VIBRATION AND BUCKLING OF COMPOSITE TWISTED PANELS SUBJECTED TO HYGROTHERMAL LOADING

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1 VIBRATION AND BUCKLING OF COMPOSITE TWISTED PANELS SUBJECTED TO HYGROTHERMAL LOADING A thess submtted n partal fulfllment of the requrements for the degree of MASTER OF TECHNOLOGY n CIVIL ENGINEERING (STRUCTURAL ENGINEERING) BY ASHISH SINGH ROLL NO-11CE31 DEPARTMENT OF CIVIL NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA ROURKELA-7698, ODISHA, MAY 13

2 VIBRATION AND BUCKLING OF COMPOSITE TWISTED PANELS SUBJECTED TO HYGROTHERMAL LOADING A thess submtted n partal fulfllment of the requrements for the degree of Master of technology n CIVIL ENGINEERING (STRUCTURAL ENGINEERING) BY ASHISH SINGH ROLL NO-11CE31 UNDER THE GUIDANCE OF Prof. Shshr Kr. Sahu DEPARTMENT OF CIVIL NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA ROURKELA-7698, ODISHA, MAY 13

3 DEPARTMENT OF CIVIL ENGINEERING NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA, ODISHA-7698 CERTIFICATE Ths s to certfy that the thess enttled, VIBRATION AND BUCKLING OF COMPOSITE TWISTED PANELS SUBJECTED TO HYGROTHERMAL LOADING submtted by ASHISH SINGH bearng Roll no. 11CE31 n partal fulfllment of the requrements for the award of Master of Technology degree n Cvl Engneerng wth specalzaton n Structural Engneerng durng sesson at the Natonal Insttute of Technology, Rourkela s an authentc work carred out by her under my supervson and gudance. To the best of my knowledge, the matter emboded n the thess has not been submtted to any other Unversty / Insttute for the award of any Degree or Dploma. Place: Rourkela Date: 31/5/13 Prof. Shshr Kr. Sahu Department of cvl Engneerng Natonal Insttute of technology Rourkela, Odsha-7698

4 ACKNOWLEDGEMENT I would lke to express my sncere grattude to my gude, Prof. Shshr Kr. Sahu, Department of Cvl Engneerng, Natonal Insttute of Technology, Rourkela for kndly provdng me an opportunty to work under hs supervson and gudance. Hs encouragement, advce, help, montorng of the work, nputs and research support throughout my studes are emboded n ths dssertaton. Hs ablty to teach, depth of knowledge and ablty to acheve perfecton wll always be my nspraton. I also express my sncere thanks to Head of the Cvl Engneerng Department Prof. N. Roy, Natonal Insttute of Technology Rourkela, for provdng valuable departmental facltes for my work. I am very thankful to all the faculty members and staffs of Cvl engneerng department who asssted me n my research, as well as n my post graduate studes. I would lke to gve thanks to all my batch mates, who have drectly or ndrectly helped me n my project work and completon of ths report. Fnally I would lke to thank my famly members for ther support durng ths perod. Ther encouragement and never endng kndness made everythng easer to acheve. Ashsh Sngh Roll no- 11ce31

5 CONTENTS Acknowledgements Contents Abstract Nomenclature Lst of tables Lst of fgures v v x x Chapter-1 INTRODUCTION Introducton 1 1. Importance of the present study 1 Chapter- REVIEW OF LITERATURE Introducton 3. Revews on twsted panels 3..1 Vbraton and bucklng of twsted panels subjected to hygrothermal Load 3.3 Revews on plates Vbraton of composte panels subjected to hygrothermal load 8.3. Bucklng of composte panels subjected to hygrothermal load 1

6 .4 Am and scope of the present studes 1 Chapter-3 THEORY AND FORMULATION The Basc Problem The proposed Analyss Assumptons of the analyss Governng Equatons Governng dfferental equatons Fnte Element Formulaton The shell element Consttutve Relatons Stran Dsplacement relatons The stffness matrx Geometrc stffness matrx [K r ] Ge Geometrc stffness matrx [K a Ge ] Element mass matrx Soluton process Computer program 31 Chapter-4 RESULTS AND DISCUSSION Introducton 3 4. Boundary condtons 3

7 4.3 Vbraton and bucklng of twsted panels Convergence study Comparson wth prevous studes Vbraton of composte plates and shells subjected to hygrothermal envronment Bucklng of composte plates and shells subjected to hygrothermal envronment Numercal Results Vbraton Results for plate Effect of angle of twsts on vbraton of pre twsted cantlever plates Effect of change n aspect rato on vbraton of pre twsted cantlever plates Bucklng results for plates Effect of angle of twsts on bucklng of pre twsted cantlever plates Effect of change n aspect rato on bucklng of pre twsted cantlever plates 59 Chapter-5 CONCLUSIONS Scope for future work 68 REFERENCES 69 v

8 ABSTRACT The twsted composte cantlever have sgnfcant applcatons n aeronautcal and aerospace ndustry cvl, naval and other hgh-performance engneerng applcatons due to ther lght weght, hgh-specfc strength and stffness, excellent thermal characterstcs, ease n fabrcaton and other mportant specaltes These structural members are often exposed to varous servce loads durng ther entre servce lfe. The presence of temperature and mosture concentraton n the envronment may greatly reduce the stffness and strength of the structures and may affect some desgn parameter such as vbraton and stablty characterstcs of the structures. To avod these typcal problems caused by vbratons and stablty, t s mportant to fnd out natural frequency, statc stablty of the composte lamnated twsted cantlever panels under hygrothermal condtons. Therefore the vbraton and stablty behavor of lamnated composte twsted panels subjected to hygrothermal condtons are studed n the present nvestgaton. A smple lamnated model s developed for the vbraton and stablty analyss of lamnated composte pre-twsted cantlever panel subjected to hygrothermal condtons. A computer program based on FEM n MATLAB envronment s developed to perform all necessary computatons. Here an eght noded soparametrc quadratc shell element wth fve degrees of freedom per node s used based on FSDT theory wth hygrothermal loadng. Element elastc stffness matrces, mass matrces, geometrc stffness matrx due to hygrothermal loads and load vectors are derved usng the prncple of mnmum potental energy. The nfluences of varous parameters such as angle of twst, aspect rato, ply-orentaton, geometry and number of layers of lamnate are studed on the vbraton and bucklng characterstcs of lamnated pre twsted cantlever panels for dfferent temperatures and mosture concentratons. Numercal results are presented to show the effects of pre-twst angles, geometry and lamnaton detals on the vbraton and bucklng characterstcs of twsted plates. Ths can be used to the advantage of talorng durng desgn of composte twsted structures. Keywords: Composte twsted panel, hygrothermal loadng, stablty v

9 NOMENCLATURE The prncpal symbols used n ths thess are presented for easy reference Englsh a, b Dmensons of the twsted panel a/b Aj, Bj, Dj and Sj b/h [B] Aspect rato of the twsted panel Extensonal, bendng-stretchng couplng, Bendng and transverse shear stffnesses Wdth to thckness rato of the twsted panel Stran dsplacement matrx for the element C, C elevated and reference mosture concentratons [D] dx, dy dv Stress-stran Element length n x and y-drecton Volume of the element E 1, E Young s modul of a lamna along and across the G1, G 13, 3 fbers, respectvely G Shear modul of a lamna wth respect to 1, and 3 [ K e ] [ K ] r G axes The elastc stffness matrx The ntal stress stffness matrx [ K ] The geometrc stffness matrx due to appled n- a Ge k x, k y, k xy plane loads Curvatures of the plate ( M x, M y and M xy ) Internal moment resultants per unt length v

10 M, M, M Intal nternal moment resultants per unt length N x x N y y xy N xy M, M, M Non-mechancal moment resultants per unt length [N] N Nx, Ny, Nxy due to mosture and temperature The shape functon matrx Shape functon at a node In-plane nternal force resultants per unt length N, N, N In-plane ntal nternal force resultants per unt x N x y N y xy N xy length N, N, N In-plane non-mechancal force resultants per unt Nxx length due to mosture and temperature Crtcal bucklng load n x-drecton { P e } The element load vector due to external transverse statc load N { P e} The element load vector due to hygrothermal forces Q x, Q y Q x, Q y and moments Transverse shears resultants. Intal transverse shear resultants t Thckness of the plate T, T Elevated and reference temperatures u, v Dsplacements of the md-plane along x and y axes, U V, W w respectvely, Dsplacements of node along x, y and z axes, respectvely Dsplacement along z axs x, y, z System of co-ordnate axes Z k, Z k 1 Bottom and top dstance of lamna from md-plane v

11 Greek α α 1, α Shear correcton factor Thermal coeffcents along 1 and axes of a lamna, respectvely β1, β Mosture coeffcents along 1 and axes of a ε x, ε y, γ xy ε x N θ, ε y N, γ xy N lamna, respectvely In-plane strans of the md-plane. Non-mechancal strans due to mosture and temperature Fber orentaton n a lamna x, y Rotatons of the plate about x and y axes ϑ 1, ϑ 1 Posson s ratos x, y Partal dervatves wth respect to x and y ρ Mass densty (ρ) k Mass densty of kth layer from md-plane η, ξ Local natural co-ordnates of an element, Local natural co-ordnates of the element at th node x, y Shear rotatons n x-z and y-z planes, respectvely ω n Rx, Ry Rxy ϕ Natural frequency The rad of curvatures n the x and y drecton Radus of twst of the twsted plate Angle of twst of the twsted panel Mathematcal Operators v

12 [ ] -1 Inverse of the matrx [ ] T Transpose of the matrx LIST OF TABLES Table 1: Convergence of non-dmensonal free vbraton frequences for cantlever twsted plate for dfferent ply orentatons at 35K temperature 47 Table : Convergence of non-dmensonal free vbraton frequences for Table 3: Table 4: Table 5: Cantlever twsted plate for dfferent ply orentatons at.1% mosture concentraton 48 Convergence of non-dmensonal crtcal loads for cantlever twsted plate for dfferent ply orentatons at 35K temperature. 49 Convergence of non-dmensonal crtcal loads for cantlever twsted plate for dfferent ply orentatons at.1% mosture concentraton 49 Comparson of non-dmensonal free vbraton frequences for cantlever twsted plates for dfferent ply orentatons (a) Angle of twst ( ) =15 5 (b) Angle of twst ( ) =45 5 Table 6: Comparson of non-dmensonal bucklng load for cantlever twsted plates for dfferent ply orentatons and angle of twst 53 x

13 LIST OF FIGURES Fg 1: Geometry of an N-layered lamnate (a): Composte twsted panel 7 (b): The lamnaton 8 Fg : Force and moment resultants of the twsted panel 3 Fg 3: Eght nodded soparametrc element 31 Fg 4: Lamnated Shell Element 36 Fg 5: Effect of temperature on non-dmensonal frequency for Φ = 3 and for angle ply lamnated pre twsted cantlever plates (a /b=1, b/t=) 54 Fg 6: Effect of mosture on non-dmensonal frequency for Φ =3 and for angle-ply lamnated pre twsted cantlever plates (a /b=1, b/t=) 55 Fg 7: Effect of temperature on non-dmensonal frequency for Φ =3 and for cross ply lamnated pre twsted cantlever plates (a /b=1, b/t=) 56 Fg 8: Effect of mosture on non-dmensonal frequency for Φ =3 and for cross ply lamnated pre twsted cantlever plates (a /b=1, b/t=) 56 Fg 9: Effect of temperature on non-dmensonal frequency for lamnated Angle ply pre twsted cantlever plate (3/-3/3/-3)s for dfferent angles of twst (a/b=1,b/t=) 57 Fg 1: Effect of mosture on non-dmensonal frequency for lamnated angle ply pre twsted cantlever plate (3/-3/3/-3)s for dfferent angle of twst (a/b=1,b/t=) 58 Fg 11: Effect of temperature on non-dmensonal frequency for lamnated cross ply pre twsted cantlever plate (/9/9/)s for dfferent angle of twst (a/b=1, b/t=) 58 Fg 1: Effect of mosture on non-dmensonal frequency for lamnated cross ply pretwsted cantlever plate (/9/9/)s for dfferent angles of twst (a/b=1, b/t=) 59 Fg 13: Effect of temperature on non-dmensonal frequency for lamnated cross-ply pretwsted cantlever plate (/9//9)s for dfferent angle x

14 of twst (a/b=1, b/t=) 6 Fg 14: Effect of mosture on non-dmensonal frequency for lamnated cross ply pretwsted cantlever plate (/9//9)s for dfferent angles of twst (a/b=1, b/t=) 6 Fg 15: Effect of temperature on non-dmensonal frequency for ϕ =3⁰ And for angle-ply. lamnated pre twsted cantlever plates (a /b=3, b/t=) 61 Fg 16: Effect of mosture on non-dmensonal frequency for ϕ =3⁰ and for angle-ply lamnated pre twsted cantlever plates (a /b=3, b/t=) 6 Fg 17: Effect of temperature on non-dmensonal frequency for ϕ =15⁰ and for cross-ply lamnated pretwsted cantlever plates (a/b=3, b/t=) 6 Fg 18: Effect of mosture on non-dmensonal frequency for Φ =15⁰ and for cross ply lamnated pre twsted cantlever plates (a /b=3, b/t=) 63 Fg 19: Effect of temperature on non-dmensonal frequency for lamnated angle ply pre twsted cantlever plate (3/-3/3/-3)s for dfferent angles of twst (a/b=3, b/t=) 64 Fg : Effect of mosture on non-dmensonal frequency for lamnated angle ply pre twsted cantlever plate (3/-3/3/-3)s for dfferent angle of twst (a/b=3, b/t=) 64 Fg 1: Effect of temperature on non-dmensonal frequency for lamnated cross ply pre twsted cantlever plate (/9//9)s for dfferent angle of twst (a/b=3, b/t=) 65 Fg : Effect of temperature on non-dmensonal frequency for lamnated cross ply pre twsted cantlever plate (/9//9)s for dfferent angle of twst (a/b=3, b/t=) 65 Fg 3: Effect of temperature on non-dmensonal frequency for lamnated cross ply pre twsted cantlever plate (/9/9/)s for dfferent angle of twst (a/b=3, b/t=) 66 x

15 Fg 4: Effect of mosture on non-dmensonal frequency for lamnated Cross ply pre twsted cantlever plate (/9/9/)s for dfferent angles of twst (a/b=3, b/t=) 66 Fg 5: Effect of temperature on non-dmensonal crtcal load for ϕ =3⁰ And for angle-ply lamnated pre twsted cantlever plates (a/b=1, b/t=) 67 Fg 6: Effect of mosture on non-dmensonal crtcal load for ϕ =3⁰ and for angle-ply lamnated pre twsted cantlever plates (a/b=1, b/t=) 68 Fg7: Effect of temperature on non-dmensonal crtcal load for ϕ =3⁰ and for cross-ply lamnated pre twsted cantlever plates (a /b=1, b/t=) 69 Fg 8: Effect of mosture on non-dmensonal crtcal load for ϕ=3⁰ and for cross-ply lamnated pre twsted cantlever plates (a /b=1, b/t=) 69 Fg 9: Effect of temperature on non-dmensonal bucklng load for lamnated angle ply pre twsted cantlever plate (3/-3/3/-3)s for dfferent angles of twst (a/b=1, b/t=) 7 Fg 3: Effect of mosture on non-dmensonal bucklng load for lamnated angle ply pre twsted cantlever plate (3/-3/3/-3)s for dfferent angles of twst (a/b=1, b/t=) 7 Fg 31: Effect of temperature on non-dmensonal bucklng load for Lamnated cross ply pre twsted cantlever plate (/9/9/)s for Dfferent angles of twst (a/b=1, b/t=) 71 Fg 3: Effect of mosture on non-dmensonal bucklng load for lamnated cross ply pre twsted cantlever plate (/9/9/)s for dfferent angles of twst (a/b=1, b/t=) 71 Fg 33: Effect of temperature on non-dmensonal bucklng load for lamnated cross ply pre twsted cantlever plate (/9//9)s for dfferent angles of twst (a/b=1, b/t=) 7 Fg 34: Effect of mosture on non-dmensonal bucklng load for lamnated cross ply pre twsted cantlever plate (/9//9)s for dfferent x

16 angles of twst (a/b=1, b/t=) 73 Fg 35: Effect of temperature on non-dmensonal crtcal load for ϕ =3⁰ And for angle-ply lamnated pre twsted cantlever plates (a /b=3, b/t=) 74 Fg 36: Effect of mosture on non-dmensonal crtcal load for ϕ =3⁰ and for angle-ply lamnated pre twsted cantlever plates (a /b=3, b/t=) 74 Fg 37: Effect of temperature on non-dmensonal crtcal load for ϕ =3⁰ And for cross-ply lamnated pre twsted cantlever plates (a /b=3, b/t=) 74 Fg 38: Effect of mosture on non-dmensonal crtcal load for ϕ=3⁰ and for cross-ply lamnated pre twsted cantlever plates (a /b=3, b/t=) 74 Fg 39: Effect of temperature on non-dmensonal bucklng load for lamnated angle ply pretwsted cantlever plate (3/-3/3/-3)s for dfferent angles of twst (a/b=3, b/t=) 77 Fg 4: Effect of mosture on non-dmensonal bucklng load for lamnated angle ply pre twsted cantlever plate (3/-3/3/-3)s for dfferent angles of twst (a/b=3, b/t=) 77 Fg 41: Effect of temperature on non-dmensonal bucklng load for lamnated cross ply pre twsted cantlever plate (/9/9/)s for dfferent angles of twst (a/b=3, b/t=) 78 Fg 4: Effect of mosture on non-dmensonal bucklng load for lamnated cross ply pre twsted cantlever plate (/9/9/)s for dfferent angles of twst (a/b=3, b/t=) 78 Fg 43: Effect of temperature on non-dmensonal bucklng load for lamnated cross ply twsted cantlever plate (/9//9)s for dfferent angles of twst (a/b=3, b/t=) 79 Fg 44: Effect of mosture on non-dmensonal bucklng load for lamnated cross ply pre twsted cantlever plate (/9//9)s for dfferent angles of twst (a/b=1, b/t=) x

17 CHAPTER 1 INTRODUCTION Composte materals are beng ncreasngly used n aeronautcal and aerospace ndustry, cvl, naval and other engneerng applcatons due to ther lght weght, hgh-specfc strength and stffness, excellent thermal characterstcs easy n fabrcaton and other mportant specaltes. Structures used n the above felds are more often exposed to hgh temperature as well as mosture. The varyng envronmental condtons due to mosture absorpton and temperature seem to have an adverse effect on the stffness and strength of the structural compostes. Ths wde range of practcal applcatons demands a fundamental understandng of ther vbratons and statc characterstcs of lamnated composte shell n dfferent temperature and mosture concentraton. 1.1 Importance of Present Study In a weght senstve applcaton such as aerospace ndustres are ntensvely nvolved n the development of advanced turbomachnery composte materals because of ther excellent propertes. Compostes are usually subjected to changng envronmental condtons durng ther servce lfe. Among dfferent envronmental condtons structures are more often exposed to hgh temperature and mosture. The effect of temperature s known as thermal effect and the effect of mosture absorpton from the atmosphere s known as hygroscopc effect. The combned effects of temperature and mosture are known as hygrothermal effect. The varyng envronmental condton due to mosture absorpton and temperature seem to have an adverse effect on stffness and strength of the twsted compostes. Heat gets conducted nto the lamnate when subjected to rse n the temperature. The lamnate consumes mosture when subjected to the most condtons. The swellng or expanson s more across the fbres of the lamna. Hygrothermal effects nduce a dmensonal change n the lamna. But the propertes of the consttuents of the lamnate s contrast, ts free movement s nhbted. Thus, deformatons and correspondng stress condtons are nduced. The nduced hygrothermal stresses s referred as resdual stresses. As, the matrx s 1 P a g e

18 more vulnerable to the hygrothermal condton than the fber, the deformaton s examned to be more n the transverse drecton of the composte materal. The rse n hygrothermal condtons decreases the elastc modul of the materal and nduces nternal ntal stresses, whch may affect the stablty as well as the safety of the structures. Hence, t s necessary to study and analyze the behavor such as bucklng and natural frequences of lamnated composte twsted panels due to the hygrothermal effect seem to be an mportant consderaton n composte analyss and desgn, whch are of practcal nterest. Composte materals are beng ncreasngly used n automotve, marne and especally weght senstve aerospace applcatons, prmarly because of the large values of specfc strength and these can be talored through the varaton of fbre orentaton and stackng sequence to obtan an effcent desgn. The optmum desgn of lamnated structures demands an effectve analytcal procedure. But the presence of varous couplng stffness s and hygrothermal loadng complcates the problem of vbraton and bucklng analyss of twsted panels for obtanng a sutable theoretcal soluton. Even the real stuaton of boundares n lamnated structures s more complex because there are many types of boundary condtons that can be called smplysupported or clamped edges. So a clear understandng about vbraton and bucklng characterstcs of the composte twsted panels s of great mportance. A comprehensve analyss of the vbraton problems of homogeneous turbo machnery blades, modelled as beams has been studed exhaustvely. Some studes avalable on the untwsted plates subjected to hygrothermal loadng and some studes are there on vbraton aspects of lamnated composte pre twsted cantlever panels. The vbraton and bucklng analyss of lamnated composte pre twsted cantlever panels subjected to hygrothermal loadng s current case of study. The problem nvolves dfferent complcated effect such as geometry, especally non-developable doubly curved surfaces, boundary condtons wth varable temperature and mosture concentratons. The above dscussed aspects need attenton and thus consttute a problem of current nterest. A thorough revew of earler works done n ths feld s an mportant requrement to reach the objectve and scope of the present nvestgaton P a g e

19 CHAPTER REVIEW OF LITERATURE.1 Introducton The vast uses of conventonal metals, ts alloys and the ever ncreasng demand of composte materals n plates and shells are the subject of research for many years. Though the nvestgatons s manly focused on vbraton and bucklng of composte twsted panels subjected to hygrothermal loadng, some relevant researches on vbraton & bucklng of composte plate subjected to hygrothermal loadng are also studed for the sake of ts relevance and completeness. The lterature revewed n ths chapter are grouped nto two parts Twsted panels Composte plate. Revews on twsted panels The twsted panels have sgnfcant applcatons aerospace, cvl, naval and other hgh-performance engneerng applcatons due to ther lght weght, hgh specfc strength, stffness, excellent thermal characterstcs, and ease n fabrcaton and other sgnfcant attrbutes. Ths range of practcal applcatons demands a fundamental understandng of ther vbraton and bucklng. Due to ts sgnfcance, a large number of references deal wth the free vbraton of twsted plates. The followng areas of analyss pertanng to the plates are covered n the revew of lterature:..1 Vbraton and bucklng of twsted panels subjected to hygrothermal load The vbraton analyss of turbomachnery blades has long been an ntensve area of research. Its practcal use can be seen n aerospace ndustry, fan and compressor blades. Knowledge of resonant frequences and mode shape s necessary to ensure a rsk free desgn of turbomachnery. To talor the structural propertes, fber for renforced compostes lamnates are 3 P a g e

20 ncreasngly used for desgnng turbomachnery blades requrng hgher strength, more durablty and less weght. An excellent survey of the earler works n the free vbraton of turbomachnery blades was carred out by Rao [1973, 1977a, and 198], Lessa [198, 1981] for both statonary and rotatng condtons. Durng the recent years, more rgorous methods of analyss have been developed based on the plate theory. Dokansh and Rawtan [] nvestgated the natural frequences and the mode shapes of a cantlevered plate mounted around a rotatng dsc. They consdered the chord-wse bendng effects, and obtan accurate results. Ansar (1975) evaluated the nonlnear modes of vbraton of a pretwsted non unform cantlevered blade of unsymmetrcal cross-secton mounted on the perphery of a rotatng dsk. Krkhope and Wlson (1976) calculated the coupled vbraton modes of a rotatng blade-dsc system by usng fnte element method. Rao and Banerjee (1977) developed a polynomal frequency equaton method to determne the natural frequences of a cantlever blade wth an asymmetrc cross-secton mounted on a rotatng dsc. Consderng the blade as a dscrete system, generalzed polynomal expressons for the slope, lnear and angular deflectons are derved, usng Myklestad expressons wth necessary modfcatons. Walker (1978) studed a conformng fnte shell element sutable for the analyss of curved twsted fan blades and appled to a number of fan blade models. The element s assumed to be a doubly curved rght helcodally shell, n whch the curvature s shallow wth respect to the twsted base plane defnng the helcods. Sreenwasamurthy and Ramamurt (198) nvestgated the effect of a tp mass on the natural frequences of a rotatng Pre-twsted cantlever plate. Sreenwasamurthy and Ramamurt (1981) used a fnte element technque to determne the natural frequency of a pre-twsted and tapered plate mounted on the perphery of a rotatng dsc. The pre-twsted plate has been dealzed as an assemblage of three nodded rectangular shell elements wth sx degrees of freedom at each node. Lessa et al. (1983) determned the Vbratonal characterstc of doubly curved shallow shells havng rectangular plan forms, clamped along one edge and free on the other three usng Rtz method wth algebrac polynomal tral functon. In (1984) they expermented on vbraton of twsted cantlever plates and summarzed the prevous and current studes. Ramamurt and Kelb (1984) presented a detaled comparson of the egen frequences of twsted rotatng plates as obtaned by usng two dfferent shape functons. 4 P a g e

21 Rao and Gupta (1987) analyzed the free vbraton characterstc of a rotatng pre twsted small aspect rato blade, mounted on a dsc at a stagger angle, are determned usng classcal bendng theory of thn shells. Qatu and Lessa (1991) presented the vbraton studes for lamnated composte twsted cantlever plates usng Rtz method wth algebrac polynomal dsplacement functon. Lew et al. (1994) presented a mathematcal model to nvestgate the effects of ntal twst on the vbratory characterstcs of cantlever shallow concal shells. Pa and Nayfeh (1994) used a new approach to develop a geometrcally exact non-lnear beam model for naturally curved and twsted sold composte rotor blades undergong large vbratons n three dmensonal space. Lakhtaka (1995) studed wave propagaton n a pezoelectrc, contnuously twsted, structurally chral medum along the axs of spralty. Lew et al. (1995) presented a computatonal nvestgaton nto the objects of ntal twst and thckness varaton on the vbratory characterstcs of cantlevered pretwsted thn shallow concal shells wth generally varyng thckness. Rand (1995) presented a expermental method for detectng the natural frequences of helcopter-lke thn-walled rotatng composte blades, and a study of ther tendency n lamnaton angles and n the rotor angular velocty. Rand & Barka (1997) studed a nonlnear formulaton for the structural behavor of ntally twsted sold and thn walled composte blades s presented. The model s desgned to handle arbtrary thck sold cross-sectons or general thn-walled geometres, and ncludes three-dmensonal out-of-plane warpng. Parh et al. (1999) nvestgated about the dynamc analyss of delamnated composte twsted plates based on a smple multple delamnated models usng fnte element method. He et al. () presented a paper about a computatonal method for characterzng the resonant frequency propertes of cantlever pre-twsted plate composed of fbre-renforced lamnated compostes. Chen and Chen (1) employed a fnte element model to nvestgate the mean square response and relablty of a rotatng composte blade wth external and nternal dampng under statonary or non statonary random exctaton. The effects of transverse shear deformaton and rotary nerta are consdered. Hu and Tsuj (1) nvestgated the vbraton analyss of lamnated cylndrcal thn panels wth twst and curvature usng Raylegh-Rtz method. Cho and Chou (1) proposed a modfed dfferental quadrature method (MDQM) for vbraton analyss 5 P a g e

22 of elastcally supported turbomachnery blades. Yoo et al. (1) derved the equatons of moton for the vbraton analyss of rotatng pre-twsted blades, derved from a modelng method whch employs hybrd deformaton varables. Hu et al. () studed a numercal method for free vbraton of a rotatng twsted and open concal shell by the energy method, based on a non-lnear stran dsplacement relatonshp of a non-rotatng twsted and open concal shell on thn shell theory. Hu et al. () nvestgated the vbraton analyss of 7twsted concal shells wth tapered thckness. Hu et al. () studed a methodology for free vbraton of a lamnated composte concal shell wth twst s proposed, n whch a stran dsplacement relatonshp of a twsted concal shell s gven by consderng the Green stran tensor on the general thn shell theory, the prncple of vrtual work s utlzed, and the governng equaton s formulated by the Raylegh Rtz procedure wth algebrac polynomals n two elements as admssble dsplacement functons. Zhu et al. () examned the modelng of torsonal vbraton nduced by extensontwstng couplng of ansotropc composte lamnates wth pezoelectrc actuators. Lee et al. () studed the fnte element method based on the Hellnger Ressner prncple wth ndependent stran s appled to the vbraton problem of cantlevered twsted plates and cylndrcal, concal lamnated shells. Kuang, Hsu () studed the egen value problem of a tapered pre-twsted orthotropc composte blade s formulated by employng the dfferental quadrature method (DQM). The Euler Bernoull beam model s used to characterze the pretwsted orthotropc composte blade. Lm (3) presented a new approach n the bendng analyss of helcodal structures wth a large non-lnear pre twst and an external lateral loadng. Yoo and Perre (3) nvestgated the modal characterstc of a rotatng cantlever plate by usng a dynamc modelng method for rectangular plates undergong prescrbed overall moton to derve the equatons of moton. Oh et al. (3) analysed the effects of pre twst and presettng on coupled bendng vbratons of rotatng thn-walled composte beams usng refned dynamc theory of rotatng blades modelled as ansotropc composte thn-walled beams, experencng the flappng-laggng-transverse shear couplng. Ln et al. (3) nvestgated about rotatng no unform pretwsted beams wth an elastcally restraned root and a tp mass Usng Hamlton s prncple derves the governng dfferental equatons for the coupled bendng bendng vbraton of a rotatng pretwsted beam wth an elastcally restraned root and a tp mass, subjected to the external transverse forces and rotatng at a constant angular velocty. Chandraman et al. (3) 6 P a g e

23 examned the free and forced vbraton of a rotatng, pretwsted blade modelled as a lamnated composte, hollow (sngle celled), and unform box-beam. The structural model ncludes transverse shear flexblty, restraned warpng and centrfugal and Corols effects. Nab and Ganesan (3) proposed the vbraton characterstcs of pre-twsted composte blades are analyzed usng a three-noded trangular cylndrcal shell element. The specfc example of glass fbre renforced plastc materal s analyzed wth ts materal dampng. The effect of dfferent parameters such as pre-twst, fbre orentaton, skew angle, taper rato and aspect rato on natural frequency and system loss factor s nvestgated. Sakar and Sabuncu (4) presented a fnte element model for the statc and dynamc stablty study of a pretwsted aerofol cross-secton rotatng blade subjected to an axal perodc force. Tsa (4) studed the Rotatng vbraton behavor of the turbne blades wth dfferent groups of blades. Hu et al. (4) studed vbraton of twsted plate Based on general shell theory and the frst order shear deformaton theory, an accurate relatonshp between strans and dsplacements of a twsted plate s derved by the Green stran tensor. Kee and Km (4) derved a general formulaton for an ntally twsted rotatng shell structures ncludng the effect of centrfugal force and Corols acceleraton to study the vbraton characterstc of ntally twsted rotatng shell type composte blades. Hu et al. (4) proposed vbraton of an angle-ply lamnated plate wth twst consderng transverse stran and rotary nerta, an analytcal method by usng Raylegh-Rtz procedure. Dokansh and Rawtan (5) used a fnte element technque s to determne the natural frequences and the mode shapes of a cantlever plate mounted on the perphery of a rotatng dsc. The plane of the plate s assumed to make any arbtrary angle wth the plane of rotaton of the dsc. Chazly (5) analyzed the Statc and dynamc analyss of wnd turbne blades usng the fnte element method. Sahu et al. (5) studed the vbraton and stablty behavour of angle ply lamnated twsted panels usng Fnte element method. Huang (6) examned the effect of number of blades and dstrbuton of cracks on vbraton localzaton n a cracked pre-twsted blade system. Sahu et al. (7) studed the bucklng and vbraton analyss of cross-ply lamnated cantlever twsted plate usng the fnte element method wth frst order shear deformaton theory. An eght noded soparametrc quadratc element s employed n the present analyss wth fve degrees of freedom per node. 7 P a g e

24 Cho et al. (7) studed the bendng vbraton control of the pre-twsted rotatng composte thn-walled beam s studed. The formulaton s based on sngle cell composte beam ncludng a warpng functon, centrfugal force, Corols acceleraton, pre-twst angle and pezoelectrc effect. Hashem et al. (9) studed a fnte element formulaton for vbraton analyss of rotatng thck plates s developed. Mndln plate theory combned wth second order stran dsplacement assumptons are appled for plate modellng. Snha and Turner (11) proposed startng wth the thn shell theory, the governng partal dfferental equaton of moton for the transverse deflecton of a rotatng pre-twsted plate s derved to determne the natural frequences of a twsted blade n a centrfugal feld. Farhad and Hasem (11) examned the aero elastc behavor of a supersonc rotatng rectangular plate n the ar medum usng the Mndln frst-order shear deformaton plate theory along wth Von Korman nonlnear terms..3 Revews on Plates The behavour of structures subjected to n-plane loads wth hygrothermal load s less understood n comparson wth structures under transverse loads. Some of the lterature coverng composte plates subjected to hygrothermal loadng s presented here..3.1 Vbraton of composte plates subjected to hygrothermal load The effect of envronment on the free vbraton of lamnated plates has been consdered earler by Whtney and Ashton (1971). They used the Rtz method to analyze symmetrc lamnates and equlbrum equatons of moton n the case of antsymmetrc angle-ply lamnates, based upon the classcal lamnated plate theory. A few results were presented for only symmetrc angle-ply lamnates. Yang and Sheh [1987] consdered vbratons of ntally stressed antsymmetrc cross-ply lamnates. Intal stresses ncluded both force and moment resultants. Rotary nerta and transverse shear effects were taken nto account. Dhanaraj and Palannathan (1989) used the sem-loof shell element to study the free Vbratonal characterstcs of composte lamnates under ntal stress, whch may also arse due to temperature. Results were presented showng how temperature affects the fundamental frequences of ant-symmetrc lamnates. Saram & Snha (199) are nvestgated the effects of 8 P a g e

25 mosture and temperature on the free vbraton of lamnated composte plates. The analyss s carred out by the fnte element method wth the quadratc soparametrc element, whch takes transverse shear deformaton nto account. The analyss also accounts for lamna materal propertes at elevated mosture concentraton and temperature. Results are presented showng the reducton n the natural frequency wth the ncrease n unform mosture concentraton and temperature for symmetrc and antsymmetrc lamnates wth smply supported and clamped boundary condtons The vbraton characterstcs of rectangular plates subjected to non-unform loadng are studed usng power seres method by Khukla et al. (1995) & dfferental quadrature method by Guterrez et al. (1999). Len- Wenchen, Chen (1988) studed vbratons of hygrothermal elastc composte plates. Parh et al. (1) presented a quadratc soparametrc fnte element formulaton based on the frst order shear deformaton theory for the free vbraton and transent response analyss of multple delamnated doubly curved composte shells subjected to a hygrothermal envronment. Rohwer et al. (1) nvestgated hgher theores for thermal stresses n layered plates. Cheng and Batra (1) studed the effect of thermal loads on mperfectly bonded lamnated composte shells, the nterfacal mperfectons necesstate that condtons requrng the contnuty of surface tractons and dsplacements between adjonng faces be sutably modfed, and nterfacal damage properly accounted for. Shen and Shen (1) examned the effect of hygrothermal condtons on the bucklng and post bucklng of shear deformable lamnated cylndrcal shells subjected to combned loadng of axal compresson and external pressure s nvestgated usng a mcro-to macro-mechancal analytcal model. Sngha et al. (1) analysed the thermal post bucklng behavour of graphte/epoxy mult-layered rectangular plates of varous boundary condtons usng the fnte element method. Vel and Batra (1) nvestgated the generalzed plane stran quas-statc thermo elastc deformatons of lamnated ansotropc thck plates by usng the Eshelby-Stroh formalsm. Shen and Shen (1) studed the nfluence of hygrothermal effects on the post bucklng of shear deformable lamnated plates subjected to a unaxal compresson s nvestgated usng a mcro-tomacro-mechancal analytcal model. Rutgerson and Bottega () presented the bucklng behavor of multlayer shells for composte structures subjected to combnatons of unform temperature change, appled external pressure, and appled and reactve crcumferental edge loads. Wu and Chu () analyzed on thermally nduced dynamc nstablty of lamnated 9 P a g e

26 composte concal shells s nvestgated by means of a perturbaton method. The lamnated composte concal shells are subjected to statc and perodc thermal loads. Patel et al. () studed the statc and dynamc characterstcs of thck composte lamnates exposed to hygrothermal envronment are studed usng a realstc hgher-order theory developed. Shen and Shen () studed the effect of hygrothermal condtons on the bucklng and post bucklng of shear deformable lamnated cylndrcal panels subjected to axal compresson s nvestgated usng a mcro-to macro-mechancal analytcal model. Nonlnear vbraton & dynamc response of smply supported shear deformable lamnated plates on elastc foundatons studed by Huang, Zheng (3). Nonlnear vbraton & dynamc response of shear deformable lamnated plates n hygrothermal condton studed by Zheng et al. (4). Rao & Snha (4) studed dynamc response of multdrectonal compostes n hygrothermal envronments. Jeyaraj et al. (9) represented numercal studes on the vbraton and acoustc response characterstcs of a fberrenforced composte plate n a thermal envronment by consderng the nherent materal dampng property of the composte materal. Intally the crtcal bucklng temperature s obtaned, followed by free and forced vbraton analyses consderng the pre-stress due to the mposed thermal envronment. Panda & Sngh (9) presented nonlnear fnte element model for geometrcally large ampltude free vbraton analyss of doubly curved composte sphercal shell panel s presented usng hgher order shear deformaton theory (HSDT)..3. Stablty of composte plate subjected to hygrothermal load The statc stablty of mechancal, cvl engneerng structures under compressve loadng has always been a mportant feld of research wth the ntroducton of steel a century ago. The studes of temperature and mosture effects on the bucklng loads of lamnates are lmted n number, and all these studes assumed perfectly ntal confguratons. Whtney and Ashton (1971) gave the frst theoretcal nvestgaton of hygrothermal effects on the bendng, bucklng and vbraton of composte lamnated plates based on the classcal lamnated plate theory. The hygrothermal effects on the bucklng of cylndrcal composte panels were studed by Snead and Palazotto (1983) neglectng the transverse shear deformaton effects, and by Lee and Yen (1989) ncludng the transverse shear deformaton effects. 1 P a g e

27 Ram and Snha (199) studed the hygrothermal effects on the bucklng of composte lamnated plates usng the fnte element method. Chao and Shyu (1996) calculated the bucklng loads for composte lamnated plates under hygrothermal envronments, where a mcro to- macromechancal analytcal model was proposed. These analyses addressed ntal bucklng problems and were based on the frst-order shear deformaton plate theory (FSDPT). Shen () studed the effect of hygrothermal condtons on the bucklng and post bucklng of shear deformable lamnated cylndrcal panels subjected to axal compresson whch s nvestgated usng a mcroto-macro-mechancal analytcal model. Egen senstvty analyss of mosture-related bucklng of marne composte panels was studed by Barton (7). He presented the elastc bucklng of rectangular, symmetrc angle-ply lamnates subjected to a unform mosture envronment. Ths nvestgaton presents an alternatve method of computng the bucklng load usng Egen values. Usng ths approach, an approxmate closed-form expresson s developed whch can be used when exact soluton are not avalable. The studes on dynamc stablty of structures are much less than n comparson to statc stablty & got a boost after Bolotn s (1964) contrbuton to the lterature. The nstablty results of thn smply supported (1995) plates are sparsely treated n the lterature. The nstablty of lamnated composte plates consderng geometrc non-lnearty s also reported usng fnte element method by Balamurugan et al. (1996). The nstablty of the elastc plates consderng geometrc non-lnearty s also reported, through a fnte element formulaton by Ganapath (). Partha & Sngha (6) studed the dynamc stablty characterstcs of smply supported lamnated composte skew plates subjected to a perodc n-plane load are nvestgated usng the fnte element approach. The formulaton ncludes the effects of transverse shear deformaton, nplane and rotary nerta. The boundares of the nstablty regons are obtaned usng the Bolotn s method and are represented n the non-dmensonal load ampltude-exctaton frequency plane. Lu (7) started a mesh-free formulaton for the statc and free vbraton analyses of composte plates s presented va a lnearly conformng radal pont nterpolaton method. The radal and polynomal bass functons are employed to construct the shape functons bearng Delta functon property. 11 P a g e

28 .4 Am and scope of the present studes A Revew of lterature shows that a lot of work has been done on the vbraton and bucklng of composte panels subjected to hygrothermal loadng. Some work has been done on vbraton of lamnated composte twsted cantlever panels. However very few study s avalable on vbraton and bucklng of lamnated composte twsted cantlever panels subjected to hygrothermal loadng. The present study s manly amed at fllng some of the lacunae that exst n the understandng of the vbraton and stablty characterstc of lamnated composte twsted cantlever panels subjected to hygrothermal loadng. The nfluence of varous parameters such as angle of twst, number of layers, lamnaton sequence, ply orentaton on the vbraton and bucklng behavour of twsted panels s studed n detal. Based on the revew of lterature, the dfferent problems dentfed for the present nvestgaton s presented as follows. Vbraton of lamnated composte twsted cantlever panels subjected to hygrothermal loadng. Bucklng of lamnated composte twsted cantlever panels subjected to hygrothermal loadng. 1 P a g e

29 CHAPTER 3 THEORY AND FORMULATON 3.1 The Basc Problem Ths chapter presents the mathematcal formulaton for vbraton and bucklng analyss of the twsted plate. The basc confguraton of the problem consdered here s a composte lamnated doubly curved twsted panel of sdes a and b as shown n Fgure. Fg. (a) Composte twsted panel The twsted panel s modelled as a doubly curved panel wth twstng curvature so that the analyss can be done for twsted plates, cylndrcal and sphercal confguratons by changng the value of the curvature. The boundary condtons are taken to be that of a cantlever, that s fxed at the left end and free at the other edges. The basc composte twsted curved panel s consdered to be composed of composte materal lamnates. n denotes the number of layers of the lamnated composte twsted panel. 13 P a g e

30 (b) The lamnaton Fg.1 Geometry of an N-layered lamnate 3. Proposed Analyss The governng equatons for the vbraton and bucklng of lamnated composte twsted Panels/shells subjected to n-plane loadng are developed. The presence of external n-plane loads nduces a stress feld n the structure. Ths necesstates the determnaton of the stress feld as a prerequste to the soluton of problems lke vbraton and bucklng behavour of pre twsted plates and shells. As the thckness of the structure s relatvely smaller, the determnaton of the stress feld reduces to the soluton of a plane stress problem. The equaton of moton represents a system of second order dfferental equatons wth perodc coeffcents of the Matheu-Hll type. The development of the regons of nstablty arses from Floquet s theory and the soluton s obtaned by Bolotn s approach usng fnte element method. The governng dfferental equatons have been developed usng the frst order shear deformaton theory (FSDT). The assumptons made n the analyss are gven below. 3.3 Assumptons of the analyss 1) The analyss s lnear wth a few exceptons. Ths mples both lnear consttutve relatons (generalzed Hooke s law for the materal and lnear knematcs) and small dsplacement to accommodate small deformaton theory. ) The composte twsted panels are of varous shapes wth no ntal mperfectons. The consderatons of mperfectons are less mportant for dynamc loadng. 3) The straght lne that s perpendcular to the neutral surface before deformaton remans straght but not normal after deformaton (FSDT). The thckness of the composte twsted panel 14 P a g e

31 s small compared wth the prncpal rad of curvature. Normal stress n the z-drecton s neglected. 4) The loadng consdered s axal wth a smple harmonc fluctuaton wth respect to tme. 5) All dampng effects are neglected. 3.4 Governng Equatons The governng dfferental equatons, the stran energy due to loads, knetc energy and formulaton of the general dynamc problem are derved on the bass of the prncple of potental energy and Lagrange s equaton Governng Dfferental Equatons The equatons of moton are obtaned by takng a dfferental element of the twsted panel as shown n fgure. Ths fgure shows an element wth nternal forces lke membrane forces N x N y andn xy, shearng forces( Q x andq y ) and the moment resultants( M x, M y and M xy ). The governng dfferental equatons for vbraton of a shear deformable lamnated composte plate n hygrothermal envronment derved on the bass of frst order shear deformaton theory (FSDT) subjected to n-plane loads are (Chandrasekhar, Sahu and Dutta). Nx x + Nxy y 1 1 Ry 1 Rx Mxy y + Qx Rx + Qy u θx = P1 + P Rxy t t Nxy x + Ny y Ry 1 Rx Mxy y + Qy Rx + Qx v θy = P1 + P Rxy t t Qx x + Qy y Nx Rx Ny Ry Nxy w w + Nx + Ny w/ y = P1 Rxy y t Mx x + Mxy y Qx = P3 θx u + P t t 15 P a g e

32 Mxy x + My y Qx = P3 θy t + P v t (1) Fgure : Force and moment resultants of the twsted panel Where N x and N x are the external loadng n the X and Y drecton respectvely Rx, Ry and Rxy dentfy the rad of curvatures n the x and y drecton and radus of twst. n zk k=1 () zk =1 P1, P, P3 = ρ k 1, z, z dz 16 P a g e

33 Where n = number of layers of lamnated composte curved panel, ( ) = mass densty of kth layer from md-plane. k 3.5 Fnte Element Formulaton For problems nvolvng complex geometrcal and boundary condtons, analytcal methods are not easly adaptable and numercal methods lke fnte element methods (FEM) are preferred. The fnte element formulaton s developed hereby for the structural analyss of sotropc as well as composte twsted panels usng a curved shear deformable shell theory The shell element The plate s made up of perfectly bonded layers. Each lamna s consdered to be homogeneous and orthotropc and made of undrectonal fber-renforced materal. The orthotropc axes of symmetry n each lamna are orented at an arbtrary angle to the plate axes. An eght-noded soparametrc quadratc shell element s employed n the present analyss wth fve degrees of freedom u, v, w, θx and θy per node as shown n Fgure. But the n-plane deformatons u and v are consdered for the ntal plane stress analyss. The soparametrc element shall be orented n the natural coordnate system and shall be transferred to the Cartesan coordnate system usng the Jacoban matrx. In the analyss of thn shells, where the element s assumed to have md-surface nodes, the shape functon of the element s derved usng the nterpolaton polynomal. For problems nvolvng complex n-plane loadng and boundary condtons numercal methods lke fnte element method (FEM) are preferred. Eght-nodded sopermetrc element s used to the present free vbraton problem. Fve degrees of freedom u, v, w, θx and θy are consdered at each node. The stffness matrx, the geometrc stffness matrx due to resdual stresses, geometrc stffness matrx due to appled n-plane loads and nodal load vector of the element are derved usng the prncple of mnmum potental energy. 17 P a g e

34 Fg. 3 Eght nodded soparametrc element The element and dsplacement feld are expressed by the shape functon N The shape functon N are defned as N = (1+ξξ )(1+ηη ) (ξξ +ηη -1)/4 =1 to 4 (3) N = 1 ξ (1+ηη )/ = 5, 7 (4) N = (1+ξξ ) (1- η )/ = 6, 8 (5) Where ξ and η are the local natural coordnates of the element and ξ and η are the values at th node. The dervatves of the shape functon are N wth respect to x and y are expressed n term of ther dervatves wth respect to ξ and η by the followng relatonshp. N,x N,y = J 1 N,ξ N,η Where [J]= X I,ξ X,η Y I,ξ Y,η [J] s the Jacoban matrx. The shell wth the ntal stresses undergoes small lateral deformatons. Frst order shear deformaton theory s used and the dsplacement feld assumes that the md 18 P a g e

35 plane normal remans straght before and after deformaton, but not necessarly normal after deformaton, so that u(x, y,z)= u (x,y)+ z θ y (x,y) v(x, y,z)= u (x,y)+ z θ x (x,y) w(x,y,z)= w (x,y) Where u, v, w and u, v, w are dsplacement n the x, y, z drectons at any pont and at the md surface respectvely. θx and θ y are the rotatons of the mdsurface normal about the x and y axes respectvely. Also x = N x, y = N y u = N u, v = N v, w = N w (6) θ x = N θ x, θ y = N θ y 3.5. Consttutve Relatons temperature, are gven by The consttutve relaton for the twsted plate, when subjected to mosture and {F} = [D] {ε} - {F N } (7) F N = Non mechancal forces due to temperature and mosture [D] and { ε } are Elastcty and stran matrx respectvely. Where {F} = { N x, N y, N xy, M x, M y, M xy, Q x, Q y } T {F N }= {Nx, Ny, Nxy, Mx, My, Mxy, Q x, Q y } T 19 P a g e