I. Construct. Steel Research 18 (1991) 89110
Parametric Study on Distortional Buckling of Monosymmetric BeamColumns
C. M. Wang Department of Civil Engineering, National University of Singapore, Kent Ridge, Singapore
C. K. Chin & S. Kitipornchai Department of CivilEngineering, The University of Queensland, St Lucia, Queensland, Australia
(Received 8 October 1990; revised version accepted 23 November 1990)
ABSTRACT The paper presents a parametric study on the elastic distortional buckling of simply supported monosymmetric 1section members under a uniform moment and an axial force. Parameters describing member length and the geometry of the section are nondimensionalised in such a way that the flange and web sizes are separated, thereby enabling clear identifica'tion of the effects on distortional buckling behaviour of the flange size, the web size and the member length. The buckling analysis is based on the RayleighRitz method with the total potential energy functional for the flanges (modelled as beamcolumns) and web (modelled as a plate) derived rigorously using the fundamental straindisplacement relationships of a threedimensional solid continuum. The reduction in the buckling moment capacity resulting from web distortion is expressed by the distortion factor, being defined as the ratio of the buckling moments with and without web distortion. Results obtained show that the effect of web distortion is more significant in members that are short, have slender webs or whose smaller flanges are in compression. A simple empirical buckling formula is proposed for predicting the moment distortion factor for monosymmetric beamcolumns having equal flange th&knesses. 89 J. Construct. Steel Research 0143974X/91/$03.50 © 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain
90
('. M. Wang, ('. K. Chin, S. Kittpornchai
NOTATION A Aw Br, B~
[D]
E G h hw 1,,, L J JT, JJ~
L M P tw II i , V i, W i
V 14'f, WH, WV¢ x y,z Yo C~
Y
3/o
F
0~T, 0 ~
Crosssectional area Web area Width of top and bottom flanges, respectively Hookean constant given by eqns (9) and (10) Young's modulus of elasticity Shear modulus of elasticity Distance between flange centroids Height of web Second moments of area about y and ~ axes, respectively Torsion constant of crosssection Torsion constant of top and bottom flanges, respectively Beam parameter  [TrXElyh2/(4GJL2)] InLength of member Applied moment: sagging positive, hogging negative Axial force: compression positive, tension negativc Polar radius of gyration about centroid Thickness of web Thicknesses of top and bottom flanges, respectively Displacements at an arbitrary point i of crosssection in X, Y, Z directions, respectively Volume of member Lateral displacements of top flange, bottom flange and web, respectively Axis along member Crosssection reference axes with origin at web midheight Shear centre position with respect to centroid Web to crosssectional area ratio = Aw/A Monosymmetry property Nondimensional buckling moment with web distortion = ML/~/EI,, GJ Nondimensional buckling moment with web distortion suppressed Nondimensional buckling moment derived from onedimensional formulation (eqn (34)) Shearing strain tensors given by eqns (14)(16) Flange thickness parameter = Tr/(TT + TB) Normal strain tensors given by eqns (11)(13) Linear and nonlinear components of strain tensor Twist displacements of top and bottom flanges, respectively
Distortional buckling of monosymmetric beamcolumns
91
Nondimensional axial force = pL2/(Tr2 Ely) Poisson's ratio Beam slenderness parameter = L/(BT + BB) liFT, rIFB, [Iw Total potential energy of top flange, bottom flange and web, respectively Degree of section monosymmetry = lyT/(IyT + IyB) P Normal stress O'xx Flange widths to thicknesses parameter = (BT+ BB)/ (TT+ TB) q, Web slenderness parameter = hw/tw f~ Flange width parameter = BT/(B T + BB)
A /}
1 INTRODUCTION The elastic flexuraltorsional buckling analysis of thinwalled Isection members is usually performed assuming that the crosssection of the member undergoes rigid crosssectional translation and rotation. When a member is fabricated with a slender web, however, or if one of the flanges is restrained against rigid crosssectional movement, the member may buckle as shown in Fig. 1, where deflection and twisting have been accompanied by a change in crosssectional shape due to web distortion. The effect of web distortion on the buckling capacity of the member can be quite significant. Web distortion is an intermediate buckling mode between local and flexuraltorsional buckling modes of failure. The distortional buckling failure mode was studied as early as 1944 by Goodier & Barton 1 who treated the web as a series of thin vertical beams, each with an equivalent modulus of E/(1  u2) where E is the Young's modulus and u the Poisson's ratio. The study led to the important conclusion that web deformation could be represented by a cubic polynomial function. Since then, investigators such as Suzuki & Okumura 2 and Kollbrunner & Hajdin 3 have used the folded plate approach while others 48 have employed more powerful finite element/finite strip techniques, based on either all plate elements or a combination of beam (flanges) and plate (web) elements for the distortional buckling analysis. More recent investigations by Bradford 9~1 included monosymmetric Ibeams under general transverse and axial loads with different boundary and restraint conditions. So far the study of distortional buckling behaviour of monosymmetric Isection members has not been satisfactorily presented because of the large number of parameters involved in describing the section. Recently, Wang & Kitipornchai 1214 proposed a new set of buckling parameters for general monosymmetric Isection members which reduces the number of
92
('. M. Wang, C. K. Chin, S. Kitipornchai y w T
,
~"
OxB
~fI
:
~
J ~
•
 ~
~'1
~
~ !
r"
7:Ji B o t t o m '
~ 
TT J t
B
~
i
t
TB
w B
Fig. 1. Monosymmetric lsection and buckled section with web distortion.
parameters through appropriate nondimensionalisation and approximation. Consequently, the buckling capacity can be determined and presented in a general format without having to resort to some arbitrarily selected beam dimensions. Thus the results for a wide range of beam dimensions can be summarised and conclusions regarding the buckling behaviour can readily be drawn. Some interesting features were highlighted, such as the observation that the flexuraltorsional buckling axial load of monosymmetric members may be increased if an appropriate moment is applied to the member. The present study continues this line of investigation by examining similar geometric parameters for the distortional buckling of simply supported monosymmetric beamcolumns/tiebeams subject to a uniform moment and an axial force. In the analysis, the RayleighRitz energy method is employed. The widely accepted straindisplacement relationships for a threedimensional solid continuum and elastic constitutive relations have been used as a basis for the derivation of the total potential energy expressions for the inplane bending and twisting of the flanges and the outofplane bending of the web. The rigorous derivation furnishes energy expressions that are slightly different to, and more accurate than, those obtained by previous investigators 15t¢' who made a number of simplifying approximations. An investigation is made into the sensitivity and critical ranges of the various geometric parameters in influencing distortional buckling of monosymmetric beamcolumns. A simple empirical formula, which retains the same familiar form as those derived for lateral buckling, is presented to aid designers in making a quick estimation of the distortional buckling capacity.
Distortional buckling of monosymmetric beamcolumns
93
2 BASIC ASSUMPTIONS AND G E O M E T R I C P A R A M E T E R S The following assumptions are made in this study: (1) local buckling is not considered; (2) prebuckling deformations may be ignored; (3) Kirchhoff's plate assumptions remain valid for the outofplane deformation of the web; (4) the material is homogeneous, elastic and isotropic; (5) loading is conservative; and (6) the member is prismatic and straight. Consider a simply supported monosymmetric Isection member of length L with a web height h w, web thickness tw, flange widths BT and BB, and flange thicknesses TT and TB (see Fig. 1). The subscripts T and B denote the top and bottom flanges respectively. The independent geometric parameters may be nondimensionalised as follows:

BT
TT ;
F
BT + BB
=


BT + B B •
TT+ TB '
(I)

TT+ TB (1)
q ' = hw , tw
a
Aw ; A
~
L BT + B B
in which Aw is the area of the web and A the total crosssectional area. The parameters f~, F and • describe the geometry and the degree of asymmetry of the flanges. ~ and a describe the web slenderness, and indicates the beam slenderness. Practical values for the above parameters lie in the ranges: 0<1 ) < 1; 10___~_< 150;
O
5<~< 15 15_<~<_80
(2)
Note that an upper limit for ~ of 15 is taken to avoid local buckling in the flanges. 5"17 The values of ~ and F are physically limited to the ranges indicated in eqn (2).
3 D I S P L A C E M E N T FUNCTIONS A simply supported monosymmetric Isection member of length L is acted upon by equal and opposite end moments M, and an axial force P along its centroidal axis as shown in Fig. 2. The buckled crosssection of the beam is shown in Fig. 1, where the top and bottom flanges translate sideways and
94
('. M. Wang, ('. K. Chin, S. Kitipornchai
twist by WT, WB and 0~T, 0~B respectively. The flange displacements may hL' assumed to vary in a halfsine wave given by ~5~' T/'X
W,r =
7TX
qlhwsin~
WB = q2hwsin L
(3) 7"~'X
"n'X
0,1 = q 3 s i n  L
0,~
= q4sinL
in which q~, qe, q~, q4 are nondimensional coefficients and x is the longitudinal coordinate.
S h e a r c e n t r e axis
2
........
Centroidal axis
,1
Fig. 2. Simply supported monosymmetric beamcolumn under uniform m¢)ment ~md axial ores.'.
A cubic displacement equation may be used to describe d e f o r m a t i o n of the web during buckling l~' as follows
a~ + a ~ + a ~ , '  7 + a 4 : 5 
Ww = hw
hw
nw
h w
sin I,
(4)
(12, a3, a 4 are nondimensional coefficients of the cubic polynomial function. Using the following compatibility conditions,
tll,
at x = L / 2 , y = h w / 2 0Ww WT = W w ,
0~'1

#v
(5)
at x = L / 2 , y =  h w / 2 WB = W W , O,T~ 
aWw 0V
(6)
Distortional buckling of monoSymmetric beamcolumns
95
the coefficients al, a2, a3, a4 can be shown to be related to the coefficients ql, q2, q3, q4 by 1
1
1
1
3
8 1
1
1
1
al
3 a2
o
a3
o
2
a4
2
ql q2
(7)
q3
1
1
q4
Thus, all the displacement functions WT, WB, WW, OxT , OxB can be described by the four coefficients ql, q2, q3, q4.
4 TOTAL POTENTIAL E N E R G Y 4.1 G e n e r a l
As the member buckles, the total stored potential energy, 18 H, is
lfv
H = mFy[ I~FB [ I~ W  ~
({eL}T[D]{eL}+ 2{o'}T{eN})dV
(8)
in which FlEaand 1IFBrepresent the total potential energy stored in the top and bottom flanges, respectively; II w the total outofplane energy stored in the web (the inplane membrane energy stored in the web is neglected); {EL},{EN}the linear and nonlinear components of the strain tensor; {tr}the cartesian components of the Cauchy stress tensor; V the volume of the member. The superscript T denotes the transpose of a matrix. The matrix [D] is the Hookean constant relating material properties. For the flanges it' is given by
[D]=
E 0 0
0 G 0
0] 0 6;
(9)
ID. ,.>
+..+
B
Z
z
,54
÷
f
I
i
t
+
~
+
+
II
i
i
I
Z
f ~  , ,
+
',..#/
+
I
i
+
+
N
Z
+
~7
4~
+
+
+
II
I,..l
7
F
i
I 
A
+
+
~
II
t~
to
t
i
I
Z
~
+
t
4
II
r
~
+
Ft
+
+
H
,.I
0
m1
I
v
"1'1 0
4
Distortional buckling of monosymmetric beamcolumns
97
4.2 Total potential energy for the flanges Since the isolated flanges are essentially rectangular beam elements, the components of the strain tensor for eyy, e~z and 3% can be ignored. Assuming small deformations and from kinematic considerations, the displacements ui, vi and wi at the point i in the top and bottom flanges may be expressed in terms of the crosssection centroidal displacement, u, and the displacements and rotations WT, VT, OxT, WB, I~B, OxBat the junctions between the flanges and the web, as Wik = Wk+ (y + ak) O~k
(17)
vik = Vk  ZOxk
(18)
Uik = u   ( Y   y o c )
bVk Z Ox
OWk aOxk Wk3x Ox
(19)
in which k = T (top flange) and B (bottom flange); 6k =  h w / 2 and + hw/2 for top and bottom flanges respectively; y, z are, respectively, the coordinates of the reference axes whose origin is at the web midheight (see Fig. 1) and Yoc the coordinate of the centroid of the crosssection. Substituting eqns (9), (11), (14), (15) and (17)(19) into eqn (8) yields the following expression for the total potential energy of the flanges
Hvk = 2 J v [ E L \ ox ! + (y  yoc) 2 \5~x2,1 +~\
0~y/
z2
+ G z2+Zz Oy + \  ~ y y /
2(Y+'~k)~Z
\ OZ /
J\
ax /
\~~x2 ) (Y+6k)2
°XXLkax !
(20)
+
7
\
ox
/
\~x/\~x!
)
/
\gZ/
98
('. M. Wang, ('. K. Chin. S. Kittpornchai
where the normal stress, or,.,., is given by ~r,., 
P
M O '  y,,,.)
A
/
(?_1)
in which 1, is the second m o m e n t of area about the gaxis through the centroid (see Fig. 1). The sign of axial force P is positive when compressive, and the m o m e n t M is positive when acting clockwise on the left hand side of the member. Note that the lengthy expression associated with the shear modulus, G , in eqn (20) relates to the torsion constant of the flange, Jk (see Ref. 20). 4 . 3 T o t a l p o t e n t i a l e n e r g y for o u t  o f  p l a n e d e f o r m a t i o n of the w e b
Since thc web is a twodimensional plane stress element, its stress tensor components ¢r=:, r,:, r,=, and strain components, "g,= and y,.: are equal to zero. Moreover, the strain c o m p o n e n t ~:..: can be neglected, ts From kinematic considerations and the assumption that deformations arc small, the displacements at the point i in the web, u,, v~and wi can be expressed in terms of the outofplane deflection of the web midsurface, ww, as ui=
iJw w
z"
i)Ww
~'i . . . .
&r
w,= Ww
i~v
(22)
Substituting eqns (10), (I 1), (12), (14) and (22) into eqn (8) yields the following expression for the total outofplane potential energy of thc web
llw = ~
]
•
/)"
\
.
.
~
av 2
l
\
i~r
/\
ov~
i 23) [ 0 e Ww ~
+ 2(1  v ) ~ . ~ , '

J dydx +
(r~ 
h,,'2
/
i~x
dvd_t /
"
in which the rigidity of the web plate is given by Er~,
Dw
(24)
1 2 ( 1  v ~)
Note that the ttange strain energy terms associated with (i/v/ax') ~, ( a v / a x ) ( a O , . / o x ) and (a:0,Jax2) 2 in eqn (20) may bc neglected due to the assumptions that prebuckling deformations are small
( O u / a x ) z, ( a v / a x ) '  ,
Distortional buckling of monosymmetric beamcolumns
99
and that warping terms in the flanges are negligible. In the work by Hancock et al. 15 and by Bradford & Waters, 16 the terms Z2(32Wk/OX2) 2, 2(y + t~k)(3Wk/OX)(OOx/OX ) and (y + 3k)2(OOx/OX) 2 in eqn (20) have also been neglected. These terms did not apply to their formulations as the normal stresses in the flanges were assumed to be constant and the web height was taken between the flange centroids. The comparative studies made indicate that their assumptions somewhat compensate for the error associated with neglecting the terms. However, in the study by Hancock et al., 15 which assumed a linear web deformation, the error was relatively significant in short members with a slender web.
5 RAYLEIGHRITZ ENERGY METHOD Applying the RayleighRitz method, the total potential energy is miniraised with respect to ql, q2, q3 and q4 as all = 0 3qi
i= 1,2,3,4
(25)
Substituting eqns (3) and (21) into (20) and eqns (4) and (21) into (23), and combining with the transformation matrix given in eqn (7), four homogeneous linear equations are derived in terms of qi, i = 1, . . . , 4. These can be collectively written as [K]4x4{Q}4xl = {0}4xl
(26)
in which the nondimensionalised displacement vector {q} = ((qJL)~v/(Elyo/GJo) (q2/L)X/(Elyo/GJo) q3 q4) T
(27)
in which GJo is the torsional rigidity of the flanges and Elyo the flexural rigidity of the flanges about the minor yaxis. For a nontrivial solution, det[K]4×4 = 0
(28)
The lowest root of the quartic equation (eqn (28)) furnishes the buckling load. The following nondimensionalised moment and axial load parameters, y and A, may be adopted y = ~
ML ,
PL 2 A = 7r2Elv
(29)
100
C. M.
Wang. C. K. Chin. S. Kitipornchai
GJ and E!v are the torsional and minor axis flexural rigidity of the crosssection, respectively. The elements of the symmetric matrix [ K] arc given in the Appendix.
6 N U M E R I C A L RESULTS
6.1 Distortion factor For a given axial force, A, the buckling moment capacity, y, of the member can be computed from eqn (28). The reduction in the buckling moment capacity due to web distortion may be measured by the ratio y/y,, which will be referred to as the (moment) distortion .factor where y,, is the buckling moment of a similar member with web distortion suppressed. The buckling moment, y,,, may be obtained by constraining 0 and w in the following manner: O,T = 0,.f~ = 0 
W'j14',
hw
VX~
[0, L I
(30)
The effect of web distortion may be studied by plotting the distortion factor, Y/7o, with respect to the six design parameters ~Q, F, ¢P, ~ , ~ and for given values of axial load, A.
6.2 Doubly symmetric beamcolumns In the case of doubly symmetric Ibeams under uniform moment only four of the six design parameters may be varied since [1 = F = 0.5. The axial load, A, is zero. Due to length limitations, only results for • = 10 will be presented. The distortion factors, y/y,,, are plotted in Fig. 3 against the beam slenderness, ~, for web area ratios a = 0.1 and 03 and web slenderness factors qs = 10, 5(I, 100 and 150. In Fig. 4, the distortion factors, 7/y,,, are plotted against the web slenderness, ~F, for web area ratios a = 01 and 0.3 and for beam slenderness factors ~: = 20, 30, 4(1 and 50. It is seen from these figures that a reduction in web area (i.e. decreased a) leads to a lower distortion factor. A reduction in the buckling capacity of up to 25% due to web distortion is possible when the member is short (say £ = 20) and has a very slender web (say xF = 150). It is evident that the distortional buckling effect is enhanced by slender webs, short beams and low web area ratios. Members with such characteristics therefore should be analysed with appropriate consideration of the effect of web distortion during buckling. For members with stocky webs ( ~ < 50) and a
Distortional buckling of monosymmetric beamcolumns //
o
.

%~

/
0.85
jy
/
.~

~,~/  ¢t:0.1
/
o.8
101
/
$
_.,,:o.~ ¢=1o
.J
/
~ t~
•
,/
Mr, L!,
0.75 0.7
yM
^ =o
2)
.1
,L. i_ ~'1
Doubly Symmetric ~ =['= 0.5 I I I I 30 40
I
I
20 50 Beam Slenderness ~=L/(BT+B e )
Fig. 3. Variation of distortion factor, Y/Yo,with beam slenderness, ~:,for doubly symmetric Ibeams.
1.0 ~
0
0.95
>~ P.

0.9
 Ot: 0 . ~ ~ ~ O ~  O~= 0.3 X~O \
0.85
._g
~I0 5
0.8
A=O
~
L,
0.75
0,7
X
!
Doubly Symmetric Q=F:O.5
0
I
I

I
30 60 90 120 Web Slenderness, @= hw/tw
50
Fig. 4. Variation of distortion factor, y/y,,, with web slenderness, ~, for doubly symmetric Ibeams.
U. M. Wang, ('. K. Chin, S. Kitipormhat
102
~.o~'/~
~  ~
I
,//
+ / // / /
.'/// "'/," ,
,
~
i

~":%"
/ // //,
0.8
~5
r
~
,
o I s 0.85 (
g
T
,
/ ,"
/
:
/
0.75 i 
/'
/
.,'
"
l + / o. 7oL_ffl
/
M p_
+ .... 
a:
A:: 0.5 ,,
L 4
i [i
A:+0.5 ?, :: 0
1
M
,~==::~++,
o. 1
P
q
,
Doubly Symmetric O : F : O . 5 I _J 1 2. . . . _L_ ~ . . . . . . J 20 30 40 50 Beam Slenderness ~= L / ( B T + B B)
Fig. 5. I n f l u e n c e o f axial fl~rce o n d i s t o r t i o n a l b u c k l i n g o f d o u b l y s y m m e t r i c b e a m  c o l u m n s .
high w e b area ratio (a > 0.3), distortional buckling is less significant. For such m e m b e r s , the reduction in the buckling m o m e n t capacity due to the effect of w e b distortion is of the order of only 3gi. The influence of the axial force on the distortional buckling m o m e n t of doubly symmetric b e a m  c o l u m n s (~1 = F = (1.5) is shown in Fig. 5, for ~, = 01 a n d ~ = 10and for axial force ratios, A =  0 . 5 , 0 a n d +0.5. It is seen that axial compression enhances the effect of distortional buckling m o m e n t while the effect is reduced if the axial force is tensile.
6.3 Monosymmetric beamcolumns For the general casc of m o n o s y m m e t r i c lbeams, 11 and F can take on various combinations depending on the top and b o t t o m flange dimensions. For presentation of results, the values of a and • are taken as 0.1 and 10 respectively. The flange thicknesses are assumed to be the same, as is often the case in practice. This implies F = 05. The influence of the axial force on the distortional buckling m o m e n t of m o n o s y m m e t r i c b e a m  c o l u m n s is shown in Figs 6 and 7 for ~ = (1.3 (smaller flange in compression) and 1) = 0.7 (larger flange in compression). It can be seen from Fig. 6 that the reduction in buckling m o m e n t capacity resulting from the effect of w e b distortion is far more significant when the compression (top) flange is the smaller flange (i.e. ~1 < 05, see curves for ~  0.3) than when it is the
Distortional buckling of monosyrnmetric beamcolumns 1.0 f f / l 1  ~
~ ct = 0.1; qb=lO
,
,
103
I
I .... ^:05
o~L~_o0 = ../S~
~
//M°n°swmme~°.S~
//o:o.= ,?~s31
~. 0.85
i
, ,¢1~ ' Jr=0.~ /
,:l/
,;sf.,
,I I
,;I / ,,,',// ,i ,"// .%'/ I / I,,'/II  ,/ /' / / I o.75 , //i I/ //71/ o=
o.,
.,, ,,';//,,'WI 20 30
,
,
40 bO Beam Slenderness, ~= L / ( B T + B B )
Fig. 6. Variation of distortion factor, 7/%, with beam slenderness, ~:, for monosymmetric b e a m  c o l u m n s , [1 = 0.3, I" = 0.5.
1.0 ~¢LI
I
I
I
I
I
I
0.95
o
0.9 .,8~
. .......
ii c
~"

Ot : 0.1 ~=10
0.85
o
A= *0.5 A=O . . . . A= 0.5
0.8
0
p ~ M p 0.75
0.7
~L~ I
Monosymmetric ~ = 0.7 ; F = 0.5 / I i I i 1 20 30 40
50
Beam Slenderness, ~ = L/(B'r+ Ba) Fig. 7. Variation o f distortion factor, 71%,, with b e a m s l e n d e r n e s s , ~, for m o n o s y m m e t r i c b e a m  c o l u m n s , fl = 0.7, F = 0.5.
('. M. Wang, C. K. ('hin, S. Kitipornchai
104
o
/
(x : 0.1 ;
~ : 30 :
l
~ :
~:
10 ;
F : 0,5
6 ~~'2
4 L
1.0
"~~. ~ ~ ~
0.8
0.6
0,4
[
85 :
 Web distortion ('~)  No web distortion ('1[o)
7 ....... i Approx. formula
~'~'~
0.2
0
7
0.2
0.4
0.6
0.8
1,0
Axial Load Ratio, A = P/PE
Fig. 8. C o m p a r i s o n of i n t e r a c t i o n curves. 7 a n d 7,,. showing influence of web distortion and
proposed approximate formula.
larger flange (i.e. II > 0.5, see curves for ~ = 0.7). This effect becomes m o r e p r o n o u n c e d as the web slenderness p a r a m e t e r ~ increases. Note that when the compression (top) flange is the larger flange ([l  0.7), the m a x i m u m reduction in the buckling m o m e n t capacity resulting from the effects of web distortion is less than 15%. M o n o s y m m e t r i c beamcolumns with I " = 1).5, q ) = 10, ¢ ~  01, qr = 85 and ~: = 30, under uniform m o m e n t and axial force for II = 03, 0.5 and 07 have been analysed. The stability criteria shown in Fig. 8 give the interaction between m o m e n t and axial force with allowance for web distortion. The influence of web distortion on buckling m o m e n t capacity may be shown by comparing interaction curves for m e m b e r s with and without web distortion. Figure 8 illustrates this comparison. Again, it is seen that the reduction in the buckling m o m e n t capacity, y, due to web distortion increases as the compression (top) flange size decreases.
6.4 Simple distortional buckling formula Studies have been m a d e of the distortion factor, y/y,,, for a wide range of parameters. To simplify the problem, the m o n o s y m m e t r i c lsections arc assumed to have equal flange thicknesses (i.e. F = 05). A regression analysis has been used to curvefit the results yielding a simple empirical formula for the distortion factor, Y/~/o. The formula may be used in
Distortionalbucklingof monosymmetricbeamcolumns
105
conjunction with the familiar closed form solution, Yo, derived from the onedimensional beam formulation, 13 3'  1  f~[(1)ot + 10(1  a)] < 1.0 7o o'¢¢ 2 
(31)
in which the factor f is given by f = 0.180  0.7591) + 1.516~ 2  14531)3 + 0.516~ 4 + g(A)
(32)
where  8 . 0 × 10  3  < g ( A )
A
(33)
= ~  ~ ( 1 . 0 5  1  1 A )  2 . 5 x 10  3
and Yo is given by ~o = r r [ a + [a 2
h~A+(1
A)
4~ '~
,/2] (34)
where a =
  u2yo A +   ~7 a (1  A ) ] R
(35)
K,=[rr2Elyh2] '/24GJL 2
(36)
B~TB + BBTB(h d)2) _d(B.~TT fl~= ~~[(hd)(~ \
12
+
BTTTd2) (37)
+ [ ( h  d  T~) 4 ( d  ~ ) 4 l ~ ]  2yo and
TT+TB
h =hw+d =
Y°=
(38)
2BB TBh + hw(hw + TT)tw 2(BB Tx + BB TB + hwtw)
h()]
(BT~ 3 TT
1+ \~/
d
(39)
(40)
106
( . M. Wang. ('. K. ('hin, ~. Kitq:t,rncha~
in which y,, is the shear centre position, r<, the polar radius el gyration about the centroid, h the distance between the flange centroids. /3. the Wagner effect and K the beam parameter. Note that cqn (;4t i~a~, I~<> roots, one with the positive sign in front of the square reel ternl ~lrid ttlc' other with the negative sign. Both roots arc feasible solutions Io~ /J :tt.% and A > A,, where A<, is the column flexuraltorsional buckling I~ad in ~lJc absence of any applied moment. In the case of doubly symmetric beamcolumns, the factor I reduces lu
f = 1103+ glA)
!41)
in which g(A) = 0 and ~A, 7rX/(I + K') if axial l:orce A : 0. The proposed empirical formula of eqn (31) has been found to predict the distortional buckling m o m e n t to within about 47/c of the rigorously calculated value for almost the entire range of the axial force A. The accuracy of the formula is demonstrated by comparing with numerical results in Fig. 8. Equation (31) may thus be used to study the various parameters influencing distortional buckling of monosymmctric beamcolumns, and may be adopted by designers for estimating the effects of web distortion in their designs.
7 CONCLUSIONS The paper presents a rigorous derivation of the total potential energy functional for the distortional buckling of monosymmetric Isection beamcolumns. In the derivation, the flanges are modelled as beamcolumns and the web as a plate. Under the simple loading condition of uniform m o m e n t and axial force, the displacement functions ma~ bc defined by halfsine waves along the m e m b e r length and by a cubic polynomial along the webheight. The RayleighRitz method then furnishes a 4 x 4 tangent stiffness matrix in which the buckling capacity can be obtained easily by solving the determinantal equation. A set of six nondimensional parameters defining the monosymmetric crosssectional geometry and m e m b e r length is introduced. These parameters enable the effects of web distortion on buckling of monosymmetric beamcolumns to be studied and quantified without having to rely on specific beam geometry. From the study, it is found that members which are: (a) relatively short (low ~), or (b) have slender webs (high ~ ) , or (c) have low web area ratio (small c~), or (d) have their smaller flanges in compression ({~< 0.5), are more susceptible to web distortion during buckling. It could be very unsafe to neglect the effects of web distortion for such members by using the conventional onedimensional beam model fl)r lateral buckling analysis.
Distortional buckling of monosymmetric beamcolumns
107
In general, the presence of a compressive axial force tends to lower the moment distortion factor while the reverse is true of tensile force. An empirical formula has been proposed for estimating the distortion factor for monosymmetric beamcolumns under combined uniform bending and axial force. The formula predicts results to within 4% of the rigorously calculated buckling moments for almost the entire range of axial force, and may thus be adopted for design specifications.
ACKNOWLEDGEMENTS The work was conducted in the Department of Civil Engineering at the University of Queensland while the first author was on sabbatical leave from the Department of Civil Engineering at the National University of Singapore. The work forms part of the project 'Stability of Beams and BeamColumns' supported by the Australian Research Council (ARC) under Project Grant No. 834 and by Tube Technology Pty Ltd (Palmer Tube Group). The authors wish to thank Mrs Suchada Chin for her assistance with some calculations and Mr Warren H. Traves of Gutteridge Haskins and Davey Pty Ltd for proofreading the manuscript.
REFERENCES 1. Goodier, J. N. & Barton, M. V., Effects of web deformation on the torsion of Ibeams. Trans., Am. Soc. Mech. Engrs, 66A (1944) 3540. 2. Suzuki, Y. & Okumura, T., Final Report, Eighth Congress, International Association for Bridge and Structural Engineering 1968, pp. 32131. 3. Kollbrunner, C. F. & Hajdin, N., Die Verschiebungsmethode in der Theorie der dunnwandigen Stabe und ein neues Berechnungs modell des Stabes mit in seinen Ebenen defomierbaren Querschnitten. Publications, International Association for Bridge and Structural Engineering, 28I1, 1968, pp. 87100. 4. Johnson, C. P. & Will, K. M., Beam buckling by finite element procedure. J. Struct. Div., ASCE, 100(ST3) (1974) 66985. 5. Hancock, G. J., Local, distortional and lateral buckling of Ibeams. J. Struct. Div., ASCE, 104(STll) (1978) 178798. 6. Bradford, M. A. & Trahair, N. S., Distortional buckling of 1beams. J. Struct. Div., ASCE, 107(ST2) (1981) 35570. 7. Bradford, M. A. & Trahair, N. S., Distortional buckling of thinweb beamcolumns. Engng Struct., 4(1) (1982) 210. 8. Chin, C. K., AIBermani, F. G. A. & Kitipornchai, S., Stability of thinwalled members having arbitrary flange shape and flexible web. Engng Struct. (in press). 9. Bradford, M. A., Distortional buckling of monosymmetric 1beams. J. Construct. Steel Res., 5(2) (1985) 12336.
108
(7. M. Wang, C. K. Chin, S. Kitipornchai
10. Bradford, M. A,, Buckling strength of deformable monosymmetric ibeams Engng Struct., 10(3) (1988) 16773. 11. Bradford, M. A., Buckling of elastically restrained beams with web distortions. ThinWalled Struct., 6(4) (1988) 287304. 12. Wang, C. M. & Kitipornchai, S., Buckling capacities of monosymmetric lbeams. J. Struct. Engng, ASCE, 112(11) (1986) 237391. 13. Wang, C. M. & Kitipornchai, S., New set of buckling parameters for monosymmetric beamcolumns/tiebeams. J. Struct. Engng, ASCE, 115(6) (1989) 1497513, 14. Kitipornchai, S, & Wang, C. M., Flexuraltorsional buckling of monosymmetric beamcolumns/tiebeams. The Structural Engineer, 66(23) (I 988) 3939. 15. Hancock, G. J., Bradford, M. A. & Trahair, N. S,, Web distortion and flexuraltorsional buckling. J. Struct. Div., ASCE, 106(ST7) (1980) 155771. 16. Bradford, M. A. & Waters, S. W., Distortional instability of fabricated monosymmetric lbeams. Comp, Struct., 29(4) (1988) 71524. 17. Kalyanaraman, V., Pekoz, T. & Winter, G., Unstiffened compression elements. J. Struct. Div., ASCE, 103(ST9) (1977) 183348. 18. Cook, R. D., Concepts and Applications of Finite Element Analysis, 2nd edn. John Wiley, New York, 1981. 19. Washizu, K., Variational Methods in Elasticity and PlasticiO,, 2nd edn. Pergamon Press, New York, 1975. 20. Love, A. E. H., A Treatise on the Mathematical Theory of Elasticity, 4th edn. Dover Publications, New York, 1944.
APPENDIX The elements of the symmetric matrix [K], after compaction and assuming /vo ~ ly are 13
klj
=r ~
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s)
113 .at
+ ~  Aa N +
2
¢~a~I'] +
24~2
(A1)
~12 ] ] v'~.[3~ 78X ~ +(12x+~r)~ar ( 3+q~]] kl~ = f 140 kl3 =  ~" 4  ~ t
5
Y~d l0
F
3(~)4
+ ~v k 35
+ Aa
42[)
(A2)
] 2
(A3)
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110
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M. Wang, C. K. Chin, S. Kitipornchal / qb2(l
[ 

4
l_
,
1+2X+/3(1I'))
+
(1

I3 2
J
l
4
~/3~(1  ~z)< ]  1") ~
2
< I + Sx)
]
140
I
+
(AI0)
in which
p

Tr+
(All)
~31"+(1 ~)3(1 [')
l,,,r+lvB TB _
I
[
c~
1
(AI2)
_
(AI3) 7rhw
/x
=
7r
TB)(B, ,, + BB)
(T+
= qbfi2xp,
(A14)
hw
Dw hw El,,()
X
y

=
p[32
Yoc
ce/x
hw
2
i
(1  u2)ix3f131"
(Al5)
[IH'(I+[3F)(1~Z)(11")(I+[3(II'))].
(Al6)
21)I{(1  ~ ) ( 1  I") 3 + lW3}/x/32q ,e + I ] / [ / . ~ W ( I + u)]
3#(1  1~)(1  I'){1 + 2X +/3(1  I')} 2 + 3tx~H'{1 
]
12
(AI7)
2X+/3'}2/]