1、ANALYSIS OF VIBRATION OF A PIPELINE SUPPORTED ON ELASTIC SOIL USING DIFFERENTIAL TRANSFORM METHODMge Balkayaa, Metin O. Kayab*, Ahmet Salameraa Department of Civil Engineering, Istanbul Technical University, 34469, Maslak, Istanbul, Turkeyb Faculty of Aeronautics and Astronautics, Istanbul Technical
2、 University, 34469 Maslak, Istanbul, TurkeyAbstractIn this paper, a simulation method called the Differential Transform Method (DTM) is employed to predict the vibration of an Euler-Bernoulli and Timoshenko beam (pipeline) resting on an elastic soil. The Differential Transform Method is introduced b
3、riefly. DTM can easily be applied to linear or nonlinear problems and reduces the size of computational work. With this method exact solutions may be obtained without any need of cumbersome work and it is a useful tool for analytical and numerical solutions. To make clear and illustrate the features
4、 and capabilities of the presented method, different problems have been solved by using the technique and solutions have been compared with those obtained in the literature. Keywords: Differential Transform Method, DTM, Elastic Soil, Vibration, Beam, Pipeline*Corresponding author. Tel.: +90 212 2853
5、110; Fax: +90 212 2852926E-mail address: kayamitu.edu.tr (Metin O. Kaya)1. IntroductionIt is obvious that the analysis, design and performance of buried structures such as pipelines require an understanding of soil-structure interaction. Soil protects and supports the buried pipelines, thus reduces
6、the risk of possible hazards that can lead to malfunction of the structure. Without utilizing the strength of the surrounding soil, buried pipelines usually cannot resist the loads and displacements to which they are subjected. Therefore, it is important to accurately evaluate soil restraint or load
7、ing on the buried pipelines so as to increase the service life of the structure. In geotechnical engineering, there are numerous studies in which the structural elements as pipelines, shallow foundations and piles are presented as a beam. On the other hand various types of foundation models such as
8、Winkler, Pasternak, Vlasov, etc. have been used in the analysis of structures on elastic foundations. The well known and widely used mechanical model is the one devised by Winkler. According to the Winkler model, the beam-supporting soil is modeled as a series of closely spaced, mutually independent
9、, linear elastic vertical springs which provide resistance in direct proportion to the deflection of the beam. In the Winkler model, the properties of the soil are described only by the parameter k, which represents the stiffness of the vertical springs. However, due to its inability to take into ac
10、count the continuity or cohesion of the soil, the Winkler model is considered as a rather crude approximation of the true mechanical behavior of the soil material. The assumption that there is no interaction between adjacent springs, also results in overlooking the influence of the soil on either si
11、de of the beam. To overcome this weakness, several two-parameter elastic foundation models have been suggested. In these models, while the first parameter represents the stiffness of the vertical spring as in the Winkler model, the second parameter is introduced to account for the coupling effect of
12、 the linear elastic springs 1.As stated before pipelines, shallow foundations and piles may be modelled as a beam. There are different types of beam model. The well knowns are Euler-Bernoulli and Timoshenko beam models. Euler-Bernoulli theory works well for slender beams. On the other hand this theo
13、ry is not applicable for moderately short and thick beams. However, Timoshenko model evaluates the effects of transverse shear deformation and rotary inertia on the dynamic behavior of beams.There are numerous studies on the vibration analysis of a beam resting on an elastic foundation. De Rosa 2 in
14、vestigated the free vibration frequencies of Timoshenko beams on two-parameter elastic foundation, in which the differential equation of motion was deduced by following the geometrical approach. Matsunaga 3 , based on the power series expansions of displacement components, derived a fundamental set
15、of equations of a one-dimensional higher order theory of deep elastic beam-columns resting on elastic foundations by using Hamiltons principle. El-Mously 4 derived explicit formulae for the fundamental natural frequencies for vibration of finite Timoshenko-beams mounted on finite Pasternak-foundatio
16、n by virtue of Rayleighs principle. Chen 5 investigated the vibration of a beam resting on an elastic foundation by using differential quadrature element method (DQEM). Chen 6 developed the DQEM free vibration analysis model of non-prismatic shear deformable beams resting on the elastic foundation.
17、Cokun 7 studied the response of an elastic beam on a two-dimensional tensionless Pasternak foundation that was subjected to a central concentrated harmonic load, in which the governing differential equations of the problem were solved by using the trigonometric-hyperbolic functions and suitable boun
18、dary and contiunity conditions. Chen et al. 8 studied on a mixed method which combined the state space method and the differential quadrature method for bending and free vibration of arbitrarily thick beams resting on a Pasternak elastic foundation. Maheswari et al. 9 used finite difference method f
19、or the solution of governing differential equations of the problem investigating the response of a moving load on an infinite beam resting on a reinforced granular bed on soft soil. Auciello and De Rosa 10 used the differential quadrature method and the integral variational formulation RayleighRitz
20、method for the dynamic analysis of a foundation beam on a two parameter elastic soil in the presence of subtangential follower force. Elfelsoufi and Azrar 11 presented a model for the investigation of buckling, flutter and vibration analyses of beams using the integral equation formulation. And rece
21、ntly Ruta 12 applied Chebyshev series approximation to solve the problem of nonprismatic Timoshenko beam resting on a two-parameter elastic foundation.In this study, the natural frequencies of a pipeline represented by a uniform beam resting on a Winkler and Pasternak soil are investigated by the Di
22、fferential Transform Method (DTM). The Differential Transform Method is a semi analyticalnumerical technique based on the Taylor series expansion method for solving differential equations. It is different from the traditional high order Taylor series method. The Taylor series method is computational
23、ly taken long time for large orders. However, with DTM, doing some simple mathematical operations on differential equations a closed form series solution or an approximate solution can be obtained quickly. This method was first proposed by Zhou 13 in 1986 for solving both linear and nonlinear initia
24、l-value problems of electrical circuits. Later, Chen and Ho 14 developed this method for partial differential equations and Ayaz 15-16 studied two and three dimensional differential transform method of solution of the initial value problem for partial differential equations. Arikolu and zkol 17 exte
25、nded the differential transform method (DTM) to solve the integro-differential equations. Recently, the second author used the DTM method successfully to handle various kinds of rotating beam problems 18-20. In order to show effectiveness of the DTM method, two different problems will be considered.
26、 The first one is Euler-Bernoulli beam resting on Winkler foundation. The second example is Timoshenko beam resting on Pasternak foundation. In the next section, the governing equations and the associated boundary conditions for the above mentioned problems will be given.2. The Equations of Motion a
27、nd Boundary ConditionsConsider Euler-Bernoulli beam resting on Winkler foundation (Figure 1). The equation of motion for this problem is given as follows. (1)42()()0wwEIkxAtwhere k is spring constant; w is deflection (m); is the mass density (kg/m3); A is the cross sectional area (m2); E is the Youn
28、gs Modulus (Pa) and I is the area moment of inertia about the neutral axis (m4). Here x is the horizontal space coordinate measured along the length of the beam and t is any particular instant of time.The associated boundary conditions handled in this paper are given as follows:fixed-fixed :at x = 0
29、, L (2)0wcantilever: at x=0 (3)0xat x=L (4)32wsimply supported-simply supported:at x=0, L (5)02xIn Figure 2, a Timoshenko beam on Pasternak foundation is shown. The governing equations for this problem are given as a system of differential equations. De Rosa 2 modelled the Pasternak foundation in tw
30、o different forms. These forms lead the same differential equation for Euler-Bernoulli beam; on the other hand it is not true for Timoshenko beam which is adequate for stocky beams. Following De Rosa 2, we will call these definitions as Model I and Model II also. More information about these definit
31、ions can be found in De Rosa 2. The first equation is the same for both Model I and Model II.(6)2 20wwkGAAkxtwhere G is the shear modulus.Model I:(7)2 20wEIkGAIkxxtwhere k is the constant of proportionality between bending moments and bending rotations.Model II:(8)2 20wwEIkGAIkxxtxwhere is the const
32、ant of proportionality between bending moments and global wrotations.Two different boundary conditions, i.e. cantilever and clamped-simply supported, are investigated for Timoshenko beam resting on Pasternak foundation:Cantileverat x=0 (9)0w, at x=L (10)xxClamped -simply supportedat x=0 (11)0wat x=L
33、 (12)xAfter completing equations of motion and associated boundary conditions, we will concentrate now about free vibration analysis of the beams resting on elastic foundations.In order to make free vibration analysis of the Euler-Bernoulli beam on the Winkler foundation, let us assume the solution
34、is in the form of a sinusoidal variation of w(x,t) with circular frequency :(13)tiexWtw)(,Substituting equation (13) into equation (1), equations of motion is expressed as follows:(14)42dEIkAxSimilarly, for the free vibration analysis of Timoshenko beam resting on Pasternak foundation, the solution
35、is assumed in the form of(15)(,)()itwxtWxe(16)(,)()ittSubstituting equations (15) and (16) into equations (6) and (7) for Model I; and equations (6) and (8) for Model II, the new form of equations of motion are expressed as follows:(17)2 20wdWkGAAWkxModel I:(18)2 20ddWEIkGAIkxxModel II:(19)2 20wddWd
36、WEIkGAIkxxx3. NondimensionalizationThe non-dimensional parameters for the Euler-beam on the Winkler foundation are defined as, , , (20)LxWEIkL4kAUsing these parameters, the nondimensional form of equation (14 ) can be written as:(21)0)1(24dNondimensional boundary conditions are as follows:fixed-fixe
37、d :at = 0, 1 (22)0dWcantilever: at = 0 (23)dat =1 (24)23Wdsimply supported-simply supported:at = 0, 1 (25)20dFollowing De Rosa 2, the nondimensional parameters of Timoshenko beam on Pasternak foundation are given as follows:(26)24 2, , , , wwkLxWkssLEIIEI1/4222, , , kGALALcrcrEIEIUsing these paramet
38、ers, nondimensional form of Equations 17-19 can be written as follows:(27) 24()0dWModel I:(28) 22420drrrsModel II:(29) 22420wddWrrsrNondimensional boundary conditions are as follows:Cantileverat =0 (30)0W, at =1 (31)ddClamped -simply supportedat =0 (32)0at =1 (33)dW4. Differential Transformation Met
39、hodThe differential transform method (DTM) is a transformation technique based on the Taylor series expansion and is a useful tool to obtain analytical solutions of the differential equations. In this method, certain transformation rules are applied and the governing differential equations and the b
40、oundary conditions of the system are transformed into a set of algebraic equations in terms of the differential transforms of the original functions and the solution of these algebraic equations gives the desired solution of the problem. It is different from high-order Taylor series method because T
41、aylor series method requires symbolic computation is expensive for large orders. The DTM is an iterative procedure to obtain analytic Taylor Series solutions of differential equations. Consider a function which is analytic in a domain D and let represent any xf 0xpoint in D. The function is then rep
42、resented by a power series whose center is located at . The differential transform of the function is given by0x xf(34)01()()!kxdfFwhere is the original function and is the transformed function.f ()FkThe inverse transformation is defined as(35)0()(kkfxxFCombining Eqs.(34) and (35) gives(36)0 0)(!)()
43、k xkkdfxxfConsidering Eq.(36), once more it is noticed that the concept of differential transform is derived from Taylor series expansion. However, the method does not evaluate the derivatives symbolically.In actual applications, the function is expressed by a finite series and Eq.(36) can xfbe writ
44、ten as follows(37)mk xkkdfxxf0 0)(!)()which means that is negligibly small. Here, the 1 0)(!)()(mk xkkdffvalue of depends on the convergence rate of the natural frequencies.Theorems that are frequently used in the transformation of the differential equations and the boundary conditions are introduce
45、d in Table 1 and Table 2, respectively.5. DTM Formulation and Solution ProcedureFirst of all we will derive DTM form of Equation (21) which models Euler-beam on the Winkler foundation. Here we quit using the bar symbol on and instead, we use . WWIf Table 1 is referred the following expression can be
46、 written easily.(38)2()2(3)4()(1)(0kkWkkIf Eq. (38) is arranged, a simple requrrence relation can be obtained as follows:(39)2(1)(4)34)WkWkThe boundary conditions can be written from Table 2 as follows:fixed-fixed :(40)(0)1(41)0kW(42)0()kcantilever: (43)()1W(44)0()0kk(45)0(1)2()ksimply supported-simply supported:(46)()20W(47)0k(48)
