network of thin thermoelastic beams

Mathematical Methods in the Applied Sciences. Vol. 16, 327-358 (1993) MOS subject classification: 73 K 05. 73 D 35, 35 A 15, 35 L 20. 35 L 40

Modelling of Dynamic Networks of Thin Thermoelastic Beams

J. E. Lagnese, G. Leugering

Department of Mathemaiics, Georgeiown University, Washington D.C. 20057 USA

and

E. J. P. G. Schmidt

Departnient of Matheniutics and Statistics. McGill University, 805 Sherhrook 3. West, Motitreal. Quebec. Canada H3A2K6

Communicated by E. Meister

We derive a distributed-parameter model of a thin non-linear thermoelastic beam in three dimensions. The beam can also be initially curved and twisted. Our main task is to formulate the non-homogeneous initial, boundary and node value problem associated with the dynamics of a network of a finite number of such beams. The emphasis here is on a distributed-parameter modelling of the geometric and kinematic node conditions. The forces and couples appearing in the boundary and node conditions can then be viewed as control variables. The analysis of the resulting control systems and their controllability and stabilizability properties is the subject of [25] and of forthcoming papers.

1. Introduction

Our goal in this paper is to develop a comprehensive distributed-parameter model for a dynamic network of thin thermoelastic beams. Models of this kind are funda- mental in the theory of stabilization and controllability of the transient behaviour of flexible structures. The structures we have in mind are multiple-link constructions that are composed of flexible beams, plates, shells or combinations of such elements. Such structures are representative of trusses, frames, robot arms, solar panels, antennae, deformable mirrors, etc. currently in use. Here we shall develop three-dimensional, non-linear models of dynamic flexible beam structures that will incorporate various joint interactions, torsional, thermal, viscous and viscoelastic effects, initial curvature and twist, and other considerations.

There is a vast literature concerning the modelling of multibody systems. However, the literature concentrates almost entirely on rigid rather than flexible elements, on finite-dimensional nodal (hence, global) modelling rather than on a spatially dependent (local) approach through distributed-parameter modelling. As a result, the interac-

0 I 70-42 14/93/050327-3252 I .oo 8 1993 by B. G. Teubner Stuttgart-John Wiley & Sons Ltd.

Received 6 August 1991 Revised 15 October 1991

328 J. E. Lagnese ct a/ .

tion/transmission and conversion of bending, torsion and axial deformation at the nodes is not taken into account. Recent engineering research strongly indicates, however, that it is important to account for such phenomena in order to understand completely transmission effects in trusses and frames and in other aggregates of this kind. See [20] and the references therein.

In order to accomplish a comprehensive theory of networks of flexible beams, we have to first concentrate on typical elements. There are many requirements from the area of applications to be taken into account. Although, in general, the structure has to be considered in three dimensions, sometimes a planar model will be sufficient. Therefore, in each situation, the elements have to be described in their own reference co-ordinate systems. The model should allow for finite displacements. Hence, the strains necessarily contain non-linearities. Further, the model should allow for twist and the coupling between twist and bending. In addition, in many applications, particularly to satellite dynamics, temperature plays an important role and the contribution of heat flux to bending is of great importance. This necessitates the incorporation of thermal effects into the modelling. In addition, weak damping is always present in the structures under consideration. Besides the dissipation caused by the coupling of heat conduction to stretching and bending, other sources of internal damping, such as Kelvin-Voigt or Boltzmann-type viscoelastic damping, should be considered. We have not been able to find in the literature a model of a single beam meeting all, or even almost all, of these requirements. There is the general theory of non-linear rods by Antman [I], and further work of Simo [35] and of Marsden et al. [36] extending and modifying the classical papers of Reissner [29-3 I ] and Love [27]. These papers emphasize the viewpoint of geometrically exact kinematic descriptions and very general constitutive relations. Besides the fact that the thermomechanical aspect is usually not dealt with, i t appears to be as difficult to step down from these models to the level of applicability in the context described above as i t is to develop the models directly in a geometrically ‘semi-exact’ sense, to be clarified in this paper. We would like to mention a preprint of Rogers [32], where models for straight, linear and isothermal beams are derived from the geometrically exact rod theory. There is also much engineering oriented work being done, for instance, by Kane, Ryan and Banerjee [21], Song and Haug [37], Thompson and Hunt [38], Brown [9], Britvec [8] and Han and Zhao [I91 (see the bibliography therein), that focuses on the linear modelling issue, and by Brink and Kratzig [7], Bathe and Bolourchi [S], Wood and Zinkiewicz [43] and Wunderlich and Obrecht [44] provid- ing a finite element analysis of non-linear beams. Even though that work is directed towards applications to begin with, it does not in most cases include thermal effects. In addition, particularly in the given context of the transient behaviour of trusses and frames, the papers do not dwell on the dynamical behaviour of the flexible system, but rather on the response of the finite element approximation to it, which, by its nature, does not reflect flexible interactions at the joints.

For these reasons and for the sake of self-consistency, we will derive the models from a continuum-mechanical viewpoint, guided by the catalogue of requirements outlined above. We will follow the approach given by Wempner [41] (see also [17]); for the benefit of the reader who has this book available, we keep the notation as close to Wempner’s as reasonable. We also refer the reader to Ciarlet’s book on mathematical elasticity [lo].

Our main purpose, however, is not to model a single beam but, rather, a network of such beams. (In a later study we will consider combinations of other elements like

Networks of Thin Thermoelastic Beams 329

plates and shells.) Modelling and analysis of Jlexible frames and trusses has been a subject of great interest since the work of Koiter [23]. The engineering literature dealing with this topic is usually related to either buckling or postbuckling under imperfection, where the theory is quasi-static, or to simulation, where we encounter the problems just mentioned. This is different from Howard’s thesis [20], where the transient behaviour of simple planar trusses and frames is investigated. The mathematical literature about this topic is very sparse. Here we can only quote the ‘Dystunder-Ciarlet’ approach to obtaining 1-d or 2-d beam or plate models from the 3-d elasticity equations by way of singular perturbation; see [13]. This approach has been followed by LeDret [26], Cimitiere ef al. [l5], Ciarlet [lo-121 and Ciarlet et al. [14] to obtain models of the junctions between various elastic bodies such as between beams or between plates. In the limit, the procedure excludes longitudinal deformation, which seems to be rather restrictive in multibeam structures. We mention the work by the last author [33], where part of the program was accomplished for string networks, and [28], where Euler-Bernoulli beam networks have first been introduced.

We are grateful to the referees for having brought the works of Ali Mehmeti and Nicaise [4], Ali Mehmeti [2, 31, and von Below [39, 401 to our attention. These papers deal with interaction problems in the context of wave equations [2,3] and parabolic equations [39,40,4], and combinations thereof. In particular, in [40] von Below discusses the eigenvalue problem for a Sturm-Liouville operator on a network of arcs. In this respect, see also [33,25], where an eigenvalue problem on a network is treated in connection with the problem of approximate controllability and strong stabilizability. See also the work of Dystunder [ls], where eigenvalue problems for multiple element structures are approached by the so-called substructuring method. Von Below’s results can possibly be extended to cover operators appearing in the theory of networks of beams as being developed in this paper. The difference between the eigenvalue problems discussed by von Below and the one related to multiple beam networks, where longitudinal as well as vertical motion i s taken into account, is that in the first case only scalar functions on the network are involved, while in the second case vector-valued functions have to be considered. Using the notations of von Below, one can derive a matrix fourth-order differential eigenvalue problem, where some of the coefficient matrices are not invertible. We have, however, not made any attempt to include eigenvalue problems in this paper, simply because of space limitations, neither have we included any results on existence and uniqueness of solutions to the variety of network problems. We want to remark though that in the linear cases existence and uniqueness of solutions can always be obtained by the exeniplaric variational method explicated in [25]. In the case of non-linear equations and non-linear interaction at the multiple nodes, existence and uniqueness results are, of course, much harder to obtain, and are to be expected to hold only locally in space. In the latter case one might lo& for a non-linear-contraction semigroupperturba- tion setting, as in [24], [4], where interaction problems are discussed involving scalar functions on a network. This is the subject of future research.

2. Modelling of a thin thermoelastic curved beam

We consider a thin beam of length 1 with a given initial curvature and torsion, and with variable doubly symmetric cross-section. To be more precise, the undeformed

330 J. E. Lagnese er a/.

beam, in its reference configuration, occupies the region

R : = {r : = ro(xl) + x2e2 + x3e3 I x l E [ O , l ] ,

(x2rX3):= -x2e2 + x3e3eA(x1)),

where ro : [0, 13 + R 3 is a smooth function representing the centreline of the beam, and the orthonormal triads el(xl) , e2(x1), e3(xl) are chosen to be smooth functions of x1 such that el is the direction of the tangent of the centreline with respect to the variable x l , i.e. (d/dxl)ro(xl) = el(xl) , and such that e2(x1) and e3(x1) span the orthogonal cross-section at x l . The meaning of the variables xi is as follows: x1 denotes the length along the undeformed centreline, x2 and x3 denote the lengths along lines orthogonal to the reference line. The set R can then be viewed as obtained by translating the reference curve ro(xl) to the position x2e2 + x3e3 within the cross-section vertical to the tangent of ro. The cross-section at x1 is defined as

A(.xl) = {x2e2 + x3e31xlel + x2e2 + x3e3€R}.

A(xl) is assumed to vary smoothly with respect to x l in a sense that will be made clear later on. In order that certain moments vanish, we will restrict our attention to cross-sections which are doubly symmetric in the sense that (x2, x3) = x2ez + x3e3cA(x1) implies that ( - x2, x3), (x2, - x ~ ) E A ( x ~ ) and, hence, - (x2, X ~ ) E A ( x , ) . With this notation, the curvatures and twist of the beam, more

precisely, of its centreline in the reference configuration, can be described using Frenet-type formulae as

K~ := e 2 * e l , l = - e l*e2 ,1 ,

K~ := e3*e1,1 = - el . e3 ,1 ,

~ : = e 3 - e 2 . ~ = - e 2 - e 3 , 1 .

The material is supposed to be thermally and elastically isotropic, and Hookean. We allow the beam to undergo large lateral displacements and ‘moderate’ rotations of its cross-sections, to be specified later. Essentially, we follow the method outlined in [41]. By definition, a particle has the initial location vector

r = r(xl, x2, x3).

whereas, after deformation, the particle has the position

R = R(x1, ~ 2 , ~ 3 ) .

Therefore, the displacenient vector is given by

V : = R - r.

An initial position on the reference line (x2 = x3 = 0), denoted by

ro = r(x 1, 0,OL is deformed to

Ro = R(xl,O, 0).

Accordingly, the displacement vector of a particle on the reference line is

w : = V(Xl,O, 0).


The tangent vector at any particle in the undeformed configuration is denoted by

where the subscript separated by a comma means spatial derivation with respect to variable x i . Accordingly, the tangent at this particle in the deformed configuration is

g . = r . 1 , I ?

Gi = R.i = gi + V,i. Consequently, we have two triads (g,), (Gi) at that particle, in the undeformed and deformed configuration, respectively. These are right-handed systems as the determi- nant of the deformation matrix (Gi)i is positive. In particular, at the reference line we have the triads

ei : = g i ( x l , 0, 0), Ei : = G i ( x l , 0, 0), i = !,2, 3.

We have

r, l = (1 - X ~ K ~ - x ~ K ~ ~ ~ + .x2'se3 - x35e2,

r.2 = e2. r.3 = e3.

Note that in the presence of an initial twist the triad gi is not orthogonal. Introducing the length si of an undeformed x,-co-ordinate line passing through (x2, xj), we have the increments

ds, = lr.l I dx l , ds2 = dx2, ds3 = dx3.

With this notation, we are able to write down the strain components as follows:

1. 1 aR aR ar ar &.. := - -._ - _._

IJ 2 ( asi asj asi asj

Note that the reference configuration already includes curvatures and twist. In the case of a straight and untwisted reference configuration, the matrix (&/asi * &/asj) reduces to the identity, whereas the matrix (aR/asi -aR/8sj) becomes the right Cnuchy-Green tensor, and the strain E just defined then is the Green-St.- Venant strain tensor. see [lo].

aR

Now

dX. ar dxi - = G . - - = g i - , asi dsi ' asi dsi

where

Hence, the components of the strain tensor can be written as

332 J. E. Lagnese et al.

1 2

1 2

1 2

~ 2 2 = - (C2 * G2 - l),

~ 3 3 =-(G3*G3 - l),

~ 2 3 = - (G2 * G3).

The last three strains are strains within the cross-section. Those are usually neglected in theories of thin beums.

The deformation of r into R will be considered as a succession of two motions: a rigid rotation carrying the initial triad e i (x l ) to an intermediate triad tii(.xl), followed by a deformation into the non-orthogonal triad Ei. The two triads Ci and Ei then differ on account of a strain E, to be specified below. We choose to orient the intermediate (right-handed) triad Ci, which serves as a moving orthonormal reference frame, such that

El =(E,.CI)ti1 =IEIICI, 62.E3 =$3*E2 (2) holds. Note that the triad ti neither is the canonical Frenet system, nor the one employed in [35]. Nevertheless, it is uniquely determined by these properties. The strain E is then defined by

The remaining strains are defined by requiring symmetry: Eij = Ej i . We may now use the strains Eij in order to express the triad Ei in terms of the moving orthonormal reference triad C i .

If we now assume small strains E i j in the sense that the products EijEkl may be ignored regardless of the magnitude of the rotation that carries the triad ei into Ci, we obtain

Eij x $ (Ei * Ej - aij). ( 5 )

However, the latter is exactly the strain cij(xl, 0,O) at the deformed reference curve. It is our goal to express the strains cij in terms of the reference strains Eij and the

strains related to the rotation. This rotation is obtained by succession rotations about the e l , e2 and e3 axes, respectively, through respective angles 01, O2 and 03. The


result is

6, = (cosO2 ~ o s @ ~ ) e , + (cosO2 sin@,)e, - s ina2e3 (6) C2 = (sin@, sin@2 C O S @ ~ - cos0 , sin@,)e,

+ (cos0, C O S @ ~ + sin@, s i n e 2 s in03)e2 + (sin@, cose2)e3

C3 = (cos0, s i n e 2 C O S @ ~ + sin@, sin@,)e,

- ( s ine l C O S @ ~ - cosOI sin@, s ina3 )e2 + (cosOI cos02)e3.

It is now possible to introduce ‘curvatures’ and ‘twist’ in the same way as with the undeformed reference curve, but now based on the moving triad Ci (see also [32]):

K2:=Cz’C1.1 = -C1.&2,1,

We assume throughout the paper that the dcformed state can be represented by means

(8)

Note that for E l < 1 we have El = C,. In (8) the function 4 is related to the solution x of the St. Venant torsion problem, called the torsion function, in the following way: x satisfies the equation Ax + 2 = 0 (A signifies the Laplace operator) over the cross- section and vanishes at the boundary of the cross-section. This function can be written as 4 , - i ( x $ + x:) with a harmonic function 4 , . With this notation, 4 above is the conjugate harmonic function to Cpl. Hence, 4 is itself harmonic. By the assumed double symmetry and the application of Schwartz’ reflection principle, 4 has the following symmetries: 4(x2, x,) = C p ( - x2, x j ) = 4(x2, - x,) = - &(- .x2, - x3) . The function y(xl) measures the amount of warping. Warping is considered small if the beam is thin and, hence, is small compared to the breadth of the beam. We note that (8) restricts the geometry of the deformed beam, and that it would be more accurate to account for higher-order terms in the variables x2, x,. However, (8) is the standard assumption in theories for thin beams, and we will adopt it in this paper. With this setting we obtain

of the deformed reference curve Ro and the triad Ei as follows:

R = Ro + ~ 2 E 2 + ~ 3 E 3 + ~ ( ~ 1 ) 4 ( - ~ 2 9 . ~ 3 ) 6 1 .

= R . l = El + X2E2.l + X3E3,1 + 7.1 421 + Y4(K262 + K3C3)r

G2 = R,2 = E2 + ~ 4 , 2 C i , (9)

G3 = R.3 = E3 + Y4.3&17

In order to obtain any practical set of equations of motions, we have to approximate the rotations. There are two stages of approximation, based on power series expansions, to be found in the literature, which have been termedfnite rotation and moderate rotation. A finite rotation approximation to any given rotation described in terms of three angles as above amounts to keeping all quadratic terms in the angles, while a moderate rotation, which has also come to be known as infinitesimal rigid displacement (see [lo], Chapter 2), reflects only the linear terms. Another way of thinking of moderate and finite rotations is to consider a one-parameter group of rotations in R3 generated by a skew-symmetric matrix R. The linear approximation is

334 J. E. Lagnese ef al.

precisely the moderate rotation, while the quadratic approximation coincides with the finite rotation. The finite rotation approximation to (6) is as follows. We have

6, = (1 - +(@: + @:))el + 03e2 - 02e3,

Cz = (0, 0, - @3)e1 + ( 1 - f ( @ f + @:))e2 + 01e-3, (10)

C 3 = (0, - @1@3)e1 - (0, - 0 2 0 3 ) e z + (1 -;(Of + @2,))e3.

If we assume that the 0; terms are small compared to 2, we obtain

C1 = e , + O3e2 - 0 , e 3 ,

CZ = (0, O2 - O3)el + e, + O1 e3,

b3 = (0, - a 1 e 3 ) e l - (0, - Q2Q3)ez + e3.

This is still to be viewed as a finite rotation. Ultimately, we want to consider the deformation of the reference triad ei into the triad E i as being composed of two rotations. This is possible if we neglect the strains E 2 2 , E 3 3 , E 2 3 . We have

El = ( 1 + EIl)(e1 + O3ez - Oze3),

E2 = e2 - (0, - 2E2, - 0, 0 2 ) e l + 01e3,

E3 = e3 - (0, - 0203)ez + (0, + 2E31 - @ 1 @ 3 ) e 1 .

We may define a vector 8 as follows:

9, := 0 1 - 0 2 0 3 ,

93 := 0 3 - 2E,, - 0 1 0 2 ,

8 2 := 0 2 + 2E3, - 0 1 0 3 ,

9:= 81el + s2e2 + 93e3.

Also, we set

0:= Ole l + 0 2 e 2 + 0 3 e 3 .

With these definitions and the definition of E l , we can rewrite (12) as

El = ( 1 + E l l ) C , = e l + W , , ,

(13)

E2 = e2 + 9xe , , (14)

E3 = e3 + 9 x e 3 .

Equations (14) express the triad Ei as a product of two rotations: afinice rotation (0) of the traid ei into Ci (see (1 l)), and a moderate rotation (8) of the cross-section at the centreline. The latter is interpreted as a moderate rotation since it approximates a rotation, as a map, by the identity plus a cross product with a vector. According to the definition of the shear strains El i , the difference g2 - (0, - O1 0,) = E3*C1 and Q3 - (0, - 0, 0,) = - E2-C1 are the shear angles between the deformed cross- section and the triad Ci. The angles 8, = E3 -e l and 93 = - E2 *e l are the total angles of rotation of the cross-section, while 91 measures the torsion of the cross-section.

We now concentrate on the curvatures K i and the twist T. Within the finite rotation


approximation (lo), we obtain

K 2 - ~2 = 0 3 . 1 + ~ 3 0 1 + ~ 0 2 + 0 2 0 1 . 1 - ~ 3 0 2 0 3 - ) ~ 2 ( 0 : + a:), K 3 - ~3 = - 0 2 . 1 - ~ 2 0 1 + ~ 0 3 + 0 3 0 1 . 1 - + ~ 3 ( @ 1 + a:)? ( 1 5 ) T - T = 0 1 . 1 - ~ 2 0 2 - ~ 3 0 3 - 0 2 0 3 . 1

- + T ( @ : + 0:) + ~ 2 O 1 O 3 - ~ 3 0 1 0 2 .

These expressions can be found in [29,30]. If we use the definition of 9 to express the components Oi in terms of the components 4, and if, in addition to the standing assumption of small reference strains, we neglect higher-order terms involving the 9;s, which is consistent with the assumption of a moderate rotation represented by 8, we find

T - T = $ ~ , ~ - K 2 $ 2 - K 3 8 3 .

We proceed to express the strain eij entirely in terms of the displacement W and the rotation 9. We will use the approximation

( r , l I z I - x 2 ~ 2 - x 3 ~ 3

and (9), (12) or (14) to compute the scalar products Gi-Gj . W e consider the beam so thin that the quantities x2 K 2 , x 3 K 3 , x2 T, x3 T, x 2 K ? , x3 K ~ , x 2 5 and x3? are small compared to unity. We then obtain the following approximations:

G1.G1 = 1 + 2 E 1 1 +2y. ,4-22~2(K2-2E, l .1 ) (1 + E l l )

As new strains, we introduce the quantities rZi and i based on the differences of the curvatures K i - K ~ , the torsions T - T, and the 'error' made in using the triad Ci instead of Ei, as follows:

336 J. E. Lagnese et a/.

Upon using (18) in (17) and neglecting products of strains (such as xk t k E i j , etc.), we find the approximations

E l , = E l , - X 2 k 2 - x3f3 + y . 1 4 ,

E l 2 = E l 2 - k 3 f - Y 4 . 2 ) r (20)

& 1 3 = E13 + t ( X 2 f + *1’4,3). The strain tensor E depends on eight unknown functions of the variable x 1 alone: the three reference strains Eij, the strains ti, f produced by ‘curvature and twist’ of the deformed beam, and the warping strains y, y , , . (It is customary to introduce y , ] as an additional strain, representing warping due to the non-uniformity of the torsion.)

What is left to be done in order to express the strains c i j in terms of the variables Wi := W-e, , ai, i = 1,2,3, and 1’ is to express the strains E i j in terms of the displacements Wi and the angles Si. To this end, we define

F i j : = t ( e i - E j + ej .Ei) - dij,

Gij := f ( e i - E j - ej*Ei).

A direct calculation then shows that

1 1 F i j + - 1 ( F p i + Wpi) ( F p j + (5,) = - (Ei * Ej - d i j ) ,

? 2 , - so that for small strains 2, we find the approximation

If we assume that E , , G 1, then

W,,.e3 = - 0 2 ,

W , , . e z = 03.

Furthermore,

P 1 = e l * W . ] = W,,, - t i2 W2 - K 3 W 3 ,

e I 2 = f(@ - 8 ) - e 3 = &( Wz.l - Q3 + K~ W 1 - rW3) ,

213 = f ( 9 - @)‘e2 = $ ( 8 2 + w3,1 + K 3 w1 + Twz),

~ 5 , ~ = - f(@ + 8 ) . e 3 = - f( w2., + G3 + ti2 W , - TW,) ,

6 1 3 = f(@ + 8 ) . e 2 = t ( 8 , - w ~ , ~ - ti3 w1 - TW,) ,

0 2 3 = - $ * e l = - 9,.

We are now, finally, in the position to write down the components of the reference strain tensor F in terms of the displacement and the rotations:

W , , , - K K ~ W ~ - K ~ W ~ + ~ { ( W ~ . ~ + 8 3 + K 2 w 1 - T w 3 ) 2

-k (92 - w3.1 - ti3 w1 - fW2)2}9

E1.7 = f( w2,1 - 8 3 + ti2 w1 - fw3) - $81(82 - W3,i - K 3 w1 - TWz), (22)

= f ( 9 2 -t w3.1 -t ti3 w1 + TW2) - a $ , ( w2.1 + 8 3 + K 2 w, - S w 3 ) .


Let AT(.wl, x2, x3) denote the difference of the temperature T ( x l , x2, x3) of the deformed state and the uniform temperature To of the reference state. As the body is supposed to be very thin, we may take the temperature difference AT to satisfy

AT = o1 (.XI + x2 o 2 ( X , + x3 e3(x1 ).

In addition, let E denote Young modulus and G the shear modulus of the beam at the point x1 on the reference curve. We have assumed from the beginning that the cross-section A(x,) is doubly symmetric with respect to the lines x2 = 0 and x3 = 0.

We introduce the moments of the cross-section as follows:

1 2 2 := ] ] A x i dA, 133 := x i dA, I := 122 + I,,,

* *

l - : = j j A b2dA.

Note that these quantities are still functions of x1 since A(x,) varies with xl. We now focus on the stress-strain relations. We, therefore, invoke the hypothesis of a Hookean material. Let sij denote the entries of the second Piola~Kirchho~stress tensor S . Under the standing assumption that the body is very thin, we have s22 = s33 = ~ 2 3 = 0. By the symmetry of the stress tensor, the three remaining entries are to be given in terms of the strains and the temperature. As for the stress-strain constitutive relation, we invoke Hooke's law and find

~ 1 1 = E ~ l l - xAT=E(ElI - ~ 2 ~ ; . 2 - ~ 3 1 ; 3 + j ' , I 4 ) - x A T ,

By definition, the deformation matrix is G := (Gl , G 2 , G2). If we apply the stress matrix S : = (sij) to G', we obtain the first Piola-Kirchhofl stress tensor T. Upon introducing the Piola-KirckhofS stress vector by

we can express T as

Also, we note that C1 is the normal of the cross-section A(xl ) in the deformed configuration, and that, therefore, = (GT)- ' e l , since el is the normal to this cross- section in the undeformed configuration. This then shows that t = Sel, as it should be.

t = ( s i i G i + s I ~ G ~ + ~ 1 3 G 3 ) ,

T = (t, s12 G1, s13 GI 1.

The resultant force F and the couple M on a cross-section are given by

F : = I j A t l r + l l d A ,

M : = I j A ( R - Ro)xt(r , , IdA.

338 J. E. Lagnese et u1.

Let f,, co and to denote the external body force, body couple and surface traction, per unit undeformed volume and surface, respectively. Then the body force P and the body couple C, are given by

C : = J J [co + (R - R o ) l x (fo - m o R ) l Ir., I dA A

where S is the curve circumscribing the cross-section. As we are mainly interested in forces and couples applied to the endpoints of the beam, we set f,, co, to equal to zero. We consider a small segment (slice) RQ of the beam located at a point Q on the undeformed centreline with length 2Ax1 in the direction of e l . The traction on the cross-section at Q - Ax, is - t( Q - AxI ) and the corresponding traction at Q + Axl is t ( Q + Axl). Let us denote the end surfaces of the slice Q, by C,, and the surface tractions by g. The body force f on this volume element is provided by the inertial force - mo R. Let us finally denote by A : B the inner product of matrices, i.e. A : B = tr(ATB), and by V6R the Jacobian of the virtual displacement

6R = 6W + x26E2 + ~ 3 d E 3 + 6 ~ 4 6 1 := 6W + Sfi. Then the principle of virtual work can be expressed as

[j[o,T:V6Rdx= [S[%f*6Rdx + j [zQg-6Rdo, (27)

for all sufficiently smooth vector fields 6R : fi -+ R3 (see [lo], p. 79, in fact, for all variations 6W, 68,6y satisfying some specified geometrical boundary conditions. Using the notation introduced above, we find that (27) is equivalent to

lQQ-y[ { j A ( t ' 6 R , l +(S12Gl'~R.2+~13Gl'~R,3)}~r,l~dA dxl 1 = { [IA (- moR).6Rlr.,I dA] dx,

Q - A

+ J J t-GRlr.,) dA - j t-bRlr, l l dA. A(Q + A ) A(Q - A)

Following Wempner [41], we introduce an auerage external force f l , an average resultantforce D and a surface action 5 reflecting the warping of the cross-sections, as follows:


6 := 61 ’ (S12 4 . 2 -l- S 1 3 4 . 3 ) G i lr,i 1 dA. J J A

We obtain the following approximations for the variations of R:

6 R = 6 W + 6 & J x R + 6 , l @ , ,

6R., = 6W,, + S&J., x R + 6 9 ~ + 6y,, + 6y4(KzG2 + K 3 e 3 ) , 6R.2 = 6 9 ~ e 2 + 6 ~ , 2 6 1 ,

6R,3 = 69 x 63 + 6 ~ , 3 & I ,

R.1 = - x2(K2C1 - T@3) - .x3(K3@1 + TCZ)

+ y.1461 + yb(K262 + K3C3).

Then, after some calculus, using these approximations, we obtain from the principle of virtual work the following balance laws, formulated for the whole beam:

F.1 + P = 0,

M,1 + C1 x F + C = 0,

Dl.1 - 5 + f l = 0.

(28)

The associated boundary conditions are - Flb = Flb, Mlb = MIL, D.1 = 0, (29)

with given boundary forces F and couples M (see [41]). In this formulation, equations (28) are the classical equations of motion. Boundary forces are not, of course, to be prescribed where geometric boundary conditions are imposed. To these equations we have to add an equation governing the flow of heat.

We have assumed that the body has, at its reference configuration, a homogeneous reference temperature To. Introducing the heat capacity c,, and relying on Newton’s law for the heat flux, we obtain

m , c , A ~ = C AT^^^ - T,U C i i i ,

A T ( : = k l b .

i i

together with the boundary conditions, e.g.

In order to explicate the system of balance and heat equations, we could use the moving reference frame C i , as it is done in Wempner’s book. However, as we have expressed the deformation of the initial triad ei into the triad Ei in terms of the initial triad, it is most natural to express all the quantities in terms of the triad ei, e.g. F = & F i e i , M = CiMie i ,


This helps us to express the balance equations more explicitly as

If the forces F and couples M are taken to be linear functions of the stresses and strains only, i.e. if we approximate t by s1 e l + sI e2 + s1 e3, and I r, , I by 1, which is justified for a thin beam with small initial curvatures and twist, then we find the approximations

Fi = j j A si dA,

We may then use the stress-strain constitutive equations (23) in (33) to obtain

F~ = E A E ~ ~ - M e l ,

F2 = 2GAE12,

F3 = 2GAE13,

MI = GI? - G ( I - J ) y ,

M2 = - E133E3 - aQ3,

M3 = E I z z E 2 + uO2,

D1 = ETY,l,

5 = G(I - J ) ( y - ?).

Relations (34) constitute a set o

(34)

eight functions of the eight strains E l I . E 1 2 , El 3,172,123,?, y, yql. The components of P and C may be approximated as

Pi = - mo Wi, Ci =. - m 0 4, (35) whereas the average force f l is


In the absence of the average body forcefl and in view of (34), the last equation of (32) reduces to the equation

- G ( I - J ) ( Y - ?) = o or, equivalently, to

G ( I - J ) ? = G(I - J ) y - ( E T y , , ) , , . (37) If the warping strain y.l is neglected and if the beam is initially straight and untwisted, then (37) implies y = S1,,, and, hence, M1 = GJ91.1, or M , = G191,1 if also 4 = 0. If the torsion is considered to be non-uniform to the effect that the warping strain y , i is present, the right-hand side of (37) can be used as an approximation for the torsion if y is still taken to be equal to Note that in this case the torsion is given by a third-order differential expression in the variable 9, (see [21]).

Before writing down the differential equations in some examples of particular interest, we derive an expression for the potential energy. Under the standard assumptions, the free energy per unit volume @(xl, x2, x3) is assumed to be of the following form:

(38) E 2

@(xl. x2, x3) = - E : , + 2G(&i2 + E : ~ ) - aEllAT.

Note that the trace of the strain tensor, which is the coefficient of the temperature difference in (38), is just E , , . The potential energy density u is then given by I[* @Ir.,l dA, (39)

and the potential energy is, therefore,

E l 2 2 8; U = j: { fi E:, + 2 G A ( E f 2 + Ef3) + - 2 2

+ -2; + - f 2 GI +-y:, ET - G(1 - J ) y ? + - ( I G - J ) y 2 2 2 2 2

- uAE, 0 , + r122 k2 e2 + XI,, t3 0 3 dx. I It is interesting to observe that in the case where no warping is present (in particular, if r = 0, J = I , y = ?), the only term representing torsion is f GI?’, whereas if warping is present and y is set equal to 91.1, torsion is represented by 4 G J 9:.1 + f E T 9:,1 (if the beam is initially straight and untwisted). As has been indicated above, it is in this case also possible to take (37) as an approximation for the torsion and to account for torsion through

This is done in [21]. It is obvious that in this case the first approximation, based on the potential energy (40), results in a fourth-order differential equation governing the torsion angle 9,, whereas the second approximation, based on (40) with (41) as the only term to represent torsion, results in a sixth-order differential equation in 9,. In

342 J. E. Lagnese et 01.

this paper we either take warping into account while keeping the body forcef,, to the extent that the approximations just mentioned do not hold, or we neglect the warping entirely.

In conclusion, if we assume our beam to be very thin, with doubly symmetric cross-section, and isotropic, and if we assume that its deformation is moderate (in the sense explained above) as well as that the forces and couples are linear in stresses and strains, then the balance equations (32) together with (34) and the heat equation (30), complemented by the appropriate boundary and initial conditions, constitute a complete set of equations governing the motion of the beam together with its flux of heat. We remark that the first three equations of (32) are determined by the axial and in-plane forces F i , and that the second three equations describe twist ( M , ) about the el-axis and bending M 2 and M 3 in the e1-e3 and el-e2 planes, respectively. The last equation of (32) accounts for warping of the cross-section. It is apparent that the initial curvatures K~ and the twist 7 introduce a linear coupling between stretching, bending, twisting and the temperature. It is also apparent that bending and twisting are non- linearly coupled to shear. Furthermore, under the Euler-Bernoulli-Kirchhoff hypothesis, namely that the cross-section remains perpendicular to the centreline, the shear strains E12, E13 are then equal to zero, implying that F 2 , F 3 are identically zero. If, in addition, the non-linearities within the expressions of the shear strains are neglected, then, under the assumption of an initially straight and untwisted beam, we obtain a decoupling of the twist and the warping from bending, even if we allow for moderate rotations in the bending planes. If we do not neglect these non-linear terms, we obtain a high-order non-linear coupling of twist and bending.

We prefer not to write down the complete set of resulting differential equations. Rather, we focus on some examples of particular interest, namely:

1. an initially straight and untwisted, non-shearable non-linear thermoelastic 3-d

2. an initially curved and twisted, shearable linear 2-d beam 3. a planar non-linear isothermal Bresse beam.

beam

3. Initially straight and untwisted, non-shearable non-linear thermoelastic 3-d beams

As we have only one spatial variable (.xl) to take into account, we will consequently replace the subscript ,1 with a prime in order to indicate spatial derivation. We set t i 2 = t i 3 = 7 = 0. By the Euler-Bernoulli-Kirchhoffhypothesis, E12 = E13 = 0. In addition, we neglect the non-linearities in the shear strains. This then implies e 1 2 = el3 = 0 and, hence,

8 3 = w;, c2 = w;, c3 = w;, r ' = 9;.

~ 1 1 = W; + +(( W;)2 + ( w;)'),

92 = - w;,

The reference strains are

~ 1 2 = ~ 1 3 = 0.

As for the body forces and couples, we find P i = - m o A W i , i = 1,2,3,

C, = - m o I & , c2 = m o ~ 3 3 it.;, c3 = - mo ii2.


We make use of the approximations (34) and insert the relations above into the balance laws (32). Then we differentiate with respect to x 1 the second and third of the equations representing the balance of angular momentum, and insert the second and third equation of the balance of forces, respectively. The result is a set of five equations governing the longitudinal displacement W,, the two lateral displacements Wz, W3, the twist 81 and the warping 7. This set has to be complemented by the equations governing the temperatures ei, and by boundary and initial conditions.

3.1. Equations qj’ motion

[Longitudinal motion]

~ ~ o / l b ~ = [ E A ( W; + f ( W;)2 + +( - ~ 6 ; .

[Vertical motion]

,noAii / , - [mo I z 2 b;]’ + Wi]” + a $ ; = [ ( E A ( Wl + f ( Wz)2

+ +( W3y) - a $ , ) W2]’.

[Lateral motion]

m o A b 3 - [mo 1 3 ) I?\]’ + Wi]” + NO; = [ ( E A ( W; + $( W Z ) ~

+ +(w;)z) - Wi]‘.

[Torsional motion]

i n o f j1 = [ G I 8; - G ( I - J)7]’.

[Warping]

m o r 7 = [EI -~ ] ‘ - G ( f - J ) ( Y - 8 ; ) .

The equation of heat conduction reduces to

m o c , . { e , + dZx2 + e 3 x 3 ) = 6“ + xzO; + ~ 3 6 ; - Toa’,kll - x Z i 2 - . x 3 i 3 ) . .

Upon taking the average and the first moments with respect to the .xi axis, we obtain a set of three 1-d heat equations for the variables 6i as follows:

[Heat conduction]

ntoc,Oz = 6; + rT, W;,

moc, ,e3 = 6; + U T ~ W ; .

(47)

These equations have to be complemented by appropriate boundary conditions. As a typical example, we choose a beam which is clamped and thermally insulated at the boundary x = 0, and to which bending moments and shear forces as well as heat are applied at the end .x = 1.

344 J. E. Lagnese et a/.

3.2. Boundary conditions

[Geometric boundary conditions]

Wi(0) = 0, i = 1,2,3,

w; (0) = W3 (0) = 0,

$1 (0) = 0, y(0) = 0.

[Insulated boundary conditions]

e;(O) = 0, i = 1,2, 3.

E A ( w; + tcc w2 + ( W;)V ( I ) - .el(/) =L - E l 3 3 W;l(l) - d , ( l ) = m 2 ,

E l z z W;(/) + a&(/) = m 3 , (49)

- [ E l 3 3 W;l]’(l) - ~ r O ; ( l ) + [ E A ( W; + +(( W;)’ + ( W3)2))

- ~ e , ] w;(I) + r n o ~ 3 3 ik3(/) =fi, + [El2; Wi]’(l) + ctfI;(l) - [ E A ( Wl + +(( W2)2 + ( W ; ) 2 ) )

- c t ~ , ] w;(/) - r n o l z z ti.;(/) = f 3 ,

[Dynamical boundary conditions]

G I $ ; ( / ) = m , , y ’ ( l ) = 0.

[Heat flux boundary conditions]

e;(U = ( P I 9

= (P2,

= cp3.

3.3. Initial conditions

wi(t = 0;) = wi0(-), & ( t = 0;) = 0,

) I ( f = 0, .) = 0,

O,(f = 0;) = O i O ( ’ ) ,

ct;,(t = 0;) = wil(*), $ , ( t = 0;) = 0,

j ( t = 0;) = 0,

2 = 1,2, 3.

As the body is supposed to be initially straight and untwisted, we have set the torsion and warping distributed along the beam at the time t = 0 to be equal zero. A twist can be exerted, however, at the free end of beam (i.e. through m, ).

If the non-linearities within the expressions of the shear strains are not neglected, then the Euler-Bernoulli-Kirchhoff hypothesis implies only that

w;-83=+8,(9,- W;) ,

9 2 + W ’ 3 = + 8 1 ( W ; + 8 3 ) .

Networks of Thin Thermoclastic Beams 345

While we still have F 2 = F 3 = 0, the moments M 2 , M 3 contain a coupling with the torsion LJ1, which results in high-order non-linear differential expressions coupling bending and torsion.

In the foregoing, we have assumed that the material is homogeneous, to the effect that the mass, the Young and shear moduli, as well as the thermal coefficient u and the reference temperature To can be treated as constant. The cross-section, however, is not constant in xl, and so neither are the moments Ii. It is clear that the general inhomogeneous material can be included without any extra effort, as long as we have isotropy. I t is important to note that the isothermal homogeneous process (42)-(46) with Oi set equal to zero conserves the total energy U + T, where in this case

GJ 1 1 2 2 2

+ - 9;" + -El-(?')' + - G ( I - J ) (9; - 7)'

The temperature-dependent system is easily seen to be dissipative. In fact, in the temperature-dependent case, we may work with T as in (52), but with U in (52) replaced with

The reason for this is the fact that the energy (40) is not purely quadratic as it contains terms mixed in strains and temperatures. This energy, however, is easily seen to be bounded above by some constant times the potential energy just defined. Therefore, dissipativity with respect to the new 'energy' is transferred to the physical energy. If only planar isothermal motion is considered, where we may look at the 1-2 or 1-3 plane (with = 0), the system above reduces to the one we have discussed in [24], which, in turn, reduces to a one-dimensional analogue to the von Karman plate if the inertia of the longitudinal motion is replaced with a constant load. If, finally, we neglect the rotational inertia = 133 = 0), the model reduces to an Euler- Bernoulli-type system.

3.1. Approximation-generalizations

When the dimensions in the x2, x3 directions are very small, the heat conduction with respect to these directions may be represented by

1 moc,.8, = 07 - uTo Wl + 2(( W2)' + ( W3)2) at

mocve2 = - p 2 e 2 + X T ~ W i ,

moc, ,83 = - 113e3 + C ~ T ~ W;.

(53)


Then the second and the third equation can be solved by variation of parameters to give

If we use this in (42)-(45) and neglect warping, we obtain:


mo A ii, = L E A ( W; + f( w ; ) ~ + f ( w ~ ) ~ ) ] ’ - ae;

[Vertical motion]

rnoAk2 - ~ 1 ~ 1 6 ’ ; + W;]” + a [’ a2( t - s) w;”(s)ds

[Lateral motion]

[Torsional motion]

mol$, = [GI 9;]’.

(54)

Thus, the heat equation we are left with is

1 at

moc,d1 = 0; - aTo W; + $( W2); + ( W3)2)

Equations (54)-(57) are now integro-partial differential equations, describing a thin thermoelastic beam in three dimensions. Upon replacing the convolution kernels e-’“ by more general functions, for instance, completely monotone kernels, we can refine the assumptions made in order to arrive at (53). Also, this could be viewed as a particular kind of a viscoelastic, in fact, thermoviscoelastic beam model. Further- more, instead of the simplifying assumption that the original heat equations (47) are replaced with (53), we could replace Fourier’s law with a more general heat flux law accounting for a long memory of the material, and end up with an even more general thermoviscoelastic model. If then kernels are chosen to approximate the Dirac distribution, we arrive at what has come to be known as the slow flow approximation, or the Kelvin-Voigt viscoelastic damping. There are other models for internal damping, in particular, the so-called structural damping of Chen and Russell [ 161 or, as an almost equivalent analogue to that, the shear diffusion damping model described in [32], which can be introduced here. It is also worthwhile noting that upon keeping non-linear terms in the representation of the forces F and moments M, one can derive beam models which account more accurately for the effect of microrotations as in the Cosserat continuum.

Networks of Thin Thennoelastic Beams 347

4. Nonlinear planar, shearable thermoelastic beams with initial curvature

Here we set W 2 , r, x2, Q3, Q1 equal to zero. Since there is no torsion, obviously neither is there warping. We have

k2 =o , k3 = - $2, i = 0,

E l 1 = W ; - ~3 W j + $( 8 2 - W’3 - ~3 W1)2,

E l 2 = 0, El3 = $(82 + w3 + k‘3 wl),

0 2 = - w3 - K 3 w1,03 = 0.

F, M are given by (34), and C, P reduce to

Pi = - ??lo wi, C 2 = - m0133 Q2, C1 = C 3 = 0.

We obtain the following system of equations:

4.1. Equations of motion


[Vertical motion]

mohk’3 = CGh(Q2 + W3 + ~3 W , ) ] ’ + ~ 3 [ E h ( W; - ~3 W3

-k i( 8 2 - w; - K3 w ~ ) ~ ) - c&].

[Shear motion]

r n o ~ 3 3 lii, = ~1~~ 9;’ + ( W ; + K~ w l ) W; - K~ w3 + Q( 82 - w; - K~ W1 12) - olel]

- Gh( Q2 + W3 + K~ W , ) - ae;.

[Heat flux]

Again, this set of equations has to be complemented by appropriate boundary and initial conditions. We choose the situation analogous to the preceeding section, where at x = 0 the beam is clamped and insulated, and at x = I it is driven by forces, moments and heat supplies.


4.2. Boundary conditions

[Geometric boundary conditions]

Wi(0) = 0, i = 1, 3, 92(0) = 0.

[Insulated boundary conditions]

Ot(0) = 0, i = 1,3.

[Dynamical boundary conditions]

Eh[ w; - K 3 w3 + &( 92 - w; - K 3 W I ) ~ ] ( 1 ) - Cdi(l) = f1 ,

4.3. Initial conditions

Wi(r = 0;) = Wio( - ) , Wi(t = 0;) = W i , ( - ) , i = 1,3,

$.(t = 0;) = 9.0, 92(t = 0;) = 921,

Oi(t = 0;) = ei,(.), i = 1,3.

From this seemingly complicated system very interesting specializations can be obtained. In particular, the isothermal linear system is exactly the system obtained by Bresse [6] in 1856, which is, in fact, more general than the Timoshenko system (this has been pointed out by Schmidt in [34]); for this reason, we should in future call the latter Bresse-Timoshenko system. We should also mention that the equations of balance for precurved and pretwisted beams have been discussed by Love [27] (also before Timoshenko), who traced the literature back to Kirchhoff in 1859. This once again shows that it might be more appropriate not to attach names to the various beam models to begin with, as many of these equations seem to have been redis- covered repeatedly. Again, let us note that the homogeneous isothermal model conserves the total energy U + T, where now

The temperature-dependent system is then easily seen to be dissipative.

Networks or Thin Thermoelastic Beams 349

4.4. Approximation-Generalizations

and replace the heat flux equations (71) with We apply the same consideration as in the preceding section to the linear equations,

nioc,03 = - p03 - aTo9;, (67) and integrate the second of these equations. Then we obtain the following system of integro-partial differential equations:

[Longitudinal motion] m0 h it., = [ Eh( W ; - ti3 W , ) - a0, 1‘ - ti3 Gh( 82 + W3 + ti3 W , )

The same remarks as in the preceeding section apply to this system. Systems of this type do not seem to have appeared in the literature. They offer the opportunity of introducing very subtle damping mechanisms into the model.

5. A list of beam models

For easier reference, we represent the beam models obtained so far in form of a compact list. We use the same labelling as in the text. We restrict ourselves to display the equations of motion.

5.1. Initially straight and untwisted, non-shearable non-linear beams


m o A it, = LEA( w; + +( w ~ ) ~ + +( w;)~)]’ - ere;.

mo A it., - [mO lZ2 +2]r + [ E I ~ ~ w;]” + [Vertical motion]

= [ ( E A ( w; + +( W2)2 + $( W3)2) - at),) W2]’.

[Lateral motion]

m o ~ k 3 - [ n t o ~ 3 3 e31t + [ E I ~ ~ W;]” +a@<

= [ ( E A ( W; + +( W ; ) 2 + +( W3)2) - at),) W3]’.

350 J. E. Lagnese el al.

[Torsional motion]

moI& = [GI9; - G(I - J)?]’.

[Warping]

morjj = [ E T y ’ ] ’ - G(I - J ) ( y - 8; ) .

[Heat conduction]

1 at a {

moc,.& = e; + U T , k;, m, c,, 8, = 6; + U T , k;.

moc,,B1 = 6: - clTo - W ; + 2 (( W;)’ + ( W ; ) 2 )

5.2. Planar isothermal approximation to model 5.1 (see [24])


m o A k l = L E A ( W; + +( w~)~) ] ’ .

[Lateral motion]

m, ~ k 3 - [ m o ~ 3 3 F31t + [ E I ~ ~ w;]” = [ ( E A ( W, + +( w;)*)) w3]‘ 5.3 . A reduction of model 5.2 to tmnsoerse motion

We replace the inertial force with respect to the longitudinal motion by a loadf,, integrate the first equation and insert the result into the second equation of 5.2. We define F = - S L f , (s) ds. In the two-dimensional analogue, this procedure leads to the Airy stress function and, hence, to the von Karman system. In the one-dimensional case we just get an axial force:

n 1 o ~ k 3 - [ m o l 3 3 k3lJ + [ E I ~ ~ W Y I ” = [ F W ~ ] ‘ .

5.4. The Rayleigh beam model

We neglect the longitudinal motion entirely:

m o ~ W 3 - [ t n , ~ , ~ k31t + [ E I ~ ~ w;]” = 0.

5.5. The Euler-Bernoulli beam model

We neglect the rotatory inertia of the cross-sections:

m o ~ k 3 + ~ ~ 1 3 3 w;]” = 0.

5.6. Non-linear planar shearable beams with initial curvature


m o h k l = W; - tc3 W , + - w3 - K~ w , ) ~ ) - Uel]t - K 3 Gh( $2 + W3 + K~ W , ) + C L [ K ~ ~ ~ ] ’ .


[Vertical motion]

ntohk'3 = [Gh(92 + W3 + ~3 Wi)]'

+K3[Eh(W1 - h ' 3 W 3 + Q ( & - w3-K3W1)2)-cle1].

[Shear motion]

m o 133 8 2 = El33 8'; + ( w; + h'3 w, ) x [Eh( W; - ~3 W3

+ +(92 - w3 - K 3 w,y) - ore,] - Gh( Q2 + W3 + K~ W,) - ae;.

[Heat flux]

1 at a { 8

m o c v i l = 0; - c~T,- W; - K 3 w3 + - ( Q ~ - W; - K~ w , ) ~

~ 1 0 ~ ~ 6 3 = e;, - a~~ 9;.

5.1. The Bresse systern


mo h it., = [Eh( W; - ti3 W3)]' - K~ Gh( $2 + W; + ti3 W,).

[Vertical motion]

mohk'3 = [Gh(92 + W; + t i3 W,)]' + K3Eh[ W ; - ~3 W3].

[Shear motion]

mol33 $2 = El33 9'; - Gh( 82 + W3 + K 3 W, ).

5.8. The Tinioshenko system

[Vertical motion]

inohii/3 = [ G h ( $2 + W ; ) ] ' .

[Shear motion]

in0133 j2 = E l 3 3 8;' - Gh( LJ2 + W3).

Damping. Damping can be introduced in all the models as outlined in the text. Possible sources of (internal) damping are:

(a) thermal damping

(b) viscous damping

(c) viscoelastic damping

(d) thermoviscoelastic damping

(e) structural or fractional power-type damping.

352 J. E. Lagnese et a1

6. Networks of beams

Once we have achieved a sufficiently rich theory for a single beam, we may proceed to develop models of networks of such beams. This is the main purpose of this part of the paper. As with the reference arc in the case of a single beam, we have now a connected graph, consisting of a set of smooth open arcs which are connected through multiple nodes, labelled with N , and which may or may not end at a simple node. This network of arcs serves then as the reference net. It would be possible to impose a static prestretching on this net. Then additional equilibrium conditions would have to be taken into account. This has been carried out for a network of strings in [33], and for Euler-Bernoulli beams in [28]. It turns out that a prestretching leads to a coupling between axial strains and bending. For simplicity, we assume that no such prestretching is present here. In the case of strings and Euler-Bernoulli beams it is sufficient to consider the centreline only, which makes the theory much simpler. In all the other cases, where we deal with shear, a joint (multiple node) between two or more beams is considerably more complicated to model. However, apart from the multiple nodes, each beam satisfies the requirements of the preceeding sections, in particular, the balance laws. This implies that each beam, individually, satisfies the equations of motion which we have derived so far. This also is easily seen from the viewpoint of energy principles; the potential and kinetic energy have to be replaced with their sum over all members. Coupling between longitudinal, transversal, lateral and torsional motion occurs at the multiple nodes. It is, for instance, obvious that in the case of a carpenter’s square, the longitudinal displacement of one beam translates to a lateral displacement of the other and vice oerso. Also, a twist of one beam results in a bending moment at the other beam, and uice uersu.

6.1, Geometric joint conditions

6.1.1. Rigid joints. Here we consider two or more beams (of length li) of the nature described in the previous sections, which are supposed to end at a joint. Let d ( N ) denote the set of indices i such that the ith beam meets at the node N . When talking about a connected structure at all, the first condition which has to be satisfied at any joint, regardless of its particular nature, is the continuity relation

W’(N) = Wj(N), Vi,j€&‘(N). (71)

In addition, we always assume that the temperature ATi are all the same at each joint.

O i ( N ) = O : ( N ) , Vi . jE&(N) , k = 1,2,3. (72)

In order to specify other features of the joints, we have to distinguish between geometrical requirements and force/moment balances at the joints. Two different types of joints will be investigated: rigid joints and pinned joints. Rigid joints are usually encountered in frames, whereas pinned joints occur in trusses. In this section we discuss rigid joints. A joint will be called rigid if the cross-sections of the beams connected to each other at the node N undergo the same ‘moderate rotation’, characterized by the vector 9, i.e.

$ ( N ) = $ j ( N ) , Vi,jE&(N). (73)


This allows the joint to rotate as a whole and to twist, but the shape of the joint before and after deformation is forced to be same. If there is no shear involved, rigidity of a joint means nothing else than that the tangents to the deformed reference curves span the same angle as the tangents to undeformed reference curves.

It would be of great interest to derive a rnutheinutically precise model of joints that are essentially rigid, but where the angle between two consecutive beam elements is allowed to switch, if the deflection exceeds a certain threshold. Joints of this type are of common occurrence in satellite structures, where arms or solar panels are to be unfolded in space.

It should be noted that flexible joint conditions, such as the ones described here, are usually not reflected in the engineering literature on frames and trusses. The reason for this is that the equations of motions are formulated in a modal representation to begin with. In effect, this means that a joint of two flexible elements is replaced with a joint described by the first two finite elements of the connecting beams, which, of course, behave like rigid bodies. As has been demonstrated by Howard in his recent thesis [20], the transmission of elastic energy in a truss or a frame can only be fully understood if the flexibility of the links is taken into account at the joints. He shows that (and how) bending energy of a truss structure (or a frame) is converted to axial energy, and oice versa, through the joints.

We should also mention the work of Brown [9] on the failure of elastic frames (see also [38]). Their results also strongly indicate the significance of a careful modelling of frames.

Example. In this section we confine ourselves to the illustration of a three-dimensional carpenter's square, a rectangular configuration of two beams. We neglect the warping.

We have two initial triads e;, e l , e ; . i = 1,2. However, in this special situation we easily see that we may choose

2 2 e ; = - e 3 , e: = e:, e: = e , .

This implies that the first joint condition (71) can be written as

w ; = - w:, w : = w;, w:= w:. The temperature conditions (72) are independent of the triads. Accordingly, (73) is then

Although the first set of conditions obviously coincides with intuition, the second set deserves further explanation. We focus on the Euler-Bernoulli framework first. Here we find ( C l j = 0, j = 2,3)

9; = - $:, 9 := $f, 9: = 9:.

8 2 = - (w; + K 3 w, -k TW2),

83 = ( W > + K Z W2 - sW3).

Hence, (73) becomes

8: = - (( wi)' + K : w: - T 2 w:), ( W A Y + IC: W : + T' W: = ( w:)'+ K : W : + T ~ W : ,

(Wi)' + .: w: - TI w: = 8:.

354 J. E. Lagnese el a/.

The interpretation of these conditions is particularily simple if the beams are initially straight ( K : = 7 i = 0). Then it is apparent that a twist 9: of one beam causes a vertical strain - ( W:)’ on the other, whereas the orthorgonality for the tangents is preserved in the 1-3 plane throughout the deformation.

6.1.2. Pinned joints. A joint is called a pinned joint if no bending waves are transmit- ted, i.e. if the bending moment of each member at the joint is equal to zero, or prescribed by a moment applied from the outside. In addition, the sum of all shear forces has to be zero, or equal to a driving force. This means that the members, individually, may perform a moderate rotation about the joint. Joints of this type are typical for truss structures. The geometrical or compatibility conditions are now (71) and (72) but not (73).

6.2. Dynamic joint conditions

6.2.1. Rigid joints. Dynamic joint conditions can be obtained by balance of force and momentum, just as in (32). Then the geometric node conditions (71)-(73) serve as compatibility conditions. The dynamic node conditions can also be obtained via the variational formulation of the boundary-node value problem associated with the structure. Then the test functions have to satisfy the geometric joint conditions (71)-(73), in addition to the boundary conditions at the simple nodes (where only one beams ends). We assume that all multiple nodes (joints) are rigid. We give a discussion only in the case of the linearized dynamics. The non-linear node conditions are far from being completely understood in the general case. They are, however, fully understood in the context of non-linear Euler-Bernoulli beam networks, but for the sake of brevity we do not elaborate on this here.

In order to be able to assign to a typical element an outward pointing normal at a multiple node N, we introduce the notation

- 1 + 1

if R’(N) = Ri(0), if R’(N) = Ri(Ii) .

& i ( N ) : =

In words, this means that if beam # i ends in the node at x = 0, then we assign - 1 to that end, and + 1 otherwise. This is, essentially, the notation used in the paper by Wittenburg [42], where networks of rigid bodies have been discussed. We will give the dynamic node conditions in the following two exemplary cases:

(1) The initially straight untwisted Euler-Bernoulli beam in R3 (2) The initially curved Bresse beam.

In addition, we will then reduce the first class to planar motion, and the second to the Timoshenko system.

6.2.2. Dynamic node conditions for a network of linear, initially straight and untwisted thermoelastic Euler-Bernoulli beams. The geometrical node conditions are (7 1) and (73), with $1 = - ( WS)’, $3 = ( W‘,)’. With the above notation, we obtain the following condition on the sum of the shear forces at N :

C q ( N ) [- ( ( E i l i 3 3 ( W;)”)’ + ctiO<)ei - ((EiIiZ2( W’,)”)’ + aiO\)e’, i

+ ( E , A , ( W i )‘ - a iBi )e i ] ( N ) = F N + mN W‘(N) , (74)


where FN is a force applied to the joint N, and mN is the mass of this joint. The bar on top of an index indicates that all values are the same for all i according to a particular node condition, and that one takes one of those values as a representative. The sum of moment at N satisfies

CEi(N)[GiIi($i)’e: - ( E i ~ i 3 3 ( ~ 3 ) ” + r i 8 \ ) e i i

+ ( E i I i 2 2 ( W ~ ) ” + d i 8 ~ ) e ~ ] ( N ) = M N + I N f i ’ ( N ) , (75) where MN are couples applied to the joint N , which has a moment of inertia IN. In the case of planar motion in the 1-3 plane, conditions (74) and (75) reduce to, respectively,

C E ~ ( N ) [ - ( ( E ~ I ~ ~ ~ ( W ~ ~ ) ) +a’O$)eiJ + ( E i A i ( W ’ , ) ’ -aiO\)e’ , ] (N) i

= F N + mN k ’ ( N ) , (76)

- C E ~ ( N ) [ ~ i 1 i 3 3 ( wi,~’ + ( N ) = M N - I N ( i+(y ( N ) . (77)

One may, however, treat the joints N as mass points, so that IN is taken to be zero. The inertia part in (74) is not to be confused with the rotary inertia of the cross-section which is present in the Rayleigh beam model. One may suppose the joint to be massless if the beams are considered as welded to each other.

Example. In the example of the carpenter’s square, with no forces applied at the joint, which is also assumed to carry no extra mass, condition (74) at N is

i

The reduction to the planar case is obvious, as is the reduction to the isothermal situation.

We remark that the node condition obtained in this case are in accordance with the results of Le Dret [26] if the inertia term in the equation governing the longitudinal motion in the vertical beam is neglected, and if the resulting longitudinal stress is expressed in terms of the weight of the vertical beam. Also, Le Dret allows the vertical beam to move out of the plane, causing a twist of the horizontal beam expressed in terms of the weight and of surface tractions. The corresponding node condition is obtained from our more general ones by neglecting the inertia term in the equation governing the torsional motion and by expressing the twist 9:, which appears in the node conditions, in terms of appropriate forces and couples.

6.2.3. Dynamic node conditions for a planar network of linear initially curved Bresse beams. Here we confine ourselves to the 1-3 plane, and proceed as mentioned above.

356 J. E. Lagnese er a/.

The balance law for the forces at N is

C c i ( ~ ) [ ~ ~ 1 ~ ~ ~ ( 9 1 ; ) 1 - a%\] ( N ) = M~ + I ~ S ~ ( N ) . i

It is obvious that, for zero initial curvature, (80) reduces to

C c i ( N ) ( ( E i h i ( Wi>,- a'8';)e'; + G i A i ( g 2 + ( W \ ) ' ) e ; ) ( N ) = FN + m N W 7 ( N ) ,

while (81) remains the same. These are the dynamic node conditions for the Bresse-Timoshenko system.

Example. Let us again consider the case of a carpenter's square as above. Condition (80) on the sum of forces at N now reads

i

E ~ ~ ~ ( ( W : ) ' - K : W : ) - ~ ' O : = G ~ ~ ~ ( ~ : + ( W : ) ' + K : W : ) ,

E ~ I ~ ~ ( ( w:)' - K: w:) - a 2 e 2 3 - - - G l h l ( 9 ; +(I+';)' + K: W i ) . (82) Condition (81) is already independent of the triads el, e2 and remains unchanged. From this the corresponding conditions in the isothermal situation, and also in the uncurved case, are easily obtained.

6.2.4. Pinned joints. As mentioned above, in the case of pinned joints we retain (71) and (72), but we do not impose (73) as geometrical/compatibility conditions. Also, for the Euler-Bernoulli system, the sum of shear forces at the node has to satisfy (74) or (76) in the 3-d or 2-d case, respectively. The conditions on the sum of the moments, however, is replaced by the much simpler condition.

[ ( & I i J 3 ( W\)" + a'Oj)e\ - ( & I i z 2 ( Wi)" + ai0i )e\ ] ( N ) = M i

[ ( E ~ I ~ ~ ~ ( w;y + aie;)l ( N ) = M L

(83)

(84) in the 2-d case. As for the Bresse-Timoshenko system we obtain the dynamic node conditions (80) together with

(85)

in the 3-d case, or

[EiIi33(92)) - d O \ ) ] ( N ) = M i .

Acknowledgements

J. E. L. was supported by the Air Force Office of Scientific Research through grant AFOSR 88-0337, G. L. by the Deutsche Forschungsgemeinschaft (DFG). Heisenbergreferat L-595-3-1, and E. J. P. G. S. by the Natural Science and Engineering Research Council Grant A7271.

Networks of Thin Thennoelastic Beams 357

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network of thin thermoelastic beams

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