B Budiansky

L B Sills

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BRIEF NOTES 
Postbuckling Eing Analysis 
L. B. Sills1 and B. Budiansky2 
I n t r o d u c t i o n 
It is well known that the buckling behavior of circular rings depends 
on the type of pressure loading (e.g., hydrostatic, dead, central). The 
main purpose of this Note is to show how the initial postbuckling 
behavior of inextensional rings under various types of uniform pres-
sure loading can be readily analyzed by the application of general 
postbuckling theory [l]3 in conjunction with suitable Lagrangian 
multipliers. Hydrostatic loading was considered in [1], with results 
that agreed with those obtained by Carrier [2] in an early analytic-
numerical study. Dead loading has recently been considered by El-
Naschie [3]. In the present Note we study the cases of constant-
magnitude central loading (C) and central loading governed by an 
inverse-square law (IS). For comparison, we will also display the re-
sults for hydrostatic loading (H), found in [1]. 
R i n g F u n c t i o n a l 
The potential energy of an inextensional ring subjected to a con-
servative pressure loading may be written as (see Fig. 1 for nota-
tion) 
El p2" /dW\2 
2R Jo \da/ H 
(1) 
where q is the magnitude of the uniform pressure that would be im-
posed by the loading system if the ring did not deform. With F = 
(EI/R)<t>, X = qR3/EI, and B(w, u) = R2A{w, v), a nondimensional 
potential energy functional may be defined as 
1 r2* /d0\2 , 
0 = - t ( — ) da+ XA(w,u) 
2 Jo \da/ 
(2) 
The precise form of A(w, v) depends on the type of pressure loading 
considered. For uniform central loading, we have 
J»2rr \[(l + w)2 + V2]1'2 ~ l\da 0 
which gives 
J*2» f 1 1 1 1 1 
\ w + -v2 wu2 + -w2v2 v4 + . . . da (3a) 
o L 2 2 2 2 J 
An inverse-square central loading gives 
B = R2 s: i - i [(1 + w)2 + v2}1'2 
so that 
A sri- w2 + - v2 + ws — wu2 — v4 2 2 8 
+ 3v2w2-w4 + . . [ d o (36) 
Finally, for hydrostatic pressure, as in [1], we have 
J*
2*- f dw 1 1 „"j 
w — v h -w2 + -v2\ da 
o L da 2 2 J 
Equation (3c) is exact; the integrands of (3a) and (3b) are nonter-
minating series. The condition of inextensionality is 
(3c) 
1 Division of Applied Sciences, Harvard University, Cambridge, Mass.; now, 
Division of Engineering, Brown University, Providence, R.I. 
2 Professor, Division of Applied Sciences, Harvard University, Cambridge, 
Mass. 
3 Numbers in brackets designate References at end of Note. 
Manuscript received by ASME Applied Mechanics Division, August, 1977; 
final revision, October, 1977. 
Fig. 1 Thin ring 
dv IV / du\2 /dw \21 
C(w, v, 8) = w + — + - \(w +—) + ( u) = 0 
da 2l\ da \da \ (4) 
and for an inextensional ring the rotation 0 is related to w and u by 
dw 
D(w, v,e)=sm9 + v = 0 
da 
(5) 
G e n e r a l T h e o r y 
The variational equation of equilibrium, together with the con-
straining relations (4) and (5) are embodied in the assertion 
Q'[U; X]5U = 0 
where 
0 = , vC — wD 
(6) 
(7) 
i/(a) and w(a) are Lagrangian multiplier functions, and U is the vector 
of state variables w,v,6,i>, and w. The prime in (6) denotes Frechet 
differentiation, and the 6(7 is an arbitrary admissible variation of the 
state variables. (Admissibility, here, is simply periodicity and an 
appropriate number of continuous derivatives.) 
By the general theory [1], the solution of (6) is 
U = Uo + £t/i + i2U2 + . (8) 
where Uo is the fundamental state, U\ is the buckling mode, suitably 
normalized, and £2£/2 + . . . is chosen to be orthogonal, in some sense, 
to U\. The postbuckling variation of the load parameter X is then 
X = Xc + fa + f
2X2 + . . . (9) 
where Xc is the critical load. In the present example, the bifurcation 
is always symmetric, with Xi = 0. The variational equation for U\ 
is 
Q c " t / 1 5( /=0 (10) 
where the subscript c denotes evaluation at the fundamental state 
corresponding to X = Xc. For Xi = 0, the equation governing Ui is 
UC"U2&U + -QC'"U1
2W=Q 
and the general formula for X2 is 
x 2 = -
where (•) = (d/d\) ( ) . 
--ilc
i"Ui4+2Qc"U2
2 
6 
Q / f / i 2 
(ID 
(12) 
S o l u t i o n s 
In the three cases (C), (IS), (H), the fundamental state Uo may be 
chosen as w0 = vo = 60 = oi0 = 0, v0 = X. With D = d/da, the differential 
equations that follow from the variational statement (10) are 
L£/i = 0 (13) 
208 / VOL. 45, MARCH 1978 Transactions of the ASME Copyright © 1978 by ASME
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B R I E F N O T E S 
where 
Ui = 
and 
L(D) 
M 
(14) 
0 
- 1 
,D 
: o 
_ _ _ i _ o . 
0 I D 2 
-D j 0 
- l : I 
- l 
- D 
"o 
0 
0 
D' 
M = Xc [_ 
in which the (2 X 2) submatrix M is given by 
D 2 - l - 2 D 1 
-2D 
for uniform central loading; 
D 2 - 3 - 2 D 1 
-2D 
for inverse-square central loading; and 
M=\, l\ 
- ] 
-D 2 J 
-D2J 
(15) 
(16a) 
r D 2 - D i 
M = Ac 
L-D -D2J 
(16c) 
for hydrostatic pressure. In the determination of equations (16), use 
was made of the fact that vc = \c. 
In each of the loading cases, the eigenvector associated with the 
lowest nontrivial buckling load may be taken as 
Ui = Re |Ke2 (17) 
where if is a (complex) column vector. Substituting (17) into equation 
(13) leads to 
L(2i)K = 0 (18) 
and so the eigenvalue Xc is the solution to the determinantal equa-
tion 
|L (2£ ) |=0 
For the three cases considered the results 
Xc = 9/2 (C) 
= 9/4 (/S)< 
= 3 (H) 
(19) 
(20) 
are recovered from (19). The solution of (18) may be conveniently 
normalized by choosing the first element of K equal to unity. This 
gives 
(21) 
' w-T 
V\ 
h 
n 
_ < » i -
= Re ' 
' 
. 
1 
ill 
-3i/2 
ki 
- -6( . 
where the values of kt are 
•• - 3 / 2 (C) 
•• 3/4 (IS) 
3 (H) 
(22) 
for the three loading cases. 
The variational statement (11) governing U2 can now be evaluated 
and its Buler equations deduced. They are 
LU2 = F (23) 
= Re • 
/ 
I 
"9/32 " 
0 
0 
9/16 
L o J 
+ 
153/32 
-9(74 
0 
-9/16 
U 0 . 
(24a) 
for uniform central loading; 
F = Re S 
for inverse-square central loading, and 
F = Re 
' 
I 
"-189/64" 
0 
0 
9/16 
- 0 -
+ 
"-99/64" 
-9(78 
0 
-9 /16 
_ 0 -
(24b) 
0 
0 
0 
9/16 
- 0 . 
+ 
9 " 
9(74 
0 
-9 /16 
. 0 _ 
' 
p4ia 
, 
(24c) 
(16i>) f° r hydrostatic pressure. The solution to equation (23) is given by 
(72 = Re 
for uniform central loading; 
/ 
1 
"-9 /16" 
0 
0 
9/4 
_ 0 _ 
+ 
" -1 /16" 
-5(732 
3£/32 
-33/16 
- 3(72 _ 
' 
e4ia 
> 
(25a) 
' 
U2 = Re " 
\ 
-9/16 
0 
0 
27/4 
L 0 J 
+ 
-5/112 ' 
-17(7112 
3(/112 
-111/112 
- 3i/7 J 
(256) 
for inverse-square central loading; and 
t/2 = Re 
for hydrostatic pressure. Equation (12) yields 
X2 = 63/32 (C) ' 
= -999/112 (IS) 
= 81/32 (II) 
' 
I 
"-9/16" 
0 
0 
0 
- 0 -
+ 
0 
-9(764 
-9£/64 
-27/16 
- -9(74 -
• 
e4ia 
; 
(25c) 
(26) 
where 
C o n c l u s i o n s 
As expected, the critical buckling load for the ring whose central 
load is governed by an inverse square law is lower than that for the 
ring with uniform central load. Furthermore, the solution for the 
former predicts that the ring is imperfection sensitive. 
This Note illustrates the way in which the general buckling theory 
[1] can easily be used to solve for the prebuckling and postbuckling 
behavior of the inextensional ring. The use of Lagrangian multipliers 
in conjunction with the general theory simplifies the calculations. 
Finally, at the suggestion of an anonymous reviewer, we make ref-
erence, for the sake of completeness, to the study of postbuckling ring 
behavior by Rehfield [4], which was executed on the basis of a cur-
vature-displacement relation that unjustifiably omits nonlinear terms. 
As a consequence, [4] predicts imperfection-sensitivity for rings under 
hydrostatic pressure, in contrast to the insensitivity shown here, and 
earlier in [1]. The same reviewer directed our attention to the suspect 
postbuckling results for central loading, different from ours, that have 
been given by El Naschie and El Nashai in [5]. At least one reason for 
this disagreement is the erroneous neglect in [5] of cubic and quartic 
terms in the energy of the loads. On the other hand, [5] does agree with 
Journal of Applied Mechanics MARCH 1978, VOL. 45 / 209 Downloaded From: https://appliedmechanics.asmedigitalcollection.asme.org on 07/01/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use
BRIEF NOTES 
our result for hydrostatic loading, because for this case the potential 
energy of loading is exactly quadratic. 
A c k n o w l e d g m e n t s 
This work was supported in part by the National Science Foun-
dation under Grant MPS76-15469, in part by the Air Force Office of 
Scientific Research under Grant AFOSR-77-3330, and by the Division 
of Applied Sciences, Harvard University. 
R e f e r e n c e s 
1 Budiansky, B., "Theory of Buckling and Postbuckling Behavior of Elastic 
Structures," Advances in Applied Mechanics, Vol. 14,1974, pp. 1-65. 
2 Carrier, G. F., "On the Buckling of Elastic Rings," Journal of Mathe-
matics and Physics, Cambridge, Mass., Vol. 26,1947, pp. 94-103. 
3 El Naschie, M.S., "The Initial Postbuckling of an Extensional Ring Under 
External Pressure," International Journal of the Mechanical Sciences, Vol. 
17, 1975, pp. 387-388. 
4 Rehfield, L. W., "Initial Postbuckling of Circular Rings Under Pressure 
Loads," AlAA Journal, Vol. 10, 1975, pp. 1358-1359. 
5 El Naschie, M. S., and El Nashai, A., "Influence of Loading Behavior on 
the Postbuckling of Circular Rings," AlAA Journal, Vol. 14,1976, pp. 266-
267. 
On Uniform Convergence of the 
Finite-Element Method 
R. W. McLay1 
I n t r o d u c t i o n 
In the author's original paper on the convergence of the finite-
element method with singularity (See Johnson and McLay [l]2) the 
uniform convergence of the displacement functions is shown with a 
bound (at least) of c ( 1 / 2 )-\ where X is a measure of the singularity in 
the problem and c is the mesh dimension on the model. It is the pur-
pose of this Note to develop a sharper bound on the uniform conver-
gence of the displacements with the concepts of completeness and the 
fundamental inequality associated with the minimum principle 
used. 
A n Inequa l i ty 
We make use of the fundamental inequality of equation (10) in [1], 
&R(U-U, V-v) < bn(U-u, V-v), where all notation is the same as in the 
original paper. Interest is centered on that portion of the finite-ele-
ment model, i?2, shown in Fig. 1. Region #2 is of finite size, has dis-
placements U,V chosen by the finite-element method and the re-
mainder of the model has displacements tJ,V fitted to the exact so-
lution by replacing the finite-element freedoms by the exact values 
at the nodal points in the same manner as is done in the proof of [1] 
(the completeness property of the finite-element functions proved 
by a Taylor's expansion). Note that each of the single strips of ele-
ments has U, V on one side and 0, V on the other. By using this model 
with the mixed functions U, U, etc., it is possible to develop a further 
inequality from the one above which has some very interesting 
properties: 
0 < OR2(U-U, V-v) + 5(S t r ip 8)(Mixed) + ^Elsewheret^"", V-v) 
< 5R2(0-U, V-V) + S(stripS)(&-u, V-v) 
+ 0E\a^here(U-U, V-v) < Mc*
1'" (1) 
Since all terms are positive and since bR2(tj-u, V-V) < Mc
2 (the second 
derivatives of the exact solution are bounded in R2) the inequality (1) 
reduces to 
1 Visiting Professor of Metallurgy and Materials Science, Cambridge Uni-
versity, England; Professor of Mechanical Engineering, University of Vermont, 
Burlington, Vt. Mem. ASME. 
2 Numbers in brackets designate References at end of Note. 
Manuscript received by ASME Applied Mechanics Division, April, 1977; final 
revision, August, 1977. 
Single / 
element 
strips \ 
* 
/N14\i/ 
U,V 
R2 
/T\L/I\i,'-
3S- Point of 
Fig. 1 
n 
"vvj 
u,v 
singularity 
Subregions 
0,V 
R'2 
fl2 and R'2 
* 
* 
/ 
/ 
/ 
/ 
/ 
/ 
/ 
/ 
/ 
/ — • 
/ 
/ 
/ 
/ 
0 < 5R2(U-U, V-V) < bR2(tJ-u, V-v) + 5(Strips)(U-u, V-v), 
< Mc2 + Mc2V-»c (2) 
where the last term in (2) is generated by the fact that each strip has 
width c. But (2) proves the convergence of the energy in R2 at a higher 
rate than that found [1], provided R2 is a finite distance from each of 
the singularities. It is not difficult to show then that 
I \^H f^l 
I Idx dx\ 
<Mc2 + Mcs-2X (3) 
with an identical statement for | |dV/dy — dv/dy\ | | 2 . From [1], the 
displacements (in R2) converge in the mean at a rate 0(CX), inde-
pendently of the singularities, and the displacements (in R2) converge 
uniformly 
|r>! — C/xl < const c i / 2 (4) 
J R'2 J d( 
!dv (5) 
for X < 1/2. More important, it is possible to use (4) to construct a 
sharper bound on the uniform convergence error in R2. 
Let us generalize equation (30) in [1] to the form 
dU_ 
H 
where R'2 is the subregion of R2 to the right of (x\,y\), the point at 
which the nodal displacement t/i is defined, and f(ij) is a function to 
be chosen. We integrate (5) to the form 
U1=-Sf(v)U(x1,v)dv (6) 
where the integral occurs on the ?/-axis. 
We choose /();) as 
f(v) 
Ut 
SU{xi,a)da 
(7) 
which, contrary to the development in [1], makes it a function of the 
problem itself. This does not replace the proof of [1] which requires 
a polynomial related to the trial functions. Instead, it builds on the 
proof of [1] with (3) and (4) to obtain 
f(v) = 
" 1 
+ 0(c1'2) (9) 
Ju(xi,o-)d<r 
which is problem dependent but is only weakly dependent on the fi-
nite-element mesh. With the assumption that Sti(xi,a)d<x ^ 0, true 
except for a special case of (xi.yi) and of course repairable with a 
minor change in the extent of R2, we know that/(ij) < M. Note again 
that M is problem dependent but that for each problem an M can be 
chosen for any point in fl2- It follows then directly from [1] and 
equation (5) 
| 0 i - t/tl < const c1 (10) 
with an identical development for | V\ - V\\. Thus the uniform con-
vergence in R2 is of higher order than that shown in [1], a result der-
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BRIEF NOTES Postbuckling Eing Analysis

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BRIEF NOTES Postbuckling Eing Analysis

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