Application of a biparametric perturbation method to large-deflection circular plate problems with a bimodular effect under combined loads

Accepted Manuscript Application of biparametric perturbation method to large-deflection problems of circular plate with bimodular effect under combined loads Xiao-Ting He, Liang Cao, Jun-Yi Sun, Zhou-Lian Zheng

PII: DOI: Reference:

S0022-247X(14)00451-X 10.1016/j.jmaa.2014.05.016 YJMAA 18518

To appear in:

Journal of Mathematical Analysis and Applications

Received date: 18 June 2013

Please cite this article in press as: X.-T. He et al., Application of biparametric perturbation method to large-deflection problems of circular plate with bimodular effect under combined loads, J. Math. Anal. Appl. (2014), http://dx.doi.org/10.1016/j.jmaa.2014.05.016

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Application of biparametric perturbation method to large-deflection problems of circular plate with bimodular effect under combined loads Xiao-Ting He1,2,*, Liang Cao1, Jun-Yi Sun1,2, Zhou-Lian Zheng1,2 1

College of Civil Engineering, Chongqing University, Chongqing 400045, PR China

2

Key Laboratory of New Technology for Construction of Cities in Mountain Area (Chongqing University), Ministry of Education, Chongqing 400030, PR China * Tel: +86-(0)23-65120898; Fax: +86-(0)23-65123511; E-mail: [email protected]

Abstract: The large deflection condition of a bimodular plate may yield a dual nonlinear problem for which the superposition theorem is inapplicable. In this paper, the bimodular Föppl-von Kármán equations of a plate under the combined action of uniformly distributed load and centrally concentrated force are solved by a biparametric perturbation method. The deflection and radial membrane stress were expanded, first, in double power series with respect to the two types of loads. However, the obtained biparametric perturbation solution shows a relatively slow convergence. By introducing a generalized load and its corresponding generalized displacement, next, the solution is expanded in single power series with respect to the generalized displacement parameter, resulting in a better convergence for the solution. A numerical simulation is also adopted to verify the correctness of the biparametric perturbation solution. The introduction of bimodular effect will change the stiffness of the plate to some extent. Especially, when the compressive modulus is greater than the tensile one, the bearing capacity of the plate will be further strengthened. Keywords: Biparametric perturbation method; Bimodulus; Combined loads; von Kármán equations; Generalized displacement

1

1 Introduction The flexible plate-like structures constructed of advanced materials have many applications in engineering practices featuring large deformation and apparently different mechanical responses in the states of tension and compression due to external loads. In theoretical analysis, it is necessary to consider the geometrical nonlinearity (large deformation) and the nonlinearity of materials (different elastic properties in tension and compression) to give a better mechanical characterization. In reality, all materials may exhibit some different elastic properties in tension and compression, but characteristics like this is often neglected due to the complexity of analysis. The materials which show apparently different moduli are known as bimodular materials [1,2], for example, ceramics, graphite, concrete and some composites. During the past decades, many works have presented some useful material models for studying bimodular materials. Among them, Ambartsumyan’s bimodular model [3,4] for isotropic materials has drawn the keenest attentions in engineering community. This model judges different moduli in tension and compression in the light of positive-negative signs of principal stresses, which is important especially for the analysis and design of structures. In structural engineering, an engineer often needs to have a comprehensive understanding for the potential development of cracks which comes mainly from the increase of principal tensile stress in a beam or plate member composing the structure. Except for some fundamental problems, however, acquiring the state of stresses in a structure quite often can only rely upon the resource of FEM analysis following an iterative strategy [5-9]. 2

In addition to the bimodular effect of materials mentioned above, the large deflection problem of thin plates is also an issue to be dealt with. It is well known that the classical Föppl-von Kármán equations consist of two nonlinear high-order partial differential equations in which two types of deformations, bending and tension, are considered. Although various kinds of methods, either analytical approaches or numerical techniques, have been developed to solve the classical von Kármán equations, a valid analytical solution applicable for a convenient structural analysis is still unavailable. Among these analytical methods, the perturbation method based on certain small parameter may realize this aim. Poincaré’s perturbation method, for example, is one of the representative analytical methods capable of solving nonlinear problems in practical application. It provides the solution for an initial or boundary value problem in the form of an asymptotic series with respect to certain parameter. This parameter either appears explicitly or is introduced artificially in the problem. In brief, the typical feature of solving von Kármán equations for thin plate using perturbation method is to expand the solution in the form of ascending powers with respect to a known parameter (load or deflection, for example). Substituting the asymptotic expansion into governing equations and corresponding boundary conditions and next decomposing them, the unknown functions in the solution may be subsequently determined. During the application of perturbation method, the choice of perturbation parameter is a key problem because the correct choice will lead directly to the asymptotic solution with better convergence. Grossly speaking, there are two basic

3

choices for perturbation parameter, one, the load; and the other, the displacement. Vincent [10] firstly adopted the external load as perturbation parameter to solve von Kármán equations; however, the convergence of the obtained solution is relatively poor. Chien [11] utilized the central deflection as a perturbation parameter to solve the same problem and found a better convergence in solution that agrees well with experimental results. Besides load and central deflection, there are several other choices for perturbation parameters, for example, a generalized displacement [12], a linear function of Poisson’s ratio [13], an average angular deflection [14] and so on. Chen and Kuang [15] discussed the differences about choice of parameters and proposed that variation principle may be used for the solution of large deflection problem with a general perturbation parameter. Under a combined load, the perturbation method based on central deflection proposed by Chien [11] will encounter difficulty because the combined loads can not be expanded with respect to central deflection. Moreover, the method using singular load as perturbation parameter proposed by Vincent [10] seems to be inappropriate since the problem we discuss here concerns two types of load, a uniformly distributed load and a concentrated load. Therefore, a so-called biparametric perturbation method is easily associated with the solution for the problem. However, there are relatively few research works in this field. Nowinski and Ismail [16] used a multi-parameters perturbation method to solve the problem of plates. Chien [17] adopted an iterative method to solve the non-linear arc length problem arisen from the design and construction of Yongjiang Railway Bridge in Ningbo, which is the first application of

4

multi-parameters perturbation method in solving beam problems. By simplifying the governing equation, He and Chen [18] obtained a biparametric perturbation solution for the same practical problem. More recently, He et al. [19] further illustrated the application of biparametric perturbation method to beams with gradient under various boundary conditions, in which two small parameters, the loads and the gradient of the bridge, are adopted. Comparing to traditional single-parameter perturbation, biparametric perturbation method has its own advantages and disadvantages. The choice for multi-parameters may reveal comprehensively the nonlinear effects caused by multi-factors. The basic idea by incorporating two perturbation parameters, one for uniformly distributed load and the other for concentrated force, may be adopted for the solution of the problem involved with a combined load, although the associated convergence problem may still need to be resolved. Recently, some analytical works concerning bimodular thin plate have been carried out. The key problem in the analysis of the bimodular plate is how to build the simplified mechanical model concerning tension and compression. The classical Kirchhoff hypothesis is firstly used to judge the existence of the elastic neutral layers of a thin plate in bending with small deflection [20]. Consequently, a series of analytical solutions in rectangular and polar coordinate systems [20,21] are derived, respectively. Based on the known flexural stiffness in small-deflection bending, the bimodular von Kármán equations are established and the single-parameter perturbation method is used to obtain the explicit expressions for stress and deflection [22]. By constructing two material parameters, a general perturbation solution under various

5

edges conditions is derived [23]. All these studies dealt with single load only, however. The same problems with bimodular effect under the combined action of a uniform distributed load and a concentrated force have not been considered. In this paper, a biparametric perturbation method is adopted to solve a bimodular large-deflection plate under the combined loads. The deflection and radial membrane stress were expanded, first, in double power series with respect to the uniformly-distributed loads and concentrated force. In viewing of the slow convergence of the obtained solution, the solution is expanded, next, in single power series with respect to the generalized displacement parameter by introducing the generalized load and its displacement. A numerical simulation is also adopted to verify the correctness of the biparametric perturbation solution. The validity of superposition theorem and the influence of bimodular effect on the stiffness of thin plates are also discussed. 2 Bimodular von Kármán equations under combined loads In this section, a circular plate in large axisymmetric deformation will be studied, in which the plate is subjected to both a uniformly distributed load and a centrally concentrated force. The bimodular von Kármán equations will be established via the modification for the classical equations, incorporated with the bimodular effect of materials. 2.1 Equilibrium equation A bimodular circular plate with thickness t and radius R is subjected to a normal

6

uniform distributed load Q and a centrally concentrated force P, as shown in Fig.1. This causes an axisymmetric deformation for the plate. Let the radial and circumferential coordinates of any point in the plate be r and θ , respectively. Note that due to axisymmetric characteristics, the coordinate θ is not shown in Fig.1. Also, let the deflection of any point be denoted by w(r ) and the radial membrane stress of the plate by σ r , N r = tσ r . The equilibrium equation of a bimodular thin circular plate under the afore-mentioned combined loads is given by

D*

d 1 d dw dw Q P , [ (r )] = N r + r+ dr r dr dr dr 2 2π r

(1)

where D* denotes flexural stiffness with bimodular effect in small-deflection bending and is given by the earlier works [20,21] D* = D + + D − =

E + t13 E −t23 + , 3[1 − ( μ + ) 2 ] 3[1 − ( μ − ) 2 ]

(2)

where, E + ( − ) and μ + ( − ) are the Young's modulus and Poisson's ratio; and a superscript “+(-)” denotes a tensile (compressive) quantity; t1 and t2 are the tensile and compressive thicknesses of the plate, respectively, as shown in Fig.1. They have been determined in the earlier work [20]

t1 =

E − [1 − ( μ + )] E + [1 − ( μ − )] + E − [1 − ( μ + )]

t , t2 = t − t1 .

(3)

It is obvious that the only difference between bimodular equilibrium equation and the classical one is just embedded in the flexural stiffness, which is the same as that in the case of small deflection.

2.2 Consistency equation Ambartsumyan’s bimodulus model [3,4] may be expressed in terms of stress and 7

strain as follows (for the case of two dimensions)

ε1 = a11σ 1 + a12σ 2 , ε 2 = a21σ 1 + a22σ 2 ,

(4)

where ε1 and ε 2 denote strain components in the principal coordinates (1,2), σ 1 and σ 2 denote the corresponding stress components, and elastic compliance

aij (i, j = 1, 2) are defined as follows, according to the sign of the principal stress + °­1/ E (σ 1 > 0) a11 = ® − ¯°1/ E (σ 1 < 0),

½ ° ° ¾ + + ­°− μ / E (σ 1 > 0) ° a21 = ® − − ° ¯°− μ / E (σ 1 < 0). ¿

+ °­1/ E (σ 2 > 0) a22 = ® − ¯°1/ E (σ 2 < 0),

­°− μ + / E + (σ 2 > 0) a12 = ® − − ¯°− μ / E (σ 2 < 0),

(5)

The geometrical relation of strain and displacement in large axisymmetric deformation is given by

dur 1 dw 2 ½ + ( ) dr 2 dr °° ¾, ur ° εθ = °¿ r

εr =

(6)

where ε r and εθ denote the radial unit elongations and circumferential one at the middle plane of the plate, respectively; ur denotes the radial displacement at the middle plane of the plate and w is the deflection as defined above. If we let circumferential membrane stress be σ θ , thus the circumferential membrane force per unit length is Nθ , Nθ = tσ θ , defined similar to the radial components N r = tσ r . Note that in the case of large deflection, any point at the middle plane of the plate is in the state of tension along two perpendicular directions, that is σ r > 0 and σ θ > 0 , which agrees with the positive cases in Eq. (5). The corresponding elastic compliance may be taken as E + and μ + . Therefore, according to the bimodular materials model, the stress-strain relation is as follows 8

1 1 ½ (σ r − μ +σ θ ) = + ( N r − μ + Nθ ) ° + ° E E t ¾. 1 1 + + εθ = + (σ θ − μ σ r ) = + ( Nθ − μ N r ) ° °¿ E E t

εr =

(7)

Considering the in-plane equilibrium relation, i.e.,

Nθ =

d (rN r ) , dr

(8)

the consistency equation may be obtained via Eqs. (6) to (8), namely,

r2

d 2 Nr dN r E + t dw 2 r + 3 + ( ) = 0. dr 2 dr 2 dr

(9)

3 Biparametric perturbation method 3.1 Non-dimensionalization

The von Kármán equations for a bimodular circular plate under the combined loads have been established, as shown in Eqs. (1) and (9). For a simple illustration of the biparametric perturbation technique, a fundamental edge restriction, i.e., a movable clamped edge for the bimodular circular plate of radius R shown in Fig.1, is considered, that is

w=

dw = N r = 0, at r = R . dr

(10)

The boundary conditions at the center are dw and N r are finite at r = 0 . dr

(11)

The following dimensionless quantities are introduced

η=

Nr R2 r2 w QR 4 PR 2 W S q p , = , = , = , = . π E +t 4 R2 t E +t 3 E +t 4

Eqs. (1) and (9) are transformed into the dimensionless forms, respectively, i.e.,

9

(12)

d2 dW 3S (1 + β ) dW 3q (1 + β ) 3 p (1 + β ) (η )= + + 2 dη dη 4K dη 16 K 16 Kη

(13)

and d2 1 dW 2 (η S ) + ( ) = 0, 2 dη 2 dη

(14)

where, E+ − E− E+ + E− , E , E + = (1 + β ) E , E − = (1 − β ) E , = E+ + E− 2 t t (1 + β )T13 (1 − β )T23 3D* T1 = 1 , T2 = 2 , K = + = 3 t t 1 − (μ + )2 1 − ( μ − )2 Et

β=

(15)

where β is the ratio of different moduli, E is an average modulus, T1 and T2 are dimensionless quantities of t1 and t2 , respectively, and K is the dimensionless parameter following D* . The boundary conditions, Eqs. (10) and (11), are also changed as, respectively W=

dW = S = 0, at η = 1 , dη

(16)

and dW and S are finite, at η = 0 . dη

(17)

3.2 Dual load parameters perturbation

If q and p are chosen as two perturbation parameters, the expressions of W = W (q, p,η ) and S = S (q, p,η ) may be written as

°½ ¾ S = S1q + S2 qp + S3 p + S 4 q + S5 q p + S6 q p + S7 qp + S8 p +  , °¿

W = W1q + W2 p + W3 q 3 + W4 q 2 p + W5 qp 2 + W6 p 3 +  , 2

2

4

3

2

2

3

4

(18)

where Wi and Si (i = 1, 2,3,...) are unknown functions of η . Substituting the expressions into the governing equations (13) and (14), and also into the boundary 10

conditions (16) and (17), we may obtain a series of decomposed differential equations and the corresponding boundary conditions used for Wi and Si (i = 1, 2,3,...) . Here the even exponential terms with respect to q and p in the expansion of W and the odd terms in S are not considered because they will be eliminated during the perturbation. Due to the use of two perturbation parameters, however, the process is relatively complicated and is described in detail as follows. Comparing the coefficients of q and p in Eq. (13), we have the differential equations for Wi (i = 1, 2) that dW1 d2 3(1 + β ) ½ (η )= 2 dη dη 16 K °° ¾, dW2 d2 3(1 + β ) ° (η )= dη 2 dη 16 Kη °¿

(19)

which may be solved under the boundary conditions

dWi ½ = 0, at η = 1 ° dη ° ¾ (i = 1, 2) . dWi is finite, at η = 0 ° °¿ dη

Wi =

(20)

Thus we obtain 1 W1 = λ (η − 1) 2 , W2 = λ (1 − η + η lnη ) , 4

(21)

3(1 + β ) . 16 K

(22)

where

λ=

Comparing the coefficients of q 2 , qp and p 2 in Eq. (14), we have the differential equations used for Si (i = 1, 2,3) that

11

½ d2 1 dW (η S1 ) + ( 1 ) 2 = 0 ° 2 dη 2 dη ° 2 dW1 dW2 d °° = 0¾ , (η S 2 ) + 2 dη dη dη ° 2 ° d 1 dW (η S3 ) + ( 2 ) 2 = 0 ° 2 dη 2 dη ¿°

(23)

which may be solved under the boundary conditions Si = 0, at η = 1

½ ¾ (i = 1, 2,3) . Si is finite, at η = 0 ¿

(24)

Using the determined Wi (i = 1, 2) , we obtain 1 2 ½ λ (−3 + 6η − 4η 2 + η 3 ) ° 96 ° 1 2 ° 2 2 S 2 = λ (22 − 27η + 5η + 18η ln η − 6η ln η ) ¾ . 72 ° 1 2 ° 2 S3 = λ (7 − 7η + 6η ln η − 2η ln η ) ° 8 ¿ S1 = −

(25)

Similarly, comparing the coefficients of q 3 , q 2 p, qp 2 and p 3 in Eq. (13), we have the differential equations for Wi (i = 3, 4,5, 6) that ½ ° ° 3(1 + β ) dW4 dW2 dW1 ° (η )= ( S1 )° + S2 4K dη dη dη ° ¾, dW5 3(1 + β ) dW1 dW2 ° (η )= ( S3 ) + S2 4K dη dη dη ° ° dW6 3(1 + β ) dW2 ° (η )= S3 ° 4K dη dη ¿

dW3 3(1 + β ) dW1 d2 (η )= S1 2 4K dη dη dη d2 dη 2 d2 dη 2 d2 dη 2

(26)

which may be solved by the boundary conditions as follows,

dWi ½ = 0, at η = 1 ° dη ° ¾ (i = 3, 4,5, 6) . dWi is finite, at η = 0 ° °¿ dη

Wi =

Using the determined Wi (i = 1, 2) and Si (i = 1, 2,3) , we obtain 12

(27)

1 ½ λ 4 (η −1)2 (73 − 76η + 45η 2 −14η3 + 2η 4 ) ° 17280 ° 1 4 2 3 4 5 2 2 3 λ [−12946 + 44385η − 55800η + 32200η − 8550η + 711η −180η (−45 + 60η − 25η + 3η )lnη]° . W4 = ° 259200 ¾ 1 4 λ [−1419 + 5500η − 7248η 2 + 3744η3 − 577η 4 + 24η 2 (88 −104η +19η 2 )lnη − 24η3 (−24 + 5η) ln2 η] ° W5 = ° 6912 ° 1 4 2 3 2 3 2 3 3 ° λ [−191+ 849η −1134η + 476η − 9η (−63 + 64η)lnη + 234η ln η − 36η ln η] W6 = 648 ¿ W3 = −

(28) Similarly, comparing the coefficients of q 4 , q 3 p, q 2 p 2 , qp 3 and p 4 in Eq. (14), we have the differential equations for Si (i = 4,5, 6, 7,8) that ½ dW1 dW3 dW dW4 dW2 dW3 d2 d2 ( η S ) 0, (η S5 ) + 1 + = + = 0, ° 4 2 2 dη dη dη dη dη dη dη dη ° 2 °° dW1 dW5 dW2 dW4 d + = 0, (η S6 ) + ¾ 2 dη dη dη dη dη ° ° d2 dW1 dW6 dW2 dW5 d2 dW2 dW6 S S + + = + = ( η ) 0, ( η ) 0, ° 7 8 dη 2 dη dη dη dη dη 2 dη dη °¿

(29)

which may be solved following the boundary conditions that, Si = 0, at η = 1

½ ¾ (i = 4,5, 6, 7,8) . Si is finite, at η = 0 ¿

(30)

Using the determined Wi (i = 1, 2,3, 4,5, 6) , we obtain 1 ½ λ5 (−599 +1554η −1778η2 +1260η3 −588η4 +182η5 −34η6 + 3η7 ) ° 483840 ° 1 5 2 3 4 5 6 ° S5 = λ [−1091295 + 2664375η − 2856455η +1854650η − 696927η +134897η −9245η ° 50803200 ° 2 3 4 5 + 420η(−777 +1260η −1260η + 693η −182η +17η )lnη] ° ° 1 S6 = λ5[−2226252 + 5091075η −5046750η2 + 2974250η3 −890496η4 + 98173η5 ° 15552000 ° 2 3 4 2 2 3 2 − 30η(44385 − 69900η + 59900η − 23166η + 2753η )lnη +1800η (−90 +180η −99η +13η )ln η]°¾ . ° 1 S7 = λ5[−16963529 + 35938125η −31330000η2 +15208125η3 − 2852721η4 ° 38880000 ° ° + 30η(−515625 + 796000η −564750η2 +129516η3 )lnη − 450η2 (8800 −14100η + 3861η2 )ln2 η ° + 27000η3 (−40 +11η)ln3 η] ° ° 1 ° S8 = λ5[−64241+122256η −83664η2 + 25649η3 −12η(6792 −9576η + 4667η2 )lnη 124416 ° 2 2 3 3 3 4 ° + 648η (−56 + 65μ)ln η −13536η ln η +1728η ln η] ¿ S4 =

13

(31) Thus, the remaining functions may be solved in a similar manner. It is assumed that the computation is ended here according to the required precision. When η = 0 , the central deflection of bimodular circular plate under the combined action of q and p at the center may be obtained, i.e., 1 73 6473 4 2 473 4 2 191 4 3 Wm = λ q + λ p − λ 4q3 − λ q p− λ qp − λ p . 4 17280 129600 2304 648

(32)

Obviously, let p = 0 or q = 0 , we will have the relations of load versus central deflection under the uniformly-distributed load ( p = 0 ) or concentrated force ( q = 0 ), respectively. For uniformly-distributed load q, the inversion for Eq. (32) when p = 0 yields 3 4K 73 3 q= Wm + Wm , 16 1+ β 360

(33)

which is the same as that in Ref. [22], and for concentrated force p, the inversion for Eq. (32) when q = 0 yields K 3 191 3 p= Wm + Wm . 16 1+ β 3456

(34)

If the ratio q / p is known in advance, which is available in practical cases, q and p may be expressed in terms of single load parameter as follows q =α f , p =γ f ,

(35)

where α and γ are constants and satisfy α / γ = q / p ; f

may be called a

generalized load. Substituting Eq. (35) into Eq. (18), we have the expansions expressed in terms of f °½ ¾. S = ( S1α + S 2αγ + S3γ ) f + ( S 4α + S5α γ + S6α γ + S7αγ + S8γ ) f +  , °¿

W = (W1α + W2γ ) f + (W3α 3 + W4α 2γ + W5αγ 2 + W6γ 3 ) f 3 +  , 2

2

2

4

3

14

2

2

3

4

4

(36)

Let δ be the generalized displacement corresponding to the load parameter f

δ = αν + γΔ ,

(37)

where ν and Δ are the displacements corresponding to uniformly distributed load and concentrated force, respectively. Via Eq. (18), ν may be computed as 1

ν = ³ Wdη 0

1 1 1 4 3 73 4 2 6473 4 2 473 4 3 = λq + λ p − λ q − λ q p− λ qp − λ p 12 4 896 5760 129600 6912

(38)

and due to the fact that, Δ = W (0) = Wm ,

(39)

we may also get Δ , as shown in Eq. (32). Thus, the relation between δ and f may be written as

δ = δ1′ f − δ 3′ f 3 ,

(40)

where 1 2 1 ½ α + αγ + γ 2 ) °° 12 2 ¾. 1 4 73 3 6473 2 2 473 191 4 ° 4 3 δ 3′ = λ ( α + αγ+ α γ + αγ + γ ) °¿ 896 4320 64800 1728 648

δ1′ = λ (

(41)

Substituting Eq. (35) into Eq. (39) or (32), we have the Δ expressed in terms of the generalized displacement

Δ = Δ1′ f − Δ′3 f 3 (42) where 1 ½ Δ1′ = λ ( α + γ ) °° 4 ¾  (43) 73 6473 2 473 191 3 ° 4 3 2 Δ′3 = λ ( α + α γ+ αγ + γ ) °¿ 17280 129600 2304 648 For the convenience of the next computation, some numerical values about δ1′ , δ 3′ ,

15

Δ1′ and Δ′3 are given in Tables 1 and 2. For any given values of

μ + and β of certain bimodular material,

μ − = μ + (1 − β ) /(1 + β ) , the dimensionless T1 and T2 as well as K may be determined via Eq. (15) and μ + / E+ = μ − / E− . Thus, the values of λ is finally obtained. For μ + = 0.3 and β = 1/ 3 (i.e. E + = 2 E − ), the variation curves of δ , Δ versus f for different values of α / γ (i.e. Eqs. (40) and (42)) are plotted in Fig.2. It is easily seen that, under the combined action of q and p, when the values of f becomes larger, δ

and Δ will decrease with the increase of

f , which is

inconsistent with the known phenomenon. It is indicated that the above solution is valid only for small p and q . For greater values of p and q , the convergence of the biparametric perturbation solution will become slow. This maybe results from two reasons. First, the convergence of the dual series with respect to two parameters is generally slower than that of single series. Second, the load but not the displacement is chosen as the perturbation parameter. Therefore, in order to improve the convergence of the solution, the generalized displacement may be chosen as perturbation parameter to solve this problem. 3.3 Generalized displacement perturbation

The generalized displacement, δ = αν + γΔ , is taken as the perturbation parameter. According to the perturbation process presented by Chien [11], the deflection, the radial membrane stress as well as the generalized load are uniformly expanded in single power series with respect to δ

16

f = f1δ + f3δ 3 +  , ½ ° W = W1δ + W3δ 3 +  , ¾ ° S = S 2δ 2 + S 4δ 4 +  , ¿

(44)

where W1 , W3 ,... and S 2 , S 4 ,... are unknown functions of η , and f1 , f3 ,... are undetermined constants. Substituting the expansion (44) into Eqs. (13), (14), (16) and (17), we may obtain a sequence of differential equations and boundary conditions for f1 , f3 ,... , W1 , W3 ,... and S 2 , S 4 ,... . f1 and W1 may be solved via the following differential equation and corresponding boundary conditions, respectively d2 dW1 3(1 + β ) γ (η )= ( + α ) f1 , 2 16 K η dη dη

(45)

and

½ ° ° ¾ 1 dW1 α ³ W1 (η )dη +γ W1 (0) = 1, is finite, at η = 0.° 0 °¿ dη

W1 =

dW1 = 0, at η = 1, dη

(46)

Meanwhile, S 2 may be obtained via the equation and boundary conditions below, that d2 1 dW (η S 2 ) + ( 1 ) 2 = 0, 2 2 dη dη

(47)

and S 2 = 0 at η =1; but is finite at η =0 .

(48)

Similarly, f 3 and W3 may be solved via the differential equation and the associated boundary conditions, respectively, that, dW3 3(1 + β ) γ 3(1 + β ) dW1 d2 (η )= ( + α ) f3 + , S2 2 16 K η 4K dη dη dη

and

17

(49)

½ ° ° ¾ 1 dW3 is finite, at η = 0.° α ³ W3 (η )dη +γ W3 (0) = 0, 0 °¿ dη

W3 =

dW3 = 0, at η = 1, dη

(50)

Subsequently, S4 may be obtained following the equation and boundary conditions of d2 dW1 dW3 (η S 4 ) + = 0, 2 dη dη dη

(51)

S 4 = 0, at η =1; S 4 is finite, at η =0 .

(52)

Thus, the next computation will proceed in a similar manner. However, it should be pointed out that the computation is cumbersome because f1 , f 3 ,... , W1 , W3 ,... and S 2 , S 4 ,... are the expressions related to α and γ . Alternatively, the solutions may be easily obtained by using the results presented in Section 3.2. Considering that Eq. (40) is the inversion of the first one of Eq. (44), we readily have

f1 =

δ′ 1 , f3 = 3 4 . δ1′ δ 1′

(53)

Substituting the first one of Eq. (44) into Eq. (36), W1 , W3 and S 2 , S 4 are easily determined as, respectively °½ ¾, W3 = (W1α + W2γ ) f3 + (W3α + W4α γ + W5αγ + W6γ ) f °¿

W1 = (W1α + W2γ ) f1

3

2

2

3

3 1

(54)

and ½° ¾. S 4 = 2( S1α 2 + S 2αγ + S3γ 2 ) f1 f 3 + ( S 4α 4 + S5α 3γ + S6α 2γ 2 + S7αγ 3 + S8γ 4 ) f14 °¿

S 2 = ( S1α 2 + S 2αγ + S3γ 2 ) f12

(55)

Thus, following the results obtained in Section 3.2, the solutions of Eq. (44) may be obtained in an easier manner and the solution is essentially a biparametric perturbation one.

18

3.4 Generalized load versus central deflection

In engineering practices, the relation of generalized load versus central deflection is of great concern to the analysis and design of the plate. According to the second expression of Eq. (44) as well as Eqs. (53) and (54), the following expression may be obtained Wm = δ1 (α , γ )δ − δ 3 (α , γ )δ 3 ,

(56)

where 3(α + 4γ ) α + 6αγ + 12γ 2

½ ° ° (57) 4 3 2 2 3 4 ¾ λα (3180α + 40664α γ + 193095α γ + 384622αγ + 228900γ ) ° δ 3 (α , γ ) = °¿ 2100(α 2 + 6αγ + 12γ 2 ) 4

δ1 (α , γ ) =

2

Via Eq. (40), i.e. δ = δ1′ f − δ 3′ f 3 , the inversion transformation of Eq. (56) yields

f = f1 (α , γ )Wm + f3 (α , γ )Wm3 (58) where ½ ° ° . 3 2 2 3¾ 2190α + 25892α γ + 106425αγ + 152800γ ° f3 (α , γ ) = °¿ 2025(α + 4γ ) 4

f1 (α , γ ) =

4 λ (α + 4γ )

(59)

When E + = 2 E − and μ + = 0.3 , a value of λ may be obtained. For the

different ratios of α / γ , the curves of the generalized load f versus central deflection Wm are plotted, as shown in Fig. 3. It is easily seen that Wm will increase with the increase of f , even when the values of f becomes larger, indicating that the convergence of the relation of f versus Wm is better than that shown in Fig. 2.

19

4 Comparisons and discussions 4.1 Comparisons with numerical results

In order to verify the correctness of the perturbation solution, the central deflection Wm of a bimodular circular plate subjected to the combined loads will be calculated via the analytical expression obtained and a numerical analysis program, respectively. A bimodular circular plate with thickness t = 100mm and radius R = 1000mm is subjected to a normal uniformly-distributed load q and a centrally concentrated force p. A layer-wise plate theory is adopted to analyze this problem, thus the location of the

unknown neutral layer should be determined, first, in the light of bimodular parameters of materials. The circular plate is made of certain glass fabric material whose tensile and compressive Young’s modulus and Poisson’s ratio are, E + = 30.38GPa ,

E − = 16.17GPa , μ + = 0.35 and μ − = 0.19 , respectively. Hence, μ + / E+ = μ − / E− approximately holds, indicating that the symmetric characteristic of the flexibility matrix is satisfied. Via Eq. (3), the tensile and compressive thicknesses of the plate may be computed and they are t1 = 40mm and t2 = 60mm , respectively. The program ABAQUS 12.0 is used to carry out the numerical simulation. During the computation, the inconsistent element C3D8i with eight nodes is adopted and the geometrical nonlinearity is considered. Due to the axisymmetric characteristics of the structure, a quarter of the plate is considered. The values of the generalized load range from 1.0 to 2.0 with an increment of 0.2, i.e., the cases f = 1.0,1.2,1.4,1.6,1.8, 2.0 are calculated, respectively. For a fixed value of the generalized load, the load values of q 20

and p are taken by ranging one of the two quantities, α or γ , from 0.0 to 1.0 with an increment of 0.1, while keeping a fixed value of 1 for the other. The results are given in a dimensionless form, as shown in Tables 3 and 4. On the other hand, the central deflection may be obtained in an analytical way. Using the known materials parameters mentioned above, β , K as well as λ may be determined and they are ȕ=0.305, K=0.251 and Ȝ=0.975, respectively. Thus, the central deflection for different values of f , α and γ are computed via Eq. (58), as shown in Tables 3 and 4, in which the relative errors between the perturbation solution presented in this paper and the numerical results are also given. For a better understanding of the differences between the analytical and numerical results, the curves of generalized load versus central deflection for different values α or γ are plotted based on the results of Tables 3 and 4, as shown in Fig. 4. It is easily found that the dashed lines are very close to the solid lines (with the maximum relative errors 7.69% while f = 1.0, α = 1 and γ = 0 , and other errors are in the range of tolerable values), showing that the biparametric perturbation solutions presented in this paper are overall valid. 4.2 Discussions

It is well known that the superposition theorem does not hold, for a nonlinear problem. Therefore, it is also interesting to compare the perturbation result, i.e. Eq. (58), with the superposition results obtained by simply summing up Eqs. (33) and (34). The following two cases are considered: for case 1, α = 1, γ = 1 is taken; for case 2, the superposed results of α = 0, γ = 1 and α = 1, γ = 0 is adopted. Besides, the 21

bimodular effect is also considered by taking μ + = 0.3 and β = 1/ 3, 0, −1/ 3 which correspond to the three cases of E + =2 E − , E + =E − and E − =2E + , respectively. Fig. 5 shows the comparison of the results; and it is seen that, (i) In the case of small deflection, the perturbation result agrees well with the superposition results, meaning that the superposition theorem holds in a linear problem. With the increase of the central deflection, however, for the three cases, E + =2 E − , E + =E − and E − =2E + , the results obtained by superposition method are uniformly

larger than those of perturbation method, and the differences will become larger as the central deflection increases further. This may be explained by considering the mechanical responses in a plate. In the case of a single load, the plate undergoes bending deformation, first. When the other load is applied, bending of the plate magnifies the deformation which inevitably causes a more severe tension. On the other hand, the increasing tension will stiffen the bending stiffness of the plate and thus will moderate the deformation. However, the pure superposition method ignores this interaction between bending and tension modes. (ii) Comparing with the case of E + = E − , the external load applied on the plate will increase when E − = 2 E + and will decrease when E + = 2 E − for the same central deflection. It indicates that the introduction of bimodular effect will change the stiffness of the plate to some extent. Especially, when the compressive modulus is greater than the tensile one, the bearing capacity of the plate will be further strengthened.

22

5 Concluding remarks

In this paper, the bimodular Föppl-von Kármán equations were established for a thin circular plate under the combined loads. These equations were solved using the dual loads and single generalized displacement as perturbation parameters, respectively. The following three conclusions can be drawn. (i) Under the combined loads, the biparametric perturbation solution based on the dual loads shows a relatively slow convergence. The convergence of the solution may be further improved by adopting the generalized displacement as a perturbation parameter. (ii) The superposition result obtained by simply summing up the solutions under single load is larger than the biparametric perturbation solution under the combined loads. This is because the former method ignores the interaction between the bending effect and tension one in a plate. (iii) The introduction of bimodular effect will change the stiffness of the flexible plate to some extent. Especially, when the compressive modulus of the material is greater than the tensile one, the bearing capacity of the bimodular plate will be further strengthened. Compared to the traditional single-parameter perturbation method, the biparametric approach apparently has shown the advantage of a diverse choice of parameters. The similar method can be employed to solve large deflection problems of a bimodular plate, even for a single loading case. The relative work is in process. Moreover, the biparametric perturbation method proposed in this paper may be 23

extended in the field of stabilization [24] and buckling [25] of von Kármán plates. Acknowledgements

This work was supported by the Natural Science Foundation of Chongqing (Grant No. cstc2013jcyjA30012) and also by the National Natural Science Foundation of China (Grant No. 51178485). We thank the reviewers for their time and effort, and for helping us improve the manuscript. References [1] R.M. Jones, Apparent flexural modulus and strength of multimodulus materials, J. Compos. Mater. 10 (1976) 342-354. [2] R.M. Jones, Stress-strain relations for materials with different moduli in tension and compression, AIAA J. 15 (1977) 16-23. [3] S.A. Ambartsumyan, A.A. Khachatryan, Basic equations in the theory of elasticity for materials with different stiffness in tension and compression, Inzh. Zhur. MTT, 2 (1966) 44-53. [4] S.A. Ambartsumyan, Elasticity Theory of Different Moduli (R.F. Wu, Y.Z. Zhang, Trans.), China Railway Publishing House, Beijing, 1986. [5] Y.Z. Zhang, Z.F. Wang, Finite element method of elasticity problem with different tension and compression moduli, Chinese J. Comput. Struct. Mech. Appl. 6 (1989) 236-245. [6] Z.M. Ye, T. Chen, W.J. Yao, Progresses in elasticity theory with different modulus in tension and compression and related FEM, Chinese J. Mech. Eng. 26 (2004) 9-14. [7] W.J. Yao, C.H. Zhang, X.F. Jiang, Nonlinear mechanical behavior of combined members with different moduli, Int. J. Nonlinear Sci. Numer. Simul. 7 (2006) 233-238. [8] X.T. He, Z.L. Zheng, J.Y. Sun, et al., Convergence analysis of a finite element method based on different moduli in tension and compression, Int. J. Solids Struct. 46 (2009) 3734-3740. [9] J.Y. Sun, H.Q. Zhu, S.H. Qin, et al., A review on the research of mechanical problems with different moduli in tension and compression, J. Mech. Sci. Technol. 24 (2010) 1845-1854. [10] J.J. Vincent, The bending of a thin circular plate, Phil. Mag. 12 (1931) 185-196. [11] W.Z. Chien, Large deflection of a circular clamped plate under uniform pressure, Chinese J. Phys. 7 (1947) 102-113. [12] H.C. Hu, On the large deflection of a circular plate under combined action of uniformly distributed load and concentrated load at the center, Chinese J. Phys. 10 (1954) 383-394. [13] R. Schmidt, D.A. DaDeppo, A new approach to the analysis of shells, plates and membranes with finite deflection, Int. J. Non-linear. Mech. 9 (1974) 409-419. 24

[14] C. Huang, Large deflection of circular plate under compound load, Appl. Math. Mech. (Engl. Ed) 4 (1983) 791-804. [15] S.L. Chen, J.C. Kuang, The perturbation parameter in the problem of large deflection of clamped circular plates, Appl. Math. Mech. (Engl. Ed) 2 (1981) 137-154. [16] J.L. Nowinski, I.A. Ismail, Application of a multi-parameter perturbation method to elastostatics, Dev. Theor. Appl. Mech. 2 (1965) 35-45. [17] W.Z. Chien, Second order approximation solution of nonlinear large deflection problem of Yongjiang Railway Bridge in Ningbo, Appl. Math. Mech. (Engl. Ed.) 23 (2002) 441-451. [18] X.T. He, S.L. Chen, Biparametric perturbation solution of large deflection problem of cantilever beams, Appl. Math. Mech. (Engl. Ed.) 27 (2006) 453-460. [19] X.T. He, L. Cao, Z.Y. Li, et al., Nonlinear large deflection problems of beams with gradient: A biparametric perturbation method. Appl. Math. Comput. 219 (2013) 7493-7513. [20] X.T. He, Q. Chen, J.Y. Sun, et al., Application of the Kirchhoff hypothesis to bending thin plates with different moduli in tension and compression, J. Mech. Mater. Struct. 5 (2010) 755-769. [21] X.T. He, X.J. Hu, J.Y. Sun, et al., An analytical solution of bending thin plates with different moduli in tension and compression, Struct. Eng. Mech. 36 (2010) 363-380. [22] X.T. He, Q. Chen, J.Y. Sun, et al., Large-deflection axisymmetric deformation of circular clamped plates with different moduli in tension and compression, Int. J. Mech. Sci. 62 (2012) 103-110. [23] X.T. He, J.Y. Sun, Z.X. Wang, et al., General perturbation solution of large-deflection circular plate with different moduli in tension and compression under various edge conditions, Int. J. Non-linear Mech. 55 (2013) 110-119. [24] I. Ryzhkova, Stabilization of von Kármán plate in the presence of thermal effects in a subsonic potential flow of gas, J. Math. Anal. Appl. 294 (2004) 462-481. [25] A. Borisovich, J. Dymkowska, C. Szymczak, Buckling and postcritical behaviour of the elastic infinite plate strip resting on linear elastic foundation, J. Math. Anal. Appl. 307 (2005) 480-495.

25

Table 1

δ1′ , δ 3′ , Δ1′ and Δ′3 for different values of γ when α = 1

γ

δ1′

δ 3′

0.0

0.083 λ

0.1

Δ1′

Δ′3

0.001 λ

4

0.25 λ

0.004 λ

4

0.143 λ

0.004 λ

4

0.35 λ

0.012 λ

4

0.2

0.223 λ

0.011 λ

4

0.45 λ

0.025 λ

4

0.3

0.323 λ

0.025 λ

4

0.55 λ

0.046 λ

4

0.4

0.443 λ

0.049 λ

4

0.65 λ

0.076 λ

4

0.5

0.583 λ

0.087 λ

4

0.75 λ

0.117 λ

4

0.6

0.743 λ

0.145 λ

4

0.85 λ

0.172 λ

4

0.7

0.923 λ

0.227 λ

4

0.95 λ

0.241 λ

4

0.8

1.123 λ

0.339 λ

4

1.05 λ

0.326 λ

4

0.9

1.343 λ

0.490 λ

4

1.15 λ

0.430 λ

4

1.0

1.583 λ

0.686 λ

4

1.25 λ

1.000 λ

4

Table 2

δ1′ , δ 3′ , Δ1′ and Δ′3 for different values of α when γ = 1

α

δ1′

0.0

1.000 λ

0.1

δ 3′

Δ1′

Δ′3

0.295 λ

4

1.000 λ

0.295 λ

4

1.051 λ

0.323 λ

4

1.025 λ

0.316 λ

4

0.2

1.103 λ

0.354 λ

4

1.050 λ

0.338 λ

4

0.3

1.158 λ

0.386 λ

4

1.075 λ

0.361 λ

4

0.4

1.213 λ

0.421 λ

4

1.100 λ

0.385 λ

4

0.5

1.271 λ

0.459 λ

4

1.125 λ

0.410 λ

4

0.6

1.330 λ

0.499 λ

4

1.150 λ

0.436 λ

4

0.7

1.391 λ

0.541 λ

4

1.175 λ

0.464 λ

4

0.8

1.453 λ

0.588 λ

4

1.200 λ

0.493 λ

4

0.9

1.572 λ

0.635 λ

4

1.225 λ

0.523 λ

4

1.0

1.583 λ

0.686 λ

4

1.250 λ

1.000 λ

4

26

Table 3

1.2

1.4

1.6

1.8

2.0

γ and f when α = 1 γ

f 1.0

Wm for different values of

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

This paper

0.240

0.332

0.419

0.503

0.581

0.655

0.725

0.791

0.853

0.912

0.968

Abaqus

0.260

0.352

0.440

0.522

0.599

0.671

0.739

0.803

0.864

0.922

0.977

Error (%)

7.69

5.68

4.77

3.64

3.00

2.38

1.89

1.49

1.27

1.08

0.92

This paper

0.286

0.394

0.495

0.589

0.677

0.759

0.836

0.907

0.975

1.038

1.097

Abaqus

0.306

0.414

0.513

0.605

0.690

0.770

0.844

0.914

0.980

1.043

1.104

Error (%)

6.54

4.83

3.51

2.64

1.88

1.43

0.95

0.77

0.51

0.48

0.63

This paper

0.331

0.453

0.566

0.670

0.766

0.854

0.936

1.012

1.083

1.150

1.213

Abaqus

0.350

0.470

0.580

0.680

0.772

0.858

0.937

1.011

1.082

1.148

1.219

Error (%)

5.42

3.62

2.41

1.47

0.78

0.47

0.11

0.10

0.09

0.17

0.49

This paper

0.376

0.511

0.635

0.747

0.849

0.943

1.029

1.109

1.183

1.253

1.318

Abaqus

0.394

0.526

0.645

0.752

0.851

0.941

1.027

1.108

1.185

1.256

1.324

Error (%)

4.57

2.85

1.55

0.66

0.24

0.21

0.19

0.09

0.17

0.24

0.45

This paper

0.420

0.567

0.699

0.819

0.927

1.025

1.115

1.197

1.274

1.346

1.413

Abaqus

0.435

0.578

0.704

0.823

0.923

1.020

1.110

1.194

1.279

1.352

1.426

Error (%)

3.45

1.90

0.71

0.49

0.43

0.49

0.45

0.25

0.39

0.44

0.91

This paper

0.462

0.620

0.761

0.887

0.999

1.101

1.194

1.279

1.359

1.432

1.502

Abaqus

0.475

0.627

0.767

0.881

0.990

1.092

1.187

1.276

1.362

1.439

1.515

Error (%)

2.74

1.12

0.78

0.68

0.91

0.82

0.59

0.24

0.22

0.49

0.86

27

Table 4

1.2

1.4

1.6

1.8

2.0

α and f when γ = 1 α

f 1.0

Wm for different values of

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

This paper

0.818

0.834

0.850

0.865

0.880

0.896

0.910

0.925

0.940

0.954

0.968

Abaqus

0.844

0.858

0.872

0.886

0.900

0.913

0.926

0.939

0.952

0.964

0.977

Error (%)

3.08

2.79

2.52

2.37

2.22

1.86

1.73

1.49

1.26

1.04

0.92

This paper

0.935

0.953

0.970

0.986

1.003

1.019

1.035

1.051

1.067

1.082

1.097

Abaqus

0.961

0.976

0.991

1.006

1.020

1.035

1.049

1.062

1.076

1.089

1.104

Error (%)

2.71

2.36

2.12

1.99

1.67

1.55

1.33

1.04

0.84

0.64

0.63

This paper

1.041

1.059

1.077

1.095

1.113

1.130

1.147

1.164

1.180

1.197

1.213

Abaqus

1.070

1.086

1.102

1.118

1.133

1.148

1.163

1.177

1.192

1.205

1.219

Error (%)

2.71

2.49

2.27

2.06

1.77

1.57

1.38

1.10

1.01

0.66

0.49

This paper

1.138

1.157

1.176

1.195

1.213

1.231

1.249

1.267

1.284

1.301

1.318

Abaqus

1.168

1.169

1.201

1.217

1.233

1.249

1.264

1.279

1.294

1.309

1.324

Error (%)

2.57

2.53

2.08

1.81

1.62

1.44

1.19

0.94

0.77

0.61

0.45

This paper

1.226

1.246

1.266

1.286

1.305

1.324

1.342

1.360

1.378

1.396

1.413

Abaqus

1.258

1.276

1.293

1.310

1.327

1.343

1.359

1.375

1.390

1.405

1.426

Error (%)

2.54

2.35

2.09

1.83

1.66

1.41

1.25

1.09

0.86

0.64

0.91

This paper

1.308

1.329

1.349

1.369

1.389

1.409

1.428

1.447

1.465

1.484

1.502

Abaqus

1.346

1.364

1.382

1.400

1.416

1.433

1.450

1.466

1.482

1.498

1.515

Error (%)

2.82

2.56

2.39

2.21

1.91

1.67

1.52

1.30

1.15

0.93

0.86

28

Fig. 1 Scheme of a bimodular circular plate under the combined loads

(a) Į=1

(c) Į=1

(b) Ȗ=1

(d) Ȗ=1

Fig. 2 Relations between į, ǻ (or Wm) and f for different values of Į/Ȗ

29

(a) Į=1

(b) Ȗ=1

Fig. 3 Relation between the generalized load f and central deflection Wm.

30

(a) α = 1

(b)

γ =1

Fig. 4 Comparison between perturbation solution and Abaqus result, the solid lines are for Abaqus results and the dashed lines for perturbation solution

31

Fig. 5 Comparison between perturbation result and pure superposition result

32