- Email: [email protected]
Lateral, local and distortional buckling of I-beams
Thin-Walled Structures 1 (1983) 289-308
Lateral, Local and Distortional Buckling of 1-Beams T. M. R o b e r t s and P. S. Jhita Department of Civil and Structural Engineering, UniversityCollege, Newport Road, Cardiff, UK
ABSTRACT A theoretical study of the elastic buckling modes of 1-section beams under various loading conditions is presented. The analysis is based on energy considerations and the energy equations governing instability are derived using plate theory to allow for distortion of the cross-section. The resulting analysis is able to predict lateral, local and distortional buckling modes, The results are compared with classical lateral buckling solutions based on beam theory.
NOTATION 2b b, d Df D,, E f(mx), g(ny) F(xi) 1 11 K. KG m.n._p.q M.M mc~
Span of beam. Width of flange. Depth of web. Flexural rigidity of flange. Flexurai rigidity of web. Young's modulus. Functions defining displacement of web. Function of variables xi. Second moment of area of beam about major axis. Second moment of area of flange about major axis. Square symmetric matrices defining eqn (19). Scalars. Bending moment. Critical moment.
289 Thin- Walled Structures 0263-8231/83/$03- 00 O AppliedSciencePublishersLtd, England, 1983. Printed in Great Britain
290
T. M. Roberts, P. S. Jhita
Mp Mx, Mr, M~y Nx, N_r, N2, Nx, Nv, N~y P Per
pbcr
P~ Pp qi tf t~ U, I,', W Uf, Vf, Wf
V Vf
vL Vw x,y,z Xi, X i O l ran
Txy
8 82 = o)
Ei Ex, Ey
Of ~.cr V O'i O'x
~'x
Critical moment based on beam theory. Plastic moment. Bending and twisting moments per unit length. Membrane forces per unit length. Membrane forces per unit length---base set. Load. Critical load. Critical load based on beam theory. Generalised load. Load causing plastic failure. Generalised displacement. Flange thickness. Web thickness. Displacement of web in the x, y and z directions. Displacement of flange in the x, y and z directions. Potential energy. Potential energy of flange. Components of potential energy of flange. Potential energy of web. Co-ordinate axes. Generalised variables. Coefficients defining absolute magnitude of displacement. Shear strain. First variation. Second variation. Delta function. Generalised strain. Strains in the x and y directions. Clockwise rotation of flange about the x-axis. Critical load factor. Poisson's ratio. Generalised stress. Stress in flange in the x-direction. Stress in flange in the x-direction--base set.
1 INTRODUCTION The elastic buckling modes of I-section beams have been classified as being either local, lateral or distortional.
Lateral, local and d~tortional buckling of 1-beams
291
Local buckling is characterised by changes in the cross-section geometry without overall lateral displacement or twist. The amplitude of the buckle may, or may not, diminish rapidly away from the point of maximum amplitude but, contrary to what the description implies, it is likely that the entire beam will be affected to some extent. Local buckling can be sub-divided into flange buckling in which only the flange buckles, web buckling in which only the web buckles, and coupled local buckling in which the flange and web buckle together. Lateral buckling involves lateral displacement and twist without local changes in the cross-section geometry. It is also possible for a member to buckle into modes which combine lateral displacement and twist, together with local changes in the crosssection geometry. This type of buckling is referred to as distortionai buckling and it has been shown ''~ that, for beams of certain dimensions, distortional buckling can lead to a significant lowering of the elastic critical load. Coupled local buckling of axially loaded I-sections has been analysed 3 by solving the equations of equilibrium together with the compatibility conditions for the individual plate members comprising the section. The section was assumed to buckle in a symmetrical coupled local buckling mode and the results of the analysis indicated a rapid decrease in a non-dimensional buckling parameter as the flange width for a particular section was increased beyond a certain limit. A finite element analysis by Rajaskaran and Murray 4gave good agreement with the preceding results. Coupled local buckling of wide flanged I-beams, simply supported and loaded by a concentrated load at mid-span, was also analysed. Although many studies have been performed separately on local and lateral buckling, few studies have been made on distortional buckling. Johnson and Will 5 used the finite element method to analyse the buckling of simply supported and cantilever I-beams, loaded at mid-span and at the end, by concentrated loads applied to the top flange. Rectangular plate elements were used to model the beams. For a stocky section, which did not exhibit distortion of the cross-section, the results were in agreement with beam theory. However, for slender cross-sections, which exhibited distortion of the cross-section, the buckling loads were significantly below those given by beam theory. The finite strip method has been used ~ to analyse the buckling of I-beams subjected to uniform moment and a small reduction in the critical moment of certain sections q due to cross-section distortion, was observed. Later, energy methods were used to obtain an approximate closed form solution for the same
292
T. M. Roberts, P. S. Jhita
problem: and the results obtained confirmed the earlier finite strip analysis. In this paper, energy methods are used to analyse local, lateral and distortional buckling of I-section beams. The analysis, although approximate, is capable of predicting with reasonable accuracy all types of buckling modes of I-sections and results are presented for stocky and slender sections subjected to a variety of loading conditions.
2 THEORY For a structure in an equilibrium state, the first variation of the total potential energy, 8V, is zero. If the structure is subjected to external forces P~, assumed stationary at the equilibrium state, the corresponding displacements being q~, the above statement can be represented by the equation 8V = - 2 Pi8qi+ f
o'iS~idvol = 0
(1
)
where o-, represents the stresses in the structure at the equilibrium state, BE,- represents the first variation in the corresponding strains, and the integration is over the volume of the structure. For stable equilibrium, the first variation corresponds to a minimum value and the second variation of the total potential energy, denoted by 8 2 V, is positive definite, ~7 i.e. positive for all admissible variations in the displacements and corresponding strains. From eqn (1), 8 2 V is given by 1
The factor of 1/2 which appears in the second variation is that which appears in a Taylor's series expansion of a function of several variables x . i.e. 1 c~-~F F(xi + 8xi) = F(xi) +OF 8 x i + - - - 8 x i S x i + t~x~ 2 ~x,c)x./
etc.
(3)
Also, since qi can usually be expressed as linear functions of displacement variables, 82 qi vanishes and ZPiS: q~ can be omitted. Critical conditions occur when 82V vanishes. 6v Since the second variation of any
Lateral, local and distortional buckling of 1-beams
x(u)
°I[
1 ! I I
293
zS
1
y(v)1" (o) x(uf) z (wf)
,,v,, (b) Fig. 1. (a) I-beam dimensions and co-ordinate axes. (b) Local flange axes and displacements.
linear function vanishes, it is apparent that it is necessary to conslder second-order strains and displacements to completely define eqn (2). The second variation of the total potential energy for the 1-beam shown in Fig. l(a) will now be derived, x, y and z define the co-ordinate axes, and u, v and w are the displacements in the x, y and z directions, respectively. To allow for distortion of the cross-section, the web is analysed as a plate. The web has thickness tw, depth d and is made of material having Young's modulus E and Poisson's ratio v. If N~, Ny and Nxy are the membrane forces per unit length, and Mx, My and Mxy are the bending and twisting moments per unit length, the first variation of the total potential energy of the web, trV,, can be expressed as s
8V,,=- ~ PiSqi+f ~ (N.~+NyS.y+N.y~'yxy)dydx -
M,~+My-'~y
-2Mxro-~y)dydx
(4)
294
T. M. Roberts, P. S. Jhita
w h e r e ~x, ~y and 3% are the membrane strains. From eqn (4), 82 V,, is given by
82V.= f ~ (N~82,~+Ny82,y+Nxy82yxy)dydx
f f
( 8Nx8% + 8NyS~y + 8NxySTxy
)dydx
8 M ~aZSw + SMy ~a"Sw - 28M,y a:Sw a--~-~'~d ) y dx
(5)
T e r m s such as Mx(OZ82w/Ox2) do not occur in eqn (5) since w can be expressed as a linear function of displacement variables and hence 82 w vanishes. T h e non-linear expressions for membrane strains in terms of displacements u, v and w are s
t[ x =
--+
0x
2 ~ cgx )
ov
ay 2 ~ ay I Ou
3G,y
=
tgv
ay + ax
- -
- -
aw cgw +
~
Ox Oy - -
(6)
For plane strain, membrane forces terms of strains in matrix form as
Nx, Ny and Nxy can
ENxlw I1 °1 N~
N~y
(1 ---v-~"
v
1
0
0
0
(l-v)/2
Ev
xV
be expressed in
Lateral, local and distortional buckling of l-beams
295
The bending and twisting moments M,, My and M~r can be expressed in terms of displacements as
[ O2w v b2w ~ Mx = - ° " U x- + -V:y )
Mr-- - o .
[ OZw
a2w \
-Ty
)
b2w
Mxy = Dw( I - v) axa----y
(8)
where Dw = Eta/12(1 - v 2) is the flexural rigidity of the web. Substituting eqns (6), (7) and (8) into eqn (5), the complete expression for the second variation of the total potential energy of the web becomes, after neglect of third-order terms, 82V~ =
~I f
[
[bbw)2+Nr(bbw~2+2Nxvabw abw] dydx Nx k bx \ by ] " b---x by
+ 2 ( 1 - v 2) J . J
\-'-~x ] + \ by ]
Ox by
+(1--/") (t98U + b~v ) 2 ] 2
+ Dw +2(1-v)
by
bx
dydx
[ [ aZbw~2 [ 8Zbw ) 2
\ OxOy ]
dydx
a28w a28w
(9)
The second variation of the total potential energy of the flanges is derived in two parts. The first part 82 V~' which is due to axial displacements and bending about the major axis is derived using beam theory, and the second part 82 V~ which is due to twisting and bending about the minor axis is derived using plate theory. The flange shown in Fig. l(b) has width bland thickness tf. The x, y and
296
T. M. Roberts,P. S. Jhita
z axes are assumed to pass through the centroid of the flange, u . v~and wf are the displacements of the flange in the x, y and z directions, and Of is the clockwise rotation of the flange about the x-axis. Considering firstly the beam action, it is assumed that the initial stress distribution in the flange prior to buckling consists of an axial stress, o'x, which is uniform over the cross-section. If the bending stress resultant about the major axis of the flange is denoted by M , the first variation of the total potential energy can then be written as
-d28Wedx 8 V ~ = - E PiSqi+ y or,,b,tfS,.dx+ y M ~
(I())
From eqn (10), 8 2V} is given by f(
1
)
~I
d-'6wf
The non-linear axial strain, ex. can be expressed in terms of displacements as duf+ l [ dwf ~-" 'x = dx
5~--~--1
(]2)
Noting also that o'x = E~x and M EIfd2wf/dx 2, where If eqn ( l !) becomes, after neglect of third-order terms, -
=
tfb3/|2.
dx
Considering now the twisting and bending of the flange the non-linear membrane strains due to twisting and bending are
2 ,ox ] ': =
Ove c~vf ~/xz
--
~z 8x
- -
(14)
Lateral, local and distortional buckling of l-beams
297
Hence, by inspection from eqn (9), 8" V~ is given by 1 82V~ = + ~ f
[ 88vf \2 f o'~tf ~ ) dzdx
+ 2(1 - v) [\ 02~vf axOz ) 2 ]dxdz
(15)
where Df = Eta~12(1- v 2) is the flexural rigidity of the flange. It is assumed that the centroid of the flange does not move in the y-direction and that the flange cross-section does not distort and hence that vf = -zOf and ¢~28vf/Oz 2 = 0. Equation (15) then becomes $2V[ = ~ + ' Of ~f
~O'xtfZ 2 ( d~-~O-------~f ) 2dzdx S [z2[d280f~2+2(1-v)(d~--~Of) 2 ~ - - - - - ~ - - )
3J dzdx
<16)
Performing the integration with respect to z gives 82Vf° - 2 12
o'x ~
dx
The second term on the right-hand side of eqn (17) is the second variation of the bending energy about the minor axis of the flange due to twist which is neglected in conventional beam theory for flexure and torsion. It may, however, be significant if the flange buckles in several waves in the longitudinal direction. The second variation of total potential energy for the whole beam is obtained by summing the contributions from eqns (9), (13) and (17), which gives 62 V = 62 V~. + 62 Vrr + 8zVm
(18)
T. M. Roberts, P. S. Jhita
298
where 82 Vf = 82 VP + 8 2 V~', and suffixes T and B denote the top and bottom flanges, respectively. Critical conditions occur when 82V = 0. It is therefore possible to simplify the expression for 8 z V by omitting all terms which are positive definite, i.e. positive for all admissible variations in displacements. The t e r m ( d S u f / d x ) 2 in eqn (13) is positive definite and it can be shown that the second integral on the right-hand side of eqn (9), which contains the partial derivatives of u and v, is positive definite for all values of Poisson's ratio --<0.5. Critical conditions occur therefore when 82V =
+
21 I [N" x'-~x ~ ,,}/OSw\" [aSw~" + NY ~-'~y ]'+ 2N*y OSwO,W]dydx Ox c3v
°-
2 f f C-x /+
+2(l-v) +
.w.w + 2 v - -3.x 2
c3v2
[~OxOY 028w)" ]dydx
O'x~T) dx"}-?I~:'dX]TandB / d a w , X"
= 0
(19)
The compatibility conditions at the junctions between the web and the flanges can be expressed as (wf)v = (W)y=.
( w f h = (w)y=~
Equation (19) can therefore be completely defined by the web displacem e n t 8w and the membrane forces Nx, Ny and Nxy in the web, and the axial
Lateral, localand distortionalbucklingof 1-beams
299
stress o'~ in the flange prior to buckling. Taking the web displacement 8w to be given by
where f(mx) and g(ny) are functions of x and y, respectively, and ,v_ are coefficients which define the absolute magnitude of 8w, eqn (19) can be reduced to the form aT[K + KG]a = 0
(22)
In eqn (22), a is the vector of coefficients t~_. K and KG are square matrices, K being derived from all the terms in eqn (19) which are i n d e p e n d e n t of Nx, Ny, Nxy and o'x, and KG being derived from all terms which are dependent upon Nx, Ny and o'x. The values of N,, Ny_,Nxy and o-~at which buckling occurs can be related to a base set Nx, Ny, Nxv and ~-x by a scalar load factor her such that
[N.. Ny. Nxy. tY,,]er = hcr[N., Ny. Nxy. &x]
(23)
Equation (22) can then be written as (24)
otT[K + hcrKG]at = 0
where Kc depends on Nx, Ny, Nxy and ~'~. Equation (24) is a complete quadratic form in the coefficients et_ and the condition for a quadratic form changing from positive definite to semi-positive definite is that D E T I K - xc,~,GI = 0
(25)
A computer program was written 9 for the solution of eqn (24). The program generates the matrices K and KG for the displacement 8w given by eqn (21 }--all integrations being carried out numerically--and solves for the lowest eigen value her, which defines the buckling or critical load, and corresponding eigen vector a, which defines the buckled shape of the beam.
3 RESULTS Several problems were analysed to investigate the significance of local and distortional buckling of I-beams. The buckling loads determined
300
T. M. Roberts, P. S. Jhita
from the present analyses are compared with those obtained for lateral buckling using beam theory.
3.1 Symmetrical I-beam subjected to uniform moment M This problem is illustrated in Fig. 2(a). The beam has a span of 2b and the displacement 8w was taken as p=13
8w =
q=3
m~rx
~ ~ a..cos " ,,=1.3.~. . . . . 0. z.2. -~--Y
(26)
Taking q = 3 allows for a cubic variation of 8w with y. The m e m b r a n e forces in the web are given by !
-y
,
Ny = N,v
= 0
(27)
where I is the second moment of area of the beam cross-section about its major axis. The stresses in the top and bottom flanges are given by (O'.+)T ; -- T
---- -- (o',).
(28)
X
M(
] )M [a)
[
t
lP
Jt
(b) Fig. 2. (a) I-beam subjected to uniform bending. (b) I-beam with a central point load.
Lateral, local and distortional buckling of 1-beams
301
The beam analysed had a relatively stocky cross-section with d/t~ = 40 and tf/t~ = 1.5. Results were obtained for various flange widths, all other dimensions being kept constant and as indicated in Fig. 3. Figure 3 shows the relationship between Mc,/M~, and the ratio bf/d where M . is the critical moment determined from the present theory and M~, is the critical moment for lateral buckling determined from beam theory. ~ For low b,/d ratios the beam buckled laterally with little web distortion and the lowest eigen value corresponding to p = 1 in eqn (26),
1-0
', M r
"J""<"
,J=
tw= 10
\
',
is
~.-~ 0'5
\
2b= 3200
\',,k
',,
I
I
0"5
1"0
!
bf/d
1-5
Fig. 3. Critical load ratios for an I-beam subjected to uniform moment.
i.e. a single wave in the longitudinal direction. As the bdd ratio was increased the buckled mode changed from lateral to coupled local with a sharp drop in the ratio M¢,/M"¢, and with the lowest eigen value corresponding to p = 5, i.e. several waves in the longitudinal direction. The buckled modes are indicated in Fig. 3. As a check on the validity of the present solution, Fig. 4 compares the results obtained with an alternative finite element solution,"' incorporating plate elements to model the web and beam elements to model the flanges. There is almost exact agreement between the present and
302
T. M. Roberts, P. S. Jhita
finite element solutions. The results are also compared with the critical moment that can be deduced by considering the flanges to act as plates, simply supported along the web flange junction and free along the outer edges, and subjected to axial compression.8 This approach gives slightly lower values than the present solution due to the influence of the web. It is worth noting that the sharp drop in the critical load ratio corresponds to an actual reduction in the absolute value of M~ as the flange width is increased beyond a certain limit. Also plotted in Fig. 3 is the ratio Mp/M~,whereMpis the plastic moment
1-0
Mcr
0-5
Met (isolated ~ \ ~ " flanges) ~
I
J
0.5
1.0
I
bf/d
1"5
b'Ig. 4. Critical load ratios--comparison with other solutions.
of resistance of the section. This curve indicates that, for the beam considered, plastic failure would occur at values of M below those at which coupled local buckling occurs. Other results obtained but not presented herein indicated that the critical moment for local buckling is almost independent of the span which is in accordance with the critical moment based on the critical stress in the isolated flanges. This substantiates the conclusion that plastic failure is likely to occur before local buckling in beams having relatively stocky cross-sections.
Lateral, local and distortional buckling of I-beams
303
3.2 Symmetrical 1-beam with a concentrated load P acting at the centroid
This problem is illustrated in Fig. 2(b). The displacement 8w was taken as given by eqn (26). The membrane forces in the web due to overall bending of the beam are given by -
I
-y
,
Ny = 0
Nxy=~(x-ve),
Nxy= - P ( x + ve)
(29)
where M is the moment at any section which can be deduced from statics. The stresses in the top and bottom flanges are as given by eqn (28), The local variations in the membrane forces in the web due to the concentrated load were neglected for this problem. Results were obtained for a beam having the same dimensions as in the previous problem. Figure 5 shows the variation of P,,/P~ with the ratio bf/d where P~r is the critical load given by the present theory, and pb is the critical load for lateral buckling given by beam theory. ~For low b f/d ratios the beam buckled laterally with little web distortion and the lowest eigen
T Per
P~r
_~.
"I""'-. 7"'LT\
d = 400 tw= 10 tf = 15 2b= 3200
'\, ,,
0"5
p~k
~'~
__p/\
",,
I
0"5
I
bf/d
1"0
Fig. 5. Critical load ratios for an I-beam with a load acting at the centroid.
304
T. M. Roberts, P. S. Jhita
value corresponding to p 1. As the flange width was increased. distortion of the cross-section became more pronounced with a decrease in the ratio P,/P~,. At higher values of bdd, coupled local buckling occurred with a rapid decrease in the ratio P,/P~r and the lowest eigen value corresponding to p = 3. Also plotted in Fig. 5 is the ratio Pp/P~rwherePr,is the load which would result in the maximum m o m e n t in the beam being equal to the plastic m o m e n t of resistance. This indicates that plastic failure would occur before distortional or local buckling. =
3.3 Synunetrical l-beam with a concentrated load P acting on the top flange
The problem is as illustrated in Fig. 2(b) except that the load P is applied on the top flange. The analysis was similar to that of the preceding section but with an approximate representation of the local membrane forces in the web due to the concentrated load. It was assumed that these local, compressive m e m b r a n e forces were concentrated along the line of action of the force from y = 0 to y = d/2. This approximation corresponds to the lowering
1'0 . . . . . . . . . . . \ Pcr
\"-.o
Finite element ~ " ' ' 0 / o,, i ef'(lO)
~
0\x\
I
0"5
bf/d
110
Fig. 6. Critical load ratios for an I-beam with a load acting on the top flange.
Lateral, local and distortional buckling o f 1-beams
305
of the load due to rotation of the cross-section about the centroid which is used in beam theory analysis. Therefore, the local membrane forces in the web wer e a p p r o x i m a t e d as Ny = - P ~ ( x = O)
(30)
O
w h e r e ~(x = 0) is a delta function which is equal to zero everywhere except at x = 0 where it is equal to unity. The results are shown in Fig. 6 and compared with an alternative finite e l e m e n t solution. 10 The present solution differs slightly from the finite e l e m e n t solution due to the approximate representation of the m e m b r a n e stresses in the web. As in the previous example, there is a transition from lateral to distortional buckling and eventually to coupled local buckling.
3.4 Distortional buckling of I-beams with slender cross-sections The preceding results indicate that the buckling modes of I-beams with stocky cross-sections are predominantly either lateral or coupled local with distortional buckling occurring only in the transition from lateral to
1.0
Mcr/Mcbr b Pcr/Pcr 0-5
-T-_ZT= _&_
~ t U n i f o r m \.
", \ ~,,,\
\,
Load at centroid-~-.\ j\ Load on top flange"\\ d = 600
moment
tw--
6
~ ~ "\.\
2b : L.800
"4 k~,,,
~"--..~
tf/tw= ~. I
0.5
I
bf/d
1.0
Fig. 7. Critical load ratios for various loading conditions.
306
T. M. Roberts, P. S. Jhita
1"0
Uniform moment
M /M r Pcr I pcbr
.-".. .......................... - " - Z // / /*' / /
i
i
0"5
/
Load at centroid Load on top flange
/ / i
/
d = 600 -,~
b f / d = 0.2
3_
t f / t w =Z, !
2b = Z.800
!
5
tw
10
Fig. 8. Critical load ratios for various loading conditions.
1.0
Mc~Mbr moment
, ~ i f o r m
Pcr/ Pbr .. " ~ / -"\ 0.5
/
, L o a d at centroid "'~.-'~"-~/~...~ -r- --r//
~ .......
_ ................
"J'-
/
Load on top flange d = 600
2b = 4800
b f / d = 0-2
tf/tw= 8
I
5
I
tw
10
Fig. 9. Critical load ratios for various loading conditions.
Lateral, localand distortionalbucklingof 1-beams
307
coupled local buckling. Also, the full plastic m o m e n t of the section is likely to be reached before either distortional or local buckling significantly reduces the critical load ratio. Figure 7 shows the critical load ratios plotted against bdd for a more slender section having d/tw = 100 and tdtw = 4. The results indicate some reduction in the critical load ratios at values of bdd at which the plastic m o m e n t of resistance is not likely to be exceeded. Similar sets of results were obtained for tf/tw = 2 but there was no significant reduction in the critical load ratios for low values of bdd. Figure 8 shows the critical load ratios for a section having bdd = 0-2 and t,/tf = 4 plotted against the web thickness. The reduction in the critical load ratio is only significant for very slender webs. The very low value of the critical load ratio shown in Fig. 8 corresponded to a local and not to a distortional buckling mode. Figure 9 shows the critical load ratios for a section having bdd = 0.2 and tw/tf = 8 plotted against the web thickness. The reduction in the critical load ratio is more pronounced but a tf/tw ratio of 8 is probably outside the range of practical significance.
4 CONCLUSIONS Although conclusions drawn from a limited number of results cannot be entirely conclusive, the indications are as follows: 1. For I-beams with stocky cross-sections the buckling modes are predominantly lateral or coupled local. Coupled local buckling, which is characterised by changes in the cross-section geometry without any overall lateral displacement, occurs theoretically in beams with high bald ratios but at loads above those at which the plastic m o m e n t of the section is reached. 2. For I-beams with slender cross-sections, particularly those with high tf/tw ratios, distortional and coupled local buckling may result in a reduction of the critical load ratios at loads below those at which the plastic m o m e n t is reached, although the reduction in the critical load ratios is unlikely to be very significant for beams of practical dimensions.
308
T. M. Roberts, P. S. Jhita REFERENCES
1. Hancock, G. J., Local distortional and lateral buckling of I-beams, Proc. ASCE, 104 (1978) ST 11, 1787-98. 2. Hancock, G. J., Bradford, M. A. and Trahair, N. S., Web distortion and flexural torsional buckling, Proc. ASCE, 106 (1980) ST 7, 1557-71. 3. Bulson, P. S., The stability of flat plates, London. Chatto and Windus, 1970. 4. Rajaskaran, S. and Murray, D. W., Coupled local buckling of wide flange beam-columns, Proc. ASCE, 99 (1973) ST 6, 1003-23. 5. Johnson, C. P. and Will, K. M., Beam buckling by finite element procedure, Proc. ASCE, 100 (1974) ST 3,669-85. 6. Langhaar, H. L., Energy methods in applied mechanics, New York, Wiley. 1962. 7. Roberts, T. M., Second order strains and instability of thin walled bars of open cross-section, Int. J. Mech. Sci.. 23 ( 1981 ), 297-306. 8. Timoshenko, S. P. and Gere, J. M., Theory of elastic stability, New York, McGraw Hill, 1961. 9. Jhita, P. S., 'Local lateral and distortional instability of I-beams', M.Sc. Thesis, University College, Cardiff, 1981. 10. Azizian, Z. G., Instability of beams and plate girders, Report, Department of Civil and Structural Engineering, University College, Cardiff, 1982.
Lateral, Local and Distortional Buckling of 1-Beams T. M. R o b e r t s and P. S. Jhita Department of Civil and Structural Engineering, UniversityCollege, Newport Road, Cardiff, UK
ABSTRACT A theoretical study of the elastic buckling modes of 1-section beams under various loading conditions is presented. The analysis is based on energy considerations and the energy equations governing instability are derived using plate theory to allow for distortion of the cross-section. The resulting analysis is able to predict lateral, local and distortional buckling modes, The results are compared with classical lateral buckling solutions based on beam theory.
NOTATION 2b b, d Df D,, E f(mx), g(ny) F(xi) 1 11 K. KG m.n._p.q M.M mc~
Span of beam. Width of flange. Depth of web. Flexural rigidity of flange. Flexurai rigidity of web. Young's modulus. Functions defining displacement of web. Function of variables xi. Second moment of area of beam about major axis. Second moment of area of flange about major axis. Square symmetric matrices defining eqn (19). Scalars. Bending moment. Critical moment.
289 Thin- Walled Structures 0263-8231/83/$03- 00 O AppliedSciencePublishersLtd, England, 1983. Printed in Great Britain
290
T. M. Roberts, P. S. Jhita
Mp Mx, Mr, M~y Nx, N_r, N2, Nx, Nv, N~y P Per
pbcr
P~ Pp qi tf t~ U, I,', W Uf, Vf, Wf
V Vf
vL Vw x,y,z Xi, X i O l ran
Txy
8 82 = o)
Ei Ex, Ey
Of ~.cr V O'i O'x
~'x
Critical moment based on beam theory. Plastic moment. Bending and twisting moments per unit length. Membrane forces per unit length. Membrane forces per unit length---base set. Load. Critical load. Critical load based on beam theory. Generalised load. Load causing plastic failure. Generalised displacement. Flange thickness. Web thickness. Displacement of web in the x, y and z directions. Displacement of flange in the x, y and z directions. Potential energy. Potential energy of flange. Components of potential energy of flange. Potential energy of web. Co-ordinate axes. Generalised variables. Coefficients defining absolute magnitude of displacement. Shear strain. First variation. Second variation. Delta function. Generalised strain. Strains in the x and y directions. Clockwise rotation of flange about the x-axis. Critical load factor. Poisson's ratio. Generalised stress. Stress in flange in the x-direction. Stress in flange in the x-direction--base set.
1 INTRODUCTION The elastic buckling modes of I-section beams have been classified as being either local, lateral or distortional.
Lateral, local and d~tortional buckling of 1-beams
291
Local buckling is characterised by changes in the cross-section geometry without overall lateral displacement or twist. The amplitude of the buckle may, or may not, diminish rapidly away from the point of maximum amplitude but, contrary to what the description implies, it is likely that the entire beam will be affected to some extent. Local buckling can be sub-divided into flange buckling in which only the flange buckles, web buckling in which only the web buckles, and coupled local buckling in which the flange and web buckle together. Lateral buckling involves lateral displacement and twist without local changes in the cross-section geometry. It is also possible for a member to buckle into modes which combine lateral displacement and twist, together with local changes in the crosssection geometry. This type of buckling is referred to as distortionai buckling and it has been shown ''~ that, for beams of certain dimensions, distortional buckling can lead to a significant lowering of the elastic critical load. Coupled local buckling of axially loaded I-sections has been analysed 3 by solving the equations of equilibrium together with the compatibility conditions for the individual plate members comprising the section. The section was assumed to buckle in a symmetrical coupled local buckling mode and the results of the analysis indicated a rapid decrease in a non-dimensional buckling parameter as the flange width for a particular section was increased beyond a certain limit. A finite element analysis by Rajaskaran and Murray 4gave good agreement with the preceding results. Coupled local buckling of wide flanged I-beams, simply supported and loaded by a concentrated load at mid-span, was also analysed. Although many studies have been performed separately on local and lateral buckling, few studies have been made on distortional buckling. Johnson and Will 5 used the finite element method to analyse the buckling of simply supported and cantilever I-beams, loaded at mid-span and at the end, by concentrated loads applied to the top flange. Rectangular plate elements were used to model the beams. For a stocky section, which did not exhibit distortion of the cross-section, the results were in agreement with beam theory. However, for slender cross-sections, which exhibited distortion of the cross-section, the buckling loads were significantly below those given by beam theory. The finite strip method has been used ~ to analyse the buckling of I-beams subjected to uniform moment and a small reduction in the critical moment of certain sections q due to cross-section distortion, was observed. Later, energy methods were used to obtain an approximate closed form solution for the same
292
T. M. Roberts, P. S. Jhita
problem: and the results obtained confirmed the earlier finite strip analysis. In this paper, energy methods are used to analyse local, lateral and distortional buckling of I-section beams. The analysis, although approximate, is capable of predicting with reasonable accuracy all types of buckling modes of I-sections and results are presented for stocky and slender sections subjected to a variety of loading conditions.
2 THEORY For a structure in an equilibrium state, the first variation of the total potential energy, 8V, is zero. If the structure is subjected to external forces P~, assumed stationary at the equilibrium state, the corresponding displacements being q~, the above statement can be represented by the equation 8V = - 2 Pi8qi+ f
o'iS~idvol = 0
(1
)
where o-, represents the stresses in the structure at the equilibrium state, BE,- represents the first variation in the corresponding strains, and the integration is over the volume of the structure. For stable equilibrium, the first variation corresponds to a minimum value and the second variation of the total potential energy, denoted by 8 2 V, is positive definite, ~7 i.e. positive for all admissible variations in the displacements and corresponding strains. From eqn (1), 8 2 V is given by 1
The factor of 1/2 which appears in the second variation is that which appears in a Taylor's series expansion of a function of several variables x . i.e. 1 c~-~F F(xi + 8xi) = F(xi) +OF 8 x i + - - - 8 x i S x i + t~x~ 2 ~x,c)x./
etc.
(3)
Also, since qi can usually be expressed as linear functions of displacement variables, 82 qi vanishes and ZPiS: q~ can be omitted. Critical conditions occur when 82V vanishes. 6v Since the second variation of any
Lateral, local and distortional buckling of 1-beams
x(u)
°I[
1 ! I I
293
zS
1
y(v)1" (o) x(uf) z (wf)
,,v,, (b) Fig. 1. (a) I-beam dimensions and co-ordinate axes. (b) Local flange axes and displacements.
linear function vanishes, it is apparent that it is necessary to conslder second-order strains and displacements to completely define eqn (2). The second variation of the total potential energy for the 1-beam shown in Fig. l(a) will now be derived, x, y and z define the co-ordinate axes, and u, v and w are the displacements in the x, y and z directions, respectively. To allow for distortion of the cross-section, the web is analysed as a plate. The web has thickness tw, depth d and is made of material having Young's modulus E and Poisson's ratio v. If N~, Ny and Nxy are the membrane forces per unit length, and Mx, My and Mxy are the bending and twisting moments per unit length, the first variation of the total potential energy of the web, trV,, can be expressed as s
8V,,=- ~ PiSqi+f ~ (N.~+NyS.y+N.y~'yxy)dydx -
M,~+My-'~y
-2Mxro-~y)dydx
(4)
294
T. M. Roberts, P. S. Jhita
w h e r e ~x, ~y and 3% are the membrane strains. From eqn (4), 82 V,, is given by
82V.= f ~ (N~82,~+Ny82,y+Nxy82yxy)dydx
f f
( 8Nx8% + 8NyS~y + 8NxySTxy
)dydx
8 M ~aZSw + SMy ~a"Sw - 28M,y a:Sw a--~-~'~d ) y dx
(5)
T e r m s such as Mx(OZ82w/Ox2) do not occur in eqn (5) since w can be expressed as a linear function of displacement variables and hence 82 w vanishes. T h e non-linear expressions for membrane strains in terms of displacements u, v and w are s
t[ x =
--+
0x
2 ~ cgx )
ov
ay 2 ~ ay I Ou
3G,y
=
tgv
ay + ax
- -
- -
aw cgw +
~
Ox Oy - -
(6)
For plane strain, membrane forces terms of strains in matrix form as
Nx, Ny and Nxy can
ENxlw I1 °1 N~
N~y
(1 ---v-~"
v
1
0
0
0
(l-v)/2
Ev
xV
be expressed in
Lateral, local and distortional buckling of l-beams
295
The bending and twisting moments M,, My and M~r can be expressed in terms of displacements as
[ O2w v b2w ~ Mx = - ° " U x- + -V:y )
Mr-- - o .
[ OZw
a2w \
-Ty
)
b2w
Mxy = Dw( I - v) axa----y
(8)
where Dw = Eta/12(1 - v 2) is the flexural rigidity of the web. Substituting eqns (6), (7) and (8) into eqn (5), the complete expression for the second variation of the total potential energy of the web becomes, after neglect of third-order terms, 82V~ =
~I f
[
[bbw)2+Nr(bbw~2+2Nxvabw abw] dydx Nx k bx \ by ] " b---x by
+ 2 ( 1 - v 2) J . J
\-'-~x ] + \ by ]
Ox by
+(1--/") (t98U + b~v ) 2 ] 2
+ Dw +2(1-v)
by
bx
dydx
[ [ aZbw~2 [ 8Zbw ) 2
\ OxOy ]
dydx
a28w a28w
(9)
The second variation of the total potential energy of the flanges is derived in two parts. The first part 82 V~' which is due to axial displacements and bending about the major axis is derived using beam theory, and the second part 82 V~ which is due to twisting and bending about the minor axis is derived using plate theory. The flange shown in Fig. l(b) has width bland thickness tf. The x, y and
296
T. M. Roberts,P. S. Jhita
z axes are assumed to pass through the centroid of the flange, u . v~and wf are the displacements of the flange in the x, y and z directions, and Of is the clockwise rotation of the flange about the x-axis. Considering firstly the beam action, it is assumed that the initial stress distribution in the flange prior to buckling consists of an axial stress, o'x, which is uniform over the cross-section. If the bending stress resultant about the major axis of the flange is denoted by M , the first variation of the total potential energy can then be written as
-d28Wedx 8 V ~ = - E PiSqi+ y or,,b,tfS,.dx+ y M ~
(I())
From eqn (10), 8 2V} is given by f(
1
)
~I
d-'6wf
The non-linear axial strain, ex. can be expressed in terms of displacements as duf+ l [ dwf ~-" 'x = dx
5~--~--1
(]2)
Noting also that o'x = E~x and M EIfd2wf/dx 2, where If eqn ( l !) becomes, after neglect of third-order terms, -
=
tfb3/|2.
dx
Considering now the twisting and bending of the flange the non-linear membrane strains due to twisting and bending are
2 ,ox ] ': =
Ove c~vf ~/xz
--
~z 8x
- -
(14)
Lateral, local and distortional buckling of l-beams
297
Hence, by inspection from eqn (9), 8" V~ is given by 1 82V~ = + ~ f
[ 88vf \2 f o'~tf ~ ) dzdx
+ 2(1 - v) [\ 02~vf axOz ) 2 ]dxdz
(15)
where Df = Eta~12(1- v 2) is the flexural rigidity of the flange. It is assumed that the centroid of the flange does not move in the y-direction and that the flange cross-section does not distort and hence that vf = -zOf and ¢~28vf/Oz 2 = 0. Equation (15) then becomes $2V[ = ~ + ' Of ~f
~O'xtfZ 2 ( d~-~O-------~f ) 2dzdx S [z2[d280f~2+2(1-v)(d~--~Of) 2 ~ - - - - - ~ - - )
3J dzdx
<16)
Performing the integration with respect to z gives 82Vf° - 2 12
o'x ~
dx
The second term on the right-hand side of eqn (17) is the second variation of the bending energy about the minor axis of the flange due to twist which is neglected in conventional beam theory for flexure and torsion. It may, however, be significant if the flange buckles in several waves in the longitudinal direction. The second variation of total potential energy for the whole beam is obtained by summing the contributions from eqns (9), (13) and (17), which gives 62 V = 62 V~. + 62 Vrr + 8zVm
(18)
T. M. Roberts, P. S. Jhita
298
where 82 Vf = 82 VP + 8 2 V~', and suffixes T and B denote the top and bottom flanges, respectively. Critical conditions occur when 82V = 0. It is therefore possible to simplify the expression for 8 z V by omitting all terms which are positive definite, i.e. positive for all admissible variations in displacements. The t e r m ( d S u f / d x ) 2 in eqn (13) is positive definite and it can be shown that the second integral on the right-hand side of eqn (9), which contains the partial derivatives of u and v, is positive definite for all values of Poisson's ratio --<0.5. Critical conditions occur therefore when 82V =
+
21 I [N" x'-~x ~ ,,}/OSw\" [aSw~" + NY ~-'~y ]'+ 2N*y OSwO,W]dydx Ox c3v
°-
2 f f C-x /+
+2(l-v) +
.w.w + 2 v - -3.x 2
c3v2
[~OxOY 028w)" ]dydx
O'x~T) dx"}-?I~:'dX]TandB / d a w , X"
= 0
(19)
The compatibility conditions at the junctions between the web and the flanges can be expressed as (wf)v = (W)y=.
( w f h = (w)y=~
Equation (19) can therefore be completely defined by the web displacem e n t 8w and the membrane forces Nx, Ny and Nxy in the web, and the axial
Lateral, localand distortionalbucklingof 1-beams
299
stress o'~ in the flange prior to buckling. Taking the web displacement 8w to be given by
where f(mx) and g(ny) are functions of x and y, respectively, and ,v_ are coefficients which define the absolute magnitude of 8w, eqn (19) can be reduced to the form aT[K + KG]a = 0
(22)
In eqn (22), a is the vector of coefficients t~_. K and KG are square matrices, K being derived from all the terms in eqn (19) which are i n d e p e n d e n t of Nx, Ny, Nxy and o'x, and KG being derived from all terms which are dependent upon Nx, Ny and o'x. The values of N,, Ny_,Nxy and o-~at which buckling occurs can be related to a base set Nx, Ny, Nxv and ~-x by a scalar load factor her such that
[N.. Ny. Nxy. tY,,]er = hcr[N., Ny. Nxy. &x]
(23)
Equation (22) can then be written as (24)
otT[K + hcrKG]at = 0
where Kc depends on Nx, Ny, Nxy and ~'~. Equation (24) is a complete quadratic form in the coefficients et_ and the condition for a quadratic form changing from positive definite to semi-positive definite is that D E T I K - xc,~,GI = 0
(25)
A computer program was written 9 for the solution of eqn (24). The program generates the matrices K and KG for the displacement 8w given by eqn (21 }--all integrations being carried out numerically--and solves for the lowest eigen value her, which defines the buckling or critical load, and corresponding eigen vector a, which defines the buckled shape of the beam.
3 RESULTS Several problems were analysed to investigate the significance of local and distortional buckling of I-beams. The buckling loads determined
300
T. M. Roberts, P. S. Jhita
from the present analyses are compared with those obtained for lateral buckling using beam theory.
3.1 Symmetrical I-beam subjected to uniform moment M This problem is illustrated in Fig. 2(a). The beam has a span of 2b and the displacement 8w was taken as p=13
8w =
q=3
m~rx
~ ~ a..cos " ,,=1.3.~. . . . . 0. z.2. -~--Y
(26)
Taking q = 3 allows for a cubic variation of 8w with y. The m e m b r a n e forces in the web are given by !
-y
,
Ny = N,v
= 0
(27)
where I is the second moment of area of the beam cross-section about its major axis. The stresses in the top and bottom flanges are given by (O'.+)T ; -- T
---- -- (o',).
(28)
X
M(
] )M [a)
[
t
lP
Jt
(b) Fig. 2. (a) I-beam subjected to uniform bending. (b) I-beam with a central point load.
Lateral, local and distortional buckling of 1-beams
301
The beam analysed had a relatively stocky cross-section with d/t~ = 40 and tf/t~ = 1.5. Results were obtained for various flange widths, all other dimensions being kept constant and as indicated in Fig. 3. Figure 3 shows the relationship between Mc,/M~, and the ratio bf/d where M . is the critical moment determined from the present theory and M~, is the critical moment for lateral buckling determined from beam theory. ~ For low b,/d ratios the beam buckled laterally with little web distortion and the lowest eigen value corresponding to p = 1 in eqn (26),
1-0
', M r
"J""<"
,J=
tw= 10
\
',
is
~.-~ 0'5
\
2b= 3200
\',,k
',,
I
I
0"5
1"0
!
bf/d
1-5
Fig. 3. Critical load ratios for an I-beam subjected to uniform moment.
i.e. a single wave in the longitudinal direction. As the bdd ratio was increased the buckled mode changed from lateral to coupled local with a sharp drop in the ratio M¢,/M"¢, and with the lowest eigen value corresponding to p = 5, i.e. several waves in the longitudinal direction. The buckled modes are indicated in Fig. 3. As a check on the validity of the present solution, Fig. 4 compares the results obtained with an alternative finite element solution,"' incorporating plate elements to model the web and beam elements to model the flanges. There is almost exact agreement between the present and
302
T. M. Roberts, P. S. Jhita
finite element solutions. The results are also compared with the critical moment that can be deduced by considering the flanges to act as plates, simply supported along the web flange junction and free along the outer edges, and subjected to axial compression.8 This approach gives slightly lower values than the present solution due to the influence of the web. It is worth noting that the sharp drop in the critical load ratio corresponds to an actual reduction in the absolute value of M~ as the flange width is increased beyond a certain limit. Also plotted in Fig. 3 is the ratio Mp/M~,whereMpis the plastic moment
1-0
Mcr
0-5
Met (isolated ~ \ ~ " flanges) ~
I
J
0.5
1.0
I
bf/d
1"5
b'Ig. 4. Critical load ratios--comparison with other solutions.
of resistance of the section. This curve indicates that, for the beam considered, plastic failure would occur at values of M below those at which coupled local buckling occurs. Other results obtained but not presented herein indicated that the critical moment for local buckling is almost independent of the span which is in accordance with the critical moment based on the critical stress in the isolated flanges. This substantiates the conclusion that plastic failure is likely to occur before local buckling in beams having relatively stocky cross-sections.
Lateral, local and distortional buckling of I-beams
303
3.2 Symmetrical 1-beam with a concentrated load P acting at the centroid
This problem is illustrated in Fig. 2(b). The displacement 8w was taken as given by eqn (26). The membrane forces in the web due to overall bending of the beam are given by -
I
-y
,
Ny = 0
Nxy=~(x-ve),
Nxy= - P ( x + ve)
(29)
where M is the moment at any section which can be deduced from statics. The stresses in the top and bottom flanges are as given by eqn (28), The local variations in the membrane forces in the web due to the concentrated load were neglected for this problem. Results were obtained for a beam having the same dimensions as in the previous problem. Figure 5 shows the variation of P,,/P~ with the ratio bf/d where P~r is the critical load given by the present theory, and pb is the critical load for lateral buckling given by beam theory. ~For low b f/d ratios the beam buckled laterally with little web distortion and the lowest eigen
T Per
P~r
_~.
"I""'-. 7"'LT\
d = 400 tw= 10 tf = 15 2b= 3200
'\, ,,
0"5
p~k
~'~
__p/\
",,
I
0"5
I
bf/d
1"0
Fig. 5. Critical load ratios for an I-beam with a load acting at the centroid.
304
T. M. Roberts, P. S. Jhita
value corresponding to p 1. As the flange width was increased. distortion of the cross-section became more pronounced with a decrease in the ratio P,/P~,. At higher values of bdd, coupled local buckling occurred with a rapid decrease in the ratio P,/P~r and the lowest eigen value corresponding to p = 3. Also plotted in Fig. 5 is the ratio Pp/P~rwherePr,is the load which would result in the maximum m o m e n t in the beam being equal to the plastic m o m e n t of resistance. This indicates that plastic failure would occur before distortional or local buckling. =
3.3 Synunetrical l-beam with a concentrated load P acting on the top flange
The problem is as illustrated in Fig. 2(b) except that the load P is applied on the top flange. The analysis was similar to that of the preceding section but with an approximate representation of the local membrane forces in the web due to the concentrated load. It was assumed that these local, compressive m e m b r a n e forces were concentrated along the line of action of the force from y = 0 to y = d/2. This approximation corresponds to the lowering
1'0 . . . . . . . . . . . \ Pcr
\"-.o
Finite element ~ " ' ' 0 / o,, i ef'(lO)
~
0\x\
I
0"5
bf/d
110
Fig. 6. Critical load ratios for an I-beam with a load acting on the top flange.
Lateral, local and distortional buckling o f 1-beams
305
of the load due to rotation of the cross-section about the centroid which is used in beam theory analysis. Therefore, the local membrane forces in the web wer e a p p r o x i m a t e d as Ny = - P ~ ( x = O)
(30)
O
w h e r e ~(x = 0) is a delta function which is equal to zero everywhere except at x = 0 where it is equal to unity. The results are shown in Fig. 6 and compared with an alternative finite e l e m e n t solution. 10 The present solution differs slightly from the finite e l e m e n t solution due to the approximate representation of the m e m b r a n e stresses in the web. As in the previous example, there is a transition from lateral to distortional buckling and eventually to coupled local buckling.
3.4 Distortional buckling of I-beams with slender cross-sections The preceding results indicate that the buckling modes of I-beams with stocky cross-sections are predominantly either lateral or coupled local with distortional buckling occurring only in the transition from lateral to
1.0
Mcr/Mcbr b Pcr/Pcr 0-5
-T-_ZT= _&_
~ t U n i f o r m \.
", \ ~,,,\
\,
Load at centroid-~-.\ j\ Load on top flange"\\ d = 600
moment
tw--
6
~ ~ "\.\
2b : L.800
"4 k~,,,
~"--..~
tf/tw= ~. I
0.5
I
bf/d
1.0
Fig. 7. Critical load ratios for various loading conditions.
306
T. M. Roberts, P. S. Jhita
1"0
Uniform moment
M /M r Pcr I pcbr
.-".. .......................... - " - Z // / /*' / /
i
i
0"5
/
Load at centroid Load on top flange
/ / i
/
d = 600 -,~
b f / d = 0.2
3_
t f / t w =Z, !
2b = Z.800
!
5
tw
10
Fig. 8. Critical load ratios for various loading conditions.
1.0
Mc~Mbr moment
, ~ i f o r m
Pcr/ Pbr .. " ~ / -"\ 0.5
/
, L o a d at centroid "'~.-'~"-~/~...~ -r- --r//
~ .......
_ ................
"J'-
/
Load on top flange d = 600
2b = 4800
b f / d = 0-2
tf/tw= 8
I
5
I
tw
10
Fig. 9. Critical load ratios for various loading conditions.
Lateral, localand distortionalbucklingof 1-beams
307
coupled local buckling. Also, the full plastic m o m e n t of the section is likely to be reached before either distortional or local buckling significantly reduces the critical load ratio. Figure 7 shows the critical load ratios plotted against bdd for a more slender section having d/tw = 100 and tdtw = 4. The results indicate some reduction in the critical load ratios at values of bdd at which the plastic m o m e n t of resistance is not likely to be exceeded. Similar sets of results were obtained for tf/tw = 2 but there was no significant reduction in the critical load ratios for low values of bdd. Figure 8 shows the critical load ratios for a section having bdd = 0-2 and t,/tf = 4 plotted against the web thickness. The reduction in the critical load ratio is only significant for very slender webs. The very low value of the critical load ratio shown in Fig. 8 corresponded to a local and not to a distortional buckling mode. Figure 9 shows the critical load ratios for a section having bdd = 0.2 and tw/tf = 8 plotted against the web thickness. The reduction in the critical load ratio is more pronounced but a tf/tw ratio of 8 is probably outside the range of practical significance.
4 CONCLUSIONS Although conclusions drawn from a limited number of results cannot be entirely conclusive, the indications are as follows: 1. For I-beams with stocky cross-sections the buckling modes are predominantly lateral or coupled local. Coupled local buckling, which is characterised by changes in the cross-section geometry without any overall lateral displacement, occurs theoretically in beams with high bald ratios but at loads above those at which the plastic m o m e n t of the section is reached. 2. For I-beams with slender cross-sections, particularly those with high tf/tw ratios, distortional and coupled local buckling may result in a reduction of the critical load ratios at loads below those at which the plastic m o m e n t is reached, although the reduction in the critical load ratios is unlikely to be very significant for beams of practical dimensions.
308
T. M. Roberts, P. S. Jhita REFERENCES
1. Hancock, G. J., Local distortional and lateral buckling of I-beams, Proc. ASCE, 104 (1978) ST 11, 1787-98. 2. Hancock, G. J., Bradford, M. A. and Trahair, N. S., Web distortion and flexural torsional buckling, Proc. ASCE, 106 (1980) ST 7, 1557-71. 3. Bulson, P. S., The stability of flat plates, London. Chatto and Windus, 1970. 4. Rajaskaran, S. and Murray, D. W., Coupled local buckling of wide flange beam-columns, Proc. ASCE, 99 (1973) ST 6, 1003-23. 5. Johnson, C. P. and Will, K. M., Beam buckling by finite element procedure, Proc. ASCE, 100 (1974) ST 3,669-85. 6. Langhaar, H. L., Energy methods in applied mechanics, New York, Wiley. 1962. 7. Roberts, T. M., Second order strains and instability of thin walled bars of open cross-section, Int. J. Mech. Sci.. 23 ( 1981 ), 297-306. 8. Timoshenko, S. P. and Gere, J. M., Theory of elastic stability, New York, McGraw Hill, 1961. 9. Jhita, P. S., 'Local lateral and distortional instability of I-beams', M.Sc. Thesis, University College, Cardiff, 1981. 10. Azizian, Z. G., Instability of beams and plate girders, Report, Department of Civil and Structural Engineering, University College, Cardiff, 1982.