Acta 5'[echaniea 33, 229--242 (1979) A C T A M E C H A N I C A @ by Springer-Verlag 1979

Thin-Walled Beams as Directed Curves

By

M. Epstein, Alber ta , Canada

With 5 Figures

(Received December 6, 1977)

Summary -- Zusammenfassung

Thin-Walled Beams as Directed Curves. A non-linear theory of thin-walled elastic beams of open cross section is formulated in terms of a space curve endowed with several director fields, one for each of the plates composing the beam. Physically meaningful con- straints among the directors are introduced and a technique for the algebraic elimination of the corresponding Lagrange multipliers from the field equations is discussed. Compatibility conditions for a minimal set of strain measures are presented.

Diinnwandige Tr~ger als gerichtete Kurven. Eine niehtlineare Theorie diinnwandiger, elastiseher Tr/iger mit offenem Querschnitt wird besehrieben durch eine Raumkurve mit ver- schiedenen Richtungsieldern; eines fiir jede Platte des Tr~gers. Physikalische Zwangsbe- dingungen zwischen den Richtungen werden eingefiihrt, und eine Technik fiir die algebraische Eliminierung der entsprechenden Lagrangeschen ~ultiplik~toren arts den Feldgleichungen wird diskutiert. Vertr~glichkeitsbedingungen ffir eine minimale Anzahl yon Verzerrungs- mal3en werden angegeben.

1. In t roduct ion

I n a recent p a p e r [1] a n o n - l i n e a r th in-wal led beam t h e o r y was p re sen ted based on the iden t i f i ca t ion of each s t ra igh t e lement of the (open) cross-sect ion with a vector. A l though such a mode l s t rong ly suggests a resemblance with a mu l t ipo la r curve theory , the app roach used in [1] consists e s sen t i a l ly in in tegra t ing th ree- d imens iona l fields. The numer ica l resul ts in [1], as well as fur ther unpub l i shed resul ts in the ine las t ic range, show t h a t the predic t ions of the t h e o r y are in r e m a r k a b l y good ag reemen t wi th expe r imen t a l d a t a for the pos t -buck l ing be- hav iour of t h in -wa l l ed beams. To the au tho r ' s knowledge, a genera l t h e o r y of th in-wal led beams covering the case of large th ree-d imens iona l de fo rmat ions does no t exist . The purpose of the p re sen t work is to develop such a t h e o r y f rom a p u r e l y one-d imens ional app roach t r ea t ing the beam as a d i rec ted curve, [2], [3], [4]. I t is somet imes s t a t ed [5] t h a t two directors in the p lane of the cross sect ion are enough to descr ibe all the s i tua t ions met in prac t ice in beam analysis . This is no t the ease, however , for th in-wal led beams [6], which are the mos t commonly used in engineer ing pract ice , so t h a t unless one is willing to t r e a t a th in-wal led beam as a shell - - t h e r e b y increas ing the complex i ty of the analys is - - a grea ter n u m b e r of di rectors has to be considered. To develop a meaningfu l theory , cer ta in cons t ra in ts have to be imposed on the directors . Such cons t ra in ts are br ief ly

i5"

0001-5970/79/0033/0229/802.80

230 M. Epstein:

reviewed in Section 2. Section 3 aims at a systematic selection of an adequate number of strain measures. This selection is achieved by using some concepts of the theory of invariants [7]. In Section 4 the field equations are obtained along the lines of Toupin's work [8], namely by postulating a Lagrangian density satisfying a principle of Euclidean invariance. Because of the constraints, the field equations will contain Lagrange multipliers. Since the number of these is quite large (roughly twice the number of elements in the cross-section) it is interesting to investigate the possibility of their algebraic elimination. In Appen- dix I a discussion of this mat ter is presented in terms of some concepts borrowed from Graph Theory. Finally, Section 5 deals with the compatibili ty eonditions.

2. Kinematics and Constraints

A thin-walled beam (Fig. 1) is composed of a number n of plates joined to- gether to form a straight prismatic or curved structural element. A cross-section of such a beam will, accordingly, be composed of n elements, here assumed to be straight. All joints are considered to be simple, i.e., they connect two elements only. A joint between, say, three elements is resolved into two simple joints.

x 3

• Fig. 1. Thin-walled beam geometry

Let the axis /~ of the beam in reference configuration be given by the para- metric equation

P : X ~ = X ~ ( S ) 0 ~ S _< L I = 1, 2, 3 (1)

where S measures length along the axis. With every straight element ~ of the constant open cross-section (Fig. 2)

a d i r ec to r /~ (S) is associated. Each director field will thus provide information about the deformation of its associated plate.

A motion is defined by specifying the coordinates z z of the deformed axis and the deformed directors d~ as functions of S and time, i.e.

x~ = x q S , t) (2a)

d~ = d~(S, t) ~ = 1 . . . . . n (2b)

Thin-Walled Beams as Directed Curves 231

Note tha t S is used as a parameter. I f a different parameter, say 8, were used for the deformed configuration, the definition of the motion should include a functional dependence between s and S.

|

5,

|

(o)

5, ~ ~ - '~ D5

(b) 54

:Fig. 2. (a) Beam cross-section, (b) Beam axis with directors

In order to get a meaningful theory of thin-walled beams it is convenient to introduce the following constraints:

d~. d e = D~. D~ = C.~ (3)

for ~ = fl and for n - - 1 suitably chosen distinct pairs c,, ft. In the physical model these pairs represent contiguous plates. The constraints are, therefore, intended to represent the in-plane inextensibility of the cross-section elements (for ~ = fl) and to avoid rigid rotation between contiguous elements [1]. Note tha t although the constraints (3) prescribe conservation of angle between contiguous elements of the cross-section, the directors need not remain coplanar, thus allowing for the global "warping" effect which is essential to explain the behavior of thin-walled beams. In the development tha t follows it is important to bear in mind tha t Eq. 3 is valid only for cr = fl or cr contiguous to #.

Note tha t for a constant cross-section the right hand side of (3) is constant, yielding by differentiation the following constraints on the derivatives of the

232

directors.

M. Epstein:

d~' �9 cTe + d~. d'~ = 0 o

where ( )' ~ ~-~ () .

Similar constraints apply, of course, to the t ime derivat ives, namely ,

where (') = -}7 ( )"

Each of the Eqs. (3), (4) and (5) represents 2 n - - 1 constraints .

(4)

(5)

3. Strain Measures

Because the directed curve approach is pure ly one-dimensional , a crucial pa r t of such nmdel is whether it can provide the necessary n u m b e r and nature of s t ra in measures needed for the descript ion of the physical s i tuat ion which it intends to represent. Let it be postulated that the state of strain at a point of the directed curve is complete ly determined b y the vectors .~', c7~ and ~/,' a t t ha t point . Assume tha t ~b is a scalar valued funct ion of the strain. Then

,p = r &, & ' ) . (6)

Assume, fur thermore , tha t ~ is f rame indifferent, i.e.

(o(~', &, d,') = $(Q~', Q&, Q&') for all Q < 0 (7)

where 0 is the full or thogonal group. In o ther words ~ is an isotropic scalar. According to a theorem by Cauehy (see e.g. [9]) this is equivalent to the possibil i ty of expressing r as a funct ion of the inner products be tween all pairs of a rguments , viz. : 2 ' �9 ~ ' , ~ ' �9 d~, ~2' �9 a l l , d~ �9 d~, d~ �9 d / , d~' �9 a l l . These inner products con- s t i tute therefore an in tegr i ty basis [7] for ~b. On the other hand, it is well known [10], [11], t ha t out of the a forement ioned integr i ty basis only ,] - - p e lements are funct ional ly independent , where ~ is the number of independen t components of the vectors involved and where p = m ( m - - 1)/2 with m = dimension of the space (m ---- 3). In our par t icular ease, because of (3) and (4), the number of independen t components is

~ ? = 3 ( 1 - i - n ~- n) - - 2(2n - - 1 ) ~ 2 n @ 5 (8)

so t ha t the number of funct ional ly independent e lements of the in tegr i ty basis is 2n + 2. I n conclusion, a n y f rame indifferent scalar funct ion of the strain is expressible as a funct ion of 2 n q- 2 sui tably chosen inner products be tween the a rguments ~' , d , , d / . These 2n q- 2 inner products are therefore suitable s t rain measures for the rod.

Wi thou t loss of general i ty, it can be assumed tha t dx and ds are contiguous and non-eolinear elements. Under these circumstances the t r iad (N', dl, d2) forms a local basis in E s. As shown in [11], and also as in tui t ively understood, a minimal

Thin-Walled Beams as Directed Curves 233

function basis for the given arguments disregarding the constraints is

7' .7', ~ . 7 ' , do,' "x', d~'dl, d~' "dl, d#'d2, an' .d2 # = 2 ..... n

Consider now the constraints. Each constraint should allow to eliminate one element from this basis. To determine which elements to remove consider the following heuristic argument. From the six products of the local basis, only ~' �9 U, 7' - ~/~ and 7' �9 el2 have to be retained, the other three products (da �9 dl, di �9 d2 and d2 �9 d2) being constrained. Each additional vector d3, da .... (~ has a prescribed magnitude and a prescribed inner product with either di, de or another d,. It will therefore be completely specified by the product d~ �9 7' (a --~ 3, ..., n). Similarly, consider d(. Since, by constraint, it is perpendicular to dl, it is only necessary to specify two components, namelyd( �9 7' and dl' �9 d2. For d2' only one component will be necessary (since d ( �9 (~1 ~ - - 6 ~ 1 t " (~2) , namely c~2' �9 7'. The same is true for all remaining d~'. Summarizing, then, a minimal function basis taking account of the constraints is given by

e = 7 ' . 7 '

~2a ~ 7 t " ~ a

#cr ~ 7 t " (~a t

0 ~ (~1 / " d 2

(9)

These are therefore adequate measures of strain for a directed curve whose directors are subjected to the constraints given by Eqs. (3) and (4). To see that these measures of strain are also adequate for the thin-walled beam one needs only to realize that e is readily interpreted as a stretch, y, as shear strain asso- ciated with element cr q~ as bending strain in element cr and 0 as an angle of twist. I t is interesting to note that only one measure of twist is necessary for the whole section (and not one for each individual element). This fact was suggested in [1] without proof.

4. Field Equations

The field equations will be derived following closely Toupin's approach [8], [12] with the addition of Lagrange multipliers for the constraints in a manner similar to that used by Maugin and Eringen [13] in their t rea tment of magnetically saturated media.

The action A associated with the rod in the time interval T ~- [6, t2] is

A T F

(lO)

where the Lagrangian density ~ is assumed to depend on the variables indicated (note the absence of higher order gradients).

234 M. Epstein:

The following action principle is postulated

T F T

F F F

(11)

where

~, ~" = external loads per unit reference length of the axis ; ~, i" - - stress vectors; ~ , ~ = momenta per unit reference length of the axis ; ~ = symmetric Lagrange multipliers associated with the constraints (3); and where the summation convention on ~, fl is understood over the pairs tha t are constrained by Eq. (3). Suspension of the summation convention will be indicated by parentheses around the indices involved.

Note that there is no need to include Lagrange multipliers associated with constraints (4), (5) because these are holonomic constraints derived from (3). I f one were to include such Lagrangian multipliers, they would combine with ;t ~z and a renaming of such combinations would eliminate them (cf. [13]).

The Euler equations and natural boundary conditions associated with (10) when ~ , Sd~, (~"~ are varied independently, are

e s (12a) - - ~ * + T ' + ~ - - ~

- ~ . - ~*~ + ~'~ ,i.~dp + ~ = o (12b)

d~ �9 d~ = C~ (12c)

* L - - ~ = 0

~ - - ~ " = 0

}

(13a) at t = t l , t = t ~

(13b)

(14a) at S = O, S = L

(t4b)

where the following notation has been introduced:

m = - - T - -

ag:

(15)

Thin-Walled Beams as Directed Curves 235

To derive integral forms of the equations of balance consider first a var ia t ion dY = d, 6d~ = 0, ~2"~ = 0 with d an a rb i t ra ry constant vector. The principle of action yields

ff t= t~ - - s = o - - ~ dS dt = 8-'-~-x

F T F T

Now consider the variat ion 8~--~ 1 ~ , ~d, = Wd,, ~A ~ = 0 where W is an a rb i t ra ry skew-symmetr ic tensor and ~ is the position vector. This yields

f X d~ : : - f f (e • 7o + & • -#) d,S dt lY" T [ "

(17) -a S = L

T T F

where

= ~ x m + d . x m o ( i s )

/ ~ = ~ X #---~ - - d,,, X p* + ~ X m + ~~, X m" - - ~' • T - - d~,' X f " (19) 8~

Finally, consider a variat ion of the form : ~ = &, &l~ = ~,, O2~z = arbi trary, to get (after taking account of the constraints)

T T F

T T F

where

E = m . ~ + m ~ . $ ~ - - ~ e (21)

The requirement of Eucl idean invariance ([8]) on ~f leads to the conditions

- - = 0 (22a)

8 ~ - - = 0 (22b)

~t

= 0 (22c)

so tha t the r ight hand sides of (16), (17) and (20) vanish and these equations can be regarded respectively as laws of balance of momen tum, momen t of m o m e n t u m and energy.

Bearing in mind the application to elastic th in walled beams, assume tha t the Lagrangian densi ty is expressible as ([14]):

= ~ - ~ f (2a)

236 M. Epstein:

with

and

(2~)

= r c<, <', s) (25)

where R ~z = Rz ~ is zero if c~ is not contiguous to ft. Assmue moreover [hat these coefficients are t ime independent. Since T and L satisfy Euclidean invarianee, it follows that to must r Therefore (see See. 3) one must have

= ~ ( e , y=, qb,, 0, S) (26)

which yields, by the chain rule,

T-- ~ ~ Or + ~<~ j_ ~o~ d~-' (27a)

_~ ~ 0r y, ~r - . . . . + dl ' de" (27b)

I t is important to verify that the condition/~ = 0 is satisfied automatically, as expected from Eq. (7). The coefficients in (27) can be given a physical inter- pretation in terms of moments of various orders of the three-dimensional stresses, by comparing with the internal virtual work expression in [1]. From such a

e-gz comparison it follows that, as a first approximation, ~ represents the axial

force, OCK is the torque and 0 ~ and e ~ o--O- O?,~ ~ are associated with the shearing force

and the bending moment of plate or respectively. Thus, for example, a logical choice of ~/" for a linearly elastic s y m m e t r i c / - b e a m could be

1 ~/y = -21 E A (e -41)24- 2-1 gAlY12 @ -~l gA2(y ~ + ~32) + "~ EI41 ~

1 GJO~ @_.~1 Ei2(c~22 @ 6p32 ) _}_ --2 (2sa)

where the web is plate No. 1 and where A is the total cross-sectional area, A1 and A2 denote the shearing area of the web and each flange respectively, I is the overall centroidal moment of inertia about the "strong axis", Ie is the larger principal moment of inertia of a flange, J is the torsional constant of the cross- section and E and G are the normal and transverse moduli of elasticity, respec- tively. Similarly, the kinetic energy density could be assumed as

Y=V~ ~o[A~ �9 ~ + I i , . i= + ~=(i=. i= + i=- i=)I (2Sb)

where o is the density.

Thin-Walled Beams as Directed Curves 237

5. Compatibility Conditions

Given the 4n - - 1 quant i t ies e, y:, ~b:, 0, C:~ as funct ions of S, wha t conditions mus t these sat isfy so t h a t t hey m a y be der ivable f rom fields x i and d: according to Eqs. (3) and (9)?

Apar t f rom the obvious l imitat ions

C(~)(~) > 0 (29a)

C<~)(,) > ] / ~ / e (29b)

C(,)(~)C(~)(~) > C~p (29c)

the nine quant i t ies e, 71, y2, q~, qb~, 0, Cn, C~ and C2~ will a lways allow to solve for the nine components of x i, d~ and d~ and thus de termine (uniquely to within a rigid body motion, [2]) a space curve with two director fields. The vectors ~' , d~ and d~ can be now regarded as forming a " m o v i n g " t r iad which serves as a local basis in E a. We can ar range the inner products among these vectors in a metr ic ma t r ix ~(1, 2) as follows:

~ ( 1 , 2 ) = d~ .~ ' d l - d l d l - d 2

d 2 - d l d2"d~

e

Y2"

Yl ~'2

Cll C12

C12 C~2

(30a)

For a n y two contiguous e lements cr and fi (i.e. for a n y edge c~fl of the tree representa t ion of Append ix I) we can similarly define

e y~ 78

~(~, fl) ~- 7a C(a)(a) Carl ~- [gab(M, fl)] (30b)

78 C ~ C(~)(~) a, b = 1, 2, 3

Now, consider a n y element , ~ say, different f rom 1 or 2. This element , e, is contiguous to some other element , say fi, in its tu rn contiguous to a different element, say c~. Assume tha t ~(~, fl) is non-singular. Then de has a unique rep- resenta t ion as

d~ = ul~ ' 4- u~d~ -]- u~d~. (31)

Since ~(~, fi) is non-singular, it has an inverse

~-~(~, ~) = [g~ fl)] (32) and one can write

U a = t a b u b (33)

where the a rguments (cr fi) have been omi t t ed for clar i ty and where

ul = de �9 x ' = 7~, (34a)

u2 = d e �9 d , , (34b)

u3 = d~- d~ = Cq~. (34c)

238 M. Epstein:

We see that us is not known directly, but it can be calculated from the quadratic equation

C(~)(Q) -~ gabUaUb -~ gn7~2 @ 2gl~yoU2 d- 2gls7~C~ (35)

Differentiating d~ and taking its inner product with ~' yields the compatibility condition.

1 e' dpo = (g~aua)' e 4- (g2aua)' r , + (g3au~,) ' Y~ ~ -~ g~aUa + g~au~r + ~aUar ~ (36)

wliere the following equation has been used

e' = 2~' �9 ~". (37)

There are n - - 2 such compatibility conditions. That these compatibility con- ditions are sufficient for the existence and uniqueness of the director fields d~ (Q = 3 . . . . . n), follows from the fact that, excluding Ce, Eqs. (31)--. (35) provide directly such unique director fields, d~ say, in terms of components on the moving triad along the unique curve determined by the first nine measures of strain. On the other hand, specification of r -----de'-~' defines for each ~o a family of director fields all having a comnLon Z' component of the first derivative. Eq. (36) simply establishes the fact that d e belongs to this family.

"6. Summary and Conclusions

A complete non-linear theory of thin-walled elastic beams has been presented based on a representation of the beam as a multi-directed curve with directors subjected to certain physically meaningful constraints. The value of such a rep- resentation lies not only in allowing a rigorous formulation of the non-linear theory of thin-walled beams but also in providing a valid physical model for the theory of multi-directed curves. Another such physical model worth exploring is the theory of multilayered beams, in which case the directors would be sub- jected to a different kind of constraints.

A minimal set of strain measures, each of them having a definite physical meaning, has been selected in such a way that objectivity is automatically en- sured and no restriction on the magnitude of displacements and rotations needs to be imposed. The compatibility conditions for the given set of strain measures have been obtained. A novel technique for the algebraic elimination of the La- grange multipliers associated with the constraints has been included as an Ap- pendix.

By a proper interpretation of the statical quantities involved, the present theory can be reduced to an earlier formulation [i] based on a three-dimensional approach. The numerical examples of [i] are therefore also valid as an illustration of the power of the present theory in dealing with highly non-linear problems. On the other hand, the generality of the present approach should provide a framework for the analysis of most practical problems related to the behaviour of thin-walled beams of open cross-section. Some of these problems will be con- sidered in a future paper.

Thin-Walled Beams as Directed Curves 239

Acknowledgement

This work has been supported by the National t~eseareh Council of Canada, through operating grant Nt~C A-4662.

Appendix I

Elimination o/"the Lagrange Multipliers

Eqs. (12b, c) have the form

{ ,t~d~ + A -- o, (A. la)

d~.d~-- Qe = 0 (A. lb)

where/~ are (independent) functions of g~ but not of A ~ and where C~e are con- stants. Note that (A. 1 b) applies only for c~ = fl and for the n - - 1 pairs of con- tiguous elements a, ft. Accordingly, A *~ are defined only for the same combinations, a fact tha t has to be taken into account when using the summation convention on Greek indices.

| |

|

V Fig. A.1. Beam cross-section

The problem considered in this Appendix is the elimination of i"~ by purely algebraic operations. To this end, it is convenient to define a graph (Fig. A. 2) associated with the cross-section (Fig. A. 1) in the following way: a) each element of the cross section is mapped into one vertex, which will be identified by the element number; and b) every joint between two elements of the cross section is mapped into an edge joining the corresponding vertices. Since multiple joints are excluded and since the cross section is open, the resulting graph is a tree, i.e., a connected graph without any circuits. As it is well known (e.g. [15]) a tree with n vertices has exactly n - - 1 edges. Also, a tree can be defined by the proper ty

�9 240 M. Epstein:

that there is one and only one pa th between every pair of its vertices. Moreover, every edge in a tree is a cut-set, i.e., its removM leaves the tree disconnected. I t is easy to see from here that the constraints (A. lb) are mutually independent since they either fix the length of one element or prescribe the angle between two otherwise disconnected sections.

2

13

Fig. A.2. The tree associated with the cross-section

As a first step towards the elimination procedure we prove the following

Theorem 1: Eqs. (A. la) imply

Proof: The theorem follows immediately from the symmet ry of #..~# and the skew symmetry of the vector product. A less obvious proposition is embodied in the following.

Theorem 2: To every edge Qa and one of its incident nodes, say a, there corresponds the equation

where T~ is the sub-tree that contains ~ determined by the removal of ~a.

Proof: Eq. (A. la) yields

Z ~ (~d~ x g~ + [~ x 3~) = 0. (A. 4) a~T 0

The argument of symmet ry of )l ~# and skew symmetry of d~ X d~ can be invoked for all ~ E To except for 5 = ~ and fl = a (because a~To), so tha t Eq. (A. g)

\ e2e

from which the theorem follows immediately. The theorem cart also be proved by finite induction starting from a pendant

vertex.

Thin-Walled Beams as Directed Curves 241

Eqs. (A. 2) and (A. 3) represent a total of 3 + 2(n - - 1) = 2n + 1 equations. Out of these, n + 1 independent equations have to be chosen so that together with the 2n - - 1 constraints (A. 1 b) they provide a total 3n independent equations in the 3n unknowns d~. Towards this goal it is useful to prove the following

Theorem 3: At every vertex, ~, the Eqs. (A. 3) associated with ~ and with each of the edges concurring at c~ and the general Eq. (A. 2) are functionally dependent.

Proof: The sub-trees T~ associated with all possible edges a~ concurring at node ~ are a disjoint cover of the whole tree minus the node ~. So, writing Eq. (A. 3) for each of those sub-trees and smnming yields

0 = Z (i s x d~). d~ = ( y • de) . d= - (7~o~ • eL). d(~) = ( y x cg~). ~. ,~UTQ

But Eq. (A. 6) is also obtainable from (A. 2), which proves the theorem. Let now the tree be structured into an arboreseence, i.e., a rooted directed

tree (Fig. A. 3). This can be achieved by singling out one vertex (the root) and assigning the outgoing direction to all edges emerging from it. This process is repeated from the new vertices so reached until the whole tree is covered. The fact tha t in an arborescenee all the vertices, except the root, have an in-degree (number of edges converging to the vertex) of 1, suggests the following

Theorem 4: Let ~ be the root of an arboreseenee and let ~a be one particular edge emerging from t h e root. For every edge, except Ca, write the Eq. (A. 3) associated with the vertex at the tail of the arrow. Then the system of n - - 2 equa- tions thus obtained and Eq. (A. 2) are mutual ly independent.

@ 2

I0 I0

II- It ~ ~ -7 Ii �9 ~ i~. 6 ql 13 13 7

2

io 5

II �9 7 13

Fig. A.3. Three possible arborescences for the same tree. Roots are circled

242 M. Epstein: Thin-Walled Beams as Directed Curves

Prool : Consider first an edge o~ emerging from the root. The corresponding Eq. (A. 3) will be the only one among the n - - 2 equations to contain /~ since, according to the process described above , the sub-trees associated in Eq. (A. 3) with the other edges will be either disjoint with T, or proper subsets of T~ not containing ~. Therefore, the Eq. (A. 3) associated with any edge emerging f rom the root is independent of the remaining n - - 3 equations. Now consider the vertex ~x. I t certainly is the root of the arborescenee f~ and therefore the pre- ceding reasoning applies to all edges emerging from it. Since in this way the whole arborescenee is eventual ly covered it follows tha t the n - - 2 equations are mutua l ly independent . Finally, consider Eq. (A. 2). I t is a vector equat ion containing ]~ and [~, which do not appear in any of the previous n - - 2 equations. Since both [q ~nd ]~ cannot be eliminated simultaneously and since (except for the trivial case n ~ 1) the 3 Eqs. (A. 2) are mutua l ly independent , the theorem follows.

Note tha t the choice of the tail, and not the tip, of the arrows is not capricious as can be seen by applying Theorem 3 to a pendan t (terminal) ve~ex.

References

[1] Epstein, M., Murray, D.W.: Three-dimensional large deformation analysis of thin walled beams. Int. J. Solids Structures 12, 867--876 (1976).

[2] Ericksen, J. L., Truesdell, C. : Exact theory of stress and strain in rods and shells. Arch. 1%ational Mech. Anal. 1, 295--323 (t958).

[3] Green, A. E., Laws, 1~. : A general theory of rods. Proc. Roy. Soc. (London) 293, 145 155 (1966).

[4] Cohen, H.: A nonlinear theory of elastic directed curves. Int. J. Eng. Sci. 4, 511--524 (1966).

[5] Antman, S. S.: The theory of rods, in: Kandbuch der Physik, Bd. Via/2 (Fliigge, S., Truesdell, C., eds.). Berlin-- Heidelberg -- New York: Springer. 1972.

[6] Vlasov, V. Z. : Thin-walled elastic beams, 2nd ed. Israel Program for Scientific Trans- lations, Jerusalem, 1961.

[7] Spencer, A. J. M. : Theory of invariants, in: Continuum Physics, Vol. 1 (Eringen, A. C., ed.). :New York: Academic Press. 1971.

[8] Toupin, R. A. : Theories of elasticity with couple-stress. Arch. Rational Mech. Anal. 17, 85--112 (1964).

[9] Truesdell, C., Noll, W.: The non-linear field theories o~ Mechanics, in: Handbuch der Physik, Bd. III/3 (Fliigge, S., ed.). Berlin--tteidelberg--New York: Springer. 1965.

[10] Chakrabarti, S. K., Wainwright, W. L.: On the formulation of constitutive equations. Int. J. Eng. Sci. 7, 601--613 (1969).

[11] Kafladar, C. B., Eringen, A. C. : 5iicropolar media I. Int. J. Eng. Sei. 9, 271--304 (1971). [12] Whitman, A.B., DeSilva, C.N.: A dynamical theory of elastic directed curves. Z.

Angew. Math. Phys. 2O, 200--212 (1969). [13] Maugin, G.A., Eringen, A.C.: Deformable magnetically saturated media I. Field

Equations. J. Math. Phys. 13, 143--155 (1972). [14] Shahinpoor, M. : Plane waves and Hadamard stability in generalized thin elastic rods.

Int. J. Solids Structures 11, 861--870 (1975). [15] Deo, N. : Graph theory with applications to engineering and computing science. Prentice-

Hall. 1974.

M. Epstein Depa~'tment o] Mechanical Engineering

The University o/Calgary Calgary, Alberta

Canada