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Procedia Materials Science 10 (2015) 793 - 802

2nd International Conference on Nanomaterials and Technologies (CNT 2014)

Cylindrical bending of elastic plates

Poonam V.Nimbolkar1, Indrajeet M.Jain2

Sinhgad institute of technology and science Narhe, Pune-411041, Maharashtra (India)

Abstract

This paper deals with the cylindrical bending of elastic and composite plates subjected to the mechanical transverse loading response under plain strain condition, a complete analytical solution is presented for the cylindrical banding of multilayered orthotropic plates with simply supported edge conditions based on Reissner-Mindlin's first order shear deformation theory (FOST). Composite material is orthotropic in nature and exhibits certain advantages of higher strength and stiffness to weight ratios, longer fatigue life, enhanced corrosion resistance etc. Laminated plate consists of homogeneous elastic laminae of arbitrary thickness. Composite laminates are widely used in construction of mechanical, aerospace, marine and automotive structures. In this formulation, a two dimensional (2D) elasticity problem reduces to one dimensional (1D) plate problem. Excel programming is used to execute the problem. Results of displacements and stresses are presented for simply supported isotropic and orthotropic plates and compared with exact and other available solutions from the literature.

© 2015Published byElsevier Ltd.Thisisan openaccess article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the International Conference on Nanomaterials and Technologies (CNT 2014) Keywords/Cylindrical bending; Laminated Composites; Plain strain; Reissner-Mindlin platetheory

1. Introduction:

A flat plate is a structural member having thickness is less than other two dimensions (length and width). Flat plates are extensively used in many engineering applications like tank bottom, floors and roof of the building, deck slabs of the bridges, turbine disks etc. Plates used for these applications are subjected to lateral loads that causing bending deformation as well as stretching. The geometry of the plate is normally defined by the middle plane which is equidistant from top and bottom of the edges of the plate. Thickness of the plate is always measured in the direction of the middle plane. The flexural strength is totally depends on the thickness of the plate. In recent years, the utilization of advanced composite materials is being used increasingly in many of structural applications such as high performance structures. A composite material is obtained by combining two or more materials so that the properties of composites are different from individual constituent material, Due to the special properties exhibited by these new materials, the conventional methods of analysis become inadequate. Very often the structures are subjected to both static and dynamic loads of various magnitudes and complexities. To investigate the actual behaviour of the structures under these loads, rigorous analysis is required to assess the strength and stability under various boundary conditions

2211-8128 © 2015 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the International Conference on Nanomaterials and Technologies (CNT 2014) doi:10.1016/j.mspro.2015.08.001

and loading cases. An effective and efficient use in structural applications requires good understanding of their static and dynamic behaviour under various types of loading conditions. It is a challenging problem to understand the dynamical behaviour of composite or sandwich plates with sufficient accuracy.

Laminated composite structures are widely used in many of engineering applications such as civil, mechanical, nuclear, aerospace, chemical industries as well as in sports and health instrument applications due to low specific density and low specific modulus. Laminated composites and sandwich structures constitute light weight with high stiffness, high structural efficiency and durability. Advanced composite materials like graphite/epoxy, boron/epoxy, Kevlar/epoxy etc. are replacing metals and alloys in the manufacturing of structural members. High ratio of inplane modulus to shear modulus of composite laminates, the shear deformation effects are obvious in the thick composite laminates and hence any analytical model should predict accurate of interlaminar stresses in the laminate. These composite materials permit the designer to 'tailor make' the structural properties through various lamination schemes to achieve the specified objectives.

Isotropy property implies that the material properties at a point are the same in all directions. However, some materials have properties that are not independent of direction and such materials are called anisotropic materials. When the material properties are different in two mutually perpendicular directions is called orthotropic materials. There are two types of orthotropy, namely material orthotropy and structural orthotropy. Material orthotropy is due to physical structure of the material e.g. wood, crystals etc., while structural orthotropy is due to fabrication methods used for making structural component e.g. reinforced concrete slabs, fiber reinforced plastics, stiffened plates etc. Ghugal and Shimpi (2002) presented A review of displacement and stress based refined theories for isotropic laminated plates is presented. A refined hybrid plate theory for composite laminate with piezoelectric laminae studied by Mitchell and Reddy (1995).Reissner and Mindlin are the firstly given FOSDT , based on the assumed stress and displacement fields by Chandrashekhar (2001).Bending analysis of a moderately thick orthotropic sector plate subjected to various loading Conditions with the help of first order shear deformation theory studied by Aghdam M and Mohammadi M (2009).Bhar et al.(2009) Significance of using higher-order shear deformation theory (HOSDT) over the first order shear deformation theory (FOSDT) for analyzing laminated composite stiffened plates is brought out using the finite element method (FEM).Vel et al.(2004) gives an Analytical solution for the cylindrical bending vibration of piezoelectric composite plates. The generalized plane deformations of linear piezoelectric laminated plates and cylindrical bending of laminated plates with piezoelectric actuators are analyzed by Vel and Batra (2000).A new theory is proposed by Pagano (1978) to define the complete stress field within an arbitrary composite laminates. The theory is based upon an extension of Reissners variational principle to laminated bodies. Kant and Swaminathan (2000) studied estimation of transverse/interlaminar stresses in laminated composites-A selected review and survey of current developments. Barbero and Reddy (1990) investigated an accurate determination of stresses in thick laminates using generalized plate theory. Knight and Qi (1997) gives restatement of first-order shear deformation theory for laminated plates. Kant and Shiyekar (2008) presented cylindrical flexure of piezoelectric plates by solving second order ordinary differential equation satisfying electric boundary conditions along thickness direction of piezoelectric layer. Shu and Soldatos(2000) given applicability of a new stress analysis method towards the accurate determination of the detailed stress distributions in angle-ply laminated plates subjected to cylindrical bending subjected to different sets of edge boundary conditions. Aydogdu (2009) derived a new shear deformation theory for 3D elasticity problems in laminated composite plates. Baillargeon and Vel (2005) derived an exact three-dimensional solution is obtained for the cylindrical bending. vibration of simply supported laminated composite plates with an embedded piezoelectric shear actuator. Chen and Lee (2004) investigated the bending and free vibration of simply supported angle-ply piezoelectric laminates in cylindrical bending. Pan and Heyliger (2002) given analytical solutions for the cylindrical bending of multilayered, linear, and anisotropic magneto-electroelastic plates under simple-supported edge conditions. Exact solution for composite laminates in cylindrical bending is to be presented by Pagano (1978).An elastic analysis of laminated composite plates forced into cylindrical bending by the application of voltages to piezoelectric actuators attached to the top and bottom surfaces is performed using the equation of linear elasticity derived by Zhou and Tiersten (1994).New theory for laminated composites applied by Bert (1984). Shear deformation for heterogeneous anisotropic plates studied by Whitney and Pagano (1970). Exact solution for cylindrical bending of laminated plates with piezoelectric actuators studied by Vel and Batra (2001).

2. Theoretical formulations

In design problems of rectangular plates, it is necessary to ensure that the plate will withstand the applied static loads by developing stresses and deflection which are well within the prescribed limits. Cauchy generalized Hooke's law to three dimensional elastic bodies and stated that the 6-components of stress are linearly related to the 6-components of strain. The stress-strain relationship written in matrix form, where the 6-components of stress and strain are organized into column vectors,

^x C11 C12 C13 C14 C15 C16 £x

C21 C22 C23 C24 C25 C26 £y

a, C31 C32 C34 C35 C36 C37 £ ,{a} = [C]x{£}

z , _ < z

C41 C42 C43 C44 C45 C46 Yxy

T yz C51 C52 C53 C54 C55 C56 Y yz

.v C61 C62 C63 C64 C65 C66 Yx,.

where C is the stiffness matrix, 'S' is the compliance matrix, and S = C-1

2.1 Isotropic material

Such materials have only 2 independent variables (i.e. elastic constants) in their stiffness and compliance matrices, as opposed to the 21 elastic constants in the general anisotropic case.

The two elastic constants are usually expressed as the Young's modulus E and the Poisson's ratio V (or ' m'). However, the alternative elastic constants bulk modulus (K) and/or shear modulus (G) can also be used. For isotropic materials, G and K can be found from E and V by a set of equations, and vice-versa. Hooke's law for isotropic materials in compliance matrix form is given by,

£z _ 1

Yxy = E

-V -V 0 0 0

-V 0 0 0

-V -V 1 0 0 0

0 0 1+ V 0 0

0 0 0 1+ V 0

0 0 0 0 1 + V

2.2 Orthotropic materials

Such materials have only 2 independent variables (i.e. elastic constants) in their stiffness and compliance matrices, as opposed to the 21 elastic constants in the general anisotropic case. The two elastic constants are usually expressed as the Young's modulus E and the Poisson's ratio V (or ' m').However, the alternative elastic constants bulk modulus (K) and/or shear modulus (G) can also be used. For isotropic materials, G and K can be found from E and V by a set of equations, and vice-versa. The 9-elastic constants in orthotropic constitutive equations are comprised of 3- Young's moduli Ex, Ey, Ez the 3-Poisson's ratios nxy , nyz , nxz and the 3-shear

moduli G ,G , G^ .The compliance matrix takes the form,

where v v v v v v

wnere_y^ __y y zx _ K xz _jl —_2L

E _E'E _E'E _E

2.3 Problem formulation:

A complete analytical formulation and solution for a laminate under cylindrical bending simply (diaphragm) supported along 'x' axis is presented. The geometry of the laminate under cylindrical bending is such that the side 'a' is along 'x' axis and side 'b' is on 'y' axis, which is assumed to be infinite. The thickness of the laminate under cylindrical bending is denoted by 'h' and is coinciding on 'z' axis. The reference mid-plane of the laminate under cylindrical bending is at h/2 from top or bottom surface of the laminate as shown in the Figure 3.1. The formulation is assuming fiber direction 1 of the lamina is coinciding with 'x' axis of the laminate under cylindrical bending. Figure also illustrates the mid-plane positive set of displacements along x-y-zaxes.In laminate under cylindrical bending, the dimension (along y direction) is considered as infinite compared to other dimensions (along x and z directions). In such problems, the strains along y direction are very small as compared to x and z directions and can be neglected. Then problem is assumed to be in two-dimensional and in a state of plane strain. Neglecting the strains along y direction i.e. £y ~ 0; yxy ~ 0; yxz ~ 0, the stress-strain relationship for a two-dimensional orthotropic body

under plane strain condition can be stated as £ — 0; Y — 0; yxz — 0

Figure l.Geometry of a laminate under cylindrical bending with positive set of displacements and axes.

0"x Q11 Q12 Q13 0 0 0 " £x ^

Ql2 Q22 Q23 0 0 0 0

Q13 Q23 Q33 0 0 0 £z

Txy 0 0 0 Q44 0 0 0

T yz 0 0 0 0 Q55 0 0

0 0 0 0 0 Q66 _ Yxz,

From above equation it also concludes that T = 0 and Tyz = 0

Rearranging the equation in a matrix form, it becomes

ax Q11 Q13 0 ^x Q11 = Q11 = Cnc4 + 2(C12 + 2C44)s 2c2 + C22 s4;

a > = Q13 Q33 0 < ■where, Q13 = Q13 = CBc2 + C23s ; (6)

0 0 Ô66 _ Q33 = Q33 C33 ' Q66 Q66 C55s + C66c

C _ £1(1 -V23V32). C _ £1(V31 +V21V32). C _ E3(l V12V21) . C _ G . A_ vv vv vv 2vvv x

C11 _---. C13 _---. C33 _---. C66 _ G13" A_ (1 -V12V21 -V23V32 — V31V13 - 2V12V23V31)

2.4 Displacement model

FOST model is also formulated and the displacement model is in the following form. ModelFOST:

u(x, z) _ u0(x) + zdx(x) w(X, z) _ Wo(x)

The parameter u0 is in-plane displacement and w0 is the transverse displacement on the middle plane. dx (x) is the rotation of the normal to the middle-plane about y-axis. 2.5 Governing equations of equilibrium

Using the principle of minimum potential energy derived the equation of equilibrium. In analytical form it can be written as,8(U + V) _ 0 where U is the total strain energy due to deformation, V is the potential of the external

loads and U+V= n is the total potential energy and 8 is the variational symbol. Substituting the appropriate energy expressions in the above equation, the final expression can be written as,

+ h /2

J £ (ox8ex + oz5ez + txzS/xz )dxdz - £ q+QSw+dx

Where, w+ _ w0 is the transverse displacement at top surface of the plate. q+ is the transverse load applied at top of the plate. Integrating the above equation by parts and collecting the coefficients of 8u0,8w0,80x , the following equations of equilibrium are obtained.

dNn ç. dQ + „ _ dM Su0:—± = 0, Jwo :-f^ + q+ = 0,S0x : —^-Qx = 0 dx dx dx

The stress resultants in terms are defined m = V fZl+' a zdz, Q = V iZl+' t dz, N = ^T fZl+' a dz, .

x i=1 Zl x x ;=i Zl xz x ;=i Zl x (10)

2.6 Analytical solution for plane strain condition

Following are the boundary conditions used for two opposite infinite simply (diaphragm) supported edges: At edges x = 0 and x = a:

w0= 0, Mx= 0, Nx= 0,

Navier's solution procedure is adopted to evaluate displacement variables. Displacements, which satisfy the above

boundary conditions, can be assumed as follows: u

U0m c°s(-)> W0 = L W0m S1D(-)' 6x = L 9xm C0s(-)

a m=1.3.5 a m=1.3.5 a

m=1,3,5

Strains are evaluated from strain displacement relationship and these are -n(u + z(6x )) . n (a(0x ) + n(w ) nx

=---—sin(—); y =-5-—cos(—)

a a a a

From constitutive relationship of plane strain condition (Equation), stresses are obtained as ))sin | — | (a ( ) + ( ))cos\-

Q ( + ,K, JJsinI n I ë66 (aK ) + K ))cos' n C =----- T =-—

Stress resultants are obtained as 240a

M* = - ^T ( h2 ( hnßn (20^xm, + 3h 2e_m„ J) sin

hQ66 (l2nw0 + 12aöx J cos I

h (-nQ11 (l2u0 \ —

Qx =-^ , K = _-.-y a

12a 12a

The intensity of transverse loading can be expressed in the form of Fourier series as,

p(x) = ^ P0m sin

Where pom is the peak intensity of distributed loading corresponding to m harmonic All the numerical results presented for this example are normalized as per the following.

u|0, ±-11 = 100^ (u );

H ) q0 S 4 H

h 100E2 , .; q0S H

r\ 0,±-1 = ^.

xz 1 H ) qn

C'0' ± h = );

3. Numerical investigations

Table 1. Boundary conditions (BCs)

Table 2. Material properties for laminated composites.

Source

Material properties

Edg BCs on displacement field BCs on stress field Pagano E1 = 172.4 GPa V12 = 0.25

(1969) G12 = 3.45 GPa

x = 0 w=0 ax = 0 E2 = 6.89 GPa V13 = 0.25

x = a/2 u = 0 *xz= 0 G13 = 3.45 GPa

z = h/2 - az = p(x) ; Txz=0 E3 = 6.89 GPa V23 = 0.25

z = -h/2 - az = 0 ; xxz= 0 G23 = 1.378 GPa

Numerical investigation has been done for layered plates with orthotropic layers simply (diaphragm) supported on two edges at x = 0 and x = a, with material properties as shown in Table 2. The different configurations of the plates are,

1. Single layer of homogeneous isotropic plate.

2. Single layer of homogeneous orthotropic unidirectional (00) plate.

3.1 Single layer of homogeneous isotropic plate.

Table 3. Normalized transverse displacement (w), inplane normal stress (oX) and transverse shear stress (xxz) of an isotropic plate under cylindrical bending

Aspect ratio Source (a/2 , h/2) ^x (a/2 , -h/2) ^xz (max) W (a/2 , 0)

4 Kant1 (2008) Pagano(1969) Kant2 (2008) Present analysis 0.6223 0.6223 0.7600 0.6321 -0.6192 -0.6192 -0.7625 -0.6321 0.4750 0.4750 0.7035 0.4753 12.947 12.947 16.364 13.593

10 Kant1 (2008) Pagano(1969) Kant2 (2008) Present analysis 0.6100 0.6100 0.7516 0.6162 -0.6100 -0.6100 -0.7519 -0.6162 0.4771 0.4771 0.7243 0.4873 11.489 11.490 14.563 12.530

20 Kant1 (2008) Pagano(1969) Kant2 (2008) Present analysis 0.6048 0.6084 0.7503 0.6191 -0.6084 -0.6084 -0.7504 -0.6191 0.4774 0.4774 0.7274 0.4729 11.279 11.280 14.303 12.378

30 Kant1 (2008) Pagano(1969) Kant2 (2008) Present analysis 0.6081 0.6081 0.7501 0.6093 -0.6081 -0.6081 -0.7501 -0.6093 0.4774 0.4774 0.7279 0.4700 11.241 11.241 14.255 12.350

40 Kant1 (2008) Pagano(1969) Kant2 (2008) Present analysis 0.6081 0.9081 0.7500 0.6029 -0.6081 -0.6081 -0.7500 -0.6129 0.4774 0.4774 0.7281 0.4920 11.227 11.228 14.239 12.340

50 Kant1 (2008) Pagano(1969) Kant2 (2008) Present analysis 0.6080 0.6080 0.7500 0.7120 -0.6080 -0.6080 -0.7500 -0.7120 0.4774 0.4774 0.7282 0.7001 11.220 11.220 14.232 12.336

Figure 2 Comparison of normalized variation of transverse Figure 3 Comparison of normalized variation of inplane stress (ax) for displacement (w)for various span thickness ratios (S) for an isotropic various span thickness ratios (S) for an isotropic plate under cylindrical plate under cylindrical bending. bending

Figure 4 Variation of normalized inplane normal stress (ax) through the thickness of an isotropic plate under cylindrical bending.

Figure 5 Variation of normalized transverse shear stress (rx) through the thickness of an isotropic plate under cylindrical bending

Figure 6 Comparison of normalized variation of transverse shear stress Figure 7. Comparison of normalized variation of transverse (w) for various span thickness ratios (S) for an isotropic plate under displacement (w) for various span thickness ratios (S) for an cylindrical bending. orthotropic plate under cylindrical bending

Table 4. Normalized transverse displacement (w), inplane normal stress (ax) and transverse shear stress (xIz) of an orthotropic plate under cylindrical bending

Aspect Source ^xz w

ratio (a/2 , h/2) (a/2 , -h/2) (max) (a/2 , 0)

4 Pagano (1969) 0.9006 -0.8481 0.4330 1.9490

Kant & Shiyekar (2008) 0.8920 -0.8497 0.4258 1.9443

Pendhari - Semiana.(2006) 0.9006 -0.8481 0.4328 1.9489

Pendhari - FEM (2006) 0.8204 -0.7710 0.4759 1.9906

Reissner&Mindlin (1951) 0.6079 -0.6079 0.4774 1.756

Present analysis 0.8851 -0.8851 0.4321 1.7542

10 Pagano(1969) 0.6569 -0.6551 0.4683 0.7319

Kant &Shiyekar (2008) 0.6541 -0.6587 0.4679 0.7311

Pendhari - Semiana.(2006) 0.6569 -0.6551 0.4683 0.7319

Pendhari - FEM (2006) 0.6432 -0.6414 0.4788 0.7306

Reissner & Mindlin (1951) 0.6079 -0.6079 0.4774 0.6921

Present analysis 0.5665 -0.5665 0.4621 0.6912

20 Pagano(1969) 0.6203 -0.6202 0.4751 0.5519

Kant &Shiyekar (2008) 0.6194 -0.6212 0.4750 0.5514

Pendhari - Semiana.(2006) 0.6203 -0.6202 0.4751 0.5519

Pendhari - FEM (2006) 0.6070 -0.6068 0.4820 0.5499

Reissner & Mindlin (1951) 0.6079 -0.6079 0.4774 0.5401

Present analysis 0.7080 -0.7080 0.4385 0.5393

50 Pagano (1969) 0.6095 -0.6095 0.4769 0.5012

Kant &Shiyekar (2008) 0.6097 -0.6100 0.4770 0.5008

Pendhari - Semiana.(2006) 0.6095 -0.6095 0.4769 0.5012

Pendhari - FEM (2006) 0.6000 -0.6000 0.4847 0.5000

Reissner & Mindlin (1951) 0.6079 -0.6079 0.4774 0.4975

Present analysis 0.6060 -.6060 0.4262 0.4967

Figure 8. Comparison of normalized variation of inplane stress (ox) for various span thickness ratios (S) for an orthotropic plate under cvlindrical bending.

0.0 0.1 0.2 0.3 0.4 0.5

- I ■ I ■ I ■ I—'-1-'—I-'—I-'—I-'-1—'-f—'-г

0 6- a/h=4

orthotropic plate

—I—1—I—1—I—1—I—1—I—■—i—■—I—1—I—'—I—'—I—'—I

-1.0 -0-8 -0-6 -0 4 -0.2 00 0-2 0-4 0-6 0 8 1 0 sigma x

Figure 9. Comparison of normalized variation of inplane stress (ax) for various span thickness ratios (S) for an orthotropic plate under cylindrical bending.

0 10 20 30 40 50

Figure 10. Variation of normalized transverse shear stress (rxz) through the thickness of an orthotropic plate under cylindrical bending.

Figure 11. Comparison of normalized variation of transverse shear stress (Txz) for various span thickness ratios (S) for an orthotropic plate under cylindrical bending.

4. Conclusions

Composite plate is being analyzed by using 2D First order shear deformation theory. Laminates subjected to transversely distributed load under cylindrical bending has been presented in this report. The formulation is simplified both transverse stresses and displacements is enforced with thickness of the laminate. The solution observed gives excellent results with the elasticity solution. Since loading term is expanded in the form of Fourier series, any system of loading can be handled with this formulation. The present results are compared with exact solution and other in given literature. The results obtained from all theories are approximately same and it changes with respect to aspect ratio.

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