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Flange buckling in a bent I-section beam
FLAXGE
BCCKLISG
IS
1.
_-1 BEST
I-SECTIOS
BEAM
IsTR~DuCTIOS
WHEN a thin walled structural member is subjected to end load and bentlitq. failure may be caused by buckling of the most severely compressed walls. ‘1’11~. rnlculation of the critical load in such cases is necessary for the design of the 111o\t The present paper deals with one a\pcct of’ t11i\ cllicient structural sections. problem, namely, flange buckling in an I-section beam bent by pure csouples illclinctl at angles between O and 00 degrees to the web. The information given has brcn
yerifird by experiment and will. together with pre\Aour work. farilitntc tllc, calculation of flange bucklin, w in an I-s;cction under any combination of aria1 thrust and bending. In the theoretical analysis. the buckled flange has been trcntctl as a flat an(i infinitely long elastic plate, cm~presscd by thrusts in its own plane antI p:~rallcl to two opposite edges. One of these edges is free, the other being fixed it) position but elastically clamped. Under these conditions the deformations follow ccluatiolls Wh en the thrusts in tllc plant of t h of the type originated by B~Y.\s (1891). plate vary linearly across its width, solution of these cquationr in tcrlll< of kr~own functions is not possible. Two approximate methods were therefor? used IO calculate the critical load, which is gi\-cn o\-er a wide range of values of the three independent variables. These are (i) The wavelength of the buckled plate. (ii) The amount of elastic clamping. (iii) The amount of variation of thrust across the plate width. The RAYLEIGFI-RITZ method was used for the bulk of the calculations. nhile n method originaled by SCII~‘ARZ (1890) has been used as an accwatr checsk at selected points, Some theoretical interest is attached to the application of th(, first method, in that the assumed deformation of the plate satisfied only tlrc geometrical edge conditions, but did not satisfy statical edge conditions. Ttlis
piring zero moment and shear. (iii) Along IX’ where ?/ = 0, (f-4)
70 I= 0, A&o, roi&io~t 3~i’3y of the plate ~baut DC (Fig. I) is rasisted the d~fo~at~on of the remainder of the r-section. These coupres are proportional. to 3z0,@ so that -_.._ -__
by caupies
due to
D 3”% _ K 3” 3$ a$ K being a constant, unit length per unit
the coup1 rotation,
Nan-dimetzsiofial fbm “Ibe above eqmtions may simplified arxd red7xCed to d~me~s~ona~ terms by using
be
I *
; = F([)sil$ _I_-where f = y/b, and h is the half wavelength in the x direction. satisfies the edge condition and when substituted in (2.1) gives CFF --&?+W=kF d,EG
p:ql-
#l3=r njf’rLtEf = xp7 k = _xl b~,‘zW, L’i&2. ftW+&~~fmb fRAYtErGR.IfIT1: 3 tenmj q&l34 fLfrrr3w-ksus fbiity md bmding. and X3 is the +&xe of it‘ at f = I, $0 that Ic is referred to the stress at the odter edge of the plate. Also, N = M, [x + (1-x) i’f = L?, ~~‘/%r2, SO that x gives the ratio of the stress at the fixed edge to the stress at the free edge. The edge condition (2.5) becomes, after substitution of (2*0), at 6 = 1, 11”’ - apl? 3T”< dashes denttting (2.4) End (2G),
(9
(8 II a) p2.F’ = 0,
d~~cre~t~at~o~l F = 0 F”
:==0,
z @PI
with
respect
to
1 5. I
(2.8)
(ii) Similarly,
(if (ii)
et
4 = i), from
{‘?_f$] .- ’
I”or
a
I)lnte
in
which
E is
~o,/.vtlrrlf,if wc vary the length MA, 01 being the number of half waves. the buckling load I; will IX li minimum at the value of X gi\,ct~ by t Iic l~li~iirnurnin Fig. 2. When the fn~~ldi~~g load is the same for 818or )i8 - 1 half wavesI its \-nluc will bc a maximumt wI1osc ratio to the previous minimum is given in Table I.
\Vhen the fixit?- E is :t function, obtainable from the tables of KIWIJ, (19G). of /L, the dotted curve4 in Fig. 3 which gi\-c E ;tgaitM bt at constant 1;. are required. Thus if the line AB is the known relation between E and ,u, whose slope ir; always negative, the minimum k will be at point Irj where .iB is a tangent to one of the family of dotted curves. BC being a line of constant ~9.we find the IniI~itl~~~n~ \-Ate of i2 from the point Q. A slightly larger
1.‘1;111~vIl11144111~ 111
\-alue of I; is obtaiiwt from minimum lint _1C‘. Now it
P
:,
I,Clll
I-.(~rtlllll
c~orrcsp0ntlin.g
with
1%
IKYIIII
the
interwction
_i on the
will be seen from Iq’ig. 3 that each of the dotted cwrves rcaclrcs a masimw~l ilcgativc slope - ( JE Ap) max. and that if this is iiumcrically less tliali the slope of the line .f B. the minimum 1; will IX inwh smaller than tlic \-aluc at Jwint P. Table II giws - ( bei J/L) rnax at \.arious E. and the error in k when the lille :ifI takes this maximum 4ope and tlic illtcrscction -1 is used in plaw of 11. Sate that each inasimum slope - (36, JC() n~az refers to a point 011 the line .I(‘. In liailrg ‘I’al)lc JI to test a particular E - p wlatiotlship, .fB. for the \.alidity of tllc ‘* intcrv’ctioll .1 .’ method, the positiotl of point _1. as ncll as the slope of tllc rclntionahip. must be known.
Thus, while the change in p is large, the error in k is small enough to justify the use of the minimum curve. lwozirl~rl the slope AB is not greater than the value given, - (3~ ‘>p) max, at the point -1 in question. The requirements for critical load calculations for x ‘f 1 are thus : (i) (ii) (iii) (il.)
Curves of Ii against p to give minimum 1~ at constant E (Fig. 2). These minimum k’s and the corresponding \-alues of 11 to be plotted against E (Fig. 3). Examination of the in Fig. 3 for r I 1. Examination slope with (Fig.
maximum
slope
of the
families
of dotted
curves
of the percentage error in L due to the use of this maximum the intersection ii in place of the true minimum point I’
3). 3.
THE
RAYLEIGH-RITZ
JIEIIIOI~
\Vhen z + I. w that R varies with [, it is not possible to solve (2. 7) in trrmi of known functim~. The critical value of k nl:ly be then be obtained by equating the potential r~~rrpy of bending to the work done by the buckling loads in the manner originated b?- RAYLEIGH (18!J1) and tbr expression for the clitical load is z/32 (1 -
O) (FF’)*_l
4 6 (1.“)“5z”
-
Ii-
Then applying Itmz’:, estwsiou assumed sh:il)e
(S.1)
(1WJ)
J uf RAYLEIGH’S
principle. \se substitute
for 1’
l?langr buckling in a beut I-section beam
RdYLEmn’s priuciple. (3.4) may be twice differentiated can be obtained by following given, for example, by Jnurners (1943). To find the coelhcient~ by Q F, and integrate, giving
in view of (335ii). Then substituting
Then integratin::
(1 -
for F,, R from (2*7),
by parts we obtain
6, p/F (I -
=pz
155
of (F,,‘F
IT) (Fn‘
F&_1
+ FnF')t+ + -
-t E
c
- j5z Fn)2 c&f] J-~(F,~~
(F,,‘)‘2fmo -+-
(Fn’Fr)~,.O
-J-
j)F,” -
p2 F=) (F”
-
8% F) df
[F(Fn"' - ,8” “‘,J]~+.
(3.3)
If it is possible to differentiate (3.4) tw-ice, we may also find the coefficients b,, by using the first orthogonality relation (3*5i), so that substituting for F in (I -
o) p2 [(I’ Pn’ + F’ I’,&+
+ c [(F F,‘)][zo
-i- IV’,”
-
P2 F,J (J”’
-
pa F) dE
we get b, [2fi2(1 = p (I -
u) (l;‘,’ E’,)‘+
+ c (Fn’f2tco
5) (P,‘ F -t Fn qzl
+ j;!fi’,/
+ c (Pm’ Fq+J
+-
fi2 1,‘,12 dg] I
p,=
-
$2 Fn) (P” -
p F) d‘$.
(S9)
Subtracting (3.8) from (3.9) we get fF)$=o = 9, from the last term in (3,8), as the condition for differentiation of (3.4), and hence f& fulfilling RAYLEIGH’S principle. This method is illustrated in Fig. 4 which shows the state of affairs at E = 9 when the assumed function F is not zero there. An analysis in normal modes leads to the graph shown, having infinite first and second derivatives at 5 = 0. A similxr result would obtain at a boundary where F’ was fixed, but if E”’ has a fixed value at a boundary the assumed P” need not take I this value. Thus while an analysis of the form F” = zlb, F”, -----leads to an infinite F”‘, this last does not occur iu (3.1) so the f critical load will not be affected. Since the geometric boundary Fig. 4. Assumed ftiri~,tioxl conditions fix F and F’, while the static ones fix P” and I+““, at .$ == 0. we are led to the same conclusion as before. The shape used in the present ease was
FI
r I
I I
I
xrhich satisfies the condition F = 0 at 5 = 0. ‘he first term wm included since exact analysis gives a linear relation between F and f when E = ,8 = 9. Thus the linear relation satisfies the conditions (23). (29) and also (2.7) which becomes &F/dE$ = 0. Substitrition of E’ =_-o, f in (3.1) gives the critical load
A&J, the work of PLAAS (1949) who considered the case when a = - 1 (pure bending), shoxvs_ an almost linear relation when c and fi are small. The first term ~2, .$ has to be deleted when E --f a, since tlmn F’ = 0 at ZJ= 0 giving a second geometrical boundary condition to be satisfied.
J. F. I)AvtDSoS
1x
Thcu writing (:$*I) in the form k I V/T, we apply Rrrz’s method, as givc~ for example by ‘I’I:xPr~I
.\lso by putti+>:!
&v]aa,
we huve Subtracting
A,, twice (3.13) from (:.I”),
- Iz bT/afl,
a,% + an %~a,
= 0, (3.13)
Am = 0.
and then substituting
for a, from @.13), gives
Then if a1 to a,,_, are known, (9.14) gives a quadratic in Is, and a, may be found from (3.13) so that a more accurate value of k may be found from the n + 1 th equation (8.14). Hence starting from “1, q, being indeterminate. we get. a series of values of k, converging towards, but not necessarily to, the true valte. The dotted lines in Fig. 2 are ~&es obtained by tl1i.i method with D = 0.82, using 3 terms (S = 2) iu (3*10). It will be seen that at any given value of c the p to give minimum k is iudependent of cx, so that in Fig. 3, the single E - Q curve sufices for all Q. The n~inimum values of k, recalculated with 4 terms (,V = 5), we plotted iu Fig. 3 and shown in Table III together with the values of UAICERand RODWWX (1948) for do5 1.
TABLE III
.iO 20 3 2 G+ll O.? G
1U.i 1 *G7 I -71 1 .&St) “CO !+GG :1..;4 CD
l*Si 1.21 1.1-l O-O%0 G-840 o-7o(t O.*iGH 0418
1.41 1.36 1.2% 1.12 G*%55 0.808 0.64% 04fS
1.5% 1.54 148 1.28 1.10 0,%36 Wi5G 0.5%
1*we! I*77 I.70 14% I*30 1.11 O*Qw! 0.65%
2.11 2.00 I*%% l-78 1.57 1.38 1.12 0.827
The diffcreuces betieen the vidues.of k with 3 and 4 terms are not greater than l%, suggesting siatisfaetory convergence. We have thus fulfilled requircmcnfx (i) atId (ii), Section 2, nud now consider (iii) aud (iv). Calculations show that for 1 > c( & - 1 the curvature and maximum slopes of the c - p eurW?x(eonstint k) are grc&cr than for CL= 1. Use of the maximum slope ( - ~c/s& max given in ‘I’ublc II shows that the percentage errors iu Is Nhcn using the minimum curve AC (I?&. 3) ure less than those in Table 11. Table II therefore gives the rnaxi]I~ul~lerrors in t due to use
157
Fltu~ge butkling in a bent I-section beam of the i&emcction d for I > z > at the point .* in question.
-
4.
1 prooidcd
the slope of
SCHWARZ
THE
AB
does not exceed the values given
METHOD
.\ n~rthod origitl~tctl by S~HW.~~~ (1890) has been used as an independent check on the results of Section ,‘: . \\‘e start with an assumed shape for the buckled plate I#, ([), which can be analysed in normal ~llodrs \o that I+~ = ,j!,b, ~liKerrllti;ll rcluatiol,
Fn, and substitute
\Ihich arc multiplied
4,
in those terms
by the critical load.
& (0, will he in the form of a particular plen~entary function whose four nrhitrary constants are also boundary conditions (!2.8), (2.9). Hence $2 = k w (f), where w the equation for the nth mode, Fniv - 2b2 F,,” + 82 F,, = ~ihence froin (2.1), ‘IIIc solution of (+l),
of the governing
(2.7) then becomes
integral, linear in k, and a COIW linear in k, since q$ satisfies the is now known. Now considering k,, Q E’,, we obtain F,, = k, w,
w=~b,w,=~b,,F,,‘,lk, 1 1
(4.2)
so that
(4.3)
& = L 7 b,, F,,/k,,.
Thus q$ is a better approximation to the first mode than $1. This method of improving a guessed shape nas used for strut problems by ENGESSER (1893) and VIANELLO (1898), for whirling shafts by STODOW (1905), and more recently by SOUTHWEI,L and others (1941), when (4.1) was solved by relaxation. niving k as a function of [, Following VIASELLO, we put & =a& to find the critical load, ~
(4.4)
Then SCFILEUSSNER (1938) has shown (see also Appendix II) that k (I) must take values above and below the first critical load provided F,, o and D are positive throughout the range of I. A more accurate method is to write
which becomes,
on substituthlg
the expanded k
s
G=l” s
‘Qi?b,,2
forms for
41 and 42,
Fn2kl/k,,dl
ci b,2 Fn2 k12 /kn2 df ’ o Q f:
(4-W
A comparison with (3.6) shows that this is the same as applying RAYLEICH'S method to the improved mode d2, without the trouble of calculating d2’ and 42”. The equivalent of (4.5) was given by SCELIXISSNER (1938) who attributed it to SCXIW.UU (1890). A less accurate method is to pul
which gives
(4.8)
158
J. F. DAVXDSOX
In applying this method evehave to solve (4-l) nith assumed values for &. The most convenient solution is to have the particular integral in integral form so that the complete solution is #a = c cash 81 + d sinh /lf + 6 (e cash ph + f sinh ,%$) (4.9)
The four arbitrary constants c, d, e, f are found from the condition that d/9must satisfy (23) and (2-Q). It is not necessary for .+I to satisfy these conditions, but merely that (6r can heexpanded in terms of normal modes as in (4.1). More accurate approxim&ions, (bar & etc. to the first mode can be obtained by substituting & (bs etc. successively in (49). Results obtained by this method are shown in Table IV, in comparison with exact results for c( = 1 (pure compression) from the calculations of BAKEB and RODEREX (1948).
TAuLI% IV Schwrz
p = 1.67, E = 56,
mtiwd compared witfa ewct values D = @32, cc = 1 (pure conapr&ov~)
k = 1.213 First integration True shape 5
First shape (Rayleigh)
f
A
Equation
(4.4)
-I
r
Second integration n Equation
(4.4)
‘\
F(t)
0
0
0
O-2
0.0921 O-278 OGiQS 0.749 lQ60
0.0642 0.212 0431 0.702 lQ60
0.4 O-6 09 16 k from equation
0
0 OQ893 0‘275 0.503 0.744 1660
0.935 1Q63 l-114 l-227 1362
0.0912 O-278 0+5Qs O-747 1QOO
l-192 l-202 l-206
1-215 1.218
(4-5)
1.213
l-'112
(4.7)
1.219
1.212
The fir& shape (6r was obtained from the RAYLEIGE-RITZ process usiug 2 terms in (3.10). &, the result of two integrations, is correct to 1%. Nevertheless, k obtained from VJANILLO’S method (4.4) has a maximum error of 3%. At the same time the results from (4.5) and (4.7) are accurate to the number of decimals used after the first integration. This latter method has, therefore, been used in comparisons with the RAYLEIGH-RITZ process when a $r 1. such a comparison is given in Table V where values of k from the RAYLEIGH~RIT~ process, lvith J terms in (?-IO), and exuct values we given. l’l~! exact values at CL= 1 we those of BAKER :tiid &UXRKXS (IQ&s), the remainder coming from (4.5). The vaIues of et (P, 50 and O-2, are the extremes required, and Jois taken from Table III. TABLE &@-i@~-Ritz
c
P
V
bracketed) compared with exact values oj k ((I = 0.32).
(dues
01= 0.0
CL= 14
CL= -
1.0
M
I.66
1.260
(1.270)
l-582
(1.593)
2.098
(2.112)
50 O*?
1.67 3-5-i
1,213
(1.224)
1.532
(1.542)
NM8
(2.059)
0.368
(O%Q)
0.754
(O-756)
1.118
(1,121)
It q-ill tpe -ew
tktt the bracketed
greater than 1%.
values, aud hence those of Table III,
have an error nowhere
l~‘lmigebuckliug in ii bent I-section beam
159
The way in which (3.10) fails to satisfy the bouudary condition?, whilst giving an upper limit for k in accordance with the argument of Section 3, is shown in Fig. 5. This gives, in :I typical cai;c, values of F and F” from the two methods. Thus F” horn the Rnr~mon-Rrrz process approaches the true values slowly, owing to (3.10) having a zero in E”’ at t = 1. This zero makes it impossible for (3.10) to satisfy the boundary condition (3.8(i)) at 6 = 1, although F” from (3.10) is approaching the true values near 5 = 1 as the number of terms is increased. 5.
EXPERIMEXTS
A preliminary experiment was carried out on an aluminium alloy I beam approximately Min. long. The section was 2$in. deep, the flange width was Sin. and the web and flanges were nominally O.llin. thick. The material had a @l O/$proof stress of about 28 tons! in2. and Young’s modulus was 4,660 tons/in2. Pure ‘bending was applied in the plane of the web by a four point loading system described in a kI previous paper (1952), using bearings described by NEAL At the ends of the (1950). specimen, the web and flanges were each clamped, although the fact that the compression flange buckled in four half waves made this effect unimportant. Buckling was detected by means of six electric strain gauges fixed to the compression flange as shown in Fig. 6. The figure also shows A/q plotted against A in the manner originated by SOUTHWELL (1932). A is the maximum strain in any gauge minus the nominal maximum strain due to bending, and is e.thus the strain due to buckling. Shape of deformatiou F and its derivative Fig. 5. q is the nominal maximum F” from RAYLEIGII-RITL and SCHWAM methods. stress. The linearity of the CI= 0, c = Xl, p = 1.37, 0 = 0.32. experimental relation between A/q and A, and the close agreement between the experimental buckling stress of 22.6 tons/in.2 (the inverse slope of the line) and the theoretical buckling stress of 22.0 tons;‘in.2, was felt to justify the use of the Southwell plot method in subsequent experiments. The theoretical buckling stress was obtained by using the data of KROLL (1943) with Fig. 3. The applied bending moment being in the plane of the web, it was first assumed that the. web was encastered by the tension flange. The ratio (moment per m~it length)/(rotation of lower edge of web) can then be obtained from the quantity S given by KROLL at various A,‘h, ILbeing the web depth. The web moment is then taken to be divided equally between the buckling flanges, so that by assuming various A, wc: can get the curve _iU in Fig. 3 of c agaiust
.J. 1’.
160 h,;O
=
p.
critical
The
of intersection
LW
I),iV:DSOS
al. which
was justified
by Table
II, ga\-c thr
load.
In a second calculation. assumed to be free from
also based on the data of Krtor.r.. the top flange I\a\ tension. thereby providing a smaller restraint to th(a
web than in actual fact.
This showed
the error of the previous
assumption.
tllat
the web was enc!astrC, to have been negligible. Three
scrir\ of tests were then carried
to the first, Ilaving flange width machined
Young’s
were bot.h Zjin., to
about
thickness, the web being about 0.11Oin. second
modulus
web,
after
of the
a third
series
“g
11 “G 4.~ X 31%
buckling
bearings
iit ~FC’IY’
’ a 11
In all eases elastic. end
Itlntc~ri:~l
depth
I
’ b-m 52 $E
The
sinlilar
j
unaltered.
with a sharp corner between the 0.030in. web and’0.060in. flanges. the
01
‘Q
0.03Oin..
reduction
necessitated
hr;1111
0.06Oin.
A fillet between web and flanges, of about 0.05Oin. radius, which remained
I
thickness For the
to about
the flanges remaining
an
tons, in.2 The section
and for the first series of tests the Ilanges
series. the web thickness
was reduced
out on
of 4,740
was
used
are
shown in Fig. 7, the design being based on the bearings used in the preliminary test, so as to allow freedom of rotation of each end of the beam. Loading, as before, was by
two equal
each forming
lever
arms,
an extension
of the
specimen, and each carrying an equal weight. In setting up, it was possible specimen about that the bending
to twist the its own axis so moments,
which
I~_..~~
0
50
100 150 100 s,R*!~( A x 10~ D”I TO DKKLING -
25
Fig. 6. “ Southwell *’ plot of strain due to buckling. Theoretical buckling stress 22.0 ton/in.?. Experimental buckling stress 2’2.6 ton/in.‘.
must always
act
in a vertical
plane,
any desired angle with the web. In Fig. 7, 0 is 60”, and the distance the bearings (i.e. the effective length of the specimen) is 17ins.
make
between
Four mirrors, visible inFig.7, were fixed in pairs to the two most highly compressed flanges. The members of each pair were an inch apart, and held by plasticine to the edges of the flanges at the centre section. Four horizontal telescopes sighting perpendicularly to the specimen then gave the rotations of the flanges. about the axis of the specimen, by deflections on a scale. The scale deflections were then used to give the buckling load by means of the Southwell plot. The results are shown in Fig.. 8, the theoretical curves being obtained from the curves in Fig. 3 using the “ intersection A ” method. The E - p curve was obtained
from the tables of KROLL (1943), treating
the web and unbuckled
flanges
Flan,oe buckling
as a series of plates. half of the bottom E. The tensile
When
161
in a bent I-section bean1
e =/=0, the mean compressive
flange was used when calculating
stress in the unbuckled
the contribution
to the fixity
stress in the top flanges was neglected.
The only plausible reason for the rather large differences between theory and experiment is the inaccuracy of the “ Southwell plot ” method. This has been considered
by
Hu,
LIWDQ~JIST and
BATDORF (1946)
and
by
Cox
(1951)
w!lo
3’ I
-
Ok
0
15
JO
60
IS
e
DlGRtLS
75
90
SCALE DEFLEXION
-
Fig. 3. Buckling moments from mirror readings. 0.110 in web 0 ; 0.030 in. web with 0.050 in. fillet 0 ; sharp corners 0. Theoretical curves-.
Fig.
-
9. “ Southwell” plots of mirror rendings. 0.030 in. web, 8 = 15.
showed, from the theory of large deflections of plates, that the Southwell plot method is likely to give a curved relationship between A/q and A, even when the initial curvature of the plate is quite modest. Such was the case with the present tests, the worst example being shown in Fig. 9, from which it is difficult to determine the critical moment within 10 prr cent, in sharp contrast with the However, the results confirm the predictions of Hu, results of Fig. 6. LUNDQUIST and BATDORF and COX, in giving an experimental critical load abore the theoretical, and to that extent are satisfactory.
ACKNOWLEGMEKTS The author would like to express his thanks to Dr. E. N. Fox discussion of Section 3, and to the Aluminium Development
sponsored
the work.
for much helpful Association who
BAKER,
.I. F. and
RODERICK. J. I\‘.
1948
Thr
Strengfh
of
~erclopment BRYAN,
G. H.
Cox, H. L. DAVIDSO>-. ~1. 1’. 1)1,X HARIYW. .I. I’.
1891
Proc.
19;1 ]I);“ . 1_
Ii. m/d .W. 269H.
1 !)
$0
I’rw.
Lond. Req.
Math.
F. HI-, P. C. LUNDQVIRT. E. E. BATDORF, S.
1898
Vibrations
2. d. Oest. Ing.
.Illq~
,Ytrrrts.
.\lr~miniunr
XI,. :j.
Sec. 22. 5$.
J’oc. A, 212,
‘W/7-ha,riraI McGraw
EN~ESSER,
Light
Assn. Res. Rq.
HO. t’11rl. (‘Cl.. ,!.
Hill). 11. Ark.
V. 45, 306.
I!)” (S<.i\
\.,,rk
Fhnpz imekling in a bent I-section beam
163
S..LC.A. Rep. Xa. 734. Phil. Trans. A, 242, 197. Rep. Dep. Mech. Eng. Stanford Univ. Calif. Theory of Sound, Vol. 1, p. 287 (London : ~Iarmillan). ,I. j. wine II. a@. Math. 135, 1 (Also Gesammelte We& 1’. 192. Paris : Gauthier Villars, 1911). Zztr Konvergenz des Engesser - l’ianelln - f’rrfnhrenn (Berlin : Teubnerj. Ge.~~l~n~lte ~~h~~~sch,e .,~b~~andZ?~n~en, ft. %I (Berlin : Springer). Proc. Roy. Sot. A, 135, t&f. Theory oJ EkMicify. p. 44.5 (Osford. (‘hendon Pr~su). R. V., ALLES L.,
~Of:TSiWELL.
U. N. DE G., Fos
alld IJOTZ, EI. STOnOLA, A.
Phil. Trans, A, 239, 488. 6’ttwm TwbineS 2nd ed.? p. 185 (Lnndon
: Constnl~le).
TKMPLE G. and BICKLEY %A?.‘ELLO,
I..
W. G.
Rayleigh’s Principle (Oxford University X. I’. D. I. 42, 14%.
Press).
BCCKLISG
IS
1.
_-1 BEST
I-SECTIOS
BEAM
IsTR~DuCTIOS
WHEN a thin walled structural member is subjected to end load and bentlitq. failure may be caused by buckling of the most severely compressed walls. ‘1’11~. rnlculation of the critical load in such cases is necessary for the design of the 111o\t The present paper deals with one a\pcct of’ t11i\ cllicient structural sections. problem, namely, flange buckling in an I-section beam bent by pure csouples illclinctl at angles between O and 00 degrees to the web. The information given has brcn
yerifird by experiment and will. together with pre\Aour work. farilitntc tllc, calculation of flange bucklin, w in an I-s;cction under any combination of aria1 thrust and bending. In the theoretical analysis. the buckled flange has been trcntctl as a flat an(i infinitely long elastic plate, cm~presscd by thrusts in its own plane antI p:~rallcl to two opposite edges. One of these edges is free, the other being fixed it) position but elastically clamped. Under these conditions the deformations follow ccluatiolls Wh en the thrusts in tllc plant of t h of the type originated by B~Y.\s (1891). plate vary linearly across its width, solution of these cquationr in tcrlll< of kr~own functions is not possible. Two approximate methods were therefor? used IO calculate the critical load, which is gi\-cn o\-er a wide range of values of the three independent variables. These are (i) The wavelength of the buckled plate. (ii) The amount of elastic clamping. (iii) The amount of variation of thrust across the plate width. The RAYLEIGFI-RITZ method was used for the bulk of the calculations. nhile n method originaled by SCII~‘ARZ (1890) has been used as an accwatr checsk at selected points, Some theoretical interest is attached to the application of th(, first method, in that the assumed deformation of the plate satisfied only tlrc geometrical edge conditions, but did not satisfy statical edge conditions. Ttlis
piring zero moment and shear. (iii) Along IX’ where ?/ = 0, (f-4)
70 I= 0, A&o, roi&io~t 3~i’3y of the plate ~baut DC (Fig. I) is rasisted the d~fo~at~on of the remainder of the r-section. These coupres are proportional. to 3z0,@ so that -_.._ -__
by caupies
due to
D 3”% _ K 3” 3$ a$ K being a constant, unit length per unit
the coup1 rotation,
Nan-dimetzsiofial fbm “Ibe above eqmtions may simplified arxd red7xCed to d~me~s~ona~ terms by using
be
I *
; = F([)sil$ _I_-where f = y/b, and h is the half wavelength in the x direction. satisfies the edge condition and when substituted in (2.1) gives CFF --&?+W=kF d,EG
p:ql-
#l3=r njf’rLtEf = xp7 k = _xl b~,‘zW, L’i&2. ftW+&~~fmb fRAYtErGR.IfIT1: 3 tenmj q&l34 fLfrrr3w-ksus fbiity md bmding. and X3 is the +&xe of it‘ at f = I, $0 that Ic is referred to the stress at the odter edge of the plate. Also, N = M, [x + (1-x) i’f = L?, ~~‘/%r2, SO that x gives the ratio of the stress at the fixed edge to the stress at the free edge. The edge condition (2.5) becomes, after substitution of (2*0), at 6 = 1, 11”’ - apl? 3T”< dashes denttting (2.4) End (2G),
(9
(8 II a) p2.F’ = 0,
d~~cre~t~at~o~l F = 0 F”
:==0,
z @PI
with
respect
to
1 5. I
(2.8)
(ii) Similarly,
(if (ii)
et
4 = i), from
{‘?_f$] .- ’
I”or
a
I)lnte
in
which
E is
~o,/.vtlrrlf,if wc vary the length MA, 01 being the number of half waves. the buckling load I; will IX li minimum at the value of X gi\,ct~ by t Iic l~li~iirnurnin Fig. 2. When the fn~~ldi~~g load is the same for 818or )i8 - 1 half wavesI its \-nluc will bc a maximumt wI1osc ratio to the previous minimum is given in Table I.
\Vhen the fixit?- E is :t function, obtainable from the tables of KIWIJ, (19G). of /L, the dotted curve4 in Fig. 3 which gi\-c E ;tgaitM bt at constant 1;. are required. Thus if the line AB is the known relation between E and ,u, whose slope ir; always negative, the minimum k will be at point Irj where .iB is a tangent to one of the family of dotted curves. BC being a line of constant ~9.we find the IniI~itl~~~n~ \-Ate of i2 from the point Q. A slightly larger
1.‘1;111~vIl11144111~ 111
\-alue of I; is obtaiiwt from minimum lint _1C‘. Now it
P
:,
I,Clll
I-.(~rtlllll
c~orrcsp0ntlin.g
with
1%
IKYIIII
the
interwction
_i on the
will be seen from Iq’ig. 3 that each of the dotted cwrves rcaclrcs a masimw~l ilcgativc slope - ( JE Ap) max. and that if this is iiumcrically less tliali the slope of the line .f B. the minimum 1; will IX inwh smaller than tlic \-aluc at Jwint P. Table II giws - ( bei J/L) rnax at \.arious E. and the error in k when the lille :ifI takes this maximum 4ope and tlic illtcrscction -1 is used in plaw of 11. Sate that each inasimum slope - (36, JC() n~az refers to a point 011 the line .I(‘. In liailrg ‘I’al)lc JI to test a particular E - p wlatiotlship, .fB. for the \.alidity of tllc ‘* intcrv’ctioll .1 .’ method, the positiotl of point _1. as ncll as the slope of tllc rclntionahip. must be known.
Thus, while the change in p is large, the error in k is small enough to justify the use of the minimum curve. lwozirl~rl the slope AB is not greater than the value given, - (3~ ‘>p) max, at the point -1 in question. The requirements for critical load calculations for x ‘f 1 are thus : (i) (ii) (iii) (il.)
Curves of Ii against p to give minimum 1~ at constant E (Fig. 2). These minimum k’s and the corresponding \-alues of 11 to be plotted against E (Fig. 3). Examination of the in Fig. 3 for r I 1. Examination slope with (Fig.
maximum
slope
of the
families
of dotted
curves
of the percentage error in L due to the use of this maximum the intersection ii in place of the true minimum point I’
3). 3.
THE
RAYLEIGH-RITZ
JIEIIIOI~
\Vhen z + I. w that R varies with [, it is not possible to solve (2. 7) in trrmi of known functim~. The critical value of k nl:ly be then be obtained by equating the potential r~~rrpy of bending to the work done by the buckling loads in the manner originated b?- RAYLEIGH (18!J1) and tbr expression for the clitical load is z/32 (1 -
O) (FF’)*_l
4 6 (1.“)“5z”
-
Ii-
Then applying Itmz’:, estwsiou assumed sh:il)e
(S.1)
(1WJ)
J uf RAYLEIGH’S
principle. \se substitute
for 1’
l?langr buckling in a beut I-section beam
RdYLEmn’s priuciple. (3.4) may be twice differentiated can be obtained by following given, for example, by Jnurners (1943). To find the coelhcient~ by Q F, and integrate, giving
in view of (335ii). Then substituting
Then integratin::
(1 -
for F,, R from (2*7),
by parts we obtain
6, p/F (I -
=pz
155
of (F,,‘F
IT) (Fn‘
F&_1
+ FnF')t+ + -
-t E
c
- j5z Fn)2 c&f] J-~(F,~~
(F,,‘)‘2fmo -+-
(Fn’Fr)~,.O
-J-
j)F,” -
p2 F=) (F”
-
8% F) df
[F(Fn"' - ,8” “‘,J]~+.
(3.3)
If it is possible to differentiate (3.4) tw-ice, we may also find the coefficients b,, by using the first orthogonality relation (3*5i), so that substituting for F in (I -
o) p2 [(I’ Pn’ + F’ I’,&+
+ c [(F F,‘)][zo
-i- IV’,”
-
P2 F,J (J”’
-
pa F) dE
we get b, [2fi2(1 = p (I -
u) (l;‘,’ E’,)‘+
+ c (Fn’f2tco
5) (P,‘ F -t Fn qzl
+ j;!fi’,/
+ c (Pm’ Fq+J
+-
fi2 1,‘,12 dg] I
p,=
-
$2 Fn) (P” -
p F) d‘$.
(S9)
Subtracting (3.8) from (3.9) we get fF)$=o = 9, from the last term in (3,8), as the condition for differentiation of (3.4), and hence f& fulfilling RAYLEIGH’S principle. This method is illustrated in Fig. 4 which shows the state of affairs at E = 9 when the assumed function F is not zero there. An analysis in normal modes leads to the graph shown, having infinite first and second derivatives at 5 = 0. A similxr result would obtain at a boundary where F’ was fixed, but if E”’ has a fixed value at a boundary the assumed P” need not take I this value. Thus while an analysis of the form F” = zlb, F”, -----leads to an infinite F”‘, this last does not occur iu (3.1) so the f critical load will not be affected. Since the geometric boundary Fig. 4. Assumed ftiri~,tioxl conditions fix F and F’, while the static ones fix P” and I+““, at .$ == 0. we are led to the same conclusion as before. The shape used in the present ease was
FI
r I
I I
I
xrhich satisfies the condition F = 0 at 5 = 0. ‘he first term wm included since exact analysis gives a linear relation between F and f when E = ,8 = 9. Thus the linear relation satisfies the conditions (23). (29) and also (2.7) which becomes &F/dE$ = 0. Substitrition of E’ =_-o, f in (3.1) gives the critical load
A&J, the work of PLAAS (1949) who considered the case when a = - 1 (pure bending), shoxvs_ an almost linear relation when c and fi are small. The first term ~2, .$ has to be deleted when E --f a, since tlmn F’ = 0 at ZJ= 0 giving a second geometrical boundary condition to be satisfied.
J. F. I)AvtDSoS
1x
Thcu writing (:$*I) in the form k I V/T, we apply Rrrz’s method, as givc~ for example by ‘I’I:xPr~I
.\lso by putti+>:!
&v]aa,
we huve Subtracting
A,, twice (3.13) from (:.I”),
- Iz bT/afl,
a,% + an %~a,
= 0, (3.13)
Am = 0.
and then substituting
for a, from @.13), gives
Then if a1 to a,,_, are known, (9.14) gives a quadratic in Is, and a, may be found from (3.13) so that a more accurate value of k may be found from the n + 1 th equation (8.14). Hence starting from “1, q, being indeterminate. we get. a series of values of k, converging towards, but not necessarily to, the true valte. The dotted lines in Fig. 2 are ~&es obtained by tl1i.i method with D = 0.82, using 3 terms (S = 2) iu (3*10). It will be seen that at any given value of c the p to give minimum k is iudependent of cx, so that in Fig. 3, the single E - Q curve sufices for all Q. The n~inimum values of k, recalculated with 4 terms (,V = 5), we plotted iu Fig. 3 and shown in Table III together with the values of UAICERand RODWWX (1948) for do5 1.
TABLE III
.iO 20 3 2 G+ll O.? G
1U.i 1 *G7 I -71 1 .&St) “CO !+GG :1..;4 CD
l*Si 1.21 1.1-l O-O%0 G-840 o-7o(t O.*iGH 0418
1.41 1.36 1.2% 1.12 G*%55 0.808 0.64% 04fS
1.5% 1.54 148 1.28 1.10 0,%36 Wi5G 0.5%
1*we! I*77 I.70 14% I*30 1.11 O*Qw! 0.65%
2.11 2.00 I*%% l-78 1.57 1.38 1.12 0.827
The diffcreuces betieen the vidues.of k with 3 and 4 terms are not greater than l%, suggesting siatisfaetory convergence. We have thus fulfilled requircmcnfx (i) atId (ii), Section 2, nud now consider (iii) aud (iv). Calculations show that for 1 > c( & - 1 the curvature and maximum slopes of the c - p eurW?x(eonstint k) are grc&cr than for CL= 1. Use of the maximum slope ( - ~c/s& max given in ‘I’ublc II shows that the percentage errors iu Is Nhcn using the minimum curve AC (I?&. 3) ure less than those in Table 11. Table II therefore gives the rnaxi]I~ul~lerrors in t due to use
157
Fltu~ge butkling in a bent I-section beam of the i&emcction d for I > z > at the point .* in question.
-
4.
1 prooidcd
the slope of
SCHWARZ
THE
AB
does not exceed the values given
METHOD
.\ n~rthod origitl~tctl by S~HW.~~~ (1890) has been used as an independent check on the results of Section ,‘: . \\‘e start with an assumed shape for the buckled plate I#, ([), which can be analysed in normal ~llodrs \o that I+~ = ,j!,b, ~liKerrllti;ll rcluatiol,
Fn, and substitute
\Ihich arc multiplied
4,
in those terms
by the critical load.
& (0, will he in the form of a particular plen~entary function whose four nrhitrary constants are also boundary conditions (!2.8), (2.9). Hence $2 = k w (f), where w the equation for the nth mode, Fniv - 2b2 F,,” + 82 F,, = ~ihence froin (2.1), ‘IIIc solution of (+l),
of the governing
(2.7) then becomes
integral, linear in k, and a COIW linear in k, since q$ satisfies the is now known. Now considering k,, Q E’,, we obtain F,, = k, w,
w=~b,w,=~b,,F,,‘,lk, 1 1
(4.2)
so that
(4.3)
& = L 7 b,, F,,/k,,.
Thus q$ is a better approximation to the first mode than $1. This method of improving a guessed shape nas used for strut problems by ENGESSER (1893) and VIANELLO (1898), for whirling shafts by STODOW (1905), and more recently by SOUTHWEI,L and others (1941), when (4.1) was solved by relaxation. niving k as a function of [, Following VIASELLO, we put & =a& to find the critical load, ~
(4.4)
Then SCFILEUSSNER (1938) has shown (see also Appendix II) that k (I) must take values above and below the first critical load provided F,, o and D are positive throughout the range of I. A more accurate method is to write
which becomes,
on substituthlg
the expanded k
s
G=l” s
‘Qi?b,,2
forms for
41 and 42,
Fn2kl/k,,dl
ci b,2 Fn2 k12 /kn2 df ’ o Q f:
(4-W
A comparison with (3.6) shows that this is the same as applying RAYLEICH'S method to the improved mode d2, without the trouble of calculating d2’ and 42”. The equivalent of (4.5) was given by SCELIXISSNER (1938) who attributed it to SCXIW.UU (1890). A less accurate method is to pul
which gives
(4.8)
158
J. F. DAVXDSOX
In applying this method evehave to solve (4-l) nith assumed values for &. The most convenient solution is to have the particular integral in integral form so that the complete solution is #a = c cash 81 + d sinh /lf + 6 (e cash ph + f sinh ,%$) (4.9)
The four arbitrary constants c, d, e, f are found from the condition that d/9must satisfy (23) and (2-Q). It is not necessary for .+I to satisfy these conditions, but merely that (6r can heexpanded in terms of normal modes as in (4.1). More accurate approxim&ions, (bar & etc. to the first mode can be obtained by substituting & (bs etc. successively in (49). Results obtained by this method are shown in Table IV, in comparison with exact results for c( = 1 (pure compression) from the calculations of BAKEB and RODEREX (1948).
TAuLI% IV Schwrz
p = 1.67, E = 56,
mtiwd compared witfa ewct values D = @32, cc = 1 (pure conapr&ov~)
k = 1.213 First integration True shape 5
First shape (Rayleigh)
f
A
Equation
(4.4)
-I
r
Second integration n Equation
(4.4)
‘\
F(t)
0
0
0
O-2
0.0921 O-278 OGiQS 0.749 lQ60
0.0642 0.212 0431 0.702 lQ60
0.4 O-6 09 16 k from equation
0
0 OQ893 0‘275 0.503 0.744 1660
0.935 1Q63 l-114 l-227 1362
0.0912 O-278 0+5Qs O-747 1QOO
l-192 l-202 l-206
1-215 1.218
(4-5)
1.213
l-'112
(4.7)
1.219
1.212
The fir& shape (6r was obtained from the RAYLEIGE-RITZ process usiug 2 terms in (3.10). &, the result of two integrations, is correct to 1%. Nevertheless, k obtained from VJANILLO’S method (4.4) has a maximum error of 3%. At the same time the results from (4.5) and (4.7) are accurate to the number of decimals used after the first integration. This latter method has, therefore, been used in comparisons with the RAYLEIGH-RITZ process when a $r 1. such a comparison is given in Table V where values of k from the RAYLEIGH~RIT~ process, lvith J terms in (?-IO), and exuct values we given. l’l~! exact values at CL= 1 we those of BAKER :tiid &UXRKXS (IQ&s), the remainder coming from (4.5). The vaIues of et (P, 50 and O-2, are the extremes required, and Jois taken from Table III. TABLE &@-i@~-Ritz
c
P
V
bracketed) compared with exact values oj k ((I = 0.32).
(dues
01= 0.0
CL= 14
CL= -
1.0
M
I.66
1.260
(1.270)
l-582
(1.593)
2.098
(2.112)
50 O*?
1.67 3-5-i
1,213
(1.224)
1.532
(1.542)
NM8
(2.059)
0.368
(O%Q)
0.754
(O-756)
1.118
(1,121)
It q-ill tpe -ew
tktt the bracketed
greater than 1%.
values, aud hence those of Table III,
have an error nowhere
l~‘lmigebuckliug in ii bent I-section beam
159
The way in which (3.10) fails to satisfy the bouudary condition?, whilst giving an upper limit for k in accordance with the argument of Section 3, is shown in Fig. 5. This gives, in :I typical cai;c, values of F and F” from the two methods. Thus F” horn the Rnr~mon-Rrrz process approaches the true values slowly, owing to (3.10) having a zero in E”’ at t = 1. This zero makes it impossible for (3.10) to satisfy the boundary condition (3.8(i)) at 6 = 1, although F” from (3.10) is approaching the true values near 5 = 1 as the number of terms is increased. 5.
EXPERIMEXTS
A preliminary experiment was carried out on an aluminium alloy I beam approximately Min. long. The section was 2$in. deep, the flange width was Sin. and the web and flanges were nominally O.llin. thick. The material had a @l O/$proof stress of about 28 tons! in2. and Young’s modulus was 4,660 tons/in2. Pure ‘bending was applied in the plane of the web by a four point loading system described in a kI previous paper (1952), using bearings described by NEAL At the ends of the (1950). specimen, the web and flanges were each clamped, although the fact that the compression flange buckled in four half waves made this effect unimportant. Buckling was detected by means of six electric strain gauges fixed to the compression flange as shown in Fig. 6. The figure also shows A/q plotted against A in the manner originated by SOUTHWELL (1932). A is the maximum strain in any gauge minus the nominal maximum strain due to bending, and is e.thus the strain due to buckling. Shape of deformatiou F and its derivative Fig. 5. q is the nominal maximum F” from RAYLEIGII-RITL and SCHWAM methods. stress. The linearity of the CI= 0, c = Xl, p = 1.37, 0 = 0.32. experimental relation between A/q and A, and the close agreement between the experimental buckling stress of 22.6 tons/in.2 (the inverse slope of the line) and the theoretical buckling stress of 22.0 tons;‘in.2, was felt to justify the use of the Southwell plot method in subsequent experiments. The theoretical buckling stress was obtained by using the data of KROLL (1943) with Fig. 3. The applied bending moment being in the plane of the web, it was first assumed that the. web was encastered by the tension flange. The ratio (moment per m~it length)/(rotation of lower edge of web) can then be obtained from the quantity S given by KROLL at various A,‘h, ILbeing the web depth. The web moment is then taken to be divided equally between the buckling flanges, so that by assuming various A, wc: can get the curve _iU in Fig. 3 of c agaiust
.J. 1’.
160 h,;O
=
p.
critical
The
of intersection
LW
I),iV:DSOS
al. which
was justified
by Table
II, ga\-c thr
load.
In a second calculation. assumed to be free from
also based on the data of Krtor.r.. the top flange I\a\ tension. thereby providing a smaller restraint to th(a
web than in actual fact.
This showed
the error of the previous
assumption.
tllat
the web was enc!astrC, to have been negligible. Three
scrir\ of tests were then carried
to the first, Ilaving flange width machined
Young’s
were bot.h Zjin., to
about
thickness, the web being about 0.11Oin. second
modulus
web,
after
of the
a third
series
“g
11 “G 4.~ X 31%
buckling
bearings
iit ~FC’IY’
’ a 11
In all eases elastic. end
Itlntc~ri:~l
depth
I
’ b-m 52 $E
The
sinlilar
j
unaltered.
with a sharp corner between the 0.030in. web and’0.060in. flanges. the
01
‘Q
0.03Oin..
reduction
necessitated
hr;1111
0.06Oin.
A fillet between web and flanges, of about 0.05Oin. radius, which remained
I
thickness For the
to about
the flanges remaining
an
tons, in.2 The section
and for the first series of tests the Ilanges
series. the web thickness
was reduced
out on
of 4,740
was
used
are
shown in Fig. 7, the design being based on the bearings used in the preliminary test, so as to allow freedom of rotation of each end of the beam. Loading, as before, was by
two equal
each forming
lever
arms,
an extension
of the
specimen, and each carrying an equal weight. In setting up, it was possible specimen about that the bending
to twist the its own axis so moments,
which
I~_..~~
0
50
100 150 100 s,R*!~( A x 10~ D”I TO DKKLING -
25
Fig. 6. “ Southwell *’ plot of strain due to buckling. Theoretical buckling stress 22.0 ton/in.?. Experimental buckling stress 2’2.6 ton/in.‘.
must always
act
in a vertical
plane,
any desired angle with the web. In Fig. 7, 0 is 60”, and the distance the bearings (i.e. the effective length of the specimen) is 17ins.
make
between
Four mirrors, visible inFig.7, were fixed in pairs to the two most highly compressed flanges. The members of each pair were an inch apart, and held by plasticine to the edges of the flanges at the centre section. Four horizontal telescopes sighting perpendicularly to the specimen then gave the rotations of the flanges. about the axis of the specimen, by deflections on a scale. The scale deflections were then used to give the buckling load by means of the Southwell plot. The results are shown in Fig.. 8, the theoretical curves being obtained from the curves in Fig. 3 using the “ intersection A ” method. The E - p curve was obtained
from the tables of KROLL (1943), treating
the web and unbuckled
flanges
Flan,oe buckling
as a series of plates. half of the bottom E. The tensile
When
161
in a bent I-section bean1
e =/=0, the mean compressive
flange was used when calculating
stress in the unbuckled
the contribution
to the fixity
stress in the top flanges was neglected.
The only plausible reason for the rather large differences between theory and experiment is the inaccuracy of the “ Southwell plot ” method. This has been considered
by
Hu,
LIWDQ~JIST and
BATDORF (1946)
and
by
Cox
(1951)
w!lo
3’ I
-
Ok
0
15
JO
60
IS
e
DlGRtLS
75
90
SCALE DEFLEXION
-
Fig. 3. Buckling moments from mirror readings. 0.110 in web 0 ; 0.030 in. web with 0.050 in. fillet 0 ; sharp corners 0. Theoretical curves-.
Fig.
-
9. “ Southwell” plots of mirror rendings. 0.030 in. web, 8 = 15.
showed, from the theory of large deflections of plates, that the Southwell plot method is likely to give a curved relationship between A/q and A, even when the initial curvature of the plate is quite modest. Such was the case with the present tests, the worst example being shown in Fig. 9, from which it is difficult to determine the critical moment within 10 prr cent, in sharp contrast with the However, the results confirm the predictions of Hu, results of Fig. 6. LUNDQUIST and BATDORF and COX, in giving an experimental critical load abore the theoretical, and to that extent are satisfactory.
ACKNOWLEGMEKTS The author would like to express his thanks to Dr. E. N. Fox discussion of Section 3, and to the Aluminium Development
sponsored
the work.
for much helpful Association who
BAKER,
.I. F. and
RODERICK. J. I\‘.
1948
Thr
Strengfh
of
~erclopment BRYAN,
G. H.
Cox, H. L. DAVIDSO>-. ~1. 1’. 1)1,X HARIYW. .I. I’.
1891
Proc.
19;1 ]I);“ . 1_
Ii. m/d .W. 269H.
1 !)
$0
I’rw.
Lond. Req.
Math.
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.\lr~miniunr
XI,. :j.
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J’oc. A, 212,
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I!)” (S<.i\
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