Maupertuis-Lagrange mixed variational formula for laminated composite structures with a refined higher-order beam theory

Pergamon

Inl. J Non-Linear Mechanics, All ngbts

Vol. 32, No. 5, pp. 989-1001, 1997 0 1997 Elsevier Science Lid reserved. Printed in Great Britain CQ20-7462/97 $17.CWl+ 0.00

PII: SOO20-7462(96)00120-5

MAUPERTUIS-LAGRANGE MIXED VARIATIONAL FORMULA FOR LAMINATED COMPOSITE STRUCTURES WITH A REFINED HIGHER-ORDER BEAM THEORY A. M. Zenkour Department of Mathematics, Faculty of Education, Kafr El-Sheikh, Egypt (Received 12 February 1996) Abstract-Maupertuis-Lagrange (M-L) mixed variational principle is obtained by using Legendre’s transformation and Lagrange’s multipliers. This principle is used to deduce the governing equations for laminated composite structures. A rational higher-order displacement-based onedimensional theory for the analysis of laminated beams is presented. This theory is established using the M-L mixed variational formula to study the vibration behavior of a symmetric laminated beam subjected to normal and tangential traction fields. Governing equations are derived without introducing a shear correction factor employed by other authors. Numerical results for natural frequencies are obtained according to the classical, first- and higher-order beam theories. A comparison is made to illustrate the practical significance of the inclusion of the effect of transverse normal strain forces (as well as the effect of transverse shear deformation and rotatory inertia forces) in the vibration analysis of beam problems. 0 1997 Elsevier Science Ltd. Keywords: variational principles, laminated beam theories, natural frequencies

1. INTRODUCTION

The importance of variational formulations in the theory of elasticity goes far beyond their use as alternate approaches to many problems. The utilization of these variational formulations has refined some classical theories of elastic laminates which have inherent limitations and are inadequate to describe the stress-strain analysis for many problems. As it is known in the variational principles of the theory of elasticity, either displacements or stresses may be taken arbitrarily. However, in the mixed variational principles both the displacements and stresses must be considered arbitrary. A variational theorem with displacement and stress as independent fields, is presented by Hellinger (see [l], and the references therein) for finite elasticity problems. A generalized version of Hellinger’s results is found in [2]. In the early 1950s Reissner [3] presented a variational theorem for finite elasticity which is based on the complementary energy, and explicitly includes boundary data, by allowing the independent variation of the surface traction. This work is an extension of an earlier result [4], which is concerned with the variational statement for linear elasticity. Hu [S] presented a variational theorem including displacements, stresses, and strains as independent fields. For finite elasticity, a variational statement with three independent fields is given by Washizu [6]. Since 1980, a series of publications have appeared concerning the variational principles in the theory of elasticity such as ([7-l 11). In [lo] Chien discussed particularly variational principles and generalized variational principles for elastic body with non-linear stress-strain relations. Chien [ll] used the high-order Lagrange multiplier method to construct a generalization of the minimum potential energy principle and complementary energy principle for geometrically non-linear elasticity. Reissner, in his papers [12-141, gives a recent variational formulation of the problem of infinitesimal elasticity. The analysis in [12] was based on a specialized version of the classical variational equation for displacements and some stresses in infinitesimal elasticity. However, in [ 13,141, he derived an analogous result from a statement of the general variational equation for displacements and allstresses. He used it to refine a shear deformation plate theory. Fares et al. [15]

Contributed by W. F. Ames 989

A. M. Zenkour

990

presented an extension of the mixed variational formula of Reissner based on Hamilton’s principle, to study the linear dynamical problems of anisotropic elastic bodies. The main purpose of this paper is, on the one hand, to present an extension of the mixed variational principle of Reissner based upon Maupertuis-Lagrange principle in which the total energy is conserved. This mixed variational principle is used to study the dynamical problems of laminated composite structures. On the other hand, it is to illustrate the practical significance of the inclusion of the effect of transverse normal strain in the vibration analysis of beam problems.

2. FORMULATION

AND SOLUTION

OF THE PROBLEM

Let us consider the fundamental mixed problem of the theory of elasticity for which the boundary conditions are Ui - U* = 0

on S,,

F* - njOij = 0

on S,,

s = s, + s,,

(1) i

where FE7are the components of the surface force prescribed over a part S, of the surface S of the body; UT are the displacement components prescribed over the remaining surface S, and nj are the components of the unit vector along the outward normal to the surface. Maupertuis-Lagrange principle [16] states that the integral w=

f2 2T dt, s fl

(2)

has a stationary value for any part of an actual trajectory, provided that the energy of the system is conserved; T + II = const. = H,

(3)

and the total variation of displacements AUi satisfies AUiJt, = AuiIt, = 0.

(4)

It should be kept in mind that the definition of the total variation of any function (r for example) deals with Ar = 6l- + FAt;

(5)

and

where AT is the total variation of I- with respect to the position and time; and X is the variation of r with respect to the position only. Using (5), the condition (4) tends to 6UiIt, = --tiiAtlt,;

8UiIt, = -riiAtI,I.

(7)

The kinetic energy T is given by: T=

where pCk)is the material density of layer k. The total potential energy consists of the deformation potential energy II, and the potential energy of the external forces II,, defined

Maupertuis-Lagrange

mixed variational formula

991

respectively by:

(9)

where U(Eij) is the potential energy of deformation per unit volume (strain energy density) and Fi are the body forces. In the mixed variational principle the stress components Pii and the displacement components Ui are taken to be independent. But the functional (2) lacks the stress components aij. So, an attempt to generate them will be done. The potential energy U(EU)for linear elastic laminates is a homogeneous quadratic function of the strain components eij and satisfies the following relation:

The above relation is called Legendre’s transformation The inverse of this transformation takes the form:

aR(aij) t%ij ’

E.’ v = ~’

R(oij)

=

DijEij

with iY(sij)as a generating function.

-

(11)

U(&ij),

where R(oij) is the generating function of the inverse Legendre’s transformation and is called the additional work (complementary energy density). It takes the form [17]: R(oij) = ai;i;ttnOijCmn,(i,j,m,ti = 1,2,3),

(12)

where a$ are the elastic constants, which depend on the symmetry of internal structure of the kth layer. Using (ll), the total potential energy takes the following form: I-I=

[Cij&ij - R(o
F*Uids.

(13)

Now, the problem is to determine the extremum of the functional (2), subjected to conditions (1) and (3). We introduce the Lagrange’s multipliers 1 and li to obtain the unconditional functional J=W+

l(T+l-I-II)+

li(Ui - u*)ds

1 dt.

(14)

Taking the total variation of the functional (14) and using Gauss’ theorem, with the help of (5) and (6), we get

+

[(nj~ij - FF)dUi] ds +

[(A, + Injaij)bui + (ui - u?)Sni] ds

+ (T + II - H)&l

)1 t2

+

li(Ui -u*)ds

At

.

(15)

fl

Since the variations 6Uiand 60, in volume V, 6aij on the surface S, and 6Uion the surface S,, are arbitrary, then the extremum condition of the functional (14) takes place when lli + lZnjaij = 0;

1 + ;1 = 0,

A. M. Zenkour

992

or A = - 1, Ai =

njOij.

Therefore, we get the M-L mixed variational formula (14) in the final form: J=

fl tz

T-rI+H+

ss

SF

S”

1

njoij(ui - UF) ds dt,

(16)

and also the extremum condition of this functional gives the dynamic equations of the kth layer: p(k)c.

=

F.

I

and the appropriate

I I

aaij Bxj

’

Hooke’s law for the problem: aR(cij) &ii= doij )

as well as the boundary conditions (1). The utilization of the M-L mixed variational formula (16) allows one to deduce rational deformation theories for laminated composite structures without introducing shear correction factors as those employed by other authors (see for example [18,19,25,26]).

3. A GENERALIZED

HIGHER-ORDER

THEORY

OF BEAM BENDING

The increasing use of laminated composite structures in several engineering applications has generated considerable interest among many researchers to formulate refined theories for the analysis of such structures. A majority of these theories deal with laminated composite plates, while a more restricted number of theories have been proposed for laminated composite beams and shells. The displacements in a laminated composite are presented as products of two sets of unknown functions, one of which is only a function of the thickness coordinate and the other is a function of the in-plane coordinates (i.e. separation of variables approach). With respect to the equivalent single-layer theories, the displacements are expanded in terms of various powers of the thickness coordinate x3 and unknown functions ui(xr , x2) [19]: ui(x19xZ~x3)

=

2

(x3)j4,

G

=

1,2,3),

j=O

where ni denotes the number of terms in the ith displacement expansion. The higher-order theories (often not exceeding third-order) are based on the above displacement expansion with nl = rr2 = 2 or n, = n2 = 3 and rr3 takes one of the values 0,l or 2 [20-261. In the first-order plate theory [27-301 it is assumed that nl = n2 = 1 and n3 = 0: ur = UY+ x,u:,

u2 = u; +

x34,

u3

=

u;.

In the classical laminate theory, it is further assumed that u: and uf are equal to -iTu~/axl respectively, to satisfy the normality condition (i.e. straight normal to and -a$/ax2, midplane remain normal to the midsurface after deformation). Now, let us consider a beam of length L, width b and constant thickness h composed of a finite number, N, of thin layers. Let the coordinate axes (x, y, z) be so chosen that x is the axial coordinate and let the y and z axes lie in the plane of the cross section and the surfaces z = -h/2 and z = h/2 are, respectively, the top and bottom surfaces of the beam. The beam surfaces are taken to be subjected to the following traction fields:

i(,,,,-5) =($0,-+);

i(x,y,;) =(?,o,+).

I

(17)

Maupertuis-Lagrange

mixed variational formula

993

Thus the boundary conditions on the beam surfaces are: Is2

=

Pl

g3

--)

=

Tl

-

b

at

b

h z

=

--;

2

(18) h b b 2 Here, we have substituted ran, o2 and cr3 in place of the conventional oXX,c,, and o,,. Let us assume that the x and z components of the displacement-based 1D field are of the form: p2

02 = --)

T2

03=-

at z=-.

u(x, z; t) = uo(x; t) + zu,(x; t), (19)

w(x,z;t) = wo(x;t) + zw,(x;t) + z2w,(x;t), 1

where the functions no and w. are the axial and transverse normal components of the displacement of points on the neutral axis of the beam and u1 is the rotation of the cross section. The functions w 1 and w2 are kinematics measures which, as will be seen, respectively, relate the extensional and flexural deformation of the beam to the transverse normal component of the displacement [31]. The displacement fields of classical beam theory (CBT) and first-order beam theory (FBT) can be obtained from the present higher-order beam theory (HBT). For example, from (19), we have (i) Classical Beam Theory (CBT): w1 = w2 = 0, (ii) First-Order

u1 = -dw,/dx;

Beam Theory (FBT): WI = w2 = 0.

The plane components

of strain are given by: El

=

F,,

=

82

=

Gz

=

E3

=

2E,,

z+z2, duo

WI

=

+2zw2,

(20)

dwo u,+-+z-+z2-. dx

dwl dx

dwz dx

The higher-order theory of beam behavior based upon (19) will be derived by application of M-L mixed variational formula (16), in which both the displacements and stresses are taken to be arbitrary. So, let us also assume that the non-vanishing stresses are of the form: \ 01 = G1 + zG2,

O3 =$I

+cZ($)](l

+$)

+G3[492]+;[c3+cd(&)](l-j$),~

(21)

o,=(G,+zGs)[1-(&)1]+S,+zS2+z2S3+z3S,,

where c, are constants, and the functions G, and S, (m = 1, . . . ,5; n = 1, . . . ,4) have dependence upon the coordinate x. It is convenient to introduce in expression (21) the following stress resultants: h/z [N1,N2,N31

= b

CM1,Mzl=b

5 -h/2

h/2 s -h/2

Ca,,az,a31dz, (22)

zCal,azldz,

I

where N1, N2 are the inplane stress resultants; M1, M2 are the bending moment resultants and N3 is the transverse shear force resultant. The physical meaning of the functions G, arises from the point that the stresses oij satisfy the stress resultants (22). Also the physical meaning of the functions S, arises from the boundary conditions (18) and that the

A. M. Zenkour

994

normal stress c2 has extremum values on the top and bottom surfaces of the beam. In this case, the stresses (21), can be obtained in the following final form: 12M1 bh3 >

Nl

o1=a+r

(23)

(T2= $+z$][l-(&)‘]+$+2(&5(&)2](1+&2) [ (24)

+$[1-2(+2)-5(&)21(1-&)~ 0~=$[1-(&)2]-$$l-3(+2)](l++2)

-F$l+ (&)I(’-$). 4. DERIVATION

(25)

OF GOVERNING

EQUATIONS

The M-L mixed variational formula (16) is used to derive the equations of motion and the constitutive equations of a laminated beam. This formula will be applied to the present problem in the absence of the body forces and the prescribed displacements (i.e. S = S,). Substituting the displacements (19), the strains (20) and the stresses (23)-(25) into the functional (16), we can get easily the total variation of this functional. In this case, the extremum condition of (16) gives the following dynamic equations: dNi duo: -& + (T, - T2) = b[Zi& + Z2iil-J h 6u . -dM1 - N3 + $T, I’ dx

(26)

+ T2) = b[12iio + 13iil],

(27) (28)

awl:

(29)

-N2-~(P,+P2)+~~(T,-T,)=b[J2)\jo+~3~l+~412],

where the inertias I, (n = 1, . . . ,5) are given by:

s h/2

1, =

p(k)Z”-l&

=

-h/2

5 k=l

s

=li + 1

pfk)z”-’ da.

zk

Also, the extremum condition gives the following constitutive equations for the kth layer: -1 a11

43

al5

6 -5 a33

bh

duo

dx

6 ja35

6 -a55 5

sym. 1

1 -1

a13

10 7

-a33

k

. k

(32)

Maupertuis-Lagrange

mixed variational formula

995

In addition to the governing equations (26)-(32), the M-L mixed variational formula indicates that the essential and the natural boundary conditions of the problem are given by: essential: specify uO,wO,ul, wl, w2, natural: specify N1,N3,A4r,~(TI

- T,),;

[

N3 +i(T,

+ Tz)

1.

We notice that the above governing equations do not contain any correction factor. This returns back to the fact that the utilization of the mixed variational formula allows one to treat the beam problems by introducing kinematics assumptions with any power of the thickness coordinate and the shear stresses which are consistencies with the surface conditions. So the rational for the shear correction factor is obviated. Moreover, equations (26)-(32) describe the deformation behavior of a beam loaded in such a manner that contact with a smooth rigid body is induced. This allows one to use these equations for contact problems involving a laminated beam with surface constrained. Furthermore, these equations require, in general, five boundary conditions along each edge of the beam. These boundary conditions give an exact description for the stress resultants at the beam edge. Finally, it should be mentioned that the present theory is of a rather complicated form, thus the question arises whether there is a practical need for such a theory. It is a question of the degree of accuracy required. So, in what follows, we shall illustrate numerically the practical significance of the inclusion of the transverse normal strain in the vibration problems of a laminated beam.

5. FREE

VIBRATION

OF AN AXIALLY

SYMMETRIC

BEAM

Here we shall calculate the natural frequencies for the free vibration analysis of a laminated beam. If the ends of the beam are completely clamped, then the boundary conditions at these ends are: u,=wO=ul=w,=wz=O,

atx=O,L.

For displacements uO,wo, u1 , w1 and w2, we choose the approximate identically satisfy the conditions (33) in the form: u. = C &F,(x) m

sin WC,

w. = 1 &F;(x)

sin ot,

u1 + W,,, = C X,F,(x)sinwt, m

(33) solutions which

(34)

w1 = 1 Y,Fk(x)sinwt, m w2 = C Z,Fa(x) sin or, m where the prime represents the differentiation with respect to x; o is the natural frequency; A,, B,, X,, Y,,,and Z, are arbitrary parameters, m is the mode number and F,(x) represents the form of vibration for bar arranged along x-axis with its ends clamped, and can be taken as [32]: F,(x) = L fi

[(COS&,X - cash p,,,x) - &,,(sinp,,,x - sinh&x)],

cos 1, - cash I, Pm = sin A, - sinh 1, ’

(35)

For this problem we shall use Ritz variational method [32]. With the help of equations (31) and (32), it is possible to get the functional (16) in terms of the displacements uo, wo. ui, w1 and w2 only. Substituting (34) into the functional J(uo, wo, u1 , wl, w2) and integrating with

A. M. Zenkour

996

respect to the time t from 0 to 2n/w. Taking the total variation of the obtained formula with respect to the parameters A,,,, B,,X,, Y,,, and Z,, we get a system of homogeneous equations in the previous five parameters. Using the condition that this system of equations has a nontrivial solution, we get the frequency equation in the 5 x 5 symmetric determinant form: IDijI=O,

i,j=1,2

,..., 5.

(36)

The coefficients Dij are given by g2 -

Dll = -

D

14

D22

=

A1144 A

13

,

_Ad2M2--nJ, L2

!d$ ((2_

A,)

_

=

=D

D

24

_.

34-

9

g) _ B1lh4;;$- ‘,),

h2D14(80B33 055

D12 = D1, = D15 = 0,

>

- 3~51~)+ As5h62;

%0A13

4OOL4 ’

where

and 0 = wh& is the frequency parameter. The forms of the frequency equations according to the classical (CBT) and first-order (FBT) beam theories may be obtained easily from (36).

6. NUMERICAL RESULTS AND CONCLUSIONS

To study the effects of transverse normal strain, transverse shear deformation and rotatory inertia forces, the natural frequencies of a single-layer anisotropic beam (structure I) and a three-layer isotropic beam (structure II), with total thickness h, are determined according to the classical, first-order and higher-order theories. The following material properties of structure I, typical of quartz crystals, are used [17): alI = az2 = 12.73, al2 = -3.67,

a33 = 9.71,

al3 = a23 = -1.49,

a44 = as5 = 19.66, a56 = al4 = -a24

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

a66 = 2(a,, - a22), = -4.23,

in (lo-’ cm2 kgff ‘).

Maupertuis-Lagrange Table 1. The frequency

parameter

0

mixed variational

(=wh&)

of an anisotropic

formula

991

beam according

to various

theories,

L/h = 2,5,8,10 m

Theory

Ljh

1

2

3

4

5

6

7

8

9

10

1.7032 1.5875 1.4508 1.3909

3.2943 2.9451 2.4879 2.1895

4.9090 4.2795 3.4721 2.7257

6.5221 5.5976 4.4599 3.0707

8.1280 6.9012 5.4573 3.2814

9.7274 8.1939 6.4642 3.4074

11.322 9.4785 7.4790 3.4829

12.914 10.757 8.5001 3.5290

14.503 12.031 9.5262 3.5578

16.090 13.301 10.556 3.5744

CCBTI

0.4570

0.9555

1.5545

2.2094

2.8880

3.5727

4.2552

4.9323

5.6031

6.2679

CBT FBT HBT

0.4367 0.4267 0.4248

0.8791 0.8280 0.8185

1.3800 1.2517 1.2176

1.9069 1.6765 1.5912

2.4416 2.0953 1.9259

2.9768 2.5084 2.2184

3.5097 2.9174 2.4693

4.0394 3.3237 2.6810

4.5659 3.7285 2.8570

5.0894 4.1326 3.0015

0.1925 0.1882 0.1862 0.1856

0.4148 0.3963 0.3840 0.3827

0.7065 0.6567 0.6194 0.6158

1.0543 0.9531 0.8748 0.8648

1.4423 1.2708 1.1380 1.1147

1.8567 1.6002 1.4028 1.3570

2.2869 1.9353 1.6667 1.5871

2.7251 2.2725 1.9286 1.8026

3.1661 2.6100 2.1884 2.0021

3.6068 2.9467 2.4463 2.1853

0.1248 0.1230 0.1221 0.1216

0.2708 0.2624 0.2566 0.2559

0.4670 0.4431 0.4243 0.4228

0.7080 0.6564 0.6140 0.6106

0.9855 0.8924 0.8160 0.8082

1.2914 1.1438 1.0241 1.0080

1.6180 1.4047 1.2345 1.2050

1.9588 1.6713 1.4452 1.3960

2.3085 1.9408 1.6553 1.5791

2.6632 2.2116 1.8642 1.7530

CCBTI CBT FBT HBT

2

5

CCBTI CBT FBT HBT

8

CCnTl CBT FBT HBT [CBT]:

10

Classical

beam theory

without

rotatory

inertia

forces.

Table 2. The dimensionless fundamental frequencies of a three-layer beam; 0 = wh&, y1 = vj = 0.3, vz = 0.3, hllh = h,lh = 0.25, h,lh = 0.5, L/h = 10

GI = %,

m Theory

GIIG

CCBTI CBT FBT HBT

2

CCnTl CBT FBT HBT

5

CCBTI CBT FBT HBT [CBT]:

15

Classical

1

2

3

4

5

6

7

8

9

10

0.1193 0.1175 0.1160 0.1136

0.2589 0.2509 0.2414 0.2372

0.4465 0.4237 0.3935 0.3874

0.6772 0.6277 0.5615 0.5524

0.9430 0.8536 0.7369 0.7225

1.2361 1.0943 0.9152 0.8917

1.5493 1.3442 1.0941 1.0565

1.8763 1.5996 1.2728 1.2150

2.2121 1.8578 1.4509 1.3658

2.5527 2.1172 1.6285 1.5082

0.1334 0.1314 0.1297 0.1270

0.2894 0.2805 0.2698 0.2652

0.4992 0.4737 0.4400 0.4331

0.7571 0.7018 0.6278 0.6176

1.0543 0.9544 0.8239 0.8078

1.3820 1.2235 1.0232 0.9970

1.7322 1.5029 1.2232 1.1813

2.0978 1.7884 1.4230 1.3584

2.4731 2.0771 1.6221 1.5270

2.8540 2.3672 1.8207 1.6863

0.1415 0.1394 0.1376 0.1347

0.3070 0.2975 0.2862 0.2813

0.5295 0.5024 0.4667 0.4594

0.8030 0.7443 0.6659 0.6551

1.1182 1.0123 0.8738 0.8568

1.4658 1.2977 1.0852 1.0574

1.8372 1.5940 1.2974 1.2529

2.2250 1.8968 1.5093 1.4408

2.6232 2.2031 1.7205 1.6196

3.0272 2.5107 1.9311 1.7886

beam theory

without

rotatory

inertia

forces.

Suppose that the total thickness of the outside layers of structure II is equal to that of the middle layer. Also all layers are assumed to have the same density. For an isotropic layer of Young’s modulus E, Poisson’s ratio v, and shear modulus G, we have 1 a,, = a*2 = a33 = -, E

a12 = al3 = az3 = --,

V

E

a44=a55=a66=p=-.

Z(l+v) E

1 G

For the sake of illustration, we have displayed the dimensionless frequencies of a singlelayer anisotropic beam according to various theories in Table 1. The dimensionless fundamental flexural frequencies of a three-layer isotropic beam according to various beam theories are presented in Table 2. The subscripts 1,2 and 3 refer to the top, middle and bottom layers.

A. M. Zenkour

998 6-

_/_. _.'.

01

: 12

:

: 3

; 4

: 5

: 6

: 7

; 8

: 9

: ' lOm

Fig. 1. Variation of w vs m of structure I for L/h = 5.

2.5..... m 2 -' _-. -.

.. .,.' ,. ,' /

-1.5..I

,' .p

1 ..

,,_;/$ ./ ,p ..g

0.5.'

O?

,y ,' : : 12

: 3

: 4

; 5

: 6

: 7

: 8

: 9

:i 1001

Fig. 2. Variation of 0 vs m of structure I for L/h = 10.

As-

4.'

\

3.5.' 3 -' 2.5.. 2 .' 1.5-. 1 .' 0.5.' 01 2

4

6

8

10

1 IJh

Fig. 3. Variation of w vs L/h of structure I for WI= 3.

The effect of rotatory inertia forces on the fundamental frequencies can be studied with the help of numerical results in Tables 1 and 2. It is seen that by ignoring the rotatory inertia forces the frequencies are only slightly increased. In Figs 1 and 2 we have plotted the dimensionless fundamental frequency of structure I vs the mode number m for L/h = 5,lO. But in Figs 3 and 4, it is vs L/h ratio for m = 3,7. Finally, in Figs 5-8 we have plotted the dimensionless fundamental frequency of structure II vs L/h ratio for different values of m in the case of G1/G2 = 5. For structure I, the numerical results show that the classical beam theory (CBT) and the first-order beam theory (FBT) overestimate the frequencies. CBT is sufficient with very few relative errors in some cases as: L/h 6 5, m -c 3

(Fig. I),

L/h > 5, m G 4

(Figs 2 and 3),

where the relative errors predicted by CBT do not exceed 6%. HBT does not provide a significant advantage over FBT in the following cases: LJh 2 8, m < 7

(Figs 2-4),

L/h G 5, m d 3

(Fig. 1).

Maupertuis-Lagrange 10 9 .. 8 -. 7 6 5 4

mixed variational formula

.. . .. .. .. --. -.m

'... '.., ;. ‘\ '!, \ \ \ \ \ ':. \ :\ \ ... '. 'L., ----__ '\ .... ---c+_

.. .. .. -.

3 .. 2 -.

999

--EBT

1 .. OJ

I

2

4

6

8

10

IJh

Fig. 4. Variation of 0 vs L/h of structure I for m = 7.

... --%I --_pBT

--

u”

OJ

1

2

4

6

8

10

ull

Fig. 5. Variation of w vs L/h of structure II for m = 3. 87 --

';.

6 .. ',..,, \ '\ \ :; \ '\ \ '.

’\

5 .' 4 .' 3 -' 2 .. 1 -.

___._._____ -..

01

I 2

4

6

8

10

IJh

Fig. 6. Variation of w vs L/h of structure II for m = 5. 12 10 .' 8 .. 6 .. 4 -' 2 .'

\. , '\\ , '.., \ '.. \\ \\ \'t \L__ ',"... .<... < +.*.;;_'i;;.;y_ a...

07

, 2

4

6

8

10

Lfb

Fig. 7. Variation of w vs L/h of structure II for m = 7.

In these cases, CBT predicts frequencies with relative error about 20% while the error by FBT is 5%. Therefore, in light of these results, the transverse normal strain has significant effect. But this effect is belittled when L/h > 5 and m < 3 where the maximum of the relative error compared with CBT or FBT does not exceed 2%. This means that the frequencies predicted by HBT are the most accurate ones.

1000

A. M. Zenkour 14 12

._-.... --I --_pBT

10

--Em

8/

01

2

4

6

8

10

IJh

Fig. 8. Variation of ci, vs L/h of structure II for m = 9

For structure II, as G1 : G2 ratio increases, the frequencies increase but the errors between all theories approximately remain constant. Therefore, the above discussion for structure I is applicable here to any value of G1 : G2 ratio. The main result of the current section shows that the higher-order beam theory (HBT), which includes the effect of transverse normal strain forces, is generally needed to provide a significant improvement in the level of accuracy over that afforded by FBT and CBT.

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7. 8. 9. 10.

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(1987).

11. W. Z. Chien, Variational principles and generalized variational principles for nonlinear elasticity with finite displacement. Appl. Math. Mech. 9 (l), 1-12 (1988). 12. E. Reissner, On a certain mixed variational theorem and a proposed application. Int. J. Numer. Meth. Engng 20, 1366-1368 (1984). 13. E. Reissner, On mixed variational formulations in finite elasticity. Acta Mech. 56, 117-125 (1985). 14. E. Reissner, On a mixed variational theorem and on shear deformable plate theory. Int. J. Numer. Meth. Engng 23, 193-198 (1986). 15. M. E. Fares, M. N. M. Allam and A. M. Zenkour, Hamilton’s mixed variational formula for dynamical problems of anisotropic elastic bodies. Solid Mech. Arch. 14 (2), 103-114 (1989). 16. F. Gantmacher, Lectures in Analytical Mechanics. Mir, Moscow (1975). 17. S. G. Lekhnitskii, Theory ofElasticity ofan Anisotropic Body. Mir, Moscow (1981). 18. K. Chandrashekhara and R. Tenneti, Non-linear static and dynamic analysis of heated laminated plates: a finite element approach. Compos. Sci. Technol. 51, 85-94 (1994). 19. J. N. Reddy, A general nonlinear third-order theory of plates with moderate thickness. Int. J. Non-Linear Mech. 25 (6), 677-686 (1990). 20. P. M. Naghdi, On the theory of thin elastic shells. Q. J. Appl. Math. 14, 369-380 (1957). 21. R. B. Nelson and D. R. Larch, A refined theory of laminated orthotropic plates. J. Appl. Mech. 41 (2), Trans. ASME, Vol. 96, Series E, pp. 177-183 (1974). 22. E. Reissner, On transverse bending of plates, including the effects of transverse shear deformation. Int. J. Solids Struct. 11, 569-573 (1975). 23. A. V. Krishna Murty, Higher order theory for vibration of thick plates. AIAA J. 15 (12), 1823-1824 (1977). 24. K. H. Lo, R. M. Christensen and E. M. Wu, A high-order theory of plate deformation: part 1, homogeneous plates; part 2, laminated plates. J. Appl. Mech. 44 (4), 663-668, 669-678 (1977). 25. J. N. Reddy, A simple higher-order theory for laminated composite plates. J. Appl. Mech. 51, 745-752 (1984). 26. J. N. Reddy, A small strain and moderate rotation theory of laminated anisotropic plates. ASME J. Appl. Me&. 54, 623-626 (1987). 27. E. Reissner, On the theory of bending of elastic plates. J. Mach. Phys. 23,184-191 (1944).

Maupertuis-Lagrange

mixed variational formula

1001

28. E. Reissner, The effects of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 12 (2) Trans. ASME, Vol. 7, pp. 69-77 (1945). 29. J. M. Whitney and N. J. Pagano, Shear deformation in heterogeneous anisotropic plates. ASSM 1. Appl. Mech. 37. 1031-1036 (1970). 30. J. N. Reddy, Energy and Variational Methods in Applied Mechanics. John Wiley, New York (1984). 31. F. Essenburg, On the significance of the inclusion of the effect of transverse normal strain in problems involving beams with surface constrains. J. Appl. Mech. 42 (1) Trans. ASME, Vol. 97, Series E, pp. 127-132 (1975). 32. L. V. Kantorovish and V. I. Krylov, Approximate Methods of Higher Analysis. Fismatgiz, Moscow (1962).