On a consistent first-order shear-deformation theory for laminated plates

PII: S1359-8368(96)00058-3

ELSEVIER

Composites Part B 28B (1997) 397-405 © 1997 Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-8368/97/$17.00

On a consistent first-order shear-deformation theory for laminated plates

Norman F. Knight Jr and Yunqian Qi Department of Aerospace Engineering, Old Dominion University, Norfolk, VA 23529-0247, USA (Received 10 Apri/ 1995; accepted 7 May 1996) This paper systematically states the consistent first-order shear-deformation theory for laminated plates recently proposed by Qi and Knight. It assumes that only in an average sense does a straight line originally normal to the midplane remain straight and rotate relative to the normal of the midplane, and in a local sense a slight displacement perturbation around the average rotated line is also permitted after deformation. Since the curved line is very shallow, the present theory still approximates linear in-plane and constant transverse displacements through the thickness just as Reissner and Mindlin's first-order shear-deformation theory does. Reissner and Mindlin's theory leads to uniform transverse shear strain distributions by employing pointwise straindisplacement relationships, and satisfies the transverse shear constitutive relationships only in an average corrected form. In contrast, Qi and Knight's theory accounts for variable transverse shear strain distributions by enforcing pointwise constitutive relationships, and relates transverse shear strains to kinematic unknowns only in a weighted-average form. Through-the-thickness transverse shear strains are thus consistent with the stress counterparts and their transverse-shear-stress-weighted-average values are just the nominal-uniform transverse shear strains which correspond to the average rotations. The new theory combines the advantages of several prevailing 2D laminated plate theories while overcoming their drawbacks. Numerical results for the cylindrical bending problem of orthotropic laminated plates exhibit excellent agreement between Qi and Knight's theory and Pagano's 3D exact elasticity results. © 1997 Elsevier Science Limited. (Keywords: A. laminates; C. analytical modelling; shear-deformation theory)

INTRODUCTION The importance of transverse shear effects for fiberreinforced laminated plates has been well recognized. Appropriate distributions of transverse shear strains through the thickness of plates should be taken into consideration in order to predict both global and local responses of structures accurately. Traditional first-order shear-deformation theory (FSDT) was proposed by Reissner I for isotropic materials and is based on the assumptions that the in-plane stresses are distributed linearly over the thickness of the plate. Transverse shear stresses exhibit parabolic distributions. These stress field assumptions are consistent with the displacement field assumptions which include linearly distributed in-plane displacements and constant transverse deflection over the plate thickness, which was later explicitly restated by Reissner 2. Such displacement field assumptions are also made by Bolle 3 and Mindlin 4. The displacement-based FSDT is more widely used today as the de facto version of FSDT, even though the displacementbased FSDT leads to uniform transverse shear strains through the plate thickness and requires a shear correction factor to accommodate parabolic transverse shear stresses. It is widely regarded that FSDT is inadequate for local parameter prediction when applied to laminated plates,

because the transverse or interlaminar stresses recovered from the constitutive relationships are not continuous through the thickness of the laminate. Many higher-order, shear-deformation theories 5-m have been proposed as attempts to account for variable distributions of transverse shear strains that FSDT fails to do. However, the continuity of transverse shear strains based on displacement assumptions contradicts the continuity requirement of transverse shear stresses at interfaces of dissimilar layers when constitutive relationships are used. The involvement of higher-order terms also makes higher-order theories computationally expensive. Layerwise theories 11 15 assume separate displacement field expansions (either first-order or higher) through the thickness of each layer and usually satisfy the continuity requirement of transverse shear stresses through the thickness of anisotropic plates. Even though layerwise theories yield better approximations for both global and local response prediction, the number of independent variables is significantly increased and makes them impractical or computationally prohibitive. One exception is the zigzag theory developed by Di Sciuva 12. The first-order, shear-deformation theory still remains the most widely used theory for anisotropic laminated plates in engineering applications owing to its simplicity and low computational cost. It is generally regarded that FSDT is

397

Shear-deformation theory for laminated plates: N. F. Knight Jr and Y. Qi adequate for global response determination (e.g. deflections, natural frequencies, buckling loads and stress resultants) if used with proper shear correction factors. Investigators 16-21 have been working on the estimation of the composite shear correction factors upon which the range of validity of FSDT is strongly dependent. Frequently, these procedures for calculating the shear correction factors are simply ignored for laminated plates and the value of 5/6 from isotropic plates is used instead. Theories which alleviate or discard the need for these factors should provide improved response if for no other reason, because the shear deformation is correctly modeled. The transverse shear strains derived directly from the displacement fields of FSDT are constant, giving rise to piecewise constant transverse shear stresses through the laminate thickness if the constitutive relationships are used. However, if the equilibrium equations are utilized, the transverse shear stresses so obtained would be piecewise quadratic through the composite thickness. In other words, FSDT gives inconsistent local response determination for plates. Recently, Qi and Knight 22 proposed a refined first-order, shear-deformation theory for laminated plates. Even though retaining the displacement assumptions of Reissner and Mindlin's FSDT, Qi and Knight's refined theory obtains variable transverse shear strains by enforcing the constitutive relationships between transverse shear stresses and strains in a pointwise manner through the thickness. The uniform transverse shear strains that are directly derived from the displacement field assumptions correspond to their transverse-shear-stress-weighted-average values based on the equivalent shear strain energy. By imposing pointwise transverse shear constitutive relationships and permitting transverse shear strains to vary, Qi and Knight's refined theory removes the inconsistency between transverse shear stresses and shear strains which exists in Reissner and Mindlin's FSDT. Without losing the simplicity of Reissner and Mindlin's FSDT, Qi and Knight's refined theory yields excellent agreement for both global and local response parameters (deflections, transverse shear strain and stress distributions) with the exact solution for cylindrical bending problems of cross-ply laminated plates given by Pagano 23. Being different from Reissner and Mindlin's FSDT in that the transverse shear stresses and strains are always pointwise consistent in terms of the constitutive relationships, Qi and Knight's theory is referred to as a consistent first-order shear-deformation theory (CFSDT). The physical explanation of the underlying assumption of Qi and Knight's CFSDT is offered, and a line of demarcation between CFSDT and FSDT is drawn in this paper. Numerical results for the cylindrical bending problem of orthotropic laminated plates using CFSDT are obtained and compared with Pagano's three-dimensional (3D) exact elasticity results.

THEORETICAL ANALYSIS Kirchhoff's classical plate theory imposes two constraints to suppress transverse shear flexibility--a line initially straight

398

C

v

/

J

/ /

Figure 1 ComparisonamongCPT, FSDT and CFSDT

and normal to the midplane before loading remains both straight and normal after loading, or no rotation relative to the normal is allowed, as indicated by the solid normal line AOB in Figure 1. Reissner and Mindlin's FSDT allows a uniform transverse shear strain by relaxing one of these two constraints. Namely, a line initially straight and normal to the midplane remains straight but not necessarily normal after deformation, and hence the line is permitted to rotate with respect to the normal of the midplane, as represented by the dashed line COD in Figure l. The relative rotation angle is the nominal-uniform transverse shear strain. Qi and Knight's CFSDT goes one step further and lifts the remaining constraint--a line initially straight and normal to the midplane may become curved and is hence not normal after deformation, as illustrated by the solid curve CEOFD in Figure 1. A line through the thickness is allowed to rotate relative to the normal of the midplane and is also allowed to become slightly curved (or deflected) around the rotated line in a manner that enables transverse shear strain to be consistent with the transverse shear stress in terms of the constitutive law at each point through the thickness. However, the radius of the curved line is so large that the in-plane displacement is still approximated as a linear function through the thickness, and the slope is the average rotation with respect to its originally undeformed position. Alternatively, Qi and Knight's CFSDT assumes that a line that is initially straight and normal to the midplane remains straight, and rotates relative to the normal of the midplane only in an average or global sense. Also in a local sense, a slight displacement perturbation around the hypothetically rotated average line is permitted. The relative rotation angle, which is the average one of the line in some sense, corresponds to some weightedaverage transverse shear strain through the thickness. The difference between the curve CEOFD and the dashed line COD is the newly introduced local displacement perturbation which enables transverse shear strain to vary to an extent that is compatible with the transverse shear stress and the constitutive relationship. Even though introducing a slight displacement perturbation around the average rotated line compared with Reissner

Shear-deformation theory for laminated plates: N. F. Knight Jr and Y. Qi Table 1

Comparison between Reissner-Mindlin's FSDT and Qi-Kni hts CFSDT for the bending problem of an isotropic beam

Reissner-Mindlin's FSDT

Qi-Knight's CFSDT

UNDERLYING ASSUMPTION: A straight line initially normal tO the midplane remains strictly straight and rotates with respect to the normal of the midplane when deformed. Its rotation with respect to the originally undeformed position is 0(=).

UNDERLYING ASSUMPTION: Only in some average sense does a straight line initially normal to the midplane remain straight and rotate with respect to the normal of the rnidplane when deformed. Also in a local sense a slight displacement perturbation around the average rotated line is allowed. Its average rotation with respect to the originally undefomed position is O(z) which is equivalent to 0(x).

KINEMATICS: The in-plane displacement is there- KINEMATICS: The in-plane displacement is therefore fore assumed to be linear through the thickness. The; strictly not straight but slightly curved through the thickness. However, it is so shallow that the in-plane displacetransverse deflection is independent of z. ment is still assumed to be linear through the thickness. The u.(=, ~) = ~(=) + • o(=) transverse deflection is independent of z. u, (=, z) = ~(=) u . ( = , z) = u(=) + z ~(=)

,,.(=, .) = ,,,(=) S T R A I N . D I S P L A C E M E N T RELATION for the transverse shear strain is satisfied pointwise through the thickness. Transverse shear strain %z(=, z) is derived directly from the kinematics assumptions and is therefore constant or uniform across the thickness

~,,(=, z) = w.= + 0 = ~ , ( = )

S T R A I N - D I S P L A C E M E N T RELATION for the transverse shear strain in a pointwise form is relaxed and only satisfied in a weighted-average sense. Transverse shear strain 7=~ (z, z) is assumed to vary across the thickness and its shear-stressweighted-average value equals the nominal-uniform trans. verse shear strain, T=z (z). h

f ,'..(=, ~)'r=(=, ~)d~

where ~'=,(=) is refered to as the nominal-uniform transverse shear strain.

-h

h

f -h

= ~'..(z) = w,. + ~

r..(,', z)dz

EQUILIBRIUM EQUATION is pointwise satis- E Q U I L I B R I U M EQUATION is pointwise satisfied. The fied. The transverse shear stress is integrated from transverse shear stress is integrated from the equilibrium the equilibrium equation along with the inplane equation along with the inplane stress. stress,

h

h

r=z(,,z)--/zEO,==dz

-~3V=(z)I1 - (h)' ]

&

&

C O N S T I T U T I V E RELATION between transverse C O N S T I T U T I V E RELATION between transverse shear shear stress and shear strain is satisfied only in an stress and shear strain is enforced pointwise through the thickaverage-corrected form. ness. h

,-.. (=, z) = a - t . . (=, z)

= I¢2(2h)G~=,(x) -h

RESULT:

RESULT:

v . . ( = , ~) = ~ . . ( = ) = w,. + o

v..(-,=)

= i(,,,,. +~)

-

....,z) = p(o,. + ,) [, -

']

h

r..v..dz

g.(=) = ~

=

F(2h)G(w,~

+

o) 2

h

Uo(Z) = ~

rx, Tz, dz =

(2h)G(w = +0-) =

-h -h

and Mindlin's FSDT, Qi and Knight's CFSDT ends up with the same form of displacement kinematics as does FSDT for linear in-plane and constant transverse displacements. However, CFSDT departs from FSDT immediately, and the difference is summarized and compared as shown in Table 1. In both theories, the in-plane stresses and strains are

treated identically and therefore are not included in the table. Both theories are used to solve a bending problem of an isotropic beam of thickness 2h, which suffices to show the preference of CFSDT. Based on the linear in-plane and constant transverse displacement fields, Reissner and Mindlin's theory as

399

Shear-deformation theory for laminated plates: N. F. Knight Jr and Y. Qi indicated in Table 1 first employs a pointwise straindisplacement relationship, obtains uniform transverse shear strain through the thickness, and leaves the transverse shear constitutive relationship satisfied only in an average corrected form (i.e. the shear correction factor k 2 is required). Since a linear form of in-plane displacement is just an approximation, its use with the pointwise straindisplacement relationship results in only an approximation of the transverse shear strain. In contrast to FSDT, CFSDT imposes the constitutive relationship between transverse shear stress and shear strain in a pointwise form through the thickness, accounts for variable distribution of transverse shear strain, and satisfies the strain-displacement relationship for transverse shear strain only in a weighted-average form (i.e. not an average corrected form, thus no shear correction factor is involved). The equilibrium equation is satisfied pointwise in both theories. When the shear correction factor in FSDT takes on an appropriate value (k 2 = 5/6 for isotropic materials), both theories lead to the same value of transverse shear strain energy U~(x) per unit length, and therefore both theories predict the same results for global response parameters such as deflection and stress resultants. Since CFSDT leads to a viable transverse shear strain distribution that is consistent with the stress counterpart through the thickness in terms of the constitutive relationship, CFSDT will successfully predict local response parameters such as the through-the-thickness transverse shear strain while the method based on the traditional FSDT fails to do so. Qi and Knight 22 applied CFSDT to the cylindrical bending problem of symmetric cross-ply laminated plates. For pure bending problems of semi-infinite, symmetric laminated plates with total thickness of 2h, the displacement field in CFSDT is assumed as

Ux(X,Z)=ZO(x )

(1)

Uz(X, z) = w(x) where u~ and u z are displacements in the x- and z-directions, respectively. Note that the transverse deflection w(x) and the average rotation angle 0(x) relative to its initial position_are independent of the thickness coordinate z. Furthermore, 0 in eqn (1) is simply replaced by 0 in Reissner and Mindlin's FSDT. The transverse shear strain obtained directly from the displacement field assumptions, eqn (1), is constant or uniform through the plate thickness as in Reissner and Mindlin's FSDT. It does not represent the actual distribution and is therefore referred to as the nominal-uniform transverse shear strain, ~xz. That is,

OUx

5,x~(X)= ~-z+

Ouz=O + w x Ox

(2)

where a subscript comma denotes differentiation with respect to the variable ensued. Thus, the term ~xz represents some average value of the through-the-thickness transverse shear strain. The transverse shear strain energy per unit length, Us(x), can be expressed in an integral form by using either the actual value of the transverse shear strain,

400

7xz, or its average value, 5%, as Us (x) = ~

_ hrxz(X, Z)"/xz(X, Z) dz ---- ~'~xz(X)

_ hrxz(X, Z) dz

(3) From eqn (3), the nominal-uniform transverse shear strain 5% can be shown to be equal to the weighted-average value of the transverse shear strain through the thickness where the weighting factor is just the corresponding actual transverse shear stress, rxz(x,z). That is,

~/xz(x) =

_ hrxz(X, Z)yxz(X, Z) dz

(4)

fh_hT"Xz(X,z) dz The actual pointwise transverse shear strain, "Yxz(X,Z), cannot be obtained directly from the strain-displacement relationship given by eqns (1) and (2). However, it satisfies the constitutive relationship pointwise or is consistent with the transverse shear stress

• xz(X, z) = c ~ x z ( x ,

z)

(5)

where G(~kz) is the transverse shear modulus for the kth layer. As shown in 24, it is appropriate to interpret the average rotation angle with respect to its initial position, 0, as the transverse-shear-stress-weighted-average one through the thickness rather than as the one determined from a linear regression of the in-plane displacement. Alternatively, the average rotation angle should be calculated from eqn (2) instead of eqn (1). For plane-strain bending problems of orthotropic laminated plates, the normalized-distribution shape of the through-the-thickness transverse shear stress and shear strain are shown by Qi and Knight ee to be independent of the span-to-thickness ratio and external loading of plates. A closed-form relationship between actual transverse shear strain "Yxz and the nominal-uniform transverse shear strain 5% can be obtained a priori. The in-plane strain ex can be obtained from eqn (1) and the strain-displacement relationship. The in-plane stress a~ can be calculated from the constitutive relationship just as in FSDT. By using a non-dimensional thickness coordinate ~" given by Z

~'=~

fE[-1,1]

(6)

the in-plane strain, ex, and in-plane stress, c% are given as

ex(X, z) = ZO,x = ~hO,x

(7)

. . a(k) ,q(k) e x = Z '.~(k)7~ Ox(X,Z)= x =~d~ll ~ l l V , x = 5 ~.h,q(k)O ~11 ,x

(8)

where Q~) is the reduced stiffness matrix component for the kth layer. Note that since all stress and strain components are assumed to be functions of both x and z (or ~') in CFSDT, their superscript (k), which usually appears in FSDT, is no longer necessary for the stress and strain components. From one of the equilibrium equations without the body force and by assuming traction-free boundary conditions,

Shear-deformation theory for laminated plates: N. F. Knight Jr and Y. Qi Table 2 Analogy among extensional, bending and transverse shear parameters Strain components Base-strain components Distribution shape functions

Extensional parameters 0 e~(e)(x, ~') = ex(x)

Bending parameters e}b)(x, ~') = [h~']~(x)

3/xz(X,

e°(x)

r~(x)

~xz(X)

h'~

f.~H.~(f)

1 1

,

0.2

Strain energy components

U~ =

Stiffness components

A=

Force or moment resultants

N , = A e ° -- 2U¢ 80

Ub

~Aie~)

_ ~Q~l)hdr

~(k)

(9)

in terms of the non-

dimensional thickness coordinate,

7xz(X,~)=h2O xx IlrrlQ~kl) d~

2Ub

H*(f)= 3Vdx)rxz(X'~)=

f 1_1 ( f r~Q~ d~) d~

Vt =

Kx

(10) em

j-i Q~)dr/

f =

7xz

4

]

H,(~) Sc~

"~ J

(14)

I_ 1H~(~lH.y(~)d~

strain and stress can be expressed,

"yxz(X,~) = "y~m(x)Hv(~)=fv%z(x)H~(~)

(12)

(15)

and

rxz(X, r) = C~ks)Txz(x, ~) =

zemn~(~) =

f'Y~xz(x!HT(r) S

(16)

The magnitudes of transverse shear stress and shear strain can be related by the effective transverse shear compliance

of the plates. That is,

S

(17)

S

Both distribution shape functions for homogeneous plates take the same special parabolic form

c55

H~(~') d~" _ 3 I t H~.(~') d~" 4 - t _(k~ lHr(~ ") d~" c55

- 1 c55 _(k)

S=

~xz(X) =f~5%(X)

1

tern(X)-- q/era(Z)-- fvTxz(X)

where S is the effective transverse shear compliance of the plates, which is actually the stress-weighted-average transverse shear compliance of the plates and defined as ~i

"Yxz

and span-to-thickness ratio. Hence, the final through-the-thickness distributions of

Hr(~') H,(~')~Sd f~ -

2Us

The ratio f~ is obviously independent of external loading

where Vx(x) is the transverse shear stress resultant. The normalized-distribution shape function of transverse shear stress H~(~') is a continuous piecewise-quadratic function through the thickness and is independent of the longitudinal coordinate x. The normalized-distribution shape function of transverse shear strain H~(~') can be defined in a similar manner by using the transverse shear constitutive relationship. Here the pointwise transverse shear constitutive relationship is imposed, giving

H~(~')= _3 f l

f 3'.tz -

f~_ H~(~') d~" (X)=

transverse shear respectively, as

c(k) 55

~~/vH.~(~')]c55h d~-

respective effective magnitudes, rxem and 7"xzem,and their respective normalized-distribution shape functions, H~(~') and H~(~-). The effective magnitude of the transverse shear strain rx~z can be related to its nominal-uniform value "Yxzby a constant ratio fv based on equivalent shear strain energy. As such,

Hence, the normalized-distribution shape function of transverse shear stress, H,(~'), can be introduced and defined as 4h

~') = ~f~H.~(~')]5%(x)

U~ = 12F(~7~)2

, [hrl2Q~)hdr

M x = Dt~x -"

Jj-,XX~l, dz rh 0

1D(~)22 "

D=

one can easily calculate the transverse shear stress from

T~z(X,z) = r~z(x, h) + fJzhoax Ox d z = where r~z(x,h) vanishes. Alternatively,

=

Transverse shear parameters

(13)

Unlike the normalized-distribution shape function of transverse shear stress H~(~'), the normalized-distribution shape function of transverse shear strain Hv(~') generally exhibits only a layerwise quadratic form and is usually discontinuous at interfaces of dissimilar layers. The through-the-thickness transverse shear stress rxz and shear strain 3'xz can be expressed as the products of their

H(~') = 1 - ~.2

(18)

and the effective magnitudes are their corresponding maxim u m values. In this case, the ratio f~ can be shown to be a constant equal to 5/4, and the effective transverse shear compliance S is just the transverse shear compliance of the material (1/c55). Both distribution shape functions defined in eqns (1 1) and (12) are normalized such that ffl_ lHr(~.) d~. = I1 lH.v(~.) d~._ - f t -1 (1 _ ~.2) d~. = 4

(19) which is why a factor 3/4 or 4/3 is present in eqns (11)(13).

401

Shear-deformation theory for laminated plates: N. F. Knight Jr and Y. Qi Table 3

Stacking sequence and stiffnesses

Number of layers

Stacking sequence

D X l 0 3 lb in

F X 10 3 lb in

F~ N 10 3 lb in

1 16 16 16

[0] [90/0] 4s

1069 459.8 832.7 279.4

33.34 18.62 17.92 15.67

33.34 23.33 23.33 23.33

[03]903]0]90]s [903/03/90/0] ~

The transverse shear strain energy per unit length, U~, is thus expressed by Us(x ) :

[I

1 1 2 ~ ~ _ 1T~z(X, ~)q/xz(X, ~)h d~'= ~F~xz(X )

(20)

1.2

'

I

'

F-

4hfv

(22)

402

'

I

CPT CFSDT

&

FSDT

I

I

I

f

/

/ /

0.6

0.4

3S The transverse shear compliance S is the equivalent of a material property and is independent of the total plate thickness, whereas the transverse shear stiffness F represents the overall ability of the plate to resist transverse shear loadings and is proportional to the plate thickness. It is demonstrated in Table 2 that the effective transverse shear stiffness F is the counterpart to the extensional stiffness A and the bending stiffness D, and is also analogously defined. All three strain components [e (e), e(? ), "Yxz] can be expressed in terms of their respective base-strain components (s °, ~x, 5'xz) and their corresponding distribution shape functions through the thickness [1, h~', fvH,(~')]. All three stiffness components (A, D, F) are defined relative to their respective base-strain components (e °, Kx, 5'xz) based on the strain energy component expressions (U¢, Ub, U0. It can be seen from 5'xz = 2U]V~ that the ratio of Us to V~ defines a uniqueness for the transverse shear base-strain-only defining the nominal-uniform transverse shear strain ~xz as the transverse shear base-strain enables a consistent analogy among the transverse shear parameters, the extensional parameters and the bending parameters. Since the selection of base-strain component and the definition of stiffness component are uniquely related, eqn (21) is the only appropriate definition for the transverse shear stiffness in order to achieve such an analogy. For laminated plates, the outer layers are well recognized to have a greater contribution to the bending stiffness than the inner layers. The inner layers, however, play more important roles in resisting the transverse shear deformation than the outer layers. The transverse shear strain distribution shape function is positive-definite, is zero at the top and bottom surfaces, and usually achieves a maximum value near the midplane of the plates. Since transverse normal strain energy is negligible in the plate where CFSDT is applied, the strain energy components given in Table 2 give the total strain energy of the laminate plate, which can be used to solve bending problems

I

/

(21)

The effective transverse shear stiffness F is related to the effective transverse shear compliance S by the following expression

/

/

0.8

(k) 2 d~" _ lc55H~(~)

'

1.0

where the parameter F represents the effective transverse shear stiffness of the plate and is expressed by

F = hf~

I

......... I

0.2

I

i0

0

I

20

30

I

I

40

50

R

(a) for Single [0] Layer 1.2

I

'

I

'

I

'

I

,

I

"---......

1.0

//

.i

/

,/

/

/

0.8

/ / / /

/ /

0.6

/ / / / / / / / / /, I

0.4

0.2 0

....... ......... I

I0

I

CPT FSDT CFSDT I

20

I

30

40

I

50

R

(b) Figure 2

for

[03/903/0/903,

Laminate

Transverse deflection comparison

if proper boundary conditions and loadings are given. If the expression of transverse shear strain energy given in Table 2 is used to estimate the shear correction factor as defined in Reissner and Mindlin's FSDT for symmetric cross-ply laminates, the shear correction factor so obtained is in exact

Shear-deformation theory for laminated plates: N. F. Knight Jr and Y. Oi

~"

1.0

1.0

0.5

0.5

....... Exact R=10 .......... Exact R= 4 CFSDT R=4,10

0.0

'~ i

¢v,

0.0

I /i'

t

)/

I I I

i i i i

-0.5

....... .....

i

-0.5

I

FSDT CFSDT Exact

i

i

-1.0 0

-1.0 0.5

0.0

i

4

2

1.0

I

I

8

xl0 -4

6

"Yxz

H 1-

(a)

1.o.

Figure 3 Normalized transverse shear stress distribution comparison for [03/90 J0/90] s laminate

R= 4 ii

I

I I I

1.0

'

/

0.5 -

I

I I

i I

\ 0

.

5

0.0

~

I

i

/

I I

-0.5

I I

0.0

- 0 . 5

-i.0 .......

;

....... F S D T

!I

.....

; Ii

, ~ I 0.0 0.5

CFSDT Exact

I

1.0

I

I

2.0

1.5

xO l -3

"Yxz

-1.0

0.0

(b)

0.5

1.0

HT Figure 4 Normalized transverse shear stress distribution comparison for [03/90 /0/90]~ laminate using FSDT (R = 4 and 10)

agreement with the factor obtained by Chow] 6 and Whitney 17. According to Noor and Burton ]8']9, the use of the composite shear correction factor proposed by C h o w ]6 and Whitney ]7 results in fairly accurate global response characteristics for a wide range of lamination and geometric parameters, which, from another point of view, indicates that CFSDT should be adequate for accurate global response prediction without using hypothetical shear correction factors.

NUMERICAL RESULTS The cylindrical bending problems of orthotropic laminated plates are solved by using CFSDT. The results are compared

5 Transverse [03/903/0/90] ~ laminate

Figure

shear

R=10 strain

distribution

comparison

for

with the exact elasticity solutions of the same problems obtained by Pagano 23. The composite layers are assumed to possess the following stiffness properties, which simulate a highmodulus graphite/epoxy composite material system EL=

25

× 106

GLT = 0 . 5 × 106 p s i

psi

ET =

106 p s i

(23)

G T T - - - - 0 . 2 × 106 p s i

gET = ~rr = 0.25 where L and T denote the longitudinal and transverse ply material directions, respectively. Four laminates are considered as indicated in Table 3. In each case, the total thickness of the laminated plates is assumed to remain the same, 0.08 in, while the span varies

403

Shear-deformation theory for laminated plates: N. F. Knight Jr and Y. Oi in order to keep the span-to-thickness ratio, R = L/(2h), to be 4, 10, 20 and 50, respectively. In Table 3, D and F are defined in Table 2, and Fa is the average transverse shear stiffness defined by h 5 f c(k) Fa=~j_h 55 dz

(24)

which agrees with Reissner and Mindlin's FSDT if a shear correction factor k 2 of 5/6 is used and stacking sequence is not taken into account. The single-layer plate represents a homogeneous case which also includes the isotropic plate as a specific one. The three 16-ply plates possess distinct bending stiffness values D in Table 3. Although having the same values for the average transverse shear stiffness F~, they have different values for the effective transverse shear stiffness F, since the latter parameter takes stacking sequence into account, just like the bending stiffness D does. For all cases of the cylindrical bending problems, the lateral load is assumed to be equally distributed over both the top and bottom surfaces of the plates as

P(x, +h) =+- ~Po s i n ( L )

(25)

where P0 = 100 psi in all examples. The semi-infinite plates are simply supported at x = 0,L. The analytical solution is readily obtained from the principle of minimum total potential energy where the transverse shear strain energy is represented by eqn (20). Further expressions for such analytical results are available in Qi and Knight 22. For the same bending problems, the exact elasticity solutions by Pagano 23 are also available and serve as benchmark solutions. Different theories are used to obtain four groups of results for transverse deflection at the midplane and half span. The following three groups of results are compared with the exact results from Pagano's exact elasticity theory: Group 1. Results by CFSDT,denoted by CFSDT Group 2. Results by FSDT with k2 = 5/6, denoted by FSDT Group 3. Results by classical plate theory,denoted by CPT Normalized values of transverse center deflection for each group with respect to the exact ones wi/we (i = 1,2,3) versus span-to-thickness ratio R are shown in Figure 2. For the single-layer case, CFSDT coincides with FSDT in terms of transverse deflection. For the multi-layer case, CFSDT always gives the best estimates of transverse deflection for laminated composites. For all examples considered here, the error in the transverse deflection prediction with CFSDT is less than 1% when the span-to-thickness ration is greater than or equal to 10. Comparisons of the normalized transverse shear stress distributions through the laminate thickness are shown in Figure 3. CFSDT assumes that the normalized distribution of transverse shear stress through the thickness remains independent of the span-to-thickness ratio, R. The validity of this assumption is verified by the 3D exact elasticity results. Two examples (R = 4 and 10) are offered in Figure 3. When the span-to-thickness ratio is 10, the assumed distribution by CFSDT (solid line) is very close to the exact distribution given by the dash-dot line in Figure 3. The

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actual distributions for thinner plates (i.e. larger values of R) will gradually converge to the one assumed by CFSDT. Comparison of the normalized transverse shear stress distribution through the plate thickness by using Reissner and Mindlin's FSDT is plotted in Figure 4. When the transverse shear stress is evaluated from the equilibrium equation combined with the piecewise linear in-plane stress, which is usually performed as a post-processing procedure, its normalized distribution (solid curve, denoted by Equ. in Figure 4) is in exact agreement with that predicted by CFSDT (solid curve in Figure 3). If one determines the transverse shear stress from the constitutive relationship combined with the uniform transverse shear strain distribution, the normalized distribution of transverse shear strain so obtained would be piecewise constant and discontinuous at layer interfaces (dashed line, indicated by Con. in Figure 4), which is far from adequate to represent the actual result. In other words, the transverse shear stress in FSDT obtained from the equilibrium equation is inconsistent with that obtained from the constitutive relationship. In contrast, the transverse shear stress in CFSDT obtained either from the equilibrium equation or the constitutive relationship is identical, in which sense the newly proposed approach is consistent. Note that the normalized distribution of the transverse shear strain is independent of the span-tothickness ratio in both FSDT and CFSDT. The actual distribution of transverse shear strain obtained from Pagano's 3D exact elasticity theory is shown in Figure 5. Just as anticipated, it exhibits great diversity through the laminate thickness and is discontinuous at the interfaces of dissimilar layers. The prediction for transverse shear strain by CFSDT can be taken as an accurate approximation when the span-to-thickness ratio R is greater than or equal to 10. FSDT, however, leads to a uniformly distributed result through the thickness and hence fails to account appropriately for the actual transverse shear effect. In conclusion, CFSDT offers a rather better approximation for transverse deflection than the FSDT and also accounts for the through-the-thickness distribution of transverse shear strain which FSDT fails to do. In its domain of validity (plate is only moderately thick, or R --> 10), CFSDT is adequate for both global response prediction and local response prediction.

SUMMARY Physically, CFSDT assumes a slight local displacement perturbation around the average rotated line compared with FSDT. Even though the newly introduced displacement perturbation does not affect the approximate forms of the displacement field, it does account for what the actual deformation really is. Since linear in-plane and constant transverse displacement fields are just approximations, CFSDT relaxes the strain-displacement relationship for transverse shear strain and satisfies it only in an average form, in favor of satisfying the constitutive relationship, the equilibrium equation, the boundary conditions and providing for continuity of transverse shear stress in a pointwise

Shear-deformation theory for laminated plates: N. F. Knight Jr and Y. Qi form. Transverse shear strain is allowed to vary through the thickness and, based on equivalent shear strain energy, its transverse-shear-stress-weighted-average value is shown to be just the nominal-uniform transverse shear strain which is derived directly from the assumed kinematics. As such, a consistent deformation field is achieved where the equilibrium equation, the constitutive relationship and the boundary conditions are simultaneously satisfied. By explaining the physical deformation more reasonably and utilizing elasticity equations more appropriately, the current CFSDT combines the advantages of the prevailing theories and also successfully overcomes their drawbacks. CFSDT retains the simplicity of FSDT while discarding need for a shear correction factor; CFSDT accounts appropriately for the variable distribution of transverse shear strain without using higher-order terms for displacement field; CFSDT satisfies the proper continuity requirement of transverse shear stress without assuming separate displacement expansions for each layer. Based on a selfconsistent deformation field, CFSDT leads to a more accurate expression for transverse shear strain energy and hence a more accurate prediction of global response. Prediction of local response can also be consistently determined from either the equilibrium equation or the constitutive relationship. CFSDT is successfully applied to the plane-strain bending problem of symmetric cross-ply laminates for which the normalized distributions of transverse shear stress and shear strain can be obtained a priori. Even though the consistent first-order shear-deformation theory generally holds within its range of validity, its application to the bidirectional bending problems of arbitrary anisotropic laminated plates is anticipated to be complicated.

4.

5. 6. 7. 8.

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

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