o a d p r e d i c t e d b y p r o p o s e d d e s i g n rules a n d p r o p o s e d Eurocode. 6
TABLE 2
2-4 2-4 2-4 3"5 3-5 3-5 3-5 2"55 2-55 2'55 2"55 2'55 2-55 3-75 3"75 3"75 3"75 3"75 3-75 3-75 3"75 3-75 3"75
1'6 1"6 1"6 2-5 2"5 2'5 2'5 3.4 3.4 3'4 3.4 3'4 3'4 6'4 6.4 6.4 6-4 8-7 8"7 8"7 8"7 8"7 8'7
0 0-1 0"5 0 0-05 0-3 0'55 0.37 0'6 0.75 1"12 1-5 1"85 0"66 0-99 1-2 1"98 0"65 0"99 1.2 1'65 2'11 2"5
0"0747 0"0747 0.0747 0-1046 0"1046 0.1046 0'1046 0.0745 0.0745 0"0745 0-0745 0"0745 0.0745 0"066 0-066 0"066 0"066 0"066 0-066 0'066 0'066 0'066 0-066
t PEX
19"4 20 24'5 43'9 45 57 60 41 46"6 46-5 51 49 47 34'7 34 38 38 31 36 35-5 35 39"4 39
10 10 10 18 18 18 18 18 18 18 18 18 18 30 30 30 30 30 30 30 30 30 30
bl
40"3 40"3 40-3 47"6 47-6 47-6 47"6 54 52'7 52.7 53"9 54 55-7 36"6 36"6 36"6 36-6 33 33 33 33-3 34"7 33
b
E-21.4-0.0 E-21-4-1"33 E-21.4-6'69 E-23"9-00 E-23-9-0"48 E-23"9-2-87 E-23.9-5.26 E-45'6-5"0 E-45"6-8-05 E-45"6-10"1 E-45-6-15 E-45-6-20"1 E-45.6-24"8 E-97"6-9'94 E-97-6-15 E-97.6-18-2 E-97-6-30 E-133-9"89 E-133-15 E-133-18.2 E-133-25 E-133-32 E-133-38
bw
(kip)
L
Qy
Specimen No. 0-90 1-03 0"89 0"87 1"05 0-87 0-85 0-95 0"88 0"89 0"84 0"88 0-94 0"90 0"99 0"89 0-90 0"88 0"83 0-88 0"91 0.84 0.82
0.879 0-853 0"953 0-851 0.830 0"656 0"895 0"-728 0-627 0"996 0-948 0"996 1.069 0-746 0"761 0-681 0.964 0-801 0"690 0-700 0"716 0"899 0-880
0.861 1.018 0"996 0"842 0'929 0"879 0"940 0-981 0"936 0"992 0"936 0-989 1.067 0"933 1"016 0-942 0"962 0"953 0-868 0"908 0"969 0-898 0-880
0.897 0-912 0"793 0.872 0"935 0"765 0.747 0"938 0-862 0.877 0.827 0.868 0'931 0.913 0.991 0"891 0"898 0.923 0"837 0.879 0'921 0-848 0.825
PEF/PEx PBs/PEx PR/PEx Pcu/PEx
Comparison of Experimental and Predicted Ultimate Load for Desmond's 2 Tests
ga
46
L. K. Seah
4.3 Inadequately edge-stiffened element An edge-stiffened element is referred to as an inadequately stiffened element when the flexural rigidity of the edge stiffener does not satisfy the requirement set out in eqn (1). In this case, the ultimate strength lies between the two limits, which is the ultimate strength of an adequately stiffened element and that of an unstiffened element. The design rules used in the proposed Eurocode 6 for determining the ultimate strength of an inadequately stiffened element are far too tedious for design purposes. Therefore, the following expression is proposed for predicting the ultimate strength of an inadequately stiffened plate:
Pinad
IR ) 1/n =Pu +(Ps--Pu) ~
(8)
where Pinad=Ultimate strength of an inadequately stiffened element; Pn = ultimate strength of an unstiffened element; Ps = ultimate strength of an adequately stiffened element; IRmi. =minimum required value of moment of inertia of the stiffener for adequacy, set by eqn (1); Is = t h e moment of inertia of the actual stiffener with area AR determined at yield stress, about the axis a - a shown in Fig. 7; n = variable index. The variable index, n is found to be equal to 3, which gives a best-fit curve through the experimental points.
5 COMPARISON OF EXPERIMENTAL AND PREDICTED ULTIMATE STRENGTH Table 3 and 4 show the comparison of the compound edge-stiffened sections and elements. Tables 2 and 5 show the comparison with Lim's 5 and Desmond's e t al. 2 experimental results of single-fold lipped channel sections. The values of the predicted ultimate strength given in the tables are based on the actual plate dimensions of the test specimens. The values of the predicted ultimate strength calculated based on the suggested design rules agree well with those obtained experimentally for both compound and simple edge-stiffened (Lim and Desmond) sections. The maximum overestimation and underestimation of the ultimate strength of compound edge-stiffened sections are 12 and 24%, respectively, and for the simple edge stiffened sections are 20 and 22%, respectively. The British Standard 9 and Ref. [8] design method agree well with the simple edge-stiffened sections but tend to overestimate the ultimate strength for the compound
PEX
35.463 39.357 36.437 32.496 52.708 50.205 44-364 42"278 70.926 70.231 67.727 58.410 56.741
Spec. No.
T0.8/1/1 T0.8/2/1 T0.8/3/1 T0.8/4/1 T1.0/1/1 T1-0/2/1 T1.0/3/1 Tl'0/4/1 TI.2/1/1 TI.2/2/1 T1.2/3/1A T1.2/3/1B T1.2/4/1
TABLE 3
1-013 0.805 0.839 0.989 0.949 0.933 0.992 1"018 0.766 0-902 0.906 0-771 0-768
1.236 0.962 0.985 1-057 1.160 1.129 1-165 1"082 0.950 1.098 1-088 0.936 0-910
1.012 0.805 0.838 0.991 0.949 0.926 0.991 1018 0.759 0.893 0.896 0.761 0.758
1.247 0.957 0-964 0.584 1.173 1.110 1.146 0'602 0.946 1-074 1.055 0.908 0.883
PEF/PEx PR/PEx Pcu/PEx PBs/PEx 39-774
PEX
T0.8/2/2B T0.8/3/2 T0-8/4/2 T1-0/1/2 T1-0/2/2 T1-0/3/2 T1-0/4/2 T1.2/1/2 T1.2/2/2 T1.2/3/2 T1.2/4/2
36.993 30.317 32.543 53-403 47.284 46.867 43.251 46.032 62.721 53.542 51.734
TO.8/2/2A 33-933
T0.8/1/2
Spec. No. 0.797 0-886 0.891 0-924 0.866 0-907 0-937 0.908 0.959 0.960 0.791 0-788 0.786
1.004 1.095 0.915 1-125 1.024 1.146 1.155 1.107 1.127 1.215 0.986 0.969 0.939
0.786 0.874 0-879 0.910 0-852 0.895 0.924 0.895 0-945 0-896 0.740 0.729 0-724
1.012 1-092 0.912 1.108 1.011 1.158 1.152 1.091 1.117 1.202 0.970 0-935 0.908
PEF/PEx PR/PEx Pc./PEx PBs/PEx
Comparison of Experimental and Predicted Ultimate Load for Top-Hat Sections
=,
L. K. Seah
48
TABLE 4
Comparison of Experimental and Predicted Ultimate Load for Channel Sections
Spec. No
PEX
C0"8/1/1 C0"8/1/2 C0"8/2/1 C0"8/2/2 C0"8/3/1 C0.8/3/2 C0-8/4/1 CO'8/4/2 C1.0/1/1 C1.0/1/2 C 1.0/2/1 C1"0/2/2 C 1-0/3/1 C1'0/3/2 C1"0/4/1 C1"0/4/2 C1-2/1/1 C 1-2/1/2 C1-2/2/1 C1"2/2/2 C1"2/3/1 C1"2/3/2 C1.2/4/1 C1'2/4/2
33'377 31'013 31"569 33-989 25"611 26.069 24.476 20"973 47"562 49"370 48"118 47-006 42-417 43'529 37"549 33"655 55"350 59-383 47.714 54"238 41.026 52"152 46-728 47"562
PEF/PEx 0"879 0.913 0"925 0-828 1.051 1'050 0.996 1.054 0"866 0-834 0-849 0-874 0"953 0"925 1.009 1.121 0-846 0-910 0-969 0'984 1"073 1"003 0-913 1"010
PR/PEx
Pcu/PEx
PBs/PEx
1"068 1"108 1"095 0-977 1'203 1.206 1"075 1-151 1.055 1.011 1-006 1-039 1"118 1"086 1"060 1-175 1"048 1"115 1'190 1'187 1"296 1-196 1.012 1.076
0"882 0.917 0-928 0-832 1"054 1"053 1"005 1"090 0"866 0.834 0.848 0"873 0-952 0"924 1-008 1-120 0"836 0"904 0"958 0"978 1"061 0-996 0"903 1"008
1.046 1"093 1-096 0-984 1"197 1-194 0-626 0-719 1"028 0"989 1"002 1-035 1"094 1-060 0-604 0"673 1"009 1"083 1"162 1"170 1"250 1"162 0-556 0.643
For Tables 2-4, the ultimate strength Pi is in kN. PEX = ultimate strength obtained from experiment. PEF = ultimate strength based on proposed design rules. PBs = ultimate strength based on British Standard. 9 edge-stiffened sections. N o t e t h a t for the i n a d e q u a t e l y stiffened sections, a l t h o u g h it will h a v e s o m e increase in l o a d - c a r r y i n g c a p a c i t y , it is n o t a l l o w e d to utilize this in the British S t a n d a r d . B e c a u s e o f this, a v e r y high u n d e r e s t i m a t i o n o f the u l t i m a t e s t r e n g t h for the i n a d e q u a t e l y stiffened sections is o b t a i n e d f r o m the British S t a n d a r d .
CONCLUSIONS T h e i n v e s t i g a t i o n r e p o r t e d h e r e s h o w s t h a t the shifting o f the stiffener effective n e u t r a l axis, d u e to the different r a t e of r e d u c t i o n in the effectiveness o f p l a t e a n d stiffeners, causes a significant r e d u c t i o n in the l o a d -
Strength of thin-walled sections
49
TABLE 5 Comparison of Experimental and Predicted Ultimate Load for Lim's~ Tests
Spec. No.
b/t
h
PEx
PEv/PEx P n s / P E x
Pa/P~x
P ~/PEx
A1 A2 A3 A4 A5 A6
122"5 125-2 124-3 124"2 125"7 122"9
0 0-092 0.17 0"262 0"333 0-428
10"12 11-86 17.64 18"69 19"58 18-55
0"88 1"06 0"85 0-83 0-79 0"84
1.009 0-851 0"578 0-880 0"846 0.916
0"942 1"107 0-837 0.879 0-847 0"916
0"882 1-139 0"845 0-841 0"801 0"853
B1 B2 B3 B4 B5 B6
62'2 63.4 63"5 63"8 63"8 63"4
0 0"097 0"173 0"26 0"34 0-418
27"61 48.49 63'18 66.74 68"96 66"07
1"20 1"01 0.87 0.84 0-82 0"87
1"230 0.694 0"535 0"942 0"923 0-982
1"223 1"023 0.912 0.926 0.913 0"977
1-203 1"017 0"863 0.828 0"810 0-859
C1 C2 C3 C4 C5 C6
84"8 87-6 87"3 86"8 87-2 87-5
0 0"088 0.169 0.253 0.334 0.408
18-11 23-84 33'08 37'82 37.37 40.95
1-03 1.09 0"96 0"85 0.86 0-79
1-092 0"825 0-600 0.911 0-923 0"859
1"059 1.121 0"930 0-903 0.921 0.859
1'028 1-140 0"965 0"851 0-858 0-793
h=bl/b. PR = ultimate strength based on Rhodes 8 design method. Pcu =ultimate strength based on proposed Eurocode. 6 3O q
av. b/t = 95.7
26I ]
X ~
23 22f 21 20 19 18
. . . . . . .
X
~ p
~
•
.°..'"
I
. . . . . . . . . . . .
II
X
....""
16 15
:14 1!
X
.....-"
17
13 12
X
I 2
............
Proposed Eurocode [ 6 ]
x
Ref. [81 Experimental
I 4
I J P x tip Fig. 10. Comparison of experimental and predicted ultimate load.
L. K. Seah
50
J
20 19 18 17 -16 - 15 - 14 - 13 12 11 10 _ 9 8 _ 7 65 4_ 3 2 1
X X S • J X
X x
,
.....
Ir . . . . . . . . . . . . . . . . . .
"-
av. b/t = 67.2 - Proposed ............ E u r ~ l e [6] ---Ref.[8] x Experimental
I 2
0
~
I 4 Ip/t4 x 105
~
I 6
Fig. 11. C o m p a r i s o n o f e x p e r i m e n t a l a n d p r e d i c t e d ultimate load. 14 13 12 11 - -
X
10 9
X
8 tJ
7 6 Proposed ............ Eurocode [6] .... Ref. [8] x Experimental
5 4 3 2 I
0
I 2
I 4
6 Ip/t4 x 105
I. 8
I 10
12
Fig. 12. C o m p a r i s o n o f e x p e r i m e n t a l a n d p r e d i c t e d u l t i m a t e load.
carrying capacity. This effect has been neglected in the past for the simple edge stiffener case. It also provides an explanation for why the top-hat and channel sections buckle in opposite directions at failure, which was observed during experiments. The experimental study unfortunately does not reveal the optimum stiffened plate width to thickness ratio (b/t). However, based on the highest
Strength of thin-walled sections
51
value of b/t ratio (i.e. equal to 97) tested, the limiting value of b/t of 90 for the compound edge stiffener in various design specifications, is at least justified. The ultimate load of a compound edge-stiffened element is higher than for a simple edge-stiffened element of the same area, when the width of the simple edge stiffener becomes excessively large. From the results obtained, it can said that the proposed design procedures can predict, with reasonable accuracy and consistency, the ultimate strength of sections with simple or compound edge-stiffened elements.
REFERENCES 1. Kloppel, K. v o n & Unger, B., Buckling and post-buckling of the plate with three sides simply supported and stiffened alongside the free edge using non-hnear buckling theory. Der Stahlbau, 10, 1969 289-299 (in German). 2. Desmond, T. P., Pekoz, T. & Winter, G., Edge stiffeners for thin-walled members. Proc. ASCE J. Struct. Div., 107 (ST2) (1981). 3. Rhodes, J., Buckling and failure of edge-stiffened plates IUTAM, Syrup. on Collapse, London, August 1982. 4. Bulson, P. S., Bulbs, lips and beads. Proc. Prof. J. M. Harvey Retiral Conf. on the Behaviour of Thin-walled Structures, ed. J. Rhodes & J. Spence. Applied Science, 1983. 5. Lim, B. S., Buckling behaviour of asymmetric edge-stiffened plates. Ph.D. thesis, University of Strathclyde, Glasgow, 1985. 6. Eurocode 3 Annex A in draft form, For the Design of Light Gauge Steel Members and Sheeting, 1989. 7. Rhodes, J., A simple micro computer finite strip analysis. Dynamics of Structures, Proc. of the session at Structures Congress '87. ASCE, 1987 pp. 2:r6-291. 8. Rhodes, J., Research into the Mechanical Behaviour of Cold-Formed Sections and Drafting of Design Rules. ECSC Contract No. 7210/SA/608, University of Strathclyde, April 1987. 9. British Standards Institution, Code of Practice for Design of Cold Formed Sections, B.S. 5950:Part 5, 1987. BSI, London, 1987. 10. Amer:ican Iron and Steel Institute, The Specification for the Desi#n of ColdFormed Steel Structural Members, August, 1986. 11. Seah, L. K., Buckling behaviour of edge stiffeners in thin-walled sections. Ph.D. thesis, University of Strathclyde, Glasgow, 1989.
ELSEVIER
Ultimate Strength of Uniformly Compression Edge-Stiffened Thin-Walled Sections L. K. Seah School of Mechanical & Production Engineering, Nanyang Technological University, Singapore 2263 (Received 24 January 1994; revised version received 4 July 1994; accepted 15 August 1994)
ABSTRACT This paper reports an investigation" into the collapse behaviour of uniformly compressed edge-stiffened thin-walled sections. The term 'edge stiffener' refers to a stiffener in the form of a single or double right-angled lip formed at the free edge of the un;~tiffened element. The primary function of the edge stiffener is to economically enhance the local buckling of an open thin-walled section. An outline of a series of tests on edge-stiffened thin-walled sections of various geometries is given. Design procedures are proposed based on the modifications of the draft Eurocode to predict the ultimate strength of such edge-stiffened thin-walled section,s. The empirical results obtained are then compared with existing experimental res~,~lts, and close agreement is found. A further comparison with an independent source of experimental results confirms the rationality of the approach.
LIST O F S Y M B O L S
AR
A stiff b bl b2 bw b 'eR
effective area of the edge stiffeners be1, be2 and including the adjacent plane element, beR; AR=(beR+ b~l +b¢2)t, to be determined at o-c=0.5ay, beR and b~l are estimated as a stiffened element and bo2 as an unstiffened element. ~Lrea of stiffeners only, equal to t. (be~ +b~2) at 1/2 yield. width of plate element to be stiffened width of first stiffener width of second stiffener width of adjacent web. effective adjacent plane element at twice yield: 31
32
D E IR I Rmin IR n
P inad Pu Ps t O'y
L. K. Seah
plate flexural rigidity Young's modulus of elasticity moment of inertia of the stiffener with area A R about axis a-a, as shown in figure (7). Minimum required value of moment of inertia of the stiffener for adequacy, set by equation (1). The moment of inertia of the actual stiffener with area A R determined at yield stress, about the axis a-a shown in figure (7) variable index Pb non-dimensional axial l o a d , / ~ - g 2 D Ultimate strength of an inadequately stiffened element Ultimate strength of an unstiffened element Ultimate strength of an adequately stiffened element plate thickness yield stress
1 INTRODUCTION Local buckling is a prominent behaviour of thin-walled sections. It causes a reduction in the stiffness of a section and a lowering of the load-carrying capacity. In order to improve these buckling characteristics, edge stiffeners are widely used to stiffen the flat compressive elements of thin-walled structural members which can be found in various engineering fields. The primary function of the edge stiffener is to conomically enhance the local buckling strength of an open thin-walled section. The buckling behaviour of edge-stiffened thin-walled sections has attracted the attention of large number of researchers over many years ~- s One of the most common types of edge stiffener used in cold-rolled sections is formed by turning the free edge of the unstiffened flange inwards or outwards to form a lip. Past investigations of this type of edge stiffener have tended to concentrate on single-fold lipped plate elements of thin-walled sections. However, for sections with high width to thickness ratios, the width required for a simple lip would be such that the lip itself would begin to have stability problems. In this paper, the collapse behaviour of compound right-angled lipstiffened sections is examined. The investigations here are confined to the case of shot, asymmetrically edge-stiffened, thin-walled channel struts, subjected to uniform end compression. The edge stiffened sections under these circumstances can fail, due to elastic local buckling of plate and stiffener, torsional buckling of the edge stiffened flange or a
Strength of thin-walled sections
33
combination of these two buckling modes, depending on the stiffener's flexural rigidity. An outline of a series of tests on outwardly-and inwardly-turned lipped channel sections of various geometries is given. Design procedures are proposed based on the draft Eurocode 6 to predict the ultimate strength of such edge-stiffened thin-walled sections. Reasonably good agreement is obtained between the empirical and experimental results.
2 EXPERIMENTAL INVESTIGATION
2.1 Tes! specimens A total of 50 specimens were manufactured and tested under uniform end compression, of which 26 specimens were outwardly-turned lipped tophat secl:ions and 24 specimens were inwardly-turned lipped channel sections, The prime objectives of the investigation were to examine the effects of adjacent elements on the behaviour of compound edge-stiffened elements and to devise a simple design criterion for sections having compound edge stiffeners. In the following sections, the tests perfomed in this investigation are described. The cross-sectional geometry, dimensions and material yield stress of all the specimens are shown in Table 1. Tensile test specimens were cut and tested from all sheets of material used, so that each specimen could be analysed on the basis of the true yield stress. The three nominal thicknesses of steel sheet used were 0.8, 1-0 and 1-2 mm. These specimens were manufac,tured by cold-folding, using a 4 ft long Oliver Pan Folder, to normal engineering tolerances. The ends of the test specimens were carefully milled and filed to ensure a fiat and plane surface perpendicular to the specimen's longitudinal axis. The ends were then tested for flatness against a surface table and any gap exceedirLg the maximum tolerance of 0.05 mm was not allowed. To ensure that both ends of the specimen did not warp during the experiment, aluminium plates of 5 mm thickness were glued onto both the open ends of the specimen, using quick-setting Araldite 2002. These aluminium plates would also help to take up any slight irregularities on the specimen ends, which may still exist after careful machining and filing. The length of the test specimens was required to be sufficiently short so that the behaviour will not be affected by the overall buckling, but long enough to reflect the local buckling behaviour of the individual component plates. These requirements, therefore, set bounds on the length of the specimens.
11'88 9'19 7'19 8"38 12.38 9'23 7'38 7'88 12 9'18 7'94 8'13 8.6
C0-8/1/1 C0"8/1/2 C0-8/2/1 C0"8/2/2 C0"8/3/1 C0"8/3/2 C0"8/4/1 C0"8/4/2 CI'0/1/1 C1"0/1/2 C1.0/2/1 C1-0/2/2
267.36 248.82 248'82 291.85 337'59 265.12 337.59 337'59 263"61 293"13 293.13 226-77 226-77
26.38 19'5 14.5 9'63 26'5 19.34 15.38 9'75 26'5 18.23 14'16 15'9 9"88
0'89 0'89 0'89 0"88 0"99 1'09 0"98 0"99 1.15 1"26 1-26 1.15 1"15
TO'8/1/1 T0"8/2/1 T0"8/3/1 T0-8/4/1 TI'0/1/1 Tl'0/2/1 T1.0/3/1 T1.0/4/1 T1.2/1/1 T1.2/2/1 Tl'2/3/1A T1.2/3/1B T1'2/4/1
75'4 76-8 75'5 75'9 76'6 76.02 74'42 74'91 76'4 76'74 76'36 75'21 75.11
No.
80.6 77.4 77.4 79"4 75"8 77.4 79-4 78.2 78"2 77-3 77"1 79"8 79"8
Spec.
try
b2
(N/mm2)
b
b1
I
I b2
t
bw
bw
b
Top-hat section
b
[
bI
Spec. No.
b2l
b3
0-81 0-79 0"81 0.79 0-8 0"81 0"79 0"78 0"96 0-96 0"96 0"96
t
TABLE 1 Specimen Dimensions (All Dimensions in mm)
75"56 74.91 75-46 76"41 75-71 74.12 75-46 75-82 75.24 75.34 75-81 75.67
bw
bt
76'15 76'59 75"6 76"15 74.84 74"83 75-59 74.22 76.46 76-69 76"56 76"38
b
21"72 21"79 18"17 18.18 13.04 13"06 8-27 6"27 21.78 21"04 18'03 18'61
b1
Channel section
bI
5'81 6'6 8-84 9.45 6"56 6.15 7'25 6'45 6'1 6'73 9"23 9"26
b2
ff y
281"7 281.7 281'7 287 281.7 281'7 281.7 281"7 304"45 304'45 304'45 304"45
(N/mm2)
ta,a
T0-8/1/2 T0.8/2/2A T0.8/2/2B T0.8/3/2 T0-8/4/2 T1.0/1/2 T1.0/2/2 T1.0/3/2 TI.0/4/2 T1.2/1/2 T1.2/2/2 T1-2/3/2 T1.2/4/2
0'87 0'87 0'897 0-86 0"887 0"99 0"98 0'98 0'99 1"13 1-16 1"15 1'15
77"79 80-17 78'58 77"4 76"99 77"4 77-4 77'4 76"99 75"2 78-58 77'4 78'18
50"4 51 50"9 49"61 50"01 50"4 50-2 49"6 50"4 50"8 50"4 50'8 50"2 25"9 20 19"63 15 9'31 26"88 19"18 15"63 9"625 24"5 19"85 15"38 9-125 9"5 9"5 7'75 8'38 12'26 9-25 7"69 8-13 11"31 9"88 7"63 8-25
11"75
248'82 248'82 267'36 248"82 248"82 337'59 337"59 337'59 337"59 226"77 263'61 226"77 226"77 C1.2/2/1 C1.2/2/2 C1.2/3/1 C1.2/3/2 C1.2/4/1 C1-2/4/2
c1-2/1/2
C1.2/1/1
cFo/4/2
C1.0/3/1 C1-0/3/2 C1.0/4/1 0'98 0"98 0"96 0"96 1"15 1"14 1'14 1'13 1-13 1"14 1"13 1"15
75"9 75'82 75'01 75"94 75"52 75-46 75"06 75'77 75'7 75-56 76"59 77"68 76"01 76"23 76"34 75-71 76"57 76"31 74-39 76'37 76-22 76"43 76"78 77"54
14'52 14'33 8-61 8"53 21"22 21"2 18-69 18'71 14'8 14"8 7"81 6-76
7"0 6-73 7"88 7'45 6"2 7"03 9-03 9"36 6"83 7"36 7"81 7'56
304"45 304"45 304"45 304"45 235"45 296-8 235"45 296"8 235"45 296"8 235"45 296"8
,g
t~
36
L. K. Seah
2.2 Test Equipment A Tinius Olsen electro-mechanical testing machine was used in applying uniform end displacements during the experimental investigation. The compressive load magnitudes and applied end-displacements were plotted automatically on the machine drum plotter with the use of a deflectometer. The test rig consisted of a special loading head and ground plates. This special loading head was designed to adapt onto the Tinius Olsen machine's top platten and was able to provide uniform compression. A linear variable differential transformer (LVDT), used as a contact probe, and a position transducer were used in conjunction to record the out-of-plane deflection of the flange. As the contact probe was moved along the longitudinal direction of the stiffened flange, the recorder would record any small out-of-plane deviations and the corresponding distance in that direction. Hence, when the probe was run along the stiffened flange, the resulting trace represented the combined component of local and torsional deflections.
2.3 Test procedures A prepared test column was placed in the Tinius Olsen testing machine. The two ends of the column, with glued aluminium end plates, were set between the ground plate and loading head, which had been fixed onto the lower and top platten of the machine, respectively. The loading head was then carefully lowered into position. The screws on the loading head were tightened. The LVDT probe was then fixed into position and moved along the entire length of the stiffened flange to ensure the whole moving frame carrying the LVDT probe was lying parallel to the test specimen. Initial imperfections along the flange were then recorded at the zero loading condition. At this point, the specimen as then pre-stressed to one-third of the theoretical buckling load, to get rid of any slackness in the set-up. It was then unloaded to the zero loading condition again. Each specimen was loaded axially by uniformly lowering the top platten of the testing machine at a rate of 0.05 in./min. At suitable loading intervals, the loading machine was stopped momentarily to take recordings of the out-of-plane deflection of the stiffened flange. The process of loading was continued until any further increase in applied displacement caused a drop in the load-carrying capacity of the specimen. For observation purposes, further loading into the post-failure range was also carried out.
37
Strenoth of thin-walled sections
3 EXPERIMENTAL RESULTS AND OBSERVATIONS The values of the experimental failure load and the deflected form of the stiffened flange for all the 50 specimens were recorded. From the experiments, it was observed that generally there are three types of failure; the plate local buckling mode, the stiffener buckling mode and the interaction of the two modes. The two principal mode are shown in Fig. 1. The local buckles formed on the web and flanges are characteristic of the plate local buckling mode, and this is the governing failure mode for sections with adequately stiffened flanges. The stiffener buckling mode is characterized by the displacement of the edge stiffeners in the direction perpendicular to the flanges and is the governing failure mode for sections with inadequately stiffe11Ledflanges. Figures 2-4 show some of the typical flange deflection variations along the length of the struts. From these figures, it is observed that sections with lip size No.4 usually failed by the stiffener buckling mode. All other lip sizes tended to be adequately stiffened and the failure was initiated by the plate local buckling mode. For failure initiated by the stiffener buckling mode, in each case some local buckles were formed prior to collapse. This
1
|--
Fig. 1. Bucklingmode of edge-stiffenedsections.
Specimen No.
: C0.8/1/2
0
\".-.-Jl
Fig. 2. Out-of-plane deflectionof flange.
38
L. K. Seah
1
Fig. 3. Out-of-plane deflection of flange.
i
m, - -
o
r I~ ~f--~"='qE~"
Specimen No. : TI.0/3/2
Fig. 4. Out-of-plane deflection of flange.
type of failure is characterized by the rapid growth of torsional deflection of the flange, with the web remaining plane and straight. The presence of the lip does help to minimize or eliminate local buckling of the flange, right up to failure by sudden collapse. For failure initiated by the plate local buckling mode, in most of these cases there were initially some torsional deflections of the flange which were caused by initial imperfections. The local buckles formed and grew very rapidly and were symmetrical with stationary node points. There is another mode of failure which is characterized by the sudden collapse of the web. Figure 4 shows a sudden increase in the flange out-of-plane displacement during the failure of the web. This usually occurred in those top-hat specimens with flanges width of 50 mm. It can be seen from this figure that the flange remains straight or with a little waving, right up to the collapse of the specimens. Figures 5 and 6 show some collapsed test specimens. The specimens shown were compressed significantly into the post-failure range and therefore tend to show deformations additional to those which first occurred at failure.
Strength of thin-walled sections
Fig. 5. Collapsed top-hat sections.
Fig. 6. Collapsed channel sections.
39
L. K. Seah
40
4 ULTIMATE STRENGTH DESIGN CONSIDERATION Generally, the ultimate strength of a thin-walled section made up of edge-stiffened elements is determined by treating the section as an assembly of stiffened and unstiffened plate elements. Stiffened elements are elements which have both unloaded edges supported, and unstiffened elements have only one edge supported. The summation of the ultimate load-carrying capacity of each component plate element, determined using the effective width approach, gives the total ultimate load-carrying capacity of the whole section. Stiffened elements can be further classified into adequately and inadequately stiffened elements, depending on the flexural rigidity of the stiffener. Therefore, the design rules for use in predicting the ultimate strength of an edge-stiffened thin-walled section are basically equations for assessment of: (i) (ii) (iii) (iv)
The The The The
stiffener adequacy. effective width of an adequately stiffened element. effective width of an inadequately stiffened element. effective width of an unstiffened element.
Also used are limiting b/t ratios for particular types of stiffener. The effective width equations in the current revised version of Eurocode 6 with some modifications will be u/sed to predict the ultimate strength of a compound edge-stiffened member. In calculating the effective section properties, the effective width is assumed to be located next to the supported edges, equally disposed between these edges for the case of a stiffened plate element. For an unstiffened plate element, the effective width is assumed to be located next to the singly-supported edge. The calculation of effective width is based on the middle line dimensions of the element.
4.1 Edge Stiffener Rigidity Requirements The addition of longitudinal stiffeners, though representing a relatively small part of a structure, substantially enhances the local buckling strength. If, however, the flexural rigidity of the stiffener is not adequate, the stiffeners will buckle in addition to the local plate buckling, this is referred to as the stiffener buckling mode. The edge stiffener must therefore possess a certain specified minimum flexural rigidity, in order to constrain an otherwise unstiffened element to behave and fail as a stiffened element. It is also known that the stiffener rigidity requirements for such an element are substantially dependent on the restraint against rotation by the adjacent element. That is to say, the difference in geometries of sections
41
Strength of thin-walled sections
will influence the stiffener rigidity requirements, this influence is clearly shown by Lim 5 in his theoretical and experimental investigation of single-fold lipped channel sections, and by Rhodes's 7 finite strip investigation. Rlhodes 7 and Lim 5 show that when the rotational restraint 'R' increased, the stiffener rigidity requirements reduced proportionally. When the b/t ratio increases, the stiffener rigidity requirements increase for a particular section geometry. To the authors' knowledge, only the current proposed Eurocode, 6 and the design rules proposed by Ref. [8], have taken into account the influence of section geometry on stiffener rigidity requirements. In the proposed Eurocode, 6 the rigidity requirements for the edge stiffener were based on an effective moment of inertia of the edge stiffener area, including the adjacent part of the plane element. The requirements are as given below:
IR >0.121 (1 + b~) (__~)2(~) 3
(1)
where IR = m o m e n t of inertia of the stiffener with area AR about axis a-a, as shown iLn Fig. 7, AR =effective area of the edge stiffeners be1; and be2 and including the adjacent plane element heR; Aa=(b~R÷ bet +b~2)t, to be determined at ac=0"5ay; b~R and b~l are estimated as a stiffened element and be2 as an unstiffened element; bw = width of adjacent web; b = width of plate element to be stiffened.
The stiffener rigidity requirements, set out by eqn (1) are different from the present British Standard 9 and AISI specification, t° which do not consider b ben
b I-
belt
i
i
beA
=
I~
&it, lit, (s) Simple lip
(b) Compound lip
Fig. 7. Edge-stiffenedelement.
b~R
_
L. K. Seah
42
the influence of section geometry on the stiffener rigidity requirements. Apart from that, the AISI specification 1° is not quite applicable to predicting the ultimate load of a compound or a double-fold edge-stiffened element, since the determination of the elastic buckling coefficient for a compound edge-stiffened element is not clear. 4.2 Adequately edge-stiffened element An edge-stiffened element is referred to as an adequately stiffened element in the proposed Eurocode 6 when the stiffener rigidity satisfies the requirement given by eqn (1). The proposed Eurocode, 6 which is in draft form at present, and possibly subjected to alterations, is used with some modifications in predicting the ultimate strength of the adequately stiffened element. The effective width equations are rewritten here for convenience: be
b
P
(2)
The reduction factor p at yield is given by, when 2y ~<0.673, p=l when 2 r > 0.673,
p --
(1_°22]
(3)
,,~y
where p ~<1, K is the buckling coefficient equal to 4 and
=
trCR
Referring to stiffened, the area of the according to
0"CR= K
12(1 --V
2)
(4)
Figure 7, when the edge stiffened element is adequately above equations are used to determined beA. The effective edge stiffeners and the adjacent plane element at yield, the proposed Eurocode, 6 is then
ARef = 0"5XAR
(6)
Strength of thin-walled sections
43
That is to say, the stiffened member does not fully behave as adequate when it is adequately stiffened. This is to account for the reduction in load-carrying capacity due to slight bending of the stiffeners caused by the Shifting of the effective stiffener neutral axis, as shown in Fig. 8. The change in the effective neutral axis position is, in turn, due to the different rate of reduction in effectiveness of the plate and stiffeners. The effect on the load-carrying capacity will be more noticeable for compound edge-stiffened members than for simple edge-stiffened members. One interesting point is that this bending induced at the lips is probably the reason why the top-hat and channel section buckle in the opposite direction at failure; this is illustrated in Fig. 8 and is also clearly shown in Figs 5 and 6 of the collapsed test specimens. As mentioned previously, the proposed Eurocode 6 imposes a reduction in the effective area of stiffener and adjacent plane element, i.e. ARe f =0"5XAR, at yield. However, this has a disadwmtage when the b/t ratio is small, because the element would then be fully effective in reality. Hence the proposed Eurocode 6 will give over-conservative results for elements with a small value of the b/t ratio. This overt-conservatism is very noticeable when 2 is less than 0.673, that is when b/t is less than about 35 for a yield stress of 280 N/mm 2, since at this lower range of b/t ratios, the element should be considered as fully effective. To overcome this disadvantage, the following design rules are suggested.
_
,I ,I
.able
i I
I I
I
I I
~". I
I
% I ~%!
I
!,
I I I I
!
II I
II
(a) Top-hat section . . . .
Initial neutral axis
. . . .
Shifted neutral axis
(b) Channel section
Fig. 8. Buckled shape of edge-stiffened sections.
L. K. Seah
44
At yield, the effective area of the stiffeners and adjacent plane element are (referring to Fig. 7): A stiff ARe f = beRt + - -
(7)
2
where Astiff=area of stiffeners only, equal to t(bel +be2) at half yield, b'eR = the effective adjacent plane element at twice yield. Figure 9 shows a comparison of the ultimate load predicted by the proposed design rules with the proposed Eurocode 6 for a given adequately compound edge-stiffened channel section with h=0.2, bw/b=l and b2/bl =0-1. In this figure, the ultimate load predicted by the current British specification is taken as the reference. The main purpose of this figure is to show the over-conservatism of the proposed Eurocode at bit ratios less than 35. The proposed design rules are in better agreement with the British specification at this low range of b/t ratios and, as the bit ratio increases, the proposed design rules become very close to the proposed Eurocode. This is further confirmed by comparing with the experimental results of Desmond et al. 2 on single-fold lipped channel sections, shown in Table 2. In this table, specimens with adequate lips and b/t less than 35 are E-21.4-6.69, E-23-9-2.87 and E-23.9-5.26. For these three specimens the proposed Eurocode underestimates by about 21-25% and the proposed rules are in much better agreement with Desmond's experimental results (about a 11-15% underestimate). 0.93 0.92 0.91 0.90 0.89 0.88 0.87 0.86 0.85 0.84 0.83 0.82 0.81 0.80 0.79 0.78 -0.77
Proposed
,t
•, ,"
try = 280 N m m "2
I
t
Eurocode
Fig. 9. C o m p a r i s o n
p
] 20
i
I I t I
....
Pee / PBS
~
P~/Pss
t
I 4O
I
I
6O
8O
I00 b/t o f u l t i m a t e l o a d p r e d i c t e d b y p r o p o s e d d e s i g n rules a n d p r o p o s e d Eurocode. 6
TABLE 2
2-4 2-4 2-4 3"5 3-5 3-5 3-5 2"55 2-55 2'55 2"55 2'55 2-55 3-75 3"75 3"75 3"75 3"75 3-75 3-75 3"75 3-75 3"75
1'6 1"6 1"6 2-5 2"5 2'5 2'5 3.4 3.4 3'4 3.4 3'4 3'4 6'4 6.4 6.4 6-4 8-7 8"7 8"7 8"7 8"7 8'7
0 0-1 0"5 0 0-05 0-3 0'55 0.37 0'6 0.75 1"12 1-5 1"85 0"66 0-99 1-2 1"98 0"65 0"99 1.2 1'65 2'11 2"5
0"0747 0"0747 0.0747 0-1046 0"1046 0.1046 0'1046 0.0745 0.0745 0"0745 0-0745 0"0745 0.0745 0"066 0-066 0"066 0"066 0"066 0-066 0'066 0'066 0'066 0-066
t PEX
19"4 20 24'5 43'9 45 57 60 41 46"6 46-5 51 49 47 34'7 34 38 38 31 36 35-5 35 39"4 39
10 10 10 18 18 18 18 18 18 18 18 18 18 30 30 30 30 30 30 30 30 30 30
bl
40"3 40"3 40-3 47"6 47-6 47-6 47"6 54 52'7 52.7 53"9 54 55-7 36"6 36"6 36"6 36-6 33 33 33 33-3 34"7 33
b
E-21.4-0.0 E-21-4-1"33 E-21.4-6'69 E-23"9-00 E-23-9-0"48 E-23"9-2-87 E-23.9-5.26 E-45'6-5"0 E-45"6-8-05 E-45"6-10"1 E-45-6-15 E-45-6-20"1 E-45.6-24"8 E-97"6-9'94 E-97-6-15 E-97.6-18-2 E-97-6-30 E-133-9"89 E-133-15 E-133-18.2 E-133-25 E-133-32 E-133-38
bw
(kip)
L
Qy
Specimen No. 0-90 1-03 0"89 0"87 1"05 0-87 0-85 0-95 0"88 0"89 0"84 0"88 0-94 0"90 0"99 0"89 0-90 0"88 0"83 0-88 0"91 0.84 0.82
0.879 0-853 0"953 0-851 0.830 0"656 0"895 0"-728 0-627 0"996 0-948 0"996 1.069 0-746 0"761 0-681 0.964 0-801 0"690 0-700 0"716 0"899 0-880
0.861 1.018 0"996 0"842 0'929 0"879 0"940 0-981 0"936 0"992 0"936 0-989 1.067 0"933 1"016 0-942 0"962 0"953 0-868 0"908 0"969 0-898 0-880
0.897 0-912 0"793 0.872 0"935 0"765 0.747 0"938 0-862 0.877 0.827 0.868 0'931 0.913 0.991 0"891 0"898 0.923 0"837 0.879 0'921 0-848 0.825
PEF/PEx PBs/PEx PR/PEx Pcu/PEx
Comparison of Experimental and Predicted Ultimate Load for Desmond's 2 Tests
ga
46
L. K. Seah
4.3 Inadequately edge-stiffened element An edge-stiffened element is referred to as an inadequately stiffened element when the flexural rigidity of the edge stiffener does not satisfy the requirement set out in eqn (1). In this case, the ultimate strength lies between the two limits, which is the ultimate strength of an adequately stiffened element and that of an unstiffened element. The design rules used in the proposed Eurocode 6 for determining the ultimate strength of an inadequately stiffened element are far too tedious for design purposes. Therefore, the following expression is proposed for predicting the ultimate strength of an inadequately stiffened plate:
Pinad
IR ) 1/n =Pu +(Ps--Pu) ~
(8)
where Pinad=Ultimate strength of an inadequately stiffened element; Pn = ultimate strength of an unstiffened element; Ps = ultimate strength of an adequately stiffened element; IRmi. =minimum required value of moment of inertia of the stiffener for adequacy, set by eqn (1); Is = t h e moment of inertia of the actual stiffener with area AR determined at yield stress, about the axis a - a shown in Fig. 7; n = variable index. The variable index, n is found to be equal to 3, which gives a best-fit curve through the experimental points.
5 COMPARISON OF EXPERIMENTAL AND PREDICTED ULTIMATE STRENGTH Table 3 and 4 show the comparison of the compound edge-stiffened sections and elements. Tables 2 and 5 show the comparison with Lim's 5 and Desmond's e t al. 2 experimental results of single-fold lipped channel sections. The values of the predicted ultimate strength given in the tables are based on the actual plate dimensions of the test specimens. The values of the predicted ultimate strength calculated based on the suggested design rules agree well with those obtained experimentally for both compound and simple edge-stiffened (Lim and Desmond) sections. The maximum overestimation and underestimation of the ultimate strength of compound edge-stiffened sections are 12 and 24%, respectively, and for the simple edge stiffened sections are 20 and 22%, respectively. The British Standard 9 and Ref. [8] design method agree well with the simple edge-stiffened sections but tend to overestimate the ultimate strength for the compound
PEX
35.463 39.357 36.437 32.496 52.708 50.205 44-364 42"278 70.926 70.231 67.727 58.410 56.741
Spec. No.
T0.8/1/1 T0.8/2/1 T0.8/3/1 T0.8/4/1 T1.0/1/1 T1-0/2/1 T1.0/3/1 Tl'0/4/1 TI.2/1/1 TI.2/2/1 T1.2/3/1A T1.2/3/1B T1.2/4/1
TABLE 3
1-013 0.805 0.839 0.989 0.949 0.933 0.992 1"018 0.766 0-902 0.906 0-771 0-768
1.236 0.962 0.985 1-057 1.160 1.129 1-165 1"082 0.950 1.098 1-088 0.936 0-910
1.012 0.805 0.838 0.991 0.949 0.926 0.991 1018 0.759 0.893 0.896 0.761 0.758
1.247 0.957 0-964 0.584 1.173 1.110 1.146 0'602 0.946 1-074 1.055 0.908 0.883
PEF/PEx PR/PEx Pcu/PEx PBs/PEx 39-774
PEX
T0.8/2/2B T0.8/3/2 T0-8/4/2 T1-0/1/2 T1-0/2/2 T1-0/3/2 T1-0/4/2 T1.2/1/2 T1.2/2/2 T1.2/3/2 T1.2/4/2
36.993 30.317 32.543 53-403 47.284 46.867 43.251 46.032 62.721 53.542 51.734
TO.8/2/2A 33-933
T0.8/1/2
Spec. No. 0.797 0-886 0.891 0-924 0.866 0-907 0-937 0.908 0.959 0.960 0.791 0-788 0.786
1.004 1.095 0.915 1-125 1.024 1.146 1.155 1.107 1.127 1.215 0.986 0.969 0.939
0.786 0.874 0-879 0.910 0-852 0.895 0.924 0.895 0-945 0-896 0.740 0.729 0-724
1.012 1-092 0.912 1.108 1.011 1.158 1.152 1.091 1.117 1.202 0.970 0-935 0.908
PEF/PEx PR/PEx Pc./PEx PBs/PEx
Comparison of Experimental and Predicted Ultimate Load for Top-Hat Sections
=,
L. K. Seah
48
TABLE 4
Comparison of Experimental and Predicted Ultimate Load for Channel Sections
Spec. No
PEX
C0"8/1/1 C0"8/1/2 C0"8/2/1 C0"8/2/2 C0"8/3/1 C0.8/3/2 C0-8/4/1 CO'8/4/2 C1.0/1/1 C1.0/1/2 C 1.0/2/1 C1"0/2/2 C 1-0/3/1 C1'0/3/2 C1"0/4/1 C1"0/4/2 C1-2/1/1 C 1-2/1/2 C1-2/2/1 C1"2/2/2 C1"2/3/1 C1"2/3/2 C1.2/4/1 C1'2/4/2
33'377 31'013 31"569 33-989 25"611 26.069 24.476 20"973 47"562 49"370 48"118 47-006 42-417 43'529 37"549 33"655 55"350 59-383 47.714 54"238 41.026 52"152 46-728 47"562
PEF/PEx 0"879 0.913 0"925 0-828 1.051 1'050 0.996 1.054 0"866 0-834 0-849 0-874 0"953 0"925 1.009 1.121 0-846 0-910 0-969 0'984 1"073 1"003 0-913 1"010
PR/PEx
Pcu/PEx
PBs/PEx
1"068 1"108 1"095 0-977 1'203 1.206 1"075 1-151 1.055 1.011 1-006 1-039 1"118 1"086 1"060 1-175 1"048 1"115 1'190 1'187 1"296 1-196 1.012 1.076
0"882 0.917 0-928 0-832 1"054 1"053 1"005 1"090 0"866 0.834 0.848 0"873 0-952 0"924 1-008 1-120 0"836 0"904 0"958 0"978 1"061 0-996 0"903 1"008
1.046 1"093 1-096 0-984 1"197 1-194 0-626 0-719 1"028 0"989 1"002 1-035 1"094 1-060 0-604 0"673 1"009 1"083 1"162 1"170 1"250 1"162 0-556 0.643
For Tables 2-4, the ultimate strength Pi is in kN. PEX = ultimate strength obtained from experiment. PEF = ultimate strength based on proposed design rules. PBs = ultimate strength based on British Standard. 9 edge-stiffened sections. N o t e t h a t for the i n a d e q u a t e l y stiffened sections, a l t h o u g h it will h a v e s o m e increase in l o a d - c a r r y i n g c a p a c i t y , it is n o t a l l o w e d to utilize this in the British S t a n d a r d . B e c a u s e o f this, a v e r y high u n d e r e s t i m a t i o n o f the u l t i m a t e s t r e n g t h for the i n a d e q u a t e l y stiffened sections is o b t a i n e d f r o m the British S t a n d a r d .
CONCLUSIONS T h e i n v e s t i g a t i o n r e p o r t e d h e r e s h o w s t h a t the shifting o f the stiffener effective n e u t r a l axis, d u e to the different r a t e of r e d u c t i o n in the effectiveness o f p l a t e a n d stiffeners, causes a significant r e d u c t i o n in the l o a d -
Strength of thin-walled sections
49
TABLE 5 Comparison of Experimental and Predicted Ultimate Load for Lim's~ Tests
Spec. No.
b/t
h
PEx
PEv/PEx P n s / P E x
Pa/P~x
P ~/PEx
A1 A2 A3 A4 A5 A6
122"5 125-2 124-3 124"2 125"7 122"9
0 0-092 0.17 0"262 0"333 0-428
10"12 11-86 17.64 18"69 19"58 18-55
0"88 1"06 0"85 0-83 0-79 0"84
1.009 0-851 0"578 0-880 0"846 0.916
0"942 1"107 0-837 0.879 0-847 0"916
0"882 1-139 0"845 0-841 0"801 0"853
B1 B2 B3 B4 B5 B6
62'2 63.4 63"5 63"8 63"8 63"4
0 0"097 0"173 0"26 0"34 0-418
27"61 48.49 63'18 66.74 68"96 66"07
1"20 1"01 0.87 0.84 0-82 0"87
1"230 0.694 0"535 0"942 0"923 0-982
1"223 1"023 0.912 0.926 0.913 0"977
1-203 1"017 0"863 0.828 0"810 0-859
C1 C2 C3 C4 C5 C6
84"8 87-6 87"3 86"8 87-2 87-5
0 0"088 0.169 0.253 0.334 0.408
18-11 23-84 33'08 37'82 37.37 40.95
1-03 1.09 0"96 0"85 0.86 0-79
1-092 0"825 0-600 0.911 0-923 0"859
1"059 1.121 0"930 0-903 0.921 0.859
1'028 1-140 0"965 0"851 0-858 0-793
h=bl/b. PR = ultimate strength based on Rhodes 8 design method. Pcu =ultimate strength based on proposed Eurocode. 6 3O q
av. b/t = 95.7
26I ]
X ~
23 22f 21 20 19 18
. . . . . . .
X
~ p
~
•
.°..'"
I
. . . . . . . . . . . .
II
X
....""
16 15
:14 1!
X
.....-"
17
13 12
X
I 2
............
Proposed Eurocode [ 6 ]
x
Ref. [81 Experimental
I 4
I J P x tip Fig. 10. Comparison of experimental and predicted ultimate load.
L. K. Seah
50
J
20 19 18 17 -16 - 15 - 14 - 13 12 11 10 _ 9 8 _ 7 65 4_ 3 2 1
X X S • J X
X x
,
.....
Ir . . . . . . . . . . . . . . . . . .
"-
av. b/t = 67.2 - Proposed ............ E u r ~ l e [6] ---Ref.[8] x Experimental
I 2
0
~
I 4 Ip/t4 x 105
~
I 6
Fig. 11. C o m p a r i s o n o f e x p e r i m e n t a l a n d p r e d i c t e d ultimate load. 14 13 12 11 - -
X
10 9
X
8 tJ
7 6 Proposed ............ Eurocode [6] .... Ref. [8] x Experimental
5 4 3 2 I
0
I 2
I 4
6 Ip/t4 x 105
I. 8
I 10
12
Fig. 12. C o m p a r i s o n o f e x p e r i m e n t a l a n d p r e d i c t e d u l t i m a t e load.
carrying capacity. This effect has been neglected in the past for the simple edge stiffener case. It also provides an explanation for why the top-hat and channel sections buckle in opposite directions at failure, which was observed during experiments. The experimental study unfortunately does not reveal the optimum stiffened plate width to thickness ratio (b/t). However, based on the highest
Strength of thin-walled sections
51
value of b/t ratio (i.e. equal to 97) tested, the limiting value of b/t of 90 for the compound edge stiffener in various design specifications, is at least justified. The ultimate load of a compound edge-stiffened element is higher than for a simple edge-stiffened element of the same area, when the width of the simple edge stiffener becomes excessively large. From the results obtained, it can said that the proposed design procedures can predict, with reasonable accuracy and consistency, the ultimate strength of sections with simple or compound edge-stiffened elements.
REFERENCES 1. Kloppel, K. v o n & Unger, B., Buckling and post-buckling of the plate with three sides simply supported and stiffened alongside the free edge using non-hnear buckling theory. Der Stahlbau, 10, 1969 289-299 (in German). 2. Desmond, T. P., Pekoz, T. & Winter, G., Edge stiffeners for thin-walled members. Proc. ASCE J. Struct. Div., 107 (ST2) (1981). 3. Rhodes, J., Buckling and failure of edge-stiffened plates IUTAM, Syrup. on Collapse, London, August 1982. 4. Bulson, P. S., Bulbs, lips and beads. Proc. Prof. J. M. Harvey Retiral Conf. on the Behaviour of Thin-walled Structures, ed. J. Rhodes & J. Spence. Applied Science, 1983. 5. Lim, B. S., Buckling behaviour of asymmetric edge-stiffened plates. Ph.D. thesis, University of Strathclyde, Glasgow, 1985. 6. Eurocode 3 Annex A in draft form, For the Design of Light Gauge Steel Members and Sheeting, 1989. 7. Rhodes, J., A simple micro computer finite strip analysis. Dynamics of Structures, Proc. of the session at Structures Congress '87. ASCE, 1987 pp. 2:r6-291. 8. Rhodes, J., Research into the Mechanical Behaviour of Cold-Formed Sections and Drafting of Design Rules. ECSC Contract No. 7210/SA/608, University of Strathclyde, April 1987. 9. British Standards Institution, Code of Practice for Design of Cold Formed Sections, B.S. 5950:Part 5, 1987. BSI, London, 1987. 10. Amer:ican Iron and Steel Institute, The Specification for the Desi#n of ColdFormed Steel Structural Members, August, 1986. 11. Seah, L. K., Buckling behaviour of edge stiffeners in thin-walled sections. Ph.D. thesis, University of Strathclyde, Glasgow, 1989.