Pergamon
PII : S00225096(98)000040
J. Mech. Phys. Solids, Vol. 46, No. 7, pp. 1155 1181, 1998 (~) 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 00225096/98 $19.00 + 0.00
THE POSTBUCKLING
RESPONSE OF A BILATERALLY COLUMN
CONSTRAINED
HERZL CHAI*
Department of Solid Mechanics, Materials and Structures, Faculty of Engineering, Tel Aviv University, R a m a t Aviv, Tel Aviv 69978, Israel
(Receit~ed 28 April 1997; in revised form 6 January 1998)
ABSTRACT
The postbuckling behavior of a linearly elastic column under a bilateral constraint is studied experimentally and analytically with an eye toward applications to corrugated sandwich structures. Under a controlled axial displacement, a rather rich sequence of events unfolds, including the formation of discrete or continuous contact zones between the column and the guiding walls and the instantaneous transition of the buckling waveform to a new equilibrium configuration due to local instability. The specific details, which depend on the system parameters as well as on the loading direction, are quantified based on the linearized differential equation of the column and without recourse to imperfections or energy considerations. For a frictionless contact, the response of the column exhibits a degree of statistical variation, but the range of this variation can be bounded. A companion elastica type analysis is also developed to account for large rotations which occur at large end shortenings. Under this condition, the ever repeating mode transition process which is predicted by the small deformation analysis ceases. Other interesting buckling characteristics are found which should be useful for the understanding of more complex contact/stability systems. ~ 1998 Elsevier Science Ltd. All rights reserved. Keywords: A. buckling, B. beams and columns, contact mechanics, stability and bifurcation, elastic materials.
1.
INTRODUCTION
This work was originally motivated by studies of buckling induced growth of subsurface interlaminar defects in layered composites. For certain disbond geometry and boundary conditions, a wavy buckling pattern may develop in the delaminated region. The interaction between the buckle and the parent laminate or substrate alters this process, which may affect the stiffness and failure characteristics of the global structure. This phenomenon has been dealt with analytically (e.g. Whitcomb, 1988 ; Chai, 1990; Giannaakopoulos et al., 1995), but only as it relates to delamination propagation. In view of the basic nature of this problem, a more general consideration seems to be warranted. The following discussion will be limited to a film/substrate type contact associated with elastic stability (even though such a distinction may be an artificial one). Siede (1958) studied the response due to longitudinal compression of an infinitely long, simply supported (along the long axis) plate which is unilaterally constrained by a Winkler foundation. Assuming a periodic deflection profile along
* Fax : 00972 6407617. Email: herzl(a eng.tau.ac.il. 1155
1156
H. CHA1
the long axis, the buckling load was found to depend on the foundation stiffness, being, in the case of a rigid constraint, 32% greater than that of the unconstrained plate. The contact regions in this case were limited to periodic lines extending along the short axis of the plate. This work was later extended by Shahwan and Waas (1991) to include material anisotropy and other boundary conditions. The more complex problem of a finite size plate which is constrained by a rigid wall was treated by Hhatake et al. (1980) using a finite element scheme coupled with the penalty method. These authors also provide in~brmation on the evolution of the contact region in the postbuckling regime. Although the majority of practical film/substrate contact problems are associated with two or threedimensional stress fields, in light of their complexity, analytically tractable solutions to onedimensional configurations would be desirable. Indeed, numerous works are available in this case, the majority of which deal with the response due to axial or transverse loading of either flat or curved beams under a unilateral constraint (e.g. Burgess, 1971 ; Soong and Choi, 1986; Steint and Wriggers, 1984; Plaut and Mroz, 1992). The progression of the film/substrate contact zone with load was established via the elastica equilibrium equation or other numerical procedures. No special consideration is given in any of these sources to the important phenomenon of local instability or bifurcation that may occur when the load becomes sufficiently large. From our perspective, the unilateral constraint problem is somewhat limited in scope because it does not generally lead to an everrepeating buckling mode transition with increasing load (as, for example, in the case of some twodimensional plate problems). An exception to this is the work of Chateau and Nguyen (1991), ~ who considered a rigid plane type constraint which is located a certain distance from the axis of the column. Their experiments show that following Euler buckling, a contact zone develops between the column and the wall. This zone spreads with load until it buckles, leading to a symmetric twobuckle configuration. Treating each individual buckle in the new configuration as an inextensional beam which deforms according to the classical elastica, these authors provide a graph showing the evolution ot+ the end rotation of the column with axial load. In this work, the response of a bilaterally constrained column under axial compression, see Fig. l, is studied experimentally and analytically. The tests show that such a constraint leads to an everrepeating sequence of local bifurcation which extends the range of behavior possible with the unilateral constraint case. In addition to resembling some postbuckling features encountered for narrow plates (with or without a unilateral support), the problem at hand is directly relevant to a range of technological applications. This includes compliant foil journal bearings (Heshmat et al., 1983), corrugated fiberboard (e.g. Johnson and Urbanik, 1989), structural core sandwich panels (e.g. Nordstrand and Carlsson, 1997) and sheet forming (Triantafyllidis and Needleman, 1980; Cao and Boyce, 1997). For such applications, the evolution with end shortening of both the axial force in the column and the transverse reaction exerted by the guiding walls are of interest. Both small and large deformation (elastica type) analyses are performed. It is shown that these analyses are capable of describing the entire postbuckling behavior, including secondary bifurcation branches.
'The author has become aware of this work during the review of this manuscript.
The postbuckling response of a bilaterally constrained column P
1157
I
~ loadin~ block
~t
specimen specimen
)
i
1/I
\ /
Fig. I. Test fixture for a bilaterally constrained column.
The buckling tests are reported in Section 2, where the phenomenological aspects of the deformation process are exposed. This information is then used to obtain analytically tractable solutions, both under small (Section 3) and large (Section 4) deformation. The correlation between the analysis and the test results is assessed in Section 5. In Section 6, the analysis is extended to geometrically imperfect or corrugated columns.
2.
EXPERIMENTAL
Tests are carried out to elucidate the response of a bilaterally constrained column under axial compression. Figure 1 shows the loading fixture used. To increase the range of elastic deformation possible, very slender beams are used. Because this entails a great sensitivity to geometric imperfections, a high degree of precision in the machining and alignment of all the relevant components of the test apparatus is employed. The test columns are cut from a polycarbonate sheet having a Young's modulus of 2.3 Gpa and a proportional limit of about 1%. Two specimen geometry are tested, i.e. (b, t, Lo, h) = (I 2, 1.05, 115, 2 mm) and (12, 0.70, 115, 2.35 mm), where b, t and L0 are the width, thickness and gauge length of the column, in that order, and h is the net gap between the wall and the surface of the column. The specimens
1158
H. CHAI
with t = 1.05 mm and t = 0.70 mm, designated as " A " and "B", respectively, allow for the formation of up to three and four buckles, respectively, with no detectable irreversible deformation. The specimen, with its clamp type end grips attached, is inserted into the gap between two parallel steel blocks which surfaces are oiled to reduce friction. This gap is then adjusted until an intimate contact with the end grips occurs. The load is applied via a rectangular block which presses down on the upper end grip without introducing bending into the specimen. The axial load, P, and the end shortening, A, are recorded during the test. The edge of the column is observed in real time with the aid of a video camera.
2.1. Test results
Figure 2 typifies the P vs A response for specimens A and B while Fig. 3 shows several corresponding video micrographs taken during the tests. The position on the P vs A curve of each print is identified by the common letters l, II, etc. Referring to Figs 2a and 3a, the deformation process can be characterized as follows: (1) Contact between the column and the surface of the lower wall is first formed in Frame II. (2) The contact points (at the center and the ends of the column) spread into line contact zones (Frame III). (3) The central contact zone buckles, leading to the instantaneous formation of two buckles (i.e. n = 2, where n denotes the number of buckles in the
600
.. 4 0 0
~.~ 200 o
i
~
/
Frame~_.~I
IV
Ii[
(a) sample A
4
~3 0 ?
<
20o
T3
U
lO0 1 ~ ~ ~ 3
O0 ~" ""
f 1 i 2
~
i 3
"IV (b) sample B
4 I J 5
End Shortening, A (mm)
Fig. 2. Typical compression load vs end shortening test records for specimens A and B; n denotes the number of waves in the column.
The postbuckling response of a bilaterally constrained cohmm
1159
. . . . . . .
,
,
,.. . . . . .
Ill
i
. . . . . . . . . . . . . . . . . . . . . . . . .
Ill IV
L
. . . . . . . . . . . . . . . . . . . . ". . . . . . . . . . . . . . . . . . . . .
........
....................................... :.
.
J
II
Vll
(a) S p e c i m e n A, (b, t. Lo, h) = (12mm, 1.05ram, 115mm, 2mm)
. . . . . . . . . .
~
. . . . . .
nln i
l
: ............. ;:: ? :~.~ ......

i~;~;~,~,,::
11 Ill IV V
(b) Specimen B, (b,t, Lo.h) = (12ram, 0.70ram, 115ram, 2.35mm)
Fig. 3. Selected video micrographs corresponding to the test specimens of Fig. 2 : the position of each frame on the corresponding P vs A record is identified by the common letters 1. I1. etc.
The postbuckling responseof a bilaterallyconstrained column
I 161
column) having five contact points (Frame IV). As shown in Fig. 2a, during this jump the end displacement remains constant but the load drops somewhat. (4) As A is further increased, the contact points once again grow into line contact zones, some of which are longer than the others (Frame V). Next the system snaps into a threebuckle configuration (Frame VI). A careful examination of the video records revealed that this snapping was due to the local buckling of the longest of the flattened segments of the column (i.e. the second from the right). In Frame VII, a cup like deflection pattern associated with plastic deformation is seen; this pattern initiates from the loaded end of the column. The behavior for the thinner specimen (Figs 2b and 3b) is quite similar except that one more buckle is formed before plastic deformation becomes apparent. Figure 2b also provides information on the response under a reversed loading. In this case the column, having three buckles (Frame II), was first unloaded until a jump to a twobuckle configuration occurred and then reloaded to a complete failure. As shown, this has resulted in a rather unique nonlinear elastic response. (The slight departure between the first and the second loading paths is likely due to frictional effects.) While the aforementioned phenomenological sequence was common to all tests, there were significant variations of a rather statistical nature in the P vs A curve among nominally identical samples. This effect is attributed to asymmetry in the buckling pattern which is evident in some of the prints in Fig. 3. To gain further information on this, a few tests were halted sometimes during the loading, and the buckle waveform manually displaced using tweezers. No change in either the load or the end displacement occurred as a result of the motion of the buckle. The latter greatly affects the onset of local instability, however, and thus the overall response of the column. This interesting phenomenon will be discussed further in Section 5, following the development of the analysis.
3.
SMALL D E F O R M A T I O N ANALYSIS
Let a flat, linearly elastic column of length L0, thickness t, width b and Young's modulus E be constrained by two rigid and frictionless planes separated by a distance h0, see Fig. 4a. The column, which initially lies along the surface of the upper wall, is clamped on both ends. Under axial compression, the column buckles. The postbuckling behavior, which is effected by the interaction of the column with the walls, is studied as a function of the end shortening, A. The fourthorder, linearized differential equation for an Euler beam under axial compression is given by
P
y ' " + k 2 y '' = O, k2 ~ E l 
12P
Ebt 3
(3.1)
where a prime denotes differentiation with respect to x, y is the lateral deflection and P and I are the compression force and the moment of inertia of the column, respectively.
1162 p
. . . . . . . . . . . . . . . . . . . , . . ?
H. CHAI L0
i ? ? ? ? t t ? t i i . . . . . . . . . . . . . . . . ~ . . . .
'1
A
(a)
T T tt h0 h
,[
(
c
?
)
~
"
?~.t?'?t'.
~
.... ~
'
= H
" ?
. . . .
""
.... a i?iiii
......
....................
:'6 ...... :~: . . . .
2""'.q
1ii
. ~. . . . .. . . . . . . . .. .. 4. / / .
L ..... ~ 
........................
(t) 

~
'
:
"
~
Fig. 4. The deformation sequenceof a guided column under axial compression.
3. I.
Precontact
Following buckling, the column separates from the upper wall. The midspan amplitude, A, continuously increases with A until contact with the lower wall occurs. The boundary conditions for this loading range (0 ~< A ~< h, see Fig. 4b) are
y(O) = 3"(0) = y(Lo) = y'(Lo) = 0
where h is the net gap between the column and the walls, i.e. h = h0  t The solution of (3.1) and (3.1.1) is given by y=
(3.1.1)
(3.1.2)
A( 1  c o s ~2~x~ 0 )
(3.1.3)
with the postbuckling load given by ; = I, where ~ is the normalized squareroot axial force, defined as
kLo = 2zt
(3.1.4)
The amplitude A is found from the following geometric compatibility condition
The postbuckling response of a bilaterally constrained column A = e.,Lo +
1163
0
(y,)2 dx
(3.1.5)
where e, is the m e m b r a n e compression strain, given by P
e.,. = E b t
(3.1.6)
and H = L o / 2 . Use of (3.1.3)(3.1.6) gives
A
2t
=
.........
1 ~ r/ ~< 1 + 0 . 7 5 / ~ 2
(3.1.7)
where
f~  h / t , ~l 
3Lo A
(~t)2
(3.1.8)
3.2.
Point contact
One r/is increased from 1 + 0.75/~ 2, the column touches the opposite wall at x = L o / 2 , see Fig. 4c. The b o u n d a r y conditions for the free standing segment o f length H ( = L0/2) are y(0) = y'(0) = y ' ( H ) = 0, Using (3.1), (3.2.1) and (3.1.5), one finds y h sin(zc?~/n)  z~gS:/n + [1 cosOzgg/n)] ? tan(~g/2n) 2 tan 2n (3.2.2) l/cos 2 ~ q = ?2 +0.75/~2 ~ tan~] (3.2.3)
O <<.~.  x / H <~ 1
y(H) = h
(3.2.1)
where, for the time being, n = I. Examination of (3.2.2) shows that once ? exceeds two, the column penetrates the walls in the n e i g h b o r h o o d of the contact points. Thus, the validity of (3.2.2) and (3.2.3) is limited to 1 ~< ? ~< 2. 3.3.
L&e contact
Guided by the experimental observations, it is assumed that a line contact develops over the segments a, b and c, defined in Fig. 4d, once ? exceeds two. The independent b o u n d a r y conditions for the free standing segment o f length H, taking into account the vanishing of the bending m o m e n t , M, along the contact zones, are
1164
H. C H A I
y(0) = y'(0) = y"(0) = y ' ( H ) = 0, The solution o f (3.1) and (3.3.1) is y h( 2 ~ X  sin 27rYe) 2g
y(H) = h
(3.3.1)
0 <~ 2  x / H ~< 1
(3.3.2)
with the axial load c o n f o r m i n g to
kH
2~ = l
(3.3.3)
Equations 3. 1.4 and 3.3.3 give
H / L o = l/;
(3.3.4)
Using (Yl.5), (3.3.2) and (3.3.4), one finds 9n/~2 ~ =  , ~2 ~_ _ _ where, for the time being, n = l. 3.4.
M o d e transition
27t2 ;
(3.3.5)
The contact zones spreads with increasing ~/until the longer o f the flat segments a, b, c buckles. The tests conclusively show that the system snaps to the next higher buckling mode, i.e. from n = 1 (Fig. 4d) to n = 2 (Fig. 4e), where n denotes the n u m b e r o f buckles or waves in the column. F r o m geometric compatibility
a+b+c+2H = L,~
(3.4.1)
or, using (3.3.4)
a+h+c
........... L0
1
2
?
(3.4.2)
As pointed out in Section 2. I, the buckle waveform m a y travel freely within the length o f the column. The associated variations in the lengths o f the contact zones (a, b, c) would affect the onset o f local instability. Nevertheless, the load at which mode transition occurs can be bounded. The lower limit obviously corresponds to a single zone, say c (i.e. a = b = 0). Equation 3.4.2 then reduces to c Lo or, using (3.1.4),  1 2 (3.4.3)
kc 2/r = c,  2
(3.4.4)
On the other hand, the upper limit for m o d e transition occurs if all the flattened segments are o f equal length, i.e. a = b = c. This gives
The postbuckling response of a bilaterally constrained column
1165
3c
Lo
 1
2
,"
or
kc
2n

~ 2
3
(3.4.5)
The most probable configuration corresponds to 2a = 2b = c. We shall refer to this configuration as "symmetric". In this case, eqn (3.4.2) reduces to
2z =
kc
~2 2
(3.4.6)
The flat segment of length c buckles as an Euler column having clamped ends so that kc/2rt = 1. Therefore, from (3.4.4)(3.4.6), the transition to the second buckling mode is bounded in the range 3 ~< g ~< 5, with "symmetric" buckling taking place at ~  4. In practice, the friction between the column and the guiding walls will act to reduce the mobility of the buckle and thus the degree of arbitrariness in the response of the column. Section 5 provides some experimental evidence concerning this inherent scatter. For presentation purposes, we shall adopt throughout this work the results for "symmetric" buckling. 3.5.
Solutions j b r multiple buckles
Let n buckles exist in a column of length L0 (Fig. 4f), with the buckling configuration being either a point contact or a line contact. For the former case, one has
H = Lo/2n, AH = A/2n
(3.5.1)
where H and AH are, respectively, the length and endtoend shortening of each free standing segment of the column. It can be easily shown that the deflection of each segment and the total end shortening are given by (3.2.2) and (3.2.3), respectively, where n denotes the total number of buckles in the column. Moreover, the transition from a point to a line contact occurs when ~ = 2n. Similarly, the results for the "symmetric" line contact case are given by (3.3.2) and (3.3.5), with the transition to the next buckling mode being at q = 4n. It can be easily shown that the axial load at the onset of node transition is bounded by 1 + 2 n ~< c ~< 1 +4n. For the "symmetric" buckling pattern considered (~ = 4n), eqn (3.4.1) reduces to 2 n ( c + H ) = L0 (3.5.2)
Multiplying both sides of (3.5.2) by k / 2 z and noting (3.3.3), the dependence of the contact zone length on the load is found to be
2n Lt~ = 1  2n/~
C
(3.5.3)
Figure 5 shows the variations with r / o f c, and c/H. The results are constructed for the choice /~ = 3. Also shown is the curve for /~ = 0 corresponding to the upper bound of the range of behavior possible. (The lower bound is represented by g = 1.) Following buckling, c~remains fixed ( = 1) until contact with the wall occurs (r/= 7.75). The point contact solution [dashed line, eqn (3.2.3)] holds true up to g = 2. Thereafter, a line contact develops [solid line, eqn (3.3.5)]. When ~ = 4, the column snaps into two buckles (n = 2) during which ~/remains fixed but ~ drops somewhat. Note that
1166
H. CHAi
d~
II 4 ?.)
d
10
h/t=0
3
..
4 ~0
r? t'q
/
@
2..
.
~
J
SS"
5
ill/i//"
2
.
?.. ~D
~
/
?)/
/
I.......... /
, / J / . ' . . ...............,"
,P ......... ,
,fi
O
..
.:,
Z
@
.......
~.~,"
, I ,
Z
, , I , L , ,
o
"%~*"
,
t
I
,
0
100
200
' \
300
(~t) 2
0 400
Normalized end shortening 1"1(= 3 L ~ ) Fig. 5. The normalized square root axial load (g) and the normalized length of the line contact zone (c/H) vs the normalized end shortening (q); h/t = 3. The dashed and the solid lines correspond to the point contact and the line contact configurations, respectively,while n denotes the total number of waves in the column. The upper curve corresponds to the limit case hit = O.
the new c o n f i g u r a t i o n is a p o i n t contact. W h e n g reaches four, a line contact forms. The next mode transition (i.e. to n = 3) occurs at g = 8, although the transition n o w is to a line contact. This process continues indefinitely unless plasticity or some other failure m e c h a n i s m intervenes. It should be noted that the response of the c o l u m n depends o n the l o a d i n g direction. If, following a j u m p from the nth to the next buckling mode, the c o l u m n is u n l o a d e d rather than loaded, one eventually returns to a lower buckling m o d e t h r o u g h a different path which describes a rather u n c o m m o n n o n l i n e a r elastic behavior. W h e n the buckle snaps, it is of interest to determine whether the new configuration is a point contact or a line contact. It can be easily shown that this choice is dictated by the sign of D in the following e q u a t i o n D  ,q2 ( n 2  1  2 n ) + 4 . 3 8 5 ( 3 n 2  l  2 n ) (3.5.4)
in which the j u m p is to a line contact if D > 0 a n d to a p o i n t c o n t a c t if D < 0. It is seen that regardless of/~, the t r a n s i t i o n is to a point contact if n = 1 a n d to a line contact if n >~ 3. F o r n = 2, the t r a n s i t i o n is to a p o i n t contact if/~ < 5.54 a n d to a line contact if,q > 5.54.
3.6. Lateral reaction
The lateral forces exerted on the c o l u m n once c o n t a c t with the walls occurs are of particular interest if the walls are flexible (as for corrugated fiberboard a n d structural
The postbuckling response of a bilaterally constrained column
1167
15
/r // /
h/
s, /
3/ ~
ig
i/I
5
/t
I
,f
jl
~0
I i i
2
.......
Small deformation analysis Large deformation analysis
0
i
[
i
i
i
i
i
i
i
i
i
[
i
i
0
50
100
150
Normalized total lateral reaction force, R (= R ~ ' ) Fig. 6. The normalized square root axial load vs normalized total lateral reaction. The orientation of the slanted lines connecting the different buckling modes depends on which was taken as three for illustration purposes.
h/t,
core sandwich plates) or when frictional effects are significant. These forces, which m a y move during the loading, are limited to a set of discrete points acting at the ends o f the free standing segments of the column. Their resultant, R, is given by R = 2nV(0) = 2 n E / y ' ( 0 ) , (3.6.1)
where V(0) is the internal shear force at each end of a segment. Using (3.2.2) and (3.3.2), one has /~ =
1 
4n 2
rcc
for point contact
(3.6.2)
tan for line contact (3.6.3)
/~ = 2n~ where
/~ =
R L0 Ph
(3.6.4)
Figure 6 (dashed lines) shows the development of R with g for a continuously increasing q. Note that the transition to higher buckling m o d e s occur along the slanted lines m a r k e d by an arrow. The orientation of these lines depend on /~, which was taken as three in the illustration. The total reaction force generally increases with 7, and it m a y exceed the axial force if n and h/Lo are sufficiently large.
1168
I
.....
H. CHAI
I
..1. ......................... J.........
(a)
....................
Iy
p .L~Y
. .... J
. . . .
~A0
N
(c)
:x
 v~o xt
Fig. 7. Notations for a bilaterally constrained column ; large deformation analysis.
4.
LARGE DEFORMATION ANALYSIS
The previous analysis becomes progressively inaccurate as the buckling wave number or the angle of rotation of the column is increased. Because this effect is more severe for a line type contact, the analysis will be limited to that case. Referring to Fig. 7 for notation, the equilibrium differential equation for a typical freestanding segment is given by
V(s) = EIO"
(4.1)
where the differentiation is with respect to the arc length, s, and O(s) and V(s) are the angle of rotation and the internal shear force, respectively. From equilibrium
V(s) = Vo cos 0  Psin 0
(4.2)
where
Z,, =_ V(s = 0)
(4.3)
Since the bending moment at the ends of the segment vanishes (i.e. M ==EIO' = 0 at s = 0 and s = L, where L is the length of the free standing segment), one has from equilibrium
V0 = P/~ (4.4)
where
ft = h / H
(4.5)
and H is the projected length of the buckle. Using (4.4), (4.2) and (4.1), one has 0" = k 2 (/~ cos 0  sin 0) (4.6)
where k is defined in eqn (3.1). The solution of (4.6) satisfying a zero bending moment as s = 0 is given by
The postbuckling response of a bilaterally constrained colunm
1169
dO ds Making use of

~//2k ~/(cos 0 + h sin 0  1)
(4.7)
dx ds
cos0,
dv dTx
sin0
(4.8)
one has for the arc length S and the horizontal and vertical distances, X and Y, respectively (see Fig. 7c)
~o,,_ (12cos/~.~.~.~O, s i _ n 0) d 0
J0~o) x / c o s O + h s l n O  1
_ x/~k ~.s,x,,'. jo (ds, dx, dy)
(4.9)
Except for the term associated with h, these integrals are identical to that for the elastica problem discussed in Timoshenko and Gere (1961). Substituting 0=q~+~, a=tan 1/~ (4.10)
and employing standard trigonometric identities for cos 0 and sin 0, one can show that  1 + c°s 0 +/~ sin 0 = 2w/] + / ~ ( sin2 } c~  s i n 2 ~ ) where, from the right hand side of (4.10) 1 Substituting sin ~ = sin ~" sin qS. one has

(4.11)
(4.12)
(4.13)
1 + cos 0 + fi sin 0 = (x/i +/~2 _ 1) cos: ~b
(4.14)
Putting this in (4.9), one has ffc~)
~1~l~OS ~ si n O)
1  sm" sin q5
= (S, X, Y)k "(1 +/~2)o.25
(4.15)
Since O(s = 0) = 0, (4.10) and (4.13) imply that qS(s = 0) =  n / 2 . The upper limit of the integrals in (4.15) is determined from the vanishing of the bending moment (and hence of 0') at both the left end (s = 0) and the midpoint (s = L/2) of the free standing segment of the column. Equations 4.7 and 4.14 thus, imply that ?(L/2) = n/2. Therefore,  ~ / 2 ~< ?(s) ~< 7r/'2, 0 ~< s ~< L/2 (4.16)
1170
H. CHAI
To evaluate (4.15), one needs to express cos0 and sin0 in terms of 4~. Using (4.10) and (4. ! 3), one finds c~ c~ c o s O = II 2 sin2 ~. sin2 q~ 2/~ sin ~ sin ~b? c~ 1 1 sin2 g. sin2 q~ , ~1 z J ,?/' 1 +/~2 (4.17) A similar expression for sin0 can be found with the aid of (4.14) and (4.17). Introducing these expansions in (4.15), one has
kS
27z
[F(~b, ~/2) + F(x/2, ~/2)] 2rr(l +/~2) ,,4
(4.18)
2~  27r(1 _}_/~2)3/4 2E(~b, ~/2) +2E(m/2, ~/2)
,
[
F(go,~/'2)F(g/2,~/2)+2hsin}.cos(o
(4.19)
k Y (l q.h2)l/4{
2~z


rr
E(4~, ~/2) + E(rt/2, ~/2)  cos 2 ~" [F(4~, ~/2) + F(rc/2, ~/2)]
o~
}
(4.20)
+
k(sx)
~/~
where F and E denote elliptic integrals of the first and second kind, respectively, defined by (e.g. Abramowitz and Stegun, 1972) F(~b, ~/'2) = f e
J,
1 
dq~
sin" ~ sin4~
(4.21)
E(q~, c~/2) = I+ X/(1  sin2 ~" :~ sin2 4~) d~b When ~b = =/2, one has
(4.22)
kL
2~
2F0t/2, :~/2)
x//l~2r
(4.23) (4.24)
kH 2~

212E(~/2, ~/2)  F(~/2, ~/2)] (1  2r)3.2
=
where r = sin2~ = 1(4.25)
The postbuckling response of a bilaterally constrained column
1171
2
"~\'
0 (s=L/2), deg.
90
, '/ , .
180
kH..X \
/~" 0
s
,
~
,
i
I
.
.
.
.
0
0.5
1.0
kH/2rc or H/L
Fig. 8. The variations with h/H of the ratio of projected length to arc length, H/L, the normalized square root axial, kH/2rr, and the midpoint rotation angle of the free standing segment of the column, O(s = L/2).
i.
Lo
'1
Fig. 9. The buckle waveform for two levels of end shortening; h/Lo = 20.15. The ends of the column for each load are indicated by tic marks.
Since 0 ~< r ~< 0.5 for all values of h, the series representation in terms o f r tor E and F converge rapidly for all/~. Their expansion for/~ < 1 leads to
kL  1 0.1875/~ 2 +0.103/~ 4 + 0 ( h 6) 2Tt kH 27~  ! 0.9375/~ 2 +0.830h 4 +0(/~ 6)
(4.26) (4.27)
These asymptotic expansions are found to be accurate to within 2.6% o f the full term solution if/~ ~< 0.6. Figure 8 shows the variations of the loading p a r a m e t e r kH/2g, the ratio H / L and the midpoint rotation, O(s = L/2), with/~. The small deformation solution ( H / L = 1) seem to hold well up to h ~ 0.1 and O(s = L/2) ~ 30'. As/~ is increased, the axial load in the column diminishes, a p p r o a c h i n g zero for/~ ~ ~ . Figure 9 shows the buckling w a v e f o r m for two midpoint rotation angles, i.e. 0 = 88 and 136 '~, for which tabulated data are available ( A b r a m o w i t z et al., 1972). These two choices, corresponding to h = 0.97 and 2.48, respectively, have identical column length, Lo; the ends of the deformed state are denoted by tic marks. As shown, the buckle tends to resemble an S shape as the end shortening is increased.
1172 0
. . . .
H. CHAI 1
I Y' ' '
2
'
0.12
/ '
/
~
/
'
~
1
0.04
O0
'
'
'
'
0.5
'
'
(=h/H)
'
'
1'.0 '
'
'
Fig. 10. The variations with the parameter h/L~of h/H and c/H at the onset of mode transition. The dashed lines are the predictions of the small deformation analysis.
4. I.
Mode transition
As for the small d e f o r m a t i o n analysis, m o d e transition is taken to occur when the flattened segment o f length c buckles as an Euler column (i.e. kc/2n = 1). F o r a column with n buckles, 2n(c + L) = Lo. This can also be written as c L0 /~ L
H
h 2n
H'
or
c
1
h L1
Lo2n
Lo Hh
(4.1 .I)
Thus, m o d e transition occurs when 2n \ t7 2,1 H} 1 (4.1.2)
F o r a column with a given h/Lo ratio, the onset of local instability can be determined as follows: starting from n = 1, c = 0 and L = Lo/2, one finds iteratively from Fig. 8 the equilibrium values of,~ and H/L. (Note at this step that h/Lo = h/2L = 0.5hH/L). Next, h is given a small increment (i.e. H is reduced), and the concurrent values of H/L and kH/2n determined from Fig. 8. Using eqn (4.1.1), c/H is then found. This incrementing process continues until eqn (4.1.2) is satisfied, at which point m o d e transition occurs. Figure I 0 shows the dependence of/~ and c/H at the onset o f m o d e transition on the p a r a m e t e r h/Lo. The results significantly depart from their small deformation counterparts (i.e. c/H = 1,/~ = 4h/Lo), shown by the dashed lines, once h/Lo is increased from a b o u t 0.05. Moreover, it is seen that no local instability is possible if h/Lo > 0.141. 4.2.
Endshortening ~s axial force
The end shortening for a column of length L0 having n buckles is given by
A = 2n L  H +
(
f
e, ds+cr,,
)
(4.2.1)
The postbuckling response of a bilaterally constrained column
1173
where ~x is given by (3.1.6) and ~,~and N are the membrane strain and the longitudinal compression force (see Fig. 7c), respectively, in the free standing segment: N
~?~  b t E 
P(cos 0 + h sin 0)
btE
(4.2.2)
Making use of (4.7), one has e~ds= ?
; fds , r, L, cosO+,s,nO,
e~d012~/5j0(~=0, ~ f i c o s 0 + / ~ s i n 0  1) ~z2t2(l +/~2) \2Jz/ j g~ ds . . . . . . . . . . . . . . . . . o 3H
,,,
d0
(4.2.3)
Using results from previous Sections, one finds
(4.2.4)
from which (4.2.5) Noting that  . . . . . . . . . 2~ 2g H eqn (4.2.5) can be expressed as (4.2.7) Figure 11 (solid lines) shows the variations with r/of c / H and c. The results, specified
kLo k H Lo k H Lo
2zt h
(4.2.6)
~. 12
@ ,,./
6
"P 4
~
~
"
~
~2 4
100
200
300
400
500
600
700
q (= 0tt)2 ) Fig. 11. The variations with the normalized end shortening of the normalized square root axial load and the normalized length of the contact zone ; h/L,~ = 20.15. The dashed and the solid lines correspond to the small and the large deformation analyses, respectively?
1174
H. CHA!
to /~ = 3 and Lo/h = 20.15, are constructed as follows: starting with n = 1 and c = 0, one finds from Fig. 8 (as described in Section 4.1) that /~ = 0.1. Introducing the corresponding values of H/L and kH/2g in (4.2.6) and (4.2.7), ? and r/can be found. Next,/~ is given a small increment and concurrent values of H/L, kH/27r, O, c, and t/ are determined. This process continues until condition (4.1.2) is met, at which point the buckle divides into two buckles, i.e. n = 2, under a constant t?. The iterative process above is then continued until the desired value of ~t is reached. Figure 11 shows that the small deformation analysis (dashed lines) holds fairly well up to the second buckling mode. Starting with n = 3, however, that analysis leads to a progressively larger error. In particular, for the present choices of/~ and Lo/h, the large deformation analysis prohibits mode transition beyond n = 2. Also, under a forcecontrolled loading, a global collapse of the column is now predicted once ? reaches 10.8.
4.3. Lateral reaction
The total lateral reaction force for n buckles is given by R = 2nV(0) = 2nPh or, using (4.2.6) and the right hand side of (3.1) /~ 2n; (kH/27r) or/~ 2n L0 (4.3.1)
h h
(4.3.2)
where /~ is defined in eqn (3.6.4). Figure 6 shows the variation o f / ? with ? for the column of Fig. 11 (i.e. Lo/h = 20.15, t~ = 3). As shown, the lateral reaction, and, consequently, the frictional forces that may act at the contact points, become significantly greater than their small deformation counterparts starting with the formation of three buckles.
5.
COMPARISON BETWEEN THEORY AND EXPERIMENTS
The line contact analysis (Section 3.5) shows that asymmetry in the buckling pattern may substantially alter the response of the column. Geometric imperfections and friction between the column and the guiding walls are important factors affecting this asymmetry. The effect of geometric imperfection is touched upon in Section 6. The role of friction appears to be highly involved, and will be discussed here only briefly. Friction reduces the mobility of the buckle waveform and thus the scatter in the response of the column. On the other hand, it leads to a gradual reduction of the axial load down the column, which increases asymmetry. The analysis shows that the relative magnitude of the lateral reaction, and thus the friction acting on the column, increases with the end shortening or the wave number of the buckle. For example, assuming a friction coefficient of 0.3, the total lateral force at the transition from n = 3 to n = 4, in the case of specimen A, is 38% of the axial load. Once the column snaps, the resulting rapid stress waves alter the existing distribution of frictional forces. The latter develop once again upon reloading, however. Because the guiding
The postbuckling response of a bilaterallyconstrained column Table 1. Mode transition loads* Axial Sample force (N) A B 0~1 88.4 7.94 2.3 2.6 2.36 1~2 Mode transition, n ~ n + l 2.3 294, 358, 386, 412, 428 198643,(508) 129,129,165 59191,(148) 3*4
1175
Pexp Ptheory
P~xp
Ptheory
70, 70.6, 82.4, 88.2, 90 71 198,(127) 29.2,29.4,41.8 21.259,(37.7)
389 1341,(1143) 230,243 116399 (339)
* The first two values for P~h~or~are the lower and upper bounds while the value enclosed by parenthesis corresponds to "symmetric" buckling.
walls in the present experiments were lubricated prior to testing, the frictionless contact analysis (Section 3) may reasonably well apply. Table 1 summarizes the snapping loads for each waveform encountered in the several tests that were conducted. The experimental loads seem fairly well contained by the lower and upper bounds of the theoretical predictions (i.e. 1 + 2 n ~< g ~< 1 +4n). One also observes that the predictions for the "symmetric" buckling (g = 4n) better correlate with the test results as compared with the upper bound. The experimental data allow yet for a more quantitative assessment of the analysis. For a line contact, one has, according to (3.3.3) and the right hand side of (3.1)
P = Ebt3rt2/(3H2)
(5.1)
This relationship was checked out by measuring P and H during a test. Determining H was proven difficult because of the smallness of the clearance between the walls and the ends of the free standing segments. To overcome this, two 7 #m metal foils were gently inserted between a given segment and the walls, one on each end, and the length between the ends of the foils measured. The true value of H was then calculated with the aid of the deflection profile given in (3.3.2). As shown in Table 2, the correlation between the experiments and the analysis [i.e. eqn (5.1)] seem satisfactory.
6.
THE CORRUGATED COLUMN
The problem treated is similar to that of Section 3 except that the column has an initial deflection (see Fig. 12) of the form h Y0 = ~ ( I  cos 27tx/Lo) (6.1)
Note that this profile coincides with the deflection of the flat column at the onset of contact with the guiding walls, a choice made for comparison purposes. For the sake of simplicity, the analysis is limited to small deformations, a shallow imperfection (i.e. h/Lo << 1), "symmetric" buckling and the first buckling mode. The governing differential equation in this case is given by
1176
H. CHA1 Table 2. C o m p a r i s o n o f p o s t  b u c k l i n g data, S a m p l e A *
H (mm) 55.1 46.8 41.1 37.8 27.5 26.1 22.8 20.6 18.8 18.1 19.3 17.3 15.6
P~xp (N)
P~xp/Pthe,,,~
Number of buckles One One One One Two Two Two Two Two Two Three Three Three
32 44 59 70 142 169 219 269 305 328 278 381 450
1.22 1.18 1.18 I. 15 1.05 0.98 1.0 0.99 1.06 1.05 1.09 1.0 1.04
* All results pertain to line contact buckling.
_P
~. c/2 ~~ae
L
.
....
.1
r ........
 ........................
q c/2 ~~..;.,~"p
Yl
' ' ~ ~ ~ y
ly o
]h
c
.,. Y ~ e ~
" "

Fig. 12. Notations for the corrugated column ; only one waveform is shown.
 Ely'~' = M ( x )
= P y + Mr, + V o x
(6.2)
where y~ is the deflection relative to Y0, Y is the total deflection (i.e. y = Y0+)q) and M0 and V0 are the bending m o m e n t and the shear force, respectively, at the ends o f the free standing segment o f the column. Using (6.1), eqn (6.2) can be written as ? 1+ky] = 2hk z cos2~zx/L, (6.3)
6.1.
P o i n t contact
Referring to Fig. 12 with c = 0, the b o u n d a r y conditions are y, (0) = y, (H) = y, (H/2) = 3,] (0) = y] (H) = 0 The solution of (6.3) and (6. I. 1) is found to be (6.1.1)
The postbuckling response of a bilaterallyconstrained column
y _
1177
h
~ ( 1  ? 2) ~ sin2 2
1 J
f
?2 I(l_
c°s ~XL" t a n 2 + sin ~ ? ~  ~?*1 t 2 (tan 2~  ~;)
O~=x/H~<
1
(6.1.2)
where g is defined in (3.1.4). Equation (6.1.2) is limited to ~ < 1.8232 (as compared with two for a flat column) because for larger ? the deflection penetrates the wall boundaries near the contact points. The end shortening is given by A = 2GH+
i
ll
}
((y')2y2) dx
(6.1.3)
where e,; is given by (3.1.6). Using (6.1), (6.1.2) and (6.1.3), the normalized end shortening is found to be
3]12?2 I 2~3 r/ = g2 q4(~.7~)2 2 g' + rc (tan2"7~c __ 2) 2
?
2x?
n2~g
tan2
1
+2(rc?slnrcg)4sin4~.tan 2
......... _4___ t a n ~
(1 c~2)
(6.1.4)
6.2.
Line contact
The boundary conditions for the free standing segment shown in Fig. 12 are Yl (c/2) = y, (H/2) = .v'~(///2) = y'(c/2) = y'(c/2) = 0 where H = Lo/2. Using this in (6.3), one finds y (h/2)

(6.2.1)
cosec sin{~c,(Yc0.5)} (1?2)g 2 sin{n?(?0.5)}
cosmrc ( 1  g 2)
+
2(.~e) (127)
(12,~) cos~( (127) ?2 (6.2.2)
?<~2~x/H~< i?
where g and e (c/Lo) are related through costa rff12c) F cos~ _~ l "4 ~2 ~L? sin ~?2?(1 _g2) tan(~?(0.5 ?))J = 0 (6.2.3)
1178
H. CHAI
4
0
0.1
c/Lo 0.2 0.3
t.P 2
I
i
i
l
I
I
I
l
I
i
0
0.5
1.0
(rl  g2) / 1~2
Fig. 13. The variations with the normalized square root axial load of the normalized axial shortening and the normalized length of the contact zone for a corrugated (dashed lines) and a flat (solid lines) column. Figure 13 (upper solid line) shows the variation o f ~ with ,; as obtained from a numerical solution of (6.2.3). F o r g = 0, ? = 1.8232, which is consistent with the results o f Section 6.1. Starting from approximately g = 0.1, the results virtually coincide with their flat column counterpart (dashed line). Consequently, k H / 2 n = 1 in that loading range. As for the flat column case, m o d e transition occurs when the flattened segment of length c buckles as an Euler column (i.e. g = 4). The end shortening is found from A = L,,~;,.+ which leads to ?/ = g2 _ 3/~2
f
tt c 2
'2
(y'): dx
ILo,'2
)
0 , ; ) 2 dx
(6.2.4)
3]~2
+ 4n(l _g2)2
{[(1 2g)nc+sinng(l 2g)]
~53sin2 rig(0.5 ~) (COS~/~) 1+ ~
COS2 ne
+n(12g)+sin2ng+8
[(l  ~2)2  c O s n e ' ( l + ?4)/~2]~ n(1 2 ) ) .................. J
(6.2.5)
Figure 13 (lower dashed line) shows the variation o f ~ with q. The latter is normalized in such a way as to yield only the end shortening due to bending. The response o f the flat (lower solid line) and the corrugated columns seem to defer mainly in the horizontal shift (i.e. q = g2+0.75/~2) needed to bring the flat column into contact with the guiding walls. 6.3.
L a t e r a l reaction
The concentrated lateral force exerted by the wall on each end o f the free standing segments o f the column, R, is given by E/y"', (0) and EIy'"~ (c/2) for the point and the
The postbuckling response of a bilaterally constrained column
1179
line contact cases, respectively. Using (6.1.2), (6.2.2) and (6.2.3), the total force is found to be
=
(~~
4g 2
1) (1  2
1
nTtan2) ng
he\
for point contact
(6.3.1)
/? 
(1
1+
for line contact
(6.3.2)
where R is defined in (3.6.4). For the line contact case, there will be an additional transverse force/unit length, q, acting along the contact zones which is due to the initial curvature of the column qUsing (6.1), one finds q Lo
P h
d2M
dx 2

E/yg"
(6.3.3)
=
2n 2  cos 2nx/Lo
g2
(6.3.4)
The total force due to the distributed pressure, F, is given by
F= 2
or
,)o
i
ci2
qdx
F  LoF hP

2n _2 sin ne
(6.3.5)
The total transverse force for the line contact problem is thus given by
/~r=(*q+F)=(~2~ i 1+ 1+
4
{
[
n(l20
2
tannC]~cc~
_]cosn~)
(6.3.6)
Figure 14 (solid line) shows the variation of Rr with g. The main contribution comes from the concentrated shear forces. Unlike for the flat column (dashed line), the transverse reaction is developed immediately upon the application of load. The effect of the initial deflection or imperfection is seen to diminish rapidly once a line contact is developed (i.e. ,7 = 1.8232).
7.
SUMMARY AND CONCLUSIONS
The response of a bilaterally constrained, linearlyelastic column under a monotonically increasing end shortening is studied experimentally and analytically. Both small and large deformation analyses are employed. The interaction of the column
1180
H. CHAi
/P 3
t~
2
I I
/// J
1
t [ t I It I I I
o
2
4
6
8
Lateral Reaction, PT
Fig. 14. The variation with the normalized square root axial load of the normalized total transverse reaction force for a corrugated (dashed line) and a flat (solid line) column.
with the rigid and frictionless walls following buckling gives rise to an interesting sequence of events, including the formation of discrete contact points between the column and the walls, their growth into line contact zones, and the snapping of the buckled column into a new waveform due to local instability. This process continuous indefinitely unless some failure mechanism intervenes. The experiments conclusively show that the system snaps to the next higher up wave number even though the new equilibrium state may not be associated with the largest possible energy release. The response of the column is greatly affected by the loading direction. Unloading a column having multiple buckles gives rise to an interesting nonlinear elastic behavior which may not be easily produced otherwise. The specifics of the column response, which are dictated by the slenderness ratio of the column, the wall to wall gap and other factors, are completely determinable from the equilibrium differential equation of the column. This is except for a certain degree of indeterminacy in the line contact regime which is due to the fact that the buckle waveform may travel freely within the length of the column. Such a movement, while inconsequential to the potential energy of the system, does affect the onset of local instability and thus the overall response of the column. Nevertheless, the range of possible variations can be bounded analytically, the predictions of which are borne out reasonably well by the experiments. The elastica analysis identifies the applicability range of the small deformation solution and exposes some new features, including cession of the mode transition process when h/H, the ratio of wall gap to [?ee standing segment length, becomes sufficiently large. A closedform asymptotic representation of the elastica solution is obtained which is accurate for all practical purposes up to h/H = 0.6. Geometric imperfections and friction between the column and the guiding walls may alter the behavior of the column. Some insight into the role of geometric imperfections can be gained from the results of the corrugated column. In addition to being potentially useful to the understanding of more complex
The postbuckling response of a bilaterally constrained column
1181
systems (such as narrow plates, with or without lateral constraints), the present onedimensional treatment is directly relevant to a range of technological applications, including corrugated fiberboard and corrugated sandwich plate structures. The relatively large area under the axial load vs end shortening curve seem to offer a viable source for energy absorption. For such applications, an incorporation o f plasticity into the analysis would be necessary. The lateral reaction which is developed between the column and the guiding walls rapidly increases with the end shortening. U n d e r certain conditions, this reaction may be of the same order of magnitude as the axial force. This m a y be of a special concern if the guiding walls are flexible, as is the case for corrugated plyboard.
REFERENCES Burgess, 1. W. (1971) The buckling of a radially constrained imperfect circular ring. hit. J. Mech. Sci. 13, 741 753. Cao, J. and Boyce, M. C. (1997) Wrinkling behavior of rectangular plates under lateral constraint. International Journal ~4[Solids and Structures 33, 153176. Chai, H. (1990) Threedimensional analysis of thinfilm debonding. International Journal q/~ f?acture 46, 23%256. Chateau, X. and Nguyen, Q. S. (1991) Buckling of elastic structures in unilateral contact with or without friction. Eur. J. Mech. A/SolMs 1, 71 89. Giannaakopoulos, A. E., Nilsson, K. F. and Tsamasphyros, G. (1995) The contact problem at delamination. Journal ~[~Applied Mechanics 62, 989996. Heshmat, H., Walowit, J. A. and Pinkus, O. (1983) Analysis of gas lubricated compliant thrust bearings. Journal ~[ Lubrication Technology 105, 638646. Hhatake, K., Oden, J. T. and Kikuchi, N. (1980) Analysis of certain unilateral problems in von Karman plate theory by a penalty method Part 2, approximation and numerical analysis. Computer Methods" #~ Applied Mechanic and Engineerin9 24, 317337. Johnson, M. W., Jr and Urbanick, T. J. (1989) Analysis of the localized buckling in composite plate structures with application to determining the strength of corrugated fiberboard. Journal of Composites Technoloqy and Research 11, l 21 127. Nordstrand, T. M. and Carlsson, L. A. (19971 Evaluation of transverse shear stiffness of structural core sandwich plates. Composite Structures 37, 145153. Plaut, R. H. and Mroz, Z. (1992) Unidirectional buckling of a pinned elastica with external pressure. International Journal ?~/Soli(Ls' and Structures 29, 20912100. Shahwan, K. W. and Waas, A. M. (19911 Elastic buckling of infinitely long specially orthotropic plates on tensionless foundations. Journal ~[Applied Mechanics 113, 396403. Siede, P. (1958) Compressive buckling of a long simply supported plate on an elastic foundation. Journal o["the Aeronautical Sciences. 382394. Soong, T. C. and Choi, Injae (1986) An elastica that involves continuous and multiple discrete contacts with a boundary, hit. J. Mech. Sci. 28, 1 10. Steint, E. and Wriggers, P. (1984) Stability of rods with unilateral constraints, a finite element solutior,. International Journal (~f Solids and Structures 19, 205211. Timoshenko, S. P. and Gere, J. M. (1961) Theo O, of Elastic Stability. McGrawHill. Triantafyllidis, N. and Needleman, A. (198(1) An analysis of wrinkling in the swift cup test. J. Eng. Mat. Tech. 102, 241248. Whitcomb, J. D. (1988) Instabilityrelated delamination growth of embedded and edge delamination, NASA Technical Memorandum 100655.