A refined shear deformation theory for bending analysis of isotropic and orthotropic plates under various loading conditions

Transcription

1 JOURNAL OF MATERIALS AND ENGINRING STRUCTURES (015) Research Paper A refined shear deformation theor for bending analsis of isotropic and orthotropic plates under various loading conditions Bharti M. Shinde*, Atteshamuddin S. Saad, Shantaram M. Ghumare Department of Civil Engineering, SRES s College of Engineering, Universit of Pune, Kopargaon-43601,Maharashtra, India A R T I C L E I N F O A B S T R A C T Article histor : Received December 014 Accepted 10 Februar 015 Kewords: shear deformation, trigonometric theor, shear correction factor, In this paper, a refined trigonometric shear deformation theor is applied for the bending analsis of isotropic and orthotropic plates under the various loading conditions. The two unknown variables are involved in the present theor. The present theor satisfies the shear stress free condition at top and bottom surface of the plates without using shear correction factors. The governing equations and boundar conditions are obtained b using the principle of virtual work. A closed form solution is obtained using Navier Solution Scheme. A simpl supported isotropic and orthotropic plate subjected to sinusoidall distributed, uniforml distributed and linearl varing loads are considered for the detailed numerical stud. The results obtained using present theor are compared with previousl published results. two variables. 1 Introduction The composite plates are widel used in the various fields of engineering like aerospace, ships, automotive and civil. Therefore, various plate theories have been developed b researchers to predict the correct bending behavior of composite plates. Kirchhoff [1] has developed a classical plate theor (CPT) for thin plate analsis, which is not suitable for the thick plate due to neglect of the shear deformation effect. Therefore Mindlin [] has developed first order shear deformation theor (FSDT) considering the effect of transverse shear deformation for the analsis of plates. But, this theor does not satisf the zero shear stress condition at the top and bottom and require a shear correction factor. Various higher order shear deformation theories have been reported in the literature, which considers the transverse shear deformation effect and satisfies the zero shear stress conditions at the top and bottom surfaces of the plates without shear correction factor. Among these higher order theories, Redd s [3] theor is most commonl used for the analsis of composite plates. Ghugal and Shimpi [4] has presented a review of such displacement and stress based refined theories for isotropic and anisotropic * Corresponding author. Tel.: address: bhartishinde1987@ahoo.co.in e-issn: X, Mouloud Mammeri Universit of Tizi-Ouzou, Algeria

2 4 JOURNAL OF MATERIALS AND ENGINRING STRUCTURES (015) 3 15 plates. Lev [5] was first to developed a refined theor using trigonometric functions in the displacement field in terms of thickness coordinate for the thick isotropic plate. Stein [6] also proposed such theor and applied to isotropic plates in the modified form. But Stein s theor does not satisf the zero shear stress conditions at the top and bottom surfaces of the plate. Touratier [7] has developed a trigonometric shear deformation theor for bending, buckling and vibration analsis of laminated composite and sandwich plates. Shimpi and Ghugal [8] have developed a laer wise trigonometric shear deformation theor for flexural analsis of two laered laminated plates. Shimpi et.al [9] proposed a trigonometric theor for static and free vibration analsis of isotropic, orthotropic and laered composite plates. Ghugal and Saad [10, 11] have developed trigonometric shear deformation theor considering the effects of transverse shear and normal deformations for bending analsis of thick isotropic and orthotropic plates. Mantari et al. [1, 13] also uses the trigonometric function in the displacement field and developed a new higher order shear deformation theor for bending analsis of isotropic, laminated composite and sandwich plates. Recentl, Neves et al. [14-16] have developed a quasi 3D higher order shear deformation theories using a sine and hperbolic sine function for static, free vibration and buckling analsis of isotropic, sandwich and functionall graded plate. Saad [17] has applied an exponential theor for the bidirectional bending analsis of the isotropic plate. This theor is further extended b Saad and Ghugal [18] for the analsis of orthotropic composite plate. Thai and Vo [19] have developed a trigonometric shear deformation theor for the bending analsis of functionall graded plates. Refined plate theor using parabolic function is developed b Shimpi and Patel [0] which involves onl two unknown variables for bending and free vibration analsis of orthotropic plates. In the present paper, a two variable plate theor using trigonometric function in the displacement field is applied for the bending analsis of isotropic and orthotropic plates. The theor is designated as two variable trigonometric shear deformation theor. This theor neglects the need of a shear correction factor. Governing equations and boundar conditions are obtained b using the principle of virtual work. A Navier s double trigonometric series technique is used to obtain the closed form solution. The present results are compared with exact solution given b Pagano [1]. Theoretical Formulation.1 The displacement field A square plate of the sides a and total thickness h as shown in Figure 1 is considered. The plate is made up of linearl elastic orthotropic material. The downward z-direction is taken positive. The plate occupies the region 0 x a, 0 b, -h/ z h/ in Cartesian coordinate sstem. A transverse load q(x, ) is applied on the upper surface of the plate. Figure 1. Orthotropic plate coordinate sstem. The displacements u in x-direction, v in -direction and w in z-direction consists of bending and shear components. (, ) h π z (, ) wb ws u( z,, ) = z z sin x π h x wb ws v( z,, ) = z z sin π h w (, ) = w(, ) + w(, ) b (, ) h π z (, ) s (1) Here u, v and w are displacements in the x, and z directions of a point having coordinates (x. and z) in the plate domain.

3 JOURNAL OF MATERIALS AND ENGINRING STRUCTURES (015) The non-zero normal and shear strain components are obtained using strain displacement relations given b Jones []. u wb h π z ws ε x = = z z sin, x x π h x v wb h π z ws ε = = z z sin, π h u v wb h π z ws γ = + = z z sin x x π h x π z ws γ z = cos h π z ws γ = cos h x (3). Constitutive Relation The constitutive relationships for the orthotropic plate can be given as, x Q11 Q εx Q1 Q ε = 0 0 Q γ Q 0 γ z 44 z Q 55 γ (4) where, Q are the plane stress reduced elastic constants taken from Jones [] ij E E E Q =, Q = µ, Q =, Q = G, Q = G, Q = G µ 1µ 1 1 µ 1µ 1 1 µ 1µ 1 (5).3 Governing equations and boundar conditions The variationall consistent governing equations of equilibrium and boundar conditions associated with the present theor can be derived using the principle of virtual work. The analtical form of principle of virtual work can be written as: a b h/ a b δε + δε + δγ + δγ + δγ dzddx qδ w ddx = 0 (6) x x z z 0 0 h/ 0 0 where δ be the arbitrar variations. Integrating Eq. (6) b parts and collecting the coefficients of δw and δ w to obtain the governing equations of equilibrium and boundar conditions associated with the present theor. The governing equations of equilibrium are as follows: b s wb wb wb ws ws ws D ( D1 + D66 ) + D + Bs ( Bs1 + Bs66 ) + Bs = q 4 x x x x wb wb wb ws ws ws Bs ( Bs1 + Bs66 ) + Bs + Ass ( Ass1 + Ass66 ) + Ass = q 4 x x x x (7) (8) where D ij,bs ij, Ass ij, Acc ij are the stiffness coefficients which are given as:

4 6 JOURNAL OF MATERIALS AND ENGINRING STRUCTURES (015) 3 15 h/ { Dij} = Qij { z } dz; ( i = j = 16,, ) h/ h/ { Bs ij, Assij} = Qij f ( z) { z, f ( z) } dz; ( i = j = 16,, ) h/ (9) where h/ { Accij} = Qij g ( z) dz ( i = j = 45, ) h/ h πz πz f ( z) = z sin and g( z) = cos (10) π h h.4 Navier solution for simpl supported plates The Navier solution scheme is used to obtain closed form solution for the bending analsis of isotropic and orthotropic plates simpl supported on all four edges. The plate is subjected to transverse load q(x, ) at upper surface i.e. z = -h/. The load is presented in double trigonometric series as, ( ) q, = q sinαxsin β (11) m= 1 n= 1 where q mn is the coefficient of Fourier expansion given as below for various static loadings. 0 ( ) q = q m= n= 1 Sinusoidall Distributed Load (SDL) mn 16q0 qmn = mnπ ( m= n= 1, 3, 5,... ) Uniforml Distributed Load (UDL) 8q0 qmn = cos mπ mnπ ( m= n= 1, 3, 5,... ) Linearl Varing Load (LVL) where q 0 is maximum intensit of distributed load at the centre of plate. The following solution form is assumed for unknown displacement variables δ wb and δ ws satisfing the boundar conditions of simpl supported plates exactl. where w = w sinαxsinβ and w = w sinαxsinβ b bmn s smn mn w and w are the unknown functions, α = mπ a and β = nπ b. Substitution this form of solution and bmn smn transverse load q(x, ) into the governing equations (7) - (8) leads to the following matrix form. (1) where elements of stiffness matrix [K] are as follows: K11 K1 wbmn qmn K K = w q 1 smn mn ( ) K = D α + D + D α β + D β, ( ) K = K = Bs α + Bs + Bs α β + Bs β, (13) (14) ( ) K = Ass α + Ass + Ass α β + Ass β + Acc α + Acc β From the solution of Eq. (13), unknown coefficients w and w can be obtained. Having obtained values of these bmn unknown coefficients one can then calculate all the displacement and stress components within the plate. Shear stresses are obtained b using constitutive relations and integrating equations of equilibrium of theor of elasticit. smn

5 JOURNAL OF MATERIALS AND ENGINRING STRUCTURES (015) Numerical Results To prove the efficienc of the present theor, it is applied for the bending analsis of isotropic and orthotropic plates subjected to various static loadings such as a) SDL b) UDL c) LVL. The following material properties are used to obtain the numerical results. Isotropic: E E1 = E = E = 10 GPa, µ 1 = µ 1 = µ = 0. 5, G1 = G13 = G3 = G = 1 ( + µ ) (15) E= 5E, µ = 05.,G = G = 05.E,G = 0.E (16) Orthotropic: The numerical results of displacements and stresses are presented in the following non-dimensional form xh w 3 4 x b h ue h a b w h E a b h u 0,,,,,0,,,, = qa = = qa qa a b h h b h a z,,, 0,0,, 0,,0,,0,0 z qa qa 0 qa h h h = = = = qa (17) 4 Discussion of Results The non-dimensional displacement and stresses obtained using present theor are compared and discussed with those obtained b the classical plate theor (CPT) of Kirchhoff [1], first order shear deformation theor (FSDT) of Mindlin [], higher order shear deformation theor (HSDT) of Redd [3], exponential shear deformation theor (ESDT) of Saad [16, 17], trigonometric shear deformation theor (TSDT) [9] and Exact elasticit solution given b Pagano [1]. 4.1 Bending analsis of simpl supported isotropic plates Comparison of maximum non-dimensional displacements and stresses at critical points for an isotropic square plate subjected to sinusoidal distributed load is shown in Table 1. The plate is made up of isotropic. The numerical results are obtained for aspect ratios (a/h) 4 and 10. The present theor and HSDT give a more accurate value of in-plane displacement than that is given b ESDT, TSDT, FSDT and CPT as compared to exact values. Through thickness distribution of in-plane displacement for aspect ratio 10 is shown in Figure. The values of in-plane normal stresses obtained using present theor and HSDT are excellent agreement with each other. 0.5 Redd [HSDT] u Figure. Thickness distribution of in-plane displacement (u ) for isotropic plate subjected to SDL at a/h = 10.

6 8 JOURNAL OF MATERIALS AND ENGINRING STRUCTURES (015) 3 15 Table 1- Comparison of displacements and stresses for the isotropic square (b = a) plate subjected to sinusoidall distributed load a/h Quantit Exact TSDT HSDT FSDT CPT 4 u w x z z u w x z z The TSDT overestimate the values of normal stresses whereas FSDT and CPT underestimate those as compared to exact values. Through thickness distribution of normal stress is shown in Figure 3.The value of in-plane shear stress obtained b present theor is in excellent agreement with the values of other refined theories. Transverse shear stresses when obtained b constitutive relations using present theor are on higher side, however, use of equilibrium equations ield more accurate results in case of present theor. For aspect ratio 10, present theor predicts exact value of transverse shear stresses. Through thickness distribution via equilibrium equation is plotted in Figure 4. The displacements and stresses of isotropic square plate subjected to uniforml distributed and linearl varing load are as shown in Table and 3 respectivel. The non-dimensional results are obtained for aspect ratio 4 and 10 and compared with the other higher order theories, FSDT, CPT and exact value. From Table and 3 it is observed that displacement and stresses obtained b the present theor are in close agreement with the other theories.

7 JOURNAL OF MATERIALS AND ENGINRING STRUCTURES (015) Redd [HSDT] x Figure 3. Thickness distribution of in-plane normal stress ( ) for isotropic plate subjected to SDL at a/h = 10. Table-Comparison of displacements and stresses for the isotropic square (b = a) plate subjected to uniforml distributed load. a/h Quantit Exact ESDT TSDT HSDT FSDT CPT 4 u w x z z x 10 u w x z z

8 10 JOURNAL OF MATERIALS AND ENGINRING STRUCTURES (015) Redd [HSDT] Figure 4. Thickness distribution of transverse shear stress ( ) for isotropic plate subjected to SDL at a/h = 10. Table 3-Comparison of displacements and stresses for the isotropic square (b = a) plate subjected to linearl distributed load. a/h Quantit Exact ESDT TSDT HSDT FSDT CPT 4 u w x z z u w x z z

9 JOURNAL OF MATERIALS AND ENGINRING STRUCTURES (015) Bending analsis of simpl supported orthotropic plates. The non-dimensional displacements and stresses for the orthotropic square plate under sinusoidall distributed load are listed in Table 4.The plate is made up of orthotropic. The examination of Table 4 reveals that, the present theor slightl overestimates the in-plane displacement and underestimates the transverse displacement. The in-plane normal stress ( x ) predicted b present theor is in good agreement with exact value, but in-plane normal stress ( ) is on the lower side. The values of in-plane shear stress obtained b all the theories theor are in excellent agreement with exact other. The transverse shear stress ( ) transverse shear stress ( z ) predicted b present theor is in excellent agreement with that of exact solution and is identical with those obtained b CPT. Through thickness distributions of in-plane displacement, in-plane normal stresses and transverse shear stress are shown in Figures 5 through 8 respectivel. The nondimensional results obtained of orthotropic plate subjected to uniforml distributed and linearl varing load b the present theor are presented in Table 5 and 6 respectivel. The results obtained for displacement and stresses for the aspect ratio 4 and 10. From Table 5 and 6 it is observed that present theor gives the results of displacement and stresses more or less similar to those obtained using other theories. Table 4-Comparison of displacements and stresses for the orthotropic square (b = a) plate subjected to sinusoidall distributed load. a/h Quantit Exact ESDT HSDT FSDT CPT 4 u w x z z u w x z z

10 1 JOURNAL OF MATERIALS AND ENGINRING STRUCTURES (015) 3 15 Table 5-Comparison of displacements and stresses for the orthotropic square (b = a) plate subjected to uniforml distributed load. a/h Quantit Exact ESDT HSDT FSDT CPT 4 u w x z z u w x z z Redd [HSDT] u Figure 5. Thickness distribution of in-plane displacement (u ) for orthotropic plate subjected to SDL at a/h = 10.

11 JOURNAL OF MATERIALS AND ENGINRING STRUCTURES (015) Table 6-Comparison of displacements and stresses for the orthotropic square (b = a) plate subjected to linearl varing load. a/h Quantit Exact ESDT HSDT FSDT CPT 4 u w x z z u w x z z Redd [HSDT] x Figure 6. Thickness distribution of in-plane normal stress ( ) for orthotropic plate subjected to SDL at a/h = 10. x

12 14 JOURNAL OF MATERIALS AND ENGINRING STRUCTURES (015) Redd [HSDT] Figure 7. Thickness distribution of in-plane normal stress ( ) for orthotropic plate subjected to SDL at a/h = Redd [HSDT] Figure 8. Thickness distribution of transverse shear stress ( ) for orthotropic plate subjected to SDL at a/h = Conclusions In the present stud, a two variable trigonometric shear deformation theor is applied for the bending analsis of isotropic and orthotropic plates. The present theor satisfies the shear stress free conditions at top and bottom surfaces of plate without using shear correction factor. From the numerical results and discussion, it is concluded that present theor is in good agreement while predicting the bending behaviour of isotropic and orthotropic plates. REFERENCES [1]- G.R. Kirchhoff, Uber das gleichgewicht und die bewegung einer elastischen scheibe. J. Reine Angew. Math. 40 (1850) []- R. D. Mindlin, Influence of rotator inertia and shear on flexural motions of isotropic, elastic plates. ASME J. App. Mech. 18 (1951) [3]- J.N. Redd, A simple higher order theor for laminated composite plates, ASME J. App. Mech. 51 (1984) [4]- Y.M. Ghugal, R.P. Shimpi, A review of refined shear deformation theories of isotropic and anisotropic Laminated Plates. J. Reinf. Plast. Compo. 1 (00) [5]- M. Lev, Mémoire sur la théorie des plaques élastique planes. J. Math. Pures Appl. 30 (1877) [6]- M. Stein, D.C. Jegl, Effect of transverse shearing on clindrical bending, vibration and buckling of laminated plates. AIAA J. 5 (1987) 13-19

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