thinwallvibs_2

Thin-Walled Structures 00 (2014) 1??

Thin-Walled

Structures

Elasticity-Based Free Vibration ofAnisotropic Thin-Walled Beams

Paul R. Heyliger

Department of Civil and Environmental Engineering, Colorado State University, Fort Collins, CO 80523, 970-491-6685,

[email protected]

AbstractThe free-vibration of anisotropic open and closed thin-walled beams is studied using approximate solutions to the three-dimensionalequations of motion of a linear elastic solid. The three displacement components are approximated using grouped polynomialsover both the cross-section coordinates and independent polynomial or trigonometric functions over the axial coordinate of severalbeams with representative boundary conditions. This method is applied to several sections with varying degrees of anisotropy andshape with both isotropic and orthotropic material properties. Results are compared with those computing using beam or shelltheories, and several new results are presented for these geometries.

Introduction

Three-dimensional solids that have one-dimension much larger than the other two and a cross-sectioncomprised of elements of that are generally long and thin are usually classified as thin-walled beams. Theirbehavior, which can combine elements of motion usually associated with axial, flexure, and torsional mo-tion, has been extensively studied ever since the pioneering work of Wagner and Pretschner [1], Bleichand Bleich [2], and Kappus [3]. As noted by Gjelsvik [4], the comprehensive theory of open thin-walledsections of Vlasov was also developed during the same period as these early works but not translated intoEnglish until 1963 [5]. By this time Goodier [6] and Timoshenko [7] had independently developed generaltheories. Additional work by Gere and Lin [8] on coupled vibration behavior outlined some of the moresignificant issues related to these components. Numerous computational models have been further devel-oped, including the dynamic stiffness matrix method of Banerjee and co-workers [9] and the finite elementmodel of Mei [10]. More recent applications to composite beams have also been considered by Kollar [11],Cortinez and Piovan [12], Piovan and Cortinez [13]. Vibration of thin-walled beams under initial stress hasbeen studied by Machado and Cortinez [14].

Benscoter [15] developed an early theory for thin-walled beams with closed cross-section, where thegeneral behavior becomes more complicated. Significant progress has been made since this time, includinga number of approaches for anisotropic beams, including those of Hodges and co-workers [16] and Songand Librescu [17]. Summaries of much of this work have been given by Murray [18], Librescu and Song[19], and Hodges [20] for beams of open and closed section under both static and dynamic response.

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Paul R. Heyliger / Thin-Walled Structures 00 (2014) 1?? 2

By far the most comprenensive studies of thin-walled beams have used beam, plate, or shell theories inwhich displacement fields are assumed over the domain of the plate that restrict the motion of the displace-ment components [21]. By their own definition, the displacement fields of all beam theories, thin-walledor not, require some sort of kinematic restriction. This usually takes the form of assumed componentsof representative displacement and/or stress over the cross-section of the beam. This is intended to allowa rapid analysis that captures the key mechanics of these solids. Approximate beam models are usuallycompared with either more detailed elasticity results or experimental evidence to assess their accuracy.It is less typical to allow for a full representation of the usual displacement components consistent withthe three-dimensional theory of elasticity. In this study, elasticity-based displacement fields are assumedover the cross-section of both isotropic and anisotropic beams. This is accomplished by applying the Ritzmethod directly to the three-dimensional equations of elasticity using power-series approximations in theform introduced by Visscher [22]. The intent of this work is to determine the level of accuracy of moresimplified models, provide an alternative means of analysis for these solids, and present numerical resultsfor thin-walled beams to provide a basis for future comparison.

Geometry and Governing Equations

We consider a three-dimensional solid whose cross-section coordinates are defined in the (x1,x3) or (x-z) plane with a much larger dimension in the x2 or y axis with a length of L. The beam is assumed to becomposed of an anisotropic material whose principal material axes are aligned with the (x,y,z) axes. Thismaterial is assumed to be originally orthotropic and having gone through a rotation in the x-y plane at anangle as measured from the positive x axis (the horizontal component of the cross-section axis). The beamis hollow with a uniform wall thickness of t.

The Weak Form

The governing equations used for this study are the three-dimensional equations of linear elasticity withan anisotropic constitutive tensor that can represent what amounts to a monoclinic material. These are notsolved explicitly at each point in the domain, but instead approximate solutions are sought for their weakform as expressed within Hamiltons Principle [23]. This is usually expressed as

0 = t

0

V{11 + 22 + 33 + 44 + 55 + 66}dVdt + 12

t0

V(u2 + v2 + w2)dVdt (1)

Here the conventional contracted notation has been used for Cauchy stress components (11 = 1, 23 =4, 1 = 11, 4 = 223, C1111=C11, C1123=C14, etc.) and it is understood that the 1, 2, and 3 directions are(x1 = x, x2 = y, and x3 = z).

The general constitutive law that links the Cauchy stress to the components of infinitesimal strain i jcan be written as

i j = Ci jklkl (2)

where the strain components are linked with the three displacement components u, v, and w in the x, y, andz directions as

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1 = 11 =ux

2 = 22 =vy 3 = 33 =

wz

4 = 23 =vz

+wy

5 = 13 =uz +

wx 6 = 12 =

vx

+uy. (3)

The general constitutive relation for the materials used in this study can be expressed in matrix form as

123456

=

C11 C12 C13 0 0 C16C12 C22 C23 0 0 C26C13 C23 C33 0 0 C360 0 0 C44 C45 00 0 0 C45 C55 0C16 C26 C36 0 0 C66

123456

(4)

These stiffness components are associated with either a naturally occurring monoclinic material with 13independent elastic constants or a rotated orthotropic material. In the latter case, the final Ci j as listed in the6x6 matrix are a function of the on-axis elastic constants and the angle of rotation of originally orthotropicproperties about the 3 or z axis. Their explicit forms are given in Reddy [24].

The Ritz Model

The computational approach used in this study relaxes many of the assumptions used in most beamtheories. Rather than introducing specific displacement fields over the beam domain, the three point-wisedisplacement components are approximated directly using the three-dimensional equations of elasticity.There is no need to represent any rotational variables or their equivalent in this theory, since each of thedisplacements is computed at every location in the beam.

The general form for the displacements can be expressed as

ui(x j, t) = uio (x j) +

Np=1

Aip(t)uip (x j) (5)

Here the Aip represent unknown constants that depend on time. The three values of i indicate that there arethree components of displacement and that three independent sets of functions can be used to approximateeach of those displacements. The o terms represent the simplest functions that satisfy the essential bound-ary conditions for that displacement direction. In the study discussed in this paper, the initial boundaryconditions will be assumed to be zero, with no initial displacement or velocity. The o terms are there-fore all equal to zero for the considered case. In these approximation equations, p represents a selectedapproximate function for each respective direction. These functions must satisfy general requirements ofindependence as outlined by Reddy [23]. By using a large number of these approximation terms for eachdisplacement component, very accurate solutions can be determined for the mode shapes and frequenciesof vibration for a beam.

In this study, polynomial series are used over the cross-section of the beam and then combined with otherpolynomials or transcendental functions over the axial length of the beam. This class of approximation hasa long and rich history in the field of mechanics. Among the first to use it for the case of vibrating solidswas Demarest [25], who used Legendre polynomials over the three Cartesian coordinate directions to find

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the natural modes of vibrating parallelepipeds. This method was further studied by Ohno [26], who alsodiscussed the splitting of the resulting eigenvalue problem into a smaller number of equivalent eigenvalueproblems using group theory. Polynomial expansions were also used in general cylinder [27] and sphere[28] vibrations. A significant and powerful expansion of this approach was introduced by Visscher [22],who expanded the three displacement components in terms of power series regardless of the shape of thesolid. This is generally written as

up(x, y, z) = xiy jzk (6)

This is a very attractive family of approximations in that it frequently yields very simple integration resultsover the domain of the solid. The nature of this integration will be discussed for specific cases in the sectionsthat follow. This type of approximation has seen extensive application in resonanting solids [29], vibratingbeams [30], and related areas such as granular media [31]. In terms of applications to solids with moregeneral beam geometry, Carrera and co-workers have recently developed a unified beam theory using amodification of this type of approximation over the beam cross section and have extensively applied it to awide array of problems [32, 33].

If the displacement constants cip are replaced with the letters c, d, and e for the three displacements(1=u, 2=v, 3=w) for any index p, it is possible to generalize the matrix form of the Ritz model applied toHamiltons principle following the assumption of general harmonic motion. Specifically, the Ritz coeffi-cients can be written as

A1p(t) = cp sint (7)

A2p(t) = dp sint

A3p(t) = ep sint

(8)

Substitution of these conditions and the general Ritz approximation into Hamiltons principle leads to thegeneralized eigenvalue problem that can be represented in matrix form as

[K11] [K12] [K13]

[K21] [K22] [K23][K31] [K23] [K33]

{c}{d}{e}

= 2 [M

11] 0 00 [M22] 00 0 [M33]

{c}{d}{e}

(9)The elements of the stiffness ([K]) and mass ([M]) matrices are given in the Appendix.

In the specific cases that follow, the (x,z) plane is taken as the cross-section of the beam and the y-axis isthe longitudinal axis of the beam. For the most part, the approximation functions in the y-direction appearas power series. For certain types of support conditions this may require some adjustment or replacement.This will be discussed for beams under simple support studied in the next section.

Computationally, using global power series over the cross-section greatly simplifies the integration re-quired for the relevant coefficient matrices. For hollow sections, for example, the integration can be com-pleted first over the outer bounding dimensions of the solid and then subtracting the same integral evaluatedover the inner ounding surfaces of the cross-section. The operations for rectangular sections are straightfor-ward. For the elliptical section, the integration requires a trigonometric substitution that can be evaluated inclosed form for any powers of x and z [31].

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One additional advantage of using global approximation functions over the beam cross-section andlength is the potential to group the specific approximations according to the physical character of the dis-placement components they are being used to represent [26, 34]. Various conditions of cross-sectionalsymmetry, axial wavelength, and grouping of the various approximation functions can drastically reducethe size of the matrix eigenvalue problem. Rather than generalizing this discussion, these features are dis-cussed for the specific geometries for which they are used in the examples that follow.

Beams with Simple Support: Semi-Analytic FEM Model

In cases where the axial (or y) dependence of the displacement functions are known, the form of theapproximation functions can be modified to reduce the dimension of the problem and fix the axial depen-dence as a given quantity. In other words, when the axial dependence is fixed the problem becomes two-dimensional in the cross-sectional variables (x,z). This allows for the introduction of a second independentmethod of analysis with which results of the present model can be compared. In the case of some beamsunder the end conditions of simple support beam, most beam theories usually require that the compponentsof transverse displacement are zero at the end-points and also that the resultant moment quantities are equalto zero. Using the elasticity theory of the present model, identical displacement conditions are imposed atthe endpoints of the beam for all points over the cross-section at all times. This implies that u(x,0,z,t) =u(x,L,z,t) = w(x,y,0,t) = w(x,L,z,t) = 0. In addition, the traction vector component along the axis ty = yyny = 0. For the types of materials considered in this study, each of these conditions is satisfied for beams upto and including orthotropic symmetry (C16 = C26 = C36 = C45 = 0) using the approximations

u(x, y, z, t) =nj=1

U j(t)uj(x, y) sin ky

v(x, y, z, t) =nj=1

V j(t)vj(x, y) cos ky

w(x, y, z, t) =nj=1

W j(t)wj (x, y) sin ky

(10)

Here k is the wave number and the capital letters for the field variables are each spatial constants asso-ciate with the individual subscript j, each of which correspond to independent shape functions as denotedby and each of which can still vary with time. When combined with conventional finite element approx-imations in two dimensions, they represent the nodal values of the displacement components. As for thegeneral Ritz method, the superscript on each function indicates that the approximation functions for each ofthe variables need not be the same. Depending on the geometry of the problem each approximation functioncan take a specific form. The variations of the primary field variables within Hamiltons Principle are thentaken as

u = ui (x, y) v = vi (x, y) w =

wi (x, y) (11)

Although solutions can be determined for the general transient case, the focus in this study is on the calcu-lation of periodic frequency for a given wave number k. Hence it is further assumed that the time dependent

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behavior for the spatial constants is in the form given by general periodic motion as

U j(t) = U j sint V j(t) = V j sint W j(t) = W j sint (12)

Here t is time and is the natural frequency of free vibration. Substitution of these functions and collectingterms allows the weak form from Hamiltons principal to be expressed in matrix form in a manner nearlyidentical to the original Ritz method:

[K11] [K12] [K13]

[K21] [K22] [K23][K31] [K23] [K33]

{U}{V}{W}

= 2 [M

11] 0 00 [M22] 00 0 [M33]

{U}{V}{W}

(13)There are two primary differences between this form and that of the general Ritz model that is the focus ofthe present study: 1) the integration need occur only over the beam cross-section, and 2) the constants (in thiscase the nodal variables) exist only over a single cross-section. This drastically reduces the computationalsize of the problem. This model is used as a method of comparison for the beams with simple support inthe sections that follow.

Results and Discussion

In the results that follow, the present method is labeled 3DRE to represent the three-dimensional Ritzelasticity structure of the analysis. This is used in all tabulated results for the examples considered. Severalgeneral shapes are considered. The first are somewhat standard beam cross-sections that are not necessarilythin-walled but are useful to study the accuracy of the present 3DRE model.

General Sections: The Reduction to Basic Beam Theories

One feature of Ritz-based methods based on power-series approxmations of the displacement compo-nents is that it allows a simple reduction to capture the behavior of many elementary theories. Han et at.[35] have published a complete assessment of the Euler-Bernoulli, Rayleigh, and Timoshenko beam the-ories under a variety of boundary conditions. These results can be then directly compared to the currentelasticity-based Ritz models to show the agreement between approaches.

In a power-series representation of the displacement vector, the displacement dependence on the cross-section coordinates x and z can be restricted to exactly match a number of beam theories for sectionsthat are symmetric about either of these axes (it is assumed for this section only that the cross-sectionsare symmetric about the z-axis): The Rayleigh theory can be recovered by setting the Poisson ratio usedto calculate the elastic stiffnesses Ci j to zero, setting i = k = 0 as a single term approximation for thedisplacement components u and w for all y, and keeping only the approximation i = 0 and k = 1 as theapproximation for v in Equation 6. In addition, the shear modulii are increased by 5-6 orders of magnitudeto reduce the influence of shear deformation. The Timoshenko theory can be simulated by using the samerestrictions as for the Rayleigh theory but using the exact shear modulii to account for first-order sheardeformation and using an appropriate shear correction coefficient.

The clamped-free beam is a very common configuration that has been extensively studied for a varietyof geometries. Han and co-workers [35] have given formulas for the characteristic frequency equations.The frequencies can be computed for any range of input parameters. For the Rayleigh theory, these includethe modulus of elasticity E, the density , the length L, and the cross-sectional area A. For the Timoshenko

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model these parameters are required in addition to the shear modulus G and the value of the shear coefficientk. This latter value has been the subject of intense study but there are fairly well accepted values for typicalcross-sections and noted by Han [35].

As an example, a cantilever beam was studied whose relative dimensions were 1 x 1 x 10 in the cross-section and length directions. Two shapes were considered: the square and the circle. The Rayleigh andTimoshenko approximations were used over the cross-section and so the Poisson ratio was assumed tobe zero. Along the axis of the beam, power series in y (y1, y2, and so on) were used for each of thedisplacement components to denote the requirement that all displacements are equal to zero at the clampedend where y=0. For each cross-sectional shape, two additional refinements were considered: the solidsection and the thin-walled closed section where the wall thickness is 1/100 of the outer dimension. Theresults gave agreement within four or five figures between the constrained elasticity theory developed hereand the Rayleigh and Timoshenko theories. The values of the shear coefficient [36, 37] were taken fromHan [35] who had summarized the work of Cowper [38].

Open Sections

The specific open thin-walled sections considered are the unsymmetric channel shown in Figure 1 andthe symmetric semicircle of Figure 2. Both of these sections have seen extensive study, and the goal of theirinclusion here is to evaluate the accuracy of earlier theories and models within the context of the presentelasticity solution.

The Doubly Unsymmetric Channel

The doubly symmetric section shown in Figure 1 was originally studied by Yaman [39] and has beenextensively used by a number of other researchers. It is an excellent example of a doubly symmetric sectionas there are no mirror planes that intersect the x-y cross-sectional plane. The material properties used werean elastic modulus E of 70 GPa, a shear modulus G of 26 GPa, and a density of 2700 km/m3. Yamanoriginally gave only the integrated section properties rather than the actual cross-sectional dimensions. Thelatter are needed to integrate the power series expansions using the present elasticity model. Since theseproperties were not given by Yaman, the integrated quantities were used to compute the leg dimensionsand thickness by minimizing the absolute difference between the area and the elements of the inertia tensor.Using this approach, the Yaman beam was determined to have a constant wall thickness of t = t2 = 1.275 (allmm), d1 = 13.29675, d3 = 25.9616, and d2 = 39.375. This gives A = 9.68 (9.68) x 105 m2, Ixx = 5073.64(5080) x 106 m4, Iyy = 22400 (22400) x 106 m4, and Ixy = 4256 (4250) x 106 m4 where the values usedby Yaman have been given in the parentheses. This gives a total absolute difference of 13.23 x 106 m4 forthe 3 values of the inertia tensor [40].

The level of agreement between competing formulations in regards to this cross-section have beensomewhat mixed. Ambrosini [41] has provided a recent summary and discussion regarding the results ofseveral researchers who have considered this cross-section, including the work of Arpaci and Bozdag [42]and Tanaka and Bercin [43]. Much of this discussion centers around the use of various elements of theinertia tensor of the beam cross section and other features related to the beam theories used to study thebeam behavior. This discussion does not apply to the present method since the actual cross-section is usedalong the the three-dimensional elasticity tensor components. But Ambrosini [41] notes the differences inthe lower frequencies among these various studies.

To assess the overall accuracy of the present method, the lowest four frequencies of the simply-supportedbeam were computed and compared with the values of Ambrosini [41] and those of Yaman [39] along withthe results from the two-dimensional semi-analytic finite element model and a fully three-dimensional finiteelement model. In the case of the two-dimensional finite element model, the wave number k can be directly

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input to extract the appropriate natural frequencies under the conditions of simple support. For the lowestfour modes, three of the frequencies are associated with the wave number k=pi. This is consistent with modalvibration patterns that possess variations along the axial length that are symmetric about the midpoint ofthe beam. The third lowest frequency is associated with the lowest anti-symmetric mode where k=2pi. Byanti-symmetric we mean about the mid-point of the beam. The mesh used to represent the cross-section forboth the two and three-dimensional finite element models has, along the three legs, 4, 12, and 8 elementdivisions and four element divisions through the thickness. The two corner regions therefore have 4 x 4meshes. A total of 128 elements with 165 nodes are used to model the complete cross-section. The natureof the semi-analytic approximation ensures that all boundary conditions are explicitly satisfied at the end-points of the beam. For the three-dimensional model, this same cross-section is used along with 40 elementdivisions along the length of the beam. Symmetry conditions were used so that only one-half of the beamlength is modeled. Hence the the boundary conditions at the simply-supported end and the mid-point of thebeam for the symmetric modes of the three-dimensional model are given by

u(x, y, 0) = v(x, y, 0) = zz(x, y, 0) = 0 (14)

w(x, y, L/2) = xz(x, y, L/2) = yz(x, y, L/2) = 0 (15)

As is usually the case in standard finite element formulations, the boundary conditions on the stress compo-nents are not satisfied pointwise but rather in an integral sense over the entire cross-section of the beam.

The results of these analyses are given in Table 1. There is good agreement between the various methodsfor the lowest frequency and also good agreement between the present methods, but significant variationsin the remaining frequencies. The modal shapes for the half-beams are also shown in Figures 3a-d.

The Thin-walled Semi-Circle

Friberg [44] was the first to study the vibration of the cantilevered beam shown in Figure 2 using Vlasovbeam theory. Since that time numerous researchers have also studied this particular cross-section, includingJun [45, 46], de Borbon and Ambrosini [47], and Petrolo and co-workers [48]. This beam section has someadvantages over the previous section in that the vibrational modes for isotropic beams can be split intogroups that reflect symmetry or asymmetry about the vertical axis that bisects the section. These two modalgroups have displacement components that are either even or odd about this axis. For symmetric modes, uis odd while v and w are even. For antisymmetric modes, u is even while v and w are odd about x. Theproperties of this section include an inner diameter di = 45 mm, a wall thickness of 4 mm, a length of 820mm, an elastic modulus of E = 68.9 GPa, a shear modulus of 26.5 GPa, and a density of 2711 kg/m3.

Two types of support conditions were considered: the fixed-free beam originally studied by Friberg [44]and the simply-supported beam of the same cross-section considered by Jun [45, 46] and de Borbon andAmbrosini [47]. In the present model, 3 terms were used in the x and y directions for the three variables(these terms include only those represented by the appropriate symmetry group) and 8 axial terms for thefixed-free beam. For the simply-supported beam, only a single axial term was used using the trigonometricapproximations from the semi-analytic model. This explicitly represents the appropriate boundary condi-tions and reduces the problem to being purely two-dimensional. The axial response is identified by the wavenumber k. Six in-plane terms were used for each of the displacement variables. For both support conditions,the frequencies were computed using both the full set of elastic constants and then repeated using the elastic

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constants that result with the assumption of zero Poisson ratio.For the fixed-free beam, the results are shown in Table 2. As a simple comparison, the bending frequen-

cies were also computed using the Rayleigh beam theory using the approach of Han [35]. These numericalvalues are the same to four significant figures with those computed by Friberg using the Vlasov beam theory[44]. The frequencies computed using the Timoshenko model of Jun [45] are in very good agreement withthe values obtained using the assumption of zero Poisson ratio. There is some discrepancy with the results ofPetrolo and co-workers [48], which is mildly surprising given that their approch uses similar power-seriesapproximations over the cross-section, albeit via forms incorporated into finite element approximations.The largest difference in frequency comes in comparison with the bending frequencies of the present modelusing the two constitutive theories. Those for zero Poisson ratio, which is a common assumption in manybeam theories, are consistently five to ten percent lower than those computed using the true elastic stiffnesstensor. The torsional frequencies are also influenced but to a much lesser extent.

The results for the simply-supported beam are given in Table 3, for which the main comparitive studyis the work of Jun [45, 46]. As before, the Rayleigh theory results can be computed using the methodologyof Han for the simply-supported beam [35] and these results are in good agreement with the Timoshenkoresults of Jun and the results of de Borbon and Ambrosini [47] with somewhat less agreement with thework of Petrolo and co-workers [48]. The present model gives frequencies that are close to those of Jun butare consistently lower by up to a single percentage. Additionally, there is little change in results using thepresent approach when the full elastic constants are used in the calculations.

Plots of the modal shapes along the length of the beam as measured at the outer diameter/thickness of thesection are shown in Figures 4-9 for the first three bending and torsional modes. The bending frequenciesare those for which Rayleigh frequencies are computer and hence give modal patterns dominated by thevertical or transverse displacement component v. The torsional frequencies are dominated by the in-planedisplacement components u and v but with a smaller warping motion out of the plane.

Closed Sections

In this section, two representative examples are studied that have been analyzed using other methods ofapproximation: the hollow monoclinic rectangular cantilevered beam and a homogeneous hollow ellipticalbeam under simple support. Both of these solids have beam-like behavior and have been studied by at leastone kinematic model that restricts the displacement field. The rectangular section has relatively thick wallswith anisotropic constitutive behavior. The elliptical section has isotropic properties with relatively thinwalls. Hence they provide a good range of comparison with the present model that does not restrict thenature of the displacement field other then truncating the number of terms in the series solution.

The Hollow Rectangle

Song and Librescu [17] considered the free vibrations of a hollow orthotropic thin-walled beam com-posed of an orthotropic solid that has been rotated about a single axis and is fixed at one end. In thenomenclature of this paper, the fibers always lie in the x-y plane with the rotation angle being measuredbetween the +x and +y axis. When = 0, the fibers are perpendicular to the long axis of the beam, andwhen = pi/2, the fibers are then along the axis of the beam. The material properties used are those of Songand Librescu and are taken as E1 = 206.843 GPa (30 x 106 psi), E2 = E3 = 5.171 GPa (0.75 x 106), 12 = 13= 23 = 0.25, G13 = G23 = 2.551 GPa (0.37 x 106 psi), and G12 = 3.1026 GPa (0.45 x 106. Here both SI andUS units have been given as the latter were used in the original study [17]. The original outer dimensionsof the rectangular section were given as 0.026416 m (1.04 inches) in the x-direction by 0.006096 m (0.24inches) in the y-direction with a thickness of 0.001016 m (0.04 inches) and a length of 0.254 m (10 inches)

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tiganiHighlight


in the y-direction. The beam is fixed at y=0. This implies that each of the displacement components is zerofor all (x,z) at y=0. The density was taken to be 1528.15 kg/m3 (14.3 x 105 lb/s2/in4).

Song and Leissa incorporated a kinematic model that enforces zero deformation in the cross-sectionalplane, allows for both primary and secondary warping, and assumes constant transverse shear strain. Sucha model is very useful in terms of computational effort but is likely to result in an overly stiff system thatoverestimates the bending frequencies. Part of the objective of this work is to assess such limits.

For the present model, the accuracy of the results is linked to the number of terms in the series usedto represent each of the three displacement components. As the number of terms increases, the computedfrequencies get lower and more accurate until an increase in the number of terms has little influence on theresults. For this geometry, an appropriate number of terms was assessed by computing the values of thelowest six frequencies for the case of = 0. It was found that a combination of nine axial terms (beginningwith y1 since the displacements must be zero at the fixed support) and five in-plane terms provides sufficientaccuracy for all of the results that follow.

For the extreme cases of = 0 and = pi/2, this beam configuration has axial, torsional, and flexuralmodes that completely uncouple. Group theory could have been used to drastically reduce the size of theunknowns for this problem by exploiting the symmetry of the resulting displacements. However, this wasnot used to maintain the same number of degrees of freedom for random angles of fiber orientation.

The axial frequencies of a bar that correspond to zero Poisson ratio approximations and a simplifieddisplacement field are given by [49]

n =(2n + 1)pic

2L(16)

Here c is the one-dimensional longitudinal wave speed given byE/. E is used to denote the elastic

modulus in the direction of the longitudinal wave and L is the length of the bar. For = 0, E = E2 and for = 90, E = E1. Similarly, the bending frequencies in the transverse (w-dominant) and lateral (u-dominant)directions can be approximated by the Rayleigh beam theory [35]. These assume axial displacements thatare linear in either z or x, respectively, and also incorporate the rotary inertia of the cross-section in thekinetic energy term. However, they do not include any shear deformation. These calculations are moreinvolved than using a single formula, but the frequencies can be easily calculated using the procedureoutlined by Han and co-workers [35]. Once again, the values of modulus of elasticity and second momentof area required by the Rayleigh theory are obvious depending on the direction of bending and can becomputed for the two extremes for the angle of fiber orientation.

The axial frequencies are given in Table 4 for the cases of = 0 and 90 degrees for the one-dimensionaland Song and Librescu theories, and for 15 degree increments for the present model for both the first andsecond modes. For all angles other than 0 or 90, the axial modes are not uncoupled from the rest of themotion. However, the modal displacements are the largest of all components and are generally constant overthe beam cross-section making them easy to identify. Two results are notable. First, there is a surprisingfour percent disagreement between the result of Song and Librescu for the first axial mode when = 0. Thismode is dominated by the elastic modulus of the matrix material, and it is odd that the theory of Song andLibrescu gives such a noticeable difference. The second result of note is the strongly nonlinear relationshipbetween the axial frequency and the angle of orientation of the fibers. Changing the angle from 0 to 45degrees results in under 25 percent increase in frequency, but an additional 45 degrees of rotation result inan increase in frequency of over a factor of 6. Hence the action of the fibers is strongly delayed until thefinal 15-20 degrees of rotation until they align with the axial motion.

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The torsional frequencies are given in Table 5. The uncoupled mode for = 90 is given by Song and Li-brescu for two conditions associated with neglecting or enforcing warping restraint within their theory. Thelatter case results in higher frequencies, which are in good agreement with the present model. These resultsalso indicate that the torsional stiffness for this material peaks near 60 degrees for the limited increments ofthis analysis.

Of most interest are the flexural modes since they are usually the easiest to excite and possess the lowestfrequency. These modes are classified according to their dominant displacement motion using the phrasestransverse bending modes and lateral bending modes. These incorporate flexural motion for which thedisplacements in the plane of the cross-section are generally in the z (transverse) or x (lateral) directions.Once again, for fiber angles of 0 and 90 degrees the flexural motion uncouples from the torsional and axialmodes, but for other angles there is significant rotation of the cross-section coupled with the flexural motionalong the lengths of the beam. The first three transverse modes are given in Table 6 and the first two lateralmodes are given in Table 7. The results of Song and Librescu are compared with the present model for 15degree increments, and values for the Rayleigh theory are also given for purposes of comparison for thelimiting cases of 0 and 90 degrees.

As expected, the resulting frequencies from the full equations of elasticity are lower than those of theone-dimensional beam model of Song and Librescu. The agreement for the lowest transverse mode is quitegood, with the beam theory results being within six percent of the elasticity results for all fiber orientations.The second transverse mode has good agreement for the lower fiber angles but overestimate the elasticityresults by over twenty percent as the fibers approach the axis of the beam.

The lateral frequencies are in good agreement for fiber orientations near 0 or 90 degrees, but for otherorientations are as much as 50 percent higher than the elasticity results. The second lateral modes are moreproblematic. Both the Rayleigh theories and elasticity theory give a frequency of over 5000 rad/sec forthis mode when the fibers are oriented at 0 degrees. The results of Song and Librescu have a frequency of4331 rad/sec. This pattern continues for all fiber orientations in that the results of Song and Librescu areroughly 10-15 percent under the elasticity results. This surprising result may be explained if the secondlateral mode identified by Song and Librescu was actually the third transverse bending mode, in which casethe comparison is far more in line with the elasticity results given in Tables 6 and 7.

Typical modal shapes for two representative fiber orientations are shown in Figures 10-13. The lowesttwo torsional modes are shown in Figure 10, and demonstrate the coupled nature of the deformation asthe fiber angle goes from 0 degrees to 60 degrees. In the former case, the torsional motion is dominantand virtually exclusive. As the fiber angle changes, the torsional motion still dominates the nature of thecross-sectional deformation but there is far more inclusion of deformation that is not strictly torsional thatcontributes to the stored energy and explains in part the increase in frequency for orientation angles between0 and 90 degrees. Similar behavior is demonstrated in Figures 11 and 12, which show the lowest twotransverse bending modes. For = 0, the motion is purely flexural. For other fiber angles, there is combinedvisible twist of the cross-section along with the dominant bending action. In the case of bending, thefibers are being reoriented to change what amounts to the longitudinal modulus of the beam, which alsoincreases the frequency. Only in the case of the lowest two lateral modes shown in Figure 13 do the modesstay visually uncoupled. This reflects the fact that the fibers are in the plane of the primary cross-sectiondeformation rather than perpendicular to the motion as is the case of transverse bending.

The original material properties of Song and Librescu were somewhat unusual for a fiber-reinforcedmaterial in the the Poisson ratios were identical and the shear moduli in the plane of the fibers were unequal.To present additional results for another material, the analysis with the Song and Librescu geometry wasrepeated for the graphite-polymer material properties tabulated by Hyer [50]: E1 = 155.0 (all GPa), E2 =

11

tiganiHighlight


E3 = 12.10, 12 = 13 = 0.248, 23 = 0.458, G13 = G12 = 4.40, G23 = 3.20. A density of = 1600 kg/m3

was used, which is typical for this class of material. The results of this analysis are given in Table 8, withthe values given for the first six frequencies for fiber orientation angles of pi/12. These are mainly given forpurposes of comparison and to serve as a benchmark for other computational models.

One crucial difference between the results using the material properties of Hyer and the general behaviorof Song and Librescu is the nature of the change in frequency of the primary torsion mode as the fiberorientation angle changes. For the material properties of Song and Librescu, this frequency peaked at =45 degrees with an increase in frequency of about 25 percent. For the properties of Hyer, this mode peaksat about 66 degrees with a frequency of 8780 rad/sec. This is a 73 percent increase over the value at =0. When the fibers align with the global x or z coordinates ( = 0 or = 90, respectively), the frequencydepends almost exclusively on the shear modulii in the 23 plane (a lower value) and then the 13 plane (ahigher value).

The Hollow Ellipse

The vibration of hollow shells of elliptic cross section, often referred to as elliptic-cylindrical (EC)shells when the wall thickness is thin, have a number of practical applications even though the numberof comparative studies is relatively small. For shells with constant thickness, Sewall and co-workers [51]provided both experimental and analytical results and Yamada and co-workers [52] gave numerical resultsusing classical shell theory. More recently Hayek and Boisvert [53] studied similar EC shells with a higher-order shell theory.

The shell geometry and constitutive properties considered in this section matches that of of an identicalshell studied by both Yamada and co-workers [52] and by Hayek and Boisvert [53]. The ratio of the semi-major axis length a to that of the semi-minor axis length of b is given as a/b=2.5, and the ratio L/a=2.198,where L is the length of the beam. The ratio of the shell thickness h to a is given by h/a=0.007325. Both endsof the cylinder are under simple support. Two cases are considered: the isotropic shell and the orthotropicshell.

A general spatial solution to this class of motion can be expressed as

u(x, y, z) =m=1

Um(x, z) sinmpiyL

v(x, y, z) =m=0

Vm(x, z) cosmpiyL

w(x, y, z) =m=1

Wm(x, z) sinmpiyL

These displacements satisfy the boundary conditions at both ends of the shell and, unlike the cantileversection, split the axial and cross-sectional dependencies of the problem. This allows frequencies to be de-termined for specified values of m. For this specific problem, the dimensionless frequencies are sought forthe isotropic case when the Poisson ratio = 0.3 and m=1. The are computed as dimensionless frequencies,and are given by [53]

2 =2a2(1 2)

E(17)

12

tiganiHighlight


where E is the modulus of elasticity.As was the case for the hollow rectangle, the numerical integration over the thin walled section can be

accomplished in closed form by evaluating the integrals of the coefficient matrices for the outer semi-axislengths and then subtracting similar integrals over the inner semi-axis lengths.

Since this section is isotropic, the vibrational modes can be separated into various modal groups thatisolate the nature of the motion. In the studies of others, the displacement functions over the beam cross-section were explicitly separated in terms of trigonometric functions that describe the circumferential vari-ation around the circumferential coordinate . This effectively makes the problem one-dimensional in thecircumferential direction of the cross-section mid-plane. Both Yamada [52] and Hayak [53] reported theirresults as a function of either symmetric or antisymmetric modes along with the circumferential wave num-ber n in the above expansion. In the present analysis, the frequencies are grouped according to the symme-tries of the three displacement components (u,v,w) in the cross-section (x,z) directions. A summary of thesegroups is given in Table 9, which enforces antisymmetry about an axis by the symbol O, indicating onlyodd polynomials for that displacement component were given in that coordinate direction. The numericalresults are presented in terms of these four groups.

As was the case for the rectangular section, the number of terms used to represent the displacementcomponents was adjusted until an increase in number of terms did not yield a significant drop in frequency.Since there is now a single axial term containing either the sine or cosine axial dependence, only the cross-section variables need adjustment. Approximations up to and including x9z7 for u, x9z8 for w, and x8z9 forv were used in the results that follow.

The dimensionless frequencies for all four groups are tabulated in Table 10 and compared with theresults of Yamada [52] and Hayak [53] along with the two-dimensional finite element model developedas part of the present study. These latter results were obtained by meshing one quarter of the ellipticalcross-section with 4 divisions in the thickness direction and 541 elements along one quadrant of the ellipse.These numbers were determined after both halving and doubling the number of divisions with little changein frequency. Hence there were a total of 2160 4-node elements and 2705 nodes used to represent thequadrant. Four different analyses were ran, each of which had boundary conditions at the 0 and 90 degreeborders of the quadrant that matched those of the grouped polynomials.

The results are all in very good agreement with the results from the elasticity model giving, with onlyone exception, the lower frequency. The average differences between the lowest 3 frequencies from eachsymmetry group is 2.5, 2.8, and 2.1 percent between the elasticity results and the higher-order shell theoryof Hayak [53] with some of the frequencies from shell theory for the third modes being closer to 4 percentdifference. This good agreement is to be expected since the shell in this case is quite thin and hence itskinematics should be captured by such a theory. Hence when the beam is isotropic, the simplified shelltheory overpredicts the elasticity frequencies by approximately three percent.

For orthotropic materials, the analysis was repeated using the material properties of Hyer [50] listed forthe rectangular section. The fibers are oriented along the beam axis ( = 90). In this case the same symmetrygroups hold as for the isotropic solid. The dimensionless frequencies in this case are given by

2 =2a2E1

(18)

These values are shown in Table 10 are are again compared with the results from the semi-analytic finiteelement model. These frequencies are significantly below those of the isotropic section. This is partiallyinfluenced by the ratio of the longitudinal modulus to that of the shear modulus. For the isotropic solid,

13


this ratio is 2.6. For the orthotropic material, the ratio is over 35. Hence the level of shear deformation isexpected to be significantly higher than that of the isotropic solid, and any simplified model would need totake this behavior into account.

Conclusions

An elasticity-based approximation scheme was used to develop estimates for the frequencies of a widevariety of thin-walled beams. In nearly every case, this method provided lower estimates of all frequenciesstudied. The primary conclusions of the application of this approach as applied to open and closed sectionsare given below.

For open thin-walled sections:

1. The frequency results of the present model for the doubly unsymmetric section of Yaman [39] are invery good agreement with two and three-dimensional finite element results and the lowest modes ofother researchers but are significantly lower for the remaining modes.

2. For the cantilevered thin-walled semi-circle with an assumed Poisson ratio of zero, the bending fre-quency results computed using the elasticity-based model are similar to but slightly lower than resultsfrom Vlasov and Timoshenko beam theories, the former of which are exactly the same for the beamstudied as the results from the Rayleigh theory. A similar pattern is true for torsional modes.

3. For the cantilevered thin-walled semi-circle with a non-zero Poisson ratio, the bending frequencyresults from the elasticity model are slightly above (roughly five percent) those computed by Vlasovand Timoshenko beam theories. There is little difference between torsional frequencies for the presentmodel in terms of influence of Poisson ratio.

4. For the simply-supported thin-walled semi-circle, the results of the elasticity model are slightly belowthose of Vlasov and Timoshenko model results regardless of the value of Poisson ratio used. Theassumption of zero Poisson ratio in the elasticity model developed here has a very small (well underone percent) influence on the simply-supported frequencies.

5. The agreement between the present elasticity model used here and a similar power series approachpresented by Petrolo and co-workers [48] is mixed, and may have been influenced by either truncatedapproximations or the influence of local axial polynomials rather than the global polynomials used inthe present study.

For closed thin-walled sections:

1. For a hollow anisotropic rectangular section whose thickness to cross-section length direction was1:6 for one wall and 1:21 for another, good agreement was found for axial and torsional modes whenfibers had 0 or 90 degree orientation angles between elasticity theory and a refined beam theory.

2. The lowest (transverse) bending frequencies of the rectangular section were also in good agreementfor all fiber orientations, with a maximum difference of about 7 percent. The second transversefrequencies had more significant differences that increased with fiber orientation, starting at 4 percent( = 0) and increasing to over 25 percent ( = 90).

3. The lateral bending frequencies of the rectangular section were in good (under 4 percent difference)agreement for the cases where the fibers were along or perpendicular to the beam axis, but for otherangles of orientation the present method gave frequencies that were significantly (over 50 percent)lower than those of the one-dimensional beam model. It is possible that the second lateral mode ofSong and Librescu [17] is actually the third transverse mode.

14


4. For the isotropic elliptical shell whose thickness to cross-section dimension ratio was 1:136, excellentagremeent was obtained with thin-shell theory results with most differences in frequency being lessthan 3 percent.

5. The dimensionless frequencies of an orthotropic elliptical beam are roughly a third of those for theisotropic beam, indicating a dramatic loss in general stiffness even when the wall thickness is small.

In addition to these conclusions, representative elasticity results were presented for anisotropic (for therectangle) and orthotropic (for the ellipse) thin-walled beams that can be used for comparison with othersimplified models.

Acknowledgment

This work was sponsored by the Mountains-Plains Consortium. The support is gratefully acknowledged.

15


References

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tional Association for Bridge and Structural Engineers, English edition 1936; 871.[3] Kappus R. Twisting failure of centrally loaded open-section columns in the elastic range. Luftfahrtforschung 1937; 14: 444-

457. Translated by J. Vanier, National Advisory Committee for Aeronautics, No. 851 (1938).[4] Gjelsvik A. The theory of thin-walled bars. John Wiley and Sons, New York, 1981.[5] Vlasov V. Thin-walled elastic beams. 2nd ed. Jerusalem: Israel Program for Scientific Translations, 1963.[6] Goodier NJ. The buckling of compressed bars by torsion and flexure. Cornell University Engineering Experiment Station,

Bulletin 27, 1941.[7] Timoshenko SP. Theory of bending, torsion and buckling of thin-walled members of open cross section. Journal of the

Franklin Institute 1945; 239: 201-219.[8] Gere, JM and Lin, Y. Coupled vibrations of thin-walled beams of open cross-section. Journal of Applied Mechanics, ASME

1958; 25: 373-378.[9] Banerjee JR, Guo S., and Howson WP. Exact dynamic stiffness matrix of a bending-torsion coupled beam including warping.

Computers and Structures 1996; 59: 613-621.[10] Mei C. Coupled vibrations of thin-walled beams of open section using finite element method. International Journal of Me-

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Solids and Structures 2001; 38: 42-43.[12] Cortinez VH and Piovan MT. Vibration and buckling of composite thin-walled beams with shear deformability. Journal of

Sound and Vibration 2002; 258: 701-723.[13] Piovan MT and Cortinez VH. Mechanics of shear-deformable thin-walled beams made of composite materials. Thin-Walled

Structures 2007; 45: 37-62.[14] Machado SP and Cortinez VH. Free vibration of thin-walled composite beams with static initial stresses and deformations.

Engineering Structures 2007; 29: 372-382.[15] Benscoter SU. A theory of torsion bending for multicell beams. ASME Journal of Applied Mechanics 1954; 21: 25-34.[16] Hodges DH, Atilgan AR, Fulton MV, and Rehfield LW. Free-vibration analysis of composite beams. Journal of the American

Helicopter Society 1991; 36: 36-47.[17] Song O and Librescu L. Free vibration of anisotropic composite thin-walled beams of closed cross-section contour. Journal

of Sound and Vibration 1993; 167: 129-147.[18] Murray NW. Introduction to the theory of thin-walled structures. Clarendon Press, Oxford, 1984.[19] Librescu L and Song O. Thin-walled composite beams: Theory and application. Springer, The Netherlands, 2006.[20] Hodges DH. Nonlinear Composite Beam Theory, AIAA, United States, 2006.[21] Hajianmaleki M and Qatu MS. Vibrations of straight and curved composite beams: A review. Composite Structures 2013;

100: 218-232.[22] Visscher WM, Migliori A, Bell TM, and Reinert RA. On normal modes of free vibration of inhomogeneous and anisotropic

elastic objects. Journal of the Acoustical Society of America 1991; 90: 2154-2162.[23] Reddy JN, Energy Principles and Variational Methods in Mechanics. Wiley, New York, 2002.[24] Reddy JN, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. CRC Press, Boca Raton, 2004.[25] Demarest HH. Cube resonance method to determine elastic constants of solids. Journal of the Acoustical Society of America

1971; 49: 768-775.[26] Ohno I. Free vibration of a rectangular parallelepiped crystal and its application to determination of elastic constants of

orthorhombic crystals. Journal of the Physics of the Earth 1976; 24: 355-379.[27] Heyliger PR. Axisymmetric free vibrations of finite anisotropic cylinders. Journal of Sound and Vibration 1991; 148: 507-

520.[28] Heyliger PR and Jilani A. The free vibrations of inhomogeneous cylinders and spheres. International Journal of Solids and

Structures 1992; 29: 2689-2708.[29] Migliori, A and Sarrao, JL. Resonant ultrasound spectroscopy. Wiley, New York, 1997.[30] Heyliger P, Ugander P, and Ledbetter H. Anisotropic elastic constants: measurement by impact resonance. ASCE Journal of

Materials in Civil Engineering 2001; 13: 356-363.[31] Heyliger PR. Ritz finite elements for curvilinear particles. Communications in Numerical Methods in Engineering 2005; 22:

335-345.[32] Carrera E, Petrolo M, and Nali P. Unified formulation applied to free vibrations finite element analysis of beams with arbitrary

cross-section. Shock and Vibrations 2010; 18: 485-502.

16


[33] Carrera E, Giunta G, and Petrolo M. Beam Structures: Classical and Advanced Theories. John Wiley and Sons, New York,2011.

[34] Mochizuki E. Application of group theory to free oscillations of an anisotropic rectangular parallelepiped. Journal of thePhysics of the Earth 1987; 35: 159-170.

[35] Han SM, Benaroya H, and Wei T. Dynamics of transversely vibrating beams using four engineering beam theories. Journalof Sound and Vibration 1999; 225: 935-988.

[36] Timoshenko SP. On the transverse vibrations of bars of uniform cross-section. Philos Mag 1922;125:125-131.[37] Timoshenko SP. On the correction for shear of the differential equation for transverse vibrations of bars of uniform cross-

section. Philos Mag 1921;41:744-746.[38] Cowper, GR. The shear coefficient in Timoshenkos beam theory. J Appl Mech 1966; 33:335-340.[39] Yaman Y. Vibrations of open-section channels: a coupled flexural and torsional wave analysis. Journal of Sound and Vibration

1997; 204: 131-158.[40] Yaman, Y., personal communication (2014). The original dimension of the thickness was 0.05 inches and the centerline leg

lengths of the channel were 0.5, 1.5, and 1 inches. These values are extremely close to the full dimensions of the cross-sectionused for purposes of comparison in this study.

[41] Ambrosini D. On free vibration of nonsymmetrical thin-walled beams. Thin-Walled Structures 2009; 47: 629-636.[42] Arpaci A and Bozdag E. On free vibration analysis of thin-walled beams with nonsymmetrical open cross-sections. Comput-

ers and Structures 2002; 80: 691-695.[43] Tanaka M and Bercin A. Free vibration solution for uniform beams of nonsymmetrical cross-section using Mathematica.

Computers and Structures 1999; 71: 1-8.[44] Friberg PO. Beam element matrices derived from Vlasovs theory of open thin-walled elastic beams. International Journal

for Numerical Methods in Engineering 1985; 21: 1205-1228.[45] Jun L, Wanyou L, Rongying S, and Hongxing H. Coupled bending and torsional vibration of nonsymmetrical axially loaded

thin-walled Bernoulli-Euler beams. Mechanics Research Communications 2004; 31: 697-711.[46] Jun L, Rongying S, and Jin X. Coupled bending and torsional vibration of nonsymmetrical axially loaded thin-walled Timo-

shenko beams. International Journal of Mechanical Sciences 2004; 46: 299-320.[47] de Borbon, F and Ambrosini D. On free vibration analysis of thin-walled beams axially loaded. Thin-Walled Structures 2010;

48: 915-920.[48] Petrolo M, Zappino E, and Carrera E. Refined free vibration analysis of one-dimensional structures with compact and bridge-

like cross-sections. Thin-Walled Structures 2012; 56: 49-61.[49] Rao SS. Vibration of Continuous Systems. Wiley, New Jersey, 2007.[50] Hyer MW. Stress Analysis of Fiber-Reinforced Composite Materials. McGraw-Hill, Boston, 1998.[51] Sewall JL, Thompson Jr WM, and Pusey CG. An experimental and analytical vibration study of elliptical cylindrical shells.

NASA TN D-6089, 1971.[52] Yamada G, Irie T, and Notoya S. Natural frequencies of elliptical cylindrical shells. Journal of Sound and Vibration 1985;

101: 133-139.[53] Hayek SI and Boisvert JE. Vibration of elliptic cylindrical shells: higher order shell theory. Journal of the Acoustical Society

of America 2010; 128: 1063-1072.

17


Table 1. Frequencies of the doubly unsymmetric channelMode Frequency

3D FEM 2D FEM [41] [39] 3DRE1 50.59 47.38 (k=pi) 51.39 51.87 48.72 (k=pi)2 66.98 64.60 (k=pi) 96.48 114.79 68.39 (k=pi)3 144.5 (k=2pi) 188.84 207.41 151.5 (k=2pi)4 159.5 158.5 (k=pi) 204.81 263.88 161.0 (k=pi)

Table 2. Frequencies of the cantilevered thin-walled semi-circular beamMode Frequency

[44] Rayleigh [45] 3DRE (=0) 3DRE [48]1 31.80 31.80 31.8 31.77 33.47 31.002 63.76 63.79 63.66 64.04 67.333 137.5 137.7 137.1 137.8 357.84 199.0 199.0 199.3 197.5 207.8 193.35 278.2 278.4 276.6 278.0 593.86 483.9 480.7 483.87 556.3 556.3 558.1 545.9 573.8 533.08 657.3 637.9 642.09 767.5 759.7 765.710 1075. 953.2 953.211 1087 1051.5 1103.1

Table 3. Frequencies of the simply-supported thin-walled semi-circular beamMode Frequency

[45] Rayleigh [46] [47] [48] 3DRE ( = 0) 3DRE 2DFEM1 2L 89.27 89.24 89.23 86.28 89.03 89.06 89.262 2L 150.4 149.66 149.74 181.0 149.9 150.0 149.03 2L 320.3 317.25 317.78 278.6 315.7 315.9 314.84 L 357.1 356.5 364.02 356.44 784.3 352.8 353.1 353.95 L 365.8 364.31 353.1 364.0 364.6 361.16 2L/3 604.1 599.4 601.3 596.17 2L/3 803.5 800.5 761.8 781.1 782.9 784.48 L/2 885.0 874.0 878.2 871.99 L 1107 1047. 1048. 1049.10 2L/5 1218. 1195. 1202. 1195.

18


Table 4. Axial frequencies of hollow rectangular section.Ply Angle Mode Type

First axial Second axial1D [17] 3DRE 1D [17] 3DRE

0 11376 11749 11384 34127 35247 3414915 11537 3459930 12191 3649745 13956 4159860 18442 5378575 35692 10095090 71947 72007 71944 215841 216021 216134

19


Table 5. Torsional frequencies of hollow rectangular section.Ply Angle Mode Type

First mode Second mode[17] 3DRE [17] 3DRE

0 4397.6 1314115 4620.1 1379530 5266.0 1566545 6302.1 1862960 7538.7 1959875 7397.9 1805590 4239/4640 4686.0 12717/14595 14371

20


Table 6. Transverse bending frequencies of hollow rectangular section.Ply Angle Mode Type

First Transverse Second Transverse Third TransverseRayleigh [17] 3DRE Rayleigh [17] 3DRE Rayleigh 3DRE

0 242.4 249 242.2 1517.2 1558 1497.5 4239.7 4106.715 253 245.8 1577 1520.3 4171.330 269 262.5 1676 1622.0 4446.645 318 306.8 1985 1888.6 5149.860 443 415.6 2786 2525.4 6730.275 765 730.2 4934 4123.1 1042390 1534.7 1502 1420.6 9595.9 8502 6706.7 26814 14810

21


Table 7. Lateral bending frequencies of hollow rectangular section.Ply Angle Mode Type

First lateral Second LateralRayleigh [17] 3DRE Rayleigh [17] 3DRE

0 848.4 871 844.5 5234.5 4331 5059.715 881 859.1 4383 5170.330 949 917.8 4657 5560.145 1236 1069.7 5490 6462.760 2182 1438.3 7554 8473.375 3916 2484.6 12333 1294490 5365.6 4583 4476.2 33106 18062 17269

22


Table 8. The first six modes of hollow rectangle with modified elastic constants.Angle 1 2 3 4 5 6

0 (Rayleigh) 362.8 1268 2268 6338 78250 363.1 1264 2223 5086 6013 743115 356.9 1244 2191 5459 5951 738830 354.8 1236 2187 5980 6471 747445 385.8 1337 2379 6519 7861 811060 489.0 1686 2992 8093 8982 1000175 779.4 2658 4517 7980 11720 1420290 1252 4094 5460 6649 15702 16697

90 (Rayleigh) 1298 4539 8118 22685 28008

23


Table 9. Modal symmetry groups for beam with elliptical cross-section.Group Displacement x z

u O1 v O

wu

2 v O Ow Ou O O

3 vw Ou O

4 v Ow O O

24


Table 10. Frequencies of hollow isotropic beam with elliptical cross-section.Ply Angle Method

Group 3DRE [53] [52] 2d FEM1 0.0706 0.0722 0.0721 0.07191 0.1285 0.1327 0.1327 0.13191 0.1631 0.1644 0.1644 0.16412 0.0762 0.0778 0.0782 0.07762 0.1098 0.1111 0.1111 0.11182 0.1810 0.1877 0.1874 0.18723 0.0763 0.0783 0.0778 0.07773 0.1087 0.1119 0.1110 0.11073 0.1793 0.1872 0.1879 0.18564 0.0702 0.0722 0.0721 0.07164 0.1291 0.1340 0.1339 0.13324 0.2026 0.2012 0.2005 0.2023

25


Table 11. Frequencies of hollow orthotropic beam with elliptical cross-section.Method

Group 3DRE 2D FEM1 0.0288 0.02921 0.0419 0.04281 0.0619 0.06312 0.0286 0.02902 0.0446 0.04552 0.0595 0.06043 0.0285 0.02893 0.0438 0.04453 0.0564 0.05744 0.0289 0.02934 0.0419 0.04314 0.0661 0.0665

APPENDIX

The elements of the matrix equations for both the Ritz model and the three-dimensional finite elementmodel are give as:

[K11]i j =V

C11 uix

uj

x+C16

uix

uj

y+C55

uiz

uj

z+C16

uiy

uj

x+C66

uiy

uj

y

dV (19)

[K12]i j =V

C12 uix

vj

y+C16

uix

vj

x+C45

uiz

vj

z+C26

uiy

vj

y+C66

uiy

vj

x

dV (20)

[K13]i j =V

C13 uix

wj

z+C45

uiz

wj

y+C55

uiz

wj

x+C36

uiy

wj

z

dV (21)

[K22]i j =V

C22 viy

vj

y+C26

viy

vj

x+C44

viz

vj

z+C26

vix

vj

y+C66

vix

vj

x

dV (22)

[K23]i j =V

C23 viy

wj

z+C44

viz

wj

y+C45

viz

wj

x+C36

vix

wj

z

dV (23)

26


[K33]i j =V

C33 wiz

wj

z+C44

wiy

wj

y+C45

wiy

wj

x+C45

wix

wj

y+C55

wix

wj

x

dV (24)M11i j =

Vui

ujdV (25)

M22i j =Vvi

vjdV (26)

M33i j =Vwi

wj dV (27)

The elements of the coefficient matrices for the two-dimensional semi-analytic finite element model aregiven as:

K11i j =V

C11 uix

uj

xcos2ky +C66ui

uj sin

2ky +C55uiz

uj

zcos2ky

dV (28)

K12i j =V

C12 uix

vjkcos2ky C66ui

vj

xksin2ky

dV = K21ji (29)

K13i j =V

C13 uix

wj

z+C55

uiz

wj

x

cos2kydV = K31ji (30)

K22i j =V

C22vi vjk2cos2ky +C44 viz vj

zsin2 ky +C66

vix

vj

xsin2ky

dV. (31)

K23i j =V

C23 viz

wj k cos2 ky C44vi

wj

zksin2ky

dV = K32ji (32)

K33i j =V

C33 wiz

wj

zcos2ky +C44wi

wj k

2sin2ky +C55wix

wj

xcos2ky+

dV (33)

27


d1

d2

d3

t2

Figure 1. The beam cross-section of the doubly unsymmetric channel. For the results presented here, the origin of the (x,y) systemwas located at the lower-left interior corner of the beam cross-section. This required integration of the coefficient matrices overfour rectangular subdomains.

28


t

di

Figure 2. The beam cross-section of the cantilever semi-circle. The origin of the (x,y) system is located at the center of thefull-circle extension of this geometry.

29


(a) Mode 1 (symmetric)

30


b) Mode 2 (symmetric)

31


c) Mode 3 (anti-symmetric)

32


d) Mode 4 (symmetric)

Figure 3. The first four modes of the doubly unsymmetric channel

33


0 0.2 0.4 0.6 0.8 1Normalized distance

-1

-0.8

-0.6

-0.4

-0.2

0

No

rmal

ized

di

spla

cem

ent

Figure 4. The first symmetric mode of the semi-circle: u (short dash), v (long dash), and w (solid).

34



-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

No

rmal

ized

di

spla

cem

ent

Figure 5. The second symmetric mode of the semi-circle: u (short dash), v (long dash), and w (solid).

35


0 0.2 0.4 0.6 0.8 1-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

Figure 6. The third symmetric mode of the semi-circle: u (short dash), v (long dash), and w (solid).

36



-1

-0.75

-0.5

-0.25

0

0.25

No

rmal

ized

di

spla

cem

ent

Figure 7. The first unsymmetric mode of the semi-circle: u (short dash), v (long dash), and w (solid).

37



-1

-0.8

-0.6

-0.4

-0.2

0

No

rmal

ized

di

spla

cem

ent

Figure 8. The second unsymmetric mode of the semi-circle: u (short dash), v (long dash), and w (solid).

38



-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

No

rmal

ized

di

spla

cem

ent

Figure 9. The third unsymmetric mode of the semi-circle: u (short dash), v (long dash), and w (solid).

39


a) = 0 b) = 60

Figure 10. The first (top) and second (bottom) torsional modes for two fiber orientations.

40


a) = 0 b) = 60

Figure 11. The first transverse bending mode from the side (top) and end (bottom) for two fiber orientations.

41


a) = 0 b) = 60

Figure 12. The second transverse bending mode from the side (top) and end (bottom) for two fiber orientations.

42


a) First mode b) Second mode

Figure 13. The lowest lateral bending modes from the side (top) and end (bottom). These modal plots are visually identical forboth = 0 and = 60.

43

thinwallvibs_2

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