Torsional-flexural waves in thin-walled open beams

Journal @Sound and Vibration (1983) 87(l), 115-141

TORSIONAL-FLEXURAL IN THIN-WALLED P.

WAVES

OPEN BEAMS

MULLER

Laboratoire de Mkanique Thkorique associe’au C.N.R.S., T66, Universite' P. et M. Curie, 75230 Paris Cedex 05, France (Received 26 October 1981, and in revised form 15 July 1982) A one-dimensional theory is developed for coupled torsional-flexural waves in thinwalled elastic beams of arbitrary open cross-section. Complex kinematical effects are fully taken into account with an emphasis on consistency. Exact equations of motion are obtained in terms of generalized stresses and generalized displacements defined by an averaging procedure. Constitutive relations accounting for flexural-torsional couplings are proposed. They include and generalize the static laws of strength of materials. General features of the dispersion are analyzed. This theory is applied to the case of a standard angle-section of which the dispersion curves are given. 1. INTRODUCTION The aim of this paper is to generalize second-order vibration theory to a case of quadruple In its present form the coupling between axial, torsional and two flexural vibrations.

theory is applied to calculations of propagation speeds and frequencies in infinite beams in order to evaluate the qualitative importance of various couplings which may occur in a wave-guide as a result of the geometry of the cross-section. Throughout this paper the general development of the theory is done for a solid beam, but it is kept in mind that the major application of the theory is to the case of non-symmetrical thin-walled beams where the strongest flexural-torsional couplings occur. Part of the Appendix is devoted to the simplifications which occur in the case of thin-walled beams. So far as static problems are concerned, the theory of Vlassov [l] may be considered, in the opinion of the author, as the net plus ultra. In a structure subjected to a dynamic loading, where packets of monochromatic waves (say A sin at), with a large spectrum of frequencies w, may occur, it is a triviality to remark that inertia forces (-Am’sin ot) play an essential qualitative role, even if the

amplitude (A) is relatively small. Thus, on the one hand, all hypotheses of constraints (say A = 0), which are asymptotic certitudes in statics, must be removed and, on the other hand, the inertia associated with every degree of freedom has to be taken into account. The most significant historical illustration of this may be the comparison between the Euler-Bernoulli and the Timoshenko theories for the bending of beams: in the second of these one takes into account both the rotatory inertia of the cross-section and the effect of shear (removal of the constraint “the cross-section remains perpendicular to the deformed central line”).

Except for a paper by Gere and Lin [2] on coupled vibrations of thin-walled beams of open cross-section (in which no constraint is removed), the scope of most studies of the dynamics of beams has been specialized in order to obtain simplified theories. This results in a priori elimination of couplings or in a priori exclusion of various effects: for instance, (i) in the study of uncoupled torsional waves, the effect of distortion-shear is usually a priori neglected [3-71 except in the paper of Bleustein and Stanley [S] where, 115 0022-460X/83/050115+27

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@ 1983 Academic Press Inc. (London) Limited

116

P. MULLER

as a counterpart of the asymptotic nature of the theory, a uniform xy- warping function is prescribed for any cross-section, including the case of a circular one; (ii) the theory of Aggarwal and Cranch [9] for coupled bending torsional waves strictly applies only to a channel-section beam ; (iii) in the theory of Ali Hasan and Barr [lo], which properly applies to the case of equal angle-section, the Euler-Bernoulli constraint is removed and account is taken of the cross-sectional distortion due to special modes of bending of the flanges up to a great degree of refinement, but the effect of distortion-shear due to torsion is not accounted for; in addition the authors have neglected some inertia terms and assumed various uncouplings. Here a matter of consistency shows up: in what follows here none of the effects which a study of the uncoupled cases has shown to be of importance at some level of approximation are excluded, the task of distinguishing among them as to relative importance being left to an analysis of dispersion. The crucial starting point of modelling belongs to the art of engineering: that is, to select the effects which will be taken into account when defining the kinematics of the problem. Once this has been done, the theory is entirely determined and one cannot find more than what has been included at this starting point. The method which is used gives rise to generalized stresses which will result from the kinematic choices without requiring any separate modelling of the distribution of stresses over the whole section. In section 2 an answer to the following question is proposed: “What may be, for a one-dimensional theory, a reasonable definition of the one-dimensional kinematics from the three-dimensional displacement field?“. The point of view initiated by Cowper [ 1 l] in his study of the bending vibration of Timoshenko beams is generalized, so as to be able to consider a priori, simultaneously, longitudinal, flexural or torsional motions, taking into account all secondary effects (such as lateral motion, various effects of shears, warping of the cross-section). Once a kinematic frame has been defined, Germain’s method of virtual power [12] is brought into play in section 3, to give simultaneously the generalized one-dimensional stresses, the generalized one-dimensional kinetics (automatically including all inertia effects) and, in terms of the former quantities, exact stress-equations of motion which are independent of any particular constitutive relation. It is worth noting that these equations could equivalently be obtained by means of averaging techniques analogous to the techniques employed by Naghdi [13] for shells or Mengi [14] for thermoelastic plates. With the field of interest restricted to the elastic case, section 4 is devoted to the question of one-dimensional constitutive relations, when longitudinal motions are uncoupled from flexural-torsional motions. The classical results from statics are preserved and the center of flexure plays a central role. The displacement-equations of motion are then displayed and commented upon in section 5 while the principal features of dispersion and couplings are analyzed in section 6 and illustrated in section 7 in the case of an angle-section. Comparison with earlier works is given in section 8. An Appendix is devoted to recalling various properties of St Venant’s warping functions which play a crucial role in this theory. 2. ONE-DIMENSIONAL GENERALIZED

DISPLACEMENTS

2.1. NOTATION A (semi-) infinite beam with arbitrary simply connected cross-section X (it may be an open thin-walled beam) is to be considered. Let Oz = Oxg be the line of centroids G

TORSIONAL-FLEXURAL

WAVES

IN

OPEN

117

BEAMS

and Gxi, Gxz the principal centroidal axes of .Z. One sets

Without any loss in generality one may assume that the non-dimensional

parameter

a = (12- 11)/I39

(2)

which is less than 1, is non-negative. In the sequel @(x r, x2) is the warping function of St Venant relative to G. This function Q, plays here a crucial role. Its properties are recalled in an Appendix. One sets I@ =

JJ@= H

dZ,

D=

JJ~{x~~~~/~x2+x~)-x2~a (3)

2.2. CENTERS OF FLEXURE, OF TORSION AND OF SHEAR OF THE CROSS-SECTION According to Trefftz’ formulae [15], the co-ordinates of the center of flexure cF of E are given by

xy=-(l/1,)

JJx2@ dE,

Z

XP =

(l/12)

JJxl@ .E

d.YC.

Within this theory it coincides with the center of torsion of E (see the Appendix). The question of the definition of these centers has been the object of various studies ([16, 171, e.g.). The reason for this choice is twofold: on the one hand, it presents the mathematical advantage of characterizing the degree of non-orthogonality of QG with respect to the centroidal co-ordinates x, ; on the other hand, it leads, in the case of thin-walled beams, to Vlassov’s shear center (see the Appendix). The effect of a possible built-in end on the location of the twist axis [18] is discarded, however (but it is worth noting that, within the framework of this theory, built-in end conditions have to be defined in some average sense-see section 3.2.5). For convenience, one can introduce the non-dimensional parameters E = D/13,

p = SI&

3 0,

q1=

Il(x3=/I@,

T12= I2(d)=IL.

(5)

ItmaybeshownthatO
JJ

x,dE=O, .x

~1x2 dE = 0.

(6)

118

P. MULLER

With the definitions

Gz(z,

t> =

(l/11)

jj-

X2f

(7)

dZ

I

f as an “expansion”

it is an identity to write the function

f(xl,

x2,

z,

t) =I%

t) +xlGl(z,

where the “remainder” f-defined

II r

t) +xzGzk,

t> +fh,

by equation (8)-trivially

jd.Z = 0,

If P

x2,2,

t),

(8)

satisfies

x,i dE = 0.

(9)

2.3.2. One-dimensional displacements With the “expansion” (8) in mind, the components of the three-dimensional (small) displacement field X(x1, x2, z, t) in the beam may accordingly be written without any loss in generality as

where, according to equations (7), &=(l/s)lj

X,dZ:

n,=(l/&)]I

x2x3d-%

P

El

n,=-(l/124

xlx,d-%

P

=

(l/Iz)

x1x1

d-z

z

&2 =

U/11)

z

x2X2 d.!S.

(11)

Note that, in agreement with equations (9), the “remainders” .J?i satisfy

(12) Furthermore,

for convenience 23(x1,

x2,

one can set 2,

t)

=

y(.&

t>@(xl,

x2)+~3(xl,

x2,

2,

t),

(13) (14)

TORSIONAL-FLEXURAL

WAVES

IN OPEN

The “overwarping function with respect to @” f&r, and (14)-necessarily satisfies

119

BEAMS

x2, t, t)-defined

by equations (13)

as a consequence of equations (12)-(14). Besides the (small) rigid body motion of X-displacement U(z, t) and rotation n(.z, t)a complete homogeneous deformation of the “microstructure” Z-lateral extensions ei(z, t), ~~(2, t) and the so-called effect of distortion-shear ~~~(2,t&and a St Venant warping of ,Z measured by y(z, t) are emphasized in this natural way.

3. ONE-DIMENSIONAL STRESS-EQUATIONS OF MOTION In this section a one-dimensional model is constructed by means of the so-called virtual power method [12]. It will result in equations of motion (section 3.2.2) in terms of the kinematics of section 2.2.2 and of generalized stresses listed in section 3.2.3. 3.1. VIRTUAL MOTIONS One has to start from a three-dimensional virtual velocity field 6(x1, x2, z) which defines the maximum degree of complexity of the kinematical effects taken into account. At a fixed instant t, one considers 171(x1,

x2,2)

Li2ix1,

x2,

63(X1,X2,

z)=

z)

=

ir,(z~-x2~ii,~z~+xlE11~z~+x2EI~2~z~,

=

~2(z)+xl~ii3(z)+x1~12(z)+x2EI2(z),

~~(Z)--Xl~2(Z)+X2~1(Z)+~(Z)~(~1,Xz).

(16)

Beside the rigid body motionAof Z-defined by the independent (virtual) velocity o(z) and (virtual) angular velocity 0(z)-one takes into account a complete homogeneous (virtual) rate of deformation of the “microstructure” .X-defined by E*r(z), E12(2),E*i2(z)and a warping of Z-defined by the St Venant warping function @(xl, x2) relative to G (see the Appendix) and its associated measure of (virtual rate of) warping T(z). As various couplings are possible, it is not desirable to exclude any of these effects a priori. The reasons for selecting these effects in particular are briefly as follows: (i) the work of Timoshenko [19] has clearly shown that the so-called effect of shear has to be included in a dynamical theory of flexure: one thus may exclude the constraints &,/a~ & = 0 and/or &Jar +fi, = 0, unlike Gere and Lin [2] who preserved, in the dynamical case, those Euler-Bernoulli constraints; (ii) the work of Vlassov [l] has called attention to the crucial role played by warping due to torsion in the couplings between flexural and torsional motions; independently of Vlassov, Gere [3] emphasized the specific impact of warping on the frequencies of vibrations and shapes of normal modes for various end conditions; (iii) Bleustein and Stanley [8] pointed out that avoiding the so-called effect of distortion shear ei2 leads to an incorrect representation of dispersion for torsional waves; moreover, in usual solid section beams, one may show that this effect is more important than warping on the dispersion of waves [20]; (iv) the effect of lateral contraction (Eli, E*2)is obviously needed for a good representation of longitudinal motions.

120

P. MULLER

Typically non-linear effects, like the so-called “helical shortening” [21, pp. 298-3041, are obviously not taken into account within the framework of this theory. 3.2.

STRESS-EQUATIONS

OF MOTION

3.2.1. The virtual power method If Uij are the components of the Cauchy stress tensor and p is the mass density per unit volume, the principle of virtual power allows one to write, in a small perturbation theory and at any instant t,

for any subdomain z1 G z s z2 of the beam and for any virtual motion ii. In this expression 4~~)is the virtual power of the external forces acting on the subdomain; so far as dispersion phenomena are the subject of interest, one may assume that the external loads act exclusively on the terminal sections (say z = 0 and z = L). The way of calculation now is routine: first one integrates by parts so as to obtain only independent quantities (oi;:,Ai, 9, .612,El,) under Ii:. Then one considers i’s which are zero in the vicinity of z1 and ~2. This yields the local one-dimensional equations of motion (section 3.2.2) in terms of generalized stresses which, for the sake of clarity, will be listed in section 3.2.3. Then, integrals I:; having been eliminated from equation (17), one considers ii’s which are zero except in the vicinity of z2 and one obtains a representation of the generalized internal contact forces acting through _Xfrom z >zl on z Cz2 in terms of the stresses (section 3.2.4). Finally, considering the domain 0 GZ s L itself, one may obtain the natural boundary conditions at the terminal sections (section 3.25). 3.2.2. One-dimensional equations of motion. In the case of a homogeneous cross-section of motion may be written as

aTl/az

=ps a2ul/at2,

al-,/at

aMl/az

- T2 = pI1 a2L&/at2,

(p = p(z)), the ten local stress-equations

=ps a2u2/at2,

aT3/az = ps a2u3/at2,

(184

akf2/a2 + Tl = pr, a2R2/at2,

akf3/az =p13(a2f23/at2+~a2~12/at2), aB/az - Q = pI@ a2y/at2,

aK12/az

aKJaz - H1 = p12 a2El/at2,

-HI2

(18b)

=p13(a2E12/at2+a

a2L&/at2),

(~Bc, d)

aK2/az - H2 = pI1 a2E2/at2.

We)

Beside the one-dimensional generalized inertia terms (right-hand sides of equations (18)) we simultaneously obtain the corresponding generalized stresses and internal contact forces, as follows. 3.2.3. One-dimensional generalized stresses The one-dimensional generalized stresses are global parameters averages as follows: Tl = Ml=

II I

x2u33

d-Z

~713 d-X

M2=-

T2 =

JJ

~1~33

.E

g23

dz’,

T3 =

dSC,

M3=

JJ L

IS P

associated with 2 via

(+33

(xl@23

WW

a

-x2al3)

a,

(19b)

TORSIONAL-FLEXURAL

B=

II L

@us3

Klz = Kl=

JJ

~1~13

WAVES

IN OPEN

121

BEAMS

(19c) Q = I~=((a~/ax,)~~,+(a~/~x~)~~,} d-Z

a

(x1uzs+w713)

JJ

HI=

dZ

H12=2

dX

u11

z

K2=

a

u12

(19d)

a,

JJ

x2u23

a,

I

We)

H2=&22dX. P

3.2.4. Generalized internal contact forces acting on 2 The cross-section X at .z2is subjected from the part z > z2 of the medium to a resultant force T, a resultant moment M (at the centroid) of the local distribution of stresses (u13, ~23, u33), a so-called bimoment B relative to G, and “double forces”: Kr2 (accounting for distortion-shear) and K1, K:! (causing gradients E ;, E; of lateral contractions). It is worth noting that none of the stresses Q, Hr2, HI and Hz contribute to the internal contact forces. This is a well known situation in media with microstructure. However, in the case of thin-walled beams, Q is usually interpreted, on the basis of a static distribution of shear-stress, as a flexural-torsional moment about G [l, p. 501. 3.2.5. End conditions End conditions are not needed for present purposes but one can indicate how they could be written down. Generalized external forces acting on an end (say z = L), denoted by script capital letters, have to be connected with the internal contact forces as follows: T,(L) = Yi, Mi(L) = JIci, B(L) = W, K&) = 2X12,K,(L) = Y&. These external forces are obviously defined from the surface force density Fi(xr, x2) acting on the terminal crosssection XL as follows:

JJ Fia, AtI= JJ JJ JJ& JJa.b x2= JJ

yi =

x2F3

&

a,

Ju3=

@F3

a,

x12=

b1F2

-xzFl)

1F2

&

JJ x

1F3

d-K

IL

IL

B=

A2=-

&

+ x2Fl)

d-Z

d-Z

x1

=

x81 d-X JJ-%

x2F2 CU.

Alternatively, displacement end conditions have to be given in terms of average displacements (11) on the end section. Regular end conditions require that .Yi or Vi, &i or 0, SB or y, Xl2 or &r2, and Xa or &u be given at z = L. For instance, at a built-in end, one has to satisfy the average-conditions Ui(L)=O, n,(L) = 0, y(L) =0, e12(L)==O, and E,(L) = 0. 3.2.6. Some comments on the one-dimensional model

In terms of the generalized stresses (equations (19a)-(19e)) and of the generalized kinematics (equations (ll), (13), (14)), equations ((18a)-(18e)) are exact: there is no truncation of any asymptotic expansion. Will the exact information obtained in terms of averaged quantities be sufficient or not? This is another question.

122

P. MULLER

4. ONE-DIMENSIONAL STRESS-STRAIN RELATIONS FOR AN ELASTIC MATERIAL

It is now convenient to write (),k for the partial derivative 8/8xk and use 0 for 8()/8x, = a(>/&?. The task now is to obtain constitutive equations relating the 14 generalized stresses (19a)-( 19e) and the 14 generalized (small) deformations

(U’l -fM,

cu;+nl), u;,

YT Y’P

.512,

n;, n;, n;,

&I,

E ;2,

S;,

E2,

(2% b) (2Oc-e)

E;.

It is crucial to note that, in the deformation (20a)-(20e), the displacements have to be understood in the global sense (equations (ll), (13), (14)), so that this theory differs from an asymptotic theory which should relate one-dimensional stresses to displacements, rotations, . . , , at the centroid. Attention is now restricted to the case of a homogeneous isotropic elastic material with Lame’s constants A, p (or Young’s modulus E and Poisson’s ratio Y). 4.1. EXACT CONSTITUTIVE RELATIONS The natural way to obtain the constitutive three-dimensional expressions gij =

Aai$k,k

+

relations

is to start from the classical

(21)

P (Xtj +Xj,i),

and then to calculate each one of the expressions (19a)-(19e), being written in the form (lo), (13): that is

the displacement

field X

where the one-dimensional generalized displacements are defined by equations (ll), (13) and (14). Taking into account the properties (12) and (15) of R1, g2, x”gand the properties of the St Venant warping function relative to G (see the Appendix), after a straightforward calculation one obtains

I T3=(A

+2p)SU;

+AS(EI+EJ+A

Ml = (A +2~.,1,0; M2=(A

+2p)12fl;-A

+A

II x2&1

P

(XI,, +*2,2>

+X2,2)

d-T

(234

a

JJX1GG.l+-~2,2, d-x

z

TORSIONAL-FLEXURAL

WAVES

IN OPEN

M3=CL(r2-~1)&:2+ll~3~n5+~(D_4)Y+CL

II

123

BEAMS

(x&2 -x2:3,1)

a,

(2%)

z

Q=~1.(11-12)&;2+~~(D-13)~$+~(13-D)~

B = (A +2/L)I,r’+CL

JJ@(%,I JJ * _ JJ JJ JJ JJz dE. JJ(*,,I JJ22,2 +%2,2)

(234

dZ

L:

K12

=

CLI3.5 ;2

+ w (12 -IdG

+ p V1-~22)Y

(xlx”3.2+x2f3,1)

+ CL

a,

z

H12 = 4/_& 12+ 2~

(X1,2+X2,1)

(234

d-Z

L

K1=&e;

JJx

+/_L

lf3.1

K2=~11e;

a

+/L

xZx”3,2

a,

r

z

H1 = (A +~/L)SE~+ASE~+ASU$ +A

(&+~2,2)dc+2cL

%,lds

z:

H2=ASc1+(A

+~/L)SE~+ASU;

+A

z

+*2,2,

ds

+ 2~

z

(23e)

CONSTITUTIVE RELATIONS 4.2. APPROXIMATE Until now the equations obtained are exact. At this stage one has to make simplifying assumptions so as to estimate as well as possible the influence of the “remainders” X1, X2, C3 in equations (23a)-(23e). For this purpose, along with all others working in the field, one can make use of the static St Venant solutions, so as to preserve the classical static results from strength of materials. This assumption may be considered as the best available, although it may be questionable at wavelengths approaching the order of beam dimensions. But one is interested here in propagation speeds, rather than in stresses which could be in error. One has to keep in mind that, due to the existence of an elastic energy for the one-dimensional model, the linear operator giving the 14 stresses (19a)-(19e) in terms of the 14 deformations (20a)-(20e) must be represented by a symmetric matrix. One can proceed step by step as follows.

4.2.1. Uncoupling of longitudinal motions The stresses T3, K1, K2, H1, Hz refer essentially to l_or@udinal motions in the beam. One may assume the influence of the “remainders” X1, X2, f3 to be negligible in the expressions for them, so that one may write K1 = /_LI~E ;, K2 = /Al& ;,

T3=(A +~/.L)SU; +AS(e1+~2), H~=(A+~/,L)SEI+AS(EZ+U;),

H2=(A +2/~)&2+AS(cr+U$).

This results in the uncoupling of the longitudinal motion.

(24’)

motion from the torsional-flexural

124

P. MULLER

4.2.2. Flexural moments and bimoment

The influences of XI, _%zin MI, M2 and B can be taken into account by assuming the classical relations Ml = EIJI;,

Mz = EI&;,

B = EL&.

(25)

This is justified so long as the distribution of stresses cll, cz2, u33 does not depart too much from St Venant’s static distribution in which crl 1cc (+33 and g22 cc (+33,so that MI, M2 and B may be obtained from u33 = E 3X3/&. 4.2.3. Distortion-shear stress HIZ One can neglect 21, X2 in the second of equations (23d) so that H12 = 4/_&c12.

(26)

4.2.4. Flexural-torsional constitutive coupling On taking expressions (23) and (24)-(26) into account the constitutive relations for the torsional stresses (M3, Kr2, Q) and shearing force (TI, T2) may be written in (symmetrical) matrix notation

A distinction has been made between the flexural rigidities fuB,the torsional rigidities tij and the coupling rigidities C,i. The aim of the present section is to give for these rigidities expres$on_s which take into account, in the best manner, the influences of the remainders X1, X2, and 23 in equations (23). When Z presents two axes of symmetry, the center of flexure (4) of E coincides with G. One is in the usual situation where torsion and flexure are uncoupled: c,i may be assumed to be zero. On the one hand, one may write TI= awS(U;

--fi,),

T2 = a2pS(U;

+f?d,

(28)

where the so-called Timoshenko coefficients (Y~and a2 account for the influence of 23 on TI, T2; the appropriate technique for the determination of al, (~2 has been given by Cowper [22]; although Cowper’s values have been obtained on the basis of St Venant’s static distribution of stresses, the dependence of crr, a2 on the frequency may be expected to be weak [23]. On the other hand, one may neglect the influence of the “remainders” in the expression of the torsional stresses and set K12

=wI3~;2

+/.LL(I~-IIM&

M,=CL(~~-I~)&~~+CLI~~~+CL(D-~~)Y,

+wU1-12h

Q=CL(II-r2)&;2+CL(D-13)~n;+~(~3-D)Y.

(2%

These expressions are the constitutive relations for a theory of torsion in which the warping of _E and the effect of distortion-shear are taken into account. Moreover it provides a generalization of the usual “static” law for non-uniform torsion [21, p. 2821 M3 = /ADO; -Q,

(30)

in which, beside St Venant’s contribution pDL?b, one recognizes the contribution (-Q) due to restrained warping, which is usually not determined by a constitutive equation,

TORSIONAL-FLEXURAL

WAVES

IN OPEN

BEAMS

125

because of the internal constraint y -0; = 0, but in fact results from the equilibrium equation Q = dB/dz (taking the third of equations (25) into account, one should obtain, in this case, the classical contribution -Q = -El,+’ = -EI&ls’, due to restrained warping, to M3). When _Epresents no symmetry, (i) For ftI, f22 and tij one preserves the values given by equations (28) and (29); (ii) one assumes that TI (resp. Tz) is independent of (Uh +01) (resp. (Vi -0,)) (this is an exact result from St Venant’s static solution when ,X presents one axis of symmetry)-accordingly, f12 = 0; (iii) one assumes TI, TZ to be independent of e i2, and thus cl1 = cZ1= 0; (iv) in the case of a non-symmetric thin-walled beam, the resultant (torsional) moment of the local distribution of shearing stresses (~13, (~23) is not identical with the moment of the resultant of the so-called membrane-shearing tension; accordingly, the total torsional moment M3 is the sum of two terms [l, p. 511: first, the fZexur&torsionaf moment, which is the contribution to MX due to the fact that the resultant of the membrane-shearing tension, which is equal, up to negligible terms, to the resultant {Tl, T2} of the local distribution of shearing stresses (~13, u&, does not pass through the center of flexure CF, and second, St Venant’s contribution MD&!;, due to the non-uniformity of the distribution of the tangential stresses over the thickness of the wall, which may as well be considered as resulting from a distribution of torsional couples per unity length along the middle-line of E;; thus, on the one hand, one identifies the flexural-torsional moment with (-Q): that is to say, in the absence of torsion, which has now to be defined by nj=,=o,

E;* = 0,

(31)

one sets, for any {T1, T2}, Q = -x?TZ+xFT1; then, from equation (27), with account taken of the hypotheses (i), (ii), (iii), whenever definitions (31) hold, this results in Cl3

C23=-(Y2SpXr;

=&-d

(32)

on the other hand, one preserves the classical law for thin-walled beams, h43+Q=/_~D.&,

(33)

by setting C12+C13=0,

C22+C23=0;

(34)

as a consequence, in the case of a terminal loading by a concentrated transverse shearing force of which the line of action passes through CF (that is to say, M3 + Q = 0) one has a zero twist 04 (state of twist-free flexure); furthermore, equation (33) is a reasonable expression of the quantity @43+Q) as a measure of coupling between the actual stressfield and St Venant’s uniform torsion stress-field (see the remark at the end of section A.4 of the Appendix). Thus, according to (i), (ii), (iii), (iv), theflexurul-torsionalconstitutive equations (27) when ,Y presents no symmetry may be written as TI=QICLWU;

-O,)+x%4G)},

T2=~2/.4W;

Kl2=d3k;2

~3=cLsbw%J~

-n2)+a2xW~

Q=-_cLS{-(YIX%I; 4.3.

CONSTITUTIVE

+Lh)-x&4Wl,

-a(Y-n;)),

+nd)+/d3{a&;2

-~,)+~,X~(~~+n,,}+~~3{-a&~2+(1-E)(Y-n~)}.

-(y-L!I,,+ey},

(35)

INEQUALITIES

The elastic energy of deformation per unit length, from which the constitutive equations (24), (25), (26) and (35) have to be derived, must be a quadratic positive definite form.

126

P. MULLER

Referring to the book by Jeffreys and Jeffreys [24, p. 1371 one is led, as well as to the classical relations from three-dimensional elasticity, 3A +2/_&>o,

p >o,

(36)

to the following restrictions on the values of Timoshenko’s a2>0,

a1>0,

coefficients: (37)

(S/G){(yl(X~)2+(y2(X~)2}<1--Q2-E.

In so far as the conditions (37) are not violated, Cowper’s technique for the calculation of al, (Ye,which rests on St Venant’s static distribution of shear-stresses, may be retained.

5. DISPLACEMENT-EQUATIONS

OF MOTION

In this section the displacement-equations of motion are established by injecting the approximate constitutive relations (24), (25), (26), (35) into the exact stress-equations of motion (18). As may be expected, these equations split up into two uncoupled sets of equations governing the longitudinal motions (section 5.1) and the flexuraltorsional motions (section 5.2), respectively. For convenience we shall use the notation CIA= Ci a2/&z2 - a2/at2, where CA is some velocity, is used, as before, a prime denotes the partial derivative relative to z. 5.1. LONGITUDINAL MOTIONS Substitution of equations (24) in the third of equations (18a) and equation (18e) leads to the three equations for el, e2 and U3: (&/S){OT -&1-

i;E2- ZXG = 0,

?Lus+v&; ? cc=

CLIP,

cz = (A + 2cL)lP,

4: =

(Il/S){cl,

-&2-

rX!?1-XG = 0,

+I/&; =o,

(S/I,,CZ,

(38) 42”

=

(S/l,,CZ,

5 =

h/p.

(39)

This is an elementary model for the propagation of longitudinal motions in which account is taken of the so-called inertia of lateral motion, and which is a generalization to arbitrary cross-sections, in a natural way, of either the initial model of Mindlin and Herrmann [25] for a circular cross-section or the three terms of Muller and Touratier for a rectangular cross-section in reference [26], where a complete study of the dispersion may be found. 5.2.

FLEXURAL-TORSIONAL

MOTIONS

5.2.1. Coupled flexural-torsional motions Flexural-torsional motions are governed by the set of seven equations resulting from the injection of equations (25), (26) and (35) in the first and second of equations (18a), and equations (18b), (18~) and (18d). Thus

TORSIONAL-FLEXURAL

WAVES

IN OPEN

127

BEAMS

{O~-W:2}E1*+(YO~~-(YC~‘=o, (I@/S){UB

-w2,)~+~I~/S~C~l-e,n~+~I~IS~~~~-E*

(40)

+{-Cfx~(UI-~2)+C~x~(U~+n,)}=o,

G= PIP, c:=(YlPIP, cz= w:=WIl)C~, w;=wI2)c:, a2PlP,

C; = HP,

w2,=(13/1@)(h)C+.

w:2 = (4S/L)C2,

(41)

5.2.2. Uncoupled flexural and torsional motions The coupling between flexural and torsional motions is governed by the quantities x7 and XT.When the cross-section has two axes of symmetry, that is to say, when x7 = xc = 0, one has a set of three uncoupled motions, as follows: (a) pure flexural motion in the principal plane (x1, z),

?JJl-c:n;=o, ?

(I,/S){O,

-w;}n,+c:v;

=o;

(42)

(~l/s){os-wT}-n,-c’,u~=o;

(43)

(b) pure flexural motion in the principal plane (x1, z),

?2u2+c;n;=o, ? (c) pure torsional motions, El&

+ a UT&

12 -

C2,(1 - E)y’ = 0,

{~T-W:*}E12+(YOT~n3-(YC~~‘=o,

(L/4){o,-w~}r+c~(l-E)n4+cuC~E;*

=o.

(44)

Equations (42) and (43) are simply Timoshenko’s beam equations for flexural motions in the two principal planes. Equations (44) govern the pure torsional motions in a theory in which the warping of the cross-section and the effect of distortion-shear are taken into account. The principal features of the dispersion may be discussed with the help of the three non-dimensional parameters (Y,E and /3 which characterize the geometry of 2 [20]. There are just two points to note here: on the one hand, for usual solid beams with two axes of symmetry, the effect of distortion-shear has an impact more important than warping on the dispersion; on the other hand, when one tends to statics, the asymptotic value of the velocity of harmonic torsional waves is the classical St Venant value C, = C&.

6. DISPERSION OF FLEXURAL-TORSIONAL

WAVES

6.1. DISPERSION EQUATION As usual, one looks for conditions under which solutions of the form {Ul,

U2,fJl,

02,

3, E12, y) = {UY, f-&i

ux,

nY,nZ,

f-G

d2,

Y “>

exp i&z -at)

(45)

may exist for equations (40). The linear homogeneous system of algebraic equations obtained in this manner for the amplitudes UT,. . . , y” has a non-trivial solution if and only if its determinant is zero: that is to say, after some elementary manipulations (lines

128

P. MULLER

and columns multiplied by hi, if w2-C:k2

0

0

w’-C;k2

0

C;k

C:k

0

C;x;k2

C:k

C;k

0

${w2-C:k2-u;}

;{02-C;k’-u;} -C:x;k

C;x:k

0

0

-Cfxfic

C:x:k

0

0

-C:x:k’

0

0

0 -C:X:

C,‘X;”

C:x;k’

0

C:x:k

-C;x:k’

0

-C;x:k

C;xyk

0

c;x:

-C:x;k

0

-c;x;

c+{w’-C:k2}

${w2-C$k2-o:2} -cd’:;

+C:(l-F,k

I

(46)

+:(l-e)k

a${u2-C:k2}

-C;k2}

=o .

k

-aC$-2

I k S

${w2-C;k2-w:}

In the domain k ~0, this polynomial of degree 7 with respect to w2 admits seven real positive roots for w (or, if preferred, for the phase velocity c =o/k). Each of them defines a branch of the dispersion curve giving w as a function of k (or c as a function of k) associated with a propagation mode {U?, . . . , y’}. A complete study of the dispersion requires a numerical resolution of equation (46); the example of an angle-section beam will be given in the next section. In the present section, the principal asymptotic features of the dispersion when k + 0 or k + 00 are described. A rapid investigation of equation (46) shows that the asymptotic expansions of its roots w2 contain only even powers of k. Furthermore, when k + 00, all these roots behave as k2 (which means finite values for velocities); when k +O one has, besides four finite roots, one root O(k’) and two roots O(k4). 6.2. FUNDAMENTAL BRANCHES When k + 0, one can distinguish the three followin roots: (a) Fl: o + 0, c = C,a k + 0, purely iY? mode; (b) F2: w + 0, c = C, P Ir/Sk + 0, purely U,” mode; (c) T: w + 0, c + C, = c,JE, purely 0,” mode. Clearly these branches are respectively the fundamental modes for flexure in the plane (x1, z), flexure in the plane (x2, z) (with dispersion of Euler-Bernoulli type when k -0) and torsion (with finite velocity of St Venant’s elementary theory when k + 0). 6.3.

CUT-OFF

FREQUENCIES

OF

UPPER

MODES;

EFFECT

OF

FLEXURAL-TORSIONAL

COUPLING

When k + 0, the four remaining branches are such that w tends to a finite value (the cut-off frequency) below which there is no propagation, while the phase velocity tends

TORSIONAL-FLEXURAL

WAVES

IN OPEN

to infinity. One may distinguish between (a) a torsion/distortion

and (b) three coupled flexural rotations/warping frequencies 017, &, 6~: are solutions of 2

w

rl1w:

modes [a?-@

0

w:

-w;

w:

772”:

02-w2y

-w:

0

mode,

mode E& + CYR~,

0 +wg =w&G?,

w

129

BEAMS

2

(47)

- y”] of which the cut-off

=o,

(48)

where ~1 and q2 are the non-dimensional parameters defined by equations expected, when 2 has two axes of symmetry (VI= q2 = 0), one has *_ UT =w1, CLIT=w2, 0, - w-f,

(5). As

the case of uncoupled flexural and torsional motions (section 5.2.2). When 1 has one 0x2, so that 77I = O-one has, besides the uncoupled “Timoshenkotype” flexural rotational mode [fig with cut-off frequency w? = wl, two coupled flexural rotation/warping modes [J2: - -y”] of which the cut-off frequencies wz and w*, are solutions of axis of symmetry-say

(dw2)4

- b/w*)2(1

+x2)

+

(x2)

+ Lx2 -

772) =

0,

(49)

*2=W~/W~=(1-E)(1+CY)/2Cr@=o.

(50)

Due to the fact that x2 may be S 1 so that J(l -x2)’ may be *(l -x2), one has to distinguish two cases in order that w2 and wy be the limits of wz and w: respectively when the coupling coefficient ~72tends to zero. (i) When the cross-section is such that x2<1, w~/w2={[1+~2+~(1-~2)~+477~1/2}~‘~, 0302

={[I

+x2-JU

-~2)~+4172l/2)~‘~,

(51)

so that the coupling (172# 0) raises the value of w2 > wy up to w: and lowers the value of w, down to w *,. (ii) When the cross-section is such that x2 > 1,

wT/w2={[1+X2-‘42-1)2+4q2]/2}1’2, w$/w2={[1+x2+&2-

(52)

1)2+4v2]/2}1’2,

so that the coupling (v2 f 0) lowers the value of w2
COMPARISON

BETWEEN

EFFECT

OF DISTORTION-SHEAR

AND

EFFECT

OF WARPING

The condition that the impact of distortion-shear on the dispersion surpasses the impact of warping is roughly expressed by w f < w :‘. When ,Z has two axes of symmetry (771= q2 = 0) one is led to the unique condition w ‘,< w c, which may be written as l-4p/(l-e)(l-ar2)>0.

(53)

As p is small, this inequality is generally satisfied. A correct theory of (uncoupled) torsional waves must take into account, beside warping, the effect of distortion-shear (equations (44)).

P. MULLER

When Z hasone axis of symmetry (qz Z 0) one has to make a distinction: (i) when x2 :> 1, distortion-shear

surpasses warping when

~-4Pl(~-~~(~--2)~~~~-~1/XZ)-J(1-1/X~)2+4r12/X~~r so that the condition is a furtiori satisfied (right-hand distortion-shear surpasses warping when

(54)

side
~-4P/(~-~~(~-~*~~~~1-(1/~~)+J~(f/x2~-13~+417~/~~~,

(ii) when x2< 1, (55)

so that it may be violated, because of the lowering of o, down to W$ due to coupling; this will be the case for angle-section beam (section 7) for which warping is more important than the effect of distortion-shear. 6.5.

ASYMPTOTIC

VELOCITIES

WHEN

k + m

Looking for the asymptotic value of the roots of equation (46) in terms of c(k) when k --, co and for the nature of the corresponding modes, one divides each line of equation (46) by k* and lets k tend to infinity. One may easily list (a) a purely J2?-mode (c + CB>, (b) a purely &-mode (c + CB), (c) a purely y*-mode (c + CB), (d) a purely EyZ-mode (c + C,), (e) a series of three coupled flexural translational-torsional modes [U?- Ux@] of which the asymptotic velocities C;“, C,*, C*, are solutions of cz-c: 0

0 C2-C::

K&

K&g

CC C$ c2-c;

= 0,

K1S(X:)2CY:/13(1 -cr2) = 2/3Yj,&/(l-02)(1

(56) +a),

K2=S(X5)2CY:/13(1-02)=2j3&/(1-02)(1+(Y). Again, when ,Y has two axes of symmetry (~1 = CT = Cl,

~2

=

cg = c,,

(57)

0),one has c*,= c,,

flexural and torsional motions (section 5.2.2). When 25 has one 0x2, so that ~2 = O- one has, besides the uncoupled Timoshenkotype flexural translational mode [Ug] with asymptotic velocity C4 = C2, two flexural translational/torsional modes [a;- V?] with asymptotic velocities CT and C% solutions of the case of uncoupled

axis of symmetry-say

(C/C,)4-(C/C~)Z(1+IY~)+(111~-K~)=0:

(58)

that is (with attention restricted to the case cyi < l), C~/C~={[(13-cr~)+~(1-a~)23-4K~]/2)”2, C;/C~={[(~+LY,)-J(~-~~)~+~KJ~}~‘~.

(59)

Note that the coupling raises the value of CT up to C*, and lowers the value of CI down to CT. 7. EXAMPLE OF AN ANGLE-SECTION In this section results are presented of a numerical solution of equation (46) for the case of a standard angle-section beam with equal flanges (see Figure 1) of A.G.S. For A.G.S. p = 2685 kg/m3, E = 6,988 X 10” Pa, h = 2.666 x 10" Pa, and z, = O-3106. The dimensions are t = l-50 x 10e3 m and I= 20.0 x 10s3m, so that S = Oa5775 x low4 m2,

TORSIONAL-FLEXURAL

WAVES IN OPEN BEAMS

131

Figure 1. Angle-section beam.

II= 0.0905195 x 1O-8 m4, 12 = O-357208 x lo-’ m4, and I3 = 0.4477275 x 1O-8 m4. Within the simplified framework of thin-walled beams (see the Appendix) one has x7 = 0, xc = -0.6796 x lop2 m, D = 0.4669 x lo-” m4, 1, = 0*1847X 10Pr2 m6, and, according to Cowper’s method [22], (YI= 0.347 and (YZ= 0.349. The values of the non-dimensional coefficients (2), (5) are LY= 0.596, E = 1.043 x lo-‘, p = 0.532, n1 = 0, and n2 = 0.894. The (dimensional) values of the cut-off frequencies (x2~) are (L): = 2.05 x lo5 Hz, w2 = 2.36 x lo5 Hz, wr = 4.70 x 10’ Hz, w,* = 5.04 x lo5 Hz, and og = 8.89 x lo5 Hz. (It is worth noting that, in a homothetical structural element of which the linear dimensions are multiplied by n, the values of the cut-off frequencies are divided by n.) The

Figure 2. Frequency spectrum in a non-dimensional form. Asymptotic nature of the modes: T, torsional (0:); D, distortion-shear (~72); Fr, flexural rotation (0:); i3, flexural translation (I?,$; W, warping (TO).

132

P. MULLER

I

1

Figure

3. Velocities

2

us. wave number

3

in a non-dimensional

form.

(dimensional) values of the characteristic velocities are C, = 322 m/s, CT = 1423 m/s, Ci = 1856 m/s, C, = 1863 m/s, Cr = 3150 m/s, C*, = 3368 m/s, and Cs = 5101 m/s. In terms of phase velocities and frequency, spectrum curves are given in a non-dimensional form in Figures 2 and 3, respectively. For the sake of clarity, the two autonomous branches of the uncoupled Timoshenko flexural motion (35) in the plane (xq, z) have been removed. 8. COMPARISON WITH EARLIER WORK As mentioned in the introduction, there are few papers concerning coupled flexuraltorsional waves in thin-walled open section beams. However, the present theory may be compared to the theories of Gere and Lin [2] and Ah Hasan and Barr [lo]. 8.1. COMPARISON WITH THE THEORY OF GERE AND LIN The present theory may be compared with the purely theoretical model of reference [2] for the angle-section case. For flexural-torsional waves in the principal plane (x1, z), with the notation of the present paper, the displacement equations of motion (equations (2)-(4) of reference [2]) may be written as Erz a4ul/az4 EI@*

a4&/az4-wD

+ps a2ul/at2+ps3C~

a*&/az*+pl;[l+

(S/r&c)*]

a2R3/at2 = 0, d*&/at*+pSx~

a*UJat*

= 0,

(60)

where U1 is now the deflection at the center of shear CF. In this model the effect of distortion-shear, and of rotatory and warping inertias are neglected. It includes the

TORSIONAL-FLEXURAL

WAVES

IN OPEN

133

BEAMS

usual “static” internal constraints aul/a2

a&/a2

-f&=0,

Using for convenience the non-dimensional the dispersion equation

- 7 = 0.

(61)

variables of Figure 3, from this one deduces

C4-{C;(~+$[1+;(,:,2])52+e}C2+[C;$~2][~+C;$~2]=0.

(62)

The two branches defined by the roots of thisequation have a good asymptotic k + 0). But, as a consequence of behaviour when k + O(C+ + C, = &JE, C = C,JI,/S the basis hypotheses, C, + COwhen k + 00 (as in the Euler-Bernoulli theory). Numerical comparison is given in Figure 4 for the angle-section beam of section 7.

Figure 4. Numerical comparison for an equal angle-section Gere and Lin; AH-B, Ali Hasan and Barr; PMl-PM2, present

8.2.

SITUATION

OF THE

PRESENT

PAPER

WITH

(section theory.

RESPECT

7). EB, Euler-Bernoulli;

TO THE

AL1 HASAN

GLl-GL2,

AND

BARR

THEORY

Reference [lo] is specifically devoted to equal angle-section beams and contains an important part on experimental results. Here the theoretical model is to be discussed in respect to the dispersion results. With the notation of the present paper, and for flexural-torsional waves in the principal plane (xi, z ), the three-dimensional displacement field is approximated by the truncated expression x1=

U1-x2f-&+Fb,

tBln(XI,X2),

x3=-n2~,+~~+[aF(z,t)/azlB3n(~1,~2),

X2=X11(13+F(Z,t)Ban(xl,xz),

(63)

where (i) the “static” internal constraint y = aR3/az is retained, (ii) the warping function @ is taken as (-x1x2), and (iii) Bi,(xl, x2) (n = 3, 5, 7, . . .) are known shape-functions associated with the antisymmetrical distortion of ,X due to various modes of bending of the flanges (see Figures 5(a) and (b)), measured by F(z, t); this is quite different from the effect of distortion-shear of X due to torsion and measured by ei2 of the present

134

P. MULLER

(a)

Cc)

(b)

Figure 5. (a, b) Antisymmetrical cross-sectional distortion due to to bending of the flanges; (c) cross-sectional distortion due to torsion.

theory (see Figure 5(c)); furthermore, it is clear that F may include some torsional rotation. Use of Hamilton’s principle on the basis of the truncated displacement field (63) makes it impossible to take into account possible properties of “remainders” either in kinetic quantities or in rigidities (for instance, El = E/(1 -Y’) instead of E or, in terms of characteristic speeds, Ck = c,/JF v instead * of C,). Indeed, for each (odd) value of n, from this method there results a system

[ii

;i

g

$jkj=O>

(64)

where D, are partial derivative operators of various orders defined in Appendix III of reference [lo]. The authors consider three so-called uncoupled motions, namely

Each one of them should give rise to a two-branch dispersion spectrum. But the authors assume various rotatory inertias to be negligible, so that each of equations (65) gives rise to a unique curve (respectively, equations (21a), (21~) and (21d) of reference [lo]). For instance, the first of equations (65) is originally a Timoshenko beam equation (with CL instead of C,). On neglecting the rotary inertia associated with &, it becomes a Rayleigh beam equation with infinite velocity as k + co (and C = CL a k instead of the Euler-Bernoulli C = CeJIJSk) as shown by equation (21a) of reference [lo]. The authors consider the couplings between the first and third and the second and third of equations (65) to be negligible so that, finally, equation (64) splits up into two types of motions: namely, coupled lateral bending/antisymmetrical cross-sectional distortion motions governed by [i,

;;;

ZJ~]

=o,

(66)

of which the dispersion is given by the biquadratic equation (24) of reference [lo]; “uncoupled” torsional motions governed by the third of equations (65) of which the dispersion curve (equation (21d) of reference [lo]) may be written, in the notation of the present paper, as C* = (E +A*Cb*c*)/(l

+A*E*),

A* =b/S(I,+S(x;)*).

(67)

Due to the difference of significance between the Er2 of the present paper and the F of reference [lo], motions governed by equation (66) cannot be compared with the present

TORSIONAL-FLEXURAL

WAVES

IN OPEN

BEAMS

135

theory. But the dispersion curve (67) for torsional motion has been drawn in Figure 4 for the sake of comparison in the case of the angle-section beam of section 7. 9. CONCLUSION

According to the basic ideas developed in the introduction, an attempt has been made to take simultaneously into account, without any a priori assumptions, all effects which may occur at a certain level of approximation defined once and for all by equations (16). Accordingly, the dynamical equations (18), routinely obtained by means of the virtual power method, are exact in this respect. But in equations (18) the kinematics of the one-dimensional model -W by defined from the three-dimensional displacement field X by the averaging procedure of equations (11) and (14): as pointed out by Cowper [ll], this constitutes, beside the removal of constraints, a second reason to disregard definitions of one-dimensional kinematics by means of asymptotic expansions of X in the case of dynamics. The approximate character of the theory is now entirely located in the question of the constitutive relations. Starting from some three-dimensional stress-strain relations, it is crucial to be assured, from the former analysis, of the exact definition of the one-dimensional kinematics upon which must depend the one-dimensional stresses (19). The method proposed, which consists in extending static laws of the strength of materials, may certainly be refined in future work: one-dimensional stresses and strains may be considered as homogenized quantities; one might be tempted by homogenization techniques to get one-dimensional stress-strain relations. But what about homogenization for dynamical problems? Every approximate theory becomes increasingly suspect as one approaches its borderlines. As already pointed out by several authors [4, 261 for such theories, it is worth noting that the upper modes are not exact, due to the fact that a greater refinement of the theory could give birth to new branches of the dispersion curves which would creep into the former ones. On the other hand, all information concerning fundamental branches may be considered as reliable for use in experimental investigations. ACKNOWLEDGMENT

Special acknowledgments go to my colleague M. Touratier for our fruitful discussions and for his technical help in the numerical study of the angle-section. REFERENCES 1. V. Z. VLASSOV 1961 Thin-Walled Elastic Beams. Jerusalem: Israel Program for Scientific Translations, second edition. 2. J. M. GERE and Y. K. LIN 1958 Transactions of the American Society ofMechanical Engineers, Journal of Applied Mechanics 25, 373-378. Coupled vibrations of thin-walled beams of open cross section. 3. J. M. GERE 1954 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 21, 381-387. Torsional vibrations of beams of thin-walled open section. 4. A. D. S. BARR 1962 Journal of Mechanical Engineering Science 4, 127-135. Torsional waves in uniform rods of non-circular sections. 5. E. J. BRUNELLE 1972 American Institute of Aeronautics and Astronautics Journal 10, 524-526. Dynamical torsion theory of rods deduced from the equations of linear elasticity. 6. 0. L. ENGSTRC~M 1974 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 41, 1041-1046. Dispersion of torsional waves in uniform elastic rods. 7. D. GAY 1978 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 45, 681-683. Influence of secondary effects on free torsional oscillations of thin-walled open section beams.

P. MULLER

136

8. J. L. BLEUSTEIN and R. M. STANELY 1970 International Journal of Solids and Structures 6, 569-586. A dynamical theory of torsion. 9. H. R. AGGARWAL and E. T. CRANCH 1967 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 34, 337-343. A theory of torsional and

coupled bending torsional waves in thin-walled open section beams. 10. S. ALI HASAN and A. D. S. BARR 1974 Journal of Sound and Vibration 32, 3-23. Linear vibration of thin-walled beams of equal angle-section. 11. G. R. COWPER 1968 Proceedings of the American Society of Civil Engineers, Journal of the Engineering Mechanics 6, 1447- 1453. On the accuracy of Timoshenko’s beam theory. 12. P. GERMAIN 1973 S.I.A.M. Journal of Applied Mathematics 25, 556-575. The method of virtual power in continuum mechanics. Part 2: Microstructure. 13. P. M. NAGHDI 1974 in Proceedings of the 7th U.S. National Congress of Applied Mechanics 3-21. New York: A.S.M.E. Direct formulation of some two-dimensional theories of mechanics. 14. Y. MENGI 1980International Journal of Solids and Structures 16,1155- 1168. A new approach for developing dynamic theories for structural elements. Part 1: Application to thermoelastic plates. 1.5. E. TREFFTZ 1935 Zeitschrift fiir angewandte Mathematik und Mechanik 15,220-225. Uber den Schubmittelpunkt in einem durch eine Einzellast gebogenen Balken. 16. N. G. STEPHEN and J. C. MALTBAEK 1979 International Journal of Mechanical Science 21, 373-377. The relationship between the centers of flexure and twist. 17. E. REISSNER 1979 International Journal of Solids and Structures 15, 41-53. Some considerations on the problem of torsion and flexure of prismatical beams. 18. J. H. ARGYRIS 1954(April) AircraftEngineering 102- 112, AppendixI?. A study of thin-walled structures such as interspar wing, cut-outs and open-section stringers. 19. S. P. TIMOSHENKO 1921 Philosophical Magazine, Series 6, 41, 744-746. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. 20. P. MULLER 1981 Compte rendu hebdomadaire des Seances de I’Academie des Sciences, Paris, Serie II 293, 533-536. Ondes de torsion dans les guides Clastiques uniformes. 21. S. P. TIMOSHENKO 1941 Strength of Materials, Part II. New York: Van Nostrand, second edition. 22. G. R. COWPER 1966 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 33, 335-340. The shear coefficient in Timoshenko’s beam theory. 23. N. G. STEPHEN 1978 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 45, 695-697.0n the variation of Timoshenko’s shear coefficient with frequency. 24. H. JEFFREYS and B. S. JEFFREYS 1956 Methods of Mathematical Physics. Cambridge: Cambridge University Press, third edition. 25. R. D. MINDLIN and G. HERRMANN 1952 in Proceedings of the 1st U.S. National Congress of Applied Mechanics 187- 191. New York: A.S.M.E. A one-dimensional theory of compressional waves in an elastic rod. and M. TOURATIER 1981 Wave Motion 3, 181-202. Dispersion of longitudinal 26. P. MULLER waves in a rectangular transversly isotropic wave-guide. 27. I.S. SOKOLNIKOFF 1956Mathematical Theory Of Elasticity. New York: McGraw-Hill, second edition. APPENDIX:

ST VENANT’S

WARPING

FUNCTIONS

A. 1. DEFINITION Let X be a simply connected domain of the plane with boundary &Z and outer normal II. The principal centroidal axes of X(Gxl, GxZ) are used so that

St Venant’s

warping

function

relative

V2@=0

in&

to G is the solution @(xl, ~2) of

a@/an=nlx2-n2xl

ona.%

Neumann’s problem b42)

TORSIONAL-FLEXURAL

WAVES

IN OPEN

137

BEAMS

which, in addition, satisfies

JJ r

(A3)

@d.Z=O.

The boundary condition may be written a@/& = (n x GP) - z VP E I5 in order to emphasize the crucial dependence of Qi on the center G. A.2. CHANGE OF CENTER: PRINCIPAL ST VENANT’S WARPING FUNCTION If H is any point with co-ordinates (hI, h2) in the principal centroidal Venant’s warping function relative to H, the solution of 02@, = 0

a&,/an

in Z;,

= (n x HP) *z

VP E 2,

@*

axes, the St

JJ r

&dZ;=O,

(A4)

is related to @ by @H =@+hlxz-hzx,.

(A3

Thus, every quantity defined with the help of St Venant’s warping function relative to H (such as B, Q, y) will a priori depend on the center H. The principal St Venunt’s warping function is the function @* which furthermore satisfies the orthogonality conditions

JJ I:

xl@* dZ = 0,

Clearly, from equation (A5) the co-ordinates

JJx2@‘* dZ = 0.

I

of the principal center H* are given by

W) A.3. CENTER OF TWIST ACCORDING TO THE PRESENT THEORY A mechanical significance for the purely geometrical point H* may be given as follows. The displacement-solution of the problem of uniform torsion of an elastic prismatic beam (of shear modulus CL)due to given opposite torsional couples V and -% acting on the terminal sections z = 1 and z = 0 has been given by St Venant and may be written as X1 = -(%‘//LD)x~z

~~3x2

+

~2x3 + bl,

X,=(%/pD)x9

X3=(%/~D)@+ualx2-u2x1+b3.

+uJxl-u$3+b& 647)

Due to the absence of boundary conditions on the displacements a, b is an indeterminate rigid body motion. Because of the warping @, one cannot satisfy rigorously rigid clamping conditions at z = 0. But equation (A7) has been obtained within the framework of St Venant’s principle: only the resultant torque acting on the terminal sections has been given; this implicitly postulates that the solution is not affected by the local distribution of surface load. Thus one may admit a “dual” flexibility to satisfy the rigid clamping conditions and cancel only the rigid body displacement (U(0); n(0)) defined by equations (11) at z = 0. A straightforward calculation leads to XI =

-(@,7lwD)(x2

- h;), X3

=

X~=WZICLD)(XI-~:),

W/CLD)@*(XI,

x2).

648)

Clearly, equations (AS) give H* as the location for the center of twist of the cross-section: that is, the point at rest around which the cross-section rotates in its own plane.

138

P. MULLER

Note that when E is symmetrical about an axis, the center of twist H* defined by equations (A7) lies on this axis, due to the parity properties of @ defined by equations (A2) and (A3) in this case. A.4. CENTER OF FLEXURE ACCORDING TO TREFFI-2’ DEFINITION Consider, for a cantilever-beam rigidly clamped at t = 0, the following two problems: (i) uniform torsion by a terminal torque V applied at z = I generating St Venant’s stress-field {u:‘:, afi}; (ii) bending by a concentrated terminal transverse force 4 applied at z = 1 generating St Venant’s stress-field {aif:, cr$:‘, r$f,‘}. The global elastic energy of the stresses is

Trefftz defined the state of torsion-free

bending by the uncoupling property

W’S’(a”’+ &f)) = W’“‘(a”‘) + W’“‘(a”‘), Due to the form of (I”), this trivially reduces to

IIp

{(@,I

-~)a::

+(@,z+xI)&} dE

= 0,

or, alternatively, with the notations (19b) and (19~) of the present theory, M~(u’~‘) + Q(acf’) = 0, or, alternatively again, with account taken of the exact equilibrium equation (18c), ~L&(u’~‘)+ (dB/dr)( u”‘) = 0. Due to u$:’ = (.LFI/l& - Z)xI f (.Fz/~I)(z - I)xz, one has

so that the state of torsion-free

bending occurs when

that is to say, when the line of action of the terminal transverse force passes through the center of f7exure C”, which coincides with the center of twist H* located by equations (A6). It is worth noting that one may interpret the quantity M3 + Q, calculated for a stress field u, as measuring the degree of stress-energetical coupling between u and St Venant’s uniform torsional stress-field. A.5.

IDENTITIES

SATISFIED

BY

#

Various identities are satisfied by @. Starting from the identity

one may write, taking equation (A2) into account and for any regular function W;

I

ax

!P(n,x2--n2x,)

ds =

VY*V@dX.

(A91

TORSIONAL-FLEXURAL

WAVES

IN OPEN

139

BEAMS

Suitable choices of q lead successively, with the aid of the divergence theorem, to (AlO)”

@,, d,Z = 0,

P

=

Y

=xt

P

=x1x2,

CE =0, JJzx&,2 JJ L +x~@,d cE=1, JJ(x4.1 xi@,1 dE =

-12,

J J txz~,l:xIcw

CD,

I

cizz = J J w:l

I

+@:A a.

(All) (A121

(A13)*

From equation (A13) one immediately deduces

a =0, JJ{~,1(~,1-x2)+~,2(~,2+x1)} (Al4)

r

and, for convenience,

one may recall the alternate form for the definition of D,

-XI@,*) d2 =13 -D. JJ(xz@,l P

(A151

The starred equalities are wrong for 0” when H* = CF # G. Note that D is independent of the center, but that

A.6. INEQUALITY SATISFIED BY D Let D be the geometrical torsional rigidity defined by equation (4). A straightforward application of Ritz’s method [27] leads to (Al7) for any regular function W. For the choice P = ~1x2 one readily obtains DsI~-(L-II)~/I~,

6418)

which may be written, with the help of E and (Y defined by equations (5.1) and (2), in the non-dimensional form &sl-a2. (A19) A.7. THIN-WALLED BEAMS OF OPEN CROSS-SECTION In the sequel, the beam degenerates into a long prismatic shell of constant thickness t small compared with any characteristic dimension. The middle line of Z is denoted by r (curvilinear abscissa s, unit tangent T, unit normal Y such that v = z x 7). M is a point along r, P is a point of 2 located by (s, u) where u is the normal co-ordinate along v. The outer normal to 8.X is n (see Figure Al). Proposition. When t tends to zero, St Venant’s warping function @ relative to G may be approximated, up to O(t) terms, by Vlassov’s sectorial area w(s) with pole G and sectorial origin such that w(s) ds = 0.

(A20)

140

P. MULLER

Figure

Al.

Thin-walled

beam of open cross-section.

Proof. Due to the relative smallness of t, one may write in E

@(S, u>=w(S)+uq(s)+(u2/2)r(S)+.

...

(A21)

The quantities W(S) and 4 (s) may be determined by examining the solution of the problem (A2) and (A3) when t tends to zero. The boundary condition (A2) leads immediately to q(s)=(vxGM).z=-GM.7.

Figure

AZ. Thin-walled

beam:

(A22)

domain

O(s).

Consider the variable domain L!(S) (see Figure A2) with outer normal N, the boundary of which is the union of a0 n&X and u. On account of the first of equations (A2), the divergence theorem leads to I Jl2

{M/&V - (iVrxZ- Nzx I)} ds = 0.

(A23)

Due to the second of equations (A2) on 80 n a.& equation (A23) in fact reduces to

(A241

TORSIONAL-FLEXURAL

WAVES

Noting that, on U, N = 7, one may write t/2 _~,2{~‘(s)+~~‘(~)+~r’(~)+...}ds=~*’z I

IN OPEN

BEAMS

(GM.v+u)du,

141

(A25)

-t/2

from which one deduces, up to O(t3) terms, w’(s) = GM *u, or, alternatively,

(AW

setting h(s) = GM *v, dw =h(s) ds.

(A27)

This is exactly formula (3.15.3) on p. 16 of Vlassov’s book [l], which defines w as the sectorial area with pole G. To complete the identification, the origin of s on r (or sectorial origin according to Vlassov [l, p. 161) has to be chosen so that condition (A20) is satisfied. One thus has the following correspondences between solid beams and thin-walled beams

(all formulae and pages in braces { } refer to Vlassov’s book [l]): (i) to the formula (A5) for the change of center corresponds the formula {(4.1), p. 21) for the change of pole; (ii) to the principal St Venant warping function @* corresponds the principal sectorial area (formulae {(7.1), p. 40)); (iii) to I* corresponds the sectorial moment of inertia J, (formula {(6.12), p. 39)); (iv) to the principal center H* = CF located by equations (A6) corresponds the principal sectorial pole located by {(7.5), p. 42); similarly, this principal sectorial pole is just the shear center (p. 52).