Dynamic instability of thin plates by the dynamic stiffness method

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 1. 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-of- freedom method to investigate the dynamic instability of stiffened plates subjected to non-uniform harmonic in-plane edge loading. Thông báo Khoa học và Công nghệ* Số 1-2013 55 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-thermo- mechanical 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. In this paper, the problem of dynamic stability of plates subjected to periodic in- plate 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. 2. 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. 1. SS a b Nx h y v x, u z,w O SS Edge a Edge b Bu ck lin g in o ne h al f-w av e Buckling in several half-waves N = N + N cos ts tx Fig. 1. 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 2 2 4 2 2x w wD w h N 0 t x          (1) in which 4 4 4 4 4 2 2 42 w w ww x x y y            (2) where w is the displacement at mid-surface in z-direction of rectangular Cartesian Thông báo Khoa học và Công nghệ* Số 1-2013 56 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: x s tN N N 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: s s crN N , t d crN N (4) 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 1 ( , , ) ( ) ( ) N m m m xw x y t Y y sin 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  2 '' 4 2 ( ) 2 ( ) ( ) 0 IV m m m m m m s cr d cr m m h Y f t Y k Y k Y D N N cosΩt k Y f t D             (6) where /mk m a (7) Equations (6) represent a system of second- order differential equations for the time 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 1,3,5,... ( ) sin cos 2 2k kk k t k tf t a b           (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 4 3 4 0 0 0 0                        (9) Thông báo Khoa học và Công nghệ* Số 1-2013 57 where 2 '' 1 4 21 ( ) 2 IV m m m 2 cr mm s d m Y 2k Y NΩ h Yk k 4 D D                 2 ''2 4 2 IV m m m 2 s cr m m m Y 2k Y N9Ω hk k Y 4 D D                  2 ''3 4 2 IV m m m 2 s cr m m m Y 2k Y N25Ω hk k Y 4 D D                 2 4 d cr m m N k Y 2 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) 2 '' 4 21 0( ) 2 IV m m m 2 cr mm s d m Y 2k Y NΩ h Yk k 4 D D              (10) 3. Dynamic stiffness method The general solution of differential equations (10) has the form 1 2 3 4 ( ) ( . ) ( . ) ( . ) ( . ) mY y C sinh c y C cosh c y C sin d y C cos d y     (11) where 1/2 2 2 1/2 2 2 1( ) 2 1( ) 2 2 cr m s d m 2 cr m s d m Nh Ωc k k D D4 Nh Ωd k k D D4                                       (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. 3.1. Generalized displacements a b Nx Nx h y, v x, u z, w O Wm1 Wm2 Wm1' Wm2' Q ym1Mym1 Q ym2Mym2 Fig. 2. Generalized displacements and generalized forces of plate. Generalized displacement vector can be expressed as     ' 1 1 ' 2 2 ( ,0) ( ,0)u ( , ) ( , ) T m m m m W x W x W x b W x b  (13) then ' ' 1 1 ' ' 2 2 ( ,0) (0); ( ,0) (0); ( , ) ( ); ( , ) ( ) m m m m m m m m W x Y W x Y W x b Y b W x b Y 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     1u K C (15) where    1 2 3 4C T C C C C and  1 0 1 0 1 0 0 K ( . ) ( . ) ( . ) ( . ) . ( . ) . ( . ) . os( . ) . ( . ) c d sinh bc cosh bc sin bd cos bd ccosh bc csinh bc d c bd d sin bd             (16) where [K1] is the shape function. 3.2. Generalized forces Generalized force vector can be expressed as     1 1 2 2 Q ( ,0) ( ,0) ( , ) ( , ) T ym ym ym ym Q x M x Q x b M 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 và Công nghệ* Số 1-2013 58 3 3 3 2 2 2 2 2 ( , ) ( , ) y y w wQ x y D y x y w wM x y D y x                       (18) The generalized force which are determined to Eqs (18) can be written     ''' 2 ' '' 2 ( , ) ( , ) ymi m m m ymi m m m Q x y D Y k Y M x y D Y k Y         (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     2Q K C (20) where  2K is the generalized stiffness matrix   11 12 13 14 21 22 23 24 2 31 32 33 34 41 42 43 44 K k k k k k k k k D k k k k k k k k             (21) Explicit expressions of the elements kij of the generalized stiffness matrix [K2] are as follows: 2 3 11 12 3 2 13 14 ); ( . 0 ( . ); 0 m m k c k c k k d d k k       3 2 31 3 2 32 3 2 33 3 2 34 . . . ( ( . ) . ( . )) ( ( . ) . ( . )) ( ( . ) . ( . )) ( ( . ) . . . . . . ( . )) m m m m k c cosh bc ck cosh bc k c sinh bc ck sinh bc k d cos bd d k cos bd k d sin bd d k sin bd         (22) 2 2 21 22 2 2 23 24 .0; ( ) 0; ( . ) m m k k k v c k k d vk       2 2 2 2 42 2 2 43 2 2 4 4 4 1 ( ( . ) . ( . )) ( ( . ) . ( . )) ( ( . ) . ( . )) ( ( . ) . ( . . . . . ) . . . . ) m m m m c sinh b c k v sinh b c k c cosh b c k v cosh b c k d sin b d k v sin b d k k d cos b d k v cos b d           By substituting Eq. (15) into Eq. (20), the generalized nodal displacements and nodal forces are related,       12 1Q K K u   Therefore,     Q D u (23) Where      12 1D K K   (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 : 4 2 2 2 2 4 2 2 2 2 m m m m m m m m rc k k a a rd k k a a                                          (25) where /r a b is aspect ratio of plate, mN represents the static critical load of plate for the m mode, and m represents the non- dimensional static critical loading factor of plate for the m mode, which is defined as 2 2/m mN b D  (26) The non-dimensional natural frequency parameter (natural frequency factor) m of plate is defined as  2 2/ /m m a h D    (27) where m is the natural frequency for the m mode of plate. Thông báo Khoa học và Công nghệ* Số 1-2013 59 3.4. Dynamic instability of the plate 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 2 2/cr crN b D  (28) The normalized load parameter is determined as * / 2(1 )d s    (29) The natural frequency of lateral free vibration of a rectangular plate loaded by a uniform in-plane force is defined as * 1m m s    (30) The non-dimensional frequency of excitation parameter is as follows 2 /Λ Ωa h D (31) 3.5. Dynamic instability of thin plates by the dynamic stiffness method Step 1. The motion equation (23) of plate would be:     Q D u (32) Step 2. 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:    * * *Q D u    (33) Step 3. 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 *Ddet 0    . Step 4. Solve dynamic stability equation *Ddet 0    (34) 4. Numerical results and discussions 4.1. Static stability and vibration problems 4.1.1. Problem 1. 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 in- plane forces along its opposite edges (Fig. 3). Nx y xSS SS SS a=b bSS Nx Nx Bu ck lin g in o ne h al f-w av e Buckling in one half-wave (m = 1) P.1(a) (b) Nx Fig. 3. 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. Natural frequency factor S ta tic c rit ic al lo ad in g fa ct or 0 1 2 3 4 5 6 0 2 4 6 8 4 2 Fig. 4. Relation m-m (plate P1, mode m=1). Thông báo Khoa học và Công nghệ* Số 1-2013 60 Natural frequency factor S ta tic c rit ic al lo ad in g fa ct or 0 1 2 3 4 5 6 0 2 4 6 8 6.2499 5 Fig. 5. Relation m-m (plate P1, mode m=2). Natural frequency factor S ta tic c rit ic al lo ad in g fa ct or 0 2 4 6 8 10 12 0 2 4 6 8 10 12 10 11.111 Fig. 6. Relation m-m (plate P1, mode m=3). It is observed from Fig. 4-6 that the lowest static critical loading factor and the free vibration frequency factors are determined 4cr  , 1 2 32; 5; 10     The lowest static critical buckling load 2 24 /crN D b The free vibration frequencies 2 2 1 2( / ) /a D h   ; 2 2 2 5( / ) /a D h   ; 2 2 3 10( / ) /a D h   Table 1. Comparison of cr and m of square plate P1. factor mode m DSM Ref. [2] Ref. [13,14] cr 1 4 4 4 1 2 2 2 2 5 5 5 m 3 10 10 10 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. 4.1.2. Problem 2. This problem considers a thin square plate P3 (a=b) with two edges simply supported and two edges clamped and compressed by uniformly distributed in- plane forces along its opposite edges for the static stability and free vibration frequency (Fig. 7). Nx y x SS a=b b Nx Nx B uc kl in g in o ne h al f-w av e ) (a) (b) SS C C P.3 Nx Fig. 7. Thin square plate P3 (SS-C-SS-C). Natural frequency factor S ta tic c rit ic al lo ad in g fa ct or 0 1 2 3 4 5 6 7 8 9 10 0 2 4 6 8 10 8.6044 2.9332 Fig. 8. Relation m-m (plate P3, mode m=1). Natural frequency factor S ta tic c rit ic al lo ad in g fa ct or 0 1 2 3 4 5 6 7 8 9 10 0 2 4 6 8 10 5.5466 7.6913 Fig. 9. Relation m-m (plate P3, mode m=2). Thông báo Khoa học và Công nghệ* Số 1-2013 61 Natural frequency factor S ta tic c rit ic al lo ad in g fa ct or 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 11.9178 10.3566 Fig. 10. Relation m-m (plate P3, mode m=3). It is observed from Fig. 7-10 that the lowest static critical buckling load factor and the free vibration frequency factors are determined 7.6913cr  ; 1 2.9332  ; 2 5.5466  ; 3 10.3566  The lowest static critical loading 2 27.6913 /crN D b The free vibration frequency 2 21 2.9332( / ) /a D h   ; 2 2 2 5.5466( / ) /a D h   ; 2 2 3 10.3566( / ) /a D h   Table 2. Comparison of cr and m of square plate P3 factor mode m DSM Ref. [2] Ref. [15] cr 2 7.6913 7.701 7.69 1 2.9332 2.935 - 2 5.5466 5.550 - m 3 10.3566 10.36 - 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. 4.2. Dynamic instability problems 4.2.1. Problem 1. 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 in- plane periodic forces along its opposite edges (Fig. 11). SS SS a=b bSSP.1 N =  N +  N cos tsx dcr cr Nx y xSS 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., 0 and 0.6, are considered. Case 1: the static load factor S = 0 0 Unstable Dimensionless excitation frequency:  d D yn am ic lo ad fa ct or :  10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 1.2    s 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 và Công nghệ* Số 1-2013 62 0 Dimensionless excitation frequency:  10 20 30 40 50 60 Principal region of dynamic instability for simply supported plate P.1 d D yn am ic lo ad fa ct or :  0 0.1 0.2 0.3 0.4 0.5 0.6 Unstable     s Fig. 13. Principal instability region for the square plate P1 (case 2, S = 0.6). Table 3. Comparison of principal region of dynamic instability for square plate P1 (case 1, S = 0). Dimensionless excitation frequency  DSM Ref. [3] Ref. [8] Ref. [11] d right left right left right left right left 0 39.478 39.478 39.46 39.46 39.46 39.46 - - 0.2 41.405 37.452 - - - - 41.384 37.433 0.4 43.246 35.311 43.00 35.32 43.16 35.37 43.224 35.292 0.8 46.711 30.579 46.56 30.78 46.54 30.73 - - 1.2 49.936 24.968 49.52 25.06 49.54 24.02 49.911 24.956 Table 4. Comparison of principal region of dynamic instability for square plate P1 (case 2, S = 0.6). Dimensionless excitation frequency  DSM Ref. [3] Ref. [8] d right left right left right left 0 24.968 24.968 25.06 25.06 25.04 25.04 0.16 27.351 22.332 27.43 22.49 27.41 22.48 0.32 29.542 19.340 29.60 19.53 29.58 19.51 0.48 31.582 15.791 31.57 15.91 31.55 15.89 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 3 and Table 4, which show a good agreement. 4.2.2. Problem 2. 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 uniformly distributed in-plane periodic forces along its opposite edges (Fig. 14). Nx y xC C SS SS a = 1.667b bP.4 N =  N +  N cos tsx dcr cr Thông báo Khoa học và Công nghệ* Số 1-2013 63 Fig. 14. Thin rectangular plate P4 (SS-C-SS- C). (mode1,2,3) Normalized frequency parameter:   * d D yn am ic lo ad fa ct or :  0.1 0.2 0.3 0.4 0 0 0.5 1 1.5 2 2.5 m=3 m=2 m=1 Fig.15.Principal instability regions for the rectangular plate P4(modes m=1,2,3) for S = 0.5. 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. 5. 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 dynamic stiffness matrices of thin plates subjected to uniformly distributed static in-plane edge loading and dynamic in- plane edge loading are established. On that 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. 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 [1] Bolotin V.V. 1964. The dynamic stability of elastic system, San Francisco, Holden-Day. [2] Yamaki N., Nagai K.1975. Dynamic stability of rectangular plates under periodic compressive forces, Report No. 288 of the Institute of high speed mechanics, Tohoku University 32 103-127. [3] Hutt J.M., Salam A.E. 1971. Dynamic instability of plates by finite element method, ASCE J. of Eng. Mech. 3 879-899. [4] Takahashi K., Konishi Y. 1988. Dynamic stability of a rectangular plate subjected to distributed in-plane dynamic force, J. of Sound Vib. 123 115-127. [5] Nguyen H., Ostiguy G.L. 1989. Effect of boundary conditions on the dynamic instability and non-linear response of rectangular plates, part I, theory, J. of Sound and Vib. 133 381-400. [6] Guan-Yuan W., Shih Y.S. 2005 Dynamic instability of rectangular plate with an edge crack, Comput. and Struct. 84 1 -10. Thông báo Khoa học và Công nghệ* Số 1-2013 64 [7] Singh J.P., Dey S.S. 1992. Parametric instability of rectangular plates by the energy based finite difference method, Comput. Methods in Appl. Mech. and Eng. 97 1 – 21, North-Holland. [8] Srivastava A.K.L. 2003. Datta P.K.,Sheikh A.H., Dynamic instability of stiffened plates subjected to non-uniform harmonic in-plane edge loading, J. of Sound and Vib. 262 1171-1189. [9] Wang S., Dawe D.J. 2002. Dynamic instability of composite laminated rectangular plates and prismatic plate structures, Comput. methods appl. Mech. and eng. 191 1791–1826. [10] Wu Lanhe, Wang Hongjun, Wang Daobin. 2007. Dynamic stability analysis of FGM plates by the moving least squares differential quadrature method, Composite Struct. 77 383–394. [11] Chakrabarti A, Sheikh A.H. 2006. Dynamic instability of laminated sandwich plates using an efficient finite element model, Thin-Walled Struct. 44 57-68. [12] Adrian G., Radu, Aditi Chattopadhyay. 2002. Dynamic stability analysis of composite plates including delaminations using a higher order theory and transformation matrix approach, International J. of Solids and Struct. 39 1949-1965. [13] Timoshenko S.P., Gere J.M. 1961 Theory of elastic stability. Tokyo: McGraw- hill, Kogakusha. [14] Timoshenko S.P., Young D.H. 1955. Vibration Problems in Engineering, D.Van Nostrand Co., [15] M.L. Gambhir. 2004. Stability analysis and design of structures, Springer Bất ổn định động tấm mỏng bằng phương pháp độ cứng động lực ThS. Huỳnh Quốc Hùng Khoa Xây dựng, trường Đại học Xây dựng Miền Trung Tóm tắt Bất ổn định động tấm mỏng chữ nhật chịu tải trọng điều hòa phân bố đều dọc theo hai biên đối diện trong mặt phẳng tấm được nghiên cứu trong bài báo này. Tác giả trình bày cách thành lập ma trận độ cứng động lực của tấm. Trên cơ sở đó, tác giả sử dụng phương pháp độ cứng động lực để phân tích ổn định tĩnh và bất ổn định động của tấm mỏng. Ranh giới miền chính bất ổn định động của tấm được xác định bằng cách áp dụng phương pháp Bolotin. Kết quả nhận được về tần số dao động tự do, lực tới hạn ổn định tĩnh và miền chính bất ổn định động được so sánh với kết quả của các nghiên cứu trước đây để khẳng định ưu điểm và độ chính xác của phương pháp độ cứng động lực. Từ khóa: Ổn định động; ổn định tĩnh; phương pháp độ cứng động lực; tấm mỏng.