Shear Deformation Effect in Flexural-torsional Vibrations of Composite Beams by Boundary Element Method (BEM)

Transcription

1 Shear Deformation Effect in Flexural-torsional Vibrations of Composite Beams by Boundary Element Method (BEM) E. J. SAPOUNTZAKIS J. A. DOURAKOPOULOS School of Civil Engineering, National Technical University, Zografou Campus, GR 157 8, Athens, Greece (Received 3 November 28 accepted 18 May 29) Abstract: In this paper a boundary element method (BEM) is developed for the general flexural-torsional vibration problem of Timoshenko beams of arbitrarily shaped composite cross-section taking into account the effects of warping stiffness, warping and rotary inertia and shear deformation. The composite beam consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli with same Poisson s ratio and are firmly bonded together. The beam is subjected to arbitrarily transverse and/or torsional distributed or concentrated loading, while its edges are restrained by the most general linear boundary conditions. The resulting initial boundary value problem, described by three coupled partial differential equations, is solved employing a boundary integral equation approach. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. All basic equations are formulated with respect to the principal shear axes coordinate system, which does not coincide with the principal bending one in a non-symmetric cross-section. To account for shear deformations, the concept of shear deformation coefficients is used. Six boundary value problems are formulated with respect to the transverse displacements, to the angle of twist, to the primary warping function and to two stress functions and solved using the Analog Equation Method, a BEM-based method. Both free and forced vibrations are examined. Several beams are analyzed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy. Key words: Boundary element method, composite beam, dynamic analysis, flexural-torsional vibration, shear deformation, twist, warping, vibrations. 1. INTRODUCTION In engineering practice, we often come across the analysis of beam structures subjected to vibratory loading. This problem becomes much more complicated if the cross-section s centroid does not coincide with its shear center (monosymmetric or asymmetric beams), leading to the formulation of the flexural-torsional vibration problem. Also, composite structural elements consisting of a relatively weak matrix material reinforced by stronger inclusions or Journal of Vibration and Control, 16(12): , 21 DOI: / SAGE Publications Los Angeles, London, New Delhi, Singapore Figures 1, 2, 4 9 appear in color online:

2 1764 E. J. SAPOUNTZAKIS and J. A. DOURAKOPOULOS of materials in contact are of increasing technological importance. Steel beams or columns totally encased in concrete, fiber-reinforced materials or concrete plates stiffened by steel beams are the most common examples. Moreover, the error incurred from the ignorance of the effect of shear deformation may be substantial, particularly as regards natural frequencies. Although there is an extended literature on the vibration problem of beams having two axes of symmetry, relatively little work has been done on the corresponding problem of coupled flexural-torsional vibrations. The theories for coupled flexural-torsional vibrations of thin-walled beams have firstly been developed by Timoshenko and Young (1955), Gere and Lin (1958) and Vlasov (1961) separately. Since then several investigations have appeared in the literature, which can be broadly classified into three groups based on the various beam theories employed in the formulations, named the Euler-Bernoulli, the Vlasov and the Timoshenko theory. According to the numerical solutions although a number of authors have investigated the flexural-torsional vibrations of homogeneous beams only few have taken into account the effects of warping stiffness (Mei, 197 Friberg, 1985 Bishop et al., 1989 Dvorkin et al., 1989 Leung, 1991, 1992 Banerjee et al., 1996 Bercin and Tanaka, 1997 Li et al., 24a) shear deformation (Bishop and Price, 1977 Banerjee and Williams, 1992 Bercin and Tanaka, 1997 Li et al., 24b) and rotary inertia (Friberg, 1985 Dvorkin et al., 1989 Leung, 1991, 1992 Bercin and Tanaka, 1997 Li et al., 24a,b Prokić, 26) restricting their formulations in thin-walled, monosymmetric or non-symmetric beams ignoring shear deformation product. Finally, Kim and Kim (25) proposed a general theory for the evaluation of the dynamic and static element stiffness matrices of non-symmetric open/closed cross-section shear deformable beams formulating their equations in the principal bending coordinate system ignoring principal shear axes one and restricting their numerical solution in thin-walled beams. Contrary to the aforementioned references concerning homogeneous beams, to the authors knowledge very little work has been done on the corresponding problem of composite beams (Banerjee, 1998 Kollar, 21 Lee and Kim, 22 Librescu et al, 23 Li et al., 24b Li and Jin, 25). In these efforts composite beams of thin-walled or laminated cross-sections have been analyzed using the refined models and employing the classical lamination and the thin-walled theories modified to include the transverse shear and the restrained warping induced shear deformations. However, in the calculation of the shear correction factors, these models do not satisfy the continuity conditions of transverse shear stress at layer interfaces and assume that the transverse shear stress along the thickness coordinate remains constant, leading to the fact that kinematic or static assumptions cannot be always valid (Reddy, 1989 Touratier, 1992 Karama et al., 23). A literature survey on the subject shows that there appears to be no work reported about the dynamic analysis of composite beams of arbitrary cross-section including the effects of warping stiffness, shear deformation, warping and rotary inertia in a unitary manner. In this investigation, an integral equation technique is developed for the solution of the aforementioned problem. The composite beam consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli with same Poisson s ratio and are firmly bonded together. The beam is subjected to arbitrarily transverse and/or torsional distributed or concentrated loading, while its edges are restrained by the most general linear boundary conditions. The solution method is based on

3 SHEAR DEFORMATION EFFECT IN FLEXURAL-TORSIONAL VIBRATIONS 1765 the concept of the analog equation (Katsikadelis, 22). According to this method, the three coupled fourth order hyperbolic partial differential equations are replaced by three uncoupled ones subjected to fictitious time-dependent load distributions under the same boundary conditions. All basic equations are formulated with respect to the principal shear axes coordinate system, which do not coincide with the principal bending ones in a non-symmetric crosssection. To account for shear deformations, the concept of shear deformation coefficients is used. Six boundary value problems are formulated with respect to the transverse displacements, to the angle of twist, to the primary warping function and to two stress functions and solved using the Analog Equation Method (Katsikadelis, 22), a Boundary Element Method (BEM) based method. The essential features and novel aspects of the present formulation compared with previous ones are summarized as follows. i. The proposed method can be applied to beams having an arbitrary composite constant cross-section and not necessarily to a thin-walled one. ii. All basic equations are formulated with respect to the principal shear system of axes, which does not necessarily coincide with the principal bending one. iii. Shear deformation effect is taken into account on the free or forced flexural-torsional vibration problem of a composite beam of a non-symmetric constant cross-section. iv. Both rotary and warping inertia are taken into account. v. The beam is supported by the most general linear boundary conditions including elastic support or restraint. vi. Previous formulations concerning composite beams of thin walled cross-sections or laminated cross-sections are analyzing these beams using the refined models. However, in the calculation of the shear correction factors, these models do not satisfy the continuity conditions of transverse shear stress at layer interfaces (traction vectors in the direction of the normal vector on the interfaces separating different materials should be equal in magnitude and opposite in direction) and assume that the transverse shear stress along the thickness coordinate remains constant, leading to the fact that kinematic or static assumptions cannot be always valid. vii. The shear deformation coefficients are evaluated using an energy approach, instead of Timoshenko s (Timoshenko and Goodier, 1984) and Cowper s (Cowper, 1966) definitions, for which several authors (Schramm et al., 1994 Schramm et al., 1997) have pointed out that one obtains unsatisfactory results or definitions given by other researchers (Stephen, 198 Hutchinson, 21), for which these factors take negative values. viii. The proposed method employs a pure BEM approach (requiring only boundary discretization) resulting in line or parabolic elements instead of area elements of the finite element method (FEM) solutions (requiring the whole cross-section to be discretized into triangular or quadrilateral area elements), while a small number of line elements are required to achieve high accuracy. Both free and forced vibrations are examined. Several beams are analyzed to illustrate the method and to demonstrate its efficiency and wherever possible its accuracy.

4 1766 E. J. SAPOUNTZAKIS and J. A. DOURAKOPOULOS Figure 1. Prismatic element of an arbitrarily shaped constant cross-section occupying region (a) subjected in bending and torsional loading (b). 2. STATEMENT OF THE PROBLEM Let us consider a prismatic beam of length l (Figure 1), of constant arbitrary cross-section of area A. The cross-section consists of materials in contact, each of which can surround a finite number of inclusions, with modulus of elasticity E j and shear modulus G j, occupying the regions j j 1 2K of the y z plane (Figure 1). The materials of these regions are assumed homogeneous, isotropic and linearly elastic. Let also the boundaries of the non-intersecting regions j be denoted by j j 1 2 K. These boundary curves are piece-wise smooth, i.e. they may have a finite number of corners. In Figure 1(a) CY Z is the principal shear system of axes through the cross-section s centroid C, while y C, z C are its coordinates with respect to the Syz system through the cross-section s shear center S, with axes parallel to those of CY Z. The beam is subjected to the combined action of the time- dependent arbitrarily distributed transverse loading p Y p Y X t, p Z p Z X t acting in the Y and Z directions, respectively and to the arbitrarily distributed time-dependent twisting moment m x m x x t (Figure 1(b)). Under the aforementioned loading, the displacement field of the beam with respect to the Syz system of axes is given as

5 SHEAR DEFORMATION EFFECT IN FLEXURAL-TORSIONAL VIBRATIONS 1767 u x y z t u x t Y x tz z C Z x ty y C x x t P S x x y z t x t z x x t x y z t x t y x x t (1a) (1b) (1c) and therefore the displacement components of the cross-section s centroid can be written as u C u x t x x t P S x C x t z C x x t C x t y C x x t (2a) (2b) (2c) where x t and x t are the time-dependent beam transverse displacements of the shear center S with respect to y and z axes, respectively, Y, Z are the angles of rotation of the cross-section due to bending, x x denotes the rate of change of the angle of twist x regarded as the torsional curvature and P S is the primary warping function with respect to the shear center S of the cross-section of the beam (Sapountzakis and Mokos, 23). Moreover, in equation (2a), u x t denotes the average longitudinal displacement of the cross-section (Attard, 1986), which for simplicity reasons is regarded as the axial displacement of the centroid for zero warping of the cross-section. According to the linear theory of beams (small deflections and rotations), the angles of rotation of the cross-section in the x-z and x-y planes of the beam subjected to the aforementioned loading and taking into account shear deformation effect satisfy the following relations x Y xz x Z xy (3a,b) while employing the stress-strain relations of the three-dimensional elasticity, the arising shear stress resultants Q z, Q y are given as Q z Q y j1 j1 j xz d j G 1 A G Z j xy d j G 1 A G Y x Y x Z (4a) (4b) where the first material is considered as reference material, xz, xy are the additional angles of rotation of the cross-section due to shear deformation (Figure 2(a)) and G 1 A G Z, G 1 A G Y are the cross-section s shear rigidities of the Timoshenko s beam theory, with A G Z Z A G 1 Z A G 1 Z j1 G j A j G 1 (5a)

6 1768 E. J. SAPOUNTZAKIS and J. A. DOURAKOPOULOS Figure 2. Displacements (a) and equilibrium in xz plane (b) of an element subjected in dynamic loading. A G Y Y A G 1 Y A G 1 Y j1 G j A j G 1 (5b) the shear areas with respect to Z, Y axes, respectively, Z, Y are the shear correction factors and Z, Y the shear deformation coefficients. It is worth here noting that the reduction of equations (4) using the shear modulus G 1 of the first material, could be achieved using any other material, considering it as reference material. Equilibrium of moments and forces in a beam element in the xz plane (Figure 2(b)) leads to Q z M Y x 1I Y Y 1 I YZ Z (6a) Q z x p Z 1 A y M x (6b) andinthexy plane to Q y M Z x 1I Z Z 1 I YZ Y (7a) Q y x p Y 1 A z M x (7b)

7 SHEAR DEFORMATION EFFECT IN FLEXURAL-TORSIONAL VIBRATIONS 1769 where 1 is the mass density of the reference material, A, I Y, I Z, I YZ are geometrical and inertia properties of the cross-section given by equations (8a) to (8d), M is the mass center defined by equations (8e), (8f) (with respect to the system of axes through shear center S), being different from the centroid C whose coordinates y C, z C are given by equations (8g), (8h) A j j1 1 j d j (8a) I Y j j1 1 j Z 2 d j (8b) I Z j j1 1 j Y 2 d j (8c) I YZ y M z M y C z C j j1 1 j YZd j j y j j d j j1 1 A j z j j d j j1 1 A E j y j E j d j 1 E j E j d j 1 j1 j1 E j z j E j d j 1 E j E j d j 1 j1 j1 j y j j d j 1 j j d j 1 j1 j1 j z j j d j 1 j j d j 1 j1 j1 (8d) (8e) (8f) (8g) (8h) where y j, z j, in the previous equations, are the coordinates of the centroid of each separate homogeneous material with respect to the shear centre of the cross section. It is worth here noting that the second and third terms of the right hand side of equations (6a), (7a) denote

8 177 E. J. SAPOUNTZAKIS and J. A. DOURAKOPOULOS the contribution of the rotary inertia. Employing equation (1a) to the strain displacement equations of the three-dimensional elasticity and ignoring the term arising from the average longitudinal displacement of the cross-section u x t, since a linear analysis is presented without axial loading, the normal strain component x can be written as and the arising bending moments M Y, M Z are given as x Y x Z Z x Y 2 x x 2 P S (9) M Y j1 M Z j E i x Zd j E 1 I E Y j1 j E i x Yd j E 1 I E Z Y x E 1I E Z YZ x Z x E 1I E Y YZ x (1a) (1b) where IY E, I Z E, I YZ E are defined as I E Y I E Z j1 j1 I E YZ j1 E j E 1 E j E 1 E j E 1 j Z 2 d j (11a) j Y 2 d j j YZd j (11b) (11c) Substituting equations (1a,b) and (4a,b) in equations (6a,b), (7a,b), eliminating the angles of rotation due to bending Y, Z and their derivatives and ignoring the inertia terms of the fourth order arising from coupling of shear deformations and rotary inertia (Thomson, 1981) we obtain the first two coupled partial differential equations of the problem of the beam under consideration subjected to the combined action of flexure and torsion as E 1 I E 4 Z x E 1I E 4 4 YZ x E1 IZ E 4 G 1 AY G 1 A 1 I 2 Z x 2 E1 IYZ E G 1 A G 1 A 1 I 2 YZ Z x E1 IYZ E 2 G 1 A G 1 A y M E 1IZ E Z G 1 AY G 1 A 2 x z M x 2 1 A z M x p Y E 1I E Z G 1 A G Y 2 p Y x 2 E 1IYZ E 2 p Z G 1 A G Z x 2 1I Z G 1 A G Y p Y 1I YZ G 1 A G Z p Z (12)

9 SHEAR DEFORMATION EFFECT IN FLEXURAL-TORSIONAL VIBRATIONS 1771 E 1 I E Y 4 x E 1I E 4 4 YZ x E1 IY E 4 G 1 A G 1 A 1 I 2 Y Z x 2 E1 IYZ E G 1 AY G 1 A 1 I YZ 2 x E1 IY E 2 G 1 A G Z 1 A y M E 1IYZ E G 1 AY G 1 A 2 x z M x 2 1 A y M x p Z E 1IY E 2 p Z G 1 A G Z x 2 E 1I E YZ G 1 A G Y 2 p Y x 2 1I Y G 1 A G Z p Z 1I YZ G 1 A G Y p Y (13) Equilibrium of torsional moments along x axis of the beam element leads to the third (coupled with the previous two) partial differential equation of the problem of the beam under consideration as E 1 C E S 4 x x 4 G 1I G 2 x t x 2 1I S x 1 A y M z M 1 C S 2 x x 2 m x y C p Z z C p Y (14) where I S is the polar moment of inertia with respect to the shear center S, defined as I S j j1 1 j y 2 z 2 d j (15) E 1 CS E and G 1It G are the cross-section s warping and torsional rigidities, respectively, 1 C S is the warping inertia with CS E, C S, It G being its warping and torsion constants, respectively, given as (Sapountzakis and Mokos, 23) C E S C S I G t j1 E j E 1 j j1 1 G j G j1 1 j P S 2 d j (16a) j P S 2 d j (16b) j y 2 z 2 y P S z z P S d j (17) y As already mentioned, equations (12), (13) and (14) constitute the governing equations of the beam subjected to the combined action of flexure and torsion taking into account shear deformation effect. These equations are also subjected to the pertinent boundary conditions of the problem, which are given as 1 x t 2 R y x t 3 1 Z x t 2 M Z x t 3 (18a,b) 1 x t 2 R z x t 3 1 Y x t 2 M Y x t 3 (19a,b)

10 1772 E. J. SAPOUNTZAKIS and J. A. DOURAKOPOULOS 1 x x t 2 M t x t 3 1 x x t x 2 M x t 3 (2a,b) at the beam ends x l, together with the initial conditions x x x x (21a,b) x x x x (22a,b) x x x x x x x x (23a,b) where R y, M Y and R z, M Z are the reactions and bending moments, with respect to y Y and z Z axes, respectively, given as R y E 1 I E Z 3 x E 1I E 3 3 YZ (24) x 3 M Y E 1 I E 2 Y x E 1I E 2 2 YZ (25) x 2 R z E 1 I E 3 Y x E 1I E 3 3 YZ (26) x 3 M Z E 1 I E Z 2 x E 1I E 2 2 YZ (27) x 2 the angles of rotation due to bending Y, Z at the beam ends x l are evaluated from Y x E 1IY E 3 G 1 A G Z x E 1IYZ E 3 (28) 3 G 1 A G Z x 3 Z x E 1I E Z G 1 A G Y 3 x E 1IYZ E 3 (29) 3 G 1 AY G x 3 while in equations (2) M t and M are the torsional and warping moments, respectively, given as (Sapountzakis and Mokos, 23) M t x t E 1 C E S 3 x x 3 G 1I G x t x (3) M x t E 1 C E S 2 x (31) x2 Finally, k k k k k k (k 1 2 3) are functions specified at the beam ends x l. Equations (18) to (2) describe the most general linear boundary conditions associated with the problem at hand and can include elastic support or restraint. It is apparent that all types of the conventional boundary conditions (clamped, simply supported, free or guided edge) can be derived from these equations by specifying appropriately these functions (e.g.

11 SHEAR DEFORMATION EFFECT IN FLEXURAL-TORSIONAL VIBRATIONS 1773 for a clamped edge it is , , ). The solution of the initial boundary value problem given from equations (12) to (14), subjected to the boundary conditions (18) to (2) and the initial conditions (21) to (23) which represents the flexural-torsional vibrations of beams, presumes the evaluation of the shear deformation coefficients Y, Z, corresponding to the principal shear axes coordinate system through the cross-section centroid C following the procedure presented in (Mokos and Sapountzakis. 25). 3. INTEGRAL REPRESENTATIONS NUMERICAL SOLUTION 3.1. For the Transverse v, w Displacements and the Angle of Twist x According to the precedent analysis, the flexural-torsional vibration problem of a beam reduces in establishing the displacement components x t, x t and x x t having continuous derivatives up to the fourth order with respect to x and up to the second order with respect to t, satisfying the hyperbolic coupled governing equations (12) to (14) inside the beam, the boundary conditions (18) to (2) at the beam ends x l and the initial conditions (21) to (23). Equations (12) to (14) are solved using the Analog Equation Method (Katsikadelis, 22) as it is developed for hyperbolic differential equations (Sapountzakis, 25). This method is applied for the problem at hand as follows. Let x t, x t and x x t be the sought solution of the aforementioned initial boundary value problem. Setting as u 1 x t x t, u 2 x t x t, u 3 x t x x t and differentiating these functions four times with respect to x yields 4 u i x 4 q i x t i (32) Equations (32) are quasi-static, that is the time variable appears as a parameter. They indicate that the solution of equations (12) to (14) can be established by solving equations (32) under the same boundary conditions (18) to (2), provided that the fictitious load distributions q i x t i are first established. These distributions can be determined using BEM as follows. The solution of equations (32) is given in integral form as (Sapountzakis, 25) u i x t l q i tu d u 3 u i x 3 du 2 u i dx x 2 d2 u u i dx 2 x d3 u l dx u 3 i where u is the fundamental solution given as u l3 r 3 3 r 2 (34) l l (33) with r x, x, points of the beam, which is a particular singular solution of the equation

12 1774 E. J. SAPOUNTZAKIS and J. A. DOURAKOPOULOS d 4 u x (35) dx4 Employing equation (34) the integral representation (33) can be written as u i x t l q i t 4 rd 4 r 3 u i x 3 3r 2 u i x 2 2r u l i x 1ru i (36) where the kernels j r, j are given as 1 r 1 2 sgn r l 2 r l r l 3 r 1 r r 4 l2 2 sgn r l l l 4 r l3 r 3 3 r 2 l l (37a) (37b) (37c) (37d) Notice that in equation (36) for the line integral it is r x, x, points inside the beam, whereas for the rest terms it is r x, x inside the beam, at the beam ends, l. Differentiating equation (36) with respect to x, results in the integral representations of the derivatives of u i as u i x t x 2 u i x t x 2 3 u i x t x 3 l l l q i t 3 rd 3 r 3 u i x 3 q i t 2 rd 2 r 3 u i x 3 l q i t 1 rd 1 r 3 u i x 3 2r 2 u i x 2 l 1r 2 u i x 2 1r u l i x (38a) (38b) (38c) 4 u i x t x 4 q i x t (38d) The integral representations (36) and (38a), when applied for the beam ends ( l), together with the boundary conditions (18) to (2) are employed to express the unknown boundary quantities u i t, u i x t, u i xx t and u i xxx t ( l) intermsof q i. This is accomplished numerically as follows.

13 SHEAR DEFORMATION EFFECT IN FLEXURAL-TORSIONAL VIBRATIONS 1775 Figure 3. Discretization of the beam interval and distribution of the nodal points. The interval l is divided into N equal elements (Figure 3), on which q i x t is assumed to vary according to certain law (constant, linear, parabolic etc). The constant element assumption is employed here as the numerical implementation becomes very simple and the obtained results are very good. Employing the aforementioned procedure for the coupled boundary conditions (18), (19) the following set of linear equations is obtained D 11 D 14 D 18 u 1 D 22 D 23 D 24 D 27 D 28 u 1 x E 31 E 32 E 33 E 34 u 1 xx E 42 E 43 E 44 u 1 xxx D 54 D 55 D 58 u 2 D 63 D 64 D 66 D 67 D 68 u 2 x E 31 E 32 E 33 E 34 u 2 xx E 42 E 43 E 44 u 2 xxx 3 3 F 3 F 4 3 q 1 q 2 (39) 3 F 3 F 4 while for the boundary conditions (2) we have E 11 E 12 E 14 E 22 E 23 E 31 E 32 E 33 E 34 E 42 E 43 E 44 u 3 u 3 x u 3 xx u 3 xxx 3 3 F 3 F 4 q 3 (4)

14 1776 E. J. SAPOUNTZAKIS and J. A. DOURAKOPOULOS where D 11, D 14, D 18, D 22, D 23, D 24, D 27, D 28, D 54, D 55, D 58, D 63, D 64, D 66, D 67, D 68, E 22, E 23, E 1 j,(j 1 2 4) are 2 2 known square matrices including the values of the functions a j a j j j j j ( j 1 2) of equations (18) to (2) 3, 3, 3, 3, 3, 3 are 2 1 known column matrices including the boundary values of the functions a 3 a of equations (18) to (2) E jk,(j 3 4, k ) are square 2 2 known coefficient matrices resulting from the values of the kernels j rj at the beam ends and F j j 3 4 are 2 N rectangular known matrices originating from the integration of the kernels on the axis of the beam. Moreover, u i u i t u i l t T (41a) ui t u i x x 2 u i t u i xx x 2 3 u i t u i xxx x 3 u i l t T x 2 T u i l t x 2 3 T u i l t x 3 (41b) (41c) (41d) are vectors including the two unknown time-dependent boundary values of the respective boundary quantities and q i q1 iqi 2 T N qi i is the vector including the N unknown time-dependent nodal values of the fictitious load. Discretization of equations (36), (38) and application to the N collocation points yields u i C 4 q i H 1 u i H 2 u i x H 3 u i xx H 4 u i xxx (42a) u i x C 3 q i H 1 u i x H 2 u i xx H 3 u i xxx (42b) u i xx C 2 q i H 1 u i xx H 2 u i xxx (42c) u i xxx C 1 q i H 1 u i xxx u i xxxx q i (42d) (42e) where C j j are N N known matrices H j j are N 2also known matrices and u i, u i x, u i xx, u i xxx, u i xxxx are time-dependent vectors including the values of u i x t and their derivatives at the N nodal points. The above equations, after eliminating the boundary quantities employing equations (39) and (4), can be written as u i T i q i T ij q j t i i j 1 2 i j (43a) u 3 T 3 q 3 t 3 (43b) u i x T ix q i T ijx q j t ix i j 1 2 i j (43c)

15 SHEAR DEFORMATION EFFECT IN FLEXURAL-TORSIONAL VIBRATIONS 1777 u 3 x T 3x q 3 t 3x (43d) u i xx T ixx q i T ijxx q j t ixx i j 1 2 i j (43e) u 3 xx T 3xx q 3 t 3xx (43f) u i xxx T ixxx q i T ijxxx q j t ixxx i j 1 2 i j (43g) u 3 xxx T 3xxx q 3 t 3xxx (43h) u i xxxx q i i (43i) where T i, T ix, T ixx, T ixxx, T ij, T ijx, T ijxx, T ijxxx are known N N matrices and t i, t ix, t ixx, t ixxx are known N 1 matrices. It is worth here noting that for homogeneous boundary conditions ( ) it is t i t ix t ixx t ixxx. In the conventional BEM, the load vectors q i are known and equations (43) are used to evaluate u i x t and their derivatives at the N nodal points. This, however, can not be done here since q i are unknown. For this purpose, 3N additional equations are derived, which permit the establishment of q i. These equations result by applying equations (12) to (14) to the N collocation points, leading to the formulation of the semidiscretized equation of motion q 1 M q 2 K q 3 q 1 q 2 q 3 f (44) with M 11 M 12 M 13 M M 21 M 22 M 23 (45) M 31 M 32 M 33 E 1 I E Z E 1 IYZ E K E 1 IYZ E E 1 IY E (46) E 1 C E S G 1I G t T 3xx p Y f p Z m x p Z y C p Y z C G 1 It G t 3xx P 11 P 12 P 13 P 14 P 21 P 22 P 23 P 24 p Yxx p Zxx p Ytt p Ztt (47) playing the role of the generalized mass matrix, stiffness matrix and force vector, respectively. In the above equations

16 1778 E. J. SAPOUNTZAKIS and J. A. DOURAKOPOULOS E1 IZ E M 11 G 1 AY G E1 IYZ E G 1 A G Z E1 IYZ E M 12 G 1 A G Z 1 A 1 I Z 1 A 1 I YZ 1 A 1 I YZ E1 IZ E G 1 AY G 1 A 1 I Z T 1xx T 21xx 1 A T 1 T 2xx T 12xx 1 A T 12 (48a) (48b) M 13 1 A E1 IYZ E y G 1 A G M E 1IZ E z Z G 1 AY G M T 3xx 1 A z M T 3 (48c) E1 IYZ E M 21 G 1 AY G 1 A 1 I YZ T 1xx E1 IY E GA G Z E1 IY E M 22 G 1 A G Z 1 A 1 I Y 1 A 1 I Y E1 IYZ E GAY G 1 A 1 I YZ T 21xx 1 A T 21 T 2xx T 12xx 1 A T 2 (48d) (48e) M 23 1 A E1 IY E y G 1 A G M E 1IYZ E z Z G 1 AY G M T 3xx 1 A y M T 3 (48f) M 31 1 A z M T 1 1 A y M T 21 M 32 1 A y M T 2 1 A z M T 12 M 33 1 I S T 3 1 C S T 3xx (48g) (48h) (48i) E 1 IY E, E 1I E Z, E 1IYZ E, E 1CS E, G 1It G are N N diagonal matrices including the values of the E 1 IY E, E 1IZ E, E 1IYZ E, E 1CS E, G 1It G quantities, respectively, at the N nodal points. Moreover, p Y, p Z, p Yxx, p Zxx, p Ytt, p Ztt and m x are vectors containing the values of the dynamic external loading and their derivatives with respect to x or to time t at these points, while P ij (i 1 2 j ) are N N diagonal matrices whose elements are given as P 11 ii E 1I E Z G 1 A G Y P 13 ii 1I Z G 1 A G Y P 12 ii E 1I E YZ G 1 A G Z P 14 ii 1I YZ G 1 A G Z (49a,b) (49c,d)

17 SHEAR DEFORMATION EFFECT IN FLEXURAL-TORSIONAL VIBRATIONS 1779 P 21 ii E 1I E YZ G 1 A G Y P 23 ii 1I YZ G 1 A G Y P 22 ii E 1I E Y G 1 A G Z P 24 ii 1I Y G 1 A G Z (5a,b) (5c,d) The associated initial conditions result from equations (5a) when combined with equations (21) to (23). Thus, we have v T 1 q 1 T 12 q 2 t 1 v T 1 q 1 T 12 q 2 (51a,b) w T 2 q 2 T 21 q 1 t 2 w T 2 q 2 T 21 q 1 (52a,b) x T 3 q 3 t 3 x T 3 q 3 (53a,b) Equation (44) can be solved numerically, using any time step integration technique, to establish the time-dependent vectors q 1, q 2, q 3. Substituting these vectors in equations (43) we obtain the displacements and their derivatives in the interior of the beam. Free vibrations: in this case it is (homogeneous boundary conditions), p Y p Yxx p Ytt p Z p Zxx p Ztt m x. Setting q 1 t Q 1 e it (54) q 2 t Q 2 e it (55) q 3 t Q 3 e it (56) equation (44) becomes K 2 M Q 1 Q 2 Q 3 (57) giving a generalized eigenvalue problem of linear algebra, from which the natural frequencies and eigenvectors Q 1, Q 2 and Q 3 are established. The eigenvectors may be used in equation (43) to establish the mode shapes of the beam For the Primary Warping Function P S The evaluation of the primary warping function S P and of the C S E, I t G constants employing only line integrals along the boundary is accomplished using BEM, as this is presented in Sapountzakis and Mokos (21, 27).

18 178 E. J. SAPOUNTZAKIS and J. A. DOURAKOPOULOS Figure 4. 3-D view (a) and cross-section (b) of the composite beam of example For the Stress Functions Y Z and Y Z The evaluation of the stress functions Y Z and Y Z is accomplished using BEM, as this is presented in Mokos and Sapountzakis (25), Sapountzakis and Mokos (27). 4. NUMERICAL EXAMPLES On the basis of the analytical and numerical procedures presented in the previous sections, a computer program has been written and representative examples have been studied to demonstrate the efficiency, wherever possible the accuracy and the range of applications of the developed method. In all the examples treated, each cross-section has been analyzed employing 3 constant boundary elements along the boundary of the cross-section, which are enough to ensure convergence for the calculation of the sectional constants, while the beam interval is divided into N 41 constant equal elements. Example 1: For comparison reasons, a cantilever mono-symmetric beam of length l 6m, with a composite cross-section consisting of three rectangular parts in contact (reference material 1: E knm 2, 1 3, 1 25tnm 3, materials 2, 3: E 2 E knm 2, 2 3 3, tnm 3 ), as this is shown in Figure 4,

19 SHEAR DEFORMATION EFFECT IN FLEXURAL-TORSIONAL VIBRATIONS 1781 Table 1. Geometric, inertia, torsion and warping constants and shear deformation coefficients of the composite cross section of example 1. IY E m 4 Y 295 IZ E m 4 Z 33 A G 739m 2 CS E m 6 I S 3 12 m 4 y C m It G m 4 z C m Table 2. Natural frequencies f i (Hz) of the beam of example 1. Without shear With shear MSC NASTRAN solid elements i deformation deformation (MSC/NASTRAN, 1999) has been studied. In Table 1 the geometrical and inertia properties of the composite crosssection are presented together with its warping, torsion constants and its shear deformation coefficients referred to its principal shear system of axes (the directions of which coincide with the principal bending ones due to the monosymmetric property of the cross-section). In Table 2 the first four natural frequencies ( f i i 2) of the aforementioned beam taking into account or ignoring shear deformation are presented as compared with those obtained from a FEM solution (MSC/NASTRAN, 1999) using 1152 solid elements. From this table, it is observed that the influence of the shear deformation effect is not significant, while the accuracy of the obtained results using the proposed method is noteworthy. Moreover, in Figure 5 the displacement components,, x of the first four mode shapes of the cantilever beam taking into account shear deformation effect are presented as compared with the 3-D views obtained from the FEM solution (MSC/NASTRAN, 1999). It is worth noting that in this figure and the following ones, regarding the free vibrations of the beam, the rotation component of the mode shapes has been multiplied by the distance between the shear center and the cross-section s centroid so that the components,, x can be directly compared. Example 2: To demonstrate the range of applications of the proposed method a slab-andbeam structure of length l 4m, with a composite cross-section consisting of a rectangular concrete C2/25 plate (reference material 1: E knm 2, tnm 3 ) stiffened by two concrete C35/45 I-section beams (material 2: E knm 2, tnm 3 ), as this is shown in Figure 6, has been studied. In Table 3 the geometrical and inertia properties of the composite cross-section are presented together with its warping, torsion constants and its shear deformation coefficients referred to its principal shear system of axes (the directions of which coincide with the principal bending ones due to the monosymmetric property of the cross-section). In Table 4 the first five natural frequencies for various boundary conditions of the aforementioned slab-andbeam structure taking into account shear deformation effect are presented as compared with the corresponding ones ignoring the aforementioned effect. The discrepancy of the obtained

20 1782 E. J. SAPOUNTZAKIS and J. A. DOURAKOPOULOS Figure 5. Displacement components (displ. y displ. z rot. x ) of the first four mode shapes and 3-D views of the FEM solution (MSC/NASTRAN, 1999) of the free vibrating beam of example 1.

21 SHEAR DEFORMATION EFFECT IN FLEXURAL-TORSIONAL VIBRATIONS 1783 Figure 6. Cross-section of the composite slab-and-beam structure of example 2. Table 3. Geometric, inertia, torsion and warping constants and shear deformation coefficients of the composite cross section of example 2. IY E 449m4 Y 335 IZ E 17182m4 Z 362 A G 4246m 2 I S 2811m4 CS E 1657m6 y C m It G 25m 4 z C 155m Table 4. Natural frequencies f i (Hz) of the beam of example 2 for various boundary conditions. Cantilever Clamped Fixed-hinged Without shear With shear Without shear With shear Without shear With shear i deform. deform. deform. deform. deform. deform results arising from the ignorance of this effect especially in higher frequencies is remarkable and necessitates its inclusion. Example 3: To demonstrate the range of applications of the proposed method and for comparison reasons, a non-symmetric beam of length l 1m, with a rectangular composite cross-section (A G m 2, It G m 4, CS E m 6, I S m 4 ) consisting of a rectangular part (reference material 1: E knm 2, tnm 3 ) stiffened by an L-section of unequal legs (material 2: E knm 2, 2 3, 2 785tnm 3 ), as this is shown in Figure 7, has been studied. Since the proposed method requires the coordinate system CY Z through the

22 1784 E. J. SAPOUNTZAKIS and J. A. DOURAKOPOULOS Figure 7. Cross-section of the non-symmetric composite beam of example 3. cross-section s centroid C to be the principal shear system of axes through C, in the first column of Table 5 the geometric, the inertia constants and the shear deformation coefficients of the examined cross-section are given with respect to an original coordinate system C Y Z,followed by the evaluation of the angle of rotation S (Mokos and Sapountzakis, 25) giving the final coordinate system CYZ and the new geometric, inertia constants and shear deformation coefficients given in the second column of the aforementioned table. In Table 6 the first five natural frequencies for various boundary conditions of the aforementioned beam taking into account or ignoring shear deformation effect are presented as compared with those obtained from a FEM solution (MSC/NASTRAN, 1999) using 165 elements. The discrepancy of the obtained results arising from the ignorance of shear deformation and the accuracy of the results are once more remarkable. Moreover, in Figure 8 the displacement components,, x of the first five mode shapes of the cantilever and the clamped beam are presented taking into account shear deformation effect. Finally, according to the forced vibrations case, in Figure 9 the time history of the displacement components,, x at the middle of the cantilever beam (Figure 9(b)) subjected to a concentrated static loading P Z 1kN applied at the same point (Figure 9(a)) and the same components at the beam s free end (Figure 9d) subjected to a uniformly distributed loading p Y 5kNm, p Z 5kNm, m x 1kNmm (Figure 9(c)) are presented, respectively, taking into account or ignoring shear deformation effect. The discrepancy of the response of the beam is pointed out once more.

23 SHEAR DEFORMATION EFFECT IN FLEXURAL-TORSIONAL VIBRATIONS 1785 Figure 8. Displacement components (displ. y displ. z rot. x ) of the first five mode shapes of the cantilever/clamped beam of example 3.

24 1786 E. J. SAPOUNTZAKIS and J. A. DOURAKOPOULOS Figure 9. Displacement components,, x of the beam of example 3 subjected to the static concentrated loading P Z applied at the middle point of the beam (a,b) and to distributed loading p Y p Z m x (c,d) taking into account or ignoring shear deformation effect. 5. CONCLUDING REMARKS In this paper a boundary element method is developed for the general flexural-torsional vibration problem of Timoshenko beams of arbitrarily shaped composite cross-section. The beam is subjected to arbitrarily transverse and/or torsional distributed or concentrated loading, while its edges are restrained by the most general linear boundary conditions. The main conclusions that can be drawn from this investigation are a. The numerical technique presented in this investigation is well suited for computer aided analysis for composite beams of arbitrary cross-section, subjected to any linear boundary conditions and to an arbitrarily dynamic loading.

25 SHEAR DEFORMATION EFFECT IN FLEXURAL-TORSIONAL VIBRATIONS 1787 Table 5. Geometric, inertia constants and shear deformation coefficients of the composite cross section of example 3. Coordinate system C Y Z Coordinate system CY Z I E Y m 4 I E Y m 4 I E Z m 4 I E Z m 4 I Y E Z m 4 IYZ E m 4 Y 1649 Y 1654 Z 1426 Z 1422 Y Z YZ y C m y C m z C m z C m S 14rad Table 6. Natural frequencies f i (Hz) of the beam of example 3 for various boundary conditions. Cantilever Clamped Fixed-hinged Without With MSC Without With MSC Without With MSC shear shear NASTRAN shear shear NASTRAN shear shear NASTRAN i deform. deform. solid elem. deform. deform. solid elem. deform. deform. solid elem b. Accurate results are obtained using a relatively small number of beam elements. c. The displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. d. In some cases, the discrepancy of the obtained results arising from the ignorance of shear deformation especially in higher natural frequencies is remarkable and necessitates its inclusion. Acknowledgements. This work has been funded by the project PENED 23. The project is cofinanced 75% of public expenditure through EC European Social Fund and 25% of public expenditure through Ministry of Development General Secretariat of Research and Technology and through private sector, under measure 8.3 of OPERATIONAL PROGRAM COMPETITIVENESS in the 3rd Community Support Program. REFERENCES Attard, M. M., 1986, Nonlinear theory of non-uniform torsion of thin-walled open beams, Thin walled Structures 4, Banerjee, J. R. and Williams, F. W., 1992, Coupled bending-torsional dynamic stiffness matrix for Timoshenko beam elements, Computers & Structures 42, Banerjee, J. R. and Williams, F. W., 1994, Coupled bending-torsional stiffness matrix of an axially loaded Timoshenko beam element, International Journal of Solids and Structures 31, Banerjee, J. R., Guo, S., and Howson, W. P., 1996, Exact dynamic stiffness matrix of a bending-torsion coupled beam including warping, Computers & Structures 59,

26 1788 E. J. SAPOUNTZAKIS and J. A. DOURAKOPOULOS Banerjee J. R., 1998, Free vibration of axially loaded composite Timoshenko beams using the dynamic stiffness matrix method, Computers & Structures, 69, Bercin, A. N. and Tanaka, M., 1997, Coupled flexural-torsional vibrations of Timoshenko beams, Journal of Sound and Vibration 27, Bishop, R. E. D., Cannon, S. M., and Miao, S., 1989, On coupled bending and torsional vibration of uniform beams, Journal of Sound and Vibration 131, Bishop, R. E. D. and Price, W. G., 1977, Coupled bending and twisting of a Timoshenko beam, Journal of Sound and Vibration 5, Cowper, G. R., 1966, The shear coefficient in Timoshenko s beam theory, ASME Journal of Applied Mechanics 33(2), Dvorkin, E. N., Celentano, D., Cuitino, A., and Gioia, G., 1989, A Vlasov beam element, Computers & Structures 33, Friberg, P. O., 1985, Beam element matrices derived from Vlasov s theory of open thin-walled elastic beams, International Journal for Numerical Methods in Engineering, 21, Gere, J. M. and Lin, Y. K., 1958, Coupled vibrations of thin-walled beams of open-cross section, Journal of Applied Mechanics, Hutchinson, J. R., 21, Shear coefficients for Timoshenko beam theory, ASME Journal of Applied Mechanics 68, Karama, M., Afaq, K. S., and Mistou, S., 23, Mechanical behavior of laminated composite beam by the new multi layered laminated composite structures model with transverse shear stress continuity, International Journal of Solids and Structure 4(6), Katsikadelis, J. T., 22, The Analog Equation Method, a boundary-only integral equation method for nonlinear static and dynamic problems in general bodies, Theoretical and Applied Mechanics 27, Kim, N. I. and Kim, M. Y., 25, Exact dynamic/static stiffness matrices of non-symmetric thin-walled beams considering coupled shear deformation effects, Thin-walled Structures 43, Kollar, L. P., 21, Flexural-torsional vibration of open section composite beams with shear deformation, International Journal of Solids and Structures 38, Lee, J. and Kim, S. E., 22, Flexural torsional coupled vibration of thin-walled composite beams with channel sections, Computers & Structures 8, Leung, A. Y. T., 1991, Natural shape functions of a compressed Vlasov element, Thin-walled Structures 11, Leung, A. Y. T., 1992, Dynamic stiffness analysis of thin-walled structures, Thin-walled Structures 14, Li, J., Shen, R., Hua, H., and Jin, X., 24a, Coupled bending and torsional vibration of axially loaded thin-walled Timoshenko Beams, International Journal of Mechanical Sciences 46, Li, J., Shen, R., Hua, H., and Jin, X., 24b, Bending-torsional coupled dynamic response of axially loaded composite Timoshenko thin-walled beam with closed cross section, Composite Structures 64, Li, J. and Jin, X., 25, Response of flexure-torsion coupled composite thin-walled beams with closed cross sections to random loads, Mechanics Research Communications 32, Librescu, L., Qin, Z., and Ambur, D. R., 23, Implications of warping restraint on statics and dynamics of elastically tailored thin-walled composite beams, International Journal of Mechanical Sciences 45, Mei, C., 197, Coupled vibrations of thin-walled beams of open-section using the Finite Element Method, International Journal of Mechanical Science 12, Mokos, V. G. and Sapountzakis, E. J., 25, A BEM solution to transverse shear loading of composite beams, International Journal of Solids and Structures 42, MSC/NASTRAN for Windows, 1999, Finite Element Modeling and Postprocessing System, Help System Index, Version 4., USA. Prokić, A., 26, On fivefold coupled vibrations of Timoshenko thin-walled beams, Engineering Structures 28, Reddy, J. N., 1989, On refined computational models of composite laminates, International Journal for Numerical Methods in Engineering 27, Sapountzakis, E. J., 25, Torsional vibrations of composite bars of variable cross section by BEM, Computer Methods in Applied Mechanics and Engineering 194, Sapountzakis, E. J. and Mokos, V. G., 21, Nonuniform torsion of composite bars by Boundary Element Method, ASCE Journal of Engineering Mechanics 127(9),

27 SHEAR DEFORMATION EFFECT IN FLEXURAL-TORSIONAL VIBRATIONS 1789 Sapountzakis, E. J. and Mokos, V. G., 23, Warping shear stresses in nonuniform torsion of composite bars by BEM, Computer Methods in Applied Mechanics and Engineering 192, Sapountzakis, E. J. and Mokos, V. G., 27, 3-D beam element of composite cross section including warping and shear deformation effects, Computers & Structures 85, Schramm, U., Kitis, L., Kang, W., and Pilkey, W. D., 1994, On the shear deformation coefficient in beam theory, Finite Elements in Analysis and Design 16, Schramm, U., Rubenchik, V., and Pilkey, W. D., 1997, Beam stiffness matrix based on the elasticity equations, International Journal for Numerical Methods in Engineering 4, Stephen, N. G., 198, Timoshenko s shear coefficient from a beam subjected to gravity loading, ASME Journal of Applied Mechanics 47, Thomson, W. T., 1981, Theory of Vibration with Applications, Prentice Hall, Englewood Cliffs, New Jersey. Timoshenko, S. and Young, D. H., 1955, Vibration Problems in Engineering, Van Nostrand, New Jersey. Timoshenko, S. P. and Goodier, J. N., 1984, Theory of Elasticity, 3rd edn, McGraw-Hill, New York. Touratier, M., 1992, A refined theory of laminated shallow shells, International Journal of Solids and Structures 29(11), Vlasov,V.Z.,1961,Thin-walled Elastic Beams, Israel Program for Scientific Translations, Jerusalem.