~ _ Computer methods in applied

mechanics and engineering

ELSEVII~'R Comput. Methods Appl. Mech. Engrg. 170 (1999) 175-207

Constrained Euler buckling: an interplay of computation and analysis

Phi l ip H o l m e s ~''b'*, Gf ibor D o m o k o s ~, J o h n S c h m i t t b, I m r e S z e b e r 6 n y i ~ ~Program in Applied and Computational Mathematics, Princeton University. Princeton, N.I 08544, USA

hDepartment o[' Alechanical and Aerospace Engineering, Princeton Universit 3. Princeton. NJ 08.q44, USA 'Department ~0 ~ Strength ~[' Materials. Technical University ¢~[ Budapest, H- 1521 Bud~q~est, Hungary

'*l)epartment ¢~1 Control En~,,i:neering and h{l~rmation Technology. Techni¢ al Uuiv~ rsi(~' ¢( Budapest. H- 1.521 Bmklpest. Hungap3'

Abstract

We consider elastic buckling of an inextensible beam with hinged ends and fixed end displacements, confined to the plane, and in the presence of rigid, fi'ictionless sidewalls which constrain overall lateral displacements. We formulate the geometrically nonlinear (Euler) problem and develop global search and path-following algorithms to find equilibria in various classes satisfying different contact patterns and hence boundary conditions. We derive complete analytical results for the case of line contacts with the sidewalls, and partial results fl.w point contact and mixed cases. The analysis is essential to understanding the numerical r,zsults, for in contrast to the unconstrained problem, we find a very rich bit'urcation structure, with the cardinality of branches growing exponentially with mode number. © 1999 F, lsevier Science S.A. All rights reserved.

1. Introduction

This work, which continues that of [19,15], arose from a desire to understand buckling of polypropylene fibers in a 'stuffer box' manufacturing environment for non-woven fabrics. The wavelengths and buckling modes selected are important in determining fabric properties. Since buckling onset and modal selection appear to be determined by the initial elastic behavior, we consider elastic (Euler) planar buckling of a beam subject to hard loading (displacement boundary conditions) and constrained to lie within rigid, parallel, frictionless sidewalls set at distances -+h from the centerline. To introduce the problem and some key phenomena, we begin by describing the outcome of an experiment in which the end displacement D of a hinged-end beam is gradually increased from, and then returned to, zero. The reader should refer to 1he load-displacement (A, D) bifurcati~m diagram of Fig. 1, which ,compares the results of theory and computation of [9] with experiment.

Initially, the load rises :vapidly until noticeable buckling into the first mode occurs (in the ideal symmetric case, A would rise with D = 0 up to the first critical load A = rr2). The beam thereafter deforms as in the classical Euler problem [10,19] until its midpoint contacts the sidewall Now the load rises sharply, and after an interval of point contact, the beam begins to 'lie down' in a line contact segment. Ultimately, the load in this (increasing length) segment exceeds the clamped-clamped Euler buckling load, and secondary buckling occurs with concomitant drop in load, leading to a solution with two contact points. Thus far, reflection-symmetry about the midpoint is (approximately) maintained. This can persist unti] the central point contacts the other wall, at a displacement coinciding with that at which the third mode (if it were 'stabilized') would simultaneously

* Corresponding author.

0045-7825199/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PlI: S 0 0 4 5 - 7 8 2 5 ( 9 8 ) 0 0 1 9 4 - 7

1 7 6 P. Holmes et al. / Comput. Methods Appl. Mech. En~;rg. 170 (1999) 175-207

7~

10

I " I I o o ,oi I 0

Fig. I. Load-displacement curves for a pinned-pinned beam: h = _+0.05; Solid cucves with circles correspond to experiments. Arrows indicate direction of loading histor)~. Numerical results are shown as solid curves. Beam shapes are indicated in insets. Modified from [9].

touch both walls. However, in Fig. 1, a jump occurs at a lower displacement to the second mode branch in a line contact state. This branch is then followed up and, reversing the displacement rate, down, but beyond the displacement at which it was joined, into a point contact regime. Contact is subsequently lost at one point, an asymmetric single point contact solution develops, the load rises at first modestly and then more rapidly, and a jump down to the symmetric first mode point contact solution occurs;. This portion of the outgoing path is then retraced to (,~, D) = (0, 0).

There are several interesting phenomena absent from the classical Euler problem: (1) hysteresis and multiple stable solutions for the ~,.ame load or displacement; (2) stabilization by contact of the second and higher modes, which are unstable for the unconstrained Euler problem; (3) the existence of point and line contact states in each mode; and (4) asymmetric states.

We now turn to the mathematical model. Assuming a uniform inextensible beam with linear constitutive law, after nondimensionalisation the equilibrium equation obtained by balancing moments in each ' free ' element of the beam is

0 " + A s i n 0 + # c o s 0 = 0 , (1)

where O(s) is the tangent angle, 0 " = daO]ds 2, A and p. are axial and lateral forces, and s E [0, 1] is the nondimensional arclength. We treat the hinged-end case, making F;q. (1) subject to zero-moment boundary conditions

0 ' (0 ) = 0 = 0 ' ( 1 ) , (2)

and the axial displacement constraint:

x( ] ) - x(0) = cos 0(s) ds = d ~ ( - 1, 1 ) . (3) )

At each contact point '~ii' /x undergoes a jump #,+ ~ - / x i equal to the (normal, lateral) constraining load at that point, so within the beam a sequence of BVPs must be solved and matched at contact points, which are themselves determined by constraint equations such as:

P. Holmes et al. / Comput. Methods Appl. Mech. Engrg. 170 (1999) 175--207 1 7 7

f[ -s I

y(s,) - y(O) = sin O(s) ds = + h , (4.) )

y(s/) - y(si ~) = sin O(s) ds = -+2h, etc. (5)

where h is the sidewall distance from the centerline. Unknowns include contact points sj and loads A, #~, and a solution is a C ~ function O(s) that may exhibit jumps in shear force O"(s) at contact points. Moments 0'(s) are, however, continuous throughout. The key to explicit solution is the first integral of (1)

( ~ t 2

~ - - -- A cos 0 + # sin 0 = const., (6)

from which a quadrature can be formed. Further details on boundary values and the construction of specific unconstrained, point and line contact solutions may be found in 19]; we recall them where necessary in subsequent sections.

Viewed as a sequence of phase-plane problems, ( 1 ) - (5) is an example of a hybrid dynamical system 1141, and relatively little is known about the bifurcations such systems can exhibit (the results of [3] for initial value problems are not applicable here). We approach the problem by considering the superposition of bifurcation (=solution branch) diagrams in (M D = 1 - d ) - s p a c e for the unconstrained and a sequence of increasingly more constrained problems involving point and line contact regions. Bifurcation points for the full problem correspond to intersections of solution branches of these distinct problems. We introduce these solution classes or sheets and their symbolic descriptive labels in Section 2, where we also describe the symmetries of tile problem. The numerical algorithms are discussed in Section 3. In Section 4 we develop complete analytical descriptions of line contact solutions, and partial results for point coF, tact solutions. Section 5 assembles these results with outputs of the numerical methods to provide a partial global description of load-displacement behavior. We summarize and note some open problems and possible future directions in Section 6.

Earlier work was done,' on constrained buckling of a linearised model by Link and Feodosyev 118,1 1]; Miersemann and Mittelmann considered the stability of (single) line contact equilibria using variational methods, and Keller et al. [17,161 analyzed some particular self-contact states, but our work appears to be the first in which the enormously rich variety of solutions to a constrained buckling problem is examined. In this respect, it recalls our earlier study of the discretized Euler strut [81]. It also provides an instructive example of interplay between analysis and numerical computation. For example, in [19], following [1 11 and our prejudices, we initially studied only symmetric states (symmetric under g3, Eq. (9) belc, w). Experiment followed by numerical computation began to reveal the asymmetric states shown in Fig. 11 of 191 and their continuations and generalizations, of. Fig. 8 below. But how complete is such a picture'? How should one choose regions and parameters for the global searches (Section 3.1) necessary to locate disconnected yet physically relevant branches? How can good starting points be found for path-following on asymmetric branches (Section 5;.2)? It soon became clear that classification of different solution types and further analysis would be needed to usefully interpret the tangle of computed branches. Such questions led to the present essay.

2. Classes of solutions

The experiment described above shows that, depending on loading conditions, equilibria with different numbers and types of contact with sidewalls can appear. We therefore develop a descriptive scheme to characterize distinct classes of boundary value problems. These will determine both how analytical solutions are obtained, and the number of variables which must be solved for in numerical algorithms. The basic idea is that of a sheet over the (A, D)-plane, which contains all solution branches to a given BVP. We first define some simple cases, and then generalize to the notion of sheet codes describing BVPs with arbitrarily many contact points.

• Sheet 0 contains all solutions to the unconstrained Euler buckling problem; i.e. to (1) - (2) with #(s) = const.

178 P. Holmes et al. / Comput. Methods Appl. Mech. Et.,grg. 170 (1999) 175-207

• Sheet PI contains all solutions to the buckling problem constrained by a single point load normal to the axis; i.e. to ( 1 ) - I 2 ) with /~(s~ ) # 0 in general, and subject tc~ the constraints:

y ( s ~ ) - y ( O p = ± h = y ( s L ) - y ( l ) and 0 ( s ~ ) = 0 . (7)

• Sheet L1 contaizls all solutions to the buckling problem having a single line contact region (possibly degenerating to a point); i.e. to ( l ) - ( 2 ) with

y ( s j ) - y ( 0 ) = ± h = y ( s 2 ) - y ( l ) and O ( s ) ~ O : s ~ [ s ~ , s 2 ] ; O ' ( s j ) = O ' ( s 2 ) = O . (8)

• The sheet code is a sequence of integers {k~, k2'k~ . . . . . k,,) denoting n contact points, the j th of which has angle 0(si) -- kj-rr. Contact patterns are specified by the symbols separating the kj 's: ( , ) indicates succeeding contacts on allernating walls, ( ' ) repeated contact on the same wall.

Without loss of generality, to eliminate rigid body motions we assume that the angle 0(0) ~ ( - v , wl- Since 0 ' (0 ) = 0 the solution of the BVP up to s~ belongs to a closed level set of (6) inside the separatrix, the maximum extent of which is 2~'; hence Ik~l ~ 2. At this point, succeeding entries are unrestricted; note that they count total rotations of the tangent vector and hence specify the n~Jmber and sense of loops.

The sheets described above can be written: 0 = (¢J), PI = (0), and LI = (0 '0) and we can generalize these to two classes of point and line contact solutions which will play an important r61e in what follows: Pn = {0, 0 . . . . . 0), and Ln = (,0'0, 0 '0 . . . . . 0 '0), the latter with the restriction that between adjacent contact points on the same wall ( ( . . . 0 '0 . . . ) ) the beam is straight. Note that, withoul this additional constraint, ( . . . 0 ' 0 . . . ) can represent two isolated contact points on the same wall, as in the ~econdary buckling described in Section 1. These do not exhaust the possibilities: for example, one can have 'coi led ' states such as (0, - 1 ) and (0'0, 1' 1), and 'mixed ' states such as (0'0, - 1 , 0 ' 0 ) . Fig. 2 illustrates the concept, and Fig. 3 shows examples of some characteristic solutions.

Equipped with the notion of sheets, we now develop a symbolic classification to distinguish individual branches of solutions which lie on a given sheet. Solutions are built fi'om elements separated by contact points or segments.

• We define an element o f order m and length l as a solution of the unconstrained Euler problem Ill) with #(s) -= 0, with m i~nternal inflection points and slope and mom,znt boundary conditions at s = 0 and s = 1 determined either a priori (in the case of line contact), or by subsequent matching (in the case ,of point contact). Examples will be constructed in Section 4.

• The branch code associated with an n-digit sheet code has n + 1 nonnegative integer entries: {m o, mj . . . . . m} . F~ach entry denotes the number of internal inflection points in the corresponding elernent; straight segments with O(s) ~ kTr are denoted by *. As above, tt:~e separating signs bear a special meaning: {,} (resp. {'}) denotes that the neighboring elements lie inside (resp. outside) the line of the wall near the contact point. See Fig. 3.

Fixing a sheet code and a subset of branch codes specifies a complex of solution branches that may contain many members interrelated by symmetry properties (see below). We identify some special cases which will be investigated in detail.

• We define the nth mode point contact complex as all those brar~ches belonging to sheet Pn =- (0, O, . . . 0). Such a branch will have codes of the type {m~, m~ . . . . . m,,} or . Imo 'm~ ' . . . 'm,,}, where m i is the order of each element.

• We define the nth mode line contact complex as all those solutions belonging to sheet Ln = (0'0, 0 '0 . . . . . 0'0). Such a branch will have a code of the type {m~, *, m~, * . . . . . *, m,,}, where m~ is the order of each 'free ' element.

• We define regular modes (of order m) as those with seqtLences of the form {m, 2m + 1 . . . . . m}, {m'2m + 1 ' . . . 'm}, or {m, *, 2m + 1, * . . . . . *, m}. Modes which are not regular are called irregular. Regular modes are the natural analogs of the classical Euler modes. See Fig. 3.

As we shall see, a very rich variety of codes are realized as solutions, but there are some restrictions. In general branch codes the mj can be any nonnegative integers, but solutions corresponding to branches belonging to Pn and Ln must make m full circuits in the phase plane of (1) betw,zen each pair of contact points, and so the interior elements, mi; 1 ,~-:-.j ~< n - 1 are necessarily odd for such sequences to be admissible. Moreover, no mixed l ine]poin t contact modes can exist with sheet codes containing entries such as ( . . . 0 '0, 0 . . . . ) (excepting

P. Holmes et al. / Comput. Methods Appl. Mech. Engrg, 170 (1999) 175 -207 179

1

/,a, o f

Fig. 2. Sheets of solutions and generalized bifurcations.

the case of a line contact segment shrinking to zero). For suppose the contrary, and consider an element with line contact at s = 5) and point contact at s = s t ~ ~. Thus, we have to solve a BVP for (1) with O(si) = O'(Sj) = 0

and 0(s).+~)= 0; O'(S~÷n) ~ 0: a contradiction, since these boundary points cannot lie on the same connected level set of (6).

Conventional bifurcations can occur among the solution branches on a given sheet: for example the sequence of pitchfork bifurcations to classical Euler modes on sheet 0; but much of our concern will be with genera l i zed

b i furca t ions which occur at points where a given branch violates a constraint. There the 'physical ' solution transfers to a second branch that coincides with the first at that point, but belongs to a different sheet, as sketched in Fig. 2. These violations may be ordered according to increasing severity, as in the following list (with examples):

* Tens ion at a s idewal l . Continuation of the (symmetric) single point contact solution for loads below the. point at which the unconstrained solution first touches the wall requires that the lateral load p.(s = :,) be directed toward the wall.

• Se l f - in tersec t ion . The endpoints of the classical unconstrained solution coincide at D = 1 (d = 0), and thereafter a looped (everted) state exists (cf. [19], Section 263, Figs. 54-55) .

• Pene t ra t i on o f a side,.vail. Continuation of an unconstrained solution for loads above the point at which it meets a point contact solution.

• Overa l l length exceeds 1. Line contact solutions, to be constructed explicitly in Section 4.2, are mathematically defined even for cases in which the sum of the absolute values of all free element lengths exceeds the beam length. See Fig. 11 and the paragraph following Eq. (21).

We regard the first and third of these as less serious, since one co,aid devise point loading and constraint

180 P. Hohnes et al. I Comput. Methods Appl. Mech. En,crg. 170 (1999) IZ5-207

SHEET CODE AND BRANCH CODES AND INTERPRETATIONS

INTERPRETATION

< >

L J

f 1

< 0 > L A

r 1

<0,0> L.--I~ J

U I

{o} {l} t J L - - - - - - I

r 1 [ 1

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i 7 r 7

{0,0}A {I'I}A

I 1 r 7

(o,z,o}s {o,5,oi

i ~ l I . . . . 7

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<0 '0> L 1

r 1

< 0'0,-1 '-1 > k _ l

""1" '1 F - - - I

{2} t J

( 1

{0,2}

f-

{0,L0}A I

I

{0,2,0}S I J

i I

{0,*,0}AR

I F

{o,*,o}sR {o,*,lH

- - 1 f 1 r

{0,*,0,*, 1 } S symmetric i I A asymmetric

R regular ~ I irregular

i i

Fig. 3. Examples of sheet and branch codes, showing point and line contact cases, with regular symmetric, regular asymmetric, and irregular modes.

schemes for which they are not violations of physicality. The second is frequently discounted, it being argued that 'slightly three-dimensional' nearby states exist arbitrarily close to those with self-intersecting loops. One could add a further, still weaker requirement: that the (physical) solution be stable. We include some remarks on necessary conditions for stability in Section 5.4.

Observe that branch codes can change only at (generalized) bifurcation points. In particular, classical uniqueness results for ODEs imply that the nodal structure (number of inflection points m) is invariant for each element between bifurcation points [4]. Moreover, for branches belonging to Pn, the label {,} can change to {'} only by the moments 0'(si) passing through zero simultaneously at all contact points, in which case the branch intersects the con'esponding line contact branch. We use this property in Section 4 below.

We end by noting the symmetries of this problem. A solution to a given BVP can be characterized by the equilibrium shape (x(s), y(s)), the axial load/l , the interior contact points {Si}jU jJ and the lateral loads {Iz(sj)} at those and the end points so, s u (each end of a line contact segment is counted separately in N). The symmetry group G has eight elements, generated by

P. Holmes et al. / Cvmput. Methods Appl. Mech. Engrg. 170 (1999) 175-207 181

gj : y ( s ) ~ - > - y ( s ) ; /z(st)~--> - #(sj)

g2 : x ( s )~ - -~ -x ( s ) ; A~--~ - A, (9)

g3 : x(s)~-->x(s) - x ( 1 ~ ; s~-~ 1 - s .

These respectively describe reflection in the x-axis, reflection in the y-axis, and reversal of arclength labeling, which effectively reflects the shape about a vertical axis at x = x ( l ) / 2 .

3. Simplex methods

We introduce our computational approach with some general remarks. The constrained buckling problem can be formulated as a multipoint BVP; however, the locations of these points in terms of arclength are not known a priori, which makes the correct formulation nontrivial. Since contact forces as well as contact point locations are unknown, the application of traditional force- or displacement-type melhods presents difficulties. In a physical experiment one either controls the load A ( 'soft ' BC) or the displacement d ( 'hard ' BC). For computation, the only relevant aspect is that there is o n e control parameter, so we expect solutions to emerge as one-dimensional branches. In this context we pose general questions of the following kinds:

(a) Find a l l solutions of the BVP in a given domain, including ones on branches disconnected from the trivial equilibrium.

(b) Given one solution, compute adjacent solutions and produce branches. (c) Determine the branch connectivity of a set of 'isolated' solutions. Even disregarding difficulties presented by the unknown boundary locations, traditional methods (e.g. [23])

are-- to the best of our knowledge--incapable of answering questions of type (a), which are of prime importance here. Physical experiments, reported in [9], indicated the existence of asymmetric states, reached via dynamic 'snap' from symmetric configurations. This behavior implies thai ptlysically relevant, stable disconnected branches exist, inaccessible by path continuation. The simplex method developed for ODEs in [6,7,12] can address such questions. Wc first review it, and then describe its application to constrained buckling.

3.1 . G l o b a l s e a r c h

To address questions of type (a) we wish to periorm a global search in an automated, algorithmic manner, with no a priori information on solutions. Results of a similar search were first published in [6] for classical Euler buckling. The method relies on two main ideas: direct integration of IVPs in the geometrical phase space context (cf. [13]), combined with the piecewise linear (PL) algorithm o f [ 1 ].

The relationship between IVPs and BVPs is nontrivial. BVPs are ostensibly more complicated, ~ince conditions are imposed at multiple points. On the other hand, the boundary conditions at the 'origin' fix some initial values, implying t ha candidate 1VP trajectories belong to a subspace of the full phase- and parameter space. We call this the Global Representation Space (GRS) of the BVE (For ODEs derived from canonical variational problems, the GRS has half the dimension of the IVP phase space, plus the number of the parameters.) The GRS provides a t y p o l o g i c a l l y f a i t h f u l embedding for all solution branches: branch intersec- tions in the GRS imply coincidence of the physical configurations corresponding to them and vice versa, so each physical configuration is uniquely represented by a point in the GRS. This property follows from standard existence and uniqueness results for ODE 1VPs [14]. In computational mechanics other spaces are often adopted for BVPs; however, none of these is topologically faithful unless it contains the GRS. Topological faith is not merely convenient for visualization and representation, it is the key idea in the global search algorithm, :since scanning the GRS systematically guarantees that a l l solutions will be found (up to discretization accuracy ~. No other space has this property.

We formulate the problem as a system of nonlinear algebraic equations, for which we must specify: • V v a r i a b l e s , i.e. initial conditions not fixed by boundary conditions at the origin, and parameters including

intermediate boundary conditions, to span the GRS, and • F . [ ~ m c t i o n s , i.e. matching conditions at the far end(s).

182 P. Holme~ et al. / Comput. Methods Appl. Mech. Engr¢. 170 (1999) 175-207

These latter become explicit functions of the variables when we apply any (convergent) forward integrator to the IVE The variables supplemented by the fixed initial conditions supply a full set of initial data, which provides, via the integrator, a forward trajectory and thus values for all relevant quantities appearing in the far end matching conditions. Since we have a one-parameter problem, we always have V - F = 1, so the integrator yields a mapping [~v_÷ Nv ~.

Following [12], we construct a simplicial mesh in the V-dimensional GRS, each n-dimensional simplex being determined by n + 1 nondegenerate vertices. We compute, via the integrator, the values of the F = V - 1 functions at each vertex and interpolate them piecewise linearly on the mesh. This defines V - 1 codimension- one hyperplanes that generically intersect in a line, which either does or does not intersect the simplex. If it does not, the simplex is discarded: if it does, the intersection is declared an approximate branch segment. The process is repeated for all simplices in the chosen domain of the GRS Jnd approximate solution branches are thus obtained as polygons. As the simplex mesh size and integrator step size converge to zero, the algorithm yields all solution branches in the chosen domain. However, both disc~retizations give rise to additional, spurious solutions, disappearing only in the aforementioned limit. We call th,~se solutions parasites ~f the first or second

kinds depending on whether they arise from ODE discretization or GRS discretization, respectively. The former can be eliminated easily, since computation time grows linearly with the ODE discretization. Refining the simplicial mesh is much more expensive: halving mesh size multiplies computational time by 2 v. We mitigate this via a multigrid approach. In the first phase, solutions are comptLted on a sparse grid and finer grids are only realized and computed in those domains where solutions were found in the first phase.

The global search algorithm is ideal for parallelization, since the operations (computation of function values, solution of the linear equation system) can be carried out independently for each simplex. A master program

subdivides the chosen GRS domain into subdomains and assigns each to a slave processor which, after completing its task, receives the next from the master module. Currently, our code utilizes PVM (Parallel Virtual Machine) software, enabling the user to regard a network of heterogeneous hardware as a single parallel machine. More details on parallelization can be found in [12].

3.2. Path following

The path-following version of the simplex method is based on th,z global search version. However, there are fundamental difference:~. Unlike global search, the goal of path continuation is to provide an ordered sequence of equilibria, approximating a solution branch of the BVE To start such a computation, an equilibrium configuration has to be identified via a V-dimensional vector. As before, we construct a simplicial mesh in the GRS, with the restriction that the given solution should lie on the surface of a simplex. In step 0, all function values at all vertices of" this simplex are computed and the linear equation system is solved, providing a line intersecting the faces of the simplex in two points. One of these points is the initial one, called the entry point,

the other is the exit point. In step l we select from the adjacent simplices the one on a face of which the exit point lies, and declare 1he latter the entry point to this new simplex. V-dimensional adjacent simplices share V vertices, so in the new simplex the V - 1 function values need be computed only at a single vertex. Tile linear equation system is then solved for this simplex, providing a line segment and a new exit point. Repetition of step I provides (approximate) points of the branch by direct recursion, with no iterative steps. A slightly different version of this continuation method is discussed in detail in [7].

Compared with othe~ ~ path continuation codes, our method has ~everal interesting features. Besides being iteration-free, it requires the solution of only one IVP to obt,ain each new branch point, minimizing computational effort. After finding an approximate solution from the linear equations, one can compute the function values at this point, thus obtaining a measure of error. This feature makes it possible to monitor the error along the computed branch. Moreover, if, whenever two solution points lie closer than the mesh size they belong to the same branch, then closed branches are computed as closed loops and vice versa.

A drawback in the current version, compared with path-li)llowing codes such as AUTO [5], is that bifurcation points are not treated directly in our algorithm, which may pick any of the emerging branches effectively at random. The proper approach to bifurcation points is to take cubes rather than simplices as computational units (the cubes are still subdivided into simplices to admit piecewise linear interpolation).

P. Holmes et al. / Comput. Methods Appl. Mech. l'ngrg 170 (1999) 175 207 183

3.3. App l i ca t ion to cons t r a i ned buck l ing: some beastx in the zoo o f ~5ranches

The key aspect of the simplex method is the definition of variables and functions. In the present application this has to be done separately for each sheet; indeed it can be regarded as the 'computational' definition of a sheet. Fig. 4 provides a full description for some sheets, which can be readily generalized for arbitrary sheets. Note that interior contact point locations and constraint forces appear among the variables.

Both the global search and the path continuation code can be applied to one sheet at a time. The difference between the two approaches is that while global search can be executed on an arbitrary sheet, regardless whether any other sheet has been computed previously, path continuation requires a starting point that can be supplied from another sheet at a generalized bifurcation, or directly from analysis (Section 4). Of course V generally differs for different sheets, so while some entries in the vector are directly inherited from the previous sheet, new ones must be added to represent physical quantities of the same equilibrium shape which did not count as variables on the previous sheet. For example, when swilching from the unconstrained problem to single point contact, the location and magnitude of the contact force need to be added. This procedure is described in detail in 19], Subsection 3.1.

As a first example, and to verify the code, we performed a global search on sheet (0) (unconstrained) in the domain - 2 0 0 < A < 200; --200 < ,a < 200; - 4 . 5 < 0 o < 4.5, with 500 grid points in each direction. Running 313 RISC processors the computation took less than 5 minutes. The results are shown in Fig. 5. The spirals appearing on the GRS appear as vertical lines on the (A, D) plane and correspond to rigid body rotations at D = l with/x ~ 0 and #2 +_ A2 = const., i.e. coincident ends. The exac! solutions computed as in Section 4.1 are visually indistinguishable, maximum errors in D for 0 < A < 240 are 0.12%.

The next sheet to be computed was (0> (single point contact). Global search was done in the 5-dimensional domain - 1 0 0 < A < +500.. - 2 2 0 < # , < +300, 0.1 < 0 , < +4.9, - 9 0 0 < # ~ < + 1000, 0.1 < s . < 0 . 9 with primary subdivisions 100, 100, 100, 120, 100. With 33 RISC processors computation on this grid took 7 days. Fig. 6(a) displays with the results in the (A, D) plane. A secondary grid was constructed, with mesh size 20% of the primary grid, in the domain where the first computation found solutions. Computations on this required

SHEET C O N F I 6 U R A T I O N V A R I A B L E S I FUNCTION(S) [ ]

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" ~ 0

2. A 3. ,u- o ¢" #'1 5. s~

1. Oo 6./-~ 2 2 .2~ 7. s~ 3. AZo t,. s~ 5. ~1

1. 8'(1)=0 2. y(1)=O

1. g (sO=O 2. y (sO= h 3. 8 ' (1)=0 /+. y(1)=O

1, g.(sl)--.0 2. y ( s l ) ; h 3. g(s2)=0 ¢. y(s )=h 5. 8'(1 )=0 6. y(1)=O

Fig. 4. Definition of variables and functions for some sheets.

184 P. Hohnes et al. I Conlput. Methods A/W~. Mech. Envrg. 170 (1999) I Z5-207

i

, ~,)

xx

\ . N N - ~ - \ / (a) 't (b)

I_.L...._.........~-------.------ ~ 2 D

Fig. 5. Scanning results on sheet (0) (unconstrained) (a) in the (a, #, 0.) global representation space: and (b) on the (k, D) phme.

soo 1-77, , ' .~-~7 "-T " '-" - ~ F ' - - ~ - ' - = ~

| / ~ ? "" .~" # / < ~ . ; . , v < : : " /

c " ~',~ ' 2;1-=: +"~ .....

-100 ~ ' ~ " [ = ; .:;.?<-~(c-. )L : . . . . ~ D ) . 5 0 0 , ~ - - ~ - ~ = 7 7 - ~ ~ - . - ~ - = - ~ - ' l

, /- : d" 1

- l oo , ~ D 0 I 1,7

Fig. 6. Global scanning results for h = 0.125 on sheet (0} (single point contact) (a) on the primary, and (b) on the secondary grid.

approximately the same time as those on the primary grid, and these refined results are shown in Fig. 6(b). Although displaying branch-like structures, this point set is not yet ordered according to physical solution branches. This was achieved by the path continuation code, using a single P166 PC, and computation of the 28

P. ttolmes et al. I Comput. Methods AppI. Mech. Engrg. 170 (1999) 175-207 185

branches shown in Fig. 7 look 14 days. The starting data for the branches (many of which are disconnected and thus otherwise inaccessible) was extracted from the global search. Finally, Fig. 8 shows 64 branches of various types on several sheets, computed by the path continuation code. Appendix A. 1 provides a complete list of the associated sheet and branch codes• After developing analytical descriptions of certain branches, we will return in Section 5 to (partially) explain this remarkably complex picture.

Devoting such resources to solution of ODEs may seem profligate, but our code, running under the PVM (Parallel Virtual Machine) system, utilized the 'dormant' computing power of 25 workstations that were otherwise largely idle. With capacity of this type growing fast, it seems reasonable to ask qualitatively new questions and develop codes that can answer them, at a quantitative cost in computation time. As noted above,

5004- ii ..... ~- /

...,(

(~'z,3,

. , I I I I I I I I

,<-- r~,~) /

7 \ I - \ / . . . . . . O,5" : I z > , - "~-,~,~3 ~ ,s / __,

x ,/ , \, I / _. ",,, {~,z)s / /// ,,"./ ~.~,s3 .e{~,~J,~ --

., ......... --->---., i ~ ...... ..../...- ",.. "~ \ / \ tl ":: "

.....<S" / \ ~I /" ;~-:~" ..( ~,.;~;,'-, , , , , , ' . ; / • /

.. \, \ I t , ; - . / . ' '

{2'z3s.@.. ../~- ,

- 4

...4-{0,~7

iL--.~ o'.4J. ,../ - / " ,.'

• ,..."

50

0

"~J' "'J~"A <~ ''~ 1 0'05 - O \ . - - ' < ;~ ' ' ' - - '

,,t-C. L" ~.j / 0,034 \"--- . . . . . _ _ _ / - -

~,o35 . . . . . .

,k..u sl+- <~"~' L~

o3A

f " t / / / / /:/,,i" [I,q5"" \ - i /,," I!~ fo,~J

- 3 5 0 I 1_. I I I / , , i 1 I I I I 0 0,5 1,0 1,5

- - - 4

m

t D

Fig. 7. 28 branches for h = 0 .125 on sheet {0) (single point contac|), computed by path continuation, and showing branch codes.

186 P. Hohnes et al. I Comput. Methods Appl. Mech. Engrg. 170 (1999) 175-207

500

-350 0 1 D

Fig. 8. 64 branches for h = 0.125, computed by path continuation. Sheet tLnd branch codes are listed in Appendix A.l.

to the best of our knowledge, no other code could have delivered these diagrams, which are in good accord with the experiments of 19].

4. Analytical results

We start by recalling the solution for the unconstrained Euler problem, and then use this to assemble line and point contact solutions of arbitrary complexity. We end with brief subsections on other classes of solutions, and a note on the relation between line and point contact branches.

4.1. Unconstrained buckling: elemental solutions

The classical Euler problem of buckling of an axially end-loaded strut of length l corresponds to solving ( l ) with # -- 0, A = ~, and boundary conditions 0 ' (0) = 0 = 0'(1). The first integral (6) becomes

0 r2

~ - - ,~ c o s 0 = - - a c o s 0o,

and the resulting quadrature

(oi/l = v/~l d0

(o) ~V s'n t T ) - s,n

may be evaluated to yield

~ [ 2 ( m + l)K(k) ] 2 l ' ( lo )

where K denotes the c o m p l e t e elliptic integral of the first kind, k = sin(0 o /2 ) is the elliptic modulus, and m + 1

P. Hohnes et al. / Comput. Methods Appl. Mech. Engrg. 170 (1999) 175-207 187

is the modal number; m/> 0 denoting the number of internal inflection points. Moreover, from (3), we have for the distance between the end points:

3= L J l , (11)

where E denotes the complete elliptic integral of the second kind. See the appendix of [9] for more details. Since we wish to use this solution to describe the state of each 'free' element, we denote the element's (arbitrary) length as l, the end point separation d, and the load ,~.

4.2. Line contact branches Ln

From (10)-( l 1 ) we may construct elemental solutions from which general line contact equilibria can be built. The key observation is that, at each end of a line contact segment in which O(s) =- 0, necessarily O(s) = O'(s) = O.

Thus, referring to Fig. 9, any element spanning end point to contact point or contact point to contact point, obeys (10)-(11) and the additional geometrical constraints

h IX tan0 o - A - A ' (12)

following from the fact that the resultant of axial and lateral forces must pass through both end points. The resultant load is ,~ = ~ / # 2 + ,~2 and defining A 2 ~¢2 , = - h ' , and using trignometrical identities, we find from (12):

r

2k \ /1-k 2 1 - 2k 2

tx - 1 2k 2 "~' A k 2 h - 2 k ' ] 1 -

Substitution into ( 10)-( 11 ) yields

4(m + 1)2(1 -- 2k2)K2(k) A =

(13)

hK(k) 12 , l = , (14)

2k \/1 - kZ(2E(k) - K(k))

which may be reduced to a single equation soluble for k. given A:

1 6 ( m + 1 )2 A - h ~ k2(l -- k2)(1 - 2k2)(2E(k) - K(k))2~C(m + 1)~Aj,(k), (15)

after which l, A, IX may be found from (13)-(14). Observe that, while A must be the same throughout all elements of the beam, including those in contact with the wall, IX is generally different at each free element end point.

Fig. 10 shows the 'master' function Ah(k ). Note that the logarithmic singularity of K(k) at k = 1 is dominated by ( 1 - k2), and so Ah(k) is smooth and vanishes at the endpoints as well as at k > k * ~0.908, where 2E(k) = K(k). Since 2E(k)< K(k) above k*, solutions in this range con'espond to 'everted' shapes with d < 0 I l l ) and one or more interior loops and self-intersections (cf. [19], Section 263, Figs. 54-55). While such

¢

~ T Fig. 9. End and interior elements between line contact segments.

188 P. Hohnes et al. / Comput. Methods Appl. Mech. En~,rg. 170 (1999) 175-207

25 U ' -- . . . . . . . .

~I0

01 0 2 0 3 0 4 Ok5 0 6 0 7 0 8 (39 1

Fig. 10. The function Ah(k), plotted for h = 0.35.

solutions cannot be reached on cont inuous branches starting from the trivial state and so were excluded in [15],

we now recognize that they can correspond to (quasi-) physical ly relevant states that penetrate the sidewalls. The

fact that l < 0 (14) for these states reflects the in terchanged order o1-" the e l emen t ' s ends, see be low and Fig. 11.

Note also that A h ( k ) has a unique m a x i m u m at Aj ....... ~ 2 . 7 5 / h 2 and local min ima at /lj,.m,, ~ -~ - 0 . 3 2 9 / h 2 and ,,'lj ....... 2 ~ - 0 . 5 1 3 / h 2.

Construct ion of a comple te line contact solution to ( 1 ) - ( 5 ) proceeds as fol lows. Pick a mode number n / > 1,

and an admissible sequence o f orders m o . . . . . m . Define

m = min{m,~, m,,., (m~ - 1 ) / 2 ; j = 1 . . . . . n - l} ,

pick an admissible axial load a ~ (m + 1)2[Abram2, A t ...... ], and solve (15) repeatedly with m = m o . . . . . m , for

reverse

rotate

lo

,7-7"-"<,\ 0 l ,, =:¢, . j

t,

k<k*

V

k=k ' (a)

s=0 s'-lo s=l-lj 1 ( .-/

¢_ . . . . & + 1-1o4, ~ _ ___ a~

(b)

Fig. I1. (al Limiting cases and looped states, arrow's indicate increasing s: (b) the convention for determining D, physical case shown (1,, +/~ < I).

P. lhdmes et al. / Cornput. Methods Appl. Mech. Engrg. 170 (1999) 175-207 189

ko . . . . . k,. From (13) - (14) , find #o . . . . . /~,,, the arclengths of the elements l~ . . . . . 1,,, and the axial lengths A o . . . . . A,, subtended by them. The total displacement is then

n 1

D = l - d = l o - A o + 2 ~ ] ( ! i - A i ) + l , , - A , , " (16) j = l

Thus far we have not used the fact that the total arclength of elements and line contact regions must equal 1, the beam's overall length. This gives a necessary condition lot physical solutions:

I t [

l 0 + 2 ~ 1/+ l,,a~=~l,, ~ 1 . (17) i=1

If (17) is satisfied with strict inequality, the balance of arclength can be filled with any distribution of line segment elements summing to 1 - l , o . i.e. line contact lengths need not be equal. Equality in (17) identifies bifurcation points at which the line contact branches intersect point contact branches on sheet Pn. Thus. for regular solutions, equality yields the critical value h = hl, of [9], Eq. (20). When l , , > 1, to formally respect the overall length constraint we define the contact regions with negative arclength and, to maintain continmty of O(s), attach them to the free elements with 180 ° vertices; see Fig. 11. Observe that, as h decreases, A h increases and / decreases for given k (14); thus, as h--+0 (sidewalls approach) a greater "central' region of any given branch becomes physical. However, for an), h > 0, (small) neighborhoods of the points k = 0, k* and 1 are non-physical, since Ill--~ ~ at those points. Indeed. from (13) - (14 ) we note the following, which correspond to solutions of (15) with a = 0:

l , A - - - ~ , ( l - A ) - ~ O as k---) 0 • (18)

1 - 2 k 2

I---~+~,A---~ ~ - - - h ~ - 0 . 8 5 3 h as k--->k*; ", (19) 2k \/1 - k 2

1, A --~ ~c, (1 - A) ---) 7: as k -+ 1 (see below) " (20)

This implies that branches reach h = 0 at integer multiples of D = ih or that they tend to D = ~ as a ~ 0 . In view of (18) - (21) , the only branches that cross a = 0 are those for which all kj are selected to pass through 1 / v'2 at this load: thus, one branch alone in each modal complex is continuous at A = 0. Also, branches achieve their maxima and minima, respectively, on the lines a = (m + l)2Ah ...... , (m + l)2A~,m,,~.2. See the examples below.

The definition of D = 1 - d implicitly supposes that the buckled state can be reached by following a continuous path from the undelormed state O(s) ~- 0 (d = 1). The Ioopecl line contact states with k E (k*, 1) are not so accessible, and we must choose a convention for labeling the ends. We take the one shown in Fig. 1 l(b), for which a < 0 as in Fig. 10. This requires that the signs of the second tbrmulae in ( 13)-(14) be reversed for k ~ (k*, 1), so that both A and 1 are positive in this range. D is then computed as before from (16). This choice seems natural if we observe that at k = k* the ends of each (infinite length) element coincide on a suitably rescaled diagram. Passing through k*, we effectively rotate the element as a rigid body so that the end s = 0 becomes tangent to the wa l l then interchange the ends (reversing the sign of l). The process is indicated in Fig. 11 (a); it is reminiscent of the rSle that the coincident end state plays in connecting branches in the classical Euler problem [6], cf. Fig. 5.

There are at most two solutions of (15) tor each a > 0 , so we have 2 "÷ ~ branches of solutions in the upper half of the (,t, D) plane for each sequence {m o, * . . . . . *, m,,}. Similarly, since there are up to tour solutions of (15) lor each ,~ < 0, we have 4 "~ ~ branches of solutions in the lower half of the(a, D) plane for each sequence. Modes that enjoy no symmetries project to 2 "+ ~ (resp. 4 '~+~) distinct branches in the (A ,D) plane, which connect in pairs at A~,m~ × and At, m~.,.2. However, regular modes are symmetric under the elements g3 or g~ g~ of G if the same root kj = k of (15) is picked for every element and all line contact segments are selected with

190 P. Hohnes et al. I Comput. Methods Appl. Mech. En<erg. 170 (1999) 175-207

equal length; moreover each 'bay ' of the beam between zeroes ot y(s) is then reflection symmetric about its midpoint. For 0 < A < (m + 1)2A h ...... there are thus precisely two symmetric regular mode solutions of each order m; they form a (A, D) branch similar to the graph of Aj,(k) itself, with a maximum at (m + I)2A~, . . . . . For (in + l)2At, mi~t-~ ~l<;O there are four symmetric regular mode ,solutions of each order m, and for (m + 1 )2Ax, m~,2 < A < (m + 1)2As, m~,~ again only two symmetric regular mode solutions. The balance of the 2" (resp. 4") asymmetric regular modes are constructed by picking all possible remaining combinations for the n + 1 kj 's.

For regular modes there are at most two distinct k 's for each a > 0 and (m + 1)2Aj, Mme< a < (m + l)2Ahm,,l,

and hence one ' long' and one 'short ' (!J - Ai)" By distributing choices of (!J - / t i ) over the two end and n - 1 interior elements, we can create only those total displacements corresponding to summing j short and 2 n - j long contributions for each 0~<j ~< 2n, i.e.: 2n + 1 branches in all, or the number of ordered sets of two non-negative integers summing to 2n. Subtracting the two symmetric ones, we therefore have: 2 n - 1 asymmetric branches in the (a , D) projection. For (m + l)2Ah,ninl <~ ,A ' ~ 0 there are four choices for ( ! s - - ~ ) ' and similar reasoning shows that there are as many distinct projected branches as there are ordered sets of four non-negative integers summing to 2n, at most two of which are odd. See Appendix A.2, in which we compute this number:

(2n + 3)(2n + 2)(2n + 1 ) - ( n + l ) n ( n - 1) 6 (22)

These numbers are useful when we come to interpret bifurcation diagrams such as Fig. 8, in Section 5 below. Meanwhile, Fig. 12 shows bifurcation diagrams for the first three regular modal complexes. Note that the asymmetric branches are spaced ' regularly ' between the symmetric branches out of whose elements they are built. These and subsequent branches were all computed using the elliptic function routines in M A T L A B .

Finally, let

m + = max{mo,m,,, (rn~ - 1) /2 ; j = 1 . . . . . n - 1}, (23)

correspond to the highest order element, and consider the sequence of modal complexes generated as m + ~ with all other m, fixed. Since a / ( m + + 1) 2 --~0 for any fixed ,t, we conclude that these complexes approach a limiting pattern. Fig. 13, shows several examples of irregular modal complexes, illustrating the points made above.

4.3. Point contact branches Pn

A second important class of solutions are those having point contact at the side walls, where the moment O'(s) ~ O. These involve incomplete elliptic functions and they are more difficult to work with, since direct

350

- 10 ,* , I , * ,0} - - {0 ,* ,1 ,* ,1 ,* ,0}

! . .

i!i: e00

"500 0.2 0 4 0.6 0 8 1 12 I 4 1.6 1 8 2 O

Fig. 12. Bifurcation diagrams for the regular n 1, 2, 3; m = 0 modal complexes {0, *, 0}, {(L *, 1, *, 0}, {0, *. 1. *, l. *, 0}: h = 0. l, physical regions shown solid. All branches for k > k* are non-physical and are omitted t?'om lhis figure.

2OO {0,*,1 )

,/ _!]~tt./-" / I . . . . { 0 , ° ,1 , * '1 }

' 7 - i ~

<7!~A ! i ik'

,flAi 10o

Jf,iJti\i o0ii! iii t - \

i i - ::17 t i t t '~

~o o.~ i , 5 ~ 2s ~ a!s O

Fig. 13. Bifurcation diagrams tbr some irregular modal complex- es: h = 0.125, physical regions shown solid. Note that the right- hand branch of {0,*, I} almost coincides with a branch of {0, *, 1, *, 1}. Branches for k >k* are again omitted.

P. ttolmes et al. I Comput. Methods Appl. Mech. Engrg. 170 (1999) 175-207 191

superpositions of individual elements do not yield global solutions; one has to simultaneously adjust .'ill the lateral loads & to match moments at each contact point. Nonetheless, we can make some progress for regular, symmetric branches, and for the single point contact case.

As shown in [9], an element with one hinged end (s = 0) and point contact at the other (s = 1) can be transformed to an unconstrained p inned-c lamped problem. Letting g, = 0 - 0~ with tan 0, = / ~ / a , the equilib- rium equation for the element becomes:

r ds"+ ~ sin ~ = O" ~ =V'A2 +/.,2,

O'(0) = 0 , g,(1) = - 0 , , (24)

with displacement constraints

x(l) = cos O(s) ds = ~ cos ~p ds - sin ~ ds = A I , (25)

3,(1) = sin 8(s) ds = sin ~0 ds + cos ~b ds = h ; (26) ) ) I

see Fig. 14. Integrating Eq. (24) as previously, we find

V a / = K(k) + F(&, k ) . (27)

where k = sin(~b(O)/2) and the second parameter ~b in the incomplete elliptic function of the first kind,

fo ~ d 'r F(&, k) = , , (28) V' l - k: sin2r

is given by

sin ~ - +~/a _-_a sin ~b - k - - V 2 ~ k 2 " (29)

Although Fig. 14 is drawn for (b E (0, r r /2 ) , note that F(&, k) is defined for all & and that F((j 'rr)/2, k) =jK(k) . Making the appropriate sign choices in (29), this allows us to deal naturally with elements having several interior inflection points and making more than one loop in the phase plane.

As in [9], we may evaluate the integrals in (25 ) - (26 ) to obtain

f f 2[E(k) + E(~b, k)] - [K(k) + f(~b, k)] cos 060 ds -

) ~- (30) V' ,,~

sin O(s)ds = _+~/- (2k 2 - 1) + (31) 0 V A '

where

L

~J

( A ---~

g , ~ s=l )v

~ J J J h

I g . . .~ 0, _> .b $:=0 X

Fig. 14. N e w v a r i a b l e s fo r po in t c o n t a c t e l e m e n t s : de f i n i t i ons o f Oe a n d g,.

192 f: Holmes et al. / Comput. Methods Appl. Mech. Engrg. 170 (1999) 175-207

f 6 r k2 E(4~, k) = v'l -- sin2r d r (32)

I

is the incomplete elliptic integral of the second kind and, as above, E((j 'rr)/2, k) = j E ( k ) . Again the sign choice in (31) depends on the number of complete half circuits described by the solution in the phase plane, as can be seen by drawing pictures analogous to Fig. 14 with additional inflection points.

From trignometrical identities and the relation (29), we have

_ _ = • ~ I . z / k 2 . " A 1 - 2k 2 sm~d~, - _+2k sin ~b\/l - sm-~b, (33)

and we may use direct moment balance or the first integral of (24) to compute the moment at the contact point:

~-2 N,,"kk c o s ( / ) d e f ~

& ( l ) = \ / 1 - 2 k 2 sin2eb 9-'(k, 4~) v 'a. (34)

Finally, using (33) we may eliminate # and ,~ from (25)-(26) and (30)-(31) to obtain a system analogous to the line contact master equation (15), in which an element is uniquely specified by the parameters (k, ~b) and its length 1:

4k2(1 - 2k2 sJ in2~) [sin &\/ / l - k 2 s ine~h{2[E(k) + E(qS, k)] - [K(k) + F(&, k)]} A = h e

+ cos ~b( 1 - 2k 2 s in:6)[ 2 a~, Ah(k ' 40 , (35)

(1 - 2k 2 sin2¢b) d~f ]r'(k, &) (36) ,,~ = /2 [K(k) + F ( & k)] 2 - 12

Here, we implicitly assume that 1 - 2k 2 sin2q5 g: 0, in which case the lateral force and subtended element length are given by

-+2ak sin <b½/;1 - k 2 sin 2&

tx = I - 2k ~ sin2q5 ' (37)

=~/1 - 2k 2 sin 2& Aj A [(1 - 2k 2 s in2qS){2[E(k) + E(qb, k)] - [K(k'/+ F(~b, k)]}

7- 4 k 2 sin q~ cos ~bVrl - k 2 sin24~]. (38)

(The case 1 - 2k 2 sin 2& # 0 is considered at the end of this section.) Note that, tbr 4' = ( j r r ) /2 , ~4~,(k, &) of (35) reduces to the function Ah(k) of (15), and using this, that (36) becomes the definition of l in (14). The lines & = (j-rr)/2 for j > 1 contain coincident point/ l ine contact cases, and coincidence of point contact with the unconstrained Euler branch occurs on & = 0 (cf. [9], Section 2.1). See Section 4.6 for more on the former coincidence.

For regular symmetric branches with single point contact (PI), the solution strategy is as follows. We know a priori that I = 1 /2 and k 0 = kj, ~b o = &., and that the moments are ~Lutomatically matched at the contact point (cf. 34)), so we need only solve the two equations:

a = A,,(k, ¢5) = 4/]k, ~b) (39)

and use (37)-(38) subsequently to compute/,t, A and hence D = l - 2A. Symmetric point contact solutions are

therefore determined by the sets on which the functions elh(k, &) and 4/~(k, ~b) intersect, projected on the (k, &)-plane. Fig. 15 shows level sets of these functions and Fig. 16 shows some branches on the (k, &)-plane, identified by their codes. These and subsequent branches were computed using the incomplete elliptic function codes in M a t h e m a t i c a . The generalization to symmetric n > 1 modes (n contact points) is immediate: one simply replaces l = 1 /2 by l = l ] (2n) and modifies the displacement to D = 1 - 2 n A , recognizing that each :interior element is equivalent to two end elements, with an inflection point on the centerline. Asymptotics near k = 1,

P. Hohnes et al. I Comput. Methods Appl. Mech. Engrg. 170 (1999) 175 - 207 193

(a)

2" / " / - ", ; ' " -

k ', 'x ' ~

o! i l :

I . 1 (I

0 0~1 0:2 0.3 0.4 0~5 0.6 0 7 0.8 0 9 1

J , " "" - ;7 z d

,I : c

(b)

-1 i 0 0.1 0.2 0.3 0,4 0.5 0,6 0 7 0,8 0.9 i

k

Fig. 15. Level sets of (a) ,,l~,(k, d~); and (b) 4/~k, ~); h = 0.125•

detailed in Appendix A.3, simplify the computation of branches near the singularities of Fig. 15(a); typical results are shown in Fig. 16(b).

The asymptotic behavior of symmetric n ~> 1-mode point contact branches as ]'~l--->~ is easily obtained by mechanical arguments. In this regime the lateral contact a n d / o r end loads k~/--> - zc on each element, and they are equal. Referring to Fig. 14 and Eq. (24), as ]~1--~ oc, the element becomes almost straight, all the curvature; being concentrated near the contact point. Using l = 1/(2n) and A = d / ( 2 n ) , via Pythagoras ' theorem this yields d = ± "J'l - 4 n 2 h 2, or

D = 1 _+_ V'I - 4 n 2 h '- . (40)

The ' - ' case corresponds to A--> -oo and the ' + ' case to the looped stale, with A--~ + ~ , and, via the definitions of 0~ and & (29), these limiting cases correspond to k = 1, ~(0) = "rr and

/ /

/1 +_ \ /1 - 4 n 2 h 2 s i n & = - / 2 (41)

respectively. Clearly, no solutions are possible for h > 1/(2n), and the branch approaches the line D ~- 1 as h ---> 1/ (2n) from below.

For asymmetric branches we must compute d i s t i n c t k/ , ckj and l/ in each element, matching moments at each n

contact point, subject to the overall constraint that E i_ o li = 1. Thus, for the single point contact case we must solve the four equations

A = tl/,(k o, &o) = "1~,(kl, ~bl )(axial force balance) , (42)

194 P. Holmes et al. / Comput. Methods Appl. Mech. En'cr, g 170 (1999) 1 7 5 - 2 0 7

6r i

5 -

3

2 -

- 1 -

i

0

(a)

T . . . . . I T ~ I

~ .~2 }. .............

' ~ {1,1}

{v]} . . . . . . . . . • - .....

.---_> \

\

\ \

{0,0} loop

0.1 i - I I J I ~ I - -

0.2 0.3 0.4 0.5 0.6 07. 0.8 0.9

(b)

0

-0.5

t

'0.g05 O,~le 0 . ~ 1 k

Fig. 16. (a) Some symmetric point contact branches in the (k, 05)-plane; h =0 .125; (b) an enlargement of the end portions of the {0, 0}+-+{1 7} and {0'0} looped branches (solid curves) compared with the asymptotics (dotted curves) described in the text and Appendix A.

"~/~ =V"kko, &o) - + - ~ ~b,) (length constraint), (43)

~ k o, qb o) = ~l~k I , ~, ) (moment balance), (44)

for the four variables k i, 4 at each A (given h), to produce asymmetric branches. Simple superposition of individual elements as in the line contact case is impossible, since moments must be matched at each contact point and the overall length constraint respected. The procedure may be generalized to n > 1 contact points, although it rapidly becomes unworkable. However, for the single point contact sheet (0), we have been able to

P. Holmes et al. I Comput. Methods App/. Mech. Engrg. 170 (1999) 175-207 1!)5

- 50

-100 ~

-150

i -2ool

250 . . . . . • - - r - -

{ 2 2 } S /

i ' , , { l , l ~ s . . . . . . . . . . . . / ', 1 O0 ~ % ,']

i {tq}s /'

°~ ?{0,0}S / 2 ~<

/' /

/, / " / / J

/ J

/ /~{0'0}(bop)

~ 7 T

?etlt~

j J

i I

/ ; / i

/ /' , :/

/ / / i / / /

/ / I

0 0.2 0 .4 0 .6 0.8 1 1.2 1.4 1.6 D

Fig. 17. Some symmetr ic and asymmetr ic point contact branches in the (A , / ) ) -p lane ; h = 0.125.

follow asymmetric branches in this ' semi-analyt ical ' manner. Fig. 17 shows several examples of both symmetric and asymmetric cases. The asymptotics described in Appendix A.3 reveal that any asymmetric solution becomes increasingly symmetric as k ~ 1.

When ,~ = 0, 01 = v [ 2 and 1 - 2 k 2 sin2q~ = 0, (25) - (27) reduce to

{2[E(k) + E(~b, k)l - [K(k) + F(~b, k)]} 2

# = h ~

/ 2

and

1 VIp~]/= K(k) + F(&, k) , with sin 2& - , (45)

2k 2

from which symmetric and asymmetric solutions can be found as above. The main point to note here is that only certain distinguished points in the set 1 - 2k 2 sin2~b = 0, on which (35)--(36) are identically satisfied with ,~ = 0, actually correspond to A = 0 solutions.

4.4. Coils and snakes

There are many other classes of solutions to both point and line contact BVPs. In this and the tbllowing section we briefly describe some of these. We first construct point contact solutions with many loops and large angle changes between contact points on opposite walls. The basic idea is to perturb the uniform circular coil obtained by taking an (arbitrarily long) element and applying moments + v at the ends, with forces ,~ = # = 0, so that the solution of (1) is simply 0 = 0(0) + us. (With p = 2wk/l, the coil of length l is closed and contains k full turns.)

Picking ]Pl > 1 ]h, so that the coil diameter is smaller than the distance between the walls, we can apply lateral tensile forces # < 0 to 'pull out' the coil until it is tangent to the walls at two points. The easiest way to

196 P. Holmes" et al. / Comput. Methods Appl. Mech. En~trg 170 (1999) 175-207

see this is in the phase portrait of (1) with A = 0, in which the solution is a segment of a ' rotary ' orbit outside the separatri×

0 p2 /.,-

~ - + p , s i n 0 - 2 (46)

of the first integral (6), travelling a distance k'rr with k < 0 (resp. k > 0) for v < 0 (resp. v > 0). As I/.tl increases from 0, the initially straight orbit deforms so that segments between - ' r r and 0 rood. 2"rr (resp. 0 and -rr rood. 2"rr) lie closer to (resp. further from) 0 ' = 0 and thus are traversed slower (resp. faster); consequently negative contributions to the integral of sin 0 outweigh positive ones and this effect increases monotonically with I/,tl. Without loss of generality take sj on the top wall and suppose 0(,,'j ) = 0 (resp. "rr) and O(s2)= k~r<~ w (resp. (k + l)'rr ~ 2"rr). To satisfy the constraint

f '~ sin O(s) ds = - 2h . (47)

(the integral term of which is 0 (k even) or - 2 ] p (k odd) when /,t = 0), we simply increase I/,tl until the growing effect of negative contributions vs. positive ones provides the match. In this construction we have essentially solved an IVE and the element length l = .% - s] emerges as part of the solution (cf. Section 4.2). Thus, possibly allowing penetration of one or both walls, we can find solution segments with any k, but unless k = + 1, the deformed coil has self-intersections+ See Fig. 18.

Arbitrarily long coils with elements in the same (CW or CCW) orientation can be built by concatenating a sequence of such IVP solutions starting at (k,v, p) and ending at (k i ~ ~T, p), with arbitrary k]+~< k, if p < 0 (resp, k]+~>k/ if v ;> 0), and the lateral forces #~ chosen so that

f i ' IsinO(s) d s = + _ 2 h , . ( 4 8 )

!

+

F E I t '

I

I E I

r i / r ' . . i '

~ ! 1 I 4

~)1

f d

sl [ ]

I I ~ . . , . ~ ' I ; ~ s~ I

- ~ - - ' - - 3 ~ . . .~ ~ ! / 5

SI $5

o

S2 $4 g3 [~ . , /

-2'~

s~

®,

0 0

/ SI

Fig+ 18. Elements of coiled point contact solutions and their phase planes. The lowest panels show a solution on branch CO, 0, 0, 1+ 0, 0t from sheet (0, I, - 4 . 4, +1).

P. Hohnes et al. / Comput. Methods Appl. Mech. Engrg. 170 (1999) IZq-207 197

as required. In this way we can construct a rather general sequence of interior elements. To attach ends, we must find a solution of (1) with ,~ = 0 satisfying

f f xl O ' ( O ) = O = O ( S l ) O's = , ~ v, and sin 0(s) ds = h (49)

)

with similar conditions at the other end. Taking # > 0 in ( l ) and examining the phase plane again, we see that this is possible (we simply have to find a solution arc connecting 0 ' =: 0 with (0, ~,)). Again, see Fig. 18. This produces an orbit of total length/,~,, = Z I' o !J' Finally, to satisfy the overall length constraint, we simply rescale by taking s~--~s/I t .... rescaling /,z, ~, and h in the process. Recalling thai our original choice of u was somewhat arbitrary, for suitable (sufficiently small) h ranges, we can therefore construct solutions belonging to sheets with codes of the form (kl . k 2, k~ . . . . . k,,) for any strictly increasing or de,:,reasing sequence of kj.

To reverse coil direction, we can double an 'end' element (49). which necessarily has an inflection point on the x axis, and insert where desired to produce interior subsequences of the form ( . . . . kJ ~,ki, kj+l , kj.~2 . . . . . k,,) with k, 1 > ki = k/+l < k i+2 o r k/ 1 < ki = ks+, :> k _:. Also, see Fig. 18.

For simplicity we have sketched this construction with A = 0, but since phase portraits depend continuously on A, we expect that each such solution will extend to a local branch in the A, D plane. See Section 5, Fig. 23 for an example.

Line contact coils and 'snakes' can be constructed by much the same strategy in the case A - 0, although they cannot be continued for )t 4= 0. We simply concatenate elements solving the pinned-pinned problem (10)-(11) with a 7 = 2h for interior and ~t = h for end elements, with any number of internal inflection points. The requisite /.z = ,~ values (compressive in this case) are obtained from (10). Fig. 19 shows examples.

4.5. M i x e d branches

Having constructed line ,contact branches of the type described ill Section 4.2, we initially believed that mixed solutions, containing both line and point contact regions tor the same loads and displacement, did not exist. But this is untrue, as the following example shows. Take a symmetric looped line contact solution on the branch {0, *, 0} of sheet (0'0) with k > k*, sufficiently small h, and an interior line contact segment of length (say) 1/2. Into this central segment we insert a single loop solution of type {0, 0} to the single point contact problem of overall length / < I / 2 , created by a small perturbation of an unconstrained looped (k > k * ) m = 1 state by applying compressive forces # > 0 at the ends and midpoint, to make the ends tangent to the wall. The length 1 must be chosen so that the relevant constraints (48) are satisfied. See Fig. 20.

Sl S2 $6 S~ Sl $5 S2

$3 $4 $4 $3

~0, ,3, ,0, ,2} ~l, ,0, ,2, ,0}

Fig. 19. Examples of line contact snakes on the sheel (0'0, 0 '0. 0'0).

f J

Fig. 20. A mixed l ine/point contact state on branch {0, *, 0 .0 , ~, (I} of sheet (2'2, I, 0'(I).

198 P. Holmes et al. / Comput. Methods Appl. Mech. En$!rg. 170 (1999) 175-207

4.6. Line contact branches as roots .for point contact branches

The symmetric single point contact branches compuled via le,,el set intersection methods in Section 4.3 smoothly cross the lines 05 = 'rr/2, 3"rr/2 . . . . . on which the function zlt,(k, ~) reduces to the line contact master function Ah(k ). This provides explicit proof of coincidence of line and point contact sheets on such lines. Examining the definitions of the variables 0 1 and 05 (of. Fig. 14), we see that passing through these lines corresponds to the gain or loss of an inflection point in the corresponding element, and transition from tangency with the wall from the inside to the outside or vice versa. For example, in Figs. 16 and 17 we see transitions from {0,0} to {1'1} and {1, 1} to {2'2}. We call continuous branches with elements {mo, m 1 . . . . . } and {m o + I 'm I + 1 ' . . . } (re.sp. { m o ' m l ' . . . } and {mo + l ,m I + 1 . . . . } related and sometimes write {0,01te--){l'l }, etc. Such transitions occur analogously for asymmetric and multi-point contact solutions and they may be related to line contact branches as follows.

When (17) is satisfied with equality the corresponding line contazt solution is critical: in transition between physical and non-physical, the (summed) length of the contact segments with the wall having shrunk to zero. Here, the line contact sheet coincides with a point contact sheet; more specifically, the branch {m o, *, m F, * . . . . . . *, m,,} coincides with endpoints of the point contact branches {m~, m~ . . . . . m,,} and {mo + l'm~ + 1 ' . . . 'm,, + l} (for k <k*) , or {mo + I,m~ + 1 . . . . . m, + 1} and { m o ' m ~ ' . . . 'm,,} (for k > k * ) . Con- tinuous dependence of solutions of BVPs, along with the analysis of Section 4.3 and the remarks above, imply that a pair of related point contact branches emanates from each such physical/non-physical transition point. These points, which are relatively easy to compute ['or line contact branches of arbitrary complexity, may therefore be used as starting data for the path-following methods of Section 3.2 or the direct calculations of Section 4.3; see Section 5.1.

5. Towards a global picture

We now return to the partial global bifurcation diagram of Fig. 8 and interpret some of the families of branches shown there in the light of the analyses of Section 4. We first discuss line contact families Ln and some of the point contact branches emanating from them. This leads us to consider point contact branch connectivities implied by bifurcations in which certain line contact branches becorae completely non-physical as h increases. We end with a brief discussion of complexity in terms of the growth in the number of solutions lbr a given load as a function of mode number (number of contact points/regions), and remarks on stability of physical line and point contact solutions.

5.1. Interpreting the bifurcation diagrams

In Fig. 8 some features can immediately be identified. The groups of branches with maxima and minima at ,~ ~ 176, -21.1 are line contact solutions of type {0, * . . . . . *, 0}, cf. Figs. 12 and 13; the 'giant ' branch with maximum above ,~-~ 700 (off scale) and minimum at A ~ -84 .2 is {1, *, 1}. In general, the regularity due to simple superposition involved in construction of line contact solutions makes these branches easy to compute and understand; moreover, they may be used as "roots' for the computation of line contact branches, either by the semi-analytical methods of Section 4.3, or the simplex path-following scheme of Section 3.2.

Allowing variations in h if necessary, we identity points at which a given line contact branch transits from physical to non-physical, i.e. a point at which (17) is satisfied with equality. As noted in Section 4.6, at each such point a branch of point contact solutions coincides with the line contact branch; the corresponding parameter values may thus be used as a starting point for iterative computation. The branches illustrated in Fig. 21 were computed in this way, using the physical/non-physical transition points on the sixteen tO, *',0} branches for ,~ < 0 (including the looped ones for k > k*) as starting data. The resulting point contact branches were followed by the methods of Section 3.2, in the course of which some of them were observed to meet a different line contact complex: {1,*, 1}. In this way the accuracy of the simplex method can be checked independently by direct computation via Section 4.3, and the conneclivity of the bifurcation diagram is further elucidated. In particular, we see that the codes along certain po:inl contact branches change as follows:

P. Holmes et al. / Comput. Methods Appl. Mech. Engrg. 170 (1999) 175-207 199

1 ooo

8o0

6 0 0 I I

I' 4 0 0 '1 I !

200 1

o~ I

-200 . . . . . . . . . . . . . J

0 0.5 1 1.5 ( a ) D

oO oo, '°°°lli !-~ 4o0 i .~ 2000 Hi' '!/, I

20~; ~ ~ooo~ "1

0 0.5 1 1.5 0 0.5 1 1.5 (b ) O (c) D

a o o v . . . . . . . . . . . , I 3 o o - - ~ - . . . . . . Y 7 -

20o! / ! 20o(- >,~ / / 100 If----'=" : ~.~ /..#-J-/i 100 ! S-~ ..... ~, ~' ./

-aoo t

-400 [ i ~ ' .... 0 O.fi 1 1 .fi 0 0.5 I 1 .fi

D D (d) (¢)

300 [- . . . . . . . . • - -

2 0 0 ( " " . . . . /

,oo . . . . . . , , /t 0 '!' ; / / I

-1 oo ',4

-200

-300

-400

0 0.5 1 1.5 D

(0

Fig. 21. Point contact branches emanating from the {0, *, 0} and {1, *, 1} line contact branches: h = 0.05. Panels (a), (b), (c) show the {0, *, 0}, {0, *, 0} looped, and {I, *, 1} line contact branches, respectively, and (d), (e), (f) show the point contact solutions bifurcating from these. Bifurcation points are indicated by open circles. Note that the vertical (a) scales differ from panel to panel.

{0'0}<-+{1, 1}6-+{2'2}, the transitions coinciding with tangencies between these branches and {0, *, 0} and {!, *, 1} line contact branches, respectively.

A second striking feature of Figs. 7 and 8 is the sequence of approximately self-similar branches {m,n}<-+{m + l'n + 1} that have near-vertical tangencies with line contact branches at their physical /non- physical transition points along the left-hand side of the diagram. (The line contact branches are not shown in Fig. 7, but compare with Fig. 8 and the discussion above.) Simple approximate expressions for these points, generalizations of the critical loads/ll, of [9], Eq. (18), can be found by expanding the function A~,(k) of (15) for small k, using [21

• "IT ,5, 4 Tf K(k) = 5- 1 + 7 + ( , (k) and E(k) = ~ 1 - ~- + ~,(k 4) , (50)

which give

4~r2(m+ l):k2( 1 9k2 ) &(k) = h--7- - - 5 - + C(k4) '

(51)

/ = ~ l + ~ - + C , ( k 4 ) , ~ = ~ l - - u + e , ( k ~ ) .

From the first of these we obtain

h,/ '~ ( 9Ah 2 ) k ~ 2 q . r ( m + 1) 1 + 16qT2(m + 1) 2

and using this and (50) in (13)-(14) and (16)-(17), for the case: of single point contact, we obtain approximations for the critical loads and displacements:

200 P. Holmes et al. / Comput. Methods Appl. Mech. En$!rg. 170 (1999) 175-207

3h2(mo + m I + 2) 2 A,,,~,.,,, -~ ~r~-(m~ ~ m~ + 2) 2 , D,,,,,.,,,, ~ 4(m~ + 1 )(m I + I )

[ g h ~ - ( m , , + m l + 2 ) 2 [ ( m c , + l ) 3 + ( m , + l ) ' ] ] × 1 + 16(m~+ l):~(mj + 1) 2 (52)

These compare quite well with the numerical results, giving A,,,,,.,,,, within 5.2% and D,,,,,~ within 3.1% figr the

case h = 0.125 of Fig. 7. (Taking only the leading term of D,,,,,.,,,, degrades the agreement to 14.9%.) Note that

the ,~,,,,.,,,, value depends only upon the sum m~ + m~, while D,,,,,.,,, depends on m o and m~ individually; thus the left-hand bends of the branches come in groups ordered by m~ + m~ = m , the D,,,~,.,,,, values in each group being ordered right to left 10, m ) , ( 1 , m - 1), (2, m - 2 ) . . . . . These simple asymptotics therefore assist our understanding of branch groupings.

Note also each of these ' leftmost ' {m, n}+-+{m + l ' n + 1} branches has a second tangency with the {m. *, n} line contact branch after its maximum (cf. Figs. 12 and 13), and that many of them form figure 8 loops in the (,~, D) plane. See Section 5.2 and Fig. 24 below.

A further feature visible in Figs. 7 and 8 is the clustering of bifurcation curves near the vertical line D =-- 1. As observed in Section 4.3 following Eq. (41), symmetric point contact branches converge on this set as h ~ncreases prior to vanishing at h = l / (2n ) .

5.2. Branch connectivity and h variation

In addition to the direct use of phys ica l /non-physica l transition points on line contact branches to start computations of point contact branches, we may also deduce some a.';pects of branch connectivity by considering how these transition points coalesce under variations of h. As a specific example, consider the {0, *, 0} complex of Fig. 12. For h = 0.125 there are two such transition points for .~ <~ 0 and two for A > 0, one on the symmetric branch and one on the asymmetric branch in each case (actually one on each of two asymmetric branches for 0 < k < k*, but the brz~nches coincide on the (A, D) plane). As h increases, these coalesce and ultimately at h = h~t ([9] Section 2.1 ~.. the remaining point on the rightmost symmetric branch coalesces with the other two on the leftmost symmetric and central asymmetric branch a! the maximum a > 0. This has the effect of stitching together the symmetric and asymmetric point contact branches. As h increases and the symmetric and asymmetric transition points coalesce, so the asymmetric point contact segment {1'1} shrinks and disappears, leaving an asymmetric branch ~0'0} which in turn disappears at a higher value of h. See Fig. 22. Further increases in h draw the surviving symmetric point contact branch , 'loser to the singular limit D ~ 1, which it reaches for h = 1/2, a~ per the remark following Eq. (41).

In general, one must allow h as well as (,~, D) variation to trace point contact states to physica l /non-physica l line contact transitional. An example is provided by the following coil state (cf. Section 4.4). Take an unconstrained first mode state with coincident ends d = 0, D - - 1 rotated so that A = 0, and decrease h to produce contact points ~s shown in the center shape sketch (3) of Fig. 23. As the contact loads #~ increase with h = 0, the resulting branch {0, 0. 0} on sheet (0, - 1) never reaches a line contact condition, since the moment never vanishes at the c~)ntact points. Allowing variations in ,~ for fixed h, we find the branch of Fig. 2.3. which for this value of h does not intersect the relevant {0, *, 0, *, 0} line contact branch on sheet (0, 0 ' - - 1, - 1 ) . However, we believe that it will do so for sufficiently small h.

We end our final menagerie with an example of several disconnecled line contact branches for relatively large h = 0.3. Fig. 24 shows symmetric and asymmetric states, some of which can be understood by reference to Fig. 7, in which h is sufficiently small that these branches are still connected to the corresponding line contact branches at the points identified asymptotically in Section 5.1. As h increases, the figure-eight loops successively separate flora the line contact branches, which become completely unphysical, and in this process each { m + l ' n + 1} segment of the {m,n}+--~{m + l ' n + l} branch vanishes. Further increase in h kills the asymmetric branches, ~.nd finally all the symmetric branches themselves vanish after collapsing onto D = 1.

_5.3. A notion ~/" complexity

For the unconstrained Euler problem the number of distinct solutions existing at a given load grows linearly, there being 2n + 1 branches, including reflections about x = 0 and the trivial solution, for nZTr 2 • A "~ (n -t- 1 )2W2

P. Holmes et al. / Comput, Methods AppI. Mech. Engrg. 170 (1999) IZ~-207 201

0

(a)

100

100 - -

6O

40

20

0

-20

-412

80

60

40

20

0

-20

-40

100

(c)

80

60

4O

2O

0

-20

-40

/'"

//

0.5 1 1.5 0 0.5 1 1.5 D D

0.5 1 1.5 D

(b)

lOO

80

60

4o

2o

o

-20 i -40

0 0.5 1 1.5 D

(d)

Fig. 22. The {0, *, 0} line contac~I (thin and dotted) and {0, 0}--~{l ' l} point contact (bold) branches, for vadous h: (a) h = 0.175: (b)

h = 0.25; (c) h := 0.30; (d) h = 0.35. On (c) and (d) line and point contact branches almosl coincide over their central portions.

for the pinned-pinned case. The great increase in complexity due to contact with the constraining walls may be quantified by noting that, considering line contact solutions alone, for sufficiently small h, and n contact regions (the nth mode), there are up to 4 "+~ distinct physical branches for each code (Section 4.2). Allowing up to M internal inflection points per element, we have M" ~ ~ distinct codes, resulting overall in

2(4M)" ~ 1

nth mode line contact branches in contrast to only two unconstrained nth mode branches. This exponentially growing function provides a lower bound for the total number of branches, since it does not count point contact branches which bifurcate from the line contact branches where the latter lose physicality, or disconnected branches. Including them roughly doubles the number.

As n increases for fixed h, fewer and fewer of these line contact solutions are physical; however, as h --~ 0 for any fixed M and n, portions of all such branches become physical. In this regard we remark that the solutions to the linearised contact problem of Feodosyev [11], which one expects to be valid in the same limit h--->0, represent only those solutions with k ~ 0, and thus only one out of the possible two roots of Eq. (15) for ,~ > 0 (no solutions for A < 0 appeal" near k = 0, so the linearised analyses misses these states entirely). One might object that h --> 0 for ' large' k leads to arbitrarily high curvatures and play, tic deformation in reality, but this limit is equivalent, under rescaling, to the beam's length going to infinity. Hence the linearised analysis misses the vast majority of physically relevant solutions in this limit.

5.4. A remark on stability

It is well-known that for the unconstrained Euler beam with A > ,'iT 2, only the first (n = 1 ) mode is stable to arbitrary small perturbations; all other mode shapes correspond to saddle points in the energy [20,21,6]. However, constraints can stabilize certain regular modes of order m = 0, provided no line contact region exceeds the critical length 2w/ , J~ for buckling of a clamped-clamped element [9]. Indeed, in [22] stability for

202 P. Hohnes et al. / Comput. Methods Appl, Mech. Evgrg. 170 (1999) 175-207

1 0 0 ~ - - T ~ r T

L ° t

L

I 1 F • l-T-T [ 7 -1- T-T-Y--F--T 1

2

5

-I L

I V

- 1 0 0 / I IA~J_L__L .L~I 11_2 LA_I i t_~_ 0 1,0 2,0

3 :! ! I

i t4 - ....... ! / - - - - , , , _ - . . . . . . . . . : 1

D

Fig. 23. Poin t c o n t a c t coi l s ta tes on b r a n c h {0, O, 0} on sheet ( 0 . - I): st-tapes s h o w n inc lude that at fl = O. See text.

the n = 1, m = 0 case is proven, and it follows that solutions for alll other n :> 1 and m = 0 are neutrally stable (one can freely adjust individual line contact lengths without changing the energy). (ln [22] implicitly only the leftmost k symmetric solution of Fig. 10, with A., # > 0, is considered; it would be interesting to treat the other (asymmetric, symmetric, and looped) solutions by the same methods.) Thus, at least some of the solutions fbund here are physically observable; in fact in the experiments reported in [9] we have seen regular n = I, 2, 3, m = 0

branches. In contrast, it follows from the instability results noted abow," for the zero end-moment unconstrained

problem, that any line contact solution having an inflection point in ,an end element, or more than one inflection point in an interior element, is unstable. Thus a necessary, but not sufficient, condition for stability of a line contact state is that it b,e regular and have a code of the form {0, *, 1, * . . . . *, 0}. Similarly, for a point: contact state, each end (resp. interior) element is equivalent to a pinned--clamped (resp. clamped-clamped) beam and is therefore unstable if it contains more than one (resp. three) interior inflection points in any such element.

6. Conclusions and comments

In this paper we continue the studies of [9,15], investigating the planar (Euler) buckling problem for a slender inextensible beam, subject to lateral displacement constraints due to #ictionless walls parallel to and equidistant from the centerline on which the (hinged) end points lie. The model takes the form of a sequence of BVPs whose precise formulalion depends upon whether contact occurs nowhere, at isolated points, or along line segments. We describe a numerical method based on simplicial decomposition of state-parameter space which yields both a global search strategy and a non-iterative path-following method. The algorithm produces bifurcation diagrams of surprising complexity, exhibiting many disconnected branches as well as 'secondary' branches linking different states.

We develop explicit analytical solutions lot line contact equilibrium states and partially-explicit solutions for

P. Holmes et al. I Comput. Methods Appl. Mech. Enxrg 170 (1999) IZ5-207 203

--T • T f T----~--- - I ~ " " l ~ V - l - - F I

I I

0

300

-150 0

( i"\ i X

- t2,2JA ~.. \~ [2,2}~ i

) ',, \,

'. \ /'~ fo, 3} .' r \ ,

/ / ',i~.- [o,s 7 I L.) //-,, 'i

.... i ,l' '~--- "" {0, ~ } I' i II / I [~ "1+}5 H \ , , I/ {C),OJ A -~///: ~.11 //

[- / ..... ,.. R. _~;_W~_ ~71: ..... 17 ......

/ - - - - ; . ~ - ; _ ~ :_Z~ 7 ---:L{ooj.s q {OlOj D l< F l<-- ( / I

I I <\, / !

' ~ / / i ' //' /

/

1,0 D

Fig 24. Some disconnected single point contact states on sheet (0); h = 0.3.

point contact states, in terms of complete and incomplete elliptic integrals, which yield load-displacement diagrams. Although we typically plot only axial load vs. displacement, lateral loads and contact point locations and moments can also be calculated. The analysis provides interpretation and feedback for the numerical results. Indeed, this work provides fin instructive example of rich interplay between analysis and numerical computation: our discovery of new branches proceeded about equally from the two approaches; a typical instance being the observation of 'regularities' in Fig. 8, which prompted the analysis of line contact solutions of Section 4.2. These in turn, via identification of physical/non-physical transition points, led to the numerical discovery of additional point contact branches, including asymmetric ones and the looped states of Fig. 21.

The morals should be clear. Path lollowing from 'known' (symmetric) states is insufficient: it required the experimental observations of [9] to direct our search for apparently-disconnected asymmetric point contact solutions, which were subsequently found numerically. But naive global computations alone lead to excess and confusion, it is unclear which branches are connected (in the full GRS, Section 3.1): classification and analytical study of (selected) solution types is necessary to understand bifurcation diagrams such as that of Fig. 8. Asymmetric point contact states are then revealed to be connected to a:~ymmetric line contact branches, which are in turn connected to symmetric line contact states and, for sufficiently small h, to symmetric point contact states.

Several open questions remain. Notably, we observe that everywhere that point and line contact branches meet, they appear to do so t angen t ia l l y (in the full parameter space, not just the (& D) plane). This is ostensibly a degenerate situation, unlike the transversal intersection of the unconstrained and point contact branches, for example. It would be of interest to prove that these l ine/point contact intersections are tangential.

One can suggest various mechanically relevant 'unfoldings' of this problem, which will make it analytically much more complex, but less degenerate. The sidewalls could be placed asymmetrically (at distances h j # h~ from the centerline), or non-parallel to the centerline, or they could be curved. Indeed, we suspect that the latter would lead to breaking of the degenerate tangential intersections of line and point contact branches, resulting in disconnected branches, as suggested by A. Ruina. This would further obstruct simple path-lbllowing, illustrating the value of our global slralegy.

204 P. Hohnes et al. / Comput. Methods Appl. Mech. En:zrg. 170 (1999) 175-207

In the experiments of [9], friction at the constraints is minimized by silicone lubrication and the use of glass walls, but even there, friction is significanl when lateral loads are sufficiently large. Inclusion of a suitable friction model would provide a further unfolding.

A c k n o w l e d g e m e n t s

This work was supported by OTKA grant F021307 (GD), US-Hungarian Joint Fund 656 /96 (GD ,and PH), and DoE Grant DE-FG02-95ER25238 (PH). John Schmitt was partially supported by a DoD Graduate Fellowship and a Wu Fellowship of the School of Engineering and Applied Science, Princeton University. Paul Seymour helped us with Eq. (22), and we thank Mrs. Ant6nia Nagykfildi for her painstaking artwork.

A p p e n d i x A. B o o k k e e p i n g and other details

,4.1. Sheet and branch codes for Fig. 8

Branch (0) {()) (0, 0) (0, 0, O) (0'0) (0'0, O) (0, 1)

i {o} 2 {]} 3 {2} 4 (3} 5

7 8 9

IO

12

13 14 15

]~ Co} 17 (I} 18 {2} 19

20

21 22

23

24

25 26 27

28

{o, o} {1'1}

CO, 2} {J'3} CI, i} {2'2}

{o,i} {1'2}

{o,o} {1'1}

Co, o} C1'1}

Co, i,o} {]'3'1}

C], ], o}

Co, ], o} {o, 2'1}

CO, 1, o} {0,2'1}

{o, l, I. o} {0, *, O}

{0, 2,0} C1'2'1} CI, *, I} {o, *, I}

{0, *, 2} (o, ,, ()}

{0, *, I, *, O}

{0, *, I, *, O} {0, *, 1, *, O}

{o,o,o}

29 {0, *, I } 30 {o, o, ]}

P. Holmes et al. / Comput. Methods Appl. Mech. Engrg. 170 (1999) 175-207 205

31 32 33 34 35 36

37 {1,~} {2'2}

10, i, 1, o} {0, I, 1, o} {0, 1, 1, o} {0, 1, J, o} {0, 1, ], o}

{0, l, J} {0, 2'2}

38 39 40 41 42 43 {0, O} 44 {I, 2}

{2'3} 45 {I, 3}

{?-'4} 46 {I, 4}

I:-"5} 47 {1,5}

{:z'6} 48 {2, 3}

{3 '4} 49 {0, 3} 50 {o, 2} 51 (o, o}

I ~'i} 52 I ;I, 2}

{2'2} 53 {0, 3} 54 'l], 1}

.Io'2} 55 1o'3} 56 {o, 4} 57 {t, 3}

{2'4} 58 {J, 2} 59 {0'1} 60 /::, 2}

1~;'3} 61 /1,4} 62 /O'O} 63 {(), 5} 64 {0, 4}

{0, *, O} {0, *, o} {0, 2, O}

{0, *. I, *, O} {0, *, 1, *, O}

A.2. N u m b e r s o f p r o j e c t e d branches

Take (m + 1)2Aj, mml < A < 0, so that there are four dist inct solut ions of (15) and hence four choices for (/j - Aj). For the nth mode we have two end and n - 1 interior e lements ; the former each contr ibutes a factor ( l j - A i) and the latter 2 0 , - A~) to the total d i sp lacement (16), g iv ing us effect ively 2n 'equivalent end e lements ' . We may freely dis tr ibute any combina t ion of the four choices over these 2n elements , subject only to the restr ict ion that no more than two of them may be selected an odd number of t imes (each pair of interior effect ive e lements must be given the same choice; odd numbers can be accommoda t ed only at the two real ends). Thus, the number of dist inct total lengths avai lab le is the same as the number of ordered combina t ions of four non-negat ive integers which sum to 2n, less those conta in ing ['our odd entries. This latter is, in turn, equal to the number of ordered combina t ions of four even numbers summing to 2n - 4 , which is the same as the number of ordered combina t ions of any four non-negat ive integers summing to n - 2. Lett ing %5~(N) denote the, number of ordered combina t ions of k non-negat ive integers summing to N, we therefore must calculate' ~ ( 2 n ) - ~4(n - 2).

206 P. Holmes et al. / Comput. Methods Appl. Mech. En~rg. 170 (1999) 175-207

T~(N) is easily found markers can be inserted

2f~(N) = ( X + k - -

\ k - I

Noting that this reduces

(N + 3)(N + 2)(N +

3~

when k = 4, we have

~ ( 2 n ) - %~(n -12) =

by observing that it is the same as the number of ways in which k - 1 distinguished among a string of N l ' s , breaking them into k groups each time. This is

l).

to

1)

(2n + 3)(2n + 2)(2n + 1)

6

as in Eq. (22). Also note that in the case of only two choices of (6 and we have

(2n + 1 )! 2q2(2n)- (2n)! l ! - 2 n + I .

(n + l )(n)(n - 1) 6

- Ai), no combinat ions need to be excluded

A.3. Point contact asymptotics near k = 1

The surface 171j,(k, 05) of Eq. (35) varies rapidly near k = 1, where there is a row of singularities (Fig. 15(a)), so that solution of (42)-( .44) is difficult in this region. However, we may use the asymptotic expressions [2]:

5 - k 2 (1 - k 2) sin 05 F(k, 05) -- 4 In(tan 05 + sec 05) 4 cos205 ' (A. 1)

1 + k 2 1 - k 2 E(k, 05) ~ ~ sin 05 + ~ In(tan 05 + sec 05), (A.2)

5 - k - ' ( 4 ) ( l - k 2) K(k) ~ 4 In = (A.3)

\/'1 k 2 4 '

3 + k 2 l - k 2 ( 4 ) E(k) - 4 + - - ~ - - In , = ,

V 1 - k ~- ~ A . 4 )

for k ~ 1. (Note that (A.1) and (A.2) are not uniformly valid: one cannot let &---~'rr/2 for fixed k ¢ l for example.) Substitution of these expressions along with the identity and approximation

1 2k 2 sin 2 05 1 - k 2 + k 2 - = cos 205,

1 + k z 1 - k ~ (A.5) ~¢~ - k2 sin2 & ~ 2 cos 05 + ~ - sec 05

into (35 ) - (36 ) and (341 provides useful approximations. In particular, use of the first of (A.5) in the moment match (34) and (44) shows that 05~ = 05o or 05j ~ - 0 5 o or 05~ ~ ~;50_+w. Substitution of the latter two into (42 ) - (43 ) results in contradictions, showing that any asymmetric single point contact solution must become increasingly symmetric as k ~ 1. The symmetric case 05~ = 05o, k~ = k o itself yields the single equation

In - --: + In(tan 05 + see d~) ,,'i---S k2[

- h- Z- (3 -- k ~ + (1 + k 2) cos 205)

× 4 4 In + In(tan 05 + sec d') \ /

P, Holmes et al. / Comput. Methods Appl. Mech. Engrg. 170 (1999) 175-207 207

+sind)(l+k,+(l_k2 ) 205)}+ qg( l + k 2 ]2 sec cos (cos 24, -- 1 )) (A.6)

solution of which yields branches in the (k, &) plane approximating the symmetric branches near k = [, corresponding to looped point contact states. Fig. 16(b) shows these asymptotic estimates compared with Mathematica numerical solutions of the full equations (39) tk~r the extreme portions of the symmetric {0, 0}eo{ l ' l } and {0'0} looped branches. Clearly, solutions of (A.6) provide excellent approximations in this range. Also, note that the limiting values of D and q~ as k---) 1 (A--* ~o) on {0, 0} coincide with those obtained by mechanical arguments in (4(/)-(41).

References

[1] E.L. AIIgower and K. Georg, Numerical Continuation Methods: An Introduction ISpringer-Verlag, Berlin, 1990). [2] RF. Byrd and M.D. Friedman, Handbook of Elliptic Integrals for Scientists and I=ngineers (Springer-Verlag, New York, 1971 ). [3] F. Casas, W. Chin, H.E, Nusse and E. Ott, Universal gruzing bifurcations in impact oscillators, Phys. Rev. E 53 (19661 134-139. [4] E.A. Coddington and N. Levinson, Theor3, of Ordinary Differential Equations (McGraw-Hill, New York, 1955). [5] E. Doedel, H.B. Keller and l.R Kemevez, Numerical analysis and control of bifurcation problems (li bifllrcation in finite dimensions,

Int. J. Bifurcation Chaos 1(3) (1991) 493 521). [6] G. Domokos, Global description of elastic bars, Zeitscbr. Angew. Math. und Mech. 74(4) (1994) T289 T291. [7] G. Domokos and Zs. Gfispfir, A global, direct algorithm for path-following and a,:tive static control of elastic bar structures, Int..1.

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[10] L. Euler, Additamentum I de curvis elasticis, methodus inveniendi lineas curvas maximi maximivi proprietate gaudentes, in: Opera Omnia l, Vol. 24 (Bousquet, Lausanne, 1744) 231 297.

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[141 J. Guckenheimer and S. Johnson. Planar hybrid systems, in: Lecture Notes in Computer Science No. 999 (Springer Verlag, 1995). [15] E Holmes, J. Schmitt and G. Domokos, Constrained Euler buckling: line c~)ntact solutions, in: Proc. IUTAM Symposium on New

Applications of Nonlinear and Chaotic Dynamics in Mechanics, Comell University, July 27-Aug 1, 1977. Kluwer, 1998. To appear. [16] J.B. Keller and J.E. F[aherty, Contact problems involving a buckled elastica, SIAM J. Appl. Math. 24 (Ic)73) 215-225. [17] J.B. Keller, J.E. Flaheny and S.I. Rubinow, Post buckling behavior of elastic tubes and rings with opposite sides m contact. SIAM J.

Appl. Math. 23 (1972) 44~.-455. [181 H. Link. Uber den geraden Knickstab mit gegrenzter Durchbiegung, Ingenieur Archiv. XXII (1954) 237-250. [19] A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity (Cambridge University Press, Cambridge, UK, 1927). Reprinted by

Dover Publications, Inc., New York. [20] J.H. Maddocks, Stabilily of nonlinearly elastic rods, Arch. Rat. Mech, Anal. 85 (1984) 311-354. [211 J.H. Maddocks, Stabilily and folds, Arch. Rat. Mech. Anal. 99 (1987) 301-328. [22] E. Miersemann and H.D. Mittelmann, A free boundary problem and stability ['or the nonlinear beam, Math. Methods Appl. Sci. 8

(1986) 516 532.

[23] E. Riks, An incremental approach to the solution of snapping and buckling problems, Int. J. Solids Strucl. 15 (1979) 529 551.