Composile SWucturrs 37 (1997) 195-203 0 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved K&263-8223/97/$17.00 ELSEVIER
Pll:SO263-8223(96)00014-7
An exact solution for the bending of thin and thick cross-ply laminated beams A. A. Khdeif
& J. N. Reddf *
“Department of Civil Engineeting, Division of Structural Mechanics, Middle East Technical University, 06531 Ankara, Turkey “Department of Mechanical Engineering, Texas A & M University, College Station, TX 77843-3123, USA
The state-space concept in conjunction with the Jordan canonical form is presented to solve the governing equations for the bending of cross-ply laminated composite beams. The classical, first-order, second-order and third-order theories have been used in the analysis. Exact solutions have been developed for symmetric and antisymmetric cross-ply beams with arbitrary boundary conditions subjected to arbitrary loadings. Several sets of numerical results are presented to show the deflected curve of the beam, the effect of shear deformation, the number of layers and the orthotropicity ratio on the static response of composite beams. 0 1997 Elsevier Science Ltd.
of which possess different hygro-thermo-mechthicknesses and geometric anical properties, orientations. The effects of shear deformation on isotropic beams was initially examined by Timoshenko [3]. A higher-order theory for short beams has been introduced by Krishna Murty [4]. This theory is in the form of a hierarchy of sets of governing equations, with each set describing the beam behavior to a certain degree of approximation. The zeroth-order approximation of this formulation corresponds to Euler-Bernoulli theory, while the first-order approximation corresponds to Timoshenko’s shear-deformation theory. Rehfield & Murthy [5] carried out the stress analysis of the beam simply supported at both ends and discussed the effects of transverse shear, non-classical axial stress and transverse normal strain on the deflections of beams. In a study by Krishna Murty [6] an improved theory for the bending analysis of beams is given. This formulation is in the form of three simultaneous ordinary differential equations. Numerical studies, with a shear modulus representative of sandwich beams, bring out the usefuhtess of the theory for the analysis of such soft-cored beams. In
INTRODUCTION The Euler-Bernoulli theory of bending is widely used for thin beams. When the beam is short or thick, or made of high-strength composite materials with a high anisotropy ratio, the theory needs some modifications to include the effect of transverse shear. Refined sheardeformation theories are needed. There has been continued research interest in the bending analysis of beams. A review of the recent advances and developments in the analysis of laminated beams with an emphasis on shear deformation is presented in Kapania & Raciti [l]. A discussion of various shear-deformation theories and a review of the various studies on the bending in beams is given. The studies have been conducted using analytical, numerical and experimental techniques. The basic governing equations pertaining to the hygro-thermal mechanical behavior of a laminated straight beam are developed in Vinson & Sierakowski [2]. The beam is considered to be made up from a number of elastic layers each *Author to whom correspondence
should be addressed. 195
196
A. A. Khdeir; J. IV Reddy
Suzuki [7] stress analysis is carried out on a short beam subjected to distributed load, taking into account the warping of the section. A new theory for beams of rectangular cross-section, which includes warping of the cross-section, is presented by Levinson [83. By satisfying the shear-free conditions on the lateral surfaces of the beam, a pair of coupled equations of motion are obtained such that no arbitrary
shear coefficient is required. It is shown that the uncoupled equation for the transverse displacement is the same as the corresponding equation in Timoshenko beam theory, provided that for the Timoshenko equation the shear coefficient is taken to be 5/6. Two fourth-order ordinary differential equations have been developed by Silverman [9] to describe the bending of a three-layer laminated sandwich beam. Formulae
(a>
1.75 -
--
HOBT
HOBT
1.50 -
-
-
CBT 1.25
G
-
FOBT CBT
-
1.00
0.75
-
0.50
-
0.25
L/h
L/h
Cc)
(4
II ---
HOBT SOBT .-.~- -- FOBT - CBT
1.25
0.00
-I
0
IO
20
30 L/h
40
_I
4 -1
HOBT
.._.....
FOBT
-
CBT
-
50 L/h
Fig. 1. (a) Maximum deflection vs length-to-height ratio of a hinged-hinged beam (O”/900/00) under uniformly distribution load. (b) Maximum deflection vs length-to-height ratio of a clamped-hinged beam (0°/900/00) under uniformly distribution load. (c) Maximum deflection vs length-to-height ratio of a clamped-clamped beam (F/90°/00) under uniformly distribution load. (d) Maximum deflection vs length-to-height ratio of a clamped-free beam (O”/90”/00) under uniformly distribution load.
197
Exact solution for bending laminated beams Table 1. Non-dimensional
mid-span
deflection
(I@ of symmetric conditions
cross-ply
(O”/900/O”)beams for various
boundary
Llh
Theory
H-H
C-H
c-c
C-F
5
HOBT SOBT FOBT CBT HOBT SOBT FOBT CBT HOBT SOBT FOBT CBT
2.412 1.896 2.146 0.646 1.096 0.959 1.021 0.646 0.665 0.659 0.661 0.646
1.952 1.655 1.922 0.259 0.740 0.622 0.693 0.259 0.280 0.273 0.276 0.259
1.537 1.379 1.629 0.129 0.532 0.442 0.504 0.129 0.147 0.142 0.144 0.129
6.824 5.948 6.698 2.198 3.455 3.135 3.323 2.198 2.251 2.235 2.243 2.198
10
50
for the deflection, bending and shear stresses of cantilever and simply supported beams have been developed. Ochoa & Kozik [lo] presented a beam theory based on Kozik’s [ll] linearly exact displacement formulation to evaluate the interlaminar transverse shear and normal stresses in laminated beams. heterogenous Gordaninejad & Ghazavi [12] presented a higher-order shear-deformable beam theory for the analysis of laminated composite beams that takes into account the parabolic distribution of shear strain through the thickness of the beam. A mixed finite-element formulation was also presented. Results were presented for single, two- and three-layer beams under uniform and sinusoidal distributed transverse loadings. A formulation, an efficient solution procedure, a microcomputer program and a graphics routine for an anisotropic symmetrically laminated beam finite element, including the effect of shear deformation, is introduced in Chen & Yang [13]. Hu et al. [14] present an approxi-
Table 2. Non-dimensional
mid-span
deflection
mate theory for the bending of cross-ply composites that is simpler than the exact theory of elasticity. To test the theory, cantilever specimens of cross-ply fiberglass were prepared and stressed in a standard Instron testing machine. The strain distributions in the end-effect region were found to be much closer to those predicted by their theory than those predicted by the Saint-Venant theory, however the deflection measurements did not agree with the Hu et al. theory. Recently the theories used in this study have been employed to investigate the vibration and buckling analysis of cross-ply laminated beams [15-171. In Khdeir & Reddy [15] exact eigenfrequencies of refined beam theories are obtained for various boundary conditions using the state-space concept. Khdeir [16] developed an analytical solution to study the transient response of antisymmetric cross-ply laminated beams with generalized boundary conditions and for arbitrary loadings. A general modal
(F@ of antisymmetric conditions
cross-ply
(0”/90”) beams for various
Llh
Theory
H-H
C-H
c-c
5
HOBT SOBT FOBT CBT HOBT SOBT FOBT CBT HOBT SOBT FOBT CBT
4.777 4.800 5.036 3.322 3.688 3.692 3.750 3.322 3.336 3.336 3.339 3.322
2.863 3.035 3.320 1.329 1.740 1.764 1.834 1.329 1.346 1.346 1.349 1.329
1.922 2.124 2.379 0.664 1.005 1.032 1.093 0.664 0.679 0.679 0.681 0.664
10
50
boundary C-F 15.279 15.695 16.436 11.293 12.343 12.400 12.579 11.293 11.337 11.338 11.345 11.293
198
A. A. Khdeil; .I. N Reddy
approach, utilizing the state form of the equations of motion and their biorthogonal eigenfunctions, is presented to solve for the equations of motion of the beam with arbitrary boundary conditions. The state-space concept in conjunction with the Jordan canonical form is used in Khdeir & Reddy [17] to obtain the exact critical buckling loads of refined composite beam theories.
In this paper analytical solutions of four beam theories are developed to study the bending behavior of cross-ply rectangular beams with arbitrary boundary conditions. The state-space concept in conjunction with the Jordan canonical form is used to solve for all displacement functions. Comparisons between refined theories and the classical theory are maintained to show the effect of shear deformation.
(a)
3.5
:
L/h 1.0
0.8
1.75
L/h=10
-
c-c
= 10
0.6
C-F
L/h = 10
HOBT
HOBT
-
-
~
2 Layers
----~-~~ -
3 Layers 4 Layers
-
w
C-H
L/h = 10
-
IO Layers
-
-
2 Layers 3 Layers -
-
4 Layers
-0.1
XL
0.1
0.3
0.5
L/h
Fig. 2. (a) Plots of deflection vs x/Lfor a hinged-hinged beam under uniformly distributed load and for different numbers of layers (O”/~Oo/O”/...). (b) Plots of deflection vs x/Lfor a clamped-hinged beam under uniformly distributed load and for different numbers of layers (O”/900/Oo/...). (c) Plots of deflection vs x/L for a clamped-clamped beam under umformly distributed load and for different numbers of layers (O”/900/Oo/...). (d) Plots of deflection vs x/Lfor a clamped-free beam under uniformly distributed load and for different numbers of layers (oO/90°/Oo/...).
199
Exact solution for bending laminated beams
(3) The cg = (4) The cg =
30r---__
25
C-F
.-~-~----
H - H
--
c-n -
Second-order Beam Theory (SOBT): 0, c, = 1, c2 = 1, c3 = 0; Third-order Beam Theory (HOBT): 0, c, = 1, c2 = 0, c3 = -(4/3)/z.
The linear strains associated placement field in eqn (1) are
c-c
20
with the dis-
HOBT
t;_,= t’F)+& ‘+&j;2’+&I;1’
o/90/0 w
15
L/h
= 10
(2)
y,; = y($‘+zy!rl,‘+z2y;; where
IO
CO)_ Ex -
5
du dx
d4
*(I)=c Y&X
1z+co
d2w dX2
3
0 10
0
20
30
40
50
pzc x
_
$3) .r
*z,
-
5 h3
(%+$I
(3)
E,‘E,
Fig. 3. Plots of the maximum deflection vs E,/E, for a (0”/90”/0”) beam under uniformly distributed load and for different boundary conditions.
dw yj!’ = Cl4+( l+c,) d, , y:y = 2&,
DERIVATION OF VARIOUS BEAM THEORIES
YXI
(2)_ -
3c3 h3
@+$ i
The displacement field in the beam is assumed to be ([is, 16, 171)
U(x, z> =u(x)+z co
[
$
+C,(Zlh)3 4(x)+
i
+c,&)
+c,z21)(x)
1
The stress-strain relations for the kth lamina in the laminate coordinate are given by a(k) x = Q::‘G (4)
7;:’ = Q:“:y,,
1
$
1
(1)
V(x, z) = 0 W(x, z) = w(x) where U, I/ and W are the displacements along the x, y and z coordinates, respectively; (u, w) denote the displacements of a point (x, y, 0) along the x and z directions, respectively; and 4 and $ are functions of x.The displacement field in eqn (1) contains the displacement fields of the Euler-Bernoulli classical beam theory, the Timoshenko first-order theory, and the secondorder theory and the third-order theory of Reddy. We have (1) The Classical Beam Theory (CBT): = - 1, Cl = 0, c2 = 0, c3 = 0; (2) ?he First-order Beam Theory (FOBT): co = 0, cr = 1, c2 = 0, c3 = 0;
where Q{F’ and Q$t) are the elastic stiffnesses transformed to the x direction. The principle of virtual displacements will be used to derive the governing equations and associated boundary conditions for the displacement field, eqn (l), and constitutive eqn (4); we have
(5) where f(x) is the distributed transverse load per unit length of the beam applied in the plane of symmetry and L is the length of the beam. Introducing the following definition of stress resultants (N,, M,, L,, 9,) = b cLk)(1, z, z*, z”) dA
(6)
A. A. Khdeir; J. N. Reddy
200 The Euler-Lagrange equations theories [15, 16, 171 associated placement field in eqn (1) are dN x dx
of the beam with the dis-
where (A,,,B,,,D,,,E~,,F,~,GI~,H~~) = 1 Q::‘
=()
(7)
(A,,, Bs,, Dm Em F,,) = i Q:“:U, z, z2, z’, z4) dA
dMx
c3
Cl -+-
h’
dx
-
dPx
-ClQx-
dx
3c3
-s,=o h3
EXACT SOLUTION
dL
-
(10)
FOR BENDING
-2c,R,=O
c2 dx
dQx
3c,
(l+c,)z+-
h’
We will write the equilibrium equations in terms of displacement quantities. Substituting eqn (3) into eqn (9) to define the stress resultants in terms of displacement quantities and then substituting eqn (9) into eqn (7) and after some algebraic manipulations, we obtain the following equilibrium equations for each theory
dS, d2 - dx dx2
(coMx+ $ =o PJ+f
HOBT with the following tions
associated
boundary
condi-
= s1( $+w’)+s2Wn’
(11)
w”” = Sj( ~‘+w”)+s,f
Natural B. C.
u” = S‘@+W’)+SgWn’
Essential B.C.
SOBT
u
NX
(12)
v = s, @+w’)+s,$ $P = s,@+w’)+s,$
c,M,+5P, h3
W“= - @+S#+SOf
c2L
coM,+
f
Un= s&+w’)+s,$
$
4
FOBT
dw
P,
dx
(8)
(13)
#%,(4+w’) wn = - @+s,f un = s,($+w’)
3c3
(l+co)Q, 7
CBT w””= s0f
Sx
(14)
U” = S,W”’
where h is the total thickness of the beam. The resultants are related to the linear strains bY NX MX LX = ii-PX
A,, B,, Q, E,,
B,, Q, E,, F,,
@I El, F,, G,,
E,, F,, G,, Hi,
where a prime on a quantity denotes ordinary differentiation with respect to x and the coefficients si are defined in Appendix A. The state-space approach in conjunction with the Jordan canonical form will be used to analyze the bending problems of cross-ply rectangular beams. In this approach, the systems of except the equation related to eqns (ll)-(14), u, will be converted into a system of first-order equations as
(15) {Z(x)} is defined
for
201
Exact solution for bending laminated beams
z,=~;z2=~‘;z3=w;z4=w’;z5=w~;
FOBT and CBT
z, = d”
(16)
etJIX= [ 1 x x2/2 x3/6 0 1 x x2/2 0 0 1
SOBT
x0001]
z,=~;z2=~‘;z3=1c/;z4=~‘;zs=w; z6=w’
(17)
(27)
For ill-conditioned problems during tion it is recommended that the relation is used (see Khdeir [IS])
computafollowing
FOBT (k) =[Ml-‘(0
Z,=~;Z2=~‘;Z3=w;Z4=w’
(18)
(28)
and the solution in eqn (25) will be modified as
CBT z, =w; Z,=w’;
{Z} = [M]erJ1”{k}+[M]etJ1”
z3=wn; Z,=w”
(19)
The non-zero elements of the matrix A are defined in Appendix B and the load vector {F) is defined as (F} = (0, 0, 0, 0, 0, sOf}* for HOBT, SOBT (20) {F} = {0, 0, 0, sOf)T for FOBT, CBT The eigenvalues of matrix each theory areHOBT A, =&=&=&=O,
A associated
(21) with
x j eerJ1s[M]-’
{F(q)} dq
(29)
where the constant vector {k} is to be determined from boundary conditions.The boundary conditions for hinged (H), clamped (C) and free (F) support conditions for the four theories are HOBT H: w=N,=M,=P,=O
1&Z,
(30)
I,
=-- &
(22)
SOBT A, =&=&=&=O,
&=\jisq+s3s.s, /I,= -&
4 dP, 3h2 dx
F: N,=M,=P,= (23)
/I, =&=123=/24=o
(24) Bearing in mind that the matrix A has in the four cases an eigenvalue with multiplicity 4, the solution to eqn (15) will be given in terms of the Jordan canonical form as (Z) = [M]eLJ1”[M] ~ ’ { Z}+[M]eLJ1” dg
SOBT H: w=N,=M,=L,=O
(25)
c: u=C#s$=w=O
F: N,=M,=L,=Q,=O
H: w=N,=M,=O
(32)
c: u=4=w=o F: N,=M,=Q,=O CBT H: w=N,=M,=O
c:
HOBT and SOBT
u=w=
$
(33) =o
x2/2x3/6 0 0 0 1 x x2/2 0 0 0
O1xOOOOOIOOOOOOe~~X 0 0 0 0 0 0 e@]
(31)
FOBT
where M is modal matrix which contains the eigenvectors and generalized eigenvectors of the matrix A. The elements of the modal matrix M are defined in Appendix C for each theory. J is the Jordan matrix and etJ1’ is block diagonal defined as
etJIX= [lx
- h2
S,+Q,= 0
FOBT and CBT
x 7 e-[J1q[M]-‘{F(q))
4
F: N,=M,= (26)
dK =O
-
dx
202
A. A. Hzdeir; J. N. Reddy
We deduce the boundary conditions at the ends x = +L/2 imposed on the state vector {Z} from eqns (30)-(33) as HOBT H: Z,=Z,=Z,=O
All laminae are assumed to be of the same thickness and made of the same orthotropic material. A value of 5/6 is used for the shear correction factor of FOBT. The deflections are non-dimensionalized as
(34)
c: z, =z3=z4=o W= F: Z,=Z,=Z,+Z,+
wAE,h2 lo2 .f0L4
(e,s,+e,s, - e7) (e,s,+e,s, +cJ
z,=o
SOBT H: Z,=Z,=Z,=O
(35)
c: z, =z,=z,=o F: Z,=Z,=Z,+Z6+PgZ3=0 e5
FOBT H: Z,=Z,=O
(36)
c: z, =z,=o F: Z,=Z,+Z,=O CBT H: Z,=Z,=O
(37)
c: z, =z,=o F: Z,=Z,=O Substituting in eqn (29) the desired combinations of edge conditions at the ends x = &L/2 defined in eqns (34)-(37). One has to solve a linear system of algebraic equations to find the constant vector {k}. The displacement functions are found from the definition of the state vector {Z} in eqns (16)-(19). The axial displacement u is determined by integrating twice the equation related to u in eqns (1 l)-( 14).
NUMERICAL RESULTS AND DISCUSSION To demonstrate the procedure used in the present study we present some numerical results for symmetric and antisymmetric crossbeams with the following ply laminated dimensionless orthotropic material properties E,l, = 25; G,* = G,, =OSE,; G,, =0.2E,; v,,=o.25
where L is the length of the beam, A is the cross-sectional area and h is the total thickness. The load is assumed to be uniformly distributed with an intensity fO. Beams with hinged-hinged clamped-hinged (C-H), clamped(H-H), clamped (C-C) and clamped free (cantilever) (C-F) boundary conditions have been considered to calculate the deflection of the beam by the state-space approach. To assess the importance of shear deformation, the numerical results for the nondimensional mid-span deflection obtained by the classical beam theory and the first-, secondand third-order beam theories have been compared for various boundary conditions in Fig. l(a)-(d) and Tables 1 and 2. Large differences between the results of the refined theories and the classical theory are noted, especially when the L/h ratio decreases, indicating the effect of shear deformation. The disagreement between refined theories is much less than the disagreement between any of them and the Euler-Bernoulli theory. In Fig. 2(a)-(d), the deflected curve of the beam using the third-order theory is shown for conditions various boundary and various lamination schemes. Two-layer (0”/90”), threelayer (0”/90”/0”), four-layer (O”/900/Oo/90”) and lo-layer (0”/90”/0”/90”/...) lay-ups are considered. It is seen that the symmetric cross-ply stacking sequence gives a smaller response than those of antisymmetric ones. In antisymmetric cross-ply arrangements increasing the number of layers for the same thickness will decrease the deflection for all boundary conditions. The effect of the orthotropicity ratio on the non-dimensional mid-span deflection of threelayer symmetric cross-ply beam is shown in Fig. 3. The third-order theory is used to predict the mid-span deflection for all boundary conditions. Increasing the orthotropicity ratio will decrease the deflection.
Exact solution for bending laminated beams
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