On free vibration of nonsymmetrical thin-walled beams

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On free vibration of nonsymmetrical thin-walled beams Daniel Ambrosini  Facultad de Ingenierı´a, Universidad Nacional de Cuyo, CONICET, Argentina

a r t i c l e in fo

abstract

Article history: Received 28 July 2008 Received in revised form 7 November 2008 Accepted 7 November 2008 Available online 31 December 2008

In this paper, a numerical–experimental study about natural frequencies of doubly unsymmetrical thinwalled and open cross-section beams is presented. The equations of motion are based on Vlasov’s theory of thin-walled beams, which is modified to include the effects of shear flexibility, rotatory inertia in the stress resultants and variable cross-sectional properties. The formulation is also applicable to solid beams, constituting therefore a general theory of coupled flexure and torsion of straight beams. The differential equations are shown to be particularly suitable for analysis in the frequency domain using a state variables approach. A simplified theory, which excludes the analysis the warping constraint, is presented leading to a more simple theory that is used for comparison purposes in this paper. Finally, free vibration experimental tests about unsymmetrical beams are presented which allow to verify the theory presented in this paper and provide good quality data that can be used for checking the accuracy and reliability of different theories. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Thin-walled beams Vlasov’s theory Frequency domain Free vibration

1. Introduction Thin-walled and open-section beams are extensively used as structural components in different structures in civil, aeronautical and mechanical engineering fields. Free vibrations of doubly symmetrical beams or beams with one axis of symmetry are widely studied, in general, by using Bernoulli–Navier theory. However, results about doubly unsymmetrical beams are rather scarce in the literature, especially those related with experimental evidences. In this case, triple coupled flexural–torsional vibrations are observed. The determination of natural frequencies and modes of vibration of undamped continuous beams and shafts is discussed in detail in Pestel and Leckie [1], who also describe the calculation of dynamic response to harmonic excitation. Ebner and Billington [2] employed numerical integration to study steady state vibrations of damped Timoshenko beams. Numerous other applications can be found in the literature concerning straight and curved beams, as well as arch and shell structures. On the other hand, the theory formulated by Vlasov [3] has been extensively used in the dynamic analysis of thin-walled, open-section beams. Nevertheless, although Vlasov’s theory for open-section beams is already firmly established, it presents some limitations, namely: (a) as in the common Bernoulli theory for flexure, it is assumed that shear strains do not contribute to the beam flexibility. Consequently, important errors should be expected in the analysis of deep beams or in the dynamic response associated to higher

 Corresponding author at: Los Franceses 1537, 5600 San Rafael, Mendoza, Argentina. Tel.: +54 2627 559947; fax: +54 261 4380120. E-mail address: [email protected]

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vibrations modes, even in the case of slender beams [4]. (b) The influence of rotatory inertia in the stress resultants is also neglected and (c) Vlasov’s fourth-order equations are valid only for beams with uniform cross-section. In previous papers [5,6], the author and co-authors proposed a modified theory, which is based on Vlasov’s formulation, but it accounts for the effects mentioned above. This formulation, using the so-called state variables approach in the frequency domain, lends itself to efficient numerical treatment, which on account of generality and precision can be very useful in a variety of applications. Other theories that also account for coupling between bending and torsion in beams are presented in Gere and Lin [7] who derive a simplified equation for uniform open-section beams and Muller [8] who formulates a general theory that include all coupling effects between the equations of motions, but it is not easy to handle in applications. Most other contributions in the field are restricted to particular cases. For example, Aggarwal and Cranch [9] and Yaman [10] deal with channel-section beams and Ali Hasan and Barr [11], with equal angle-sections. Regularly, new theories and approaches are presented in the literature and many comparisons were performed between them. Sometimes, some discussions and controversies are founded. As an example, one can see the critiques made by Arpaci and Bozdag [12] about the paper of Tanaka and Bercin [13]. In this paper, for instance, it will be demonstrated that Arpaci and Bozdag [12] fell in similar mistakes to those criticised to Tanaka and Bercin [13]. However, in most cases, it is notable the lack of experimental data that is essential to compare with experimental results in order to obtain reliable conclusions about the accuracy and applicability of different theories. Tanaka and Bercin [13] extend the approach of Bishop et al. [14] to study triple coupling of uniform beams using

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Mathematica. The governing differential equations of authors [13] exhibit a confusion of co-ordinate system and this confusion was clarified by Arpaci and Bozdag [12]. However, the equations presented in the last paper [12] correcting those presented in [13] are verified comparing results with a case that neglects the same terms of Tanaka and Bercin [13] making it impossible to verify the accuracy of the two theories. In the last few years, valuable papers in the area were published. Li et al. [15] present a dynamic transfer matrix method of determining the natural frequencies and mode shapes of axially loaded thin-walled Timoshenko beams. Prokic [16] presents five governing differential equations for coupled bending–torsionalshearing vibrations by using the principle of virtual displacements. Chen and Hsiao [17] analyzed the coupled vibration of thin-walled beams with a generic open section induced by the boundary conditions, using the finite element method. Finally, some valuable papers about thin-walled curved beams should be mentioned; Yoon et al. [18] and Piovan and Cortı´nez [19], among others, investigated the free or forced vibrations of thin-walled curved beams. In this paper, a numerical study based on equations developed in previous papers [5,6] is presented and the results of Arpaci and Bozdag [12] are discussed. Moreover, the field matrix of the state variables approach is presented for the case in which the warping is neglected. These equations lead to a more simple theory that is used for comparison purposes in this paper. Finally, results of experimental tests are shown because it is the way in which the results obtained by different theories should be verified. These experimental results can be used for checking the accuracy and reliability of calculation methods and procedures.

motion for free and forced vibrations may be found, as well as comparisons with other continuum formulations and a thorough discussion of the definition of shear coefficients. The present paper is more oriented to practical applications and only the derivation of the differential equations for free vibrations in the state variables method is given. In Fig. 1, A represents the centroid and O the shear centre. For the case of free vibrations, the physical model is formed by the following three, fourth-order partial differential equations in the generalised displacements x, Z, and y: ! ! " # q4 x q3 gmx q3 x q2 gmx dJy ðzÞ þ 2   E J y ðzÞ dz qz4 qz3 qz3 qz2 ! ! 4 3 3 dJ y ðzÞ q x q x q gmx q2 gmx    rJ y ðzÞ r dz qt2 qz2 qt2 qzqt2 qzqt2 ! þ rF T ðzÞ "

3

2

q3 gmy q4 Z   rJ x ðzÞ 2 qz2 qt qzqt2

!

!

þ rF T ðzÞ

!

"

3

#

2

q gmy dJ ðzÞ q Z  r x dz qzqt2 qt 2

q2 Z q2 y þ ax 2 ¼ 0, 2 qt qt #

q4 y

q3 y dJ j ðzÞ

E J j ðzÞ 4 þ 2 3 qz qz

dz

 rJ j ðzÞ

!

(1b)

q4 y qz2 qt2

! 2 dJ j ðzÞ q y q2 x q2 Z 2q y þ r F ðzÞ a  a þ r y x T dz qzqt 2 qt 2 qt 2 qt 2 3

2. Theory  GJd ðzÞ

Following Vlasov’s convention, the left-handed rectangular global co-ordinates system (x, y, z) shown in Fig. 1 was adopted. The associated displacements are designated x, Z, and z. The basic concepts needed to introduce the effects of shear strains, rotatory inertia and variable cross-sectional properties within the framework of Vlasov’s theory are described by the authors in previous papers [5,6], in which a complete derivation of the equations of

!

(1a)

q4 Z q gmy q3 Z q gmy dJx ðzÞ   þ 2 E J x ðzÞ dz qz4 qz3 qz3 qz2

r

2.1. Equations of motion

q2 x q2 y þ ay 2 ¼ 0, 2 qt qt

q2 y dJ ðzÞ qy ¼ 0. G d dz qz qz2

(1c)

In these equations, FT is the cross-sectional area, Jx and Jy are the second moments of area of the cross-section in relation to the centroidal principal axes, Jj is the sectorial second moment of area, Jd is the torsion modulus, and ax and ay are the co-ordinates of the shear centre. r denotes the mass density of the beam material. E and G are the Young’s and the shear modulus, respectively. Finally, gmx and gmy represent the mean values of shear strains over a cross-section z ¼ constant and r 2 ¼ a2x þ a2y þ

Jx þ Jy . FT

(2)

The system (1a)–(1c) represents a general model of nonuniform beams that take into account triply coupled flexural–torsional vibrations. It must be pointed out that the longitudinal vibration equation related to the generalised displacement z (Fig. 1) is non-coupled with the rest of the system (1a)–(1c) and it was not taken into consideration in the analysis. In the case that the longitudinal vibrations are of interest, this equation can be treated independently. 2.2. State variables method

Fig. 1. Definition of terms.

Using the Fourier transform, an equivalent system with twelve first-order partial differential equations with twelve unknowns, in the frequency domain, is obtained. The scheme described above is known in the literature as ‘‘state variables approach’’. Six geometric and six static unknown quantities are selected as components of the state vector v: displacements x and Z, bending rotations fx and fy, normal shear stress resultants Qx and Qy, bending moments Mx and My, torsional rotation y and its spatial

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derivative y0 , total torsional moment MT and bimoment B: 0

T

vðz; oÞ ¼ fZ; fy ; Q y ; M x ; x; fx ; Q x ; M y ; y; y ; M T ; Bg ,

general theory of beams applicable to solid as well as thin-walled beams. (3)

in which, the total torsional moment is given by M T ¼ Hj þ Hk ,

2.3. Simplified theory (4)

0

with Hk ¼ GJdy ¼ Saint Venant torsion moment. The system is

qv ¼ Av, qz

631

(5)

in which A is the system matrix given by

In order to compare the numerical results to those presented by Yaman [10] and Arpaci and Bozdag [12], which is one of the motivations of the present paper, a simplified theory is presented. This model excludes the analysis of the warping constraint, reducing the system (5) to ten first-order partial differential equations with ten unknowns. In this case, the vector of state is

(6)

in which k0 x and k0 y denote the Cowper’s shear coefficients and 0

By ¼ rJj o2  GJd .

v ðz; oÞ ¼ fZ; fy ; Q y ; Mx ; x; fx ; Q x ; My ; y; M T gT ,

(8)

(7)

The components of the 12-dimensional state vector v are designated ‘‘state variables’’. In the frequency domain, the state variables depend on the frequency o and the longitudinal coordinate z. For simplicity, the same notation is being used for the state variables and their Fourier transforms, since the domain can usually be identified by the indication of the function arguments. For example, Z(z, t) and Z(z, o) refer to the y-displacement in the time domain and to its Fourier transform, respectively. It is important to note that the present formulation constitutes a

in which M*T denotes the torsional moment, in this case given by 0

MT ¼ GJ d y ,

(9)

The new system is

qv ¼ A v , qz

(10)

in which A* is the system matrix given by

(11)

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3. Numerical procedure and boundary conditions

O

z h

System (5) may be easily integrated using standard numerical procedures, such as the fourth-order Runge–Kutta method, the predictor–corrector algorithm or other approaches. In order to solve the two-point value problem encountered both in the determination of natural frequencies and in dynamic response calculations, the latter must be transformed to an initial value problem as shown, for example, by Ebner and Billington [2]. The procedure is normally applied in the transfer matrix method [1]. Natural frequencies are determined by means of the well-known Thomson’s method. The classical boundary conditions are considered in this paper: clamped, free or simply supported. Clamped boundary

x ¼ Z ¼ 0; fx ¼ fy ¼ y ¼ 0; y0 ¼ 0.

cy C cz

(12)

ξ

y

Free boundary Q y ¼ Q x ¼ 0; M x ¼ M y ¼ M T ¼ 0; B ¼ 0.

ν

(13)

Fig. 2. Cross-section of the beam of the Example 1. Yaman’s co-ordinate system.

Hinged boundary

x ¼ Z ¼ 0; Mx ¼ My ¼ 0; y ¼ 0; B ¼ 0.

(14)

O

h

ξ

4. Numerical results and discussion

A As stated by Arpaci and Bozdag [12] in the theory developed by Tanaka and Bercin [13], although the flexural displacement and the offsets of the shear centre are determined with respect to the axes which are perpendicular and parallel to the web of unsymmetrical channel, the equations are formulated as if the principal axes are used. Effectively, in Eqs. (1)–(3) of Tanaka and Bercin [13] the product of inertia terms are not included. However, because Tanaka and Bercin [13] do not give the value of product of inertia for the cross-section considered by them in the numerical example, the comparison carried out by Arpaci and Bozdag [12] was not made for the same crosssection. Moreover, although all properties of the unsymmetrical beams used in examples 1 and 2 of Arpaci and Bozdag [12] are given in this reference, only the relative error between the two papers was presented and consequently it was not possible to compare with the theory presented in this paper. In this sense, these authors [12] fall into the same problem criticized in [13]. On the other hand, Arpaci and Bozdag [12] present another example to show good agreement between their results and those obtained by Yaman [10] using the wave propagation approach. However, Arpaci and Bozdag [12] surprisingly use Yaman’s results in which the product of inertia terms and the warping constraint are not included. In this way, it is impossible to verify if the theory presented by Arpaci and Bozdag [12] leads to right natural frequencies in the case of nonsymmetrical open cross-section beams. For this reason, the complete example presented by Yaman [10] is developed next. The theoretical model used by Yaman [10], whose crosssection is presented in Fig. 2, has the following geometric and material properties: L ¼ 1.00 m In ¼ 5.08  109 m4 Inx ¼ 4.25  109 m4 G0 ¼ 7.11 1012 m6 cz ¼ 9.09  103 m E ¼ 7.00  1010 N/m2

ax

ay

x 0

η

y Fig. 3. Cross-section of the beam of the Example 1. Principal centroidal axes.

G ¼ 2.60  1010 N/m2 A ¼ 9.68  105 m2 Ix ¼ 2.24  108 m4 h ¼ 38.10  103 m J ¼ 5.20  1011 m4 cy ¼ 10.43  103 m r ¼ 2700 kg/m3.

In Yaman’s notation [10] L is the length of the beam, A the crosssectional area, In and Ix are the second moments of area of the cross-section in relation to centroidal axes perpendicular and parallel to the web of the channel beam, Inx the product of inertia, G0 the warping constant about shear centre, J the torsion constant and cz and cy are the eccentricities between the centroid and the shear centre. The mechanical properties have the same notation of this paper. In order to use the theory presented in this paper, the second moments of area of the cross-section in relation to the centroidal principal axes (Fig. 3) must be determined as well as the coordinates of the shear centre. Then, the following well-known equations are applied: tg2a0 ¼

2IZx , Ix  IZ

(15a)

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J x ¼ IZ cos 2 a0 þ Ix sen2 0  IZx sen 2a0 ,

(15b)

J y ¼ Ix cos 2 a0 þ IZ sen2 a0 þ IZx sen 2a0 .

(15c)

Table 1 Natural frequencies (Hz) of the beam of the example 1: Case a. Reference

Mode

Moreover, it can be demonstrated that ax ¼ cz cos a0  cy sen a0 ,

(16a)

ay ¼ ðcy cos a0 þ cz sen a0 Þ.

(16b)

633

Arpaci and Bozdag [12] Yaman [10] This paper

1

2

3

4

5

6

47.00 45.49 45.47

73.39 69.91 69.88

102.36 101.73 101.72

145.34 149.82 149.47

154.53 154.79 154.80

206.70 207.43 207.25

Then, the geometric and the mechanical properties used in order to apply the theory presented in this paper are: l ¼ 1.00 m Jx ¼ 4.09  109 m4 Jy ¼ 2.34  108 m4 Jxy ¼ 0 Jj ¼ 7.11 1012 m6 Jd ¼ 5.20  1011 m4 ax ¼ 6.50  103 m ay ¼ 12.22  103 m FT ¼ 9.68  105 m2 h ¼ 38.10  103 m E ¼ 7.00  1010 N/m2 n ¼ 0.346 r ¼ 2700 kg/m3. The boundary conditions for this example are considered simply supported. Three cases were analyzed. Case a: Stiffness coupling terms and effects of warping constraint are ignored As the product of inertia terms couples the flexural vibration in the two directions considered, Yaman [10] named the product EInx as ‘‘coupling stiffness’’ and, in first place, it presents the natural frequencies for the case in which the stiffness coupling terms and additionally the effects of warping constraint are ignored. In this case, the natural frequencies have the same error as those obtained applying the theory presented by Tanaka and Bercin [13]. Obviously, these results cannot be used to verify a theory of nonsymmetrical thin-walled beams. It is important to note that Arpaci and Bozdag [12] present this case in order to show the good agreement between their results and those obtained by Yaman [10]. Although [12] indicates a value different from zero for the product of inertia, it is clear that [10] do not take into account this term in this case. It is surprising to note that Arpaci and Bozdag [12] have in it an error similar to that in [13] and this adds confusion to the topic. In the context of this paper, the simplified theory presented in Section 2.3 was used. Moreover, in order to ignore the stiffness coupling and to compare the results with those of Yaman [10], the following values were considered: Ix ¼ In, Iy ¼ Ix and Ixy ¼ 0. Table 1 shows the results obtained. The frequencies corresponding to Modes 1, 3, 5 and 6 are torsion-dominated resonance frequencies and those corresponding to the Modes 2 and 4 are bending-dominated resonance frequencies. There is complete agreement, for this case, between the simplified theory presented in this paper and the wave propagation theory [10]. Case b: Only effects of warping constraint are ignored In this case, the simplified theory presented in Section 2.3 was used. Moreover, the values of Ix and Iy that correspond to centroidal principal axes were considered. Table 2 shows the results obtained.

Table 2 Natural frequencies (Hz) of the beam of the example 1: Case b. Reference

Mode

This paper. Case a This paper. Case b

1

2

3

4

45.47 45.28

69.88 60.27

101.72 102.18

149.47 155.10

Table 3 Natural frequencies (Hz) of the beam of the example 1: Case c. Reference

Yaman [10] This paper Difference (%)

Mode 1

2

3

4

51.87 51.39 0.9

114.79 96.48 16.0

207.41 188.84 9.0

263.88 204.81 22.4

As demonstrated by Yaman [10] the consideration of the coupling stiffness is found to affect the bending-dominated resonant frequencies but it has no significant effect on the values of the torsion-dominated frequencies. Case c: General case. All couplings and warping constraint effects are considered In this case, the general theory presented in Section 2.1 was used. Table 3 shows the results obtained. The frequencies corresponding to Modes 1 and 3 are torsion-dominated resonance frequencies and those corresponding to Modes 2 and 4 are bending-dominated resonance frequencies. It may be seen that the first natural frequency is in close agreement. There are some discrepancies, however, in the higher modes especially in the flexural-dominated ones. In this context, it would be useful to check the values in Table 3 against experimental observations.

5. Experimental model 5.1. Case study Sometimes comparing numerical results obtained with different theories leads to an undefined problem and another point of view is necessary to solve differences and show the accuracy of these theories. As pointed out in Section 1, experimental evidences about doubly unsymmetrical beams are rather scarce in the literature. Moreover, taking into account the controversial point discussed in this paper, series of free vibration test were carried out about doubly unsymmetrical aluminum beams.

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Beams with cross sections shown in Fig. 4 were tested. Two different lengths were applied: 2 m (Beams 1 and 2) and 1.5 m (Beams 3 and 4). Moreover, two support conditions were used: fixed–free and fixed–fixed. For space reasons, only the results for Beam 1 fixed–free and fixed–fixed are shown in this section, but all beams tested show qualitatively the same results.

5.2. Experimental set-up and instrumentation

In order to carry out the system identification and determination of natural frequencies, the peak-picking method was used [21]. This method has enough accuracy for this type of elements. Procedures and filters to avoid aliasing and leakage were applied [21]. As an example, accelerations registered by a transducer and the corresponding frequency spectrum are presented in Fig. 6 for the case of Beam 1 fixed–fixed. Then, the geometric and measured mechanical properties used in order to apply the theory presented in this paper are:

l ¼ 2.00 m Jx ¼ 3.024  108 m4

20 mm

40 mm

Three KYOWA AS-GB accelerometers were used to measure the dynamic response of the beams. A dynamic strain amplifier— KYOWA DPM-612B—amplified the signal generated by the accelerometers. A data acquisition board Computerboards PCMDAS16D/16 of 100 kHz was mounted on a notebook computer in order to record and process the signals by means of the program HP VEE 5.0 [20]. The beams were excited with a hammer blow at different points in order to excite different modes. A global view of one test is presented in Fig. 5. Three channels were used in all tests and the signals were sampled at 500 Hz in each channel.

5.3. Dynamic response

e = 2 mm

100 mm

e = 1.6 mm

10 mm

25 mm

10 mm

76 mm Fig. 4. Cross-section of the beams tested: (a) Beams 1 and 3; (b) Beams 2 and 4.

Fig. 5. Experimental set-up and instrumentation.

Fig. 6. Dynamic response measured: (a) waveform (time); (b) magnitude spectrum.

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635

Table 4 Natural frequencies (Hz) of Beam 1. Freq.

1 2 3 4 5

Mode

1 1 2 2 3

Flexural–vertical Torsional–flexural Flexural–vertical Torsional–flexural Flexural–vertical

Test (Hz)

5.2 12.6 28.4 – 69.4

This paper

SAP2000 [22] 60 elements

SAP2000 [22] 900 elements

Freq. (Hz)

Diff. (%)

Freq. (Hz)

Diff. (%)

Freq. (Hz)

Diff. (%)

5.3 12.0 28.5 50.9 67.3

1.9 4.8 0.4 – 3.0

5.15 9.9 26.5 41.9 59.1

1.0 21.4 6.7 – 14.8

5.18 12.06 28.54 48.7 66.38

0.4 4.3 0.5 – 4.4

Fixed–free. Numerical–experimental comparison.

Table 5 Natural frequencies (Hz) of Beam 1. Freq.

1 2 3 4

Mode

1 1 2 3

Flexural–vertical Flexural–lateral Flexural–vertical Flexural–vertical

Test (Hz)

24.5 49.1 63.5 122.5

This paper

SAP2000 [22] 60 elements

SAP2000 [22] 900 elements

Freq. (Hz)

Diff. (%)

Freq. (Hz)

Diff. (%)

Freq. (Hz)

Diff. (%)

24.8 47.9 63.4 119.1

1.2 2.4 0.2 2.8

22.2 40.5 55.2 102.8

9.4 17.5 13.1 16.1

24.7 44.9 61.9 114.1

0.8 8.6 2.5 6.9

Fixed–fixed. Numerical–experimental comparison.

Jy ¼ 4.595  107 m4 Jxy ¼ 0 Jj ¼ 2.838  1011 m6 Jd ¼ 4.270  1010 m4 ax ¼ 2.67  102 m ay ¼ 1.30  102 m FT ¼ 3.12  104 m2 E ¼ 4.50  1010 N/m2 n ¼ 0.25 r ¼ 2650 kg/m3.

Considering that, in this case, the Vlasov’s hypothesis about the slenderness of the beam are fulfilled, a large shear coefficient was used in order to neglect the shear flexibility (k0 x ¼ k0 y ¼ 2000; see Eq. (6)) and to approximate the solution corresponding to the classical Vlasov’s theory plus rotatory inertia. Moreover, in order to have an additional comparison of the theory proposed in this paper, for the case of doubly unsymmetrical thin-walled beams, the results for the cantilever beam are compared with a finite element solution. The finite element software SAP2000N [22] was used in this comparison, modeling the beam by means of 60 and 900 rectangular shell elements of four nodes. The same mechanical properties were used in this case. The results are presented in Table 4 for the case of Beam 1 fixed–free and in Table 5 for the case of Beam 1 fixed–fixed. In both cases a coarse (60 elements) and a refined (900 elements) FEM shell model was used. In Table 4 it must be noted that the frequencies corresponding to Modes 2 and 4 are torsion-dominated resonance frequencies and those corresponding to Modes 1, 3 and 5 are bendingdominated resonance frequencies. It may be seen that the second torsion-dominated resonance frequency (Mode 4) could not be captured in the tests. In Table 5 it must be noted that the frequencies corresponding to Modes 1–4 are bending-dominated resonance frequencies. It is important to note that the differences between the experimental results and those obtained with the theory

presented in this paper are lower than 5% in all cases. Moreover it is contended that the accuracy of the shell FEM solution and this theory are comparable. Obviously, the accuracy of the FEM solution could be improved adding elements, but with a higher computational cost.

6. Conclusions In this paper, the equations of motion for thin-walled, variable open cross-section beams have been presented within the socalled state variables approach in the frequency domain. The equations take into account the influence of shear flexibility and rotatory inertias which are neglected in the original Vlasov’s theory. The equations enable the analysis of practical problems using direct numerical integration in conjunction with techniques routinely applied in transfer matrix analyses. In addition, they may be resorted to in order to numerically evaluate transfer matrices or stiffness matrices for open-section beam elements. Moreover, the proposed theory can also be used for solid beams in which coupling between bending and torsion occurs. Moreover, the field matrix of a simplified model that excludes the warping constraint is also presented. A discussion related to vibration of doubly unsymmetrical channel cross-section beams is presented. In this sense, relevant topics discussed in recent works [10,12,13] are pointed out and clarified. Experimental studies show very good agreement between the results obtained with the theory presented in this paper and those obtained in the tests. Moreover, it is contended that the accuracy of this theory is comparable to those of shell FEM solutions.

Acknowledgements The author wishes to thank the collaboration of Profs. Jorge Riera and Rodolfo Danesi, the help received from technical staff, Mr. Eduardo Batalla and Mr. Daniel Torielli, during the development and preparation of the tests and Ms. Amelia Campos for the

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