Asymptotic analysis of Poisson's equation in a thin domain and its application to thin-walled elastic beams and tubes

*Correspondence to: J. M. Vian8 o, Departamento de Matematica Aplicada, Universidad de Santiago deCompostela, 15706 Santiago de Compostela, Spain

CCC 0170—4214/98/030187—40$17.50 Received 12 October 1996( 1998 by B. G. Teubner Stuttgart—John Wiley & Sons Ltd.

Mathematical Methods in the Applied Sciences, Vol. 21, 187—226 (1998)MOS subject classification: 35 J 05, 35 B 40, 73 K 05

Asymptotic Analysis of Poisson’s Equation in aThin Domain and Its Application to Thin-walledElastic Beams and Tubes

Jose M. Rodrıguez1 and Juan M. Vian8 o2, *

1 Departamento de Metodos Matematicos y de Representacion, Universidad de La Corun8 a,Campus da Zapateira s/n. 15071 La Corun8 a, Spain2 Departamento de Matematica Aplicada, Universidad de Santiago de Compostela,15706 Santiago de Compostela, Spain

Communicated by B. Brosowski

We study the limit behaviour of solution of Poisson’s equation in a class of thin two-dimensional domains,both simply connected or single-hollowed, as its thickness becomes very small. The method is based ona transformation of the original problem into another posed on a fixed domain, obtention of a prioriestimates and convergence results when thickness parameter tends to zero. As an important application ofabstract results we obtain the limit expressions for functions appearing in elastic beam theories as torsionand warping functions. In this way, we provide a mathematical justification and a correct definition oftorsion, warping and Timoshenko functions and constants that should be used in the open and closedthin-walled elastic beam theories. ( 1998 by B. G. Teubner Stuttgart—John Wiley & Sons Ltd.

Math. Meth. Appl. Sci., Vol. 21, 187—226 (1998)(No. of Figures: 2 No. of Tables: 2 No. of Refs: 50)

1. Introduction

Let ut\R2 be a domain (open and connected set) with one of its characteristicdimensions (the thickness) of order t'0 which is assumed to be very small whenit is compared with the other one. We call it a thin domain or a ‘profile’. The mostsimple example of such domain is a rectangle of small thickness t, ut"(a, b)](!t/2,t/2). More general typical examples of thin domains ut take the shape of a T, L,H, I, . . . that serve us to define the typical ‘open profiles’ (‘open’ is referred hereto simple connectness). By contrast, the ‘closed profiles’ are domains taking theshape of hollowed thin domains as a single-hollowed circle, a multi-hollowed rec-tangle and so on.

In what follows, we denote by xt"(xt1, xt

2) a generic point of u6 t and by ct the

boundary of ut, which is assumed to be smooth enough. We also note by lt"(lt1, lt

2)

the outward unit normal vector on ct. Differential operators /xta , 2/xta xtb , /lt,...are denoted, respectively, by ta, tab, tl , ... . In the same way, we use the follow-ing notations for the differential operators of Laplace, gradient, divergence androtational:

*ttt"t11

tt#t22

tt , +tut"(t1ut, t

2ut),

divt f t"t1

f t1#t

2f t2, f t"( f t

1, f t

2),

curlt f t"t1f t2!t

2f t1.

Given at"(at1, at

2) and bt"(bt

1, bt

2) in R2 we denote by at · bt the euclidean scalar

product and by D at D the euclidean norm of at . We also denote by D ut D the area(measure) of ut .

Usual norms and seminorms in Sobolev spaces Hm (ut) are noted, respectively, asE · E

m, ut and D · Dm, ut . The same notation is also used for the usual product norms in

product space [Hm (ut)]p.Space H (divt, ut) is defined by

H (divt, ut)"Mf t3[¸2 (ut)]2 : divt f t3¸2 (ut)N,

endowed with the norm

D f t Ddivt,ut"[ D f t D20, ut#Ddivt f t D2

0, u t]2 .

The main goal of this work is to study the behaviour, as t becomes very small, of thesolution tt, /t and gt of the following problems associated to the Poisson’s equation inut with Dirichlet or Neumann boundary conditions:

!*t tt"F t in ut, tt"0 on ct , (1.1)

!*t /t"divt f t in ut, tl/t"!f t · lt on ct, Pu5

/t dxt"0, (1.2)

!*t gt"Gt in ut, tl gt"0 on ct, Put

gt dxt"0, (1.3)

where the right-hand side members are supposed to satisfy

F t3¸2 (ut), G t3¸2 (ut), Put

Gt dxt"0, f t3H (divt, ut ). (1.4)

In this work we restrict ourselves to a particular class of open and closed single-hollowed profiles ut having a smooth centreline defined by a plane curve of class C2

and whose boundary is smooth enough: only four corners are allowed in the openprofile and no corners in the closed profile (see below). We call these domains ‘withoutjunctions’ using the terminology of [11, 19]. For this kind of domains the methodo-logy of analysis of problems (1.1)—(1.3) when tP0 is based on introducing a change ofvariable to a reference domain that transforms problems (1.1)— (1.3) on other ones

188 J. M. Rodrıguez and J. M. Vian8 o

Math. Meth. Appl. Sci., Vol. 21, 187—226 (1998) ( 1998 by B. G. Teubner Stuttgart—John Wiley & Sons Ltd.

which have the thickness t as an explicit small parameter. Then we apply a modifica-tion of classical asymptotic techniques [20, 19] in order to obtain the t-limit problems.

If ut is a domain ‘with junctions’, then it can be described as a union of severaldomains ‘without junctions’, ut"Zp

i/1ut

i, where ut

iWut

jO0 if iOj is allowed and it

is called a ‘junction’. For this kind of domains the same methodology can be adaptedusing the technique suggested by [11, 19] who propose to transform the originalproblem to a reference domain with unitary thickness using a multiple change ofvariable compatible on the ‘junctions’. This case is considered in [29, 33, 34].Application of these results to thin-walled beam theories can be found in [35].

Our interest in this kind of problems comes from elasticity theories of thin-walledbeams, where functions solving problems as (1.1)— (1.3) appear either directly (forexample, torsion and warping functions) or through constants defined by means ofthese functions (torsion, Timoshenko or warping constants). We refer to [46, 26, 50]for an introduction to these models of elastic beams. This interest justifies devoting thesecond part of this paper for obtaining the limit values of torsion, warping andTimoshenko’s functions and constants using the abstract results obtained in the firstpart. In this way we obtain a justification (or new definition) of classical values ofconstants and functions currently used in the most known theories of thin-walledbeams (see [26, 50]).

2. Change of variable

In order to analyse the behaviour of problems (1.1)— (1.3) when tP0 we considertheir variational formulations. Let » t (ut ) be the following space:

» t (ut)"Gut3H1(ut ): Pu5

ut dxt"0H . (2.1)

Problems (1.1)— (1.3) have, respectively, the following variational formulation:

tt3H10

(ut ) ,

Put

+ttt · +tut dxt"Put

Ftutdxt, ∀ ut3H10(ut) , (2.2)

/t3» t (ut) ,

Put

+t/t · +tut dxt"!Pu t

f t · +t utdxt, ∀ ut3» t (ut) , (2.3)

gt3» t (ut ) ,

Put

+tgt · +tutdxt"Put

Gtutdxt, ∀ ut3» t (ut) . (2.4)

In what follows we suppose that u 5 is a profile with a centreline defined bya parametrization:

c : s3[a, b]P c (s)"(c1(s), c

2(s))3R2 (2.5)

Asymptotic Analysis of Poisson’s Equation in a Thin Domain 189

Math. Meth. Appl. Sci., Vol. 21, 187—226 (1998)( 1998 by B. G. Teubner Stuttgart—John Wiley & Sons Ltd.

such that

c3C2 ([a, b], R2), D c@ (s) D*c0'0, for all s3[a, b]. (2.6)

We suppose that c is one-to-one, thus if c (s1)"c (s

2), s

1, s

23 (a, b), then s

1"s

2. Let

T (s)"c@ (s), s3[a, b], be the tangent vector to curve c. We define the unit tangent andnormal vectors to the centerline c as:

m (s)"T (s)/ DT (s) D, n (s)"(!m2(s), m

1(s)). (2.7)

Let u be the reference domain

u"(a, b)](!1/2, 1/2). (2.8)

Then, for 0(t)1, the domains ut considered in this paper are defined by

ut"Ut (u), (2.9)

where Ut : uPR2 is given by

Ut (s, x)"c (s)#txh (s) n (s), (s, x)3u. (2.10)

We assume that function h : [a, b]PR` is known and it satisfies

h3C1 ([a, b]), 0(k1)h (s))k

2, s3[a, b]. (2.11)

Then the thickness of ut in the direction n(s) is given by th(s). We remark thatfor the case of constant thickness we can always assume that h,1. In fact, if h,k(k constant), it is enough to change h by hI ,1 and t by tI"kt.

Next we differentiate the two following cases:(i) ut is an open profile (‘curved rectangle’ with variable thickness without

junctions):

c(a)Oc (b). (2.12)

In this case, for t small enough, Ut is a C1-diffeomorphism.(ii) ut is a closed profile (a ‘closed single-hollowed thin domain’ with variable

thickness and without junctions):

c (a)"c (b), c@ (a)"c@(b). h (a)"h (b). (2.13)

In this case we suppose that n (s) is the unit normal vector outward to the domainenclosed by c. We denote ct

0the ‘outer boundary’ of ut (i.e. ct

0is the boundary of the

unbounded connected component of R2!ut) and ct1

the ‘inner boundary’(ct

1"ct!ct

0).

For sake of brevity, in the next we refer to these domains as open profile (c satisfies(2.12)) an closed profile (c satisfies (2.13)), respectively.

We introduce the following change of variable:

Ut : (s, x)3uPUt (s, x) "(xt1, xt

2)3ut,

xta"Uta (s, x)"ca (s)#txh (s)na (s), (a"1, 2). (2.14)



The Jacobian matrix of this transformation is

JUt (s, x)"AsUt

1(s, x)

xUt

1(s, x)

sUt

2(s, x)

xUt

2(s, x)B

"A¹

1(s)#tx (h@ (s)n

1(s) ¹

2(s)#tx (h@ (s)n

2(s)

#h (s) n@1(s)) #h (s) n@

2(s))

th (s)n1(s) th (s) n

2(s) B (2.15)

Then we have

detJUt (s, x)"td0(t) (s, x), (2.16)

where

d0(t) (s, x)"h (s) [ DT (s) D#txh (s) n@ (s) · m (s)]. (2.17)

We use the following identification:

ut :utPR % u (t) : uPR,

u(t) (s, x)"ut (Ut (s, x))"ut (xt1, xt

2), (s, x)3u . (2.18)

Then we have the following equality, which we shall use frequently:

Put

ut dxt1dxt

2"Pu

td0(t)u (t) ds dx. (2.19)

It is useful to keep in mind the following elementary relations:

su (t) (s, x)"t

1ut (xt

1, xt

2)

s't

1(s, x)#t

2ut (xt

1, xt

2)

s't

2(s, x),

xu(t) (s, x)"t

1ut (xt

1, xt

2)

s't

1(s, x)#t

2ut (xt

1, xt

2)

x't

2(s, x),

t1ut (xt

1, xt

2)"

1

det J Ut (s, x)[

su (t)

x't

2!

xu (t)

s't

2] (s, x), (2.20)

t2ut (xt

1, xt

2)"

!1

det J Ut (s, x)[

su (t)

x't

1!

xu (t)

s't

1] (s, x) .

For ut3¸2 (ut) the corresponding function u (t) belongs to ¸2 (u) and the trans-formation ut3¸2 (ut) % u (t)3¸2 (u) is an isomorphism of vector spaces. Using thechange of variable (s, x)3uP (xt

1, xt

2)"Ut (s, x)3ut we also obtain the following

isomorphisms:

ut3H1 (ut) % u (t)3»1 (u)

ut3H10(ut) % u (t)3»1

0(u) (2.21)

ut3» t (ut) % u (t)3» (t) (u)

where spaces »1 (u) and »10(u) are defined below according to the shape of ut :

(a) ut is an open profile:

»1 (u)"H1 (u), »10(u)"H1

0(u), (2.22)



(b) ut is a closed profile:

»1 (u)"Mu3H1 (u) :u (a. · )"u (b, · ) on [!1/2, 1/2]N,

»10

(u)"Mu3»1 (u) :u ( · , !1/2)"u ( · , 1/2)"0 on [a, b]N . (2.23)

The space » (t) (u) is then given by

» (t) (u)"Gu3»1 (u) : Pu

d0(t)u dsdx"0H . (2.24)

We also introduce the following space:

Hx(u)"Mu3¸2 (u):

xu3¸2 (u),

u ( · ,!1/2)"u ( · , 1/2)"0 on (a, b)N (2.25)

endowed with the following norms:

Du Dx, w

"D xu D

0, u , EuEx, u"[ D u D2

0, u#D xu D2

0, u]1@2 . (2.26)

The space Hx(u) is naturally identified to the space H1

0[(!1/2, 1/2); ¸2 (a, b)] and,

by consequence, we have that the two norms D · Dx, u and E · E

x, u are equivalent onH

x(u) (see [5]).

Let t (t), / (t) and g (t) be the functions obtained from solutions of (2.2)— (2.4)through transformation (2.18). Then, we obtain that t (t), / (t) and g (t) are the uniquesolutions of the following variational problems:

t (t)3»10(u),

t~2 At~2

(t (t), u)#At0(t (t), u)#Bt

0(t (t), u)"¸t

1(u), (2.27)

∀ u3»10

(u),

/ (t)3» (t) (u),

t~2 At~2

(/ (t), u)#At0(/ (t), u)#Bt

0(/ (t), u)"¸t

2(u), (2.28)

∀ u3» (t) (u),

g (t)3» (t) (u),

t~2 At~2

(g (t), u)#At0(g (t), u)#Bt

0(g (t), u)"¸t

3(u), (2.29)

∀ u3» (t) (u),

where we have defined the following bilinear and linear forms:

At~2

(/, u)"Pu

d1

(t)

d0(t)

x/

xu dsdx ,

At0

(/, u)"Pu

h2

d0(t)

s/

su dsdx , (2.30)

Bt0

(/, u)"!Pu

d2

(t)

d0(t)

(s/

xu#

x/

su) dsdx, (2.31)



¸t1

(u)"Pu

d0(t) F (t)udsdx, (2.32)

¸t2(u)"!Pu

M f1(t) [

suhn

2!t~1

xu (¹

2#thx

sn2#th@xn

2)]

!f2(t) [

suhn

1!t~1

xu (¹

1#thx

sn1#th@xn

1)]N dsdx, (2.33)

¸t3(u)"Pu

d0

(t) G (t)u ds dx, (2.34)

with

d1(t) (s, x)"D T (s)#th (s) xn@ (s) D2#t2 [h@ (s)]2 x2 , (2.35)

d2(t) (s, x)"h (s) h@ (s) x. (2.36)

Remark 2.1. When the domain has constant thickness, which means h@,0, we haved2(t) (s, x)"0 and then bilinear form Bt

0(· , · ) does not appear in (2.27)—(2.29).

3. Asymptotic analysis of Dirichlet problem

In this section we shall study the Dirichlet problem (2.27) when tP0. We shall needthe two following lemmas:

Lemma 3.1. If family (F (t))t;0

is bounded in ¸2 (u) independent of t, then solution ofproblem (2.27) satisfies

t (t) P 0 in H1 (u). (3.1)

Proof. Because of ut is bounded and its thickness is of order t, Poincare’s inequality(see [27, Theorem 1.2.5]) gives us

Dut D0,ut)ct D +tut D

0, ut for all ut3H10(ut) (3.2)

where ct is a constant of order t, that is, 0(ct)Ct, C a constant independent of t.Using (3.2) and (2.2) we have

Ett E21, ut"D tt D2

0,ut#D +ttt D20, ut)[1#(ct)2] D +ttt D2

0, ut

"[1#(ct)2] Put

Ftttdxt)[1#(ct )2] DF t D0, ut E tt E

1, ut .

Thus, we obtain:

Ett E1, ut)[1#(ct)2] D F t D

0, ut . (3.3)

Using the change of variable (2.14) and boundness of family F (t) independently of t,we obtain

DF t D20, ut"t Pu

d0(t) [F (t)]2 ds dx)Ct, (3.4)



and then

Ett E1, ut)C Jt . (3.5)

Applying again (2.14) and (2.20) we obtain

t~1 Pu

[xt (t)]2 dsdx#t Pu

[st (t)]2 ds dx)C D tt D2

1, ut . (3.6)

From (3.5)— (3.6) and Poincare’s inequality we deduce

Et (t) E1, u)C. (3.7)

Now we can use (3.7) to extract a subsequence t (m) weakly convergent in H1 (u).Let t3

0be its limit. From (2.27) and (3.7) we deduce

m~2 At~2

(t (m), t (m)))C, (3.8)

and then xt (m) converges strongly to zero in ¸2 (u). Consequently

xtI

0"0. Since

tI03»1

0(u), we have tI

0"0. From uniqueness of limit we deduce that all the sequence

t (t) converges weakly to zero in H1 (u). Now we obtain that "t"¸t1(t (t))!Bt

0(t (t),

t(t)) converges to zero and from (2.27) we have

t~2At~2

(t (t), t(t))#At0(t(t), t(t)) P 0, (3.9)

from which (3.1) is concluded. K

Remark 3.1. Result of Lemma 3.1 can be also obtained applying the technique of [20,19]. We also remark that from (3.9) we obtain t~1

xt (t)P0 strongly in ¸2 (u).

Lemma 3.2. If family (F (t))t;0

is bounded in ¸2 (u) independent of t then there existsC'0 independent of t such that

E t~2t (t) Ex, u)C. (3.10)

Proof. Let us suppose that (3.10) is not true. Then for each n3N there exists tn'0

such that E t~2n

t (tn) E

x, u'n (necessarily tnP0, because of Lemma 3.1). Let us

consider

/n"t~2n

t (tn)

E t~2n

t (tn) E

x, u.

We have /n3»10(u), E /n E

x, u"1. Then, it is possible to extract a subsequence /k

weakly convergent in Hx(u). Let /I be its limit. Having in mind that »1

0(u) is dense in

Hx(u), if we divide (2.27) by E t~2

kt (t

k) E

x, u and we pass to the limit, we obtain thatx/I "0. Like /I 3H

x(u), /I becomes zero in [a, b]]M!1/2, 1/2N and then we have

/3 "0. From uniqueness of the limit we deduce that it is all the sequence /n whichconverges weakly to zero in H

x(u). Now we deduce from (2.27)

t~2n

Atn~2

(t (tn), /n)#At

n

0(t (t

n), /n)"¸

tn

1(/n)!Bt

n

0(t (t

n), /n),



and dividing last equation by E t~2n

t (tn) E

x, u , we obtain

Atn~2

(/n, /n))¸

tn1

(/n)

E t~2n

t (tn) E

x, u

#

2

E t~2n

t (tn) E

x, u Pu

d2

(tn)

d0(tn)st (t

n)

x/n ds dx.

Taking into account (3.1) we obtain that the right-hand side of last inequalityconverges to zero, so

x/nP0 strongly in ¸2 (u). From that we deduce /nP0

strongly in Hx(u), which is contradictory to E/n E

x, u"1. K

We are now able to proof the following theorem about limit Dirichlet problem:

Theorem 3.1. If F (t) P FI in ¸2 (u) when tP0 then

t~1 t (t) P 0 in H1 (u), (3.11)

t~2 t (t) P t3 in Hx (u), (3.12)

where tI is the unique solution of the following variational problem:

tI 3Hx(u),

Pu

xt3

xu

DT Dh

ds dx"Pu

FI uh DT Ddsdx, ∀u3Hx(u). (3.13)

Proof. From Lemma 3.2 we deduce the existence of a subsequence m~2t (m) weaklyconvergent in H

x(u). Let tI be its limit. If we pass to the limit in (2.27) and we have in

mind (3.1) and density of »10(u) in H

x(u), we deduce that tI is the unique solution of

(3.13) (the uniqueness of solution of (3.13) is a consequence of Hx(u)-ellipticity of the

bilinear form A0~2

associated with the limit problem and the Lax—Milgram lemma).From uniqueness of limit all the sequence converges, so it only remains to prove thatthis convergence is strong. With this aim we use (2.27) again and we deduce thefollowing equality:

At~2

(t~2 t (t)!tI , t~2 t (t)!t3 )#At0(t~1t (t), t~1 t (t))

"t~2 At~2

(t (t), t~2t (t))#At0(t (t), t~2t (t))#At

~2(tI , tI !2t~2t(t))

"¸t1(t~2t (t))#At

~2(tI , tI !2t~2t (t))!Bt

0(t~1 t (t), t~1t (t))""t.

(3.14)

Writing

Bt0(t~1t (t), t~1t (t))"!2 Pu

d2

(t)

d0

(t)st(t) (t~2

xt (t)) dsdx, (3.15)

we can see that "t converges to zero if we use (3.1), (3.10), (3.13) and the fact that tI isthe weak limit of t~2t (t) in H

x(u). Then we have proved that the two terms of first

line of equality (3.14) converge to zero, from which we deduce (3.11) and (3.12). K



4. Asymptotic analysis of Neumann’s problem

In this section we shall study problems (2.28) and (2.29) when tP0. As in previoussection we shall begin proving some auxiliary results.

Lemma 4.1. ¹here exists C'0, independent of t, such that

Eu E1, u)C Du D

1, u , ∀ u3» (t) (u) . (4.1)

Proof. Let us suppose that lemma is not true. Then there exists a subsequence,/n3» (t

n) (u), such that E/n E

1, u"1 and D /n D1, u(1/n. By compacity arguments, we

deduce the existence of a subsequence, /nk , strongly convergent in H1 (u) to a con-

stant, c0. As we have E/n E

1, u"1, we obtain E c0E1, u"1 and then c

0"D u D~1@2.

From uniqueness of limit we deduce that it is all the sequence, /n, which convergesto c

0.

As /n3» (tn) (u), we have

Pu

d0(tn) /n dsdx"0, (4.2)

and having in mind that 0(tn)1, we obtain that there exists a subsequence tn

kP t

0,

t0*0. Taking limits in (4.2) as n

kPR, we have the two following possibilities, both

absurd:

(i) t0'0: 0"c

0 Pu

d0(t0) dsdx"

c0

t0Kut

0 K'0,

(ii) t0"0: 0"c

0 Pu

h DT Ddsdx'0. K

Lemma 4.2. If family (f (t))t;0

is bounded in [¸2 (u)]2 independent of t, then there existsC'0, independent of t, such that

E/ (t) E1, u)C. (4.3)

Proof. As we have done to obtain (3.6), we apply the change of variable x"Ut (s, x)and we obtain

t~1 Pu

[x/ (t)]2 dsdx#t Pu

[s/(t)]2 dsdx)C D /t D2

1, ut . (4.4)

From (2.3) we deduce

D/t D1, ut)J D f t

1D 20, ut#D f t

2D20, ut" D f t D

0, ut . (4.5)

Finally, using again change of variable xt"Ut (s, x) as in (3.4), we obtain

D f t D20, ut"D f t

1D20, ut#D f t

2D20, ut)Ct. (4.6)

It is enough to combine (4.4)—(4.6) and (4.1) to obtain (4.3). K



Lemma 4.3. If family (G (t))t;0

is bounded in ¸2 (u) independent of t, then there existsC'0, independent of t, such that

E g (t) E1, u)C. (4.7)

Proof. It is analogous to that of Lemma 4.2, where we obtain instead of (4.4)—(4.6)

t~1 Pu

[xg (t)]2 dsdx#t Pu

[sg (t)]2 ds dx)C D gt D2

1, ut ,

D gt D21, ut)D Gt D

0, ut D gt D0, ut , (4.8)

DGt D20, ut)Ct.

Moreover, we have

D gt D20, ut)Ct D g (t) D2

0, u , (4.9)

so we finish the proof using (4.1). K

We are now able to prove the following three theorems which constitute the centralresults of this article. The limit problems are posed in a subspace of H1 (a, b) noted¼ (a, b) defined as follows:

(i) If ut is an open profile:

¼ (a, b)"Gu3H1 (a, b): Pb

a

uh D T Dds"0H. (4.10)

(ii) If ut is a closed profile:

¼ (a, b)"Gu3H1 (a, b): u (a)"u (b), Pb

a

uh D T Dds"0H . (4.11)

Theorem 4.1. If f (t) P f3 in [¸2 (u)]2 then

/ (t) P /I in H1 (u), (4.12)

t~1 x/ (t) P (!f3 · n) h in ¸2 (u), (4.13)

where /I is the unique solution of the following limit problem:

/3 3¼ (a, b),

Pb

a

s/I

su

h

D T Dds"P

b

aGA!P

1@2

~1@2

fI1

dxB n2

#AP1@2

~1@2

fI2

dxB n1H h

su ds, ∀u3¼ (a, b). (4.14)

Proof. From (4.3) and (2.28) we deduce

D t~1 x/ (t) D

0, u)C. (4.15)



Then, from (4.3) and (4.15) we deduce that there exists a subsequence, still noted / (t)by convenience, satisfying the following weak convergences:

/(t) N /I in H1 (u), (4.16)

t~1x/ (t) N hI in ¸2 (u). (4.17)

For an arbitrary function u3¼ (a, b), let u (t) (s, x)"u(s)#ct where ct is a realconstant such that u(t)3» (t) (u). Taking u (t) as function test in (2.28) and passing tothe limit we obtain that /I is a solution of (4.14). Problem (4.14) has a unique solution(consequence of ¼ (a, b)-ellipticity of associated bilinear form), so the limit is uniqueand then all the sequence (and not a subsequence) converges to /I .

Let us consider now a function like u (t) (s, x)"tu (s, x)#ct, where u3H1 (u) andct3R such that u (t)3» (t) (u). We use it as test function in (2.28) and we obtain in thelimit

Pu GhID T Dh

#( fI2¹

1!fI

1¹

2)H

xu dsdx"0, ∀ u3H1 (u). (4.18)

Let k3¸2 (u) be the function

k"hIDT Dh

#( fI2¹

1!fI

1¹

2). (4.19)

From (4.18) we have, in the distributional sense, xk"0, that is, k does not depend

on x. Taking now in (4.18) u (s, x)"xs (s), with s3H1 (a, b), we obtain

Pb

a

k (s)s (s) ds"0, ∀s3H1 (a, b). (4.20)

Consequently k"0 and from (4.19) we have

hI "(!f3 · n) h. (4.21)

As hI is now determined in a unique way, we deduce that convergence in (4.17) isvalid for the whole sequence and not only for a subsequence. Last step is to prove thestrong convergence of both sequences. We observe that the following inequality isverified:

C G Kt~1 x/ (t)!hI K

2

0, u#K s / (t)!

s/I K

2

0, uH)t~2 At

~2(/ (t), /(t))#At

0(/ (t)!/I , / (t)!/I

#Pu

hI (hI !2t~1 x/ (t))

D T Dh

dsdx""t . (4.22)

Using now (2.28) we deduce that

"t"¸t2(/ (t))!Bt

0(/ (t), / (t))#At

0(/I , /I !2/(t))

#Pu

hI (hI !2t~1 x/ (t))

D T Dh

ds dx. (4.23)



Taking limits in (4.23) we obtain "tP0, if we only have in mind (4.14), (4.16), (4.17)and (4.21). Then from (4.22) and (4.1), we deduce that convergence (4.16) and (4.17) arestrong and the proof is finished. K

The following theorem improves last convergence result for problem (1.2) when ut

is an open profile and Rt"curlt f t3¸2 (ut ).

Theorem 4.2. ¸et ut be an open profile and let us suppose that f t is such that Rt"

curlt f t3¸2 (ut) and R(t) P RI in ¸2 (u) when tP0. ¹hen

(i) If f (t) · T P hIT

in ¸2 (u) and f (t) · n P hInin ¸2 (u), we obtain

/(t) P /I in H1 (u), (4.24)

t~1 x/(t) P!hI

nh in ¸2 (u), (4.25)

where /I is uniquely determined by the following relations:

x/I "0,

s/I "!hI

T, P

b

a

/I h DT D ds"0. (4.26)

(ii) If t~1 f (t) · [T#thxn@#th@xn] P hIsin ¸2(u) and f (t) · nPhI

nin ¸2 (u), then

t~1 / (t) P /I in H1 (u), (4.27)

where /I is uniquely determined by the following relations:

x/I "!h3

nh,

s/I "

xkJ

D T Dh

!hIs, Pu

/I h D T Dds dx"0, (4.28)

and where kJ is the unique solution of the following variational problem:

kJ 3Hx(u),

Pu

xkJ

xu

D T Dh

ds dx"Pu

RI uh DT Dds dx, ∀u3Hx(u). (4.29)

Proof. Let us consider pta"ta/t#f ta (a"1, 2). Then we have divt pt"0 in ¸2 (ut)and pt. lt"0 in H~1@2 (ct). Applying div-curl theorem from [15, Theorem 3.1], wededuce the existence of an unique function kt3H1

0(ut ) such that pt

1"t

2kt,

pt2"!t

1kt. Therefore,

t1kt"!t

2/t!f t

2, t

2kt"t

1/t#f t

1, (4.30)

and

!*t kt"Rt in ut, kt"0 on ct . (4.31)

We now can apply Theorem 3.1 to kt and we obtain

t~1k (t) P0 in H10(u), t~2 k (t) P kJ in H

x(u), (4.32)



where kJ is the unique solution of (4.29). With the change of variable xt"Ut (s, x)(using (2.20)), we obtain from (4.30):

s/ (t)"!t

d2(t)

d0(t)

sk (t)#t~1

d1

(t)

d0(t)

xk (t) (4.33)

!f (t) · [T#thxn@#th@xn],

x/ (t)"!t

h2

d0(t)

sk (t)#t

d2(t)

d0(t)

xk (t)!t (f (t) · n) h. (4.34)

Now using (4.32) in (4.33) and (4.34), with Lemma 4.1 and the fact that/(t)3» (t) (u) we conclude (i) and (ii). K

Remark 4.1. Proof of Theorem 4.2 and relations (4.33)— (4.34) allow us to improve theresults of some particular cases. For example, if we have t~1 f (t) · nPhI

Nin ¸2 (u),

then t~2 x/(t)P!hI

Nh in ¸2 (u).

Remark 4.2. Case (i) of Theorem 4.2 gives us result analogous to Theorem 4.1, butcase (ii) constitutes an important improvement as we shall see later for the warpingfunction in a rectangle (see Remark 6.1).

Remark 4.3. This result has not an analogous if ut is a closed profile, becauseboundary conditions at inner boundary ct

1of the stream function kt associated to

function /t depend on function /t itself (see [15, Theorem 3.1]), so we cannot useTheorem 3.1 as we have done for a simply connected domain.

We shall finish this section analysing problem (1.3), already studied in [19] fora straight rectangular domain with constant thickness from which following proof isadapted.

Theorem 4.3. If G (t) P GI in ¸2 (u) when tP0 then

g (t) P gJ in H1 (u), (4.35)

t~1 xg (t) P 0 in ¸2 (u), (4.36)

where gJ is the unique solution of the following problem:

gJ 3¼ (a, b),

Pb

a

sgJ

su

h

DT Dds"P

b

aAP

1@2

~1@2

GI dxBuh DT Dds, ∀ u3¼ (a, b). (4.37)

Proof. From (4.7) and (2.29) we deduce

D t~1 xg (t) D

0, u)C, (4.38)

and therefore there exists a subsequence, still noted g (t), satisfying following weakconvergences:

g(t) N gJ in H1 (u), t~1 xg (t) N hI in ¸2 (u). (4.39)



As in Theorem 4.1, we take u (t) (s, x)"u (s)#ct as test function in (2.29), whereu3¼ (a, b) and ct3R such that u(t)3»(t) (u). Then, taking limits in (2.29) we obtainthat g8 is the unique solution of (4.37). Finally we consider the inequality

C GD t~1 xg (t) D2

0, u#D sg(t)!

sg8 D2

0, uH)t~2 At

~2(g (t), g(t))#At

0(g (t)!gJ , g (t)!gJ ) :""t . (4.40)

Using (2.29) we deduce the following equality:

"t"¸t3(g (t))!Bt

0(g(t), g(t))#At

0(gJ , gJ !2g (t)). (4.41)

Then "t converges to zero, as we easily conclude if we have in mind (4.37) and (4.38).Using this in (4.40) with Lemma 4.1 we obtain (4.35) and (4.36). K

Remarks 4.4. It is easy to check that proofs included in sections 3 and 4 are still true iffunctions c and h are changed by two families ct and ht convergent to c and h in spacesC2 ([a, b], R2) and C1 ([a, b]), respectively.

5. Typical functions and constants depending on the cross-sectionin elastic beam theories

A prismatic beam of length ¸ and cross-section ut is a solid occupying the volume)t"ut](0, ¸) in the euclidean space Oxt

1xt2x3. The essential geometric property of

a beam is that area of cross-section ut is much smaller than ¸. Mathematical analysisof elasticity model for this type of solids has undergone a considerable development oflate, mainly due to the use of asymptotic method on the three-dimensional modelhaving the area of the cross-section as a small parameter (see [46] for a survey). In thisway the most well-known models in elastic beam theories: Bernoulli—Navier,Saint—Venant, Timoshenko, Vlassov, etc., are justified and, in some cases, generalized.

These models are mainly based on functions and constants (warping, torsion,Timoshenko, area bimoments, etc.) which depend solely on the geometry of thecross-section. The functions are each a solution of Poisson’s equation with differentboundary conditions, posed on ut. We illustrate this fact with the simple model ofSaint-Venant for a cantilevered beam subjected to a torsion moment M applied in thefree end ut]M¸N. In what follows, without loss of generality, we assume that thesystem of axes Oxt

1xt2x3

is a principal system of inertia and therefore the followingproperties are satisfied:

Pu

xt1dxt

1dxt

2"Put

xt2dxt

1dxt

2"Put

xt1xt2dxt

1dxt

2"0. (5.1)

Also, we suppose that the material whose beam )t is made of is an isotropic andhomogeneous with Young’s modulus E and Poisson ratio l. Then the Saint-Venanttheory give us the displacements ut

i(xt

1, xt

2, x

3) (i"1, 2, 3), and stresses pt

ij(xt

1, xt

2, x

3)

(i, j"1, 2, 3), in the following form (see [25, 46]):

ut1(xt

1, xt

2, x

3)"!atxt

2x3,

ut2(xt

1, xt

2, x

3)"atxt

1x3,



ut3(xt

1, xt

2, x

3)"atwt (xt

1, xt

2),

pt31

(xt1, xt

2, x

3)"at t

2tt (xt

1, xt

2), (5.2)

pt32

(xt1, xt

2, x

3)"!at t

1tt (xt

1, xt

2),

ptab (xt1, xt

2, x

3)"pt

33(xt

1, xt

2, x

3)"0 (a, b"1, 2).

In (5.2) at is the following constant

at"2 (1#l)M

EJ t, (5.3)

where Jt is the torsional constant (or torsional rigidity) of ut (see (5.15) below fora precise definition). Functions ut (warping function) and tt (torsion function) areknown from classical theories and they depend only on shape of cross-section ut (see(5.4) and (5.5)).

By applying asymptotic methods in three-dimensional elasticity other functionsand constants of the same kind are introduced by [43—45] in order to describe the firstand second order asymptotic models taking account bending, torsion, Timoshenko,Poisson and Vlassov’s effects for the beam. These functions and constants are listed ina precise way at the end of this section.

When the thickness of the cross-section (which is of the same order as t) is very smallwith respect to the other one (i.e. ut is a thin profile) )t is referred as a thin-walledbeam. This kind of structures is widely used: bridges, hydraulic pipelines, body oraircraft or a rocket, lateral surface of a ship, etc.

The models of elastic beams previously mentioned must be reviewed whenever theyare used for thin-walled elastic beams (with dimensions of the cross-section of differentorder of magnitude). The main reason why these thin-walled structures need anindependent treatment arises from the fact that both the shear deformation and stresscannot be neglected as in the case of a solid cross-section.

A method which is proved to be adapted to obtain the general models for elasticthin-walled beams has been introduced by [34, 35]. It essentially consists in obtainingthe limit behaviour of displacements ut

iand stresses pt

ijwhen thickness t becomes

small. For doing it, it is essential to know the limit behaviour as t becomes small offunctions as wt, tt and so on, which solves the Poisson’s equation in ut. This is thefundamental motivation of this work whose consequences for thin-walled beamtheories are referred in the paper [38]. Similar treatment for thin-walled beams ‘withjunctions’ (i.e. with cross-section of shape on T, L, H, ...) is carried out in [34, 35].

Next we list the functions and constants whose limit behaviour as tP0 is required.As we have already mentioned, some of these functions and constants are alreadyknown from the classical theories, and others have been introduced or redefined by[43, 46]. We recall that Oxt

1xt2x3

is supposed to be a principal system of inertia.

(i) ¼arping function. It is the unique solution of the problem:

!*t wt"0 in ut,

tlwt"xt2lt1!xt

1lt2

on ct , (5.4)

Put

ut dxt1dxt

2"0.



(ii) Saint-»enant’s torsion function or Prandtl ’s potential function. It is the uniquefunction satisfying the following conditions:

t1tt"!t

2wt!xt

1, t

2tt"t

1wt!xt

2, (5.5)

tt"0 on ct0.

Function tt is constant on each connected component of the boundary of ut (see[15]). It is easy to check that if the warping function wt is smooth enough then tt issolution of

!*ttt"2 in ut,

tt"0 on ct0, (5.6)

tlt"!(t2wt#xt

1)lt

1#(t

1wt!xt

2)lt

2on ct!ct

0.

If ut is simply connected then the torsion function is the unique solution of

!*t tt"2 in ut , (5.7)

tt"0 on ct .

(iii) Function gtb (b"1, 2), is the unique solution of

!*tgtb"!2xtb in ut,

tl gtb"0 on ct , (5.8)

Put

gtb dxt1dxt

2"0.

(iv) Function htb (b"1, 2), is the unique solution of

!*t htb"2xtb in ut ,

tl htb"!

2+a/1

'tba lta on c, (5.9)

Put

htb dxt1dxt

2"0,

where functions 'tab, (a, b,"1, 2), are defined by

't11

(xt1, xt

2)"!'t

22(xt

1, xt

2)"[(xt

1)2!(xt

2)2]/2 , (5.10)

't12

(xt1, xt

2)"'t

21(xt

1, xt

2)"xt

1xt2.

(v) Above functions allow us to define the following constants depending only onthe geometry of ut :

Ita"Put

(xta)2 dxt1dxt

2(a"1, 2), (5.11)

Hta"1

2 Put

xta [(xt1)2#(xt

2)2] dxt

1dxt

2(a"1, 2), (5.12)



Ht3"

1

4 Put

[(xt1)2#(xt

2)2]2 dxt

1dxt

2, (5.13)

Iwt

a "2 Put

xta wt dxt1dxt

2(a"1, 2), (5.14)

Jt"!Put

(xt1t1tt#xt

2t2tt) dxt

1dxt

2, (5.15)

ot"It1#It

2!Jt

It1#It

2

, (5.16)

Jtw"Put

(wt )2 dxt1dxt

2, (5.17)

It t

1"!Put

(xt2)2 t

2ttdxt

1dxt

2, Itt

2"Put

(xt1)2 t

1ttdxt

1dxt

2, (5.18)

C0, tw

"Jtw!

(Iwt

1)2

4It1

!

(Iw t

2)2

4It2

, (5.19)

¸g t

ab"Put

xta gtbdxt1dxt

2, ¸

ht

ab"Put

xta htbdxt1dxt

2(a, b"1, 2), (5.20)

Kgt

ab"2+k/1

Put

'tak tk gtbdxt1dxt

2,

Kht

ab"2+k/1

Put

'tak tk htbdxt1dxt

2(a, b"1, 2). (5.21)

Constants Jt and C0, tw

are called torsion constant and warping constant, respectively.If ut is simply connected, using (5.7) the constants J t, Itt

1and Itt

2can be written in the

following easier way:

Jt"2 Put

tt dxt1

dxt2,

Itt

1"2 Put

xt2ttdxt

1dxt

2, Itt

2"!2 Put

xt1ttdxt

1dxt

2. (5.22)

Using variational formulation of problems (5.8) and (5.9) we obtain (see [2, 28,43, 46]):

Kgt

ab"2¸ht

ba , Kht

ab"!Put

+t hta · +thtb dxt1dxt

2(a, b"1, 2). (5.23)

From symmetry properties of the solutions of (5.4)— (5.9) (see [2, 28]) we deduce:

(i) If Oxt1

is an axis of symmetry of ut then

It t

1"Iwt

1"¸

gt

12"¸

gt

21"¸

ht

12"¸

ht

21"Kgt

12"Kgt

21"Ht

2"0. (5.24)



(ii) If Oxt2

is an axis of symmetry of ut then

It t

2"Iut

2"¸

gt

12"¸

gt

21"¸

ht

12"¸

ht

21"Kgt

12"Kgt

21"Ht

1"0. (5.25)

(iii) If Oxt1

and Oxt2

are two axes of symmetry of ut then we have (5.24), and (5.25)and moreover:

K ht

12"Kht

21"0. (5.26)

Moreover, we shall consider two additional constants which depend not only onthe geometry of transversal section but also on the material through Poisson’scoefficient l. They are the generalized warping constant

Cl, tw

"Jtw!

[(1#l) Iwt

1#lIt t

1]2

4(1#l)2 It1

!

[(1#l) Iwt

2#lIt t

2]2

4(1#l)2 It2

, (5.27)

and the ¹imoshenko’s matrix (see [45])

¹ tab"!

1

I tb G(1#l) ¸gt

ab#l¸h t

ab#l

2(1#l)[(1#l) Kg t

ab#lKh t

ab#lHt3dab]

!

1

2(1#l) J t[(1#l) Iwt

a #lIt t

a ] [(1#l) Iw t

b #l It t

b ]H , (a, b"1, 2).

(5.28)

We also introduce the new ¹imoshenko’s constants

kK ta"2(1#l) I taD ut D¹ taa

(a"1, 2). (5.29)

In [45, 46], it is justified that this constant generalizes the classical Timoshenko’sconstant (see [12]).

We are now able to apply results from sections 3 and 4 to the functions andconstants depending on geometry of cross-section ut in order to obtain the asymp-totic values corresponding to thin-walled beams with thickness of order t. Obviously,we must differentiate open and closed profiles, that is, open or closed thin-walledbeams.

6. Convergence results for geometry cross-section functions and constants ofopen thin-walled beams

Next we assume that ut is an open profile defined by (2.10) and we introduce thefollowing functions that will appear in a natural way:

c (s)"1

2 Ps

a

[c@1

(r) c2(r)!c

1(r) c@

2(r)] dr, (6.1)

b (s)"12

[c21

(s)#c22(s)]"1

2Dc (s) D2. (6.2)

Function c is called sectorial area in engineering literature (see [26, 50]).



Proofs also naturally differentiate two cases: c@"0 and c@O0. We remark thatc@"0 if and only if c is a straight line passing through the origin. In fact, equation2c@"c@

1c2!c

1c@2"0 is equivalent to (c

1/c

2)@ (s)"0 if c

2(s)O0 and to (c

2/c

1)@(s)"0

if c1(s)O0. Then there exist two constants r

1and r

2such that r

1c1(s)#r

2c2(s)"0

for all s3[a, b], that is, c is a straight line passing through the origin of coordinatesand ut is a straight rectangle with variable thickness.

In what follows, we also assume that Oxt1xt2

is a principal system of inertia of ut,that is, property (5.1) is satisfied. In this case, if ut is a straight rectangle with variablethickness (c@"0) and we suppose a natural parametrization ( D T (s) D"1), we deducec (s)"(s, 0) or c (s)"(0, s). In what follows, if we find ourselves in this case, we shallassume c(s)"(s, 0). If we also suppose that h is an even function (h (s)"h(!s)) then[a, b]"[!S/2, S/2], where S is the length of the rectangle.

Let us consider functions wt, tt, gtb and htb solution, respectively, of problems (5.4),(5.7)— (5.9). Applying Theorems 3.1, 4.2 and 4.3 to these problems we obtain thefollowing result:

Theorem 6.1. ¸et ut be an open profile defined by (2.10) where Oxt1xt2

is a principalsystem of inertia. ¹hen

(i) ¹orsion function tt, solution of (5.7), satisfies

t~1t (t) P 0 in H10(u), t~2 t (t) P tI in H

x(u), (6.3)

where

tI (s, x)"(14!x2) h2 (s). (6.4)

(ii) ¼arping function wt, solution of (5.4), satisfies

(a) If c@I0

w (t) P wJ in H1 (u), t~1 xw (t) P wJ * in ¸2 (u), (6.5)

where

wJ (s, x)"wJ (s)"2 Cc(s)!:bach DT D ds

:bah D T D ds D , (6.6)

wJ * (s, x)"wJ * (s)"!b@ (s) h (s)/ DT (s) D . (6.7)

(b) If c@,0 (and then c (s)"(s, 0), s3[a, b]):

t~1 w (t) P wJ in H1 (u), (6.8)

where

wJ (s, x)"!sxh (s). (6.9)

(iii) Function gta , solution of (5.8), satisfies

ga (t) P gJ a in H1 (u), t~1 xga (t) P 0 in ¸2 (u) (a"1, 2), (6.10)



where

gJ a (s, x)"gJ a (s)"2 Ps

aCDT (r) Dh (r) P

r

a

ca (p) h (p) D T (p) D dpD dr#Ca (a"1, 2),

(6.11)

being Ca3R such that :ba

gJ a h DT D ds"0.

(iv) Function hta , solution of (5.9), satisfies

ha (t) P hI a in H1 (u), t~1 xha (t) P hI *a in ¸2 (u) (a"1, 2), (6.12)

where

hI a (s, x)"hI a (s)"!Ps

a

hMTa (r) dr

#

:ba[:s

ahI¹a (r) dr] h (s) D T (s) D ds

:ba

h D T D ds(a"1, 2). (6.13)

hI *a (s, x)"hI *a (s)"!h (s)hIna (s), (6.14)

and

hIn1"1

2[c2

1(s)!c2

2(s)]n

1(s)#c

1(s)c

2(s)n

2(s),

hIn2"c

1(s) c

2(s)n

1(s)#1

2[c2

2(s)!c2

1(s)]n

2(s),

hIT1

"12

[c21(s)!c2

2(s)]¹

1(s)#c

1(s)c

2(s)¹

2(s),

hIT2

"c1(s) c

2(s)¹

1(s)#1

2[c2

2(s)!c2

1(s)]¹

2(s). (6.15)

Proof. Convergence of case (i) is obtained from Theorem 3.1 with F t"2, that isF (t)"2. Similarly, convergence of case (ii) is given by Theorem 4.2 (part (i) if c@I0,part (ii) if c@,0) with f t (xt)"(!xt

2, xt

1). The same theorem with f t (xt)"('ta1, 'ta2)

proves case (iv). Finally Theorem 4.3 with Gt"!2xta, that is, G(t) (s, x)"!2ca (s)!2txh (s)na (s) provides convergence (iii). K

By taking c (s)"(s, 0), s3[!S/2, S/2] and h,1 we obtain the following particu-lar and interesting case, already proved in [28]:

Corollary 6.1. If ut is the rectangle ut"(!S/2, S/2)](!t/2, t/2) we obtain thefollowing convergences where u"(!S/2, S/2)](!1/2, 1/2):

(i) ¹orsion function tt satisfies

t~1t (t)P0 in H10(u), t~2t (t)PtI in H

x(u), (6.16)

where

tI (s, x)"14!x2 . (6.17)

(ii) ¼arping function wt satisfies:

t~1w (t)PwJ in H1 (u), (6.18)



where

wJ (s, x)"!sx. (6.19)

(iii) Function gta satisfies

ga (t) P gJ a in H1 (u), t~1 xga (t) P 0 in ¸2 (u) (a"1, 2), (6.20)

where

gJ1

(s, x)"gJ1(s)"

1

3s3!

S2

4s, gJ

2(s, x)"0. (6.21)

(iv) Function hta satisfies

h1(t) P hI

1in H1 (u), t~1

xh1(t) P 0 in ¸2 (u), (6.22)

h2(t) P hI

2in H1 (u), t~1

xh2(t) P

1

2s2 in ¸2 (u), (6.23)

where

hI1

(s, x)"hI1(s)"!1

6s3, hI

2(s, x)"0. (6.24)

Remark 6.1. Last corollary improves Theorem 4.1 (see Remark 4.2). In fact from(6.18)—(6.19) we deduce wt (xt)"!xt

1xt2#ft (xt), with t~1 f (t)P0 in H1 (u), but

from Theorem 4.1 we obtain w (t)P0 in H1 (u).

Now we shall apply Theorem 6.1 to study the asymptotic behaviour of torsion,warping and Timoshenko’s constants for an open profile ut when t becomes verysmall. The following theorems summarizes them for curved and straight open profile,respectively.

Theorem 6.2. ¸et ut be an open profile defined by (2.10) where Oxt1xt2

is a principalsystem of inertia and c@O0. ¹hen the geometric constants for ut satisfy (a, b"1, 2):

Ita"tI0a#O (t3), I0a"Pb

a

c2a h DT Dds, (6.25)

D ut D"t D u0 D, D u0 D"Pb

a

h D T Dds, (6.26)

Hta"tH0a#O (t3), H0a"1

2 Pb

a

ca (c21#c2

2) h D T Dds, (6.27)

Ht3"tH0

3#O (t3), H0

3"

1

4 Pb

a

(c21#c2

2)2 h DT Dds, (6.28)

Iwt

a "tIw0

a #o(t), Iw0

a "2Pb

a

cawJ h DT D ds, (6.29)



Itt

a "t3 It0

a #o (t3), Ito

1"2 Pu

c2tI h D T Dds dx,

Ito

2"!2 Pu

c1tI h DT Dds dx, (6.30)

Jt"t3J0#o (t3), J0"1

3 Pb

a

h3 DT D ds, (6.31)

Jtw"tJ0

w#o (t), J0

w"P

b

a

wJ 2 h D T Dds, (6.32)

¸gt

ab"t¸g0

ab#o(t), ¸g0

ab"Pb

a

ca gJ b h DT Dds, (6.33)

¸ht

ab"t¸h0

ab#o(t), ¸h0

ab"Pb

a

ca hI b h DT Dds, (6.34)

Kgt

ab"tKg0

ab#o(t), Kg0

ab"2¸h0

ba , (6.35)

Kht

ab"tKh0

ab#o(t), Kh0

ab"!Pb

aChI na hI

nb#hITa hI

TbD T D2 D h DT D ds, (6.36)

Cl, tw

"tC0, 0w

#o (t), C0, 0w

"J0w!

(Iw0

1)2

4I01

!

(Iw0

2)2

4I02

, (6.37)

¹ tab"t~2¹0ab#o (t~2), ¹ 0ab"(1#l) Iw0

a Iw0

b2I0b J0

, (6.38)

where functions wJ , tI , gJ b, hITb are defined in ¹heorem 6.1.

Proof. All equalities are obtained by changing of variable on original definition ofcorresponding constants (see (5.11)—(5.21)) and using Theorem 6.1 if it is necessary. Weillustrate this technique with two examples:

I ta"Put

(xta)2 dxt1dxt

2"t Pu

d0(t) (s, x) [ca (s)#txh(s) na (s)]2 ds dx

"t Pu

h (s) E T (s) D#txh (s) n@ (s) · m (s) D [ca (s)#txh (s) na (s)]2dsdx

"t Pb

a

c2a h DT D ds#O (t3).

Having in mind (6.3) and (6.4) we obtain

Jt"2 Put

ttdxt1dxt

2"2t Pu

d0(t)t(t) ds dx

"2t3 Pu

tI h DT ds dx#o (t3)"1

3t3 P

b

a

h3 D T D ds#o (t3).



We remark that hypothesis c@O0 implies that I0a and J0 are not zero in (6.37)and (6.38). K

Corollary 6.2. ¹he new ¹imoshenko’s constants (kK ta , (a"1, 2)) for an open profile ut

with curves centreline (c@O0) and Oxt1

xt2

a principal system of inertia satisfy

(i) If Iw0

a O0 then

kK ta"t2 kK 0a#o (t2), kK 0a"2 (1#l) I0aD u0 D¹0aa

"

4 (I0a )2 J0

D u0 D (Iw0

a )2(6.39)

(ii) If Iwt

a "o (t2) and (1#l) ¸g0

aa#2l¸h0

aaO0 then

kK ta"kK 0a#o (1), kK 0a"!2 (1#l) (I0a )2

D u0 D [(1#l) ¸ g0

aa#2l¸h0

aa]. (6.40)

Proof. (i) If Iw0

a O0, from (5.28) and (6.38) we deduce

kK ta"2 (1#l) ItaD ut D¹ taa

"

2 (1#l) tI0a#O (t3)

t~1 D u0 D¹ 0aa#o (t~1)"t2 kK 0a#o (t2).

(ii) If Iwt

a "o (t2), from (5.28) and Theorem 6.2 we have

¹ taa"!

1

I ta G(1#l) ¸gt

aa#2l ¸h t

aa#l2

2 (1#l)(Kh t

aa#Ht3)H#o (1). (6.41)

Using expressions (6.15), from (6.36) and (6.28), we obtain

Kh0

aa"!H03. (6.42)

Consequently, we have

Kh t

aa#Ht3"t (Kh0

aa#H03)#o (t)"o (t), (6.43)

and passing to the limit in (6.41) we have

¹ taa"¹I 0aa#o (1), ¹I 0aa"!

(1#l) ¸g0

aa#2l¸h0

aaI0a

Finally (6.40) is obtained as in case (i). K

Remark 6.2. Condition I wt

a "o (t2) is satisfied for example if Oxta is an axis of sym-metry, because then Iwt

a "0. In general, if Iw0

a "0 we have Iw t

a "O (t3) which it isenough to obtain (6.40).

Example. (¹orsion, warping and ¹imoshenko’s constants for an arc profile.) In order toillustrate above results we consider an open profile with constant thickness (h,1)where centreline is an arc of circumference of amplitude h (see Fig. 1). The followingparametrization guarantees that system of axes is principal of inertia.

c (s)"(R cos s!Rc0, R sin s), s3[!h/2, h/2], c

0"

2

hsin

h2. (6.44)



Fig. 1

After some computation from Theorem 6.2 and Corollary 6.2 we obtain thattorsion, warping and Timoshenko’s constants of this profile satisfy:

Jt"Rht3

3#o (t3), (6.45)

C0wt"

R5 t

12 Ch3!6e2

R2(h!sin h)D#o (t), (6.46)

kK t1"

!2 (1#l)m2

(1!c20) ml!2n (1#l)

#o (1) , (6.47)

kK t2"

t2

12 [12e!Rh sin h

2]2#o (t2), (6.48)

where

e"2R2 sin h

2!h cos h

2h!sin h

,

m"

1

2 A1#sin h

h B!c20, n"

1

2#

sin hh

#A1

3!

6

h2B sin2h2

. (6.49)

Expressions (6.45) and (6.46) of torsion and warping constants agree with classicalones (proposed, for example, by [26]). They are then mathematically justified for ourresults. In the same way, expressions of Timoshenko’s constants are deduced bya rigorous procedure that, before now, it was not clear anywhere.

The difference found in (6.8) for the case of a straight centreline brings outimportant differences for geometry constants. In fact, we have

Theorem 6.3. ¸et ut be an open profile with a straight centreline (c@,0) given byc (s)"(s, 0), s3[a, b]. ¹hen the geometric constants for ut satisfies

It1"tI0

1, I0

1"P

b

a

s2 h ds; It2"t3 I0

2, I0

2"

1

12 Pb

a

h3 ds, (6.50)

D ut"t D u0 D , D u0 D"Pb

a

hds, (6.51)



Iwt

1"It t

1"¸

gt

12"¸

gt

21"¸

ht

12"¸

ht

21"Kgt

12"Kg t

21"Ht

2"0, (6.52)

Iwt

2"t3 Iw0

2#o (t3), Iw0

2"!

1

6 Pb

a

sh3 ds, (6.53)

Itt

2"t3 It0

2#o (t3), It0

2"!

1

3 Pb

a

sh3 ds, (6.54)

¸gt

11"t¸g0

11#o (t), ¸

g0

11"P

b

a

sgJ1h ds, (6.55)

¸h t

11"t¸h0

11#o (t), ¸

h0

11"P

b

a

shI1h ds, (6.56)

¸gt

22"o (t3); ¸

h t

22"t3¸h0

22#o (t3), ¸

h0

22"

1

24 Pb

a

s2 h3 ds, (6.57)

Jt"t3 J0#o (t3), J0"1

3 Pb

a

h3 ds, (6.58)

Jtw"t3 J0

w#o (t3), J0

w"

1

12 Pb

a

s2 h2 ds, (6.59)

Ht1"tH0

1#t3 H2

1, H0

1"

1

2 Pa

b

s3 h ds, H21"

1

24 Pb

a

sh3 ds, (6.60)

Ht3"tH0

3#t3 H2

3#t5 H4

3, H0

3"

1

4 Pb

a

s4 h ds,

H23"

1

24 Pb

a

s2 h3 ds, H43"

1

320 Pb

a

h5 ds , (6.61)

Kh t

11"tKh0

11#o (t), Kh0

11"!

1

4Pb

a

s4 h ds, (6.62)

Kh t

22"tKh0

22#t3 Kh2

22#o (t3),

Kh0

22"!

1

4 Pb

a

s4 h ds, Kh2

22"

7

24 Pb

a

s3 h2 ds, (6.63)

Kh t

12"Kht

21"t4 Kh0

12#o(t4),

Kh0

12"!

1

48 Pb

a

s2 hh@ [4h2#2sh#s2 h@] ds , (6.64)

Cl, tw

"t3Cl, 0w

#o (t3), Cl,0w

"J0w!

[(1#l) Iw0

2#lIt0

2]2

4(1#l)2 I02

, (6.65)

¹ t11"¹0

11#o(1), ¹0

11"!

(1#l) ¸g0

11#2l ¸

h0

11I01

, (6.66)



¹ t12"t¹0

12#o (t), ¹0

12"!

l2 Kh0

122(1#l) I0

2

, (6.67)

¹ t21"t3¹0

21#o (t3), ¹0

21"!

l2 Kh0

122(1#l) I0

1

, (6.68)

¹ t22"¹0

22#o (1), ¹0

22"!

1

2(1#l) I02G4(1#l)l¸h0

22

#l2 CKh2

22#H2

3D!1

J0 C(1#l) Iw0

2#lIt0

2 D2

H . (6.69)

Proof. In this case we have c (s)"(s, 0) so we also have

xt1"s, xt

2"txh (s),

T (s)"(1, 0), n (s)"(0, 1),

d0

(t) (s, x)"h (s), d1(t) (s, x)"1#t2 (h@ (s))2x2 , (6.70)

d2(t) (s, x)"h (s) h@ (s) x.

Since Oxt1

is an axis of symmetry we have (6.52). For other constants it is enough tofollow the same procedure than in Theorem 6.2 taking into account the particularproperties (6.70) and also that c@"0 what gives wJ explicitly defined by (6.9). However,in order to obtain (6.57) and (6.63) we need to do ‘better’. Since g

2(t) is solution

of (2.29) with G (t) (s, x)"!2txh (s), we obtain from Theorem 4.3 the followingconvergences:

t~1 g2(t) P 0 in H1 (u), t~2

xg2(t) P 0 in ¸2 (u).

Then

¸gt

22"t2 Pu

xh2 (s) g2(t) (s, x) ds dx"o (t3) . (6.71)

To obtain ¸h t

22we shall use Remark 4.1. Taking f t"('t

21, 't

22) in Theorem 4.2 and

using (4.32)—(4.34) we obtain

kJ (s, x)"(x2!14) sh2 (s),

t~1 sh2(t) P xsh (s)#1

2xs2 h@ (s), t~1

xh2(t) P 1

2s2 h (s), (6.72)

from which we deduce

t~1 h2(t) P hI in H1 (u), hI (s, x)"1

2s2 h (s) x, (6.73)

and, finally, we have

¸h t

22"t2 Pu

xh2 (s) h2(t) (s, x) ds dx"

t3

24 CPb

a

s2 h3 (s) dsD#o (t3). (6.74)



Using variational formulation of (5.9) and notation (2.33) with f t"('t21

, 't22

) weobserve that

Kh t

22"!Put

D +t ht2D2dxt"Put

f t · +t ht2

dxt

"!t¸t2(h

2(t))"!

1

2 Pu

s2 xh2(t) dsdx#t2 Pu5 Cxsh2

sh2(t)

!x2 shh@ xh2(t)#

1

2x2 h2

xh2

(t)D ds dx. (6.75)

Using now (2.28) with u (s, x)"s2 h (s)x as test function (condition u3» (t) (u) issatisfied), we obtain

Pu

s2 xh2

(t) dsdx"t2 G¸t2(s2 hx)#Pu

[2x2 shh@ xh2(t)

!2xsh2 sh2(t)] ds dxH. (6.76)

An easy computation using (2.33) and (6.73) gives us

¸t2

(s2 hx)"t~1

2 Pb

a

s4 hds!5t

24 Pb

a

s2 h3 ds,

Pu

x2 shh@xh2(t) ds dx"

t

24 Pb

a

s3 h2 h@ ds#o (t),

Pu

x sh2sh2(t) dsdx"

t

24 Pb

a

[2s2 h3#s3 h2h@] ds#o (t),

Pu

x2 h2xh2(t) dsdx"

t

24 Pb

a

s2 h3ds#o(t).

Then, with (6.75) and (6.76) we obtain

Kh t

22"!

t

4 Pb

a

s4 h (s) ds#7t3

24 Pb

a

s2 h3 (s) ds#o(t3), (6.77)

and the proof is complete. K

For Timoshenko’s constants in this case we have

Corollary 6.3. ¸et ut be an open profile with a straight centreline (c@,0) given byc (s)"(s, 0), s3[a, b]. ¹hen new ¹imoshenko’s constants for ut satisfy the followingasymptotic formulae where constants are defined in ¹heorem 6.3:

(i) If ¹011O0 then

kK t1"kK 0

1#o (1), kK 0

1"

2 (1#l) I01

D u0 D¹011

, (6.78)



(ii) If ¹022O0 then:

kK t2"t2kK 0

2#o(t2), kK 0

2"

2 (1#l) I02

D u0 D¹022

. (6.79)

Proof. Using properties and values of constants given in last theorem, it is enough torepeat the same arguments as in Corollary 6.2. We remark that (6.42) is still true andas consequence (6.43) is now replaced by the following equality which allows us toprove (6.79):

Kh t

22#Ht

3"t3 (Kh2

22#H2

3)#o (t3). (6.80)

Remark 6.3. Condition ¹011O0 is satisfied, for example, in the usual case that h is an

even function (h (s)"h(!s)) as it can be easily verified. Condition ¹022O0 is always

satisfied, with the exception of special values of l which solve the second orderequation in l, ¹0

22"0.

For some particular cases, previous results can be improved because several constantsvanish. In the next section we shall see it for the particular case of a straight rectangle.

7. Torsion, warping and Timoshenko’s constants for a thin straight rectangular profile

In this section we restrict ourselves to a particular and important case: ut isa rectangular domain ut"(!S/2, S/ 2)](!t/2, t /2). First results of this kind aboutasymptotic behaviour of torsion, warping and Timoshenko’s constants were obtainedby [2, 28, 30] (see also [46]). Using the notation of this paper, this case correspond tohave c(s)"(s, 0), h (s)"1, s3[!S/2, S/2]. Then, from Theorem 6.3, we immediatelyobtain

Corollary 7.1. For the straight rectangle ut"(!S/2, S/2)](!t/2, t/2) we have

Jt"St3

3#o (t3), (7.1)

Cl, t

w"Jt

w"

S3 t3

144#o (t3). (7.2)

Expressions (7.1) and (7.2) provide us a mathematical justification of classicalformulae for torsion and warping constants of a rectangular profile currently used inengineering (see [26, 50, 23]).

In the same way, from Corollary 6.3, we obtain the following expression forTimoshenko’s constants:

Corollary 7.2. For the straight rectangle ut"(!S/2, S/2)](!t/2, t/2) the new¹imoshenko’s constants satisfy

kK t1"

10 (1#l)12#15l

#o (1), (7.3)

kK t2"

!2(1#l)2l (1#3l) A

t

SB2#o (t2). (7.4)



Formula (7.3) is a slight modification of classical value of Timoshenko’s constantgiven by (see [12])

k"10 (1#l)12#11l

. (7.5)

It also agrees with the numerical experiments performed by [45] who calculated kK t1

for several values of t using a scheme of finite elements in order to calculate thegeometry functions and constants of ut.

Calculus of new Timoshenko’s constants can be improved using the fact thatfunctions gt

1and gt

2can be computed in an explicit way. In fact, an easy computation

give us

Theorem 7.1. Functions gt1

and gt2, solution of problem (5.8) in ut"(!S/2,

S/2)](!t/2, t/2) are the following:

gt1

(xt1, xt

2)"1

3(xt

1)3!1

4S2xt

1, (7.6)

gt2

(xt1, xt

2)"1

3(xt

2)3!1

4t2xt

2. (7.7)

We can now use last theorem to improve approximation values of constants whichappear in definition of new Timoshenko’s constants. We obtain:

Theorem 7.2. For ut"(!S/2, S/2)](!t/2, t/2) we have

Jt"St3

3#o (t3), Cl, t

w"Jt

w"

S3 t3

144#o (t3), (7.8)

I t1"

S3 t

12, It

2"

St3

12, Ht

3"

S5 t

320#

S3 t3

288#

St5

320, (7.9)

Iwt

1"Iwt

2"Itt

1"Itt

2"¸

gt

12"¸

gt

21"Kgt

12"Kgt

21"0,

¸ht

12"¸

ht

21"Kht

12"Kht

21"H t

1"H t

2"0, (7.10)

¸gt

11"!

S5 t

60, ¸

gt

22"!

St5

60, (7.11)

Kgt

11"!

S5 t

240#

S3 t3

144, Kgt

22"

S3 t3

144!

St5

240, (7.12)

¸ht

11"!

S5 t

480#

S3 t3

288, ¸

ht

22"

S3 t3

288!

St5

480, (7.13)

Kht

11"!

S5 t

320!

S3 t3

288#

7St5

2880#o (t5) , (7.14)

Kht

22"!

S5 t

320#

7S3 t3

288#o (t3) . (7.15)



Proof. Formulae (7.8) are already obtained in Corollary 7.1. Values of (7.9) come aftersome computation from (6.50) and (6.61). As ut has two axes of symmetry we have(7.10) (see (5.24)—(5.26)). From Theorem 7.1 and (5.20)— (5.21) we obtain (7.11) and(7.12). In order to prove (7.13) we use equality (5.23). We obtain (7.15) from (6.63) withh,1.

Proof of expression (7.14) is more laborious. Definition (5.21) gives

Kht

11"Put

['t11

t1ht1#'t

12t2ht1] dxt .

With the change of variable xt"'t (s, x)"(s, tx) we obtain

Kht

11"

t

2 Pu

s2 sh1(t) ds dx!

t3

2 Pu

x2 sh1(t) ds dx

#t Pu

xsxh1(t) dsdx. (7.16)

Using (2.28) with / (t)"h1(t), we obtain

t~2 Pu

xh1(t)

xuds dx#Pu

sh1(t)

sudsdx

"!Pu C1

2(s2!t2x2)

su#xs

xuD ds dx, ∀ u3H1 (u), (7.17)

where we remark that in last equation we can consider u3H1 (u) and not onlyu3» (t) (u) because it appears there only derivatives of u and for any function inH1 (u), we can obtain a function in » (t) (u) only adding an appropriate constant.Taking in (7.17) u"1

2sx2 and u"1

6s3 we obtain respectively

Pu

xsxh1(t) ds dx"

St4

320!

5S3 t2

576!

t2

2 Pu

x2 sh1(t) ds dx , (7.18)

1

2 Pu

s2 sh1

(t) ds dx"!

S5

320#

S3 t2

576. (7.19)

By substituting (7.18)— (7.19) into (7.16) we obtain

Kht

11"!

S5 t

320!

S3 t3

144#

St5

320!t3 Pu

x2 sh1(t) ds dx. (7.20)

Let us consider the function

/t (xt)"ht1(xt)#(xt

1)3/6 . (7.21)

By applying Theorem 4.2 with f t (xt1, xt

2)"(!1

2(xt

2)2, xt

1xt2), we have

t~2 / (t) P /I in H1 (u), /I (s, x)"A!x2

2#

1

12B s. (7.22)



From (7.21) and (7.22) we deduce

!t3 Pu

x2 sh1(t)ds dx"

S3 t3

288!

St5

1440#o (t5). (7.23)

Now formula (7.14) is a consequence of (7.20) and (7.23). This ends the proof. K

We can now use Theorem 7.2 to compute the new Timoshenko’s constants improv-ing formulae (7.3) and (7.4). We have from last theorem and definitions (5.28)— (5.29):

Corollary 7.3. New ¹imoshenko’s constants of rectangle ut"(!S/2, S/2)](!t/2,t/2) satisfy

kK t1"

10 (1#l)212#[27!5 ( t

S)2] l#[15!5 ( t

S)2!2( t

S)4#o (( t

S)4)] l2

, (7.24)

kK t2"

10 (1#l)212#[27!5 (S

t)2] l#[o ((S

t)2)!15 (S

t)2] l2

. (7.25)

Remark 7.1. We note that (7.3) and (7.4) are a ‘first order’ approximation when tP0of (7.24) and (7.25), respectively.

Numerical experiences using a finite element method in order to solve problems(5.4)—(5.9) in ut and numerical quadrature formulae to approximate geometry con-stants of ut (see [48, 45, 28]), prove that there exists some values of Poisson’scoefficient and relative dimensions of the rectangle (l, S/ t) for which Timoshenko’sconstant kK t

2is ‘very high’. These values of Poisson’s coefficient and relative dimensions

of the rectangle are called critical values (see [45, 28, 46]). One possible reason can bethat denominator of (7.25) vanishes for these values. To try to approximate them weshall do the following hypothesis about term o ((S/t2)) which appears in (7.25):

o AAS

tB2

B"p1

S

t#p

0#p

~1

t

S#2 . (7.26)

If hypothesis (7.26) holds, we can neglect the smallest terms (remember that tP0)and suppose that a good approximation of (7.26) is

oAAS

tB2

BKp1

S

t#p

0, (7.27)

that is, we suppose then

kK t2K

10 (1#l)212#[27!5 (S

t)2] l#[p

0#p

1(St)!15 (S

t)2] l2

. (7.28)

In order to calculate values of p0

and p1

we interpolate function kK t2

in the values ofS/t considered in numerical experiences by [45] and by [28]. If we interpolate in tableof section 6.6 of [45] for values S/t"2 and S/t"3 we obtain

p0"3.521, p

1"18.656. (7.29)



Table 1. Critical S/t (l)

l t/S S/t

0·01 0·06471 15·45380·05 0·14497 6·89810·10 0·20379 4·90690·20 0·28150 3·55240·25 0·31023 3·22340·30 0·33475 2·98730·33 0·34788 2·87470·40 0·37476 2·66830·45 0·39140 2·55490·50 0·40631 2·4612

Table 2. Critical l (S/t)

t/S S/t l

0·020 50·000 0·000960·040 25·000 0·003830·050 20·000 0·005980·100 10·000 0·023800·200 5·000 0·096190·250 4·000 0·154000·300 3·333 0·231220·333 3·003 0·296170·400 2·500 0·478200·406 2·463 0·49892

Now we can give the critical values of pair (l, S/t) only solving the polynomial ofdenominator of (7.28) in variables l and S/t, respectively. We obtain

Theorem 7.3. Given a ratio S/t'0 the corresponding critical Poisson’s coefficientprovided by approximation (7.28)— (7.29) is

l"!(27!5 (S

t)2)![(27!5 (S

t)2)2!48 (p

0#p

1(St)!15 (S

t)2)]1@2

2 (p0#p

1(St)!15 (S

t)2 )

(7.30)

(if 0(l(1/2).Reciprocally, given l3 (0, 1/2) the corresponding critical value of ratio S/t is

S

t"

p1l2#[p2

1l4#20l (1#3l) (12#27l#p

0l2)]1@2

10l (1#3l). (7.31)

Critical values obtained with (7.30)— (7.31) agree with those obtained by [45, 28] inthe numerical experiences. For example, from (7.30) we deduce that in order to obtainl3 (0, 1/2) is necessary to have S /t'2.4612 (see Tables 1 and 2). In [45] a valueS/t"2.5 is proposed after numerical test.



In this way, Theorem 7.3 provides an asymptotic justification about behaviour ofTimoshenko’s constant kK t

2and about critical values of Poisson’s coefficient for

rectangular sections detected by [45, 28].

8. Convergence results for geometry functions and constants ofclosed thin-walled beams

In this section, results concerning closed profiles obtained in section 4 are applied togeometry functions defined by (5.4)—(5.9) which ut is a single-hollowed domain (closedprofile) in order to obtain mathematically justified limit functions and constants to beused in models of closed thin-walled elastic beams (tubes).

We use the same notations as in above sections, taking into account that in this casecondition (2.13) holds. As before, we shall suppose that Oxt

1xt2

is a principal system ofinertia for ut and we shall also suppose that parametrization is oriented in such wayn (s) is in each point the outward unit normal to the domain enclosed by c. We denoteby " the area of this domain. Abstract results applied to the warping and torsionfunctions leads us to introduce the following space:

HIx(u)"Mu3¸2 (u):

xu3¸2 (u), u (· , 1/2)"0 on (a, b)N, (8.1)

endowed with equivalent norms D · Dx, u and E ·E

x, u defined by (2.26). With theseconditions we have the following results:

Theorem 8.1. ¹he geometry functions of a closed profile ut satisfy(i) ¹orsion function tt :

t (t)P0 in H1 (u), t~1 t (t)PtI in HIx(u), (8.2)

where

tI (s, x)"2"

:ba

DT D

hds A

1

2!xB. (8.3)

(ii) ¼arping function wt :

w (t)PwJ in H1 (u), t~1 xw (t)PwJ * in ¸2 (u), (8.4)

where

wJ (s, x)"wJ (s)"2c (s)!2"

:ba

DT D

hds P

s

a

D T Dh

dr#D,

wJ * (s, x)"wJ * (s)"!

b@ (s) h (s)

D T (s) D, (8.5)

and

D"

1

:ba

h D T D ds C2"

:ba

DT D

hds P

b

aAP

s

a

D T Dh

drB h DT D ds!2 Pb

a

ch D T DdsD . (8.6)



(iii) Function gta :

ga (t) P gJ a in H1 (u), t~1 xga (t) P 0 in ¸2 (u), (8.7)

where

gJ a (s, x)"gJ a (s)"2Ps

aCD T (r) Dh (r) P

r

a

ca (p) h (p) DT (p) DdpD dr

#C1a Ps

a

D T (r) Dh (r)

dr#C2a , (8.8)

with

C1a"!

2 :ba

[ DT (r) Dh (r) : r

aca (p)h (p) DT (p) Ddp] dr

:ba

DT (r) Dh (r) dr

, (8.9)

and C2a a constant such that :bagJ a h DT D ds"0.

(iv) Function htaha (t) P hI a in H1 (u), t~1

xha (t) P hI *a in ¸2 (u), (8.10)

where

hI a (s, x)"hI a (s)"!Ps

a

(#a · T ) dr#D1a Ps

a

D T (r) Dh (r)

dr#D2a , (8.11)

hI *a (s, x)"hI *a (s)"!(#a (s) · n (s)) h (s),

with

#1"(1

2[c2

1!c2

2], c

1c2), (8.12)

#2"(c

1c2, 12

[c22!c2

1]), (8.13)

D1a":ba(#a · T ) ds

:ba

DT D

hds

, (8.14)

and D2a a constant such that :ba

hI a h DT D ds"0.

Proof. To obtain (8.4)—(8.14) it is enough to apply Theorem 4.1 to each problem (justas in Theorem 6.1) and use that c (b)"".

To obtain (8.2)—(8.3) we use (5.5) and the change of variable xt"Ut (s, x). From(2.20) we deduce

st (t)"

sw (t)

td2(t)

d0

(t)!t~1

xw (t)

d1(t)

d0(t)!t2 hh@x2

!c · (T#thxn@#th@xn), (8.15)

xt (t)"

sw (t)

th2

d0

(t)!t~1

xw (t)

t2 d2(t)

d0(t)

!th (c · n)!t2 h2 x .



Now it is enough to apply (8.4) to conclude.

As in the case of open profiles, we derive from the last theorem the followingasymptotic behaviour of geometry constants of a single-hollowed thin domain:

Theorem 8.2. ¸et ut be a single-hollowed domain (closed profile) with Oxt1xt2a principal

system of inertia. ¹hen the geometry constants I ta , D ut D, Hta , Ht3, Iwt

a , ¸g t

ab , ¸ht

ab , Kg t

ab, J tw

satisfy the same expressions as in ¹heorem 6.2. Moreover, for the other constants, wehave

Jt"tJ0#o (t), J0"

4"2

:ba

DT D

hds

, (8.16)

It t

1"tIt0

1#o(t), It0

1"

2":ba

DT D

hds P

b

a

c22c@1ds , (8.17)

It t

2"tIt0

2#o(t), It0

2"

2":ba

DT D

hds P

b

a

c21c@2ds , (8.18)

Cl, tw

"tCl, 0w

#o (t), Cl, 0w

"J0w!

[(1#l) Iw0

1#lIt0

1]2

4(1#l)2 I01

!

[(1#l) Iw0

2#lIt0

2]2

4(1#l)2 I02

, (8.19)

Kh t

ab"tKh0

ab#o (t),

Kh0

ab"!Pb

a

[hI @a hI @b1

D T D2#(Ha · n) (Hb · n)] h D T Dds, (8.20)

¹ tab"¹0ab#o (1),

¹0ab"!

1

I0b G(1#l)¸g0

ab#l¸h0

ab

#

l2

Kg0

ab#l2

2(1#l)(Kh0

ab#H03dab)

!

1

2(1#l)J0[(1#l) Iw0

a #lIt0

a ] [(1#l) Iw0

b #lIt0

b ]H . (8.21)

Proof. We proceed as in Theorem 6.2. It is enough to use the definition of constantswith the change of variable (2.10) and the convergences of Theorem 8.1. We onlyremark that for (8.16) we use (5.15) and (8.2)—(8.3). K

Corollary 8.1. ¸et ut be a closed profile and Oxt1xt2

a principal system of inertia. If¹0aaO0 then the new ¹imoshenko’s constants of ut satisfy

kK ta"kK 0a#o (1), kK 0a"2 (1#l) I0aD u0 D¹ 0aa

(a"1, 2). (8.22)



Fig. 2

Example (¹he circular ring). To illustrate the previous results we consider the casewhen ut is a thin circular ring of ratio R and thickness t. Its parametrization is then

c (s)"AR sins

R, R cos

s

RB, h (s)"1, s3[!nR, nR]. (8.23)

In this case we can easily prove that wt"0. Consequently, after some easy calculus,from Theorems 8.1, 8.2 and Corollary 8.1, we obtain that torsion, warping andTimoshenko’s constants for the circular ring ut satisfy

J t"tJ0#o (t), J0"2n R3, (8.24)

Cl, tw

"tCl, 0w

#o (t), Cl, 0w

"0, (8.25)

kK ta"kK 0a#o (1), kK 01"kK 0

2"

1#l2#l

. (8.26)

Special symmetry of this domain allows us to compute explicitly functions tt , wt, g taand hta and also torsion, warping and Timoshenko’s constants. Let us considerexterior ratio and interior ratio of circular ring and its cocient:

R%"R#

t

2, R

*"R!

t

2, m"

R*

R%

. (8.27)

With an easy computation we obtain the following exact functions and constants:

tt (xt)"!12

[(xt1)2#(xt

2)2!R2

%], (8.28)

wt (xt)"0, (8.29)

gt1

(xt)"1

4 C(xt1)2#(xt

2)2!3 (R2

*#R2

%)!

3R2*R2

%(xt

1)2#(xt

2)2D xt

1, (8.30)

gt2

(xt)"1

4 C(xt1)2#(xt

2)2!3 (R2

*#R2

%)!

3R2*R2

%(xt

1)2#(xt

2)2D xt

2, (8.31)

ht1

(xt)"1

4 CR2*#R2

%!((xt

1)2#(xt

2)2)#

R2*R2

%(xt

1)2#(xt

2)2D xt

1, (8.32)

ht2

(xt)"1

4 CR2*#R2

%!((xt

1)2#(xt

2)2)#

R2*R2

%(xt

1)2#(xt

2)2D xt

2, (8.33)



Jt"n2

(R4%!R4

*), (8.34)

Cl, tw

"0, (8.35)

kK t1"kK t

2"

6(1#l)2 (1#m2)2

[7#12l#4l2] (1#m2)2#4m2 [5#6l#2l2]. (8.36)

Now it is easy to check that values (8.24)—(8.26) are (as we could expect) a ‘firstorder’ approximation in t of exact values (8.34)— (8.36).

In this way, Theorem 8.2 gives a mathematical jusitifcation of values of constants oftorsion, warping and Timoshenko that should be used in elastic thin-walled tubes.Empiric formulae can be found in [26, 12, 50].

Acknowledgements

This work is part of the Human Capital and Mobility Program ‘Shells: Mathematical Modeling andAnalysis, Scientific Computing’ of the Commission of the European Communities (Contract No.ERBCHRXCT 940536) and also of Project ‘Analisis asintotico y simulacion numerica en vigas elasticas’ ofDireccion General de Investigacion Cientıfica y Tecnica (DGICYT) of Spain (PB92-0396). We express ourgratitude to J. A. Alvarez-Dios for his suggestions to improving the English version of this work.

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Asymptotic analysis of Poisson's equation in a thin domain and its application to thin-walled elastic beams and tubes