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

Authors: Utkan Mutman
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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 parameterk, 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 Figure1 is considered in this study. The equation of motion for this problem is given as follows:
(1)∂2∂x2(EI(x)∂2w∂x2)+k(x)w+ρA(x)(∂2w∂t2)=0,
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(2)w=∂w∂x=0atx=0,L.(b)
For cantilever beam (clamped-free) the end conditions are(3a)w=∂w∂x=0atx=0,(3b)∂2w∂x2=∂3w∂x3=0atx=L.(c)
For simply supported beam (pinned-pinned) the end conditions are(4)w=∂2w∂x2=0atx=0,L.(d)
For clamped-simply supported (pinned) beam the end conditions are(5a)w=∂w∂x=0atx=0,(5b)w=∂2w∂x2=0atx=L.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(6)w(x,t)=W(x)eiωt,
where ω is the circular frequency for the vibration. Substituting (6) into (1), equations of motion become as follows:
(7)d2dx2(EI(x)d2Wdx2)+kW=ρAω2W.This equation can be rearranged as(8)∂4w∂x4+2EI(x)′EI(x)∂3w∂x3+EI(x)′′EI(x)∂2w∂x2+k(x)EI(x)w+ρA(x)EI(x)(∂2w∂t2)=0,
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]
(9)x-=xL,W¯=WL,λ=kL4EI,ω¯=ωρAk.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, Nis a nonlinear operator, Bis a boundary operator, Γis the boundary of the domain Ώ, 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:
(10)H(v,p)=(1-p)[L(v)-L(u0)]+p[L(v)+N(v)-f(r)]=0or(11)H(v,p)=L(v)-L(u0)+pL(u0)+p[N(v)-f(r)]=0,
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
(12)H(v,0)=L(v)-L(u0)=0,H(v,1)=L(v)+N(v)-f(r)=0.The changing process ofp 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:
(13)v=v0+pv1+p2v2+p3v3+⋯.The approximate solution ofL(u)+N(u)=f(r), r∈Ω can be obtained as
(14)u=limp→1v=v0+v1+v2+⋯.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 Figure2.Figure 2
Beam with a linearly varying width.The variable width is formulated as(15)b(x)=b0(1-αx),
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(16)b(x)=b0e-αx,
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,
(17)A(x)=b(x)h,I(x)=b(x)h312.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
(18)A(x)=b0h(1-αx)=A0(1-αx),EI(x)=Eb0h312(1-αx)=EI0(1-αx),
where A0 and I0 are the cross-sectional area and moment of inertia of the section at the origin, respectively. Inserting (18) into (8)
(19)∂4w∂x4-2αEI0EI0(1-αx)∂3w∂x3+k(x)EI0(1-αx)w+ρA0EI0(∂2w∂t2)=0.This equation can be rewritten as(20)∂4w∂x4-2α(11-αx)∂3w∂x3+k(x)EI0(11-αx)w+ρA0EI0(∂2w∂t2)=0.Employing (6)
(21)Wiv-2αξ(x)W′′′+k(x)EI0ξ(x)W-ρA0EI0ω2W=0.Equation (21) can be made nondimensional in view of (9) as follows:
(22)W¯iv-2α¯ξ(x¯)W¯′′′+λ(ξ(x¯)-ω¯2)W=0,
where
(23)x-=xL,W¯=WL,λ=kL4EI0,ω¯=ωρA0k,ξ(x-)=11-α-x-,α-=αL.By the application of HPM, the following iteration algorithm is obtained:(24a)W-0iv-u0iv=0,(24b)W-1iv+u0iv-2α-ξ(x-)W-0′′′+λ(ξ(x-)-ω-2)W-0=0,(24c)W-niv-2α-ξ(x-)W-n-1′′′+λ(ξ(x-)-ω-2)W-n-1=0,n≥2.


### 4.2. Formulation for Exponential Variation

Employing (16) into (17), variable cross-sectional areas and variable stiffness become
(25)A(x)=b0he-αx=A0e-αx,EI(x)=Eh312b0e-αx=EI0e-αx.Inserting (25) into (8)
(26)∂4w∂x4-2α∂3w∂x3+α2∂2w∂x2+k(x)EI0e-αxw+ρA0EI0(∂2w∂t2)=0.Employing (6)
(27)Wiv-2αW′′′+α2W′′+k(x)EI0eαxW-ρA0EI0ω2W=0.Equation (21) can be made nondimensional in view of (9) as follows:
(28)W¯iv-2α¯W¯′′′+α¯2W¯′′+λ(eα¯x¯-ω¯2)W=0.By the application of HPM, the following iteration algorithm is obtained:(29a)W-0iv-u0iv=0,(29b)W-1iv+u0iv-2α-W-0′′′+α-2W-0′′+λ(eα-x--ω-2)W-0=0,(29c)W-niv-2α-W-n-1′′′+α-2W-n-1′′+λ(eα-x--ω-2)W-n-1=0,n≥2.


### 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:(30)W-0=Ax-3+Bx-2+Cx-+D.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:(31)W¯=dW¯dx-=0atx-=0,1.Clamped-free (cantilever) beam:(32a)W¯=dW¯dx-=0atx-=0,(32b)d2W¯dx-2=d3W¯dx-3=0atx-=1.Pinned-pinned (simply supported) beam:(33)W¯=d2W¯dx-2=0atx-=0,1.Clamped-pinned beam:(34a)W¯=dW¯dx-=0atx-=0,(34b)W¯=d2W¯dx-2=0atx-=1.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:(35)[M(ω-)]{A}={0},
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.


## 4.1. Formulation for Linear Variation

Employing (15) into (17), variable cross-sectional areas and variable stiffness become
(18)A(x)=b0h(1-αx)=A0(1-αx),EI(x)=Eb0h312(1-αx)=EI0(1-αx),
where A0 and I0 are the cross-sectional area and moment of inertia of the section at the origin, respectively. Inserting (18) into (8)
(19)∂4w∂x4-2αEI0EI0(1-αx)∂3w∂x3+k(x)EI0(1-αx)w+ρA0EI0(∂2w∂t2)=0.This equation can be rewritten as(20)∂4w∂x4-2α(11-αx)∂3w∂x3+k(x)EI0(11-αx)w+ρA0EI0(∂2w∂t2)=0.Employing (6)
(21)Wiv-2αξ(x)W′′′+k(x)EI0ξ(x)W-ρA0EI0ω2W=0.Equation (21) can be made nondimensional in view of (9) as follows:
(22)W¯iv-2α¯ξ(x¯)W¯′′′+λ(ξ(x¯)-ω¯2)W=0,
where
(23)x-=xL,W¯=WL,λ=kL4EI0,ω¯=ωρA0k,ξ(x-)=11-α-x-,α-=αL.By the application of HPM, the following iteration algorithm is obtained:(24a)W-0iv-u0iv=0,(24b)W-1iv+u0iv-2α-ξ(x-)W-0′′′+λ(ξ(x-)-ω-2)W-0=0,(24c)W-niv-2α-ξ(x-)W-n-1′′′+λ(ξ(x-)-ω-2)W-n-1=0,n≥2.


## 4.2. Formulation for Exponential Variation

Employing (16) into (17), variable cross-sectional areas and variable stiffness become
(25)A(x)=b0he-αx=A0e-αx,EI(x)=Eh312b0e-αx=EI0e-αx.Inserting (25) into (8)
(26)∂4w∂x4-2α∂3w∂x3+α2∂2w∂x2+k(x)EI0e-αxw+ρA0EI0(∂2w∂t2)=0.Employing (6)
(27)Wiv-2αW′′′+α2W′′+k(x)EI0eαxW-ρA0EI0ω2W=0.Equation (21) can be made nondimensional in view of (9) as follows:
(28)W¯iv-2α¯W¯′′′+α¯2W¯′′+λ(eα¯x¯-ω¯2)W=0.By the application of HPM, the following iteration algorithm is obtained:(29a)W-0iv-u0iv=0,(29b)W-1iv+u0iv-2α-W-0′′′+α-2W-0′′+λ(eα-x--ω-2)W-0=0,(29c)W-niv-2α-W-n-1′′′+α-2W-n-1′′+λ(eα-x--ω-2)W-n-1=0,n≥2.


## 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:(30)W-0=Ax-3+Bx-2+Cx-+D.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:(31)W¯=dW¯dx-=0atx-=0,1.Clamped-free (cantilever) beam:(32a)W¯=dW¯dx-=0atx-=0,(32b)d2W¯dx-2=d3W¯dx-3=0atx-=1.Pinned-pinned (simply supported) beam:(33)W¯=d2W¯dx-2=0atx-=0,1.Clamped-pinned beam:(34a)W¯=dW¯dx-=0atx-=0,(34b)W¯=d2W¯dx-2=0atx-=1.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:(35)[M(ω-)]{A}={0},
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.,EIis 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 Table1, 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.
Method
ω
1
ω
2
ω
3
ω
4
ω
5
HPM
9.92014
39.4911
88.8321
157.9168
246.7421
DTM [1]
9.92014
39.4911
88.8321
—
—
DQEM [7]
9.92014
39.4913
89.4002
—
—
Exact solution [1]
9.92014
39.4911
88.8321
—
—The first five natural frequencies for clamped-clamped beam and cantilever (clamped-free) beam are presented in Tables2 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.
Method
ω
1
ω
2
ω
3
ω
4
ω
5
HPM
22.3956
61.6809
120.908
199.862
298.557
DTM [1]
22.3733
61.6728
120.903
199.859
298.556
DQEM [7]
22.3956
61.6811
120.910
199.885
298.675Table 3
Normalized free vibration frequencies of cantilever beam resting on the Winkler foundation.
Method
ω
1
ω
2
ω
3
ω
4
ω
5
HPM
3.65546
22.0572
61.7053
120.906
199.862
DTM [1]
3.65546
22.0572
61.7053
120.906
199.862
DQEM [7]
3.65544
22.0572
61.7057
120.911
199.894Excellent 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 Table4.Table 4
Normalized free vibration frequencies of clamped-pinned beam resting on the Winkler foundation.
Method
ω
1
ω
2
ω
3
ω
4
ω
5
HPM
15.451
49.975
104.253
178.273
272.033As 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 parameterα. 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.
α
0.00
0.10
0.20
0.30
0.40
0.50
ω
1
3.65546
3.77785
3.91814
4.08113
4.27363
4.50571
ω
2
22.0572
22.2779
22.5271
22.8129
23.1475
23.5506
ω
3
61.7053
61.9182
62.1616
62.4458
62.7867
63.2104
ω
4
120.9061
121.1194
121.3645
121.6528
122.0024
122.4434
ω
5
199.8620
200.0755
200.3213
200.6117
200.9661
201.4172Table 6
Normalized free vibration frequencies of clamped-pinned beam resting on the Winkler foundation with linearly varying flexural stiffness.
α
0.00
0.10
0.20
0.30
0.40
0.50
ω
1
15.4506
15.5615
15.6801
15.8069
15.9427
16.0879
ω
2
49.9749
50.0781
50.1878
50.3052
50.4316
50.5685
ω
3
104.2525
104.3556
104.4655
104.5836
104.7121
104.8544
ω
4
178.2725
178.3756
178.4853
178.6036
178.7333
178.8785
ω
5
272.0328
272.1358
272.2455
272.3640
272.4944
272.6418Table 7
Normalized free vibration frequencies of clamped-clamped beam resting on the Winkler foundation with linearly varying flexural stiffness.
α
0.00
0.10
0.20
0.30
0.40
0.50
ω
1
22.3956
22.3922
22.3777
22.3477
22.2958
22.2120
ω
2
61.6809
61.6752
61.6541
61.6114
61.5380
61.4183
ω
3
120.9075
120.9009
120.8775
120.8302
120.7485
120.6144
ω
4
199.8620
199.8549
199.8302
199.7803
199.6939
199.5512
ω
5
298.5572
298.5499
298.5243
298.4729
298.3834
298.2351Table 8
Normalized free vibration frequencies of simply supported beam resting on the Winkler foundation with linearly varying flexural stiffness.
α
0.00
0.10
0.20
0.30
0.40
0.50
ω
1
9.9201
9.9217
9.9210
9.9170
9.9084
9.8932
ω
2
39.4911
39.4928
39.4970
39.5047
39.5166
39.5340
ω
3
88.8321
88.8340
88.8398
88.8511
88.8697
88.8986
ω
4
157.9168
157.9189
157.9257
157.9389
157.9612
157.9966
ω
5
246.7421
246.7443
246.7516
246.7661
246.7907
246.8302Variation 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 parameterα, 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.
α
0.00
0.25
0.50
0.75
1.00
ω
1
3.65546
3.99564
4.46148
5.08383
5.89198
ω
2
22.0572
22.6410
23.4038
24.4544
25.9407
ω
3
61.7053
62.2689
63.0286
64.1348
65.8187
ω
4
120.9061
121.4706
122.2407
123.3890
125.1956
ω
5
199.8620
200.4266
201.2020
202.3738
204.2517Table 10
Normalized free vibration frequencies of clamped-pinned beam resting on the Winkler foundation with exponentially varying flexural stiffness.
α
0.00
0.25
0.50
0.75
1.00
ω
1
15.4506
15.7228
16.0355
16.4684
17.1543
ω
2
49.9749
50.2233
50.5095
50.9401
51.7103
ω
3
104.2525
104.4999
104.7874
105.2357
106.0716
ω
4
178.2725
178.5190
178.8070
179.2648
180.1380
ω
5
272.0328
272.2787
272.5671
273.0313
273.9289Table 11
Normalized free vibration frequencies of clamped-clamped beam resting on the Winkler foundation with exponentially varying flexural stiffness.
α
0.00
0.25
0.50
0.75
1.00
ω
1
22.3956
22.3562
22.2409
22.0853
21.9787
ω
2
61.6809
61.6245
61.4646
61.2498
61.1004
ω
3
120.9075
120.8450
120.6698
120.4372
120.2857
ω
4
199.8620
199.7961
199.6124
199.3705
199.2211
ω
5
298.5572
298.4893
298.3002
298.0528
297.9060Table 12
Normalized free vibration frequencies of simply supported beam resting on the Winkler foundation with exponentially varying flexural stiffness.
α
0.00
0.25
0.50
0.75
1.00
ω
1
9.9201
9.8928
9.8179
9.7493
9.7849
ω
2
39.4911
39.4736
39.4584
39.5456
39.9202
ω
3
88.8321
88.8182
88.8204
88.9564
89.4478
ω
4
157.9168
157.9048
157.9164
158.0782
158.6327
ω
5
246.7421
246.7312
246.7485
246.9262
247.5200Variation 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).


## 5.1. Constant Stiffness Case

As the first example, Euler beam of constant stiffness (i.e.,EIis 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 Table1, 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.
Method
ω
1
ω
2
ω
3
ω
4
ω
5
HPM
9.92014
39.4911
88.8321
157.9168
246.7421
DTM [1]
9.92014
39.4911
88.8321
—
—
DQEM [7]
9.92014
39.4913
89.4002
—
—
Exact solution [1]
9.92014
39.4911
88.8321
—
—The first five natural frequencies for clamped-clamped beam and cantilever (clamped-free) beam are presented in Tables2 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.
Method
ω
1
ω
2
ω
3
ω
4
ω
5
HPM
22.3956
61.6809
120.908
199.862
298.557
DTM [1]
22.3733
61.6728
120.903
199.859
298.556
DQEM [7]
22.3956
61.6811
120.910
199.885
298.675Table 3
Normalized free vibration frequencies of cantilever beam resting on the Winkler foundation.
Method
ω
1
ω
2
ω
3
ω
4
ω
5
HPM
3.65546
22.0572
61.7053
120.906
199.862
DTM [1]
3.65546
22.0572
61.7053
120.906
199.862
DQEM [7]
3.65544
22.0572
61.7057
120.911
199.894Excellent 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 Table4.Table 4
Normalized free vibration frequencies of clamped-pinned beam resting on the Winkler foundation.
Method
ω
1
ω
2
ω
3
ω
4
ω
5
HPM
15.451
49.975
104.253
178.273
272.033As 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 parameterα. 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.
α
0.00
0.10
0.20
0.30
0.40
0.50
ω
1
3.65546
3.77785
3.91814
4.08113
4.27363
4.50571
ω
2
22.0572
22.2779
22.5271
22.8129
23.1475
23.5506
ω
3
61.7053
61.9182
62.1616
62.4458
62.7867
63.2104
ω
4
120.9061
121.1194
121.3645
121.6528
122.0024
122.4434
ω
5
199.8620
200.0755
200.3213
200.6117
200.9661
201.4172Table 6
Normalized free vibration frequencies of clamped-pinned beam resting on the Winkler foundation with linearly varying flexural stiffness.
α
0.00
0.10
0.20
0.30
0.40
0.50
ω
1
15.4506
15.5615
15.6801
15.8069
15.9427
16.0879
ω
2
49.9749
50.0781
50.1878
50.3052
50.4316
50.5685
ω
3
104.2525
104.3556
104.4655
104.5836
104.7121
104.8544
ω
4
178.2725
178.3756
178.4853
178.6036
178.7333
178.8785
ω
5
272.0328
272.1358
272.2455
272.3640
272.4944
272.6418Table 7
Normalized free vibration frequencies of clamped-clamped beam resting on the Winkler foundation with linearly varying flexural stiffness.
α
0.00
0.10
0.20
0.30
0.40
0.50
ω
1
22.3956
22.3922
22.3777
22.3477
22.2958
22.2120
ω
2
61.6809
61.6752
61.6541
61.6114
61.5380
61.4183
ω
3
120.9075
120.9009
120.8775
120.8302
120.7485
120.6144
ω
4
199.8620
199.8549
199.8302
199.7803
199.6939
199.5512
ω
5
298.5572
298.5499
298.5243
298.4729
298.3834
298.2351Table 8
Normalized free vibration frequencies of simply supported beam resting on the Winkler foundation with linearly varying flexural stiffness.
α
0.00
0.10
0.20
0.30
0.40
0.50
ω
1
9.9201
9.9217
9.9210
9.9170
9.9084
9.8932
ω
2
39.4911
39.4928
39.4970
39.5047
39.5166
39.5340
ω
3
88.8321
88.8340
88.8398
88.8511
88.8697
88.8986
ω
4
157.9168
157.9189
157.9257
157.9389
157.9612
157.9966
ω
5
246.7421
246.7443
246.7516
246.7661
246.7907
246.8302Variation 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 parameterα, 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.
α
0.00
0.25
0.50
0.75
1.00
ω
1
3.65546
3.99564
4.46148
5.08383
5.89198
ω
2
22.0572
22.6410
23.4038
24.4544
25.9407
ω
3
61.7053
62.2689
63.0286
64.1348
65.8187
ω
4
120.9061
121.4706
122.2407
123.3890
125.1956
ω
5
199.8620
200.4266
201.2020
202.3738
204.2517Table 10
Normalized free vibration frequencies of clamped-pinned beam resting on the Winkler foundation with exponentially varying flexural stiffness.
α
0.00
0.25
0.50
0.75
1.00
ω
1
15.4506
15.7228
16.0355
16.4684
17.1543
ω
2
49.9749
50.2233
50.5095
50.9401
51.7103
ω
3
104.2525
104.4999
104.7874
105.2357
106.0716
ω
4
178.2725
178.5190
178.8070
179.2648
180.1380
ω
5
272.0328
272.2787
272.5671
273.0313
273.9289Table 11
Normalized free vibration frequencies of clamped-clamped beam resting on the Winkler foundation with exponentially varying flexural stiffness.
α
0.00
0.25
0.50
0.75
1.00
ω
1
22.3956
22.3562
22.2409
22.0853
21.9787
ω
2
61.6809
61.6245
61.4646
61.2498
61.1004
ω
3
120.9075
120.8450
120.6698
120.4372
120.2857
ω
4
199.8620
199.7961
199.6124
199.3705
199.2211
ω
5
298.5572
298.4893
298.3002
298.0528
297.9060Table 12
Normalized free vibration frequencies of simply supported beam resting on the Winkler foundation with exponentially varying flexural stiffness.
α
0.00
0.25
0.50
0.75
1.00
ω
1
9.9201
9.8928
9.8179
9.7493
9.7849
ω
2
39.4911
39.4736
39.4584
39.5456
39.9202
ω
3
88.8321
88.8182
88.8204
88.9564
89.4478
ω
4
157.9168
157.9048
157.9164
158.0782
158.6327
ω
5
246.7421
246.7312
246.7485
246.9262
247.5200Variation 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.



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*Source: 721294-2013-04-18.xml*