Buckling analysis of thick plates using refined trigonometric shear deformation theory

Transcription

1 JOURNAL OF MATERIALS AND ENGINEERING STRUCTURES 2 (2015) Reearch Paper Buckling analyi of thick plate uing refined trigonometric hear deformation theory Sachin M. Gunjal *, Rajeh B. Hajare, Attehamuddin S. Sayyad, Mana D. Ghodle Department of Civil Engineering, SRES College of Engineering, Savitriai Phule Pune Univerity, Kopargaon , Maharahtra, India A R T I C L E I N F O A B S T R A C T Article hitory : Received 26 march 2015 Accepted 12 jully 2015 Keyword: Shear Deformation Trigonometric function Critical uckling load In thi paper, a refined trigonometric hear deformation plate theory i applied for the uckling analyi of thick iotropic quare and rectangular plate. The theory involve only two unknown, a againt three in firt order hear deformation theory and other higher order theorie. The theory involve inuoidal function in the in-plane diplacement. The tranvere diplacement involve ending and hear component. Governing equation and oundary condition of the theory are otained uing the principle of virtual work. A imply upported iotropic rectangular plate ujected to uniaxial and iaxial compreion i conidered for the detailed numerical tudy. Reult of critical uckling load for imply upported iotropic rectangular plate are compared with thoe of other refined theorie. 1 Introduction The rectangular plate are ued in engineering tructure are often ujected to normal, compreive and hearing load acting in the middle plane of plate. Under certain condition uch load can reult in plate uckling. Buckling intaility of plate i a grate practical importance. In many cae the failure of plate element may e attriuted to uckling intaility and not to the lack of their trength. Therefore plate uckling analyi preent an integral part of general analyi of tructure. The importance of uckling i the initiation of deflection pattern which if the load are further increae aove their critical value rapidly lead to vary large lateral deflection. It lead to large ending tree and eventually to complete failure of plate. The well-known claical plate theory due to Kirchhoff [1] predict good reult for the uckling of thin plate only, ecaue, the tranvere hear deformation i neglected. The tranvere hear deformation effect i more pronounced in thick plate. The diplacement aed firt order hear deformation theory conidering the effect of tranvere hear deformation i developed y Mindlin [2] which required hear correction factor. The limitation of claical plate theory and firt order hear deformation theory led the development of higher order hear deformation theorie. Reddy * Correponding author. Tel.: addre: gunjalachin20@gmail.com e-issn: X, Mouloud Mammeri Univerity of Tizi-Ouzou, Algeria

2 160 JOURNAL OF MATERIALS AND ENGINEERING STRUCTURES 2 (2015) theory [3] i one of the well-known higher order hear deformation theory. Shimpi and Patel [4] have developed two variale plate theory for the tatic and dynamic analyi of thick plate. Kim et al. [5] extended thi theory for the uckling analyi of iotropic and orthotropic plate uing Navier olution technique wherea Thai and Kim [6] ued Levy type olution for the uckling analyi of thick plate uing two variale plate theory. Ghugal and Pawar [7] applied hyperolic hear deformation theory for the uckling and viration analyi of orthotropic plate. Sayyad [8] and Sayyad and Ghugal [9] applied an exponential hear deformation theory for uckling analyi of thick iotropic and orthotropic plate. Recently Sayyad and Ghugal [10] i developed trigonometric hear and normal deformation theory for the uniaxial and iaxial uckling analyi of iotropic and laminated compoite plate. Sayyad et al. [11] applied nth order hear deformation theory for the cylindrical ending of orthotropic plate. A trigonometric hear deformation theory ued in the preent tudy wa developed y Thai and Vo [12] which extended y Shinde et al. [13] for the ending analyi of iotropic and orthotropic plate. The theory involve only two unknown variale which are three in cae of firt order hear deformation theory and other higher order hear deformation theorie cited in the literature. The theory i variationally conitent and doe not require hear correction factor. The governing equation and oundary condition are otained uing the principle of virtual work. The analytical olution for imply upported oundary condition i otained uing Navier olution technique. The numerical reult of critical uckling load are compared with exiting literature. 2 Mathematical formulation of the preent theory Conider a rectangular plate made up of iotropic material having length a in x-direction, width in y-direction, and thickne h in z-direction. The z-direction i aumed poitive in downward direction. The plate occupie a region 0 x a, 0 y, -h/2 x h/2 in Carteian coordinate ytem. The goal of uckling analyi of plate i to determine the critical uckling load. Following aumption are made in the preent analyi. 1. Prior to loading, a plate i ideally flat and all the applied external load act trictly in the middle plane of the plate. 2. State of tre i decried y the equation of linear plane elaticity. Any change in the plate dimenion are neglected prior to uckling. 3. All the load applied to plate are dead load, they are not change either in magnitude or in direction when the plate deform. 4. The plate ending i decried y the refined trigonometric hear deformation theory. The diplacement field of the preent theory can e written a: w h π z w uxyz (,, ) = z z x π h x w h π z w vxyz (,, ) = z z y π h y wxy (, ) = w( xy, ) + w( xy, ) (1) where u, v and w denote the diplacement in x, y and z-direction repectively. The w and w are the ending and hear component of the tranvere diplacement. The trigonometric function i aigned according to the hear tre ditriution through the thickne of the plate. The train aociated with the preent theory are otained uing traindiplacement relationhip from theory of elaticity. u w h π z w ε x = = z z x x π h x v w h π z w ε y = = z z y y π h y u v w h π z w γ xy = + = 2z 2 z y x x y π h x y u w πz w v w πz w γ xz = + = co, γ xz = + = co z x h x z y h y (2)

3 JOURNAL OF MATERIALS AND ENGINEERING STRUCTURES 2 (2015) The train correponding to train are a follow: E E w w w w h π z w w σ x = 2 ( ε x + µε y) = 2 z + µ z + µ + + µ 1 µ 1 µ x y x y π h x y E E w w w w h π z w w σ y = 2 ( µε x + ε y) = z µ z µ in µ , 1 µ 1 µ x y x y π h x y 2 E E w w h π z w τxy = γ xy = z + + in, 21 ( + µ ) ( 1+ µ ) x y x y π h x y E E πz w E E πz w τxz = γ xz = co, τ yz = γ yz = co h x h y ( µ ) ( µ ) ( µ ) ( µ ) where E, G and µ are Young modulu, hear modulu and Poion ratio repectively. The principle of virtual work i ued to otain the governing equation and oundary condition aociate with the preent theory. The analytical verion of principle of virtual work i: ( σε x x + σε y y + τ γ xy xy + τ γ yz yz + τ γ xz xz ) dv + N xx + 2N δ 0 v + N δ yy w da = 2 (4) dv A 2 w w w x xy y (3) where δ e the variational operator. Integration Eq. (4) y part and etting coefficient of following governing equation (Euler-Lagrange equation) of equilirium are otained. δw and δw zero, the Eh w w w Eh 24 w w w x x y y 12 1 π x x y y ( µ ) ( µ ) w w w w w w + Nxx + 2N Nyy N 2 ++ xx + N + Nyy 2 = x xy y x xy y Eh 24 w w w Eh 48 6 w w w π x x y y 12 1 π π x x y y ( µ ) ( µ ) Eh w w w w w w ( + µ ) x y x xy y x Nxx N Nyy 2 Nxx N w + N yy = 0 2 xy w y (5) (6) 3 Staility analyi of imply upported plate uing Navier olution Buckling analyi of imply upported rectangular plate i otained uing Navier olution technique. The plate i ujected to in-plane compreive force which are uniformly ditriuted along the edge. The plate ujected to uniaxial and iaxial compreion i hown in Fig. 1. All other load acting on plate are aumed to e zero. Fig. 1- A imply upported rectangular plate ujected to in-plane compreive force: (i) uniaxial compreion along x- direction (ii) uniaxial compreion along y-direction (iii) iaxial compreion

4 162 JOURNAL OF MATERIALS AND ENGINEERING STRUCTURES 2 (2015) The diplacement variale ( ) trigonometric form: w,w which atify the aove oundary condition can e expreed in the doule (, ) in m π x mn in n π y w xy w and w ( xy, ) wmn in m π x in n π y = = a a In-plane compreive force are aumed a follow, (7) N = kn, N = kn and N = 0 (8) xx 1 0 yy 2 0 xy Sutitution of equation (7) into the governing equation (5) - (6) lead to the following equation K11 K12 N11 N12 wmn 0 N0 K21 K = 22 N21 N 22 wmn 0 where element of matrix [K ij ] and matrix [N ij ] are a follow: (9) K K Eh m π m n π n π = a a 2 ( µ ) Eh 24 m π m n π n π = K = π a a 2 ( µ ) K Eh 48 6 m π m n π n π = π π a a 2 ( µ ) (10) 4 Numerical reult and dicuion N = N = N Eh m π n π ( µ ) + a m π n π = N = k + k a A imply upported iotropic quare/rectangular plate i conidered for the detail numerical tudy. The critical uckling load i otained for uniaxial compreion along x-direction, uniaxial compreion along y-direction and iaxial compreion. The plate ha following material propertie. E = 210 GPa, µ = 0. 3 and G = E 21 ( + µ ) (11) The following non-dimenional form i ued to preent critical uckling load. N cr 2 an0 3 = (12) Eh The non-dimenional diplacement and tree otained uing preent theory are compared and dicued with thoe otained y the claical plate theory (CPT) of Kirchhoff [1], firt order hear deformation theory (FSDT) of Mindlin [2], higher order hear deformation theory (HSDT) of Reddy [3] and trigonometric hear and normal deformation theory (TSDT) of Sayyad and Ghugal [10]. The non-dimenional critical uckling load in cae of uniaxial compreion along x- direction i compared in Tale 1, in cae of uniaxial compreion along y-direction i compared in Tale 2 and in cae of iaxial compreion i compared in Tale 3. The numerical reult are otained for rectangular plate (/a = 1.0, 1.5, 2.0, 2.5, 3.0, 3.5 and 4.0) with variou apect ratio (a/h = 5, 10, 20, 50 and 100). Variation of non-dimenional critical uckling load with repect to a/h i hown in Fig. 2 and 3 wherea with repect to /a i hown in Fig. 4 through 6. When plate i ujected to uniaxial compreion along x-direction, critical uckling load i decreaed with increae in /a ratio wherea increaed with increae in a/h ratio. When plate i ujected to uniaxial compreion along y-direction, critical

5 JOURNAL OF MATERIALS AND ENGINEERING STRUCTURES 2 (2015) uckling load i increaed with increae in /a ratio and a/h ratio. It i pointed out from Tale 3 that, when plate i ujected to iaxial compreion, critical uckling load i exactly half of uniaxial compreion for quare plate (/a = 1.0). Tale 1- Comparion of non-dimenional critical uckling load for imply upported iotropic rectangular plate under uniaxial compreion along x direction. /a (k 1 k 2 ) a/h Theory (1, 0) 5 Preent Sayyad and Ghugal [10] Reddy [3] Mindlin [2] Kirchhoff [1] Preent Sayyad and Ghugal [10] Reddy [3] Mindlin [2] Kirchhoff [1] Preent Sayyad and Ghugal [10] Reddy [3] Mindlin [2] Kirchhoff [1] Preent Sayyad and Ghugal [10] Reddy [3] Mindlin [2] Kirchhoff [1] Preent Sayyad and Ghugal [10] Reddy [3] Mindlin [2] Kirchhoff [1] Fig. 2- Variation of non-dimenional critical uckling load with repect to apect ratio (a/h) when quare plate (/a = 1) i ujected to uniaxial compreion along x or y direction

6 164 JOURNAL OF MATERIALS AND ENGINEERING STRUCTURES 2 (2015) Fig. 3- Variation of non-dimenional critical uckling load with repect to apect ratio (a/h) when quare plate (/a = 1) i ujected to iaxial compreion Tale 2- Comparion of non-dimenional critical uckling load for imply upported iotropic rectangular plate under uniaxial compreion along y direction. /a (k 1 k 2 ) a/h Theory (0, 1) 5 Preent Sayyad and Ghugal [10] Reddy [3] Mindlin [2] Preent Sayyad and Ghugal [10] Reddy [3] Mindlin [2] Preent Sayyad and Ghugal [10] Reddy [3] Mindlin [2] Preent Sayyad and Ghugal [10] Reddy [3] Mindlin [2] Preent Sayyad and Ghugal [10] Reddy [3] Mindlin [2]

7 JOURNAL OF MATERIALS AND ENGINEERING STRUCTURES 2 (2015) Tale 3- Comparion of non-dimenional critical uckling load for imply upported iotropic rectangular plate under iaxial compreion. /a (k 1 k 2 ) a/h Theory (1, 1) 5 Preent Sayyad and Ghugal [10] Reddy [3] Mindlin [2] Kirchhoff [1] Preent Sayyad and Ghugal [10] Reddy [3] Mindlin [2] Kirchhoff [1] Preent Sayyad and Ghugal [10] Reddy [3] Mindlin [2] Kirchhoff [1] Preent Sayyad and Ghugal [10] Reddy [3] Mindlin [2] Kirchhoff [1] Preent Sayyad and Ghugal [10] Reddy [3] Mindlin [2] Kirchhoff [1] Fig. 4- Variation of non-dimenional critical uckling load with repect to /a ratio when plate i ujected to uniaxial compreion along x-direction for a/h = 5.

8 166 JOURNAL OF MATERIALS AND ENGINEERING STRUCTURES 2 (2015) Fig. 5- Variation of non-dimenional critical uckling load with repect to /a ratio when plate i ujected to uniaxial compreion along y-direction for a/h = 5. Fig. 6- Variation of non-dimenional critical uckling load with repect to /a ratio when plate i ujected to iaxial compreion for a/h = 5. 5 Concluion In thi paper, a refined trigonometric hear deformation theory i developed for the uckling analyi of iotropic rectangular plate. The theory involve two unknown againt three in cae of firt order hear deformation theory. The theory atifie the hear tre free condition at top and ottom urface of the plate without uing hear correction factor. The critical uckling load i otained for imply upported iotropic rectangular plate ujected in-plane compreive force. It i concluded from the tudy that, the preent theory i in excellent agreement while predicting the uckling ehavior of rectangular plate. The critical uckling load for quare plate when plate i ujected to iaxial compreion i exactly half when compared to uniaxial compreion. It i alo oerved that the critical uckling load for quare plate i ame in cae of uniaxial compreion.

9 JOURNAL OF MATERIALS AND ENGINEERING STRUCTURES 2 (2015) REFERENCES [1]- G.R. Kirchhoff, Uer da gleichgewicht und die ewegung einer elatichen cheie, J. Reine Angew. Math. 40 (1850) [2]- R.D. Mindlin, Influence of rotatory inertia and hear on flexural motion of iotropic, elatic plate, ASME J. App. Mech. 18 (1951) [3]- J.N. Reddy, A imple higher order theory for laminated compoite plate, ASME J. Appl. Mech. 51 (1984) [4]- R.P. Shimpi, H.G. Patel, A two variale refined plate theory for orthotropic plate analyi, Int. J. Solid Struct. 43(22) (2006) [5]- S.E. Kim, H.T. Thai, J. Lee, Buckling analyi of plate uing the two variale refined plate theory, Thin Walled Struct. 47 (2009) [6]- H.T. Thai, S.E. Kim, Levy-type olution for uckling analyi of orthotropic plate aed on two variale refined plate theory, Compoite Struct. 93 (2011) [7]- Y.M. Ghugal, M.D. Pawar, Buckling and viration of plate y hyperolic hear deformation theory, J. Aero. Eng. Tech. 1(1) (2011) [8]- A.S. Sayyad, Flexure of thick orthotropic plate y exponential hear deformation theory, Lat. Am. J. Solid Struct. 10(2013) [9]- A.S. Sayyad, Y.M. Ghugal, Bending and free viration analyi of thick iotropic plate y uing exponential hear deformation theory. Appl. Comput. Mech. 6 (2012) [10]- A.S. Sayyad, Y.M. Ghugal, On the uckling of iotropic, tranverely iotropic and laminated compoite rectangular plate. Int. J. Struct. Sta. Dyn. 14(7) (2014) 1-32 [11]- A.S. Sayyad, S.M. Ghumare, S.T. Saane, Cylindrical ending of orthotropic plate trip aed on nth-order plate theory, J. Mater. Eng. Struct. 1(2) (2014) [12]- H.T. Thai, T.P. Vo, A new inuoidal hear deformation theory for ending, uckling and viration of functionally graded plate, Appl. Math. Model. 37(5) (2013) [13]- B.M. Shinde, A.S. Sayyad, S.M. Ghumare, A refined hear deformation theory for ending analyi of iotropic and orthotropic plate under variou loading condition, J. Mater. Eng. Struct. 2(1) (2015) 3-15