Dynamic instability of thin plates by the dynamic stiffness method

Dynamic instability of thin rectangular plates subjected to uniform in-plane harmonic compressive load applied along two opposite edges are investigated in this paper. The dynamic stiffness method (DSM), as a consequence the dynamic stiffness matrices, is used to analyze the free vibration, the static stability, and dynamic instability of thin plates under different boundary conditions.

Thông báo Khoa học Công nghệ* Số 1-2013 54 DYNAMIC INSTABILITY OF THIN PLATES BY THE DYNAMIC STIFFNESS METHOD Master Hung Quoc Huynh Faculty of Civil Engineering, Central University of Construction Abtract: Dynamic instability of thin rectangular plates subjected to uniform in-plane harmonic compressive load applied alon g two opposite edges are investigated in this paper The dynamic stiffness method (DSM), as a consequence the dynamic stiffness matrices, is used to analyze the free vibration, the static stability, and dynamic instability of thin plates under different boundary conditions The boundaries of the dynamic instability principal regions are obtained using Bolotin’s method Results obtained such as free vibration frequencies, static buckling critical load and dynamic instability principal regions are compared with the results previously published to ascertain the validity of the method Keywords: Dynamic stability; static stability; dynamic stiffness method; plate Introduction Various plate structures are widely used in aircraft, ship, bridge, building, and some other engineering activities In many circumstances, these structures are exposed to dynamic loading Plate structures are often designed to withstand a considerable in-plane load along with the transverse loads The dynamic instability of thin rectangular plates under periodic in-plane loads has been investigated by a number of researchers The dynamic stability of rectangular plates under various in-plane periodic forces was studied by Bolotin [1], as well as by Yamaki and Nagai [2] Hutt and Salama [3] demonstrated the application of the finite element method to the dynamic stability of plates subjected to uniform harmonic loads Takahasi and Konishi [4] studied the dynamic stability of a rectangular plate subjected to a linearly distributed load such as pure bending or a triangularly distributed load applied along the two opposite edges using harmonic balance method Nguyen and Ostiguy [5] considered the influence of the aspect ratio and boundary conditions on the dynamic instability and non-linear response of rectangular plates Guan-Yuan Wu and Yan-shin Shih [6] investigated the effects of various system parameters on the regions of instability and the non-linear response characteristics of rectangular cracked plates using incremental harmonic balance (IHB) method The dynamic instability behaviour of rectangular plates under periodic in-plane normal and shear loadings was studied by Singh and Dey [7] using energy-based finite difference method Srivastava et al [8] employed the nine-noded isoparametric quadratic element with five degree-offreedom method to investigate the dynamic instability of stiffened plates subjected to non-uniform harmonic in-plane edge loading Thông báo Khoa học Công nghệ* Số 1-2013 In this paper, the problem of dynamic stability of plates subjected to periodic inplate load along two opposite edges is studied by the dynamic stiffness method The problem is solved by the dynamic stiffness method in order to investigate the efficiency and the reliability of this method for solving above-mentioned problems The boundaries of the dynamic instability principal regions are obtained using Bolotin’s method The dynamic stability equation is solved to plot the relationship of the parameters of load, natural frequency, frequency of excitation from the computational program by Matlab Results obtained, such as free vibration frequencies, static buckling critical load, and principal regions of dynamic instability, are compared with the results previously published to ascertain the validity of the method Dynamic stability analysis Assume that a rectangular plate with length a, width b, and thickness h is subjected to uniform harmonic in-plane loads Nx applied along the two opposite boundaries Both unloaded edges can be simply supported (SS) or clamped (C) A Cartesian co-ordinate system (x, y, z) is introduced as shown in Fig Nx = Ns + Nt cos t Nx x, u lf-w ave O y v h a in o ne ling Edge b b Bu ck SS z,w Edge a SS The dynamic instability analysis of composite laminated rectangular plates and prismatic plate structures was determined by Wang and Dawe [9] using the finite strip method Wu Lanhe et al [10] analyzed the dynamic stability of thick functionally graded material plates subjected to aero-thermomechanical loads, using a novel numerical solution technique, the moving least squares differential quadrature method The dynamic instability of laminated sandwich plates subjected to in-plane edge loading was studied by Anupam Chakrabarti and Abdul Hamid Sheikh [11] using the proposed finite element plate model based on refined higher order shear deformation theory Dynamic stability analysis of composite plates including delaminations were performed by Adrian G Radu and Aditi Chattopadhyay [12] using a higher order theory and transformation matrix approach 55 Buckling in several half-waves Fig Rectangular plate subjected to dynamic inplane loads The equations of motion for generally isotropic plates are given by Timoshenko [13], and can be reduced to the following set of equations 2w 2w  N 0 x t2 x2 (1) 4w 4w 4w  w 2 2  x x y y (2) D4w h in which where w is the displacement at mid-surface in z-direction of rectangular Cartesian Thông báo Khoa học Công nghệ* Số 1-2013 coordinates, t is the time, and  is the mass density per unit volume The flexural rigidity is defined as D = Eh3/12(1-2 ) in which E is Young’s modulus and  is Poisson ratio In the above equation, the in-plane load factor Nx is periodic and can be expressed in the form: Nx  Ns  Nt cosΩt (3) where Ns is the static portion of Nx, Nt is the amplitude of the dynamic portion of Nx, and  is the frequency of excitation The lowest critical static buckling load Ncr may be expressed interns of Ns and Nt as follows: (4) N s   s N cr , N t   d N cr where s and  d are static and dynamic load factors, respectively The transverse deflection function w, satisfying the geometric boundary conditions, can be written as N w( x, y, t )   Ym ( y )sin m 1 m x f (t ) a (5) where m is the number of half-waves (normal spatial mode in x-direction), a is the length of plate in x-direction, f(t) are unknown functions of time, and Ym(y) are functions to be determined in order to satisfy the equation of motion (1) By substituting Eq (5) into Eq (1), the following fourth order ordinary differential equations are obtained  h  Ym f (t )  YmIV  k m2 Ym''  k m4 Ym D (  N   d N cr cosΩt )   s cr k m Ym  f (t )  D   where km  m / a (6) (7) Equations (6) represent a system of secondorder differential equations for the time 56 functions with periodic coefficients of the standard Mathieu-Hill equations, describing the instability behavior of the plate subjected to a periodic in-plane compressive load The analysis of a given structural system for dynamic stability implies the determination of boundaries between the stable and unstable regions The dynamic instability boundaries are determined using the method suggested by Bolotin [1] The stability and instability of their solution depends on the parameters of the system The boundaries between stable and unstable regions in the parameter space are formed by periodic solutions of period T and 2T, where T = 2/ The principal instability region (first instability region) is usually the most important in dynamic stability analysis, because of its width as well as due to structural damping, which often neutralize higher regions The boundaries of the principal instability region with period of 2T are of practical importance and their solution can be achieved in the form of Fourier series  f (t )  k t k t    bk cos  ak sin  2   k 1,3,5,  (8) where ak and bk are vectors independent of time Substitution of equations (8) into equations (6) leads to an eigenvalue system for the dynamic stability boundary 1 4  4 2 4 0 4 3 4         0 (9) Thông báo Khoa học Công nghệ* Số 1-2013 where 1  YmIV 3.1 Generalized displacements  2km2 Ym''  YmIV Mym2 Q ym2 a y, v  h Wm2 Fig Generalized displacements generalized forces of plate  25Ω  h  s N cr     km4   km  Ym    D D    and Generalized displacement vector can be expressed as  d Ncr kmYm D It has been shown by Bolotin [l] that solutions with period 2T are the ones of greatest practical importance, and that as a first approximation the boundaries of the principal regions of dynamic instability can be determined from element (1, 1) of determinant (9) YmIV  2km2 Ym'' (10)    km4  Ω  h  (  s   d ) N cr km2  Ym  D D   Dynamic stiffness method uT  Wm1 ( x,0) Wm' ( x,0) Wm ( x, b) Wm' ( x, b) (13)  then Wm1 ( x,0)  Ym (0); Wm' ( x,0)  Ym' (0); Wm ( x, b)  Ym (b ); Wm' ( x, b )  Ym' (b) (14) The generalized displacement vector {u} can be determined by substituting Eqs (14) into Eqs (13) taking into account (11) and evaluating it at y=0 and y=b, then Eq (13) can be rewritten in matrix form (15) u   K1 C where CT  C1 C2 C3 C4  and The general solution of differential equations (10) has the form (11) where 1/2  c  k    h  Ω   (    ) N cr k    m s d m    D  D      1/2    N  d  k m2    h  Ω   (  s   d ) cr km2  D   D    b ' Wm2 3  YmIV  2km2 Ym''  C3 sin( d y )  C4 cos( d y) Nx z, w  2km2 Ym'' Ym ( y )  C1sinh(c y )  C 2cosh (c y ) x, u ' Wm1 Wm1  9Ω  h  s N cr     km4   km  Ym    D D    4  Mym1 Q ym1 O Nx    km4  Ω  h  (  s   d ) Ncr km2  Ym  D D  2  57 (12) where C1, C2, C3 and C4 are the coefficients to be determined from the four boundary conditions, edge a at y = 0, and edge b at y = b 1     c d   K1   sinh(bc ) cosh(bc ) sin(bd ) cos(bd )    (bc ) c.sinh(bc ) d.cos(bd ) d.sin(bd ) ccosh (16) where [K1] is the shape function 3.2 Generalized forces Generalized force vector can be expressed as QT   Qym1 ( x,0) M ym1 ( x,0) Qym ( x, b ) M ym ( x, b )  (17) The Kirchhoff shear force Qy(x,y) and the bending moment My(x,y) of the plate along the line y=constant are [15] Thông báo Khoa học Công nghệ* Số 1-2013  3 w  w  Qy ( x, y )   D    x y   y  2w 2w  M y ( x, y )   D     x   y (18) The generalized force which are determined to Eqs (18) can be written Qymi ( x, y )   D Y  k Y    ''  M ymi ( x, y )   D  Ym   kmYm  ''' m ' m m (19) The generalized force vector {Q} can be determined by substituting Eqs (19) into Eq (17) taking into account (11) and evaluating it at y=0 and y=b, then Eq (17) can be rewritten in matrix form (20) Q   K C where  K  is the generalized stiffness matrix  k11 k12 k k  K   D  k21 k22 31 32  k k  41 42 k13 k23 k33 k43 k14  k24  k34   k44  (21) Explicit expressions of the elements kij of the generalized stiffness matrix [K2 ] are as follows: 58 By substituting Eq (15) into Eq (20), the generalized nodal displacements and nodal forces are related, Q   K  K1 1 u Therefore, Q   D u (23) Where  D   K  K1 1 (24) Matrix [D] in equation (24) is the required dynamic stiffness matrix With the dynamic stiffness matrix being available, the vibration, static stability and dynamic stability problems of the plate structures can be solved 3.3 Static stability and vibration of the plate Two parameters c and d of the dynamic stiffness matrix [D] for solving the static stability and vibration problem are determined as follows :  c  k         r  k  m   m   m m  a  a     r  d   km2    m2    km2  m  a  a  (25) k11  (c.km2  c3 ); k12  where r  a / b is aspect ratio of plate, N m k14  represents the static critical load of plate for the m mode, and  m represents the non- k13  (d  d km ); k31  (c cosh(bc )  c.km2.cosh(bc )) k32 (c3.sinh(bc )  c.km2.sinh(bc )) k33  (d3.cos(bd )  d.km2.cos(bd )) k34 (d3.sin(bd )  d.km2.sin(bd )) k21  0; k22  (km2.v c2) k23  0; k24  (d2  vk m2) k41  (c2 sinh(b.c)  km2 v.sinh(b.c)) k42  (c2 cosh(b.c)  km2 v.cosh(b.c)) k43  (d sin(b.d )  km2 v.sin(b.d )) 2 k44  (d cos(b.d )  km v.cos(b.d )) (22) dimensional static critical loading factor of plate for the m mode, which is defined as  m  N m b2 /  D (26) The non-dimensional natural frequency parameter (natural frequency factor) m of plate is defined as  m  m a /   h / D (27) where m is the natural frequency for the m mode of plate Thông báo Khoa học Công nghệ* Số 1-2013 59 Step Solve dynamic stability equation load  *   d / 2(1   s ) parameter is (29) The natural frequency of lateral free vibration of a rectangular plate loaded by a uniform in-plane force is defined as (30)  m*  m   s (34) Numerical results and discussions 4.1 Static stability and vibration problems 4.1.1 Problem An example is investigated for the static stability and natural vibration analysis of a thin square plate P1 (a=b) with all four edges simply supported and compressed by uniformly distributed inplane forces along its opposite edges (Fig 3) Nx Nx (a) P.1 The non-dimensional frequency of excitation parameter is as follows Λ  Ωa2  h / D a=b y (b) 3.5 Dynamic instability of thin plates by the dynamic stiffness method Q   D  u  * * * (33) Step Derive the dynamic stability equation For any displacemant {u *} to become infinitely large, [D*] must vanish and this condition means that every other displacemant in the plate must also tend to infinity Therefore, for dynamic instability the condition is det D*   Nx Nx Buckling in one half-wave (m = 1) Fig Thin square plate P1 (SS-SS-SS-SS) The dynamic stability equation (34) is solved by plotting the relationship m-m using Matlab program, which determines the static critical loading factors m and the free vibration frequency factors m Static critical loading factor Step Apply the constraints as dictated by the boundary conditions Apply boundary conditions of the problem to eliminate degeneracy of the dynamic stiffness matrix Equation (32) has the form: b SS (31) Step The motion equation (23) of plate would be: (32) Q   Du x SS Buckling in one half-wave The normalized determined as det  D*   SS For analyzing the dynamic stability, two parameters c and d of the dynamic stiffness matrix [D] are determined as in Eq (12) The non-dimensional static critical loading factor cr of plate is defined as (28)  cr  N cr b /  D SS 3.4 Dynamic instability of the plate 4 2 Natural frequency factor Fig Relation m-m (plate P1, mode m=1) 8 Results obtained in the present analysis are compared with those of Yamaki and Nagai [2] and Timoshenko [13,14] in Table 1, which shows a good agreement 6.2499 Natural frequency factor Fig Relation m-m (plate P1, mode m=2) Static critical loading factor 60 12 10 11.111 4.1.2 Problem This problem considers a thin square plate P3 (a=b) with two edges simply supported and two edges clamped and compressed by uniformly distributed inplane forces along its opposite edges for the static stability and free vibration frequency (Fig 7) Nx 2 Natural frequency factor x C 10 10 12 Nx (a) SS P.3 Fig Relation m-m (plate P1, mode m=3) b SS C It is observed from Fig 4-6 that the lowest static critical loading factor and the free vibration frequency factors are determined a=b y Nx Nx (b)  cr  , 1  2; 2  5; 3  10 ) The lowest static critical buckling load Fig Thin square plate P3 (SS-C-SS-C) 10 Static critical loading factor N cr  4 D / b2 The free vibration frequencies 1  2( / a ) D /  h ; 2  5( / a ) D /  h ; mode m 1 DSM Ref [2] 10 10 Ref [13,14] 10 2.9332 Natural frequency factor 10 Fig Relation m-m (plate P3, mode m=1) 10 Static critical loading factor Table Comparison of cr and m of square plate P1 factor 8.6044 3  10( / a ) D /  h cr m Buckling in one half-wave Static critical loading factor Thông báo Khoa học Công nghệ* Số 1-2013 7.6913 5.5466 Natural frequency factor 10 Fig Relation m-m (plate P3, mode m=2) Thông báo Khoa học Công nghệ* Số 1-2013 12 4.2 Dynamic instability problems 11.9178 10 10.3566 0 10 Natural frequency factor 12 14 16 4.2.1 Problem This problem concerns the dynamic stability of a thin square plate P1 (a=b) with all four edges simply supported and compressed by uniformly distributed inplane periodic forces along its opposite edges (Fig 11) Nx =  sNcr + dNcr cos  t Fig 10 Relation m-m (plate P3, mode m=3) SS Nx The lowest static critical loading N cr  7.6913 D / b The free vibration frequency 1  2.9332( / a ) D /  h ; 2  5.5466( / a ) D /  h ; 3  10.3566( / a ) D /  h Table Comparison of cr and m of square plate P3 P.1 b SS a=b  cr  7.6913 ; 1  2.9332 ; 2  5.5466 ; 3  10.3566 x SS It is observed from Fig 7-10 that the lowest static critical buckling load factor and the free vibration frequency factors are determined SS Static critical loading factor 14 61 y Fig 11 Thin square plate P1 (SS-SS-SS-SS) By solving the dynamic stability Eq (34), we obtain the boundaries of the principal dynamic instability regions, which are presented in the non-dimensional frequency of excitation parameter () versus dynamic load factor (d) amplitude plane Two values of the static load factor  s , i.e., and 0.6, are considered Case 1: the static load factor  S = Results obtained in the present analysis are compared with those of Yamaki and Nagai [2] and Timoshenko [15] in Table 2, which shows a good agreement Unstable 0.8  s Ref DSM Ref [2] [15] 7.6913 7.701 7.69 2.9332 2.935 5.5466 5.550 10.3566 10.36 - Dynamic load factor: d 1.2 mode factor m cr m 0.6 0.4 0.2 0 10 20 30 40 50 Dimensionless excitation frequency:  60 Fig 12 Principal instability region for the square plate P1 (case 1, S = 0) Case 2: the static load factor  S = 0.6 Thông báo Khoa học Công nghệ* Số 1-2013 0.5 Unstable 0.4 Fig 13 Principal instability region for the square plate P1 (case 2, S = 0.6)  s Dynamic load factor: d Principal region of dynamic instability for simply supported plate P.1 0.6 62 0.3 0.2 0.1 0 10 20 30 40 50 Dimensionless excitation frequency:  60 Table Comparison of principal region of dynamic instability for square plate P1 (case 1, S = 0) d 0.2 0.4 0.8 1.2 Dimensionless excitation frequency  DSM Ref [3] right left right left 39.478 39.478 39.46 39.46 41.405 37.452 43.246 35.311 43.00 35.32 46.711 30.579 46.56 30.78 49.936 24.968 49.52 25.06 Ref [8] right 39.46 43.16 46.54 49.54 Ref [11] right 41.384 43.224 49.911 left 39.46 35.37 30.73 24.02 left 37.433 35.292 24.956 Table Comparison of principal region of dynamic instability for square plate P1 (case 2, S = 0.6) d 0.16 0.32 0.48 Dimensionless excitation frequency  DSM Ref [3] right left right 24.968 24.968 25.06 27.351 22.332 27.43 29.542 19.340 29.60 31.582 15.791 31.57 Results obtained in the present analysis are compared with those of Hutt and Salam [3], Srivastava, Datta and Sheikh [8], and Chakrabarti and Sheikh [11] in Table and Table 4, which show a good agreement Ref [8] right 25.04 27.41 29.58 31.55 left 25.06 22.49 19.53 15.91 left 25.04 22.48 19.51 15.89 uniformly distributed in-plane periodic forces along its opposite edges (Fig 14) Nx =  sNcr +  dNcr cos t x C Nx 4.2.2 Problem An example is investigated for the dynamic stability of a thin rectangular plate P4 with two edges simply supported and two edges clamped and compressed by SS P.4 C a = 1.667b y SS b Thông báo Khoa học Công nghệ* Số 1-2013 Fig 14 Thin rectangular plate P4 (SS-C-SSC) (mode1,2,3) m=2 0.3 basis, the dynamic stability equation is established to analyze the problem of static stability, vibration and dynamic stability of thin plates by the dynamic stiffness method m=3 0.2 0.1 0 m=1 0.5 1.5 Normalized frequency parameter:   * Dynamic load factor: d 0.4 63 2.5 Fig.15.Principal instability regions for the rectangular plate P4(modes m=1,2,3) for S = 0.5 Research results obtained such as free vibration frequencies, static critical buckling load and principal regions of dynamic instability for the plates by the dynamic stiffness method are compared with the results previously published to be in a good agreement Thus in the analysis of plates structural one can use the dynamic stiffness method as a reliable and efficient tool References Fig 16 Principal instability regions for the rectangular plate P4 (mode m=1,2,3) for S = 0.5 of Ref [5] The plots of the principal region of dynamic instability for the rectangular plate P4 for three modes (m=1,2,3) in Fig 15 are compared and found to be in a very good agreement with the results of Nguyen and Ostiguy [5] in Fig 16 Conclussion In the paper, the dynamic stiffness method has been developed to analyze the thin plates and to consider the effect of in-plane dynamic forces on static stability, vibration and dynamic stability of such plates The 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định ưu điểm độ xác phương pháp độ cứng động lực Từ khóa: Ổn định động; ổn định tĩnh; phương pháp độ cứng động lực; mỏng  ... paper, the problem of dynamic stability of plates subjected to periodic inplate load along two opposite edges is studied by the dynamic stiffness method The problem is solved by the dynamic stiffness. .. principal regions of dynamic instability for the plates by the dynamic stiffness method are compared with the results previously published to be in a good agreement Thus in the analysis of plates structural... 3) Nx Nx (a) P.1 The non-dimensional frequency of excitation parameter is as follows Λ  Ωa2  h / D a=b y (b) 3.5 Dynamic instability of thin plates by the dynamic stiffness method Q   D