Article published by EDP Sciences and available at or

ESAIM: Proceedings Elasticite, Viscoelasticite et Contr^ole Optimal Huitiemes Entretiens du Centre Jacques Cartier URL: http://www.emath.fr/proc/vol.2/ ESAIM: Proc, Vol. 2, 1997, 145{152 A BENDING AND STRETCHING ASYMPTOTIC THEORY FOR GENERAL ELASTIC SHALLOW ARCHES JOSE A. ALVARE-DIOS AND JUAN M. VIA ~NO Abstract. We present a bending model for a shallow arch, namely the type of curved rod where the curvature is of the order of the diameter of the cross section. The model is deduced in a rigorous mathematical way from classical tridimensional linear elasticity theory via asymptotic techniques, by taking the limit on a suitable re-scaled formulation of that problem as the diameter of the cross section tends to zero. This model is valid for general cases of applied forces and material, and it allows us to calculate displacements, axial stresses, bending moments and shear forces. The equations present a more general form than in the classical Bernoulli-Navier bending theory for straight slender rods, so that exures and extensions are proved to be coupled in the most general case. Key words: elasticity, shallow arches, asymptotic methods, curved rods Mathematics subject classication: 73K5, 73C2, 35C2, 35Q72 1. Introduction Strictly speaking, we call a rod any three-dimensional solid occupying the volume generated by a planar connected domain when its mass center is dragged orthogonally along a space curve (the axis or centerline of the rod). The plane domain is called the cross section of the rod. Its essential characteristic is the fact that the diameter of the cross section should be much smaller than the length of the centerline (a ratio of 1 : 1 or less is admitted). We shall say the rod is straight whenever its axis is a straight line segment a straight rod with constant cross section is usually called a prismatic rod: otherwise we shall just talk about curved rods. In this work we are concerned with a particular class of curved rods. More to the point, we call weakly curved rods or shallow arches those characterized by the fact that the curvature of their centerline should have the same order of magnitude than the diameter of the cross section, both being much smaller than their length. A more precise denition will be given later on in the work. The results contained in this work were already announced in [4, 5]. More precisely, here we undertake the aim of obtaining an asymptotic approximation to the tridimensional linear elasticity model,for weakly curved rods. This work is then a continuation and extension of previous results by [2,3] on the asymptotic approximation of slender straight linear elastic rods. Below we summarize up the utilized technique: (a) By means of a change of variable and a change of scale of displacements and stresses, the linearized three-dimensional elasticity problem posed in the curved beam is converted into an equivalent re-scaled problem posed in a straight reference beam (see[2, 3]). (b) Then we aim to study the limit behaviour of unknowns (displacements and stress) of the re-scaled problem, as the diameter of the cross section goes Article published by EDP Sciences and available at http://www.edpsciences.org/proc or http://dx.doi.org/1.151/proc:19971

146 JOSE A.ALVARE-DIOS AND JUAN M. VIA ~ NO to zero. Thus wecharacterize a limit problem which yields the approximated model in the reference domain. It is at this stage that the weak curvature assumption plays a crucial role. (c) By undoing the change of variable in the limit model, we obtain the corresponding model for the actual curved rod. This procedure has already been used in earlier works, most notably in [6, 7], where this method leads to the mathematical justication of the twodimensional equations of a linear elastic shallow shell. 2. Posing the three-dimensional problem Greek indices will take values in the set f1 2g, whereas Latin indices will take values in f1 2 3g. The summation convention on repeated indices will also be used. Let be an open connected bounded set on plane Ox 1 x 2,having area equal to A and a Lipschitzian boundary. We shall suppose that Ox 1 x 2 is a principal system of inertia for. For all " such that <" 1andforany given L>(" L) we dene: " = " " = @ " " = " ( L) (1) ; " = " fg ; " L = " flg ; " = " ( L) (2) and wenote@ " = @=@x ", @ " 3 = @ 3 = @=@x 3 and for any function depending only on x 3 we shall note by,,, ::: the corresponding derivatives of. We shall leave out superscript " when " =1: = 1 ;=; 1 ; =; 1 ::: A generic point in " will be noted by x " =(x " 1 x" 2 x 3). For every " we dene a curve C " in the space Ox " 1 x" 2 x 3 of the form C " = f " (x 3 )=( " 1(x 3 ) " 2(x 3 ) x 3 ) 2 IR 3 : x 3 2 [ L]g (3) where " (x 3) are given functions verifying " 2 C3 [ L]. Let (t " n " b " ) be the Frenet trihedron associated to the curve C " ([1]). From now onwe suppose that n " belongs to C 1 [ L], so that the curve C " is smooth. This hypothesis is satised if " 1 and " 2 do not vanish at the same time (which is equivalent to the fact that the curvature of C " be strictly positive forany x 3 2 [ L]). The case where C " has null curvature points (e. g. a straight chunk) can be treated in the same fashion, provided that we suppose that along these points we have the same degree of smoothness as before, with t " n " b " appropriately chosen. We dene the map ' " : " ; ' " ( " )= ^ " IR 3 in the following manner: ' " (x " )=( " 1(x 3 ) " 2(x 3 ) x 3 )+x " 1n " (x 3 )+x " 2b " (x 3 ) (4) and we assume that ' " is a C 1 -dieomorphism (which can be proved if " is small enough for functions " herein considered). The body having f^ " g ; as its reference conguration is called a curved rod of axis C ". Let us bear in mind that the area of the cross section is of the order " 2. A generic point of f^ " g ; will be denoted by ^x " =(^x " i )='" (x " ), and we set ^@ " i = @=@^x ". i Rod ^ " is considered to be made of an elastic material having Young's modulus E " and Poisson's ratio ". The rod is acted on by volume forces ^f " 2 [L 2 (^ " )] 3 and surface ones ^g " 2 [L 2 (^; " )] 3. Both ends ^; " = '" (; " ), ^; " = L '" (; " ) are supposed to be clamped. Other limit conditions could have L been tackled with a similar approach. Denoting by ^u " =(^u "):f^ " i g ; ;

A BENDING AND STRETCHING ASYMPTOTIC THEORY 147 IR 3 the displacement eld and by ^ " =(^ " ):f^ " ij g ; ; IR 9 s the stress eld, we have the following mixed variational problem in linear elasticity: ^ " 2 (^ " ):=f^ " =(^ " ij ) 2 [L2 (^ " )] 9 : ^ " ij =^ " ji g (5) ^u " 2 V (^ " ):=f^v " =(^v " i ) 2 [H 1 (^ " )] 3 : ^v " =on^; " [ ^; " Lg (6) f 1+" ^ " ^" E " ij ; " E " ^" pp ij g^ " ij d^x " = ^e " ij(^u " )^ " ^ " ij d^x " for all ^ " 2 (^ " (7) ) ^ " ^ " ij ^e" ij (^v" ) d^x " " = ^f ^ " i ^v" i d^x" + ^g " ^; " i ^v" i d^a" for all ^v " 2 V (^ " ) (8) where are the linear stress tensor components. ^e " ij(^u " )= 1 2 ( ^@ " i ^u " j + ^@ " j ^u " i ) (9) 3. Equivalent formulation to the three-dimensional problem on the reference open set In order to study the behaviour of (^ " ^u " ) when the area of the cross section becomes small, we follow the same trail as in the asymptotic analysis of straight rods ([2, 3]) and the analogous work on shallow shells ([6, 7]): changing the variable to a xed open set, scaling of unknowns, assuming certain hypotheses on forces and passing to the limit. For all x =(x i ) 2 we setx " = " (x) =("x 1 "x 2 x 3 ) 2 " et ^x " = ' " (x " )=' " ( " x) and we dene the scaled displacement and stresses u(") =(u i (")) : ; IR 3 et (") =( ij (")) : ; IR 9 s by the following equations: u (")(x)="^u " (^x " ) u 3 (")(x)=^u " 3(^x " ) (")(x) =" ;2^ " (^x" ) 3 (")(x) =" ;1^ " 3 (^x" ) 33 (")(x) =^ 33(^x " " (1) ): We now suppose that there exist constants E, not depending on ", functions f i 2 L 2 (), g i 2 L 2 (;) and 2 C 3 [ L] independent of ", and an arbitrary real number t, such that: E " = " t E " = " t (2) ^f (^x " " )=" t+1 f (x) ^f 3(^x " " )=" t f 3 (x) for all ^x " = ' " ( " x) x2 (3) ^g " (^x " )=" t+2 g (x) ^g " 3(^x " )=" t+1 g 3 (x) for all ^x " = ' " ( " x) x2 ; (4) " (x 3 )=" (x 3 ) for all x 3 2 [ L]: (5) Hypothesis (5) implies that the curvature of C " is of the order ", which can be considered to be a denition of a weakly curved rod. With (1){(5), problem (5){(8) changes into a problem posed in the xed domain of the following form: (") 2 () u(") 2 V () a ((") )+" 2 a 2 ((") )+" 4 a 4 ((") )+b( u(")) + a # (" )((") )+b # (" )( u(")) = for all 2 () (6) b((") v)+b # (" )((") v)=f (v)+f # (" )(v) for all v 2 V ()

148 JOSE A.ALVARE-DIOS AND JUAN M. VIA ~ NO where the bilinear forms a ( ), a 2 ( ), a 4 ( ) and the linear form F () are given by 1 a( ) = E 33 33 dx (7) 2(1 + ) a 2 ( ) = 3 3 ; E E ( 33 + 33 ) dx (8) 1+ a 4 ( ) = E ; E dx (9) b( v) = ; ij e ij (")(v) dx (1) F (v) = ; f i v i dx ; g i v i da: (11) ; and forms a # (" )( ), b # (" )( ) and F # (" )() represent \remainders" of order 2 with respect to " ([4]). 4. Convergence when " tends to zero: first order general model The main result in this work is contained in the following theorem, the proof of which can be found in [4]. We introduce the following notations, depending on the geometric characteristics of the cross section and the centerline, supposing for simplicity that 1 2 (x 3) 6=, for all x 3 2 [ L]: b (x 3 )=(x 3 )=[(1(x 3 )) 2 +(2(x 3 )) 2 ] 1=2 (1) b 1(x) =x 1 b 1 (x 3 ) ; x 2 b 2 (x 3 ) b 2(x) =x 1 b 2 (x 3 )+x 2 b 1 (x 3 ): (2) Theorem 1. Under hypotheses (1){(5), we have the following results: i) When " tends to zero, families (u(")) "> and ((")) "> satisfy u i (") u i in H 1 () 33 (") 33 in L2 () (3) " 3 (") in L 2 () " 2 (") in L 2 (): (4) ii) Element u = lim " u(") is a generalized Bernoulli-Navier eld, namely u is of the form u (x) = (x 3 ) u 3(x) = 3 (x 3 ) ; b (x) (x 3) 2 H 2 ( L) 3 2 H 1 ( L): (5) iii) Element 33 2 L2 () is of the form 33 = E( 3 ; b + ): (6) iv) Element ( 1 2 3 ) 2 [H 2( L)]2 H 1 ( L) is the only solution to the following variational problem in ( L): ; L L m b dx 3 + n 3 3 dx 3 = L L n 3 dx 3 = L F dx 3 ; L M b dx 3 for all ( ) 2 [H 2 ( L)]2 (7) F 3 3 dx 3 for all 3 2 H 1 ( L)

where A BENDING AND STRETCHING ASYMPTOTIC THEORY 149 m b = b 33 d = ;EI b I b = b b d (8) n 3 = 33 d = EA( 3 + ) (9) F i = f i d + g i d M b = b f 3 d + b g 3 d: (1) v) For the bending moment components m b (") and the shear force components q (") when " tends to zero we have: m b (") := b 33(") d ; m b in L2 ( L) (11) q (") := 3 (") d*q = m b + n3 + M b in L 2 ( L) weakly: (12) Sketch of the proof: We begin by establishing that the sequence (u(")) "> is bounded in [H 1 ()] 3, and that sequences ( 33 (")) ">, " ( 3 (")) "> and " 2 ( (")) "> are bounded in L 2 () independently of ". Therefore we can extract weakly convergent subsequences, and we show that these limits satisfy equations (5){(7) passing to the limit in variational equations (6). Then we establish the strong convergence. We follow a similar pattern to the straight rod case ( = ) [2, 3, 5] but several diculties crop up, most remarkably to take remainders a # (" ), b # (" ) andf # (" ) into account, and to obtain the generalized Bernoulli-Navier displacement form and consequently existence and uniqueness of the solution of problem (7). It is then that we resort to the techniques in [6]. Particularly, the following lemma is of the utmost importance: Lemma 2. Let " 2 C 3 ( L IR 3 ) be such that its Frenet trihedron (t " n " b " ) is a positively oriented orthonormal basis of IR 3 and satises Frenet equations for curvature " and torsion " ([1]). If " saties (3.5) then for all "> we have: fa " g a = 1+" 2 s (" ) (a 2 IR arbitrary), (13) " = " c + " 2 s 1 (" ) (14) " = d + " 2 s 2 (" ) (15) n " 1 = b 1 + " 2 s 3 (" ) (16) n " 2 = b 2 + " 2 s 4 (" ) (17) n " 3 = " h 1 + " 2 s 5 (" ) (18) b " 1 = ;b 2 + " 2 s 6 (" ) (19) b " 2 = b 1 + " 2 s 7 (" ) (2) b " 3 = " h 2 + " 2 s 8 (" ) (21) t " 1 = "ft 1 + " 2 s 9 (" )g (22) t " 2 = "ft 2 + " 2 s 1 (" )g (23) t " 3 = t 3 + " 2 s 11 (" ) (24) where s i, i = :: 11 are uniformly bounded constants on ">: sup <" max x 3 2[ L] js i (" )(x 3 )j < +1 (25)

15 JOSE A.ALVARE-DIOS AND JUAN M. VIA ~ NO and also c = q ( 1 )2 +( 2 )2 (26) h 1 = ; 1 1 + 2 2 h 2 = 1 2 ; 2 1 c c (27) t = t 3 =1 (28) d = 1 2 ; 2 1 c 2 : (29) 5. The obtained model Problem (7) is equivalent to the following dierential problem: E(I b ) ; AE[ ( 3 + )] = F + M b in ( L) (3) ;AE[ 3 + ] = F 3 in ( L) i () = i (L) = () = (L) =: The preceding equations amount to being the base of the general rst order model for the curved rod, which is valid regardless of the applied forces, and has been obtained with no a priori hypotheses. The equations are written in terms of the limit quantities on the reference rod in order to obtain the approximate model for ^ " (">) we have to come back to quantities ^u " and ^ " 33 dened in ^ " by(cf.(1)): ^u " (^x" )=" ;1 u (x) ^u" 3 (^x" )=u 3 (x) ^" 33 (^x" )=33 (x): (31) From (5) and (6) we obtain that ^u " and ^ 33 " are of the following form: ^u " (^x" )=^ " (x 3) ^u " 3 (^x" )= ^ " 3 (x 3) ; b " (^x" )^ " (x 3) (32) ^ 33(^x " )=E " [^ 3 " (x 3 ) ; b " (^x " )^ " (x 3 )+ " (x 3 )^ " (x 3 )] (33) b " 1 (^x" )=x " 1b 1 (x 3 ) ; x " 2b 2 (x 3 ) b " 2 (^x" )=x " 1b 2 (x 3 )+x " 2b 1 (x 3 ) (34) where ^ " = ";1, ^ 3 " = 3. Therefore ^ " and ^ " 3 satisfy dierential equations of the same typeas(3): E " (I b " ) ; E " A " [ " (^ " 3 + " ^" )] = F " +(M b " ) in ( L) (35) ;E " A " [^ 3 " + " ^" ] = F 3 " in ( L) (36) ^ " i () = ^ " i (L) =and ^ " () = ^ " (L) =: (37) where F " i = f " " i d " + I b " = b " " " g " i d " M b " = " b " f " 3 d " + b " " g" 3 d " (38) b " (39) We remark that the model here found is indeed a generalization of simpler models found in literature on shallow arch theory. In this way wehave simultaneously justied a priori assumptions of a geometrical nature by showing the \limit displacement"eld ^u " is a Bernoulli-Navier eld in the sense (32). As in the shallow shell case studied in [6], no confusion arises between variables x " 2 " and ^x " 2 ^ ", carefully distinguished throughout the de-scaling process.

A BENDING AND STRETCHING ASYMPTOTIC THEORY 151 6. Some examples Further discussion of shallow arch theorywould involve the study of how well the obtained model ties in with classical theories. However we have not met with the same success as in classical straight rod theories (see [2, 3,4] among others). The long and short of it is that whenever we have tried to look into classical shallow arch theory, we feel the authors make use of simplifying a priori hypotheses wherever they meet lengthy orunwieldy calculations, which makes the whole theory appear rather mystifying from the mathematician's viewpoint, though perhaps not so from the engineer's. Nevertheless we shall verify that our asymptotic curved rod theory is consistent with asymptotic straight rod theory. Thus we shall examine the particular case of a planar centerline. Let us so suppose that 2 " for example, so that C " is a plane curve. Then the simplied model for the curved rod would be written as: with E " I " 1(^ " 1) (4) ; E " A " [ " 1 (^ " 3 + " 1 ^" 1 )] = F " 1 +(M " 1) in ( L) ^ " 1() = ^ " 1(L) = ^ " 1 () = ^ " 1 (L) = (1) ;E " A " [^ " 3 + " 1 ^" 1 ] = F " 3 in ( L) ^ " 3() = ^ " 3(L) = (2) E " I " 2(^ " 2) (4) = F " 2 +(M " 2) in ( L) ^ " 2 () = ^ " 2 (L) = ^ " 2 () = ^ " 2 (L) = (3) I " = (x " ) 2 d M " " = x " f 3 " d + x " g 3 " d: (4) " " Thus exure ^ 2 " is now decoupled from ^ 1 " and stretch ^ 3 ", and the last two are coupled themselves. In the case of a straight rodwewould have 1 "(x" 3 )=a" x " 1,wherea" is a nonzero constant in general, and equals zero if system O^x " 1^x" 2^x" 3 is principal of inertia. In this last case we would have: E " I " 1(^ " 1) (4) = F " 1 +(M " 1) in ( L) ^ " 1 () = ^ " 1 (L) = ^ " 1 () = ^ " 1 (L) = (5) E " I " 2 (^ " 2 )(4) = F " 2 +(M " 2 ) in ( L) ^ " 2() = ^ " 2(L) = ^ " 2 () = ^ " 2 (L) = (6) ;E " A " ^" 3 = F " 3 in ( L) ^ " 3() = ^ " 3(L) =: (7) Consequently we get the classical Bernoulli-Navier exion-extension theory for straight rods (see [2, 3]). 7. Conclusions By taking the limit in the linear three-dimensional elasticity problem, we have obtained a one-dimensional model (a new one, to our knowledge) generalizing classical Bernoulli-Navier theory for straight rods, to the case

152 JOSE A.ALVARE-DIOS AND JUAN M. VIA ~ NO of weakly curved rods, also furnishing a justied denition for the latter: the curvature is of the order of the diameter of the cross section. Acknowledgements This work is part of the project \Shells: Mathematical Modeling and Analysis, Scientic Computing" of the Programme \Human Capital and Mobility" of the E. E. C.(Contract No. ERBCHRXCT94536) and the DGICYT project \Analisis asintotico y simulacion numerica en vigas elasticas" (PB 92-396). References [1] J. J. STOKER, Dierential Geometry (John Wiley & sons, New York), 1989. [2] A. BERM UDE and J. M. VIA ~ NO, Une justication des equations de la thermoelasticite des poutres a section variable par des methodes asymptotiques, R. A. I. R. O., Analyse Numerique 18, 347{376, 1984. [3] L. TRABUCHO and J. M. VIA ~NO, Mathematical modelling of rods, in: Handbook of Numerical Analysis, P. G. Ciarlet and J. L. Lions eds., vol IV (North Holland, Amsterdam), 1996. [4] J. A. ALVARE-DIOS and J. M. VIA ~NO, Mathematical justication of a onedimensional model for general elastic shallow arches (to appear). [5] J. A. ALVARE-DIOS and J. M. VIA ~ NO, On a bending and torsion asymptotic theory for linear nonhomogeneous anisotropic elastic rods, J. Asympt. Anal., 7, 129-158, 1993. [6] P. G. CIARLET and B. MIARA, Justication of the two-dimensional equations of a linearly elastic shallow shell, Comm. Pure Appl. Math., Vol. XLV, 327-36, 1992. [7] P. G. CIARLET and J.-C. PAUMIER, A justication of the Marguerre-von Karman equations, Comput. Mech. 1, 177{22, 1986. Jose A. Alvarez-Dios, Juan M. Via~no: Departamento de Matematica Aplicada, Universidad de Santiago de Compostela, 1576 Santiago de Compostela, Spain.