Free Vibration Analysis of an Euler Beam of Variable Width on the Winkler Foundation Using Homotopy Perturbation Method

Abstract

Homotopy Perturbation Method (HPM) is employed to investigate the vibration of an Euler beam resting on an elastic foundation. The beam is assumed to have variable stiffness along its length. HPM is an easy-to-use and very efficient technique for the solution of linear or nonlinear problems. HPM produces analytical approximate expression which is continuous in the solution domain. This work assures that HPM is a promising method for the vibration analysis of the variable stiffness Euler beams on elastic foundation. Different case problems have been solved by using the technique, and solutions have been compared with those available in the literature.

1. Introduction

In geotechnical engineering, it is widely seen in application that pipelines, shallow foundations, and piles are modeled as a beam in the analysis procedure. There are also various types of foundation models such as Winkler, Pasternak, and Vlasov that have been used in the analysis of structures on elastic foundations.

The most frequently used foundation model in the analysis of beam on elastic foundation problems is the Winkler foundation model. In the Winkler model, the soil is modeled as uniformly distributed, mutually independent, and linear elastic vertical springs which produce distributed reactions in the direction of the deflection of the beam. The winkler model requires a single parameter k, representing the soil properties. However, since the model does not take into account either continuity or cohesion of the soil, it may be considered as a rather crude representation of the soil.

As stated before pipelines, shallow foundations and piles can be modeled as a beam. Hence, produced results may find various application areas. There are also different beam types in theory. The well-known is the Euler-Bernoulli beam which is suitable for slender beams. For moderately short and thick beams, the Timoshenko beam model has to be used in the analysis.

Vibration of a constant stiffness Euler beam on elastic foundation was studied previously by Balkaya et al. [1] and Ozturk and Coskun [2]. Balkaya et al. [1] used Differential Transform Method while Ozturk and Coskun [2] used HPM in their studies. Avramidis and Morfidis [3] studied bending of beams on three-parameter elastic foundation. De Rosa [4] analyzed free vibration of Timoshenko beams on two-parameter elastic foundation. Matsunaga [5] studied vibration and buckling of deep beam columns on two-parameter elastic foundations. El-Mously [6] determined fundamental frequencies of the Timoshenko beams mounted on the Pasternak foundation. Chen [7, 8] analyzed vibration of beam resting on an elastic foundation by the differential quadrature element method (DQEM). Coşkun [9] investigated the response of a finite beam on a tensionless Pasternak foundation subjected to a harmonic load. Chen et al. [10] developed a mixed method for bending and free vibration of beams resting on a Pasternak elastic foundation. Maheshwari et al. [11] studied the response of beams on a tensionless extensible geosynthetic-reinforced earth bed subjected to moving loads. Auciello and De Rosa [12] developed different approaches to the dynamic analysis of beams on soils subjected to subtangential forces.

2. The Equations of Motion and Boundary Conditions

An Euler beam resting on the Winkler foundation shown in Figure 1 is considered in this study. The equation of motion for this problem is given as follows: 𝜕²𝜕𝑥²(𝑥)𝜕𝐸𝐼²𝑤𝜕𝑥²(𝑥)(𝑥)𝜕+𝑘𝑤+𝜌𝐴²𝑤𝜕𝑡²=0, (1) where 𝑘 is the spring constant, 𝑤 is deflection, 𝜌 is the mass density, 𝐴 is the cross-sectional area, EI is the beam stiffness, and 𝐼 is the area moment of inertia about the neutral axis. The deflection is a function of both space and time, that is, 𝑤=𝑤(𝑥,𝑡) in which space variable 𝑥 is measured along the length of the beam and 𝑡 represents any particular instant of time.

Figure 1 

Representation of a beam on the Winkler foundation.

Due to the support conditions at both ends of the beam, different conditions have to be imposed to the obtained solution to determine unknowns included in final approximation produced by HPM. These conditions are given as follows.(a) For clamped-clamped beam the end conditions are 𝑤=𝜕𝑤𝜕𝑥=0at𝑥=0,𝐿. (2)(b) For cantilever beam (clamped-free) the end conditions are 𝑤=𝜕𝑤𝜕𝜕𝑥=0at𝑥=0,₂𝑤𝜕𝑥²=𝜕³𝑤𝜕𝑥³=0at𝑥=𝐿. (3a)(3b)(c) For simply supported beam (pinned-pinned) the end conditions are 𝜕𝑤=²𝑤𝜕𝑥²=0at𝑥=0,𝐿. (4)(d) For clamped-simply supported (pinned) beam the end conditions are𝑤=𝜕𝑤𝜕𝜕𝑥=0at𝑥=0,𝑤=²𝑤𝜕𝑥²=0at𝑥=𝐿. (5a)(5b)

Now, free vibration analysis of the beams with variable stiffness resting on elastic foundations will be formulated.

A solution is assumed as the following form to formulate the analysis of the presented problem by the separation of variables 𝑤(𝑥,𝑡)=𝑊(𝑥)𝑒^𝑖𝜔𝑡, (6) where ω is the circular frequency for the vibration. Substituting (6) into (1), equations of motion become as follows: 𝑑²𝑑𝑥²(𝑥)𝑑𝐸𝐼²𝑊𝑑𝑥²+𝑘𝑊=𝜌𝐴𝜔²𝑊. (7)

This equation can be rearranged as 𝜕⁴𝑤𝜕𝑥⁴(𝑥)+2𝐸𝐼^′(𝑥)𝜕𝐸𝐼³𝑤𝜕𝑥³+𝐸𝐼(𝑥)^′′(𝑥)𝜕𝐸𝐼²𝑤𝜕𝑥²+𝑘(𝑥)(𝑥)𝑤+(𝑥)𝐸𝐼𝜌𝐴(𝑥)𝜕𝐸𝐼²𝑤𝜕𝑡²=0, (8) where (⋅)^′ denotes total derivative with respect to 𝑥. The governing equation is now rewritten in a nondimensional form. This procedure is provided from [1] in which a constant stiffness beam was analyzed. The notation is maintained in this study for the comparison purposes. The nondimensional parameters for the Eulerbeam on the Winkler foundation are defined as [1] 𝑥𝑥=𝐿,𝑊𝑊=𝐿,𝜆=𝑘𝐿⁴,𝐸𝐼𝜔=𝜔𝜌𝐴𝑘. (9)

Using these parameters, nondimensional form of the equations and formulation procedures are explained in the following sections.

3. Homotopy Perturbation Method

In recent years, efforts towards application of analytical approximate solution techniques for nonlinear problems have increased. In these studies, He’s Homotopy Perturbation Method [13–18] can be considered as one of the most promising methods for nonlinear problems. The HPM provides an analytical approximate expression as the solution for the problems which are continuous in the solution domain. The technique is applied to an equation of the form 𝐿(𝑢)+𝑁(𝑢)=𝑓(𝑟),𝑟∈Ω with boundary conditions 𝐵(𝑢,𝜕𝑢/𝜕𝑛)=0, 𝑟∈Γ, where 𝐿 is a linear operator, N is a nonlinear operator, B is a boundary operator, Γ is the boundary of the domain '𝛺, and f(r) is a known analytic function. HPM defines a homotopy as 𝑣(𝑟,𝑝)=Ω×[0,1]→𝑅 which satisfies the following inequalities: 𝐻𝑣,𝑝=1−𝑝𝐿(𝑣)−𝐿𝑢₀+𝑝𝐿(𝑣)+𝑁(𝑣)−𝑓(𝑟)=0 (10)

or 𝐻𝑣,𝑝=𝐿(𝑣)−𝐿𝑢₀+𝑝𝐿𝑢₀+𝑝𝑁(𝑣)−𝑓(𝑟)=0, (11) where 𝑟∈Ω and 𝑝∈[0,1] are imbedding parameters, 𝑢₀ is an initial approximation which satisifies the boundary conditions. From (10)-(11), we have 𝐻(𝑣,0)=𝐿(𝑣)−𝐿𝑢₀𝐻()(𝑣)(𝑣)(𝑟)=0,𝑣,1=𝐿+𝑁−𝑓=0. (12)

The changing process of 𝑝 from zero to unity is that of 𝑣(𝑟,𝑝) from 𝑢₀ to 𝑢(𝑟). In topology, this deformation 𝐿(𝑣)−𝐿(𝑢₀) and 𝐿(𝑣)+𝑁(𝑣)−𝑓(𝑟) are called homotopic. The method expresses the solution of (10)-(11) as a power series in 𝑝 as follows: 𝑣=𝑣₀+𝑝𝑣₁+𝑝²𝑣₂+𝑝³𝑣₃+⋯. (13)

The approximate solution of 𝐿(𝑢)+𝑁(𝑢)=𝑓(𝑟), 𝑟∈Ω can be obtained as 𝑢=lim_𝑝→1𝑣=𝑣₀+𝑣₁+𝑣₂+⋯. (14)

The convergence of the series in (14) has been proven in [13–18].

4. HPM Formulation and Solution Procedure for Presented Problem

In this study, a variable stiffness is assumed for the beam considered. The variation is applied by changing the width of the beam along its length. Two different variations in stiffness are assumed. These are linear and exponential variations.

In the linear variation, beam width varies in a linear manner along the beam’s length which is shown in Figure 2.

Figure 2 

Beam with a linearly varying width.

The variable width is formulated as 𝑏(𝑥)=𝑏₀(1−𝛼𝑥), (15) where the dimension of α is [1/𝐿]. For the exponential variation, beam width varies exponentially along the beam’s length as shown in Figure 3.

Figure 3 

Beam with an exponentially varying width.

Variable width of the beam in this case is 𝑏(𝑥)=𝑏₀𝑒^−𝛼𝑥, (16) where α has the same dimension as in linear case. By the use of variable width, both cross-sectional area and moment of inertia are variable, 𝐴(𝑥)(𝑥)𝐼(𝑥)(𝑥)ℎ=𝑏ℎ,=𝑏³.12 (17)

In the following sections, formulation procedures are given separately for each case.

4.1. Formulation for Linear Variation

Employing (15) into (17), variable cross-sectional areas and variable stiffness become 𝐴(𝑥)=𝑏₀ℎ(1−𝛼𝑥)=𝐴₀(),(𝑥)1−𝛼𝑥𝐸𝐼=𝐸𝑏₀ℎ³()121−𝛼𝑥=𝐸𝐼₀(1−𝛼𝑥), (18) where 𝐴₀ and 𝐼₀ are the cross-sectional area and moment of inertia of the section at the origin, respectively. Inserting (18) into (8) 𝜕⁴𝑤𝜕𝑥⁴−2𝛼𝐸𝐼₀𝐸𝐼₀()𝜕1−𝛼𝑥³𝑤𝜕𝑥³+𝑘(𝑥)𝐸𝐼₀()𝑤+1−𝛼𝑥𝜌𝐴₀𝐸𝐼₀𝜕²𝑤𝜕𝑡²=0. (19)

This equation can be rewritten as 𝜕⁴𝑤𝜕𝑥⁴1−2𝛼𝜕1−𝛼𝑥³𝑤𝜕𝑥³+𝑘(𝑥)𝐸𝐼₀1𝑤+1−𝛼𝑥𝜌𝐴₀𝐸𝐼₀𝜕²𝑤𝜕𝑡²=0. (20)

Employing (6) 𝑊^𝑖𝑣−2𝛼𝜉(𝑥)𝑊^′′′+𝑘(𝑥)𝐸𝐼₀𝜉(𝑥)𝑊−𝜌𝐴₀𝐸𝐼₀𝜔²𝑊=0. (21)

Equation (21) can be made nondimensional in view of (9) as follows: 𝑊^𝑖𝑣−2𝛼𝜉(𝑥)𝑊^′′′+𝜆𝜉(𝑥)−𝜔²𝑊=0, (22) where 𝑥𝑥=𝐿,𝑊𝑊=𝐿,𝜆=𝑘𝐿⁴𝐸𝐼₀,𝜔=𝜔𝜌𝐴₀𝑘,𝜉(𝑥)=11−𝛼𝑥,𝛼=𝛼𝐿. (23)

By the application of HPM, the following iteration algorithm is obtained:𝑊₀^𝑖𝑣−𝑢₀^𝑖𝑣=0,𝑊₁^𝑖𝑣+𝑢₀^𝑖𝑣−2𝛼𝜉(𝑥)𝑊₀^′′′+𝜆𝜉(𝑥)−𝜔²𝑊₀=0,𝑊_𝑛^𝑖𝑣−2𝛼𝜉(𝑥)𝑊^′′′_𝑛−1+𝜆𝜉(𝑥)−𝜔²𝑊_𝑛−1=0,𝑛≥2. (24a)(24b)(24c)

4.2. Formulation for Exponential Variation

Employing (16) into (17), variable cross-sectional areas and variable stiffness become 𝐴(𝑥)=𝑏₀ℎ𝑒^−𝛼𝑥=𝐴₀𝑒^−𝛼𝑥,(𝑥)ℎ𝐸𝐼=𝐸³𝑏12⁰𝑒^−𝛼𝑥=𝐸𝐼₀𝑒^−𝛼𝑥. (25)

Inserting (25) into (8) 𝜕⁴𝑤𝜕𝑥⁴𝜕−2𝛼³𝑤𝜕𝑥³+𝛼²𝜕²𝑤𝜕𝑥²+𝑘(𝑥)𝐸𝐼₀𝑒^−𝛼𝑥𝑤+𝜌𝐴₀𝐸𝐼₀𝜕²𝑤𝜕𝑡²=0. (26)

Employing (6) 𝑊^𝑖𝑣−2𝛼𝑊^′′′+𝛼²𝑊^′′+𝑘(𝑥)𝐸𝐼₀𝑒^𝛼𝑥𝑊−𝜌𝐴₀𝐸𝐼₀𝜔²𝑊=0. (27)

Equation (21) can be made nondimensional in view of (9) as follows: 𝑊^𝑖𝑣−2𝛼𝑊^′′′+𝛼²𝑊^′′+𝜆𝑒^𝛼𝑥−𝜔²𝑊=0. (28)

By the application of HPM, the following iteration algorithm is obtained:𝑊₀^𝑖𝑣−𝑢₀^𝑖𝑣=0,𝑊₁^𝑖𝑣+𝑢₀^𝑖𝑣−2𝛼𝑊₀^′′′+𝛼²𝑊₀^′′+𝜆𝑒^𝛼𝑥−𝜔²𝑊₀=0,𝑊_𝑛^𝑖𝑣−2𝛼𝑊^′′′_𝑛−1+𝛼²𝑊^′′_𝑛−1+𝜆𝑒^𝛼𝑥−𝜔²𝑊_𝑛−1=0,𝑛≥2. (29a)(29b)(29c)

4.3. Solution Procedure

An initial approximation may be chosen as a cubic polynomial with four unknown coefficients. There exist four boundary conditions, that is, two at each end of the column, in the presented problem. Hence, an initial approximation of the following choice may be employed: 𝑊₀=𝐴𝑥³+𝐵𝑥²+𝐶𝑥+𝐷. (30)

In the computations, twenty iterations are conducted and four boundary conditions for each case are rewritten by using the final approximation of iteration. Each boundary condition produces an equation containing four unknowns spread from the initial approximation. These nondimensional boundary conditions are as follows.

Clamped-clamped beam: 𝑑𝑊=𝑊𝑑𝑥=0at𝑥=0,1. (31)

Clamped-free (cantilever) beam:𝑑𝑊=𝑊𝑑𝑥=0at𝑑𝑥=0,₂𝑊𝑑𝑥²=𝑑³𝑊𝑑𝑥³=0at𝑥=1. (32a)(32b)

Pinned-pinned (simply supported) beam: 𝑑𝑊=²𝑊𝑑𝑥²=0at𝑥=0,1. (33)

Clamped-pinned beam:𝑑𝑊=𝑊𝑑𝑥=0at𝑥=0,𝑑𝑊=²𝑊𝑑𝑥²=0at𝑥=1. (34a)(34b)

Hence, four equations in four unknowns may be written with respect to the boundary conditions of the problem. These equations can be represented in matrix form as follows: [𝑀(𝜔)]{𝐴}={0}, (35) where {𝐴}=⟨^{𝐴𝐵𝐶𝐷}⟩^𝑇. For a nontrivial solution, determinant of coefficient matrix must be zero. Determinant of coefficient matrix yields a characteristic equation in terms of 𝜔. Positive real roots of this equation are the normalized free vibration frequencies for the case considered.

5. Numerical Results

5.1. Constant Stiffness Case

As the first example, Euler beam of constant stiffness (i.e., 𝐸𝐼 is constant) with different boundary conditions is investigated. For the sake of comparison, all the values are set to unity such as 𝐼=𝐸=𝐴=𝜌=1; hence, 𝜆=1, according to previous studies [7]. Both algorithms given for linear and exponential variations lead to constant stiffness when 𝛼=0.

In Table 1, first three normalized free vibration frequencies for simply supported (pinned-pinned) beam are compared with previous available results in the literature and the exact solution. Excellent agreement is observed with the exact solution.

Table 1 

Normalized free vibration frequencies of simply supported beam resting on the Winkler foundation.

The first five natural frequencies for clamped-clamped beam and cantilever (clamped-free) beam are presented in Tables 2 and 3, respectively. They are compared with the available solutions.

Table 2 

Normalized free vibration frequencies of clamped-clamped beam resting on the Winkler foundation.

Table 3 

Normalized free vibration frequencies of cantilever beam resting on the Winkler foundation.

Excellent agreement is observed with previous available results for both cantilever and clamped-clamped beams. Clamped-pinned beam was not included in the studies used for comparison. Hence, only the result obtained from HPM is tabulated for this case in Table 4.

Table 4 

Normalized free vibration frequencies of clamped-pinned beam resting on the Winkler foundation.

As one can see, perfect agreement is obtained for constant stiffness case. This issue is mainly due to constant coefficient governing equation. In the following sections, variable stiffness cases are investigated.

5.2. Linearly Varying Stiffness

Linearly varying beam width results in a linearly varying flexural stiffness. The variation is based on parameter α. A number of case studies are conducted with respect to parameter α, and the results are given in Tables 5, 6, 7, and 8.

Table 5 

Normalized free vibration frequencies of cantilever beam resting on Winkler foundation with linearly varying flexural stiffness.

Table 6 

Normalized free vibration frequencies of clamped-pinned beam resting on the Winkler foundation with linearly varying flexural stiffness.

Table 7 

Normalized free vibration frequencies of clamped-clamped beam resting on the Winkler foundation with linearly varying flexural stiffness.

Table 8 

Normalized free vibration frequencies of simply supported beam resting on the Winkler foundation with linearly varying flexural stiffness.

Variation of normalized free vibration frequencies (𝜔) with respect to nondimensional variation parameter 𝛼 for each beam are also given in Figures 4, 5, 6, 7, and 8.

Figure 4 

Variation of normalized first mode frequency with respect to normalized variation coefficient (linear variation in stiffness).

Figure 5 

Variation of normalized second mode frequency with respect to normalized variation coefficient (linear variation in stiffness).

Figure 6 

Variation of normalized third mode frequency with respect to normalized variation coefficient (linear variation in stiffness).

Figure 7 

Variation of normalized fourth mode frequency with respect to normalized variation coefficient (linear variation in stiffness).

Figure 8 

Variation of normalized fifth mode frequency with respect to normalized variation coefficient (linear variation in stiffness).

5.3. Exponentially Varying Stiffness

Exponentially varying beam width results in an exponentially varying flexural stiffness. The variation is again based on parameter α, and a number of case studies are also conducted with respect to parameter α. Results are given in Tables 9, 10, 11, and 12.

Table 9 

Normalized free vibration frequencies of cantilever beam resting on the Winkler foundation with exponentially varying flexural stiffness.

Table 10 

Normalized free vibration frequencies of clamped-pinned beam resting on the Winkler foundation with exponentially varying flexural stiffness.

Table 11 

Normalized free vibration frequencies of clamped-clamped beam resting on the Winkler foundation with exponentially varying flexural stiffness.

Table 12 

Normalized free vibration frequencies of simply supported beam resting on the Winkler foundation with exponentially varying flexural stiffness.

Variation of normalized free vibration frequencies (𝜔) with respect to nondimensional variation parameter 𝛼 for each beam are also given in Figures 9, 10, 11, 12, and 13.

Figure 9 

Variation of normalized first mode frequency with respect to normalized variation coefficient (exponential variation in stiffness).

Figure 10 

Variation of normalized second mode frequency with respect to normalized variation coefficient (exponential variation in stiffness).

Figure 11 

Variation of normalized third mode frequency with respect to normalized variation coefficient (exponential variation in stiffness).

Figure 12 

Variation of normalized fourth mode frequency with respect to normalized variation coefficient (exponential variation in stiffness).

Figure 13 

Variation of normalized fifth mode frequency with respect to normalized variation coefficient (exponential variation in stiffness).

6. Conclusions

In this study, HPM is introduced for the free vibration analysis of variable stiffness Euler beams on elastic foundations. As a demonstration of application of the method, firstly the case studies for which previous results were available are chosen. In these studies, constant stiffness Euler beams were considered, and first analyses are conducted for constant stiffness case for comparison with the available results. HPM produced results which are in excellent agreement with the previously available solutions that encourage the application of the method for variable stiffness beams. To represent a variation in stiffness, a rectangular beam with varying width is considered, and two types of variation are taken into consideration. These are, namely, linear and exponential changes. Such varying widths lead to linearly and exponentially varying stiffnesses. The analyses are expanded for variable stiffness cases. HPM also produced reasonable results for the vibration of variable stiffness beams which show the efficiency of the method. For the variable stiffness problems, the governing equation is a differential equation with variable coefficients, and it is not easy to obtain analytical solutions for these types of problems. However, it is easy to put those variable parameters into the iteration algorithm of HPM, and the results can be obtained after performing some iterations with the method. The results obtained in this study point out that the proposed method is a powerful and reliable method in the analysis of the presented problem.