FACTA UNIVERSITATIS Series: Mechanical Engineering Vol. 19, No 2, 2021, pp. 285 - 305 https://doi.org/10.22190/FUME200629040T © 2021 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND Original scientific paper A NEW C0 THIRD-ORDER SHEAR DEFORMATION THEORY FOR THE NONLINEAR FREE VIBRATION ANALYSIS OF STIFFENED FUNCTIONALLY GRADED PLATES Received June 29, 2020 / Accepted October 03, 2020 Corresponding author: Hoang Lan Ton-That Department of Civil Engineering, HCMC University of Architecture, 196 Pasteur, District 3, HCMC, Vietnam. E-mail: lan.tonthathoang@uah.edu.vn Hoang Lan Ton-That Department of Civil Engineering, HCMC University of Architecture, Vietnam Abstract. Nonlinear free vibration of stiffened functionally graded plates is presented by using the finite element method based on the new C0 third-order shear deformation theory. The material properties are assumed to be graded in the thickness direction by a power-law distribution. Based on the Von Karman theory and the third-order shear deformation theory, the nonlinear governing equations of motion are derived from the Hamilton’s principle. An iterative procedure based on the Newton-Raphson method is employed in computing the natural frequencies and mode shape. The comparison between these solutions and the other available ones suggests that this procedure is characterized by accuracy and efficiency. Key Words: Nonlinear Free Vibration, Functionally Graded Material, Stiffened Plate, Third-order Shear Deformation Theory 1. INTRODUCTION The plates with stiffeners are often used in several fields of engineering such as medical, weapon, nuclear reactor construction, aerospace, etc. to improve stiffness of the structures. Many different ways are applied to analyzing plate structures in general as well as stiffened plate structures in particular; they are listed as Rayleigh-Ritz method [1], finite difference method [2, 3], finite element method (FEM) [4-15], constraint method [16, 17], mesh-free method [18-20], semi analytical finite defference method [21, 22], finite strip method [23-25], boundary element method [26, 27], integral transform approach [28], etc. The most important issue of this type of structure is the connection between the plate and the stiffeners. For example, Peng et al [19] used the first order shear deformation theory as well as the element-free Galerkin method to study the compatibility conditions between the plate and the stiffeners when they work together, etc. Evidently numerical methods are essential for calculating stiffened structures, in which the FEM is the most popular because of 286 H.L. TON-THAT its efficiency and stability. Chattopadhyay et al. as well as Holopainen [29, 30] analyzed nonlinear static of composite stiffened plates based on the first order shear deformation theory and the FEM method or used a new finite model for a linear free vibration analysis of stiffened plates, which is based on the nine-node quadrilateral element related to mixed interpolation of tensorial components. Functionally graded materials with two constituents e.g. ceramic and metal from ceramic surface to metal surface are widely applied. In recent years, many surveys have been carried out in the area of functionally graded plates. The thermoelastic deformations and vibration behaviors with exact solutions were given by Vel and Batra [31]. Stress-driven nonlocal elasticity for nonlinear vibration characteristics of carbon/boron-nitride hetero-nanotube subject to magneto-thermal environment was firstly introduced by Sedighi and Malikan in [32]. The nonlinear vibration and static deflection problems of actuated hybrid nanotubes based on the stress-driven nonlocal integral elasticity was studied in [33] by Ouakad et al. On the other hand, Qian et al. [34] also analyzed this kind of structures based on the meshless local Petrov-Galerkin (MLPG) method. Thau and Choi showed the bending and free vibration behaviors of functionally graded plates based on the first-order shear deformation theory. Furthermore, concerning the MLPG method, the high- order shear and normal deformation plate theory was used to analyze thick functionally graded plates by Gilhooly et al. Some papers of Liew reviewed the meshless methods for composite plate/shell structures. A review of Jha involved the listing of studies for this structure. Further, Reddy proposed a general formulation related to the third-order shear deformation plate theory and the finite element model. An isogeometric analysis and a collocation method employing the shear deformation theory were also applied to the analysis of functionally graded plates by Valizadeh et al. or Ferreira et al., Zhang and Zhou, Prakash as well as Singha who also proposed a formulation to study linear and nonlinear behaviors of the functionally graded plates with respect to the physical neutral surface. An efficient three-node finite shell element for linear and nonlinear analyses of composite structures was also given by Marinkovíc et al. [14, 35]. Shi's third-order shear deformation theory with its necessary stability was first used for a functionally graded plate structure analysis in thermal environment by Bui et al. [7]. Besides, the C0 type of this theory was also used in the analysis of functionally graded skew plates by Ton [12]. And now, the C0 type of Shi’s theory is applied to analyzing stiffened functionally graded plate structures. With the third-order shear deformation theory, we recognize that it is widely used because it does not need shear correction factors while it gives accurate transverse shear stresses. But with low-order finite elements such as four-node quadrilateral element, the need of C1 continuous approximation for the displacement fields in the third-order shear deformation theory causes some impediments. To overcome these shortcomings, the third-order shear deformation theory is a revised form which only requires C0 continuity for displacement fields. In the C0 third-order shear deformation theory, two additional variables are joined, and thence the first derivative of transverse displacements is only required, respectively. The body of this paper is organized into four Sections. In Sect.2, finite element formulation based on the C0 new third-order shear deformation theory for stiffened functionally graded plates is presented. Several examples are subsequently presented in Sect.3. The paper ends with some concluding remarks in the last Section. A New C0-TSDT for Nonlinear Free Vibration Analysis of Stiffened Functionally Graded Plates 287 2. FINITE ELEMENT FORMULATION Let us consider a stiffened functionally graded plate with geometry as plotted in Fig. 1a. The bottom and top faces of plate are to be fully metallic and ceramic, respectively. The mid-plane of the plate is xy-plane, while the z-axis is perpendicular to the xy-plane. (a) (b) Fig. 1 (a) The stiffened functionally graded plate and (b) the variation of volume fraction The volume fraction of ceramic (Vc) and metal (Vm) are formulated in Eq. (1) and the variation of volume fraction for several volume fraction coefficients of a functionally graded plate using the power-law distribution is plotted by Fig.1b. 288 H.L. TON-THAT 1 2 n c z V h   = +    1 m c V V= − 0n  (1) where z is the thickness coordinate variable with -h / 2 z h / 2  as well as c, m and n represent the ceramic, metal constituents and the non-negative volume fraction gradient index, respectively. All values of E, ,  and  that vary through the thickness of plate are also formulated as below 1 ( ) ( ) 2 n m c m z E z E E E h   = + − +    (2) 1 ( ) ( ) 2 n m c m z z h   = + − +        (3) 1 ( ) ( ) 2 n m c m z z h   = + − +        (4) 1 ( ) ( ) 2 n m c m z z h   = + − +        (5) According to the new theory of Shi [36], a three-dimensional displacement field ( , , )u v w was given as below 3 3 0 0,2 2 5 4 1 5 ( , , ) ( , ) ( , ) ( , ) 4 43 3 x x u x y z u x y z z x y z z w x y h h     = + − + −         (6) 3 3 0 0,2 2 5 4 1 5 ( , , ) ( , ) ( , ) ( , ) 4 43 3 y y v x y z v x y z z x y z z w x y h h     = + − + −         (7) 0 ( , , ) ( , )w x y z w x y= (8) This three-dimensional displacement field can be expressed in terms of the C0 third- order shear deformation theory and seven unknown variables as follows 3 3 0 2 2 1 5 5 4 ( , , ) ( , ) ( , ) ( , ) 4 43 3 b s x x u x y z u x y z z x y z z x y h h     = + − + −          (9) 3 3 0 2 2 1 5 5 4 ( , , ) ( , ) ( , ) ( , ) 4 43 3 b s y y v x y z v x y z z x y z z x y h h     = + − + −          (10) 0 ( , , ) ( , )w x y z w x y= (11) It can be seen that the present theory is composed of seven unknowns including three axial and transverse displacements, and four rotations due to the bending and shear effects. The strain-displacement relations based on the small strain assumptions can be given as follows A New C0-TSDT for Nonlinear Free Vibration Analysis of Stiffened Functionally Graded Plates 289 3 0, , , , ,2 3 0, , , , ,2 3 0, 0, , , , , , , ,2 1 5 (5 ) ( ) 4 3 1 5 (5 ) ( ) 4 3 1 5 (5 5 ) ( 4 3 s b s b x x x x x x x x x s b s b x y y y y y y y y y y s s b b s s xy y x x y y x x y y x x y y x x yz xz u z z h v z z h u v z z h −  + + + +    −   + + + +        −    = + + + + + + + +                                 , 2 , 2 2 , 2 ) 5 1 5 ( ) 4 4 5 1 5 ( ) 4 4 b b y y x s b s b y y y y y s b s b x x x x x w z h w z h                +     −    + + + +            −     + + + +                   (12) or matrix form ( ) ( ) ( ) ( ) ( )0 1 3 2 3 20 0 0 0 z z z                  = + + +                                (13) with the membrane strains obtained from ( ) ( ) ( ) 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 L NL u w wx x v w x wy y yu v w w y x y x                        = + = +                      +              (14) The bending strains are given by ( ) , , 1 , , , , , , (5 ) 1 (5 ) 4 (5 5 ) s b x x x x s b y y y y s s b b x y y x x y y x  +   = +   + + +           ( ) , , 3 , ,2 , , , , 5 3 s b x x x x s b y y y y s s b b x y y x x y y x h          + −   = +   + + +   (15) And the shear strains are basically written by ( ) , 0 , 5 1 4 4 5 1 4 4 s b y y y s b x x x w w        + +   =    + +    ( )2 2 5 s b y y s b x x h       +−   =   +   (16) The membrane, bending and shear strains can be then expressed as ( ) 4 1 1 L L i i i= = B q ( ) 4 1 1 1 2 NL NL i i i= = B q ( ) 4 1 2 1 i i i= = B q ( ) 4 3 3 1 i i i= = B q (17) 290 H.L. TON-THAT ( ) 4 0 4 1 i i i= = B q ( ) 4 2 5 1 i i i= = B q (18) in which , 1 , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 i x L i i y i y i x N N N N     =        , , 2 , , , , , , 0 0 0 5 0 0 1 0 0 0 0 5 0 4 0 0 0 5 5 i x i x i i y i y i y i x i y i x N N N N N N N N     =        (19) , , 3 , ,2 , , , , 0 0 0 0 0 5 0 0 0 0 0 3 0 0 0 i x i x i i y i y i y i x i y i x N N N N h N N N N     = −        , 4 , 5 1 0 0 0 0 4 4 5 1 0 0 0 0 4 4 i y i i x N N     =         (20) 0 0 0 0 1 0 15 0 0 0 1 0 1 0h   = −      0 0 1 0 0 , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 T NL i i x i y w w x y w w y x N N         =                 (21) where Ni, Ni,x and Ni,y are called the shape function and two derivatives of it in x-direction and y-direction. The global stiffness matrix is computed by L NL G = + +K K K K (22) The element linear stiffness matrix * * , ( ) e T T L e i j i s j d  = + K B D B S D S (23) with 1 2 3( ) ( ) ( ) L T T T i i i i  =   B B B B 4 5( ) ( ) T T i i i  =   S B B (24) And the element nonlinear stiffness matrix ** ** ** , 1 1 2 2 e T T T NL e Li Lj NLi Lj NLi NLj d    = + +     K B D B B D B B D B (25) where   T Li i i =B B S 1 0 T NL NLi i  =  B B * ** * 0 0 s   =     D D D (26) A New C0-TSDT for Nonlinear Free Vibration Analysis of Stiffened Functionally Graded Plates 291 On the other hand, the element geometric stiffness matrix , ( ) e T G e i j d  = K G NG (27) with , , 0 0 0 0 0 0 0 0 0 0 0 0 i x i i y N N   =     G x xy xy y N N N N   =     N (28) Let us consider that the addition of the stiffener is in the x-direction and by transforming three Eqs. (6-8) as follows 3 3 0 2 2 1 5 5 4 ( , , ) ( , ) ( , ) ( , ) 4 43 3 b s st st xst xst u x y z u x y z z x y z z x y h h     = + − + −          (29) ( , , ) 0 st v x y z = (30) 0 ( , , ) ( , ) st st w x y z w x y= (31) Only the plate elements having an edge coinciding with the stiffener are considered; the establishment of formulation is quite similar. The global stiffness matrix for the stiffener is given by st Lst NLst Gst = + +K K K K (32) The correlation of the displacements between the plate and the stiffener are presented in matrix form as 00 0 00 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 st st ss xxst s y bb xxst b y uu e e v ww                              =                              (33) where ( ) / 2 s e h h= + is the eccentricity between the plate and the stiffener, respectively. Besides, the mass matrix of plate element is shown / 2 / 2 ( ) ( ) e e h T T T T e V S h z dV z dz dS −   = =       M N L LN N L L N  (34) with 292 H.L. TON-THAT 3 3 2 2 3 3 2 2 1 5 5 4 1 0 0 4 43 3 1 5 5 4 0 1 0 4 43 3 0 0 1 0 0 z z z z xh h z z z z yh h       − −               = − −                L (35) and exactly the same way for the stiffener element / 2 / 2 ( ) ( ) st est est st h T L T L est st st st st V S h z dV z dz dS −   = =         M N L L N N L L N  (36) with 3 3 2 2 1 5 5 4 1 0 0 4 43 3 0 0 0 0 0 0 0 1 0 0 z z z z xh h       − −           =         L (37) For vibration of the stiffened functionally graded plate, the equation can be described as ( ) ( ) st st + +M M q + K K q = 0 (38) 2 ( ) ( ) st st  + − +   K K M M q = 0 (39) 3. NUMERICAL RESULTS In this section, the numerical solutions for the nonlinear free vibration analysis of stiffened functionally graded plates are presented. Not only the fully simply supported but also the fully clamped boundary conditions are used in this paper. The fully simply supported boundary conditions (SSSS) for this procedure 0 0 0 s b y y v w  = = = = , at 0,x a= and 0 0 0 s b x x u w  = = = = , at 0,y b= (40) and the fully clamped boundary conditions (CCCC) 0 0 0 s b s b x x y y u v w    = = = = = = at 0,x a= and 0,y b= (41) A New C0-TSDT for Nonlinear Free Vibration Analysis of Stiffened Functionally Graded Plates 293 Table 1 Material properties of the plate and the stiffener Material I E = 3x107 Pa, ν = 0.3 and ρ = 2820 kg/m3 Material II E = 211x109 Pa, ν = 0.3 and ρ = 7830 kg/m3 Material III Em = 322.7 x 109 Pa, m = 0.28, m = 2370 kg/m3, Ec = 207.79 x 109 Pa, c = 0.28, c = 8166 kg/m3 3.1. Verification Firstly, the fully simply supported rectangular plate with two stiffeners as depicted in Fig. 2 is studied in order to verify reliability and validity of the proposed method. The material properties are material I as E = 3x107 Pa, ν = 0.3 and ρ = 2820 kg/m3 in Table 1 for both the plate and the stiffener. The first five natural frequencies are given in Table 2 and compared with the solutions of Peng et al. [19] as well as the results from Ansys software. It can be seen that the values presented in this paper have a good agreement with the references. Fig. 2 The geometric properties of the stiffened rectangular plate with two stiffeners perpendicular to each other Next, the other fully simply supported rectangular plate with one stiffener in the middle is considered. The geometric properties of the plate are a = 0.6m, b = 0.41m and h = 0.00633m. 294 H.L. TON-THAT Table 2 The comparison of first five natural frequencies (Hz) of the fully simply supported rectangular with two stiffeners Results Mode 1 2 3 4 5 Ansys 0.0812 0.0849 0.1035 0.1090 0.1292 L. X. Peng et al. [19] 0.0816 0.0856 0.1000 0.1028 0.1311 Present 0.0819 0.0861 0.1055 0.1104 0.1320 Table 3 The comparison of first three natural frequencies (Hz) of the fully simply supported rectangular with one stiffener Results Mode 1 2 3 Mukherjee et al. [37] 257.05 272.10 524.70 Harik et al. [38] 253.59 282.02 513.50 Aksu et al. [2] 254.94 269.46 511.64 Dayi Ou et al. [39] 258.79 273.89 527.29 Present 259.47 283.72 525.30 Furthermore, the geometric properties of the stiffener along edge b of plate are hs = 0.0222m and bs = 0.001277m. The material properties are material II with E = 211x10 9 Pa, ν = 0.3 and ρ = 7830 kg/m3 as Table 1 for both the plate and the stiffener. The first three natural frequencies based on the proposed method are compared with the others related to Mukherjee et al. [37], Harik et al. [38], Aksu et al. [2] and Dayi et al. [39]. From Table 3, it is interesting to note that the obtained numerical solutions match very well with the others. The last example in this section is related to the nonlinear free vibration analysis for a fully simply supported functionally graded Si3N4/SUS304 square plate with a = b = 0.4m and thickness h = 0.005m. The material properties are material III with Em = 322.7 x 10 9 Pa, m = 0.28, m = 2370 kg/m 3, Ec = 207.79 x 10 9 Pa, c = 0.28, c = 8166 kg/m 3. The nonlinear to linear frequency ratios NL/L with n = 2 as given in Table 4 are compared with the results of Shen [40]. Once again, the accuracy and efficiency of the proposed method are proved by the very small errors between the results of two methods. Table 4 The nonlinear to linear frequency ratio NL/L of the fully simply supported square functionally graded plate n = 2 wmax / h Results 0.0 0.2 0.4 0.6 0.8 1 H. S. Shen [40] 1.00 1.021 1.081 1.174 1.293 1.432 Present 1.00 1.020 1.079 1.169 1.278 1.414 A New C0-TSDT for Nonlinear Free Vibration Analysis of Stiffened Functionally Graded Plates 295 3.2. Verification of the nonlinear free vibration of functionally graded plate with one stiffener By adding one stiffener in the middle as illustrated in Fig. 3 for the functionally graded Si3N4/SUS304 square plate as example above with a/h = 10 and 20, the nonlinear to linear frequency ratios NL/L are calculated. Fig. 3 The stiffened functionally graded plate with one stiffener in the middle The correlations of geometry between the plate and the stiffener are introduced as bs = a/30, hs = 5h or bs = a/50, hs = 5h. Two types of boundary condition and six values of n (0, 0.5, 1, 2, 5 and 10) are also used in this example. The numerical results based on this proposed method are given in Tables 5-12 and displayed in Figs. 4-7. We have found out that the nonlinear to linear frequency ratios NL/L decrease with increasing the volume fraction coefficient n. This order does not change when we change ratio a/h or the boundary conditions. Besides, the first four mode shapes for fully simply supported stiffened functionally graded plate with case a/h = 10, bs = a/30 and n = 2 are also depicted in Fig. 8. Table 5 Nonlinear to linear frequency ratio NL/L of (SSSS) square FGM stiffened plate with one stiffener (bs = a/30, hs = 5h and a/h = 10) a/h n (SSSS) wcentral /h 0.2 0.4 0.6 0.8 1.0 10 0 1.0211 1.0491 1.0828 1.1213 1.1543 0.5 1.0197 1.0461 1.0777 1.1139 1.1536 1 1.0191 1.0443 1.0748 1.1095 1.1475 2 1.0184 1.0426 1.0717 1.1048 1.1411 5 1.0178 1.0410 1.0688 1.1005 1.1350 10 1.0176 1.0404 1.0678 1.0988 1.1328 296 H.L. TON-THAT Table 6 Nonlinear to linear frequency ratio NL/L of (SSSS) square FGM stiffened plate with one stiffener (bs = a/30, hs = 5h and a/h = 20) a/h n (SSSS) wcentral /h 0.2 0.4 0.6 0.8 1.0 20 0 1.0210 1.0473 1.0772 1.1107 1.1470 0.5 1.0194 1.0432 1.0706 1.1012 1.1344 1 1.0183 1.0407 1.0667 1.0956 1.1269 2 1.0173 1.0384 1.0629 1.0902 1.1195 5 1.0164 1.0364 1.0594 1.0851 1.1127 10 1.0160 1.0354 1.0578 1.0828 1.1048 Table 7 Nonlinear to linear frequency ratio NL/L of (CCCC) square FGM stiffened plate with one stiffener (bs = a/30, hs = 5h and a/h = 10) a/h n (CCCC) wcentral /h 0.2 0.4 0.6 0.8 1.0 10 0 1.0032 1.0114 1.0242 1.0415 1.0625 0.5 1.0029 1.0106 1.0227 1.0390 1.0590 1 1.0027 1.0100 1.0217 1.0374 1.0566 2 1.0025 1.0095 1.0206 1.0355 1.0539 5 1.0024 1.0089 1.0195 1.0337 1.0511 10 1.0023 1.0087 1.0190 1.0329 1.0501 Table 8 Nonlinear to linear frequency ratio NL/L of (CCCC) square FGM stiffened plate with one stiffener (bs = a/30, hs = 5h and a/h = 20) a/h n (CCCC) wcentral /h 0.2 0.4 0.6 0.8 1.0 20 0 1.0019 1.0066 1.0138 1.0236 1.0358 0.5 1.0013 1.0050 1.0112 1.0197 1.0304 1 1.0009 1.0042 1.0098 1.0176 1.0275 2 1.0007 1.0036 1.0086 1.0157 1.0248 5 1.0005 1.0030 1.0075 1.0140 1.0224 10 1.0003 1.0027 1.0070 1.0132 1.0212 Table 9 Nonlinear to linear frequency ratio NL/L of (SSSS) square FGM stiffened plate with one stiffener (bs = a/50, hs = 5h and a/h = 10) a/h n (SSSS) wcentral /h 0.2 0.4 0.6 0.8 1.0 10 0 1.0213 1.0502 1.0875 1.1288 1.1793 0.5 1.0196 1.0471 1.0822 1.1230 1.1683 1 1.0188 1.0455 1.0792 1.1184 1.1619 2 1.0181 1.0438 1.0761 1.1137 1.1551 5 1.0176 1.0424 1.0734 1.1093 1.1489 10 1.0175 1.0420 1.0725 1.1079 1.1468 A New C0-TSDT for Nonlinear Free Vibration Analysis of Stiffened Functionally Graded Plates 297 Fig. 4 The effects of n on the nonlinear to linear frequency ratio NL/L for case SSSS with a/b = 1, a/h = 10 and bs = a/30, hs = 5h. Fig. 5 The effects of n on the nonlinear to linear frequency ratio NL/L for case SSSS with a/b = 1, a/h = 20 and bs = a/30, hs = 5h. 298 H.L. TON-THAT Fig. 6 The effects of n on the nonlinear to linear frequency ratio NL/L for case CCCC with a/b = 1, a/h = 10 and bs = a/30, hs = 5h. Fig. 7 The effects of n on the nonlinear to linear frequency ratio NL/L for case CCCC with a/b = 1, a/h = 20 and bs = a/30, hs = 5h. A New C0-TSDT for Nonlinear Free Vibration Analysis of Stiffened Functionally Graded Plates 299 Table 10 Nonlinear to linear frequency ratio NL/L of (SSSS) square FGM stiffened plate with one stiffener (bs = a/50, hs = 5h and a/h = 20) a/h n (SSSS) wcentral /h 0.2 0.4 0.6 0.8 1.0 20 0 1.0211 1.0486 1.0819 1.1198 1.1614 0.5 1.0194 1.0449 1.0756 1.1104 1.1486 1 1.0187 1.0430 1.0722 1.1054 1.1416 2 1.0179 1.0411 1.0690 1.1005 1.1348 5 1.0173 1.0395 1.0660 1.0959 1.1285 10 1.0170 1.0388 1.0647 1.0940 1.1258 Mode 1 Mode 2 Mode 3 Mode 4 Fig. 8 The first four mode shapes of the stiffened functionally graded plate with one stiffener Table 11 Nonlinear to linear frequency ratio NL/L of (CCCC) square FGM stiffened plate with one stiffener (bs = a/50, hs = 5h and a/h = 10) a/h n (CCCC) wcentral /h 0.2 0.4 0.6 0.8 1.0 10 0 1.0035 1.0127 1.0272 1.0467 1.0702 0.5 1.0032 1.0118 1.0255 1.0438 1.0661 1 1.0030 1.0113 1.0244 1.0420 1.0634 2 1.0028 1.0107 1.0232 1.0399 1.0604 5 1.0027 1.0101 1.0219 1.0378 1.0573 10 1.0026 1.0099 1.0215 1.0371 1.0563 300 H.L. TON-THAT Table 12 Nonlinear to linear frequency ratio NL/L of (CCCC) square FGM stiffened plate with one stiffener (bs = a/50, hs = 5h and a/h = 20) a/h n (CCCC) wcentral /h 0.2 0.4 0.6 0.8 1.0 20 0 1.0022 1.0078 1.0166 1.0284 1.0431 0.5 1.0016 1.0063 1.0138 1.0242 1.0371 1 1.0013 1.0056 1.0126 1.0221 1.0341 2 1.0011 1.0049 1.0114 1.0202 1.0313 5 1.0009 1.0044 1.0103 1.0184 1.0287 10 1.0008 1.0041 1.0098 1.0176 1.0276 3.3. Nonlinear free vibration of a functionally graded plate with two stiffeners The last example is related to the analysis of functionally graded Si3N4/SUS304 square plate with two stiffeners perpendicular to each other in the middle as illustrated in Fig. 9. Fig. 9 The stiffened functionally graded plate with two stiffeners Table 13 Nonlinear to linear frequency ratio NL/L of (SSSS) square FGM stiffened plate with two stiffeners (bs = a/30, hs = 5h and a/h = 10) a/h n (SSSS) wcentral /h 0.2 0.4 0.6 0.8 1.0 10 0 1.0225 1.0476 1.0765 1.1083 1.1381 0.5 1.0219 1.0459 1.0735 1.1038 1.1362 1 1.0211 1.0450 1.0719 1.1013 1.1328 2 1.0205 1.0440 1.0702 1.0987 1.1291 5 1.0202 1.0432 1.0686 1.0963 1.1257 10 1.0201 1.0429 1.0681 1.0954 1.1245 A New C0-TSDT for Nonlinear Free Vibration Analysis of Stiffened Functionally Graded Plates 301 Table 14 Nonlinear to linear frequency ratio NL/L of (SSSS) square FGM stiffened plate with two stiffeners (bs = a/30, hs = 5h and a/h = 20) a/h n (SSSS) wcentral /h 0.2 0.4 0.6 0.8 1.0 20 0 1.0222 1.0468 1.0736 1.1022 1.1323 0.5 1.0217 1.0456 1.0713 1.0987 1.1276 1 1.0210 1.0447 1.0699 1.0966 1.1246 2 1.0201 1.0438 1.0683 1.0942 1.1214 5 1.0200 1.0429 1.0667 1.0918 1.1181 10 1.0199 1.0425 1.0660 1.0907 1.1166 Table 15 Nonlinear to linear frequency ratio NL/L of (CCCC) square FGM stiffened plate with two stiffeners (bs = a/30, hs = 5h and a/h = 10) a/h n (CCCC) wcentral /h 0.2 0.4 0.6 0.8 1.0 10 0 1.0024 1.0084 1.0180 1.0308 1.0467 0.5 1.0022 1.0079 1.0170 1.0292 1.0444 1 1.0021 1.0076 1.0164 1.0283 1.0430 2 1.0020 1.0073 1.0158 1.0273 1.0415 5 1.0019 1.0070 1.0152 1.0263 1.0401 10 1.0019 1.0069 1.0150 1.0259 1.0395 Table 16 Nonlinear to linear frequency ratio NL/L of (CCCC) square FGM stiffened plate with two stiffeners (bs = a/30, hs = 5h and a/h = 20) a/h n (CCCC) wcentral /h 0.2 0.4 0.6 0.8 1.0 20 0 1.0015 1.0047 1.0097 1.0164 1.0246 0.5 1.0012 1.0040 1.0085 1.0145 1.0221 1 1.0010 1.0036 1.0078 1.0135 1.0207 2 1.0009 1.0033 1.0072 1.0125 1.0193 5 1.0008 1.0030 1.0066 1.0116 1.0180 10 1.0007 1.0028 1.0063 1.0112 1.0174 Table 17 Nonlinear to linear frequency ratio NL/L of (SSSS) square FGM stiffened plate with two stiffeners (bs = a/50, hs = 5h and a/h = 10) a/h n (SSSS) wcentral /h 0.2 0.4 0.6 0.8 1.0 10 0 1.0224 1.0481 1.0793 1.1144 1.1527 0.5 1.0216 1.0462 1.0755 1.1087 1.1448 1 1.0212 1.0459 1.0736 1.1056 1.1405 2 1.0208 1.0448 1.0716 1.1025 1.1361 5 1.0205 1.0439 1.0698 1.0997 1.1320 10 1.0202 1.0436 1.0692 1.0988 1.1306 302 H.L. TON-THAT Table 18 Nonlinear to linear frequency ratio NL/L of (SSSS) square FGM stiffened plate with two stiffeners (bs = a/50, hs = 5h and a/h = 20) a/h n (SSSS) wcentral /h 0.2 0.4 0.6 0.8 1.0 20 0 1.0220 1.0475 1.0761 1.1074 1.1410 0.5 1.0214 1.0460 1.0734 1.1032 1.1350 1 1.0211 1.0451 1.0718 1.1007 1.1315 2 1.0207 1.0442 1.0701 1.0980 1.1278 5 1.0204 1.0433 1.0684 1.0955 1.1242 10 1.0200 1.0429 1.0678 1.0945 1.1227 Table 19 Nonlinear to linear frequency ratio NL/L of (CCCC) square FGM stiffened plate with two stiffeners (bs = a/50, hs = 5h and a/h = 10) a/h n (CCCC) wcentral /h 0.2 0.4 0.6 0.8 1.0 10 0 1.0026 1.0094 1.0201 1.0344 1.0521 0.5 1.0024 1.0087 1.0188 1.0323 1.0491 1 1.0023 1.0084 1.0181 1.0312 1.0473 2 1.0022 1.0080 1.0173 1.0299 1.0454 5 1.0021 1.0076 1.0166 1.0286 1.0436 10 1.0020 1.0075 1.0163 1.0282 1.0296 Table 20 Nonlinear to linear frequency ratio NL/L of (CCCC) square FGM stiffened plate with two stiffeners (bs = a/50, hs = 5h and a/h = 20) a/h n (CCCC) wcentral /h 0.2 0.4 0.6 0.8 1.0 20 0 1.0017 1.0055 1.0114 1.0193 1.0291 0.5 1.0013 1.0046 1.0099 1.0170 1.0258 1 1.0012 1.0042 1.0091 1.0158 1.0241 2 1.0010 1.0039 1.0084 1.0147 1.0225 5 1.0009 1.0035 1.0078 1.0136 1.0210 10 1.0008 1.0033 1.0075 1.0131 1.0203 The parameters to be changed are given as exactly the same as in the previous example. Once again, the numerical results based on this proposed method are given in Tables 13- 20. Furthermore, the first four mode shapes for a fully simply supported stiffened functionally graded plate with case a/h = 10, bs = a/30 and n = 2 are also depicted in Fig. 10. A New C0-TSDT for Nonlinear Free Vibration Analysis of Stiffened Functionally Graded Plates 303 Mode 1 Mode 2 Mode 3 Mode 4 Fig. 10 The first four mode shapes of the functionally graded plate with two stiffeners 4. CONCLUSIONS An efficient numerical method based on the new C0 third-order shear deformation theory with respect to the Shi theory is firstly developed for a nonlinear free vibration analysis of stiffened functionally graded plates. The Shi's third-order shear deformation theory with its necessary stability is then a revised form which only requires C0 continuity for displacement fields. In this C0 third-order shear deformation theory, two additional variables are joined, and thence the first derivative of transverse displacements is only required, respectively. Furthermore, the functionally graded materials with excellent characteristics of ceramic in corrosive resistances combined with the great toughness of metals in absorb energy and plastically deform, lead to outstanding advanced materials that can withstand extreme conditions of reality. This is even more wonderful if they are reinforced by stiffeners. From the above notions, this paper aims to provide mechanical information for this type of structure. In each case of the study with different data, the achieved results are found to agree well with the solutions of other numerical methods. Based on this proposed method, the present numerical solutions show a more stable procedure than others. And its applicability has been clearly shown in the section above. 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