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 easytouse 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.
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
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 wellknown is the EulerBernoulli 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. [
An Euler beam resting on the Winkler foundation shown in Figure
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.
For clampedclamped beam the end conditions are
For cantilever beam (clampedfree) the end conditions are
For simply supported beam (pinnedpinned) the end conditions are
For clampedsimply supported (pinned) beam the end conditions are
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
This equation can be rearranged as
Using these parameters, nondimensional form of the equations and formulation procedures are explained in the following sections.
In recent years, efforts towards application of analytical approximate solution techniques for nonlinear problems have increased. In these studies, He’s Homotopy Perturbation Method [
or
The changing process of
The approximate solution of
The convergence of the series in (
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
Beam with a linearly varying width.
The variable width is formulated as
Beam with an exponentially varying width.
Variable width of the beam in this case is
In the following sections, formulation procedures are given separately for each case.
Employing (
This equation can be rewritten as
Employing (
Equation (
By the application of HPM, the following iteration algorithm is obtained:
Employing (
Inserting (
Employing (
Equation (
By the application of HPM, the following iteration algorithm is obtained:
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:
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.
Clampedclamped beam:
Clampedfree (cantilever) beam:
Pinnedpinned (simply supported) beam:
Clampedpinned beam:
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:
As the first example, Euler beam of constant stiffness (i.e.,
In Table
Normalized free vibration frequencies of simply supported beam resting on the Winkler foundation.
Method 






HPM  9.92014  39.4911  88.8321  157.9168  246.7421 
DTM [ 
9.92014  39.4911  88.8321  —  — 
DQEM [ 
9.92014  39.4913  89.4002  —  — 
Exact solution [ 
9.92014  39.4911  88.8321  —  — 
The first five natural frequencies for clampedclamped beam and cantilever (clampedfree) beam are presented in Tables
Normalized free vibration frequencies of clampedclamped beam resting on the Winkler foundation.
Method 






HPM  22.3956  61.6809  120.908  199.862  298.557 
DTM [ 
22.3733  61.6728  120.903  199.859  298.556 
DQEM [ 
22.3956  61.6811  120.910  199.885  298.675 
Normalized free vibration frequencies of cantilever beam resting on the Winkler foundation.
Method 






HPM  3.65546  22.0572  61.7053  120.906  199.862 
DTM [ 
3.65546  22.0572  61.7053  120.906  199.862 
DQEM [ 
3.65544  22.0572  61.7057  120.911  199.894 
Excellent agreement is observed with previous available results for both cantilever and clampedclamped beams. Clampedpinned beam was not included in the studies used for comparison. Hence, only the result obtained from HPM is tabulated for this case in Table
Normalized free vibration frequencies of clampedpinned beam resting on the Winkler foundation.
Method 






HPM  15.451  49.975  104.253  178.273  272.033 
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.
Linearly varying beam width results in a linearly varying flexural stiffness. The variation is based on parameter
Normalized free vibration frequencies of cantilever beam resting on Winkler foundation with linearly varying flexural stiffness.

0.00  0.10  0.20  0.30  0.40  0.50 


3.65546  3.77785  3.91814  4.08113  4.27363  4.50571 

22.0572  22.2779  22.5271  22.8129  23.1475  23.5506 

61.7053  61.9182  62.1616  62.4458  62.7867  63.2104 

120.9061  121.1194  121.3645  121.6528  122.0024  122.4434 

199.8620  200.0755  200.3213  200.6117  200.9661  201.4172 
Normalized free vibration frequencies of clampedpinned beam resting on the Winkler foundation with linearly varying flexural stiffness.

0.00  0.10  0.20  0.30  0.40  0.50 


15.4506  15.5615  15.6801  15.8069  15.9427  16.0879 

49.9749  50.0781  50.1878  50.3052  50.4316  50.5685 

104.2525  104.3556  104.4655  104.5836  104.7121  104.8544 

178.2725  178.3756  178.4853  178.6036  178.7333  178.8785 

272.0328  272.1358  272.2455  272.3640  272.4944  272.6418 
Normalized free vibration frequencies of clampedclamped beam resting on the Winkler foundation with linearly varying flexural stiffness.

0.00  0.10  0.20  0.30  0.40  0.50 


22.3956  22.3922  22.3777  22.3477  22.2958  22.2120 

61.6809  61.6752  61.6541  61.6114  61.5380  61.4183 

120.9075  120.9009  120.8775  120.8302  120.7485  120.6144 

199.8620  199.8549  199.8302  199.7803  199.6939  199.5512 

298.5572  298.5499  298.5243  298.4729  298.3834  298.2351 
Normalized free vibration frequencies of simply supported beam resting on the Winkler foundation with linearly varying flexural stiffness.

0.00  0.10  0.20  0.30  0.40  0.50 


9.9201  9.9217  9.9210  9.9170  9.9084  9.8932 

39.4911  39.4928  39.4970  39.5047  39.5166  39.5340 

88.8321  88.8340  88.8398  88.8511  88.8697  88.8986 

157.9168  157.9189  157.9257  157.9389  157.9612  157.9966 

246.7421  246.7443  246.7516  246.7661  246.7907  246.8302 
Variation of normalized free vibration frequencies (
Variation of normalized first mode frequency with respect to normalized variation coefficient (linear variation in stiffness).
Variation of normalized second mode frequency with respect to normalized variation coefficient (linear variation in stiffness).
Variation of normalized third mode frequency with respect to normalized variation coefficient (linear variation in stiffness).
Variation of normalized fourth mode frequency with respect to normalized variation coefficient (linear variation in stiffness).
Variation of normalized fifth mode frequency with respect to normalized variation coefficient (linear variation in stiffness).
Exponentially varying beam width results in an exponentially varying flexural stiffness. The variation is again based on parameter
Normalized free vibration frequencies of cantilever beam resting on the Winkler foundation with exponentially varying flexural stiffness.

0.00  0.25  0.50  0.75  1.00 


3.65546  3.99564  4.46148  5.08383  5.89198 

22.0572  22.6410  23.4038  24.4544  25.9407 

61.7053  62.2689  63.0286  64.1348  65.8187 

120.9061  121.4706  122.2407  123.3890  125.1956 

199.8620  200.4266  201.2020  202.3738  204.2517 
Normalized free vibration frequencies of clampedpinned beam resting on the Winkler foundation with exponentially varying flexural stiffness.

0.00  0.25  0.50  0.75  1.00 


15.4506  15.7228  16.0355  16.4684  17.1543 

49.9749  50.2233  50.5095  50.9401  51.7103 

104.2525  104.4999  104.7874  105.2357  106.0716 

178.2725  178.5190  178.8070  179.2648  180.1380 

272.0328  272.2787  272.5671  273.0313  273.9289 
Normalized free vibration frequencies of clampedclamped beam resting on the Winkler foundation with exponentially varying flexural stiffness.

0.00  0.25  0.50  0.75  1.00 


22.3956  22.3562  22.2409  22.0853  21.9787 

61.6809  61.6245  61.4646  61.2498  61.1004 

120.9075  120.8450  120.6698  120.4372  120.2857 

199.8620  199.7961  199.6124  199.3705  199.2211 

298.5572  298.4893  298.3002  298.0528  297.9060 
Normalized free vibration frequencies of simply supported beam resting on the Winkler foundation with exponentially varying flexural stiffness.

0.00  0.25  0.50  0.75  1.00 


9.9201  9.8928  9.8179  9.7493  9.7849 

39.4911  39.4736  39.4584  39.5456  39.9202 

88.8321  88.8182  88.8204  88.9564  89.4478 

157.9168  157.9048  157.9164  158.0782  158.6327 

246.7421  246.7312  246.7485  246.9262  247.5200 
Variation of normalized free vibration frequencies (
Variation of normalized first mode frequency with respect to normalized variation coefficient (exponential variation in stiffness).
Variation of normalized second mode frequency with respect to normalized variation coefficient (exponential variation in stiffness).
Variation of normalized third mode frequency with respect to normalized variation coefficient (exponential variation in stiffness).
Variation of normalized fourth mode frequency with respect to normalized variation coefficient (exponential variation in stiffness).
Variation of normalized fifth mode frequency with respect to normalized variation coefficient (exponential variation in stiffness).
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.