Free vibration analysis of timoshenko beams under various boundary conditions

Abstract

Keywords: Free Vibrations of Timoshenko Beams; Lagrange Equations; Lagrange Multipliers.

1. Introduction

Vibrations of beams are of considerable interest to the engineers designing mechanical and structural systems. Many researchers have investigated the free vibration analysis of beams having various boundary conditions and based on the Bernoulli-Euler beam theory (for example [1-4]). The well-known Bernoulli-Euler beam theory states that plane sections remain plane 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 *

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

Free Vibration Analysis of Timoshenko Beams… Timoshenko beam theory, the normality assumption of the Bernoulli-Euler theory 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. Lee and Schultz [5] applied the pseudospectral method to the eigenvalue analysis of Timoshenko beams and axisymmetric circular Mindlin plates. In [5], clamped, simply, free and sliding boundary conditions of Timoshenko beams are treated and numerical results are presented for different thickness-to-length ratios. Zhou [6] used the Rayleigh-Ritz method for the free vibration of multi-span beams. In [6], the static Timoshenko beam functions which are composed of a set of transverse deflection functions and a set of rotational angle functions are developed as the trial functions. Rossi et al. [7] have solved analytically the problem of free vibrations of beams carrying elastically mounted concentrated masses. Farghaly [8] has investigated the natural frequencies and the critical buckling load coefficients for multi-span Timoshenko beam. In the present study, the free vibration of Timoshenko beams is analyzed by using the Lagrange equations with the trial functions in the polynomial form denoting the deflection and the rotation of the cross-section of the beam. The constraint conditions of the supports are taken into account by using Lagrange multipliers. The convergence study is based on the numerical values obtained for various numbers of polynomial terms. In the numerical examples, the first eight eigenvalues of the Timoshenko beams are determined for the different thickness-to-length ratios. 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 and width b , having rectangular cross-section depicted in Fig. 1. 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.

Figure 1. (a) Clamped-clamped, (b) clamped-pinned, (c) pinned-pinned, (d) clamped-free, (e) free-free Timoshenko beams, (f) cross-section of the beams

The Timoshenko beam theory is based on the following displacement fields u x ( x, z , t ) = − zψ ( x, t )

where u z ( x, t ) is the transverse displacement of a point on the beam reference plane and ψ ( x, t ) is the rotation of a normal to the reference plane about y-axis. The strains and stresses in the Timoshenko beam theory are ex = − z

where E is the Young’s modulus, G is the transverse shear modulus and ks is a constant that accounts for non-uniform shear stress distribution through the thickness. The strain energy of the beam in Cartesian coordinates is L/2

With the help of Eqs. (2a) and (4), the strain energy of the beam at any time can be expressed as 2

L/2 L/2 ⎛ dψ ( x, t ) ⎞ ⎛ du z ( x, t ) ⎞ 1 1 − ψ ( x, t ) ⎟ dx Dxx ⎜ ⎟ dx + ∫ ks Axz ⎜ ∫ 2 −L / 2 dx 2 dx ⎝ ⎠ ⎝ ⎠ −L / 2

Dxx and Axz in Eq. (6) can be expressed as follows; Dxx = EI ( x ), Axz = GA( x)

where I ( x) and A( x) are the moment of inertia and the area of the cross-section. Rewriting Eq. (5) at any time in terms of the above expression gives

1. L/2 ⎡

⎢ ⎥ dx . EI ( x ) k GA ( x ) ψ ( x , t ) + − ⎜ ⎟ ⎜ ⎟ s 2 − L∫/ 2 ⎢ ⎝ dx ⎠ ⎝ dx ⎠ ⎥⎦ ⎣

It follows from Eq. (1) that the velocities take the form vx =

du x ( x, z , t ) d ψ ( x, t ) du ( x, t ) , vz = z = −z . dt dt dt

where ρ ( z ) is the mass of the beam per unit volume. Substituting Eq. (9) into Eq. (10) leads to T=

2 2 L/2 h/2 ⎡ ⎛ du z ( x, t ) ⎞ ⎤ b 2 ⎛ d ψ ( x, t ) ⎞ ⎢ + z ⎜ ⎟ ⎜ ⎟ ⎥ ρ ( z ) dz dx . 2 − L∫/ 2 − h∫/ 2 ⎢ ⎝ dt ⎠ ⎝ dt ⎠ ⎥⎦ ⎣

Free Vibration Analysis of Timoshenko Beams… By defining the following cross-sectional inertial coefficients h/2

L/2 L/2 ⎛ du ( x, t ) ⎞ ⎛ dψ ( x, t ) ⎞ 1 1 JA ⎜ z ⎟ dx + ∫ J D ⎜ ⎟ dx . ∫ 2 − L / 2 ⎝ dx ⎠ 2 − L / 2 ⎝ dx ⎠

The J A and J D expressions are given as follows by using the moment of inertia I and the area A of the cross-section; J A = ρ A( x), J D = ρ I ( x ) .

2 2 L/2 ⎡ ⎛ du z ( x, t ) ⎞ ⎛ d ψ ( x, t ) ⎞ ⎤ 1 ⎢ ρ A ( x ) ρ I ( x ) + ⎜ ⎟ ⎜ ⎟ ⎥ dx . 2 − L∫/ 2 ⎢ ⎝ dt ⎠ ⎝ dt ⎠ ⎦⎥ ⎣

the potential and kinetic energy of the beam can be written at any time as 2 2 1/ 2 ⎡ ⎛ d w ( x1 , t ) ⎞ ⎤ 1 EI ( x1 ) ⎛ dψ ( x1 , t ) ⎞ ⎢ k GA ( x ) L ψ ( x , t ) + − ⎜ ⎟ ⎜ ⎟ ⎥ dx1 , s 1 1 2 −1/∫ 2 ⎢ L ⎝ dx1 ⎠ ⎝ dx1 ⎠ ⎥⎦ ⎣ 2 2 1/ 2 ⎡ ⎛ dw ( x1 , t ) ⎞ ⎛ dψ ( x1 , t ) ⎞ ⎤ 1 T = ∫ ⎢ ρ A ( x1 ) L3 ⎜ ⎟ + ρ I ( x1 ) L ⎜ ⎟ ⎥ dx1 . 2 −1/ 2 ⎢ dt ⎝ dt ⎠ ⎝ ⎠ ⎥⎦ ⎣

It is known that some expressions satisfying geometrical boundary conditions are chosen for w ( x1 , t ) and ψ ( x1 , t ) and by using the Lagrange equations, the natural boundary conditions are also satisfied. Therefore, by using the Lagrange equations and by assuming the transverse displacement w( x1 , t ) and the rotation of cross-sections ψ ( x1 , t ) to be representable by a linear 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 ( x1 , t ) and ψ ( x1 , t ) are approximated by space-dependent polynomial terms x10 , x11 , x12 ,...., x1M and time-dependent generalized displacement coordinates Am (t ) and Bm (t ) . Thus M

The constraint conditions of the supports are satisfied by using the Lagrange multipliers. Therefore, it is not necessary at first for these functions to satisfy the geometrical boundary conditions. As it is known, there is no need for these functions to satisfy the natural boundary conditions. However, if the functions are chosen to satisfy the natural boundary conditions, rate of convergence also increase. The constraint conditions of the beams are given as follows: For the clamped-clamped beam (Fig. 1a)

w x1S1 , t = 0, w x1S2 , t = 0, ψ x1S1 , t = 0, ψ x1S2 , t = 0 ,

and there is no constraint conditions for the free-free beam (Fig. 1e). In Eqs. (21a-d), x1Si denotes the location of the i th support. The Lagrange multipliers formulation of the considered problem necessities the construction of the Lagrangian functional. The Lagrangian functional of the problem is obtained as follows: L = I + Lm

Lm = α1 w x1S1 , t + α 2 w x1S2 , t + β1 ψ x1S1 , t + β 2 ψ x1S2 , t ,

In Eqs. (23a-d), α i , βi quantities are the Lagrange multipliers which are the support force reactions and support moment reactions in the considered problem. The Lagrange equations are given as follows; ∂L d ∂L − =0, ∂Ω k dt ∂Ω k

where the overdot stands for the partial derivative with respect to time, N is the number of the Lagrange multipliers and

Ω 2 M +1 = α1 , Ω 2 M + 2 = α 2 , Ω 2 M + 3 = β1 , Ω 2 M + 4 = β 2

Ω 2 M +1 = α1 , Ω 2 M + 2 = α 2 , Ω 2 M + 3 = β1 , Ω 2 M + 4 = 0

Ω 2 M + 1 = α1 , Ω 2 M + 2 = α 2 , Ω 2 M + 3 = 0 , Ω 2 M + 4 = 0

Ω 2 M +1 = α1 , Ω 2 M + 2 = β1 , Ω 2 M + 3 = 0 , Ω 2 M + 4 = 0

Ω 2 M +1 = 0 , Ω 2 M + 2 = 0 , Ω 2 M + 3 = 0 , Ω 2 M + 4 = 0 .

The time-dependent generalized displacement coordinates for the free vibration of the beam can be expressed as follows:

In Eqs. (26a-b), Am and Bm are complex variables containing a phase angle. Dimensionless amplitudes of the displacement and normal rotation of a cross-section of the beam can be expressed as follows; M

and by using Eq. (24), the following simultaneous sets of linear algebraic equations are obtained which can be expressed in the following matrix forms

where [ A] and [ B ] are the coefficient matrices obtained by using Eq. (24) and In Eq. (30), ( x k −1 )′ is the first derivative of the x k −1 and the vector { D} in Eq. (29) is defined by

k = M + 1, ...., 2M , m = M + 1, ...., 2M k = 1, 2 , ...., M ,

The size of matrices [ A] and [ B ] is (2M + N ) × (2M + N ) and the size of vector { D} is (2 M + N ) . The total number of unknown coefficients is (2 M + N ) . Again, the number of equations which can be written by using Eq. (24) is (2M + N ) , which is given in matrix form by Eq. (29). Therefore, the total number of these equations is equivalent to the total number of unknown coefficients and these unknowns can be determined by using above-mentioned equations. The eigenvalues (characteristic values) λ are found from the condition that the determinant of the system of equations given by Eq. (29) must vanish. Moreover, the other components of matrices [ A] and [ B ] are obtained from the boundary conditions and the other

components of the vector { D} are given in appendix at the end of the paper.

3. Numerical Results

The first eight eigenvalues of the Timoshenko beam with clamped-clamped, clamped-pinned, pinned-pinned, clamped-free, free-free boundary conditions are given in Tables 2-6 for the different thickness-to-length ratios. In order to compare the obtained results with the existing results, the classical solutions based on the Bernoulli-Euler beam theory and the results of the Pseudospectral method given in the Ref. [5] are added to the tables. Convergence study of the Timoshenko beam with pinned-pinned boundary conditions is carried out for h / L = 0.05 and the results are given in Table 1. It is not necessary to give the E, G and A values of the beam in the calculations. Referring to the relationship between the E and G as

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 , the Poisson’s ratio ν = 0.3 and different thicknessto-length ratios ranging from h / L = 0.002 to 0.2 are considered. Table 1. The convergence study of the first eight dimensionless frequency parameters λi of the pinned-pinned Timoshenko beams for h/L=0.05.

It is observed from the Table 1 that, the natural frequencies decrease as the number of the polynomial 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 approached from above. From here on, the number of the polynomial terms M is taken as 18 in all of the numerical investigations. It can be deduced that the results obtained from the present study are in good aggrement with those of Lee and Schultz [5] as given in the Tables 2-4. It is known 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. As seen from the Tables 2-6, the eigenvalues of the beams decrease with the increase of thickness-tolength ratio. For example, the fifth eigenvalue of the pinned-pinned beam is 15.7066 for h / L = 0.002 , it is 15.6996 for h / L = 0.005 , 15.6749 for h / L = 0.01 , 15.5784 for h / L = 0.02 , 14.9926 for h / L = 0.05 , 13.6131 for h / L = 0.1 and 11.2219 for h / L = 0.2 . However, the two solutions are very close to each other for small values of h / L . For instance, the differences between the results of the two theories are very small when h/L is less than 0.02. Moreover, the difference of the value of the eigenvalue of the classical beam theory and the Timoshenko beam theory increases for increasing mode numbers. For example, while the value of the eigenvalue of the clamped-clamped Timoshenko beam for h / L = 0.2 is 89 percent of the classical theory for the first mode, it is 70 for the fourth mode, and 51 for the eighth mode.

Table 2. The comparison study of the first eight dimensionless frequency parameters λi of the pinned-pinned Timoshenko beams for different thickness-to-length ratios.

Methods

Classical 3.14159 6.28319 9.42478 12.5664 15.7080 18.8496 21.9911 25.1327 Solution h/L=0.002 Present

3.14158 6.28310 9.42449 12.5657 15.7066 18.8473 21.9875 25.1273 h/L=0.005

3.14153 6.28265 9.42298 12.5621 15.6997 18.8352 21.9684 25.0988 h/L=0.01

3.14133 6.28106 9.41761 12.5494 15.6749 18.7926 21.9011 24.9988 h/L=0.02

3.14053 6.27471 9.39632 12.4994 15.5784 18.6282 21.6443 24.6227 h/L=0.05

3.13498 6.23136 9.25537 12.1813 14.9926 17.6810 20.2447 22.6862 h/L=0.1

3.11568 6.09066 8.84052 11.3431 13.6132 15.6790 17.5705 19.3142 h/L=0.2

Table 3. The comparison study of the first eight dimensionless frequency parameters λi of the clamped-clamped Timoshenko beams for different thickness-to-length ratios.

Methods

Classical 4.73004 7.85320 10.9956 14.1372 17.2788 20.4204 23.5619 Solution

Table 4. The comparison study of the first eight dimensionless frequency parameters λi of the free-free Timoshenko beams for different thickness-to-length ratios.

Methods

Classical 4.73004 7.85320 10.9956 14.1372 17.2788 20.4204 23.5619 26.7035 Solution h/L=0.002 Present

7.85304 10.9952 14.1362 17.2770 20.4174 23.5575 26.6970 h/L=0.005

4.72982 7.85217 10.9928 14.1311 17.2678 20.4022 23.5341 26.6630 h/L=0.01

4.72918 7.84908 10.9843 14.1131 17.2350 20.3483 23.4516 26.5436 h/L=0.02

4.72659 7.83679 10.9508 14.0426 17.1078 20.1415 23.1394 26.0979 h/L=0.05

4.70873 7.75404 10.7332 13.6040 16.3550 18.9813 21.4834 23.8654 h/L=0.1

4.64849 7.49719 10.1255 12.5076 14.6682 16.6358 18.4375 20.0959 h/L=0.2

Free Vibration Analysis of Timoshenko Beams… Table 5. The first eight dimensionless frequency parameters λi of the clamped-free Timoshenko beams for different thickness-to-length ratios.

Methods

10.9960 14.1371 h/L=0.002 10.9949 14.1360 h/L=0.005 10.9921 14.1301 h/L=0.01 10.9820 14.1093 h/L=0.02 10.9423 14.0283 h/L=0.05 10.6861 13.5309 h/L=0.1 10.5733 12.6524 h/L=0.2 10.1580 12.4559

Table 6. The first eight dimensionless frequency parameters λi of the clamped-pinned Timoshenko beams for different thickness-to-length ratios.

Methods

h/L=0.002 13.3508 h/L=0.005 10.2074 13.3458 h/L=0.01 10.1992 13.3283 h/L=0.02 10.1668 13.2595 h/L=0.05 9.9562 12.8306 h/L=0.1 9.3658 11.7583 h/L=0.2 8.0743 9.7860

4. Conclusions

The free vibration of the Timoshenko beams have been investigated for different thickness-tolength ratios. The obtained eigenvalues for the Timoshenko beams having various boundary conditions are compared with the previously published results. Using the Lagrange equations with the trial functions in the polynomial form and satisfying the constraint conditions by the use of Lagrange multipliers is a very good way for studying the free vibration characteristics of the beams. Numerical calculations have been carried out to clarify the effects of the thickness-tolength ratio on the eigenvalues of the beams. It is observed from the investigations that the results of the classical and the Timoshenko beam theory are very close to each other for small values of h / L . However, as the thickness-to-length ratio becomes larger, the results of the classical theory and the Timoshenko beam theory differ from each other significiantly. 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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