外文翻译对在同一平面上矩形板可变刚的解析

1、Analytical solution nonrectangular plate with in-plane Variable stiffnessTian-chong YU， Guo-jun NIE， Zheng ZHONG， Fu-yun CHU(School of Aerospace Engineering and Applied Mechanics, Tongji University,Shanghai 200092, P.R. China)Abstract: The bending problem of a thin rectangular plate with in-plane va

2、riable stiffness is studied. The basic equation is formulated for the two-opposite-edge simply supported rectangular plate under the distributed loads. The formulation is based on the assumption that the flexural rigidity of the plate varies in the plane following a power form，and Poissons ratio is

3、constantA fourth-order partial differential equation with variable coefficients is derived by assuming a Levy-type form for the transverse displacement. The governing equation can be transformed into a Whittaker equation，and an analytical solution is obtained for a thin rectangular plate subjected t

4、o the distributed loadsThe validity of the present solution is shown by comparing the present results with those of the classical solutionThe influence of in-plane variable stiffness on the deflection and bending moment is studied by numerical examplesThe analytical solution presented here is useful

5、 in the design of rectangular plates with in-plane variable stiffness. Keywords: in-plane variable stiffness ,power form，Levy-type solution，rectangular plateChinese Library Classification 03432010 Mathematics Subject Classification 74B051 IntroductionThe term” variable stiffness” implies that the st

6、iffness parameters vary spatially throughout. The structure1Functionally graded materials (FGMs) are inhomogeneous composites，in which. the mechanical properties vary smoothly with the position to meet the predetermined functional. performanceThe structures composed of the FGMs are of variable stiff

7、nessThere are extensive literatures on the bending，vibration，and fracture of the FGM structures2-9The deformation of a functionally graded beam was studied by the direct approach10 An efficient and simply refined theory was presented for the buckling analysis of functionally graded plates by Thai an

8、d Choi11Jodaei et a112 dealt with the threedimensional analysis of functionally graded annular plates using the statespace based differential quadrature method(SSDQM)Wen and A1iabadi13investigated functionally graded plates under static and dynamic loads by the local integral equation method(LIEM)Th

9、ere are also some works on the FGM shells and cylinders14-17However，most of the studies on the FGMs deal with material stiffness varying along the thickness directionThe studies on the plates with inplane variable stiffness are quite fewShang 18 studied the rectangular plates with bidirectional line

10、ar stiffness with two opposite edges simply supported and the other two edges arbitrarily supported under the distributed loadsYang 19 investigated the structural analysis of the plates with unidirectionally varying flexural rigidity by the Galerkin line methodLiu et a1.20 analyzed the free vibratio

11、n of a Functionally graded isotropic rectangular plate with inplane material in homogeneity using the Levy-type so1utionUymaz et a121 Considered the functionally graded plates with properties Varying in an inplane direction based on a fivedegreeof-freedom shear deformable plate theory with different

12、 boundary conditionsIn this paper，the Levy-type solution22-23is presented for the bending of a thin rectangular plate with inplane variable stiffness under the distributed load2 Basic equationsConsider a thin rectangular plate of length A and with B with in-plane Variable stiffness，as shown in Fig.1

13、. Introduce a Cartesian coordinate system 0XYZ such that 0XA，0YBWe assume. That the flexural rigidity of the plate D=D(X,Y)is a function of X and Y. The governing equation of the plate with inplane Variable stiffness can be obtained asWhere W is the transverse displacement，V is Poissons ratio，and Q

14、is the normal pressure on The plateIt is assumed that the flexural rigidity of the plate varies only along the Y-direction according to the following power form:Where Y and P are two material parameters describing the in homogeneity of D, Do is the flexural rigidity at，Y=0, and Db is the flexural ri

15、gidity at Y=b. In this case, Eq. (1) can be reduced to3 SolutionThe rectangular plate is assumed to be simply supported along two opposite edges parallel to the Y-direction. To solve the governing equation with the prescribed boundary conditions, a generalized Levy-type approach is employed aswhere

16、Ym(y) is an unknown function to be determined.Substituting Eq. (4) into Eq. (3) yields the following differential equation:The solution of the above equation consists of two parts, i.e,the general solution Ymo of the homogeneous differential equation and the particular solution Ymo of the no homogen

17、eous differential equationThus，the solution of Eq.(5) can be expressed as . 3.1 General solution of governing equation (5)To find the solution of Eq. (5), we first consider the following fourth-order homogeneous differential equation with variable coefficients: The above equation can be transformed

18、into a Whittaker equation, and the solution can be expressed aswithfor p1 and p1/vfor p=1，andfor p=1/vWk,g(t) is the Whittaker function，is the exponential integral function, I(t) and Kg(t) are the gth-order modified Bessel functions of the first and second kinds, respectively, and c1, c2, c3, and c4

19、 are the constants to be determined.3.2 Particular solution of governing equation (5)From the solution of the homogeneous differential equation (7), the solution of the nonhomogeneous differential equation (5) can be expressed asWhere cl(t), c2(t), c3(t), and c4(t) satisfy the following equations:fr

20、om which we obtainwhereThen. we havewhere C1, C2,C3, and C4 are the constants to be determined.By substituting Eq. (15) into Eq. (12), the particular solution of Eq. (5) can be written aswhereFrom Eqs.(8) and (16)，the solution of Eq.(5) can be expressed asThe coefficients dl, d2, d3, and d4 are dete

21、rmined from the boundary conditions.In view of Eqs.(4) and (17)，the transverse displacements of the rectangular plate can be expressed as3.3 Boundary conditionsIt is assumed that the two opposite edges parallel to the y-direction are simply supported,and the other two edges have arbitrary boundary c

22、onditions such as free, simply supported, or clamped conditions.The boundary conditions for the remaining two edges(Y=0 and Y=b) are given as follows:The simply supported boundary conditions (S) areThe clamped boundary conditions (C) areThe free boundary condition 8(F) areWe consider the simply supp

23、orted conditions in Eq. (19) as an exampleThus, we haveFrom Eq. (22), dl, d2, d3, and d4 can be determined, and the results are given in Appendix A.4 Results and discussionIn this section, we make numerical studies on the static behavior of a rectangular plate with in-plane variable stiffness under

24、a uniform pressure (a=b=1 m,v=0.3, Do=100 Nm,Db=500 kNm, and q=100 kN/m2). The variation of the bending stiffness is described by Eq. (2). For different parameters p in Eq. (2), we calculate the deflection w and the bending Moments MX and MY along the y-direction (at x=a/2) of a four-edge simply sup

25、ported rectangular plate and a two-opposite-edge simply supported plate with the other two edges clamped, respectively. The results are shown in Figs.2-7, and the following observations can be made:(i) The present results for a homogeneous plate (p=0) are exactly coincident with those obtained from

26、the classical plate theory, which verifies the correctness of the present solution.(ii) If the flexural rigidities Do and Db satisfy Do<Db, the deflection of the plate with the stiffness parameter p >0 is smaller than that of the plate with P <0 (see Figs. 2 and 5). This is because the aver

27、age bending rigidity of the plate with p >0 is larger than that of the plate with p <0. It can also be seen that the maximum deflection of the plate is no longer at the center of the plate when p0.(iii) It can be observed from Figs.2 and 5 that the deflections of the homogeneous plates with th

28、e stiffness Do and Db are the upper and lower limits of the deflections of a plate with variable stiffness from Do to Db.(iv) For different stiffness parameters p, the variations of the bending moments MX and MY with the coordinate y (at x=a/2) shown in Figs.3, 4, 6, and 7 are similar. However, the

29、maximum magnitudes of the bending moments M and My decrease with the increase of P(when p >0), and they increase with the increase of the absolute value of p (when p <0). The maximum bending moments of the plate with different stiffness parameters p do not occur at the same point.5 Conclusions

30、An exact solution of the bending problem is obtained for a thin rectangular plate subjected to the distributed loads by assuming that the flexural rigidity is of the power form, and Poissons ratio is a constant. The validity of the present results is shown by comparing the present results with those

31、 of the classical solution. The influence of the variable parameters on the deflection and bending moments is studied by numerical examples. The obtained solution can be used to assess the validity and accuracy of various approximate theoretical and numerical models of plates with in-plane variable

32、stiffness.References1 Tatting, B. F. and Girdle, Z. Analysis and Design of Variable Stiffness Composite Cylinders, PhD. dissertation, Virginia Polytechnic Institute and State University, 1-200 (1998)2 Cheng, Z. (a. and Batra, R. C. Three-dimensional thermo elastic deformations of a functionally grad

33、ed elliptic plate. Composites: Part B, 31, 97-106 (2000)3 Chakrabortya, A., Gopalakrishnana, S，and Reddy, J. N. A new beam finite element for the analysis of functionally graded materials. International Journal of Mechanical Sciences, 45, 519-539 (2003)4 Dai, K. Y., Liu, G. R., Lim, K. M., Han, X.,

34、and Du, S. Y. A mesh free radial point interpolation method for analysis of functionally graded material (FGM) plates. Computational Mechanics, 34,213-223 (2004)5 Ferreira, A. J. M., Batra, R. C., Roque, C. M. C., Qian, L. F., and Martins, P. A. L. S. Static analysis of functionally graded plates us

35、ing third-order shear deformation theory and a meshes method. Composite Structures, 69, 449-457 (2005)6 Yu, T. and Zhong, Z. Vibration of a simply supported functionally graded piezoelectric rectangular plate. Smart Materials and Strictures, 15, 1404-1412 (2006)7 Shang, E. T. and Zhong, Z. Closed-fo

36、rm solutions of three-dimensional functionally graded plates. Mechanics of Advanced Materials and Structures, 15, 355-363 (2008)8 Zhong, Z. and Cheng, Z. (a. Fracture analysis of a functionally graded strip with arbitrary distributed material properties. International Journal of Solids and Structure

37、s, 45, 3711-3725 (2008)9 Chen, S. P. and Zhong, Z. Three-dimensional elastic solution of a power form functionally graded rectangular plate. Journal of Mecha epics and MEMS, 1(2), 349-358 (2009)10 Birsan, M，Altenbach, H., Sadowski, T., Eremeyev, V. A., and Pietas, D. Deformation analysis of function

38、ally graded beams场the direct approach. Composites: Part B, 43, 1315-1328 (2012)11 Thai, H. T. and Choi, D. H. An efficient and simple refined theory for buckling analysis of functionally graded plates. Applied Mathematical Modeling, 36, 1008-1022 (2012)12 Jodaei, A., Jalal, M., and Yes, M. H. Free v

39、ibration analysis of functionally graded annular plates be state-space based differential quadrature method and comparative modeling be ANN. Composites: Part B, 43, 340-353 (2012)13 Wen, P. H. and Aliabad, M. H. Analysis of functionally graded plates勿mesh less method: a purely analytical formulation

40、. Evgiv,eering Artalysi。with Boundary Elemi arts, 36, 639-650 (2012)14 Liew, K. M., Zhao, X., and Lee, Y. Y. Post buckling responses of functionally graded cylindrical shells under axial compression and thermal loads. Composites: Part B, 43, 1621-1630 (2012)15 Malekzadeh, P., Fiouz, A. R., and Sahra

41、ouian, M. Three-dimensional free vibration of functionally graded truncated conical shells subjected to thermal environment. Internatiov.al Journal of Pressure Vessels aced Piping, 89, 210-221 (2012)16 Sadeghi, H., Baghani, M., and Naghdabadi, R. Strain gradient elasticity solution for functionally

42、graded micro-cylinders. International Journal of Engineering Science, 50, 22-30 (2012)17 Zenkour, A. M. Dynamical bending analysis of functionally graded infinite cylinder with rigid core. Applied Mathematics and Computation, 218, 8997-9006 (2012)18 Shang, X. C. Exact solution on a problem of bendin

43、g of double-direction rectangular elastic plates of variable rigidity (in Chinese). Journal of Lanzhou University (Natural Sciences), 27(2), 24-32(1991)19 Yang, J. The structural analysis of plates with unidirectional varying rigidity on Gale kin line method (in Chinese). Journal of Wuhan, Institute

44、 of Chemical Techs logy, 18(1), 57-60 (1996)20 Liu, D. Y., Wang, C. Y., and Chen, W. Q. Free vibration of FGM plates with in-plane material in homogeneity. Composite Struct,92, 1047-1051 (2010)21 Uymaz, B., Aided, M., and File, S. Vibration analyses of FGM plates with in-plane material in homogeneit

45、y by Ritz method. Composite Structures, 94, 1398-1405 (2012)22 Badeshi, M. and Saudi, A. R. Levy-type solution for buckling analysis of thick functionally graded rectangular plates based on the higher-order shear deformation plate theory. Applied Mathematical Modeling, 34, 365于3673 (2010)23 Thai, H.

46、 T. and Kim, S. E. Levy-type solution for buckling analysis of orthotropic plates based on two variable refined plate theories. Crocoites Structures, 93, 1738-1746 (2011)对在同一平面上矩形板可变刚度的解析天创宇，聂国军，郑钟，傅润珠（学院航天工程与应用力学，同济大学，上海200092，P.R.中国）摘要：对平面可变刚度矩形薄板弯曲问题进行了研究。制订的两个相反边缘的简支撑矩形板的分布荷载下的基本方程。该方案是根据板的抗弯刚度变

47、化的假设，即在平面上，电源形式，和泊松比是恒定的。对一个四阶变系数偏微分方程的推导假设型形式的横向位移。控制方程可以转化为一个维特卡尔方程的解析解，得到矩形薄板进行该解决方案的的分布式载荷该方案有效性通过比较该平面上的偏转和弯曲力矩的可变刚度研究结果。这里提出的数值实例对解析矩形板面内可变刚度的设计是有用的。关键词：平面变刚度，电源形式，征费型的解决方案，矩形板1引言“变刚度”意味着整个结构具有的刚度参数在空间上1 的变化。梯度复合材料（功能梯度材料）在功能上有非均匀复合材料的力学性能，这种平滑变化的位置，以满足预定的功能。梯度材料是由可变刚度组成的。关于弯曲，振动，断裂FGM结构上有广泛的文

48、献资料2-9。由Thai 和Choi提出了一个对功能梯度板的弯曲问题梁变形的直接的高效、简单地、精炼的理论研究方法10。A1 Jodaei12使用基于状态空间微分积分的方法（SSDQM）在功能上对环形板的等级的三维分析处理。Wen和A1iabadi13 用微积分方程法（LIEM）研究了功能梯度装甲板在静态和动态负载下的变形，此外还有一些关于FGM外壳和圆柱体14-17的著作。然而，大多数功能梯度材料的研究是与材料刚度沿变厚度方向变化有关的。对于板在平面上的可变刚度的研究相当少。Shang18研究了矩形板的双向非线性刚度有两个相反得边简支和另外两条边任意支持下的分布式载荷。杨19研究了板的单方向

49、变化分析抗弯刚度的结构。Liu20分析了自由振动A1。功能梯度材料各向同性矩形板在平面材料同质化利维型中解决方案。21等A1 Uymaz。考虑功能梯度板的属性在不同的平面方向基于五度自由剪切变形板理论有不同的边界条件。本文，利维型的解决方案22-23提出了一个薄的矩形弯曲平面分布式负载下可变刚度的钢板。2基本方程考虑一个薄的矩形板的长度A和宽度B的平面变刚度，如图1所示。0-XYZ直角坐标系  0XA，0YB我们认为。板的抗弯刚度D = D（X，Y）是X和Y的在平面变刚度板的控制方程下的函数可以得到其中，W是横向位移，V是泊松比，和Q是正常的压力该板块。据推测，不同的板的抗弯刚度仅沿

50、Y方向，根据以下的电压形式：其中D是Y和P两个均匀材料参数的函数，D0表示弯抗弯刚度，Y = 0，Db是在Y=b的下的弯曲刚性方程。由（1）可以得到3解决方案假设沿平行于Y方向上的两个相对边的矩形板为简支撑。为了满足与指定的的边界条件方程，广义的利维型方法为YM（Y）是一个未知的待定函数。  代入。（4）代入式（3）式可以得到以下差分方程：上述方程的解决方案由两部分组成，即，齐次微分方程的一般解YMO没有齐次微分方程的特解YMO。因此，可以表示为方程（5）的解。3.1一般解决方案的控制方程（5）要找到解决方案的公式。（5），我们首先考虑以下四阶齐变系数微分方程：上面的公式可

51、以转化为一个维特卡尔方程，并将该方程可以表示为同对P1和p1 /v对于p = 1，和对于p = 1/v.Wk，G（t）是惠特克函数，是指数积分函数，I（t）的和Kg（t）的g阶修正贝赛尔函数的第1和第2种，分别为为c1，c2的和c3和c4是待定常数。3.2控制方程的特解（5）由方程（7）的齐次微分方程的解可知，该解决方案可以表示为等式（5）的非齐次微分 在那里，c1（t），c2（t），C3（t），和c4（t）满足下面的公式：从中我们得到那里然后，我们得到那里，C1，C2，C3，和C4是待确定的常数。通过代入。 （15）代入式。 （12），方程的特解。 （5）可以写成那里由式（8）和（

52、16），方程（5）的解可以表示为由边界条件确定的系数d1，D2，D3和d4。在式（4）和（17），的矩形板的横向位移，可以表示为3.3边界条件这是假设的两个相对的边平行于y方向的简支和其他两个边有任意边界条件，如免费，简单的支持，或钳制条件。剩下的两个边（的边界条件为：Y =0和Y= b）的由下式给出：简支的边界条件（S）边界条件（C）自由边界条件8（F）我们认为，支持简单的条件式。（19）作为一个例子。因此，我们有从式（22），D1，D2，D3，和d4可确定，其结果列在附录A中4结果与讨论在本节中，我们进行了数值研究矩形板的平面可变刚度在均匀的压力下（A = B = 1 M，V = 0.3，

53、DO = 100牛米，DB = 500 KNM的静态行为，和q = 100 kN/m2的）。所述的弯曲刚度的变化由方程（2）可知。对于不同的参数p的方程（2），我们计算的偏转和弯曲的力矩MX和MY沿y方向（在x = a / 2）的四边缘简支矩形板和两个相对的边缘与简支板其他两个边固定。其结果示于图2，图7，可以做出以下的观察：（一）目前的结果为均质钢板（P = 0）是完全一致的经典板理论，这验证了本方案的正确性获得。（ii）若弯曲刚性D0和Db满足D0<Db,的板的挠度与刚度参数对P> 0与P <0（参见图2和图5）。这是因为平均弯曲刚性板具有p> 0大于板带p <

54、;0。它也可以看出，板的最大挠度不再是在板的中心，当p0。（），可观察到从图2和图5的均匀的板的挠度与刚度执行和Db的变刚度的板的挠度从做到Db的上限和下限。（）对于不同刚度参数p，弯矩MX和MY与坐标y（在x = a/ 2）的变化示于图3图4，6和7是相似的。然而，弯矩M和我的减少与增加的P（当p> 0）的最大幅值，和它们与的p（当p<0）的绝对值的增加而增加。该板具有不同的刚度参数的最大弯矩P不可发生在同一点上。 5结论  通过分布载荷的假设可以得到对弯曲刚性的电压形式为矩形薄板的弯曲问题的精确解，泊松比是一个常数。目前的研究结果的有效性，通过比较

55、目前的结果与那些以前的解决方案。上的偏转和弯曲力矩的可变参数的影响进行了研究，通过数值例子。将所得到的结果可以用来评估板的平面内可变刚度的各种近似的理论和数值模型的有效性和准确性。参考文献1 Tatting, B. F. and Girdle, Z. Analysis and Design of Variable Stiffness Composite Cylinders, PhD. dissertation, Virginia Polytechnic Institute and State University, 1-200 (1998)2 Cheng, Z. (a. and Batra,

56、R. C. Three-dimensional thermo elastic deformations of a functionally graded elliptic plate. Composites: Part B, 31, 97-106 (2000)3 Chakrabortya, A., Gopalakrishnana, S，and Reddy, J. N. A new beam finite element for the analysis of functionally graded materials. International Journal of Mechanical S

57、ciences, 45, 519-539 (2003)4 Dai, K. Y., Liu, G. R., Lim, K. M., Han, X., and Du, S. Y. A mesh free radial point interpolation method for analysis of functionally graded material (FGM) plates. Computational Mechanics, 34,213-223 (2004)5 Ferreira, A. J. M., Batra, R. C., Roque, C. M. C., Qian, L. F.,

58、 and Martins, P. A. L. S. Static analysis of functionally graded plates using third-order shear deformation theory and a meshes method. Composite Structures, 69, 449-457 (2005)6 Yu, T. and Zhong, Z. Vibration of a simply supported functionally graded piezoelectric rectangular plate. Smart Materials and Strictures, 15, 1404-1412 (2006)7 Shang, E. T. and Zhong, Z. Closed-form solutions of three-dimensional functionally graded plates