- Email: [email protected]
Through-thickness stresses in curved composite laminates and sandwich beams
Composites Science and Technology 61 (2001) 1501–1512 www.elsevier.com/locate/compscitech
Through-thickness stresses in curved composite laminates and sandwich beams R.A. Shenoi*, W. Wang School of Engineering Sciences, University of Southampton, Southampton, SO17 1BJ, UK Received 21 September 2000; accepted 7 March 2001
Abstract An elasticity-theory-based approach is developed for delamination and flexural strength of curved layered composite laminates and sandwich beams. The governing equations in this case are derived from the results of curved orthotropic beam on an elastic foundation under flexural loading. The approach ensures an accurate description of the through-thickness and in-plane stresses in curved laminate beams. The solutions for various geometrical configurations are provided. The effects of various parameters, such as stacking sequence of the laminate, thickness of the skin, the curvature radius are also studied. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: A. Layered structures; B. Mechanical properties; C. Delamination; C. Laminates; C. Sandwich
1. Introduction Tee joints, curved sandwich beams and top-hat stiffeners are widely used in many structural applications. The mechanical behaviour of such structural elements has received a fair amount of attention. Up to now, most work in this area has focused on experimental and numerical analyses [1–3]. There are few theoretical analyses relating to the strengths of curved, layered laminated beams/panels or sandwich beams. Designers with composites are familiar with the conventional treatment of laminates such as the classical laminated-plate theory (CPT) and thin-shell theory [4]. In these classic treatments of curved shells, the assumption of a state of plane stress in the constitutive relations eliminates the possibility of rigorous calculation of interlaminar stresses. However, there is increasing awareness that the low values of through-thickness tensile and shear strength of FRP, compared to in-plane strengths, can significantly affect the performance. Moreover, although many high order, refined laminated plate theories and some approximate methods are available [5–7], their application to curved beams is limited [8]. Most of these theories * Corresponding author. Tel.: +44-23-80592316; fax: +44-2380593299. E-mail address: [email protected] (R.A. Shenoi).
or methods retain parts of the assumptions of classical plate and thin shell theory, and only incorporate the influence of shear deformation etc. on plate deflection in composite laminates. So while they can compute transverse shear stresses and give more accurate results for the in-plane stresses and displacements, they are unable to deal with through-thickness stresses. Pagano [9,10] studied the multi-ply composite laminates using linear elastic theory, and provided an exact solution for composite laminates in cylindrical bending. However, the composite laminates he considered were all flat and subjected to only transverse load condition. Lekhnitskii [11] gave a general solution to the problem of ‘bending of a plane curved rod by moments’. He studied the curved orthotropic beam but did not consider the response of a rod on an elastic foundation. The response of a curved beam without a foundation is quite different from that with a foundation. It can only sustain load condition such as pure bending or a pair tension/ compression forces at the ends, and it cannot sustain the circumferential force without radial restraints, so the solution of Lekhnitskii cannot be directly transformed to analyse the curved layered composite beam. Most recently reported work on curved beams deals with plane-strain applications of classical laminatedplate theories [12–14]. The extension of classical theories to curved beams is based on tight assumptions. For
0266-3538/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(01)00035-5
1502
R.A. Shenoi, W. Wang / Composites Science and Technology 61 (2001) 1501–1512
Mx,Nx,Qx
Nomenclature k m
p q t u; v A,B,D E; ; M; N
elastic modulus of foundation cross-ply stacking ratio signifying the ratio of the thickness sum of odd number plies to that of even number plies reaction force of elastic foundation distributed pressure thickness of beam displacement components in the x and y directions(or and r directions) stiffness matrixes Young’s modulus, shear modulus, Poisson’s ratio bending moment, axial force, shear force
example, Wu [12], Gibson [13] and Chandler [14] adopt the same model which retains the assumption that the stress normal to the cross-section of beam distributes linearly. This is acceptable when the longitudinal elastic modulus of composite beam does not change through thickness, such as a laminated beam composed of many layers oriented in the same direction. This assumption is not valid, however, for a general layered composite beam, in which the stiffness properties vary drastically from layer to layer. Stress components are thus likely to be discontinuous in their variation. Therefore, the method and results obtained by Wu [12], Gibson [13] and Chandler [14] cannot be used to analyse a general curved laminated beam. Tolf [15] analysed homogeneous transverse isotropy condition in detail, but in his discrete model, he took the laminate as a one ply fibre material by one ply matrix material. Lu [16] provided a solution of homogeneous anisotropy, but as with Tolf [15], his model and solution cannot be used to study layered composite curved beam and curved sandwich beam. In the recent past, Wisnom [17–19] has contributed much research work on experimental analyses and numerical approaches for interlaminar failure and flexural strength of composite laminate. The focus of this work has been towards understanding delamination failure in curved laminated beams. Interlaminar normal and shear stresses, acting either singly or interactively can lead to delamination as reported by Wisnom [17–19], thus affecting structural integrity. It is important to know through-thickness stresses even well below the delamination limit value, because these could have an interactive effect on failure under in-plane stress to some extent [12]. One of the key features of a curved (as distinct from a straight) laminated beam is the presence of not so insignificant
QðijkÞ R; Ro ; Ri Vf ; ; l
; ; " "0 f g
x
bending moment, in-plane force and shear force in x-direction stiffness matrix component of the kth layer radius of midplane, outer radius, inner radius Volume fraction coefficients in the expressions of reaction forces anisotropy ratio ratio of inner radius to outer radius of the beam (=Ri =Ro ) normal stress and shear stress, strain mid-plane strain the curvature of midplane curvature change of the curved beam stress function
through-thickness tensile stresses. These can significantly affect the performance of curved composite beams due to the low values of through-thickness tensile strength. Therefore, it is imperative to obtain the stress distribution through the thickness more accurately when analysing the behaviour of curved composite beam. The through-thickness stress in curved composite laminate or sandwich beam is studied in this paper by an elasticity-theory-based approach. Section 2 develops a model for characterising linear-static flexural behaviour of a curved beam on an elastic foundation by classical beam theory. The radial reaction force of elastic foundation and internal forces within the beam are obtained from the governing equations. These results are then used in Section 3 as boundary conditions for an elasticity solution for an orthotropic beam on an elastic foundation. Then a general solution for curved layered composite beam is achieved in Section 4. Sections 5 and 6 show the application of this approach in curved composite laminates and curved sandwich beams respectively. The effects of key variables such as the stacking sequence of the laminate, the curvature radius and the thickness of the skin of sandwich beam on stresses distribution within a curved layered beam or sandwich beam are also studied.
2. Flexural behaviour of a curved composite beam on an elastic foundation Consider a curved beam of radius R resting on an elastic foundation modulus k, as shown in Fig. 1. Assuming that the reaction forces in the foundation are normal to the axis of the beam and proportional at every point to the radial deflection of the beam v at the point, the following is obtained:
R.A. Shenoi, W. Wang / Composites Science and Technology 61 (2001) 1501–1512
1503
Fig. 1. Curved beam on an elastic foundation. (a) Beam geometry; (b) forces on a small element.
p ¼ kv
ð1Þ
The equilibrium equation of a curved beam can be obtained as: k
dv dq 1 dMx d3 Mx 2 ¼ dx dx R dx dx3
ð2Þ
Thickness of the laminate can be incorporated into the equations: 1 t dv t 1 dq 1 dMx k 1 þ 1 3 R 2R d 2R R d R d ¼
1 d3 Mx R3 d3
ð3Þ
In order to obtain the solution to the above differential equation, it is necessary to obtain a relation between the moment Mx and curvature change x . This is well documented for beams made from isotropic materials [20]. In the case of laminated composites beam, the relationship is obtained from the classical laminated plate equation [21]: 2
3 2 N A 4 5 ¼ 4 M B
32 3 j B "0 j 5 4 5 j D
ð4Þ
case of a cylindrical shell with large width (in z-direction) is considered, and the applied forces and boundary conditions are both uniform in cylinder axial direction (z-direction). The problem here could then be restricted to a curved beam of unit width, and it is assumed "z=0. A further assumption is made in that the transverse pressure load q is constant. Then, from Eqs. (4) and (5), the following is true:
Nx ¼ A11 "0x þ B11 x ð6Þ Mx ¼ B11 "0x þ D11 x While considering the effect of in-plane deformation on the change of curvature of circular beam or cylindrical shell, the relation between the radial deflection v and curvature x is pertinent: x ¼
Aij ; Bij ; Dij ¼
ð t=2
QðijkÞ 1; y; y2 dy
ð5Þ
t=2
Because of the tension-bending coupling and bendingtorsion coupling, a simple expression of Mx and x cannot be obtained easily; this is needed in solving the differential Eq. (2). Our approach is to simplify the problem and consider only one of the in-plane directions, says the x-direction. This could be achieved if the
ð7Þ
Incorporating this in Eq. (6) and then substituting in Eq. (3), the following is obtained: 1 B211 d5 v 2 B211 d3 v D þ D 11 11 R5 A11 d5 R5 A11 d3
k 1 B211 dv þ þ D11 R R5 d A11 ¼
where
1 d2 v v þ R2 d2 R2
1 dq 1 B11 dNx 1 B11 d3 Nx þ 3 þ 3 R d R A11 d R A11 d3
ð8Þ
Two conditions will be considered in the following. (1) Nx=constant (N0) This is a common occurrence in the case of sandwich beams subjected to pure bending, where the axial force in the skin is a constant throughout the span. Also noting the previous assumption of constant pressure q. Thus, in this condition, the right part of Eq. (8) is zero, and the equation turns out to be:
1504
R.A. Shenoi, W. Wang / Composites Science and Technology 61 (2001) 1501–1512
d5 v d3 v dv ¼0 þ 2 3 þ 2 5 d d d
ð9Þ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u R4 k where: ¼ u þ1 t B2 D11 11 A11
ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 u Rk u1 Using ¼ 1 and =u t2 B2 D11 11 A11 2
The results for Mx and Qx can be deduced to be: Mx ¼
The general solution of the homogeneous differential Eq. (9) is: v ¼ K0 þ ðK1 cosh þ K2 sinhÞcos þ ðK3 cosh þ K4 sinhÞsin
ð10Þ
From this, the reaction force of elastic foundation can be deduced as: p ¼ 0 þ ð1 cosh þ 2 sinhÞcos þ ð3 cosh þ 4 sinhÞsin
2
ð11Þ
rffiffiffiffiffiffiffiffiffiffiffi 8 1 > > < ¼ 2 rffiffiffiffiffiffiffiffiffiffiffi where > þ 1 > : ¼ 2 and where Ki and i are constants to be determined from boundary conditions. (2) Nx ¼ g0 þ g1 cos This is the case when a curved sandwich beam is subjected to a pair of tension/compression forces at two ends. Again, the right hand side of Eq. (8) is zero. Thus, the governing differential equation and solutions for the lateral deflection and reaction forces will be identical to Eqs. (9), (10) and (11) respectively. For a laminate symmetric with respect to the midplane, the case becomes much simpler, because the tension-bending coupling does not exist, i.e. Bij ¼ 0. If the curved beam is symmetric with respect to the axis ¼ 0, then in the equations (10) and (11), only even-function parts remain: v ¼ K0 þ K1 coshcos þ K4 sinhsin
ð12Þ
p ¼ 0 þ 1 coshcos þ 4 sinhsin
ð13Þ
B11 B2 1 h Nx D11 11 2 K0 A11 A11 R
þ 2K4 coshcos 2K1 sinhsin B11 dNx 1 1 B2 Qx ¼ 3 D11 11 A11 d R R A11 h 2 2 2K1 2K4 coshsin i þ 2K1 2 2K4 2 sinhcos
i
ð15Þ
ð16Þ
When Nx ¼ N0 is a constant, then from the equilibrium considerations of beam element, Qx= 0. It can be seen that the above equation is satisfied under any value of only when: K1 ¼ K4 ¼ 0. Thus, for constant Nx, and also considering the existence of distributed pressure acting downwards on the inner surface of curved beam q0 and the height of beam t, the following results can be deduced: 8 B11 B211 K0 > > ¼ N D M 2 > x x 11 > > A11 A11 R < t t N ¼ k R þ q R K x 0 0 > 2 2 > > > > : p ¼ kK0 Qx ¼ 0
ð17Þ
This conclusion shows that even in a general case where: (a) tension-bending coupling occurs as a result of the laminate layer; and (b) the beam is subjected to inplane forces Nx, bending moments Mx and the shear force Qx still is zero. Consequently, the case of a curved beam laid on elastic foundation as a problem in which stresses are independent of polar angle can still be considered pertinent and whose results can be used in analysing curved laminated beams and the skins/faces of curved sandwich beams. These results will be used in the following parts as force boundary conditions when the same problem is analysed by theory of elasticity to obtain an accurate general solution of a layered composite beam on elastic foundation.
Thus, from Eq. (8): 1 h x ¼ 2 K1 2 þ 2K4 K1 2 þ K1 coshcos R i þ K4 2 2K1 K4 2 þ K4 sinhsin þ K0 ð14Þ
3. Solution for a curved orthotropic beam on an elastic foundation Introducing a stress function in polar coordinates ðr; Þ, we can write stresses to be [22]:
R.A. Shenoi, W. Wang / Composites Science and Technology 61 (2001) 1501–1512
8 1 @ 1 @2 > > > > r ¼ r @r þ r2 @2 > > < @2
¼ 2 > @r > > > @ 1 @ > > : r ¼ @r r @
ð18Þ
8 pffiffi > 3 R1 l < 3¼ o pffiffi > 4 R1þ l : 4 ¼ o
1505
ð24Þ
Where,
The above equation can satisfy the equilibrium equation. For the problem under consideration, the stress components do not depend on and, in addition, r
0 (i.e. constant in-plane force and hydrostatic load). For the cylindrical material anisotropy, a compatibility equation can be obtained as [16]: d4 2 d3 l d2 l d ¼0 þ 3 2 2 þ 3 dr4 r dr r dr r dr
ð19Þ
pffiffi pffiffi pffiffiffi pffiffiffi 1 þ l 1 l1 1 l 1 l1 ¼ pffiffiffi ffiffi ffiffi p p pffiffiffi lþ1 l 1 1 lþ1 1þ l 1 pffiffi pffiffiffi kK0 þ q0 1 l 1 l1 R2 o 1pffiffil pffiffi 3 ¼ Ro p ffiffi ffi 2M þ kK R2 qR2 1 þ l 1 l 1 0 0 o i pffiffi p ffiffi ffi 1 þ l 1 l1 R2 kK0 þ q0 o 1þpffiffil 4 ¼ pffiffiffi Ro pffiffi l1 2M0 þ kK0 R2o qR2i 1 lþ1 ð25Þ
E2 l¼ E1
ð20Þ
where E1 and E2 are the elastic moduli in the in-plane and through-thickness directions respectively. The general solution to this ordinary differential equation is: ¼ C1 þ C2 r2 þ C3 r1þ
pffiffi l
þ C4 r1
pffiffi l
The coefficients of the stress formulae can then be obtained as: pffiffiffi pffiffiffi 3 1 l 4 2C2 ¼ kK0 1 þ l pffiffi 3 R1 l ð26Þ C3 ¼ o pffiffi 1þ l C4 ¼ 4 Ro
ð21Þ Correspondingly, the solution of stresses r and as:
from which stress components can be obtained: (
pffiffi pffiffi pffiffiffi pffiffiffi
r ¼ 2C2 þ 1 þ l C3 r l1 þ 1 l C4 r l1 pffiffi pffiffi pffiffiffipffiffiffi pffiffiffipffiffiffi
¼ 2C2 þ 1 þ l lC3 r l1 1 l lC4 r l1
ð22Þ The appropriate boundary conditions can be used to determine the coefficients in the above expressions. From Eq. (17) and the geometry and considering the direction of the forces, we have:
8 h pffiffiffi pffiffiffi i > 3 1 l 4 > > r ¼ kK0 1 þ l > > > plffiffi1 > > pffiffiffi > > 3 r > þ 1þ l > > Ro > > > pffiffil1 > < pffiffiffi 4 r þ 1 l Ro > > plffiffi1 > h > pffiffiffi pffiffiffi i pffiffiffi pffiffiffi > > 3 r 3 1 l 4 þ 1þ l l > ¼ kK0 1þ l > > > Ro > p ffiffi > l1 > > p ffiffi ffi p ffiffi ffi > r > 4 > 1 l l : Ro ð27Þ
Because of the plane-strain assumption, the equations of Hook’s Law of anisotropic material can be expressed.
t
r ¼ p ¼ p0 ¼ kK0 r¼Rþ : 2 t
r ¼ q ¼ q0 r¼R : 2 Ð Rþ2t t t þ q R K t dr ¼ N ¼ N0 ¼ k R þ 0 0 R2 2 2 Ð Rþ2t
rd ¼ M ¼ M0 R t 2
ð23Þ Substituting the expressions of stress components (22) into the above boundary condition (23), then it can be noted that the third equation in (23) is automatically satisfied when the first and second equations are satisfied. This equation group can be solved and the value of coefficients C2 ; C3 and C4 can be determined. Using the notation:
8 1 0 > > "r ¼ 0 r 120 > > E2 E1 > > < 021 1 " ¼ 0 r þ 0 > E E > 2 1 > > 1 > > : "r ¼ r 212
ð28Þ
where: E1 ; 1 13 31 12 þ 13 32 ¼ ; 1 13 31
E2 1 23 32 21 þ 23 31 ¼ 1 23 32
E01 ¼
E02 ¼
012
021
ð29Þ
1506
R.A. Shenoi, W. Wang / Composites Science and Technology 61 (2001) 1501–1512
Combining Eqs. (27), (28) and (29) with Cauchy equation in polar coordinates [19]: @ur ur 1 @u ; " ¼ þ ; @r r r @ 1 @ur @u u r ¼ þ r @ @r r
"r ¼
ð30Þ
By integrating the above expression and noting the relation:
012 E01
¼
021 E02 ,
E0
l ¼ E20 , the solution of displacement can 1
be obtained as:
pffiffiffi pffiffiffi i 1 021 h 3 1 l 4 rþ 1 þ l kC ur ¼ 0 E01 E02
plffiffi pffiffiffi 0 pffiffiffi 1 Ro r 21 pffiffiffi 1 þ l 0 l þ l 0 3 E1 Ro E2 l
pffiffil pffiffiffi 0 pffiffiffi 1 Ro r 21 pffiffiffi 1 l 0 þ l l 0 4 þfðÞ E1 Ro E2 l pffiffiffi 1 1 h 3 u ¼ 0 þ 0 kC0 1 þ l E1 E2 ð pffiffiffi i 1 l 4 r fðÞd ð31Þ The arbitrary function fðÞ in the above equations can be determined by displacement boundary condition. Here it should be noted that the above anisotropic solution cannot be extended to the isotropic case only taking l ¼ EE21 ¼ 1, though convergence to the isotropic solution can be achieved by putting l close to 1. As is known, the isotropic material case of l ¼ 1 actually corresponds to a double root of the characteristic equation arising from the generalised differential Eq. (19), leading to the logarithmic term in the expression of stress components and stress function. In homogeneous isotropic materials, as to the problem of plane symmetrical about the axis, the stress distribution is [22]: 8 a > > < r ¼ r2 þ bð1 þ 2lnrÞ þ 2c a
¼ 2 þ bð3 þ 2lnrÞ þ 2c > > r : r ¼ 0
8
R2 R2 Ro > > > a ¼ 2 o i 2 ðq0 p0 Þ 2ln b > > Ri Ri Ro > > > 2 2 > Ri > 2 2 < 2 Ri Ro M0 Ro Ri R2o p0 2R2o R2i ln ðq0 p0 Þ Ro b¼ > 2 Ro 2 > > Ri R2o 4R2o R2i ln2 > > > Ri > > a > > : 2c ¼ p0 2 bð1 þ 2lnRo Þ Ro ð34Þ
4. General solution for curved composite laminates and sandwich beams Based on the above solutions, further analysis can be done to obtain a general solution for the problem. Assuming there are n layers in curved composite beam, where every layer has the elastic coefficients: Eð1iÞ ; Eð2iÞ ; ð12iÞ and ð21iÞ and where the superscript ðiÞ means this variable is related to the ith layer. The boundary conditions to be satisfied in every layer are as follows: 8 r ¼ ri1 : r ¼ rði1Þ > > < r ¼ r : ¼ ðiÞ r r Ð ri i ði Þ
dr ¼ N > r > Ð ri1 : i ðiÞ ri1 rdr ¼ M
ð35Þ
where rðiÞ is normal tension stress in the interface between ith layer and ði þ 1Þth layer. In addition, following on from Eq. (17), there exist the following relations: 8 ðiÞ ðiÞ ði1Þ > < N ¼ r ri r ri1 ! ðiÞ ðiÞ2 4Cð0iÞ B B ð36Þ ð i Þ ði Þ 11 11 ðiÞ > : M ¼ ðiÞ N D11 ðiÞ ðiÞ 2 ð i1 Þ A11 A11 ðr þ r Þ
ð32Þ
The displacement is: 8 1h 1 > u ¼ ð1 þ Þa þ 2ð1 Þbrlnr ð1 þ Þ > r > > E r i < br þ 2ð1 Þcr þ gðÞ > ð > > > : u ¼ 4br gðÞd E
The arbitrary function gðÞ in the above equations can also be determined by boundary condition. If same boundary conditions as in Eq. (23) are assumed, then the constants in the expressions (33) can be determined as:
ð33Þ
In fact, every layer in the curved composite beam can be considered as an orthotropic beam or isotropic beam, with the interaction forces between adjacent layers taken as the reaction force of a kind of ‘elastic foundation’ — p and the water pressure — q. This is because in a mathematical context, rði1Þ and q, rðiÞ and p have the same meaning. They are only constants in the formulae, under no-circumferential-dependence condition. Then as in the analysis for the curved orthotropic beam, here too the known boundary conditions can be imposed:
R.A. Shenoi, W. Wang / Composites Science and Technology 61 (2001) 1501–1512
r ¼ ri¼0 ¼ Ri : r ¼ q0 r ¼ ri¼N ¼ Ro : r ¼ p0
ð37Þ
Therefore, in the analysis of the whole laminate, there exit unknown variables rðiÞ i¼1;2;;n1 and Cð0iÞ i¼1;2;;n . The number of these variables is n þ ðn 1Þ ¼ 2n 1 in all. Here the displacement compatibility condition on the interface can be represented as:
ðiþ1Þ ur ðri Þ ¼ uðriÞ ðri Þ i ¼ 1; 2; ; n 1 ð38Þ uðiþ1Þ ðri Þ ¼ uðiÞ ðri Þ i ¼ 1; 2; ; n 1 uðriÞ ðrÞ; uðiÞ ðrÞ are as expressed in Eqs. (31) or (33), according to that the layer is orthotropic beam or isotropic beam under consideration. Apart from these, for the case of curved beam on an elastic foundation under no-circumferential-dependence condition, there is the global force boundary condition: n X NðiÞ ¼ N0 ¼ kRo C0 q0 Ri
ð39aÞ
i¼1
And for the case of curved beam subjected to pure bending, there is another global force boundary condition: 4Cð0nÞ B11 B211 N0 D11 M0 ¼ A11 A11 ðRi þ Ro Þ2
ð39bÞ
As can be seen, there are: 2 ðn 1Þ þ 1 ¼ 2n 1 equations altogether, just the same as the number of unknown variables. Thus the problem can be solved. However, it should be noted that this general solution for curved layered composite beam on elastic foundation is under no-circumferential-dependence condition. Note also that rðiÞ i¼1;2;;n are the tensile through thickness stresses between every two adjacent layers interface tension stress.
5. Application: effect of laminate stacking sequence on stresses Shenoi and Wang [23] have studied and examined the effects of material- and geometry-related variables of a single-layer curved orthotropic beam and the elastic foundation on stresses and displacements. Here the effects of the above parameters and the stacking sequence of curved layered composite beam and elastic modulus of core material in curved sandwich are examined. In order to investigate the effects of stacking sequence of layered composite beam, an individual layer of unidirectional fibrous composite material is considered to possess the following (corresponding to a general Eglass/epoxy composite):
EL ¼ 38:6 GPa ET ¼ 8:27 GPa
1507
GLT ¼ 4:14 GPa
GTT ¼ 2:76 GPa LT ¼ 0:26
TT ¼ 0:5
Vf ¼ 0:45
Where L signifies the direction parallel to the fibres, T the transverse direction, and LT is the Poisson ratio measuring strain in the transverse direction under uniaxial normal stress in the L direction. And it is assumed that the material characteristics through thickness direction are the same as those in the transverse direction totally. These properties are used in appropriate contexts in the equations for calculating stresses and displacements (20), (29) and (33). For example, for a UD ply with 0 orientation, then E1 ¼ EL and E2 ¼ ET ; for a ply with 90 orientation, E1 ¼ ET and E2 ¼ ET , which means isotropic in the xy plane. Similar analogies can be given for other properties or for other layups. Several separate stacking configurations are considered, namely: 1. [0 /0 ] and [90 /90 ] — Describing orthortropic and isotropic beam respectively. 2. [0 /90 ] and [90 /0 ] — Two bidirectional (coupled) laminates with the layers being of equal thickness. 3. [0 /90 /0 ] — A symmetric 3-ply laminate with cross-ply stacking ratio m ¼ 1:0 4. [0 /90 /0 /90 ] and [90 /0 /90 /0 ] — Two anisymmetric 4-ply laminates both with m ¼ 1:0. 5. [0 /90 /0 /90 /0 ] — A symmetric 5-ply laminate with the layers being of equal thickness. In each case, the curved beams have the same curvature radius: Ri ¼ 30 mm, Ro ¼ 36 mm, so the thickness of the beam is t=6 mm. The curved beam is subjected to pure bending, the bending moment is M=1000 N/m. The functions of prime interest in the present case are radial stress r (the through-thickness stress of the curved layered composite beam) and circumferential stress (the in-plane stress of the curved layered composite beam). Under the no-circumferential-dependence condition, shear stress equals zero. The solutions for r and are illustrated in Figs. 2–6. In each figure, abscissa is the value of stress and ordinate is the normalized thickness y ¼ y=t. The stresses distribution in [0 /0 ] layered beam are shown in Fig. 2. The results of [90 /90 ] layered beam are very similar to Fig. 2. This correspond to the result for curved orthotropic beam on an elastic foundation [23], which showed that anisotropy ratio l has no significant effect on the stress distribution in the curved beam. The maximum through-thickness stress occurs on the inner side of the midplane of beam, very close to the midplane. Its value is 7.60 MPa.
1508
R.A. Shenoi, W. Wang / Composites Science and Technology 61 (2001) 1501–1512
Fig. 3 gives the results for [0 /90 ] and [90 /0 ] — two bidirectional (coupled) laminate cases. In each case, the layers are of equal thickness. The results show that there
is a big difference between these two coupled laminates, not only in the distribution of through-thickness stress and in-plane stress, but also in their values. The maximum
Fig. 2. Stresses distribution along thickness unidirectional laminate. (a) Through-thickness stress distribution; (b) in-plane tension stress distribution.
Fig. 3. Stresses distribution along thickness in cross-ply laminate. (a) Through-thickness stress distribution; (b) in-plane tension stress distribution.
Fig. 4. Stresses distribution along thickness in symmetric 3-ply laminate. (a) Through-thickness stress distribution; (b) in-plane tension stress distribution.
R.A. Shenoi, W. Wang / Composites Science and Technology 61 (2001) 1501–1512
1509
Fig. 5. Stresses distribution along thickness in antisymmetric 4-ply laminate. (a) Through-thickness stress distribution; (b) in-plane tension stress distribution.
Fig. 6. Stresses distribution along thickness — stacking sequence [0 /90 /0 /90 /0 ]. (a) Through-thickness stress distribution; (b) in-plane tension stress distribution.
absolute values of r and in [0 /90 ] case are both more than 10 higher than those in [90 /0 ] case separately. This difference will become bigger as the ratio ð¼ Ri =Ro Þ decreases. In a flat layered composite beam, this distinguishing difference does not exist. In these two laminates, the maximum through-thickness stresses are both occur in the 0 layer. A similar result can also be found in the analyses of [0 /90 /0 ], [0 /90 /0 /90 ] and [90 /0 /90 /0 ] laminates. The stresses distribution in these case are showed in Figs. 4 and 5. Three laminates have the same stacking ratio (m ¼ 1:0). It is reasonable that the stress distributions are different among these cases due to their different stacking sequence, but the maximum absolute values of stress in each laminate are quite different. The maximum values of through-thickness and in-plane stress
are 6.79 and 192 MPa in [0 /90 /0 ] laminate, 8.54 and 261 MPa in [0 /90 /0 /90 ] laminate, 8.11 and 240 MPa in [90 /0 /90 /0 ] laminate each. The maximum through-thickness stresses in three cases all occur in the 90 layer and close to the midplane of laminate. Fig. 6 shows the results for symmetric 5-ply stacking case. The value of maximum through-thickness stress in [0 /90 /0 /90 /0 ] laminate curved beam is close to that in [0 /90 /0 ] case, only very a little higher. The above results indicate that the stacking sequence of layered composite beam does have an effect on stresses. In two-ply (coupled) case, stacking sequence [90 / 0 ] is ‘‘better’’ than [0 /90 ]; in 4-ply case, stacking [90 / 0 /90 /0 ] is ‘‘better’’ than [0 /90 /0 /90 ]. Under the condition of the same stacking ratio, apparently the [0 / 90 /0 ] is the ‘‘best’’ stacking sequence among the above
1510
R.A. Shenoi, W. Wang / Composites Science and Technology 61 (2001) 1501–1512
Fig. 7. Stresses distribution along sandwich panel. (a) Through thickness stress distribution; (b) in-plane tension stress distribution.
stacking cases. It should also be noticed that in theory a unidirectional (UD) stacking is not a better one; the maximum value of through-thickness stress (7.60 MPa) in the UD case is more than 10 higher than that of [0 / 90 /0 ] case, although its maximum value of in-plane stress is lower (178 MPa). This indicates the possibility of identifying an ‘optimal stacking sequence’ for a curved layered composite beam.
6. Application: curved sandwich beams In order to investigate the stress distribution in curved sandwich beams, two separate geometrical configurations are considered first: (1) Thick skin-sandwich panel with ts =tc ¼ 14 1 (2) Thin skin-sandwich panel with ts =tc ¼ 10 The skin is a unidirectional cylindrical shell with the fibres oriented in the circumferential direction. The other geometrical variables of curved sandwich beam are the same as those of curved layered composite beam studied in the previous section. The material properties of the skin are the same as the above. The elastic constants of the isotropic core material of sandwich beam are: E ¼ 1:103 GPa
¼ 0:3
The curved sandwich beam is also subjected to pure bending, the bending moment is: M ¼ 1000 N=m Fig. 7 shows the results for these two kinds of curved sandwich beam. As can be seen, a significant throughthickness stress gradient exists in the skin, but the max-
imum through-thickness stress occurs in the core. It is sometimes at the interface between core and the inner skin, sometimes in the core but near to that interface (as shown in Fig. 7a). It can also be noted that the throughthickness stress changes only very slightly in the core; it keeps a high level nearly through the whole thickness of the core. This explains why in Gibson and Chandler’s experiment [13], delamination sometimes was seen to occur between either skin or core, and sometimes between both skins and the core. From the trends in Fig. 7a and b, it is known that in a thin-skin sandwich beam, there is a smaller throughthickness stress than that in a thick-skin sandwich beam, simultaneously there has the bigger in-plane stress which is reasonable. As a comparison with the numerical research results, the above approach is applied again to analyse two samples once used in Smidt’s research work [24,25]. The magnitudes of curved sandwich beam sample are: width 95 mm, inner radius of core 166 mm, skin thickness tf=2 mm, 10 mm, core thickness tc=50 mm. The skin material is GRP ES= 18.1 GPa, core material is H60 (Divinycell) EC=55 MPa. Fig. 8a and b shows the results for these two curved sandwich beam samples. Smidt analysed these two samples by the finite-element method. In his numerical analyses the maximum values of through-thickness stresses are 1.14 and 0.95 MPa for the thin and thick skin cases respectively, which are both located at the inner interface. As can be seen, the theoretical results — of about 1.16 and 9.85 MPa — coincide very well with the numerical results. It should be pointed out that, although there is a smaller through-thickness stress in a thick-skin sample, the thickness of thick skin is five times of the thin skin and the distance between the centrelines of the faces increased up to 15.4%, which leads to bigger general bending stiffness of sandwich beam
R.A. Shenoi, W. Wang / Composites Science and Technology 61 (2001) 1501–1512
1511
Fig. 8. Through-thickness tensile stress distribution in a curved sandwich beam.
and much bigger weight of course. If the total thickness of sandwich beam (tf+tc) is kept constant, then thick skin structure poses the bigger through-thickness stress, as is shown in the previous application. In order to investigate the effect of radius of the curved beam on stresses, another geometrical configuration of the curved layered composite beam is studied: Ri ¼ 18 mm, Ro ¼ 24 mm. The thickness of the beam is still h ¼ 6 mm. The curved beam is still subjected to pure bending, the bending moment is M ¼ 1000 N/m. Here the [0 /90 /0 ] stacking sequence is again considered. The results are as follows as shown in Fig. 4. As would be expected, the radius of curved beam has big effect on the stresses, especially on the value of through-thickness stress, which is similar to the result for curved orthotropic beam [23]. As can be seen, when the thickness is stable and outer radius becomes a third smaller, the maximum of through-thickness stress in beam increases by 50%, simultaneously the maximum in-plane stress becomes a little bigger.
7. Concluding remarks In conclusion, a new approach to define the elasticity solutions for curved composite beam consisting of arbitrary numbers of orthotropic or isotropic layers in nocircumferential-dependence case is presented. Since the solutions are exact, within the assumptions of linear elasticity, there need be no strict distinction between thick and thin curved beams. Also there is no limitation on whether the skin of sandwich panel is thin or thick.
The effect of stacking sequence and radius of curvature of a curved layered composite beam on the distribution and value of through-thickness stress is chiefly investigated. Curved sandwich beams with thin and thick skin are also studied. The results show that the stacking sequences have a significant effect on the delamination and in-plane tensile failure to some extent. The radius of curvature of the beam also has a large effect on the through-thickness stress, which is consistent with the results for a single layer curved orthotropic beam. The biggest through-thickness stress in a curved sandwich beam always occurs at the interface between inner skin and core or in the core but very close to that interface. The approach could be used for the optimal design of curved laminated and sandwich beams and plates.
References [1] Shenoi RA, Hawkins GL. The formulation of a curved composite brick finite element using a layer-wise theory. Proceedings of ICCM-12, Whistler, Canada, Vol. v. 1994. p.149–156. [2] Hawkins GL, Holness JW, Dodkins AR, Shenoi RA. The strength of bonded Tee-joints in FRP ships. Plastics Rubber and Composites Processing and Applications 1993;19:279–84. [3] Shenoi RA, Hawkins GL. An investigation into the performance characteristics of top-hat stiffener to shell plating joints. Composite Structures 1995;30:109–21. [4] Reissner E, Stavsky YS. Bending and stretching of certain types of heterogeneous aeolotropic elastic plates. Journal of Applied Mechanics 1961;28:402–8. [5] Whitney JM. Effect of transverse shear deformation on the bending of laminated plates. Journal of Composite Materials 1969;3:534–47.
1512
R.A. Shenoi, W. Wang / Composites Science and Technology 61 (2001) 1501–1512
[6] Reddy JR. A simple higher order theory for laminated composite plates. Journal of Applied Mechanics 1984;51(12):745–52. [7] Bhate SR, Nayak UN, Patki AV. Deformation of composite beam using refined theory. Computers and Structures 1995; 54(3):541–6. [8] Chang Fu-Kuo. The strength analysis of wooden bends. Journal of Reinforced Plastics and Composites 1986;5(10):239–52. [9] Pagano NJ. Exact solution for composite laminates in cylindrical bending. Journal of Composite Materials 1969;3(3):398–411. [10] Pagano NJ. Analysis of the flexure test of bidirectional composites. Journal of Composite Materials 1967;1(4):336–442. [11] Lekhnitskii. Theory of elasticity of an anistropic body. Moscow: Mir, 1981 [12] Wu YS, Longmuir AJ, Chandler HW, Gibson AG. Delamination of curved composite shells due to through-thickness tensile stress. Plastics Rubber and Composites Processing and Applications 1993;19(1):39–46. [13] Gibson AG, Chandler HW, Longmuir AJ, Wilcox JAD. Delamination and failure of curved composite laminates. In: Proceedings of Structural Materials In Marine Environment, The Royal Society, London, UK, May 1994. p. 88–98. [14] Chandler HW, Longmuir AJ, McRobbie S, Wu YS, Gibson AG. Tensile delamination failure of curved laminates of single and double-skinned construction. In: Proceedings of 2nd International Conference on Deformation and Fracture of Composites, UMIST, Manchester, UK, March 1993. p. 29–31. [15] Tolf G. Stress in a curved laminated beam. Fibre Science and Technology 1983;19:243–67.
[16] Lu TJ, Xia ZC, Hutchinson JW. Delamination of beams under transverse shear and bending. Materials Science and Engineering 1994;A188:103–12. [17] Wisnom MR. 3-D Finite element analysis of curved beams in bending. Journal of Composite Material 1996;30(11):1178– 90. [18] Wisnom MR, Jones MI. Delamination due to interaction between curvature induced interlaminar tension and stresses at terminating plies. Composite Structure 1995;32:615–20. [19] Kaczmarek K, Wisnom MR, Jones MI. Edge delamination in curved (04/456)s glass-fibre/epoxy beams loaded in bending. Composite Science and Technology 1998;58:155–61. [20] Hetenyi M. Beams on elastic foundation — theory with applications in the fields of civil and mechanical engineering. The University of Michigan Press, 1946. [21] Shenoi RA, Wellicome JF, editors. Composite materials in maritime structures, ocean technology series. Cambridge University Press, 1993. [22] Timoshenko S. Theory of elasticity. New York and London: McGraw-Hill Book Company, 1934. [23] Shenoi RA, Wang W. Flexural behaviour of a curved orthotropic beam on an elastic foundation. Journal of Strain Analysis for Engineering Design (in press). [24] Smidt S. Curved sandwich beams and panels: theoretical and experimental studies. Report No.93-10, Royal Institute of Technology, Stockholm, Sweden. [25] Smidt S. Bending of curved sandwich beams, a numerical approach. Composite Structures 1996;34:279–90.
Through-thickness stresses in curved composite laminates and sandwich beams R.A. Shenoi*, W. Wang School of Engineering Sciences, University of Southampton, Southampton, SO17 1BJ, UK Received 21 September 2000; accepted 7 March 2001
Abstract An elasticity-theory-based approach is developed for delamination and flexural strength of curved layered composite laminates and sandwich beams. The governing equations in this case are derived from the results of curved orthotropic beam on an elastic foundation under flexural loading. The approach ensures an accurate description of the through-thickness and in-plane stresses in curved laminate beams. The solutions for various geometrical configurations are provided. The effects of various parameters, such as stacking sequence of the laminate, thickness of the skin, the curvature radius are also studied. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: A. Layered structures; B. Mechanical properties; C. Delamination; C. Laminates; C. Sandwich
1. Introduction Tee joints, curved sandwich beams and top-hat stiffeners are widely used in many structural applications. The mechanical behaviour of such structural elements has received a fair amount of attention. Up to now, most work in this area has focused on experimental and numerical analyses [1–3]. There are few theoretical analyses relating to the strengths of curved, layered laminated beams/panels or sandwich beams. Designers with composites are familiar with the conventional treatment of laminates such as the classical laminated-plate theory (CPT) and thin-shell theory [4]. In these classic treatments of curved shells, the assumption of a state of plane stress in the constitutive relations eliminates the possibility of rigorous calculation of interlaminar stresses. However, there is increasing awareness that the low values of through-thickness tensile and shear strength of FRP, compared to in-plane strengths, can significantly affect the performance. Moreover, although many high order, refined laminated plate theories and some approximate methods are available [5–7], their application to curved beams is limited [8]. Most of these theories * Corresponding author. Tel.: +44-23-80592316; fax: +44-2380593299. E-mail address: [email protected] (R.A. Shenoi).
or methods retain parts of the assumptions of classical plate and thin shell theory, and only incorporate the influence of shear deformation etc. on plate deflection in composite laminates. So while they can compute transverse shear stresses and give more accurate results for the in-plane stresses and displacements, they are unable to deal with through-thickness stresses. Pagano [9,10] studied the multi-ply composite laminates using linear elastic theory, and provided an exact solution for composite laminates in cylindrical bending. However, the composite laminates he considered were all flat and subjected to only transverse load condition. Lekhnitskii [11] gave a general solution to the problem of ‘bending of a plane curved rod by moments’. He studied the curved orthotropic beam but did not consider the response of a rod on an elastic foundation. The response of a curved beam without a foundation is quite different from that with a foundation. It can only sustain load condition such as pure bending or a pair tension/ compression forces at the ends, and it cannot sustain the circumferential force without radial restraints, so the solution of Lekhnitskii cannot be directly transformed to analyse the curved layered composite beam. Most recently reported work on curved beams deals with plane-strain applications of classical laminatedplate theories [12–14]. The extension of classical theories to curved beams is based on tight assumptions. For
0266-3538/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(01)00035-5
1502
R.A. Shenoi, W. Wang / Composites Science and Technology 61 (2001) 1501–1512
Mx,Nx,Qx
Nomenclature k m
p q t u; v A,B,D E; ; M; N
elastic modulus of foundation cross-ply stacking ratio signifying the ratio of the thickness sum of odd number plies to that of even number plies reaction force of elastic foundation distributed pressure thickness of beam displacement components in the x and y directions(or and r directions) stiffness matrixes Young’s modulus, shear modulus, Poisson’s ratio bending moment, axial force, shear force
example, Wu [12], Gibson [13] and Chandler [14] adopt the same model which retains the assumption that the stress normal to the cross-section of beam distributes linearly. This is acceptable when the longitudinal elastic modulus of composite beam does not change through thickness, such as a laminated beam composed of many layers oriented in the same direction. This assumption is not valid, however, for a general layered composite beam, in which the stiffness properties vary drastically from layer to layer. Stress components are thus likely to be discontinuous in their variation. Therefore, the method and results obtained by Wu [12], Gibson [13] and Chandler [14] cannot be used to analyse a general curved laminated beam. Tolf [15] analysed homogeneous transverse isotropy condition in detail, but in his discrete model, he took the laminate as a one ply fibre material by one ply matrix material. Lu [16] provided a solution of homogeneous anisotropy, but as with Tolf [15], his model and solution cannot be used to study layered composite curved beam and curved sandwich beam. In the recent past, Wisnom [17–19] has contributed much research work on experimental analyses and numerical approaches for interlaminar failure and flexural strength of composite laminate. The focus of this work has been towards understanding delamination failure in curved laminated beams. Interlaminar normal and shear stresses, acting either singly or interactively can lead to delamination as reported by Wisnom [17–19], thus affecting structural integrity. It is important to know through-thickness stresses even well below the delamination limit value, because these could have an interactive effect on failure under in-plane stress to some extent [12]. One of the key features of a curved (as distinct from a straight) laminated beam is the presence of not so insignificant
QðijkÞ R; Ro ; Ri Vf ; ; l
; ; " "0 f g
x
bending moment, in-plane force and shear force in x-direction stiffness matrix component of the kth layer radius of midplane, outer radius, inner radius Volume fraction coefficients in the expressions of reaction forces anisotropy ratio ratio of inner radius to outer radius of the beam (=Ri =Ro ) normal stress and shear stress, strain mid-plane strain the curvature of midplane curvature change of the curved beam stress function
through-thickness tensile stresses. These can significantly affect the performance of curved composite beams due to the low values of through-thickness tensile strength. Therefore, it is imperative to obtain the stress distribution through the thickness more accurately when analysing the behaviour of curved composite beam. The through-thickness stress in curved composite laminate or sandwich beam is studied in this paper by an elasticity-theory-based approach. Section 2 develops a model for characterising linear-static flexural behaviour of a curved beam on an elastic foundation by classical beam theory. The radial reaction force of elastic foundation and internal forces within the beam are obtained from the governing equations. These results are then used in Section 3 as boundary conditions for an elasticity solution for an orthotropic beam on an elastic foundation. Then a general solution for curved layered composite beam is achieved in Section 4. Sections 5 and 6 show the application of this approach in curved composite laminates and curved sandwich beams respectively. The effects of key variables such as the stacking sequence of the laminate, the curvature radius and the thickness of the skin of sandwich beam on stresses distribution within a curved layered beam or sandwich beam are also studied.
2. Flexural behaviour of a curved composite beam on an elastic foundation Consider a curved beam of radius R resting on an elastic foundation modulus k, as shown in Fig. 1. Assuming that the reaction forces in the foundation are normal to the axis of the beam and proportional at every point to the radial deflection of the beam v at the point, the following is obtained:
R.A. Shenoi, W. Wang / Composites Science and Technology 61 (2001) 1501–1512
1503
Fig. 1. Curved beam on an elastic foundation. (a) Beam geometry; (b) forces on a small element.
p ¼ kv
ð1Þ
The equilibrium equation of a curved beam can be obtained as: k
dv dq 1 dMx d3 Mx 2 ¼ dx dx R dx dx3
ð2Þ
Thickness of the laminate can be incorporated into the equations: 1 t dv t 1 dq 1 dMx k 1 þ 1 3 R 2R d 2R R d R d ¼
1 d3 Mx R3 d3
ð3Þ
In order to obtain the solution to the above differential equation, it is necessary to obtain a relation between the moment Mx and curvature change x . This is well documented for beams made from isotropic materials [20]. In the case of laminated composites beam, the relationship is obtained from the classical laminated plate equation [21]: 2
3 2 N A 4 5 ¼ 4 M B
32 3 j B "0 j 5 4 5 j D
ð4Þ
case of a cylindrical shell with large width (in z-direction) is considered, and the applied forces and boundary conditions are both uniform in cylinder axial direction (z-direction). The problem here could then be restricted to a curved beam of unit width, and it is assumed "z=0. A further assumption is made in that the transverse pressure load q is constant. Then, from Eqs. (4) and (5), the following is true:
Nx ¼ A11 "0x þ B11 x ð6Þ Mx ¼ B11 "0x þ D11 x While considering the effect of in-plane deformation on the change of curvature of circular beam or cylindrical shell, the relation between the radial deflection v and curvature x is pertinent: x ¼
Aij ; Bij ; Dij ¼
ð t=2
QðijkÞ 1; y; y2 dy
ð5Þ
t=2
Because of the tension-bending coupling and bendingtorsion coupling, a simple expression of Mx and x cannot be obtained easily; this is needed in solving the differential Eq. (2). Our approach is to simplify the problem and consider only one of the in-plane directions, says the x-direction. This could be achieved if the
ð7Þ
Incorporating this in Eq. (6) and then substituting in Eq. (3), the following is obtained: 1 B211 d5 v 2 B211 d3 v D þ D 11 11 R5 A11 d5 R5 A11 d3
k 1 B211 dv þ þ D11 R R5 d A11 ¼
where
1 d2 v v þ R2 d2 R2
1 dq 1 B11 dNx 1 B11 d3 Nx þ 3 þ 3 R d R A11 d R A11 d3
ð8Þ
Two conditions will be considered in the following. (1) Nx=constant (N0) This is a common occurrence in the case of sandwich beams subjected to pure bending, where the axial force in the skin is a constant throughout the span. Also noting the previous assumption of constant pressure q. Thus, in this condition, the right part of Eq. (8) is zero, and the equation turns out to be:
1504
R.A. Shenoi, W. Wang / Composites Science and Technology 61 (2001) 1501–1512
d5 v d3 v dv ¼0 þ 2 3 þ 2 5 d d d
ð9Þ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u R4 k where: ¼ u þ1 t B2 D11 11 A11
ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 u Rk u1 Using ¼ 1 and =u t2 B2 D11 11 A11 2
The results for Mx and Qx can be deduced to be: Mx ¼
The general solution of the homogeneous differential Eq. (9) is: v ¼ K0 þ ðK1 cosh þ K2 sinhÞcos þ ðK3 cosh þ K4 sinhÞsin
ð10Þ
From this, the reaction force of elastic foundation can be deduced as: p ¼ 0 þ ð1 cosh þ 2 sinhÞcos þ ð3 cosh þ 4 sinhÞsin
2
ð11Þ
rffiffiffiffiffiffiffiffiffiffiffi 8 1 > > < ¼ 2 rffiffiffiffiffiffiffiffiffiffiffi where > þ 1 > : ¼ 2 and where Ki and i are constants to be determined from boundary conditions. (2) Nx ¼ g0 þ g1 cos This is the case when a curved sandwich beam is subjected to a pair of tension/compression forces at two ends. Again, the right hand side of Eq. (8) is zero. Thus, the governing differential equation and solutions for the lateral deflection and reaction forces will be identical to Eqs. (9), (10) and (11) respectively. For a laminate symmetric with respect to the midplane, the case becomes much simpler, because the tension-bending coupling does not exist, i.e. Bij ¼ 0. If the curved beam is symmetric with respect to the axis ¼ 0, then in the equations (10) and (11), only even-function parts remain: v ¼ K0 þ K1 coshcos þ K4 sinhsin
ð12Þ
p ¼ 0 þ 1 coshcos þ 4 sinhsin
ð13Þ
B11 B2 1 h Nx D11 11 2 K0 A11 A11 R
þ 2K4 coshcos 2K1 sinhsin B11 dNx 1 1 B2 Qx ¼ 3 D11 11 A11 d R R A11 h 2 2 2K1 2K4 coshsin i þ 2K1 2 2K4 2 sinhcos
i
ð15Þ
ð16Þ
When Nx ¼ N0 is a constant, then from the equilibrium considerations of beam element, Qx= 0. It can be seen that the above equation is satisfied under any value of only when: K1 ¼ K4 ¼ 0. Thus, for constant Nx, and also considering the existence of distributed pressure acting downwards on the inner surface of curved beam q0 and the height of beam t, the following results can be deduced: 8 B11 B211 K0 > > ¼ N D M 2 > x x 11 > > A11 A11 R < t t N ¼ k R þ q R K x 0 0 > 2 2 > > > > : p ¼ kK0 Qx ¼ 0
ð17Þ
This conclusion shows that even in a general case where: (a) tension-bending coupling occurs as a result of the laminate layer; and (b) the beam is subjected to inplane forces Nx, bending moments Mx and the shear force Qx still is zero. Consequently, the case of a curved beam laid on elastic foundation as a problem in which stresses are independent of polar angle can still be considered pertinent and whose results can be used in analysing curved laminated beams and the skins/faces of curved sandwich beams. These results will be used in the following parts as force boundary conditions when the same problem is analysed by theory of elasticity to obtain an accurate general solution of a layered composite beam on elastic foundation.
Thus, from Eq. (8): 1 h x ¼ 2 K1 2 þ 2K4 K1 2 þ K1 coshcos R i þ K4 2 2K1 K4 2 þ K4 sinhsin þ K0 ð14Þ
3. Solution for a curved orthotropic beam on an elastic foundation Introducing a stress function in polar coordinates ðr; Þ, we can write stresses to be [22]:
R.A. Shenoi, W. Wang / Composites Science and Technology 61 (2001) 1501–1512
8 1 @ 1 @2 > > > > r ¼ r @r þ r2 @2 > > < @2
¼ 2 > @r > > > @ 1 @ > > : r ¼ @r r @
ð18Þ
8 pffiffi > 3 R1 l < 3¼ o pffiffi > 4 R1þ l : 4 ¼ o
1505
ð24Þ
Where,
The above equation can satisfy the equilibrium equation. For the problem under consideration, the stress components do not depend on and, in addition, r
0 (i.e. constant in-plane force and hydrostatic load). For the cylindrical material anisotropy, a compatibility equation can be obtained as [16]: d4 2 d3 l d2 l d ¼0 þ 3 2 2 þ 3 dr4 r dr r dr r dr
ð19Þ
pffiffi pffiffi pffiffiffi pffiffiffi 1 þ l 1 l1 1 l 1 l1 ¼ pffiffiffi ffiffi ffiffi p p pffiffiffi lþ1 l 1 1 lþ1 1þ l 1 pffiffi pffiffiffi kK0 þ q0 1 l 1 l1 R2 o 1pffiffil pffiffi 3 ¼ Ro p ffiffi ffi 2M þ kK R2 qR2 1 þ l 1 l 1 0 0 o i pffiffi p ffiffi ffi 1 þ l 1 l1 R2 kK0 þ q0 o 1þpffiffil 4 ¼ pffiffiffi Ro pffiffi l1 2M0 þ kK0 R2o qR2i 1 lþ1 ð25Þ
E2 l¼ E1
ð20Þ
where E1 and E2 are the elastic moduli in the in-plane and through-thickness directions respectively. The general solution to this ordinary differential equation is: ¼ C1 þ C2 r2 þ C3 r1þ
pffiffi l
þ C4 r1
pffiffi l
The coefficients of the stress formulae can then be obtained as: pffiffiffi pffiffiffi 3 1 l 4 2C2 ¼ kK0 1 þ l pffiffi 3 R1 l ð26Þ C3 ¼ o pffiffi 1þ l C4 ¼ 4 Ro
ð21Þ Correspondingly, the solution of stresses r and as:
from which stress components can be obtained: (
pffiffi pffiffi pffiffiffi pffiffiffi
r ¼ 2C2 þ 1 þ l C3 r l1 þ 1 l C4 r l1 pffiffi pffiffi pffiffiffipffiffiffi pffiffiffipffiffiffi
¼ 2C2 þ 1 þ l lC3 r l1 1 l lC4 r l1
ð22Þ The appropriate boundary conditions can be used to determine the coefficients in the above expressions. From Eq. (17) and the geometry and considering the direction of the forces, we have:
8 h pffiffiffi pffiffiffi i > 3 1 l 4 > > r ¼ kK0 1 þ l > > > plffiffi1 > > pffiffiffi > > 3 r > þ 1þ l > > Ro > > > pffiffil1 > < pffiffiffi 4 r þ 1 l Ro > > plffiffi1 > h > pffiffiffi pffiffiffi i pffiffiffi pffiffiffi > > 3 r 3 1 l 4 þ 1þ l l > ¼ kK0 1þ l > > > Ro > p ffiffi > l1 > > p ffiffi ffi p ffiffi ffi > r > 4 > 1 l l : Ro ð27Þ
Because of the plane-strain assumption, the equations of Hook’s Law of anisotropic material can be expressed.
t
r ¼ p ¼ p0 ¼ kK0 r¼Rþ : 2 t
r ¼ q ¼ q0 r¼R : 2 Ð Rþ2t t t þ q R K t dr ¼ N ¼ N0 ¼ k R þ 0 0 R2 2 2 Ð Rþ2t
rd ¼ M ¼ M0 R t 2
ð23Þ Substituting the expressions of stress components (22) into the above boundary condition (23), then it can be noted that the third equation in (23) is automatically satisfied when the first and second equations are satisfied. This equation group can be solved and the value of coefficients C2 ; C3 and C4 can be determined. Using the notation:
8 1 0 > > "r ¼ 0 r 120 > > E2 E1 > > < 021 1 " ¼ 0 r þ 0 > E E > 2 1 > > 1 > > : "r ¼ r 212
ð28Þ
where: E1 ; 1 13 31 12 þ 13 32 ¼ ; 1 13 31
E2 1 23 32 21 þ 23 31 ¼ 1 23 32
E01 ¼
E02 ¼
012
021
ð29Þ
1506
R.A. Shenoi, W. Wang / Composites Science and Technology 61 (2001) 1501–1512
Combining Eqs. (27), (28) and (29) with Cauchy equation in polar coordinates [19]: @ur ur 1 @u ; " ¼ þ ; @r r r @ 1 @ur @u u r ¼ þ r @ @r r
"r ¼
ð30Þ
By integrating the above expression and noting the relation:
012 E01
¼
021 E02 ,
E0
l ¼ E20 , the solution of displacement can 1
be obtained as:
pffiffiffi pffiffiffi i 1 021 h 3 1 l 4 rþ 1 þ l kC ur ¼ 0 E01 E02
plffiffi pffiffiffi 0 pffiffiffi 1 Ro r 21 pffiffiffi 1 þ l 0 l þ l 0 3 E1 Ro E2 l
pffiffil pffiffiffi 0 pffiffiffi 1 Ro r 21 pffiffiffi 1 l 0 þ l l 0 4 þfðÞ E1 Ro E2 l pffiffiffi 1 1 h 3 u ¼ 0 þ 0 kC0 1 þ l E1 E2 ð pffiffiffi i 1 l 4 r fðÞd ð31Þ The arbitrary function fðÞ in the above equations can be determined by displacement boundary condition. Here it should be noted that the above anisotropic solution cannot be extended to the isotropic case only taking l ¼ EE21 ¼ 1, though convergence to the isotropic solution can be achieved by putting l close to 1. As is known, the isotropic material case of l ¼ 1 actually corresponds to a double root of the characteristic equation arising from the generalised differential Eq. (19), leading to the logarithmic term in the expression of stress components and stress function. In homogeneous isotropic materials, as to the problem of plane symmetrical about the axis, the stress distribution is [22]: 8 a > > < r ¼ r2 þ bð1 þ 2lnrÞ þ 2c a
¼ 2 þ bð3 þ 2lnrÞ þ 2c > > r : r ¼ 0
8
R2 R2 Ro > > > a ¼ 2 o i 2 ðq0 p0 Þ 2ln b > > Ri Ri Ro > > > 2 2 > Ri > 2 2 < 2 Ri Ro M0 Ro Ri R2o p0 2R2o R2i ln ðq0 p0 Þ Ro b¼ > 2 Ro 2 > > Ri R2o 4R2o R2i ln2 > > > Ri > > a > > : 2c ¼ p0 2 bð1 þ 2lnRo Þ Ro ð34Þ
4. General solution for curved composite laminates and sandwich beams Based on the above solutions, further analysis can be done to obtain a general solution for the problem. Assuming there are n layers in curved composite beam, where every layer has the elastic coefficients: Eð1iÞ ; Eð2iÞ ; ð12iÞ and ð21iÞ and where the superscript ðiÞ means this variable is related to the ith layer. The boundary conditions to be satisfied in every layer are as follows: 8 r ¼ ri1 : r ¼ rði1Þ > > < r ¼ r : ¼ ðiÞ r r Ð ri i ði Þ
dr ¼ N > r > Ð ri1 : i ðiÞ ri1 rdr ¼ M
ð35Þ
where rðiÞ is normal tension stress in the interface between ith layer and ði þ 1Þth layer. In addition, following on from Eq. (17), there exist the following relations: 8 ðiÞ ðiÞ ði1Þ > < N ¼ r ri r ri1 ! ðiÞ ðiÞ2 4Cð0iÞ B B ð36Þ ð i Þ ði Þ 11 11 ðiÞ > : M ¼ ðiÞ N D11 ðiÞ ðiÞ 2 ð i1 Þ A11 A11 ðr þ r Þ
ð32Þ
The displacement is: 8 1h 1 > u ¼ ð1 þ Þa þ 2ð1 Þbrlnr ð1 þ Þ > r > > E r i < br þ 2ð1 Þcr þ gðÞ > ð > > > : u ¼ 4br gðÞd E
The arbitrary function gðÞ in the above equations can also be determined by boundary condition. If same boundary conditions as in Eq. (23) are assumed, then the constants in the expressions (33) can be determined as:
ð33Þ
In fact, every layer in the curved composite beam can be considered as an orthotropic beam or isotropic beam, with the interaction forces between adjacent layers taken as the reaction force of a kind of ‘elastic foundation’ — p and the water pressure — q. This is because in a mathematical context, rði1Þ and q, rðiÞ and p have the same meaning. They are only constants in the formulae, under no-circumferential-dependence condition. Then as in the analysis for the curved orthotropic beam, here too the known boundary conditions can be imposed:
R.A. Shenoi, W. Wang / Composites Science and Technology 61 (2001) 1501–1512
r ¼ ri¼0 ¼ Ri : r ¼ q0 r ¼ ri¼N ¼ Ro : r ¼ p0
ð37Þ
Therefore, in the analysis of the whole laminate, there exit unknown variables rðiÞ i¼1;2;;n1 and Cð0iÞ i¼1;2;;n . The number of these variables is n þ ðn 1Þ ¼ 2n 1 in all. Here the displacement compatibility condition on the interface can be represented as:
ðiþ1Þ ur ðri Þ ¼ uðriÞ ðri Þ i ¼ 1; 2; ; n 1 ð38Þ uðiþ1Þ ðri Þ ¼ uðiÞ ðri Þ i ¼ 1; 2; ; n 1 uðriÞ ðrÞ; uðiÞ ðrÞ are as expressed in Eqs. (31) or (33), according to that the layer is orthotropic beam or isotropic beam under consideration. Apart from these, for the case of curved beam on an elastic foundation under no-circumferential-dependence condition, there is the global force boundary condition: n X NðiÞ ¼ N0 ¼ kRo C0 q0 Ri
ð39aÞ
i¼1
And for the case of curved beam subjected to pure bending, there is another global force boundary condition: 4Cð0nÞ B11 B211 N0 D11 M0 ¼ A11 A11 ðRi þ Ro Þ2
ð39bÞ
As can be seen, there are: 2 ðn 1Þ þ 1 ¼ 2n 1 equations altogether, just the same as the number of unknown variables. Thus the problem can be solved. However, it should be noted that this general solution for curved layered composite beam on elastic foundation is under no-circumferential-dependence condition. Note also that rðiÞ i¼1;2;;n are the tensile through thickness stresses between every two adjacent layers interface tension stress.
5. Application: effect of laminate stacking sequence on stresses Shenoi and Wang [23] have studied and examined the effects of material- and geometry-related variables of a single-layer curved orthotropic beam and the elastic foundation on stresses and displacements. Here the effects of the above parameters and the stacking sequence of curved layered composite beam and elastic modulus of core material in curved sandwich are examined. In order to investigate the effects of stacking sequence of layered composite beam, an individual layer of unidirectional fibrous composite material is considered to possess the following (corresponding to a general Eglass/epoxy composite):
EL ¼ 38:6 GPa ET ¼ 8:27 GPa
1507
GLT ¼ 4:14 GPa
GTT ¼ 2:76 GPa LT ¼ 0:26
TT ¼ 0:5
Vf ¼ 0:45
Where L signifies the direction parallel to the fibres, T the transverse direction, and LT is the Poisson ratio measuring strain in the transverse direction under uniaxial normal stress in the L direction. And it is assumed that the material characteristics through thickness direction are the same as those in the transverse direction totally. These properties are used in appropriate contexts in the equations for calculating stresses and displacements (20), (29) and (33). For example, for a UD ply with 0 orientation, then E1 ¼ EL and E2 ¼ ET ; for a ply with 90 orientation, E1 ¼ ET and E2 ¼ ET , which means isotropic in the xy plane. Similar analogies can be given for other properties or for other layups. Several separate stacking configurations are considered, namely: 1. [0 /0 ] and [90 /90 ] — Describing orthortropic and isotropic beam respectively. 2. [0 /90 ] and [90 /0 ] — Two bidirectional (coupled) laminates with the layers being of equal thickness. 3. [0 /90 /0 ] — A symmetric 3-ply laminate with cross-ply stacking ratio m ¼ 1:0 4. [0 /90 /0 /90 ] and [90 /0 /90 /0 ] — Two anisymmetric 4-ply laminates both with m ¼ 1:0. 5. [0 /90 /0 /90 /0 ] — A symmetric 5-ply laminate with the layers being of equal thickness. In each case, the curved beams have the same curvature radius: Ri ¼ 30 mm, Ro ¼ 36 mm, so the thickness of the beam is t=6 mm. The curved beam is subjected to pure bending, the bending moment is M=1000 N/m. The functions of prime interest in the present case are radial stress r (the through-thickness stress of the curved layered composite beam) and circumferential stress (the in-plane stress of the curved layered composite beam). Under the no-circumferential-dependence condition, shear stress equals zero. The solutions for r and are illustrated in Figs. 2–6. In each figure, abscissa is the value of stress and ordinate is the normalized thickness y ¼ y=t. The stresses distribution in [0 /0 ] layered beam are shown in Fig. 2. The results of [90 /90 ] layered beam are very similar to Fig. 2. This correspond to the result for curved orthotropic beam on an elastic foundation [23], which showed that anisotropy ratio l has no significant effect on the stress distribution in the curved beam. The maximum through-thickness stress occurs on the inner side of the midplane of beam, very close to the midplane. Its value is 7.60 MPa.
1508
R.A. Shenoi, W. Wang / Composites Science and Technology 61 (2001) 1501–1512
Fig. 3 gives the results for [0 /90 ] and [90 /0 ] — two bidirectional (coupled) laminate cases. In each case, the layers are of equal thickness. The results show that there
is a big difference between these two coupled laminates, not only in the distribution of through-thickness stress and in-plane stress, but also in their values. The maximum
Fig. 2. Stresses distribution along thickness unidirectional laminate. (a) Through-thickness stress distribution; (b) in-plane tension stress distribution.
Fig. 3. Stresses distribution along thickness in cross-ply laminate. (a) Through-thickness stress distribution; (b) in-plane tension stress distribution.
Fig. 4. Stresses distribution along thickness in symmetric 3-ply laminate. (a) Through-thickness stress distribution; (b) in-plane tension stress distribution.
R.A. Shenoi, W. Wang / Composites Science and Technology 61 (2001) 1501–1512
1509
Fig. 5. Stresses distribution along thickness in antisymmetric 4-ply laminate. (a) Through-thickness stress distribution; (b) in-plane tension stress distribution.
Fig. 6. Stresses distribution along thickness — stacking sequence [0 /90 /0 /90 /0 ]. (a) Through-thickness stress distribution; (b) in-plane tension stress distribution.
absolute values of r and in [0 /90 ] case are both more than 10 higher than those in [90 /0 ] case separately. This difference will become bigger as the ratio ð¼ Ri =Ro Þ decreases. In a flat layered composite beam, this distinguishing difference does not exist. In these two laminates, the maximum through-thickness stresses are both occur in the 0 layer. A similar result can also be found in the analyses of [0 /90 /0 ], [0 /90 /0 /90 ] and [90 /0 /90 /0 ] laminates. The stresses distribution in these case are showed in Figs. 4 and 5. Three laminates have the same stacking ratio (m ¼ 1:0). It is reasonable that the stress distributions are different among these cases due to their different stacking sequence, but the maximum absolute values of stress in each laminate are quite different. The maximum values of through-thickness and in-plane stress
are 6.79 and 192 MPa in [0 /90 /0 ] laminate, 8.54 and 261 MPa in [0 /90 /0 /90 ] laminate, 8.11 and 240 MPa in [90 /0 /90 /0 ] laminate each. The maximum through-thickness stresses in three cases all occur in the 90 layer and close to the midplane of laminate. Fig. 6 shows the results for symmetric 5-ply stacking case. The value of maximum through-thickness stress in [0 /90 /0 /90 /0 ] laminate curved beam is close to that in [0 /90 /0 ] case, only very a little higher. The above results indicate that the stacking sequence of layered composite beam does have an effect on stresses. In two-ply (coupled) case, stacking sequence [90 / 0 ] is ‘‘better’’ than [0 /90 ]; in 4-ply case, stacking [90 / 0 /90 /0 ] is ‘‘better’’ than [0 /90 /0 /90 ]. Under the condition of the same stacking ratio, apparently the [0 / 90 /0 ] is the ‘‘best’’ stacking sequence among the above
1510
R.A. Shenoi, W. Wang / Composites Science and Technology 61 (2001) 1501–1512
Fig. 7. Stresses distribution along sandwich panel. (a) Through thickness stress distribution; (b) in-plane tension stress distribution.
stacking cases. It should also be noticed that in theory a unidirectional (UD) stacking is not a better one; the maximum value of through-thickness stress (7.60 MPa) in the UD case is more than 10 higher than that of [0 / 90 /0 ] case, although its maximum value of in-plane stress is lower (178 MPa). This indicates the possibility of identifying an ‘optimal stacking sequence’ for a curved layered composite beam.
6. Application: curved sandwich beams In order to investigate the stress distribution in curved sandwich beams, two separate geometrical configurations are considered first: (1) Thick skin-sandwich panel with ts =tc ¼ 14 1 (2) Thin skin-sandwich panel with ts =tc ¼ 10 The skin is a unidirectional cylindrical shell with the fibres oriented in the circumferential direction. The other geometrical variables of curved sandwich beam are the same as those of curved layered composite beam studied in the previous section. The material properties of the skin are the same as the above. The elastic constants of the isotropic core material of sandwich beam are: E ¼ 1:103 GPa
¼ 0:3
The curved sandwich beam is also subjected to pure bending, the bending moment is: M ¼ 1000 N=m Fig. 7 shows the results for these two kinds of curved sandwich beam. As can be seen, a significant throughthickness stress gradient exists in the skin, but the max-
imum through-thickness stress occurs in the core. It is sometimes at the interface between core and the inner skin, sometimes in the core but near to that interface (as shown in Fig. 7a). It can also be noted that the throughthickness stress changes only very slightly in the core; it keeps a high level nearly through the whole thickness of the core. This explains why in Gibson and Chandler’s experiment [13], delamination sometimes was seen to occur between either skin or core, and sometimes between both skins and the core. From the trends in Fig. 7a and b, it is known that in a thin-skin sandwich beam, there is a smaller throughthickness stress than that in a thick-skin sandwich beam, simultaneously there has the bigger in-plane stress which is reasonable. As a comparison with the numerical research results, the above approach is applied again to analyse two samples once used in Smidt’s research work [24,25]. The magnitudes of curved sandwich beam sample are: width 95 mm, inner radius of core 166 mm, skin thickness tf=2 mm, 10 mm, core thickness tc=50 mm. The skin material is GRP ES= 18.1 GPa, core material is H60 (Divinycell) EC=55 MPa. Fig. 8a and b shows the results for these two curved sandwich beam samples. Smidt analysed these two samples by the finite-element method. In his numerical analyses the maximum values of through-thickness stresses are 1.14 and 0.95 MPa for the thin and thick skin cases respectively, which are both located at the inner interface. As can be seen, the theoretical results — of about 1.16 and 9.85 MPa — coincide very well with the numerical results. It should be pointed out that, although there is a smaller through-thickness stress in a thick-skin sample, the thickness of thick skin is five times of the thin skin and the distance between the centrelines of the faces increased up to 15.4%, which leads to bigger general bending stiffness of sandwich beam
R.A. Shenoi, W. Wang / Composites Science and Technology 61 (2001) 1501–1512
1511
Fig. 8. Through-thickness tensile stress distribution in a curved sandwich beam.
and much bigger weight of course. If the total thickness of sandwich beam (tf+tc) is kept constant, then thick skin structure poses the bigger through-thickness stress, as is shown in the previous application. In order to investigate the effect of radius of the curved beam on stresses, another geometrical configuration of the curved layered composite beam is studied: Ri ¼ 18 mm, Ro ¼ 24 mm. The thickness of the beam is still h ¼ 6 mm. The curved beam is still subjected to pure bending, the bending moment is M ¼ 1000 N/m. Here the [0 /90 /0 ] stacking sequence is again considered. The results are as follows as shown in Fig. 4. As would be expected, the radius of curved beam has big effect on the stresses, especially on the value of through-thickness stress, which is similar to the result for curved orthotropic beam [23]. As can be seen, when the thickness is stable and outer radius becomes a third smaller, the maximum of through-thickness stress in beam increases by 50%, simultaneously the maximum in-plane stress becomes a little bigger.
7. Concluding remarks In conclusion, a new approach to define the elasticity solutions for curved composite beam consisting of arbitrary numbers of orthotropic or isotropic layers in nocircumferential-dependence case is presented. Since the solutions are exact, within the assumptions of linear elasticity, there need be no strict distinction between thick and thin curved beams. Also there is no limitation on whether the skin of sandwich panel is thin or thick.
The effect of stacking sequence and radius of curvature of a curved layered composite beam on the distribution and value of through-thickness stress is chiefly investigated. Curved sandwich beams with thin and thick skin are also studied. The results show that the stacking sequences have a significant effect on the delamination and in-plane tensile failure to some extent. The radius of curvature of the beam also has a large effect on the through-thickness stress, which is consistent with the results for a single layer curved orthotropic beam. The biggest through-thickness stress in a curved sandwich beam always occurs at the interface between inner skin and core or in the core but very close to that interface. The approach could be used for the optimal design of curved laminated and sandwich beams and plates.
References [1] Shenoi RA, Hawkins GL. The formulation of a curved composite brick finite element using a layer-wise theory. Proceedings of ICCM-12, Whistler, Canada, Vol. v. 1994. p.149–156. [2] Hawkins GL, Holness JW, Dodkins AR, Shenoi RA. The strength of bonded Tee-joints in FRP ships. Plastics Rubber and Composites Processing and Applications 1993;19:279–84. [3] Shenoi RA, Hawkins GL. An investigation into the performance characteristics of top-hat stiffener to shell plating joints. Composite Structures 1995;30:109–21. [4] Reissner E, Stavsky YS. Bending and stretching of certain types of heterogeneous aeolotropic elastic plates. Journal of Applied Mechanics 1961;28:402–8. [5] Whitney JM. Effect of transverse shear deformation on the bending of laminated plates. Journal of Composite Materials 1969;3:534–47.
1512
R.A. Shenoi, W. Wang / Composites Science and Technology 61 (2001) 1501–1512
[6] Reddy JR. A simple higher order theory for laminated composite plates. Journal of Applied Mechanics 1984;51(12):745–52. [7] Bhate SR, Nayak UN, Patki AV. Deformation of composite beam using refined theory. Computers and Structures 1995; 54(3):541–6. [8] Chang Fu-Kuo. The strength analysis of wooden bends. Journal of Reinforced Plastics and Composites 1986;5(10):239–52. [9] Pagano NJ. Exact solution for composite laminates in cylindrical bending. Journal of Composite Materials 1969;3(3):398–411. [10] Pagano NJ. Analysis of the flexure test of bidirectional composites. Journal of Composite Materials 1967;1(4):336–442. [11] Lekhnitskii. Theory of elasticity of an anistropic body. Moscow: Mir, 1981 [12] Wu YS, Longmuir AJ, Chandler HW, Gibson AG. Delamination of curved composite shells due to through-thickness tensile stress. Plastics Rubber and Composites Processing and Applications 1993;19(1):39–46. [13] Gibson AG, Chandler HW, Longmuir AJ, Wilcox JAD. Delamination and failure of curved composite laminates. In: Proceedings of Structural Materials In Marine Environment, The Royal Society, London, UK, May 1994. p. 88–98. [14] Chandler HW, Longmuir AJ, McRobbie S, Wu YS, Gibson AG. Tensile delamination failure of curved laminates of single and double-skinned construction. In: Proceedings of 2nd International Conference on Deformation and Fracture of Composites, UMIST, Manchester, UK, March 1993. p. 29–31. [15] Tolf G. Stress in a curved laminated beam. Fibre Science and Technology 1983;19:243–67.
[16] Lu TJ, Xia ZC, Hutchinson JW. Delamination of beams under transverse shear and bending. Materials Science and Engineering 1994;A188:103–12. [17] Wisnom MR. 3-D Finite element analysis of curved beams in bending. Journal of Composite Material 1996;30(11):1178– 90. [18] Wisnom MR, Jones MI. Delamination due to interaction between curvature induced interlaminar tension and stresses at terminating plies. Composite Structure 1995;32:615–20. [19] Kaczmarek K, Wisnom MR, Jones MI. Edge delamination in curved (04/456)s glass-fibre/epoxy beams loaded in bending. Composite Science and Technology 1998;58:155–61. [20] Hetenyi M. Beams on elastic foundation — theory with applications in the fields of civil and mechanical engineering. The University of Michigan Press, 1946. [21] Shenoi RA, Wellicome JF, editors. Composite materials in maritime structures, ocean technology series. Cambridge University Press, 1993. [22] Timoshenko S. Theory of elasticity. New York and London: McGraw-Hill Book Company, 1934. [23] Shenoi RA, Wang W. Flexural behaviour of a curved orthotropic beam on an elastic foundation. Journal of Strain Analysis for Engineering Design (in press). [24] Smidt S. Curved sandwich beams and panels: theoretical and experimental studies. Report No.93-10, Royal Institute of Technology, Stockholm, Sweden. [25] Smidt S. Bending of curved sandwich beams, a numerical approach. Composite Structures 1996;34:279–90.