Free vibration analysis of elastically supported timoshenko beams

Abstract

Keywords: Timoshenko beam theory; Lagrange equations; Power series.

1. Introduction

Many studies have been carried out related with the problem of free vibration of beams with elastically supports based on the Euler-Bernoulli beam theory (EBT) or Timoshenko beam theory *

Sorumlu Yazar/Corresponding Autor: e-posta: kocaturk@yildiz.edu.tr, tel: (0212) 259 70 70 / 2775

(TBT). The well-known Euler-Bernoulli beam theory (EBT) states that plane sections remain plane and perpendicular to the central axis of the beam after deformation, regarding transverse shear strain to be neglected. Although this theory is very useful for slender beams and columns, it does not give accurate solutions for thick beams. In the Timoshenko beam theory (TBT), the normality assumption of the Euler-Bernoulli theory (EBT) is relaxed and a constant state of transverse shear strain with respect to the thickness coordinate is included. The Timoshenko beam theory requires shear correction factors to compensate for the error due to this constant shear stress assumption. The problem of free lateral vibration of an axially loaded Euler-Bernoulli beam with intermediate elastic supports and concentrated masses is considered using Green function method by Kukla [2]. H. K. Kim and M. S. Kim [3] presented a method to find accurate vibration frequencies of beams with elastic supports using Fourier series. Nallim and Grossi [4] presented a simple variational approach based on the use of the Rayleigh-Ritz method with the characteristic orthogonal polynomial shape functions for the determination of free vibration frequencies of beams with several complicating effects within the frame of Euler-Bernoulli beam theory. Lee and Schultz [5] applied the pseudospectral method to the eigenvalue analysis of Timoshenko beams. Zhou [6] used the Rayleigh-Ritz method for the free vibration of multi-span Timoshenko beams. Farghaly [7] has investigated the natural frequencies and the critical buckling load coefficients for a multi-span Timoshenko beam. Banerjee [8] investigated the free vibration analysis of axially loaded Timoshenko beams by using the dynamic stiffness method. The free vibration of Timoshenko beams with internal hinge and subjected to axial tensile load is carried out by Lee et. al. [9]. A dynamic investigation method for the analysis of Timoshenko beams which takes into acount shear deformation is proposed by Auciello and Ercolano [10]. In [10], the solution of the problem is obtained through the iterative variational Rayleigh-Ritz method. The free vibration of Timoshenko beams having classical boundary conditions, which was satisfied by Lagrange multipliers, was investigated for different thickness-to-length ratios by Kocatürt and Şimşek [11]. In the present study, the free vibration of elastically supported Timoshenko beams is analyzed by using the Lagrange equations with the trial functions in the power series form denoting the deflection and the rotation of the cross-section of the beam. The convergence study is based on the numerical values obtained for various numbers of power series terms. In the numerical examples, the first three eigenvalues of the Timoshenko beam are calculated for various values of stiffness of translational and rotational springs. The accuracy of the results is established by comparison with previously published accurate results for the free vibration analysis of the Timoshenko beams.

2. Theory And Formulations

Consider a straight uniform single-span Timoshenko beam of length L , depth h , width b , having rectangular cross-section as shown in Fig. 1, where K i and R i are the translational and rotational spring constant. A Cartesian coordinate system (X, Y, Z) is defined on the central axis of the beam, where the X axis is taken along the central axis, with the Y axis in the width direction and the Z axis in the depth direction. Also, the origin of the coordinate system is chosen at the mid-point of the total length of the beam. Although, it is possible to take lots of point supports at arbitrary points, in the numerical investigations here, it will be considered that the beam is supported at the two end points, where the parameters K i and R i are taken to have the same values at all the supports denoted by K i = K and R i = R .

Figure 1. Considered Timoshenko beams with (a) the first type and (b) the second type of translational and rotational springs. The Timoshenko beam theory is based on the following displacement fields; U (X, Z, t) = -Z Φ (X, t) W (X, Z, t) = W (X, t) ,

where W (X, t) is the transverse displacement of a point on the beam reference plane and Φ (X, t) is the rotation of a normal to the reference plane about y-axis. According to the Timoshenko beam theory (TBT), the elastic strain energy of the beam at any instant is expressed as an integral in Cartesian coordinates as follows L

2 2 ⎛ d Φ (X, t) ⎞ ⎛ d W (X, t) ⎞ ⎤ 1 2 ⎡ U = ∫ ⎢ EI(x) ⎜ - Φ (X, t) ⎟ ⎥ d X , ⎟ + k s GA(x) ⎜

2. L ⎢

where E is the Young’s modulus, G is the transverse shear modulus, I(X) is the moment of inertia, A (X) is the area of the cross-section and k s is a constant that accounts for non-uniform shear stress distribution through the thickness. Kinetic energy of the beam at any instant is L

2 2 ⎛ d W(X, t) ⎞ ⎛ d Φ (X, t) ⎞ ⎤ 1 2 ⎡ K e = ∫ ⎢ρ A (X) ⎜ ⎟ + ρ I(X) ⎜ ⎟ ⎥ dX ,

2. L ⎢

where ρ is the mass of the beam per unit volume. Additive strain energy of the translational and rotational spring is given by the following formulas, respectively FT =

where X Si denotes the location of the i th support, K i and R i are the spring constant of the translational and rotational springs at the both ends, respectively. By introducing the following non-dimensional parameters x=

the potential and kinetic energy of the beam at any instant can be written as

2 2 1/2 ⎡ ⎛ dw (x, t) ⎞ ⎤ 1 E I(x) ⎛ d φ (x, t) ⎞ ⎢ + k G L A(x) (x, t) φ ⎜ ⎟ ⎥ dx ∫ ⎢ L ⎜⎝ d x ⎟⎠ s 2 -1/2 ⎝ dx ⎠ ⎥⎦ ⎣

2 2 1/2 ⎡ ⎛ d φ (x, t) ⎞ ⎤ ⎛ d w (x, t) ⎞ 1 3 ⎢ ⎥ dx . ρ L A (x) + ρ L I (x) ⎜ ⎟ ⎜ ⎟ ∫ ⎢ 2 -1/2 ⎝ dt ⎠ ⎝ d t ⎠ ⎥⎦ ⎣

Additive strain energy of the translational and rotational springs can be written in terms of the non-dimensional quantities in the following equations: FT =

It is known that some expressions satisfying geometrical boundary conditions are chosen for w (x, t) and φ (x, t) , and by using the Lagrange equations, the natural boundary conditions are also satisfied. Therefore, by using the Lagrange equations and by representing the transverse displacement w (x, t) and the rotation of cross-sections φ (x, t) in terms of a series of admissible functions and adjusting the coefficients in the series to satisfy the Lagrange equations, approximate solutions are found for the displacement and the rotation functions. For applying the Lagrange equations, the trial functions w (x, t) and φ (x, t) are approximated by space-dependent

polynomial terms x 0 , x1 , x 2 ,...., x M -1 and time-dependent displacement coordinates a m (t) and

The time-dependent generalized coordinates for the free vibration of the beam can be expressed as follows: a m (t) = a m ei ω t

Dimensionless amplitudes of the displacement and normal rotation of a cross-section of the beam can be expressed as follows: M

The functional of the problem is L = K e - (U + V + FT + FR ) .

where the overdot stands for the partial derivative with respect to time and q k = ak

By introducing the following non-dimensional parameters λ2 =

ρA ω 2 L4 k G A L2 I K L3 R L , β= s , µ= , κ i = i , θi = i 2 EI EI AL EI EI

and by using Eq. (17), the following simultaneous sets of linear algebraic equations are obtained which can be expressed in the following matrix forms; ⎡ [ A ] [ B] ⎤ ⎧⎪{a }⎫⎪ ⎡ [ E ] [ 0] ⎤ ⎧⎪{a }⎫⎪ ⎪⎧{0}⎪⎫ 2 ⎢ ⎥ ⎨ ⎬−λ ⎢ ⎥ ⎨ ⎬=⎨ ⎬ ⎢⎣ [ C] [ D ] ⎥⎦ ⎩⎪ b ⎭⎪ ⎢⎣ [ 0] [ F] ⎥⎦ ⎩⎪ b ⎭⎪ ⎪⎩{0}⎭⎪

where [ A ] , [ B] , [ C] , [ D ] , [ E ] and [ F] are the coefficient matrices obtained by using Eq. (17) and 0 ⋅5

A km = ∫ β (x k -1 )′(x m-1 )′d x + κ1 (−0 ⋅ 5) k -1 (−0 ⋅ 5) m-1 + κ 2 (0 ⋅ 5) k -1 (0 ⋅ 5) m-1 -0⋅5

D km = ∫ ⎡⎣β ( x k -1 ) (x m-1 ) + (x k -1 )′(x m-1 )′⎤⎦ d x + θ1 (−0 ⋅ 5) k -1 (−0 ⋅ 5) m-1 + θ2 (0 ⋅ 5) k -1 (0 ⋅ 5) m-1 -0 ⋅5

The eigenvalues (characteristic values) λ are found from the condition that the determinant of the system of equations given by Eq. (21) must vanish.

3. Numerical Results

In order to investigate the influence of stiffness of the supports on the free vibration characteristics of Timoshenko beams, the first three eigenvalues of Timoshenko beam with the first and the second type of translational and rotational springs (Fig. 1) are calculated for h/L=0.005 and three dimensional plots of Tables 2, 3, 4, 5, 6 and 7 are provided in Figs. 2, 3, 4, 5, 6 and 7 to illustate how the frequency parameters change with the spring constants. The stiffness parameters κi and θi are taken as having the same values at all the supports denoted by

κ1 = 1 ⋅ 108 , κ 2 = κ, θ1 = θ, θ2 = 0 for the beam with the second type of the springs. It is possible to simulate infinite support stiffness by setting the translational or rotational stiffness coefficient equal to 1 ⋅ 108 at all the supports for comparing the obtained results with the existing results of the classically supported Timoshenko beams. Therefore, comparison study of the pinned-pinned ( κ1 = κ 2 = 1 ⋅ 108 , θ1 = θ2 = 0 ) and clamped-clamped

( κ1 = κ 2 = 1 ⋅ 108 , θ1 = θ2 = 1 ⋅ 108 ) Timoshenko beam with the classical solutions based on the Euler-Bernoulli beam theory [1] and the results of the Pseudospectral method given in the Ref. [5] is carried out, and the results are given in Tables 1b and 1c. Also, by setting the translational and rotational stiffness coefficients equal to zero at all the supports, a completely free beam situation can be obtained. Moreover, the convergence is tested in Table 1a by taking the number of terms M = 6, 8, 10, 12, 14, 16, 18 . It is not necessary to give the values of E, G and A of the beam in the calculations. Relationship between E and G is as follows: G=

where ν is the Poisson’s ratio. In all of the following calculations, the rectangular cross-sectional beams with shear correction factor k s = 5 / 6 are considered and, the Poisson’s ratio is taken ν = 0.3 . Table 1a. The convergence study of the first six dimensionless frequency parameters λ i of the pinned-pinned Timoshenko beam for h/L=0.01

It is shown that the convergence with respect to the number of the power series terms is excellent in the considered cases. As it is observed from the Table 1a, the frequency parameter decreases as the number of the power series terms increases: It means that the convergence to the exact value is from above. Namely, by increasing the number of the polynomial terms, the exact value can be approached from above. It should be remembered that energy methods always overestimate the fundamental frequency, so with more refined analyses, the exact value can be

Free Vibration Analysis of Elastically Supported… approached from above. From here on, the number of the power series terms M is taken as 16 in all of the numerical investigations, namely the size of the determinant is 32 × 32 . Table 1b. Comparison study of the first six dimensionless frequency parameters λ i of the pinned-pinned Timoshenko beam

Methods

Table 1c. Comparison study of the first six dimensionless frequency parameters λ i of the clamped-clamped Timoshenko beam.

Methods

Table 2. Variation of the first frequency parameter of the Timoshenko beam with the first type of translational and rotational spring parameters ( κ1 = κ 2 = κ, θ1 = θ2 = θ ) for h/L=0.005. λ1

Figure 2 Plot of the first frequency parameter of the Timoshenko beam with the first type of translational and rotational spring parameters ( κ1 = κ 2 = κ, θ1 = θ2 = θ ) for h/L=0.005

Free Vibration Analysis of Elastically Supported… Table 3. Variation of the second frequency parameter of the Timoshenko beam with first type of translational and rotational spring parameters ( κ1 = κ 2 = κ, θ1 = θ2 = θ ) for h/L=0.005 λ2

Figure 3 Plot of the second frequency parameter of the Timoshenko beam with the first type of translational and rotational spring parameters ( κ1 = κ 2 = κ, θ1 = θ2 = θ ) for h/L=0.005

Table 4. Variation of the third frequency parameter of the Timoshenko beam with the first type of translational and rotational spring parameters ( κ1 = κ 2 = κ, θ1 = θ2 = θ ) for h/L=0.005 λ3

Figure 4 Plot of the third frequency parameter of the Timoshenko beam with the first type of translational and rotational spring parameters ( κ1 = κ 2 = κ, θ1 = θ2 = θ ) for h/L=0.005

Free Vibration Analysis of Elastically Supported… Table 5. Variation of the first frequency parameter of the Timoshenko beam with the second type of translational and rotational spring parameters ( κ1 = 1 ⋅ 108 , κ 2 = κ, θ1 = θ, θ2 = 0 ) for h/L=0.005 λ1

Figure 5. Plot of the first frequency parameter of the Timoshenko beam with the second type of translational and rotational spring parameters ( κ1 = 1 ⋅ 108 , κ 2 = κ, θ1 = θ, θ2 = 0 ) for h/L=0.005

Table 6. Variation of the second frequency parameter of the Timoshenko beam with second type of translational and rotational spring parameters ( κ1 = 1 ⋅ 108 , κ 2 = κ, θ1 = θ, θ2 = 0 ) for h/L=0.005 λ2

Figure 6. Plot of the second frequency parameter of the Timoshenko beam with the second type of translational and rotational spring parameters ( κ1 = 1 ⋅ 108 , κ 2 = κ, θ1 = θ, θ2 = 0 ) for h/L=0.005

Free Vibration Analysis of Elastically Supported… Table 7. Variation of the third frequency parameter of the Timoshenko beam with second type of translational and rotational spring parameter ( κ1 = 1 ⋅ 108 , κ 2 = κ, θ1 = θ, θ2 = 0 ) for h/L=0.005 λ3

Figure 7. Plot of the third frequency parameter of the Timoshenko beam with the second type of translational and rotational spring parameters ( κ1 = 1 ⋅ 108 , κ 2 = κ, θ1 = θ, θ2 = 0 ) for h/L=0.005

It can be deduced that the results obtained from the present study are in good agreement with those of Lee and Schultz [5] as given in the Tables 1a-b-c. It should be remembered that, the eigenvalues obtained by using first order or higher order beam theories are lower than the corresponding eigenvalues obtained by the classical beam theory. It is observed from the Tables 1b-c that, the difference in the frequencies of the Euler-Bernoulli and Timoshenko beams becomes significant with increase of the mode numbers. For example, the value of the frequency of the pinned-pinned beam based on the classical theory is 3.14159 and of the pinned-pinned Timoshenko beam (for h / L = 0.05 ) 3.13499 for the first mode while they are 18.8496 and 17.6809 for the sixth mode. Also, as it is known and can be deduced from Table 1 that, with increase in the ratio of h/L , the dimensionless frequencies of the Timoshenko beams decreases compared with the frequencies of Euler-Bernoulli beams. However, the two solutions are very close to each other for small values of h/L (i.e. h/L=0.005) as seen from Tables 1a-b-c. It is seen from the tables and the figures that, translational springs are much more effective on the frequency parameters than rotational springs. For example, in Table 3, when the spring parameter κ is taken constant value of 100 and the parameter θ is changed from 100 to 108 , the frequency parameter λ 2 changes from 2.23329 to 3.17326 but, while the parameter θ is taken as 100 and the parameter κ is changed from 100 to 108 , the frequency parameter λ 2 changes from 2.23329 to 6.42666 . Increment in the values of parameters κ and θ is more effective on the first frequency parameter of the beam than the second and third frequency parameters. For instance, for the beam with the first type of the springs, when the parameters κ and θ are both changed from 1 ⋅ 100 to 108 , the first frequency parameter λ1 changes from 1.18562 to 4.72962 , namely, λ1 becomes four times greater in this change. On the other hand, λ 2 and λ 3 increase approximately 3.5 and

4. Conclusions

The free vibration of elastically supported Timoshenko beams have been investigated for different support stiffnesses. To compare the obtained results with the previously published results, the frequency parameters of pinned-pinned and clamped-clamped Timoshenko beams, which are special cases of the present problem are calculated. Using the Lagrange equations with the trial functions in the power series form and satisfying the constraint conditions by the use of very stiff springs is a very good way for studying the free vibration characteristics of the elastically supported beams. Numerical calculations have been carried out to clarify the effects of support stiffnesses on the free vibration characteristics of the considered beams. It is observed from the investigations that, all of the obtained results are very accurate and may be useful to other researchers so as to compare their results.

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