Generalized warping effect in the dynamic analysis of beams of arbitrary cross section

Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Generalized warping effect in the dynamic analysis of beams of arbitrary cross section I.C. Dikaros, E.J. Sapountzakis n, A.K. Argyridi School of Civil Engineering, National Technical University, Zografou Campus, GR-157 80, Athens, Greece

a r t i c l e i n f o

abstract

Article history: Received 9 September 2015 Received in revised form 11 January 2016 Accepted 11 January 2016 Handling Editor: W. Lacarbonara

In this paper a general formulation for the nonuniform warping dynamic analysis of beams of arbitrary simply or multiply connected cross section, under arbitrary external loading and general boundary conditions is presented taking into account the effects of rotary and warping inertia. The nonuniform warping distributions are taken into account by employing four independent warping parameters multiplying a shear warping function in each direction and two torsional warping functions, respectively, which are obtained by solving the corresponding boundary value problems, formulated exploiting the longitudinal local equilibrium equation. A shear stress “correction” is also performed in order to improve the stress field arising from the employed kinematical considerations. Ten initial boundary value problems are formulated with respect to the displacement and rotation components as well as to the independent warping parameters and solved using the Analog Equation Method, a Boundary Element Method based technique in combination with an appropriate time integration scheme. The warping functions and the geometric constants including the additional ones due to warping are evaluated employing a pure BEM approach. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Nonuniform warping Shear lag Flexure Torsion Shear deformation Dynamic analysis

1. Introduction In engineering practice, the dynamic analysis of beam-like members under flexure is frequently encountered. Besides, many conditions occurring in engineering practice, such as rotating machinery, wind and traffic loads, blast and earthquake forces, require taking into account inertial effects in the dynamic analysis of beams. The inclusion of such effects, especially through distributed mass models, which are considered to be more reliable, requires a rigorous analysis. In the vast majority of these cases, Euler–Bernoulli beam theory assumptions are adopted, and when shear deformation effect is non-negligible these assumptions are relaxed by using Timoshenko beam theory. However, both theories maintain the assumption that plane cross sections remain plane after deformation. By maintaining this assumption, the formulation remains simple; however, it fails to capture the well-known shear lag phenomenon, which was reported long ago (e.g., [1–3]) and observed in many structural members (e.g., beams of box-shaped cross section, folded structural members, beams made of materials weak in shear). Shear lag is associated with a significant modification of normal stress distribution, especially near the joint of the various cross-sectional components [4], due to non-uniform distribution of shear warping (Fig. 1d and e) [5].

n

Corresponding author. Tel.: þ 30 210 7721718. E-mail addresses: [email protected] (I.C. Dikaros), [email protected] (E.J. Sapountzakis), [email protected] (A.K. Argyridi).

http://dx.doi.org/10.1016/j.jsv.2016.01.022 0022-460X/& 2016 Elsevier Ltd. All rights reserved.

Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

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I.C. Dikaros et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Fig. 1. Normal stress distribution due to flexure (a,b), primary torsional warping (c), shear warping (d,e) and secondary torsional warping (f).

In up-to-date regulations, the significance of shear lag effect in flexure is recognized; however, to simplify the analysis, the effective width concept [6–8] is recommended. This simplifying approach may fail to capture satisfactorily the actual structural behavior of the member, because the influence of the shear lag phenomenon is not constant along the beam length. Apart from the geometrical configuration of the cross section, it depends also on the type of loading [9]. Therefore, it is necessary to include the effects of nonuniform shear warping distribution in both the static and the dynamic analysis. Generalizing the shear lag analysis problem, considerations similar to those made for flexure could be also adopted for the problem of torsion. It is well known that when a beam undergoes general twisting loading under general boundary conditions, it is led to nonuniform torsion. This problem has been extensively examined in the literature (e.g., [10,11]) and its major characteristic is the presence of normal stress due to primary torsional warping (Fig. 1c). In an analogy with Timoshenko beam theory, when shear deformation is of importance, the so-called secondary torsional moment deformation effect (STSDE) [12,13] has to be taken into account as well. Thus, it could be concluded that the additional secondary torsional warping due to STSDE (Fig. 1f) can cause effects similar to shear lag in flexure, i.e., a modification of the initial normal stress distribution. It is noted that because of the complicated nature of torsion, simplified concepts such as effective width cannot be applied to take this behavior into account. Toward investigating shear lag effects, the inclusion of nonuniform warping in beam element formulations based on socalled higher-order beam theories [14] is of increased interest due to their important advantages over more refined approaches [15–17]. Beam elements are practical and computationally efficient and offer better insight into the structural phenomena, since they permit their isolation and independent investigation. Furthermore, because of easy parameterization of all necessary data, beam elements are more convenient for parametric analyses than more refined models which, in most cases, require the construction of multiple models. The dynamic analysis of the generalized nonuniform warping problem has received relatively less attention than the static one. Some researchers have employed the assumptions of TTT (Thin Tube Theory) and the nonuniform torsion theory with an additional degree of freedom for nonuniform warping that is the rate of angle of twist [18–29] or an independent warping parameter [30,31]. Di Egidio et al. [32] describe the non-linear warping of open cross-section thin-walled beams in terms of the flexural and torsional curvatures and investigate the dynamics of thin-walled, open cross-section beams [33]. Minghini et al. [34] take into account shear strain effects due to both nonuniform bending and torsion in the analysis of thin walled cross sections. Some other researchers have employed the nonuniform torsion theory for beams of arbitrarily shaped cross sections with the rate of the angle of twist as an additional degree of freedom [35–42]. Sapountzakis and Tsipiras [43] have taken into account nonuniform torsion in the analysis of beams of arbitrarily shaped doubly symmetric cross sections using a secondary warping function. On the other hand Li et al. [44] developed an element that includes the effect of warping using a displacement field that is assumed to be cubic in the axial direction and quadratic in the transverse direction. Moreover, to the authors’ knowledge, a BEM procedure for the dynamic analysis of the general nonuniform warping problem of beams of arbitrary cross sections has not been reported in the literature. In this study, which is an extension of a previous work of the first two authors [45,46] at the dynamic regime, a general boundary element formulation for the dynamic nonuniform warping analysis of beams of arbitrary cross section, taking into Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

I.C. Dikaros et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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account shear lag effects due to both flexure and torsion, is presented. The beam cross section (thin- or thick-walled) is homogeneous and can surround a finite number of inclusions. The beam is subjected to the combined action of arbitrarily distributed or concentrated dynamic axial loading and transverse loading, as well as to bending, twisting, and warping moments. Nonuniform warping distributions are taken into account by using four independent warping parameters multiplying a shear warping function in each direction [45,46] and two torsional warping functions, which are obtained by solving corresponding boundary value problems. It is worth here noting that the stress field arising from the preceding kinematical considerations leads to the violation of the longitudinal local equilibrium equation and the corresponding boundary condition [47] due to inaccurate representation of shear stresses. In this study, to keep the simplicity of the formulation to the highest possible level, the increment of the number of unknowns in global equilibrium is avoided and a correction of the stress field is performed with the aid of the longitudinal local equilibrium equation. Ten initial-boundary value problems are formulated with respect to the displacement and rotation components and the independent warping parameters and solved using the analog equation method (AEM) [48], a BEM-based method in combination with an appropriate time integration scheme (Mean Acceleration Method [49]). The warping functions, the additional geometric constants due to warping, and the elementary ones are evaluated with a pure BEM approach, i.e., only boundary discretization of the cross section is used. After establishment of the kinematical components, the aforementioned boundary discretization can be used to evaluate through BEM the normal and shear stress components at any arbitrary point of each cross section. The essential features and novel aspects of the present formulation are summarized as follows:

 The cross section is an arbitrarily shaped thin- or thick-walled one. The formulation does not stand on the assumption of a thin-walled structure, and therefore the cross section's warping rigidities are evaluated exactly (in a numerical sense).

 An improved stress field is adopted with the aid of the longitudinal local equilibrium equation, leading to a better evaluation of stress components without increasing the global degrees of freedom.

 The beam is subjected to arbitrary external loading and is supported by the most general boundary conditions, including elastic support or restraint.

Fig. 2. Prismatic beam under axial–flexural–torsional dynamic loading (a) of an arbitrary cross section occupying the two dimensional region Ω (b).

Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

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 The proposed formulation is suitable for the investigation of flexural and torsional shear lag effects in beams of closed or open cross sections.

 The proposed method uses a BEM approach (requiring boundary discretization for the cross-sectional analysis) resulting

   

in line or parabolic elements instead of area elements of the FEM solutions (requiring the whole cross section to be discretized into triangular or quadrilateral area elements), whereas a small number of line elements are required to achieve high accuracy. The use of BEM permits the accurate computation of derivatives of the field functions (e.g., stresses) which is very important in the nonuniform warping analysis of beams. The generalized warping theory of beams presented in [45,46] is extended at the dynamic regime. The necessity to include warping inertial and generalized warping effects is investigated in both thin- and thick-walled open- and closed-shaped cross section bars. Both free and forced vibrations of bars are investigated obtaining free vibration characteristics and both kinematic and stress results in the time domain, respectively.

2. Statement of the problem 2.1. Displacement, strain and stress components Consider a prismatic beam of length l, of arbitrary cross section of area A. The homogeneous isotropic and linearly elastic material of the beam's cross section, with modulus of elasticity E, shear modulus G, Poisson's ratio ν and mass density ρ occupies the two dimensional multiply connected region Ω of the yz plane and is bounded by the Γ j ðj ¼ 1; 2; …; KÞ boundary curves, which are piecewise smooth, i.e. they may have a finite number of corners. In Fig. 2 CXYZ is the principal bending coordinate system through the cross section's centroid C, while yC , zC are its coordinates with respect to Sxyz reference system of axes through the cross section's center of twist S. The beam is subjected to the combined action of the arbitrarily distributed or concentrated time-dependent axial loading px ¼ px ðX; tÞ along X direction, transverse loading py ¼ py ðx; tÞ, pz ¼ pz ðx; tÞ along the y, z directions, respectively, twisting moments mt ¼ mt ðx; tÞ along x direction, bending moments mY ¼ mY ðx; tÞ, mZ ¼ mZ ðx; tÞ along Y, Z directions, respectively, as well as warping moments mφP ¼ mφP ðx; tÞ, mφS ¼ mφS ðx; tÞ, S S S S mφP ¼ mφP ðx; tÞ, mφP ¼ mφP ðx; tÞ (Fig. 2), which will be later defined in Section 2.2. CY CY CZ CZ Under the action of the aforementioned general loading and of possible restraints, the beam is leaded to flexural, axial and/or torsional vibrations. In order to take into account nonuniform warping distributions which are responsible for shear lag effects due to both flexure and torsion, four additional time-dependent degrees of freedom (warping parameters), namely ηx ðx; t Þ, ηY ðx; t Þ, ηZ ðx; t Þ, ξx ðx; t Þ multiplying a shear warping function in each direction (ηY , ηZ ) and two torsional warping functions (ηx , ξx ), respectively, are employed [45,46]. These additional DOFs describe the “intensities” of the corresponding cross sectional warping along the beam length at any time instant, while these warpings are defined by corresponding warping functions (φPCY ðy; zÞ, φPCZ ðy; zÞ, φPS ðy; zÞ, φSS ðy; zÞ), depending only on the cross sectional configuration. Thus, the “actual” deformed configurations of the cross section due to primary (in each direction) shear and primary, secondary torsional warpings are given as ηY ðx; t ÞφPCY ðy; zÞ, ηZ ðx; t ÞφPCZ ðy; zÞ, ηx ðx; t ÞφPS ðy; zÞ and ξx ðx; t ÞφSS ðy; zÞ at any position and time instant. It is worth here noting that the shear stresses generated by the above displacement considerations exhibit an inconsistency concerning the non-vanishing of tractions τxn on the lateral surface of the beam [45,47]. This inconsistency may be responsible for non-negligible errors in estimated normal stress values and thus in the present study it is removed by performing a suitable shear stress correction, which is discussed in detail in [45,46]. Within the context of the above considerations, the displacement components of an arbitrary point of the beam at an arbitrary time instant are given as uðx; y; z; t Þ ¼ uP ðx; y; z; t Þ þ uS ðx; y; z; t Þ ¼ uðx; t Þ þ θY ðx; t ÞZ θZ ðx; t ÞY þ ηx ðx; t ÞϕPS ðy; zÞ þ ηY ðx; t ÞϕPCY ðy; zÞ þηZ ðx; t ÞϕPCZ ðy; zÞ þξx ðx; t ÞϕSS ðy; zÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} primary

(1a)

secondary

vðx; y; z; t Þ ¼ vðx; t Þ  zθx ðx; t Þ

(1b)

wðx; y; z; t Þ ¼ wðx; t Þ þ yθx ðx; t Þ

(1c)

where u, v, w are the axial and transverse beam displacement components with respect to the Sxyz system of axes, while uP , uS denote the primary and secondary longitudinal displacements, respectively. Moreover, vðx; t Þ, wðx; t Þ describe the deflection of the center of twist S, while uðx; t Þ denotes the “average” axial displacement of the cross section. θY ðx; t Þ, θZ ðx; t Þ are the angles of rotation due to bending about the centroidal Y, Z axes, respectively; ηx ðx; t Þ, ξx ðx; t Þ are the independent warping parameters introduced to describe the nonuniform distribution of primary and secondary torsional warping, while ηY ðx; t Þ, ηZ ðx; t Þ are the independent warping parameters introduced to describe the nonuniform distribution of primary warping due to shear. φPS ðy; zÞ, φSS ðy; zÞ are the primary and secondary torsional warping functions with respect to the center of twist S [13], while φPCY ðy; zÞ, φPCZ ðy; zÞ are the primary shear warping functions with respect to the centroid C. Finally, it holds that Z ¼ z zC , Y ¼ y yC . Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

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Employing the displacement expressions given from Eq. (1) to the strain–displacement relations of the threedimensional elasticity for small displacements and the Hooke's stress–strain law, the non-vanishing components of the Cauchy stress tensor are obtained as     (2a) σ xx ¼ E u;x þ θY;x Z  θZ;x Y þηx;x φPS þE ηY;x φPCY þ ηZ;x φPCZ þξx;x φSS |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} primary

secondary

h      i h  i τxy ¼ G γ PZ Z ;y þ φPCY;y þ γ PY Y ;y þ φPCZ;y þγ Px  z þ φPS;y þG γ SZ φPCY;y þ γ SY φPCZ;y þ γ Sx φPS;y þ φSS;y þ Gγ Tx φSS;y |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflffl{zfflfflfflffl} primary

tertiary

secondary

h      i h  i τxz ¼ G γ PZ Z ;z þ φPCY;z þ γ PY Y ;z þφPCZ;z þγ Px yþ φPS;z þG γ SZ φPCY;z þ γ SY φPCZ;z þ γ Sx φPS;z þ φSS;z þ Gγ Tx φSS;z |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflffl{zfflfflfflffl} primary

(2b)

(2c)

tertiary

secondary

  where E ¼ Eð1  vÞ=½ð1 þvÞð1 2vÞ. If the plane stress hypothesis is undertaken then E ¼ E= 1  v2 holds [10], while E is   frequently considered instead of E (E  E) in beam formulations [10]. The first consideration has been followed throughout the paper, while any other reasonable expression of E could also be used without any difficulty. Furthermore, (  ),i denotes differentiation with respect to i, while ∇  ð UÞ;y iy þ ð U Þ;z iz is the gradient operator and iy , iz the unit vectors along y, z axes, respectively. γ Px ¼ θx;x , γ Sx ¼ ηx θx;x , γ Tx ¼ ξx  ηx þθx;x , γ PY ¼ v;x  θZ , γ SY ¼ ηY  v;x þ θZ , γ PZ ¼ w;x þθY and γ SZ ¼ ηZ w;x θY are “average” shear strain quantities (ηY and ηZ are notated in an opposite way with respect to [45,46]). In Eqs. (2b, 2c) the primary shear stress terms correspond to the St. Venant shear stresses (τPxy , τPxz ) [14], while the rest of the terms to warping shear stresses (τSxy , τSxz , τTxy , τTxz ) arising from the use of independent warping parameters. The above stress components (2) could be employed in order to derive the global equations of motion. However, as stated above, attention should be paid to the fact that the terms γ SY φPCY;i , γ SZ φPCZ;i , γ Tx φSS;i ði ¼ y; zÞ are not capable of representing an acceptable shear stress distribution, leading to violation of the longitudinal local equilibrium equation and the corresponding zero-traction condition on the lateral surface of the beam. This observation reveals an analogy with Timoshenko beam theory and nonuniform torsion theory with STSDE [12,13]. As a consequence, discrepancies in stress components near fixed supports or concentrated loads may arise [12,13,45,46,50]. In the present study, a correction of stress components is performed without increasing the number of global kinematical unknowns, following the analysis presented in [45,46]. To this end, three additional warping functions φSCY ðy; zÞ, φSCZ ðy; zÞ, φTS ðy; zÞ are introduced in expressions (2b,2c), which are modified as h  i   τxy ¼ G γ PZ ΦPCY;y þγ PY ΦPCZ;y þ γ Px z þ φPS;y þ G γ SZ ΦSCY;y þγ SY ΦSCZ;y þ γ Sx ΦSS;y þGγ Tx ΦTS;y (3a) |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflffl{zfflfflfflffl} 

primary

secondary

tertiary

     τxz ¼ G γ PZ ΦPCY;z þ γ PY ΦPCZ;z þ γ Px y þφPS;z þ G γ SZ ΦSCY;z þ γ SY ΦSCZ;z þ γ Sx ΦSS;z þ Gγ Tx ΦTS;z |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflffl{zfflfflfflffl} primary

secondary

(3b)

tertiary

where ΦPCY ¼ Z þ φPCY

(4a)

ΦSCY ¼ φPCY þφSCY

(4b)

ΦPCZ ¼ Y þφPCZ

(4c)

ΦSCZ ¼ φPCZ þ φSCZ

(4d)

ΦSS ¼ φPS þ φSS

(4e)

ΦTS ¼ φSS þφTS

(4f)

Warping functions (4) can be determined by formulating boundary value problems exploiting the longitudinal local equilibrium equation and the associated boundary condition, which will be discussed in next section. 2.2. Global equations of motion In order to establish the differential equations of motion, the principle of virtual work δW int þδW mass ¼ δW ext including virtual work due to inertial forces is employed, where Z   σ xx δϵxx þ τxy δγ xy þ τxz δγ xz dV δW int ¼

(5)

(6a)

V

Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

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I.C. Dikaros et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Z δW mass ¼

V

  ρ u ;tt δu þv ;tt δv þw ;tt δw dV

Z δW ext ¼

Lat

(6b)

  t x δu þ t y δv þ t z δw dF

(6c)

In the above equations, δ(  ) denotes virtual quantities; t x , t y , t z are the components of the traction vector applied on the lateral surface of the beam including the end cross sections, denoted by F and V is the volume of the beam. Moreover, the stress resultants of the beam can be defined as Z (7a) N ¼ σ xx dΩ Ω

Z MY ¼

σ xx Z dΩ

(7b)

σ xx Y dΩ

(7c)

σ xx φPS dΩ

(7d)

σ xx φSS dΩ

(7e)

σ xx φPCY dΩ

(7f)

σ xx φPCZ dΩ

(7g)

 τxy ΦPCY;y þ τxz ΦPCY;z dΩ

(7h)

Ω

Z MZ ¼

Ω

Z M φP ¼ S

Ω

Z M φS ¼ S

Ω

Z M φP ¼ CY

Ω

Z M φP ¼ CZ

Q Py ¼

Z  Ω

Q Sy ¼  Q Pz ¼

Ω

 τxy ΦSCY;y þ τxz ΦSCY;z dΩ

(7i)

Z   τxy ΦPCZ;y þ τxz ΦPCZ;z dΩ

(7j)

Ω

Q Sz ¼  M Pt ¼

Z 

Ω

Z  Ω

 τxy ΦSCZ;y þτxz ΦSCZ;z dΩ

(7k)

Z h    i τxy φPS;y  z þτxz φPS;z þ y dΩ

(7l)

Ω

MSt ¼  M Tt ¼

Z  Ω

Z  Ω

 τxy ΦSS;y þ τxz ΦSS;z dΩ

(7m)

 τxy ΦTS;y þ τxz ΦTS;z dΩ

(7n)

where M i ( i ¼ Y; Z) are the bending moments and M i (i ¼ φPS ; φSS ; φPCY ; φPCZ ) are the warping moments (bimoments). Q ji (i ¼ y; z j ¼ P; S) are the primary and secondary parts of total shear forces Q i (i ¼ y; z). It is noted that the secondary shear forces are also referred to as bishear stress resultants [5,51] since they equilibrate the corresponding warping moments (bimoments). Similarly, M jt (j ¼ P; S; T) are the primary, secondary and tertiary parts of total twisting moment M t . Moreover, the definitions of the geometric constants of the beam are presented in the Appendix. Using the expressions of the strain components, the definitions of the stress resultants (Eq. (7)) and applying the principle of virtual work (Eq. (5)), the differential equations of motion in terms of the kinematical components can be written as E Au;xx þ

ρΑu;tt |fflffl{zfflffl}

¼ px

(8a)

inertial contribution

     G APY þ ASY v;xx  θZ;x þ GASY ηY;x þ G DΦS

ΦS CY S

 DΦS

ΦT CY S



ηx;x  θx;xx



þGDΦS

CY

  v;tt  zC θx;tt ¼ py |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

ΦTS ξx;x þ ρA

(8b)

inertial contribution

Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

I.C. Dikaros et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎

     G APZ þ ASZ w;xx þ θY;x þ GASZ ηZ;x þG DΦS

CZ

DΦS

ΦSS

CZ

ΦTS

  ηx;x  θx;xx

þGDΦS

CZ

7

  w;tt þ yC θx;tt ¼ pz |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ΦTS ξx;x þ ρA

(8c)

inertial contribution

    E I ZZ θΖ;xx  G APY þ ASY v;x  θZ þ GASY ηY þG DΦS

ΦS CY S



 DΦS

ΦT CY S

ηx  θx;x



þGDΦS

CY

ΦTS ξx þ

ρI ZZ θZ;tt |fflfflfflffl{zfflfflfflffl}

¼ mZ

(8d)

¼ mY

(8e)

inertial contribution



   E I YY θY;xx þG APZ þ ASZ w;x þ θY GASZ ηZ  G DΦS

CZ

ΦSS

 DΦS

CZ

  ηx  θx;x ΦT S

 GDΦS

CZ

ΦTS ξx þ

ρI YY θY;tt |fflfflfflfflffl{zfflfflfflfflffl} inertial contribution

  E  I φP

CZ

φPS ηx;xx þ I φPCZ φPCZ ηZ;xx þ I φPCZ φSS ξx;xx



  þ GASZ ηZ w;x  θY

 þ G DΦS

CZ

ΦSS

 DΦS

CZ

ΦTS

  ηx  θx;x þ GDΦS

CZ

ΦTS ξx

  þ ρ I φP φP ηx;tt þI φP φS ξx;tt þ IφP φP ηZ;tt ¼ mφP CZ S CZ CZ CZ CZ S |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

(8f)

inertial contribution

  E I φP

CY

φPS ηx;xx þ I φPCY φPCY ηY;xx þ I φPCY φSS ξx;xx



 þ G DΦS

  þ GASY ηY  v;x þ θZ

CY

ΦSS

 DΦS

CY

 ΦTS

 ηx  θx;x þ GDΦS

CY

ΦTS ξx

  þ ρ I φP φP ηx;tt þ I φP φS ξx;tt þ IφP φP ηY;tt ¼ mφP CY S CY CY CY CY S |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

(8g)

inertial contribution

           G I Pt þ I St þ I Tt θx;xx þ G I St þ I Tt ηx;x  GI Tt ξx;x þ G DΦS ΦS  DΦS ΦT ηY;x  v;xx þ θZ;x þ G DΦS ΦS DΦS ΦT U CY S CY S CZ S CZ S       ηZ;x  w;xx θY;x þρ A  zC v;tt þyC w;tt þ I p θx;tt ¼ mt |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

(8h)

inertial contribution

 E I φP φP ηx;xx þI φP S

      þ G I St þI Tt ηx  θx;x  GI Tt ξx þ G DΦS ΦS  DΦS ΦT ηY v;x þ θZ CY S CY S       þ G DΦS ΦS DΦS ΦT U ηZ w;x  θY þ ρ I φP φP ηx;tt þ I φP φP ηY;tt þ I φP φP ηZ;tt ¼ mφP S S CY S CZ S S CZ S CZ S |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

S

CY

φPS ηY;xx þ I φPCZ φPS ηZ;xx



(8i)

inertial contribution



 E  I φP

CY



   þ GI t ξx ηx þ θx;x þ GDΦS ΦT ηY  v;x þ θZ CY S     þ GDΦS ΦT ηZ  w;x  θY þρ I φS φS ξx;tt þI φP φS ηY;tt þ I φP φS ηZ;tt ¼ mφS CZ S S S CY S CZ S S |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

φSS ηY;xx þ I φPCZ φSS ηZ;xx þ I φSS φSS ξx;xx

 T

(8j)

inertial contribution

where the externally applied dynamic loads are related to the components of the traction vector applied on the lateral surface of the beam t x , t y , t z as Z pi ðx; t Þ ¼ t i ds; i ¼ x; y; z (9a) Γ

Z mt ðx; t Þ ¼

Γ

t z y t y z ds

(9b)

Z mY ðx; t Þ ¼

Γ

t x Z ds

(9c)

Z mZ ðx; t Þ ¼ 

Γ

t x Y ds

(9d)

Z mi ðx; t Þ ¼

Γ

t x ðiÞ ds;

i ¼ φPS ; φSS ; φPCY ; φPCZ

(9e)

The above governing differential equations (Eq. (8)) are also subjected to the initial conditions (x A ð0; lÞ) uðx; 0Þ ¼ u0 ðxÞ

(10a)

u;t ðx; 0Þ ¼ u0;t ðxÞ

(10b)

vðx; 0Þ ¼ v0 ðxÞ

(10c)

v;t ðx; 0Þ ¼ v0;t ðxÞ

(10d)

θZ ðx; 0Þ ¼ θZ0 ðxÞ

(10e)

θZ;t ðx; 0Þ ¼ θZ0;t ðxÞ

(10f)

wðx; 0Þ ¼ w0 ðxÞ

(10g)

w;t ðx; 0Þ ¼ w0;t ðxÞ

(10h)

Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

8

I.C. Dikaros et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎

θY ðx; 0Þ ¼ θY0 ðxÞ

(10i)

θY;t ðx; 0Þ ¼ θY0;t ðxÞ

(10j)

θx ðx; 0Þ ¼ θx0 ðxÞ

(10k)

θx;t ðx; 0Þ ¼ θx0;t ðxÞ

(10l)

ηx ðx; 0Þ ¼ ηx0 ðxÞ

(10m)

ηx;t ðx; 0Þ ¼ ηx0;t ðxÞ

(10n)

ξx ðx; 0Þ ¼ ξx0 ðxÞ

(10o)

ξx;t ðx; 0Þ ¼ ξx0;t ðxÞ

(10p)

ηY ðx; 0Þ ¼ ηY0 ðxÞ

(10q)

ηY;t ðx; 0Þ ¼ ηY0;t ðxÞ

(10r)

ηZ ðx; 0Þ ¼ ηZ0 ðxÞ

(10s)

ηZ;t ðx; 0Þ ¼ ηZ0;t ðxÞ

(10t)

together with the corresponding boundary conditions of the problem at hand, which are given as a1 u þα2 Nb ¼ α3

(11a)

β1 v þ β2 V by ¼ β3

(11b)

γ 1 wþ γ 2 V bz ¼ γ 3

(11c)

β1 θZ þ β2 M bZ ¼ β3

(11d)

γ 1 θY þγ 2 M bY ¼ γ 3

(11e)

β~ 1 ηZ þ β~ 2 M bφP ¼ β~ 3

(11f)

γ~ 1 ηY þ γ~ 2 M bφP ¼ γ~ 3

(11g)

δ1 θx þ δ2 M bt ¼ δ3

(11h)

δ1 ηx þ δ2 M bφP ¼ δ3

(11i)

δ~ 1 ξx þ δ~ 2 M bφS ¼ δ~ 3

(11j)

CZ

CY

S

S

at the beam ends x ¼ 0; l, where Nb , V by , V bz , M bZ , M bY , M bφP , M bφP , Mbt , MbφP , M bφS are the reaction forces, moments and CY CZ S S warping moments respectively. ~ ~ Finally, αk ; βk ; βk ; β k γ k ; γ k ; γ~ k ; δk ; δk ; δ k (k ¼ 1; 2; 3) are functions specified at the boundaries of the beam (x ¼ 0; l). The boundary conditions (11) are the most general boundary conditions for the problem at hand, including also the elastic support. 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. for a clamped edge it is α1 ¼ β1 ¼ β1 ¼ β~ 1 ¼ γ 1 ¼ γ 1 ¼ γ~ 1 ¼ δ1 ¼ δ1 ¼ δ~ 1 ¼ 1, α2 ¼ α3 ¼ β2 ¼ β3 ¼ β2 ¼ β3 ¼ β~ 2 ¼ β~ 3 ¼ γ 2 ¼ γ 3 ¼ γ~ 2 ¼ γ~ 3 ¼ δ2 ¼ δ3 ¼ δ2 ¼ δ3 ¼ δ~ 2 ¼ δ~ 3 ¼ 0). 2.3. Warping functions due to shear ΦPCY ; ΦPCZ ; ΦSCY ; ΦSCZ The analysis described in the previous section presumes that the warping functions ΦPCY , ΦPCZ , ΦSCY , ΦSCZ with respect to the centroidal principal bending system CXYZ are already established. According to the procedure presented in [45,46], examining each primary warping function ΦPCi (i ¼ Y; Z) separately, a boundary value problem in each direction is formulated as P

∇2 ΦCY ¼ Z P

ΦCY;n ¼ 0

on the boundary

(12a) (12b)

Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

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9

P

∇2 ΦCZ ¼ Y

(13a)

P

  P P where ΦCY ¼  I YY =APZ ΦPCY ; ΦCZ

ΦZ;n ¼ 0 on the boundary (13b)   ¼  I ZZ =APY ΦPCZ , while (  ),n indicates the derivative with respect to the outward norP

mal vector to the boundary n (directional derivative). After establishing ΦCi (i ¼ Y; Z), geometric constants APi (i ¼ Y; Z) can be evaluated employing definition (A.1i) as APY ¼

ðI ZZ Þ2 DΦP ΦP

(14a)

ðI YY Þ2 DΦP ΦP

(14b)

Z

APZ ¼

Y

Z

Y

Moreover, after evaluating ΦPCi (i ¼ Y; Z), warping functions φPCi (i ¼ Y; Z) can be established through Eqs. (4a and 4c). Subsequently, φSCi (i ¼ Y; Z) employed in order to correct shear stresses are indirectly computed through ΦSCi (i ¼ Y; Z) which can be established by formulating the following boundary value problem in each direction ~ Ci ¼ φPCi ; ∇2 Φ S

S ΦCY

~ SCY ¼Φ

ði ¼ Y; ZÞ

(15a)

~ SCi;n ¼ 0 on the boundary Φ    S P P ~S φP =AZ ; ΦCZ ¼ Φ CZ þY I φP φP =AY



(15b) S ΦCY

 I φP



S

S S and þ Z I φP ¼ while it holds that φP =AZ ΦCY ; ΦCZ ¼ CY CY CZ CZ CY CY   S  I φP φP =ASY ΦSCZ . After establishing ΦCi , the geometric constants ASi (i ¼ Y; Z) can be evaluated employing definition (A.1i) CZ

CZ

as  ASY

¼

2

I φP

CZ

φPCZ

ASZ

DΦS

S CZ ΦCZ

¼

 I φP

2

CY

φPCY

DΦS

(16)

S CY ΦCY

2.4. Warping functions due to torsion φPS ; ΦSS ; ΦTS Similarly with the previous section, the torsional warping functions are established independently as follows. φPS is given by the following boundary value problem (St. Venant problem of uniform torsion) as ∇2 φPS ¼ 0 φPS;n ¼ zny ynz

on the boundary

(17a) (17b)

It is noticed that in order for φPS to be the primary torsional warping function, the analysis has to be carried out with respect to the center of twist S, which is initially specified according to the procedure presented in [11]. Following the strategy presented in the previous section, ΦSS can be established by formulating the following boundary value problem as S

∇2 ΦS ¼ φPS S

ΦS;n ¼ 0 on the boundary   S S where ΦS ¼  I φP φP =I t ΦSS . Employing Eq. (Α.1i), I St is evaluated as S S  2 I φP φP S S S It ¼ DΦS ΦS S

(18a) (18b)

(19)

S

while, after evaluating ΦSS ; φSS can be established through Eq. (4a). The final boundary value problem for ΦTS is constructed as ~ S ¼ φSS ∇2 Φ T

while it holds that

T ΦS

~ TS ¼Φ

þφPS



I φS φS =I St S S



~ TS;n ¼ 0 Φ and

T ΦS

(20a)

on the boundary (20b)  T T T ¼  I φS φS =I t ΦS . Finally, employing definition (A.1i), I t is evaluated as S S  2 I φS φS S S T It ¼ (21) DΦT ΦT 

S

S

Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

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10

It is remarked that all boundary value problems (12), (13), (15), (17), (18), (20) have solution, since the condition R   P ~S P S ~T i Γ ;n ds ¼ 0, (i ¼ Φj ; Φ j ; φx ; Φx ; Φ x , j ¼ Y; Z, m; n ¼ 1; …; M) [50] is satisfied. It is also worth here noting that, since the problems (12), (13), (15), (17), (18), (20) have Neumann type boundary conditions, each warping function contains an arbitrary integration constant indicating a parallel displacement of the cross section along the beam axis. The evaluation of these constants is performed as presented in [11].

3. Integral representations – Numerical solution 3.1. Evaluation of kinematical components According to the precedent analysis, the flexural–torsional dynamic analysis of beams of arbitrary composite cross section including generalized warping effects reduces in establishing the components uðx; t Þ, vðx; t Þ, wðx; t Þ, θx ðx; t Þ, θZ ðx; t Þ, θY ðx; t Þ, ηx ðx; t Þ, ηY ðx; t Þ, ηZ ðx; t Þ and ξx ðx; t Þ having continuous derivatives up to the second order with respect to x at the interval ð0; lÞ and up to the first order at x ¼ 0; l and up to the second order with respect to t, satisfying the initial-boundary value problem described by the coupled governing differential equations of equilibrium (Eq. (8)) along the beam, the initial conditions (Eq. (10)) and the boundary conditions (Eq. (11)) at the beam ends x ¼ 0; l. Eqs. (8), (10), (11) are solved using the Analog Equation Method [48]. According to this method, let uðx; t Þ, vðx; t Þ, wðx; t Þ, θx ðx; t Þ, θZ ðx; t Þ, θY ðx; t Þ, ηx ðx; t Þ, ηY ðx; t Þ, ηZ ðx; t Þ and ξx ðx; t Þ be the sought solutions of the aforementioned problem. Setting as u1 ðx; t Þ ¼ uðx; t Þ, u2 ðx; t Þ ¼ vðx; t Þ, u3 ðx; t Þ ¼ wðx; t Þ, u4 ðx; t Þ ¼ θZ ðx; t Þ, u5 ðx; t Þ ¼ θY ðx; t Þ, u6 ðx; t Þ ¼ ηY ðx; t Þ, u7 ðx; t Þ ¼ ηZ ðx; t Þ u8 ðx; t Þ ¼ θx ðx; t Þ, u9 ðx; t Þ ¼ ηx ðx; t Þ and u10 ðx; t Þ ¼ ξx ðx; t Þ and differentiating with respect to x these functions two times, respectively, yields ui;xx ¼ qi ðx; t Þ;

ði ¼ 1; …; 10Þ

(22)

Eq. (22) are quasi-static, i.e. the time variable appears as a parameter and they indicate that the solution of Eqs. (8), (10), (11) can be established by solving Eq. (22) under the same boundary conditions (Eq. (11)), provided that the fictitious load distributions qi ðx; t Þ (i ¼ 1; …; 10) are first established. These distributions can be determined using BEM. Following the procedure presented in [52] and employing the linear element assumption for the load distributions qi along the L internal elements (Fig. 3), the integral representations of the displacement components ui (i ¼ 1; …; 10) and their derivatives with respect to x when applied to the beam ends (0; l), together with the boundary conditions (11) are employed to express the unknown boundary quantities ui , ui;x in terms of qi (i ¼ 1; …; 10). It is noted here that since fictitious loads qi (i ¼ 1; …; 10) are not defined on the boundary points, special care is needed for the extreme elements in order to avoid the coincidence of the first and the last node with the beam ends (Fig. 3). In this case appropriate shape functions for discontinuous elements [53] have to be employed for the integration of the kernels. Thus, the following set of 40 algebraic equations is obtained Ed ¼ b

(23)

where  d ¼ d1 h b¼ 0

α3

0

β3

0

d2 γ3

d3 0

d4

β3

d5

d6

γ3

0

0

d7 β~ 3

d8 0

d9 γ~ 3

d10 0

δ3

T

(24) 0

δ3

0

δ~ 3

iT

(25)

The matrix E is rectangular 40 ½10ðL þ 1Þ þ40 known coefficient containing the values of kernels at the beam ends, the ones resulting from the integration of kernels on the axis of the beam, as well as the values of functions aj , βj , βj , β~ j , γ j , γ j , γ~ j , δj , δj , δ~ j (j ¼ 1; 2) of Eq. (11); α3 , β3 , β3 , β~ 3 , γ3 , γ3 , γ~ 3 , δ3 , δ3 , δ~ 3 are 2h 1 known column matrices including the values of the iT ^i u ^ i;x functions a3 , β3 , β3 , β~ 3 , γ 3 , γ 3 , γ~ 3 , δ3 , δ3 , δ~ 3 of Eq. (11). Finally di ¼ qi u are generalized unknown vectors, where h iT ^ i ¼ ui ð0Þ ui ðlÞ ; ði ¼ 1; …; 10Þ (26a) u h ^ i;x ¼ ui;x ð0Þ u

ui;x ðlÞ

iT

;

ði ¼ 1; …; 10Þ

(26b)

are the two unknown boundary values of the respective boundary quantities and h vectors including iT Lþ1 qi ¼ q1i q2i ⋯ qi ði ¼ 1; …; 10Þ are vectors including the L þ 1 unknown nodal values of the fictitious loads. Discretization of the integral representations of the unknown quantities ui (i ¼ 1; …; 10) inside the beam (x A ð0; lÞ) and application to the L þ 1 collocation nodal points yields ui ¼ Hdi ; ui;x ¼ H;x di ; ui;xx ¼ qi ;

ði ¼ 1; …; 10Þ ði ¼ 1; …; 10Þ ði ¼ 1; …; 10Þ

(27a) (27b) (27c)

Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

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11

Fig. 3. Discretization of the beam in linear continuous elements.

where H, H,x are ðL þ 1Þ ðL þ 5Þ known coefficient matrices and ui , ui;x , ui;xx are vectors including the values of ui ðxÞ and their derivatives at the L þ 1 nodal points. Applying Eq. (8) to the L þ 1 collocation points and employing Eq. (27), 10 ðL þ 1Þ differential equations of motion are formulated as Md;tt þ Kd ¼ f

(28)

where M; K; f are generalized mass and stiffness matrix and force vector, respectively. More specifically, M which incorporates the inertial contribution investigated in the present study is formulated as

Eq. (28) with Eq. (24) form a system of 10 ðL þ1Þ þ 40 equations with respect to the generalized unknown vector d. The system of differential equations of motion (28) can be solved employing any efficient solver. In the present study the Mean Acceleration Method [49] has been applied, performing proper static condensations to remove unknowns with zero inertial counterparts. Having established the solution for the fictitious load distributions qi and the unknown boundary quantities ^ i, u ^ i;x (i ¼ 1; …; 10), the displacements and their derivatives in the interior of the beam are computed using Eq. (27). u 3.1.1. Free vibrations For the case of free vibrations, a computational strategy similar to the ones presented in [36] regarding the boundary stress resultant terms is employed. Thus, employing the aforementioned procedure up to Eq. (28) and taking into account that px , py , pz , mt , mY , mZ , mφP , mφS , mφP , mφP along with a3 , β3 , β3 , β~ 3 , γ 3 , γ 3 , γ~ 3 , δ3 , δ3 , δ~ 3 of Eqs. (8), (11), respectively vanish, S CY CZ S Eqs. (23), (28) are formulated as Md;tt þ Kd ¼ 0

(30)

where K and M are generalized stiffness and mass matrices with respect to the generalized unknown vector d. Setting d ¼ De  iωt , the following typical generalized eigenvalue problem is obtained   K ω2 M D ¼ 0 (31) which can be readily tackled through any efficient solver. 3.2. Evaluation of warping functions The integral representations and the numerical solution for the evaluation of the kinematical components presented in the previous section assume that the constants defined in Eqs. (Α.1g–A.1j) are already computed. According to the theoretical considerations of Sections 2.3, 2.4, the evaluation of the aforementioned constants reduces in establishing warping Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

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P ~ SCi , φP , ΦSS , Φ ~ TS (i ¼ Y; Z) at any interior point of the domain Ω of the cross section having continuous partial functions ΦCi , Φ S derivatives up to the second order with respect to y; z, satisfying the boundary value problems described by Eqs. (12), (13), (15), (17), (18), (20). The numerical solution of these problems is similar and is accomplished using BEM within the context

Fig. 4. Cross sections of the beams of examples 1 (a) and 2 (b) along with the three types of boundary conditions (c–e).

Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

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13

Table 1 Discrepancies (%) of natural frequencies between GWBT- Solid FEM [55] and TBT [55]- Solid FEM [55] of the clamped beam of example 1 for various lengths. l ¼2.5 m

Natural frequency type

1st v 2nd v 3rd v 4th v 1st w 2nd w 3rd w 4th w 1st θx 2nd θx 3rd θx 4th θx

l ¼3.5 m

l ¼4.5 m

l ¼ 5.5 m

GWBT

TBT

GWBT

TBT

GWBT

TBT

GWBT

TBT

 0.04 0.07 0.16 0.23 0.77 1.10 1.16 1.24  0.64 0.17  1.14 0.49

11.46 9.25 7.43 6.02 4.35 3.36 3.38 3.19 54.61 30.71 12.06  5.02

 0.21  0.14  0.08  0.02 0.44 0.71 0.77 0.80  1.35  0.11  1.99 0.10

12.48 11.00 9.50 8.12 5.61 3.80 3.36 14.34 54.29 36.00 22.40  39.43

 0.39  0.34  0.29  0.25 0.26 0.48 0.56 0.58  0.22  0.30 0.14  0.19

12.96 11.96 10.83 9.68 6.99 4.63 3.78 3.17 52.42 37.73 26.93 17.69

 0.59  0.56  0.52  0.48 0.17 0.33 0.41 0.44  0.39  0.46  0.38  0.43

13.23 12.52 11.67 10.75 8.21 5.59 4.39 3.59 49.93 38.03 29.09 21.57

Table 2 Discrepancies (%) of natural frequencies between GWBT- Solid FEM [55] and TBT [55]- Solid FEM [55] of the cantilever beam of example 1, for various lengths. l ¼2.5 m

Natural frequency type

1st v 2nd v 3rd v 4th v 1st w 2nd w 3rd w 4th w 1st θx 2nd θx 3rd θx 4th θx

l ¼3.5 m

l ¼4.5 m

l ¼5.5 m

GWBT

TBT

GWBT

TBT

GWBT

TBT

GWBT

TBT

 0.15  0.09  0.02 0.05 0.13 0.54 0.56 0.54 0.13  0.06 0.00 0.21

13.39 11.39 9.16 7.05 9.88 2.90 2.09  0.47 52.49 31.13 16.11 1.42

 0.27  0.24  0.20  0.15 0.07 0.29 0.36 0.37 0.20  0.17  0.16  0.12

13.58 12.48 11.03 9.41 11.46 5.17 3.35 1.68 42.86 31.13 21.44 11.73

 0.42  0.41  0.38  0.34 0.04 0.17 0.24 0.26 0.21  0.25  0.27  0.33

13.66 12.97 12.01 10.83 12.30 7.04 4.62 2.88 36.37 29.58 23.22 16.24

 0.61  0.60  0.58  0.56 0.02 0.09 0.16 0.18 0.29  0.40  0.35  0.55

13.70 13.24 12.57 11.70 12.77 8.49 5.82 3.93 31.75 27.62 23.58 18.39

of the method of subdomains [53,54]. According to this method, the second Green's identity Z     GðP; Q Þ ∇2 iðQ Þ  ∇2 GðP; Q ÞðiðQ ÞÞ Ω Z    dΩ ¼ GðP; qÞ i;n ðqÞ   G;n ðP; qÞðiðqÞÞ ds;

Γ

P ~ SCj ; φPS ; Φ SS ; Φ ~ TS i ¼ Φ Cj ; Φ

j ¼ Y; Z

(32)

with P ðy; zÞ, Q ðξ; ηÞ being arbitrary points of the domain (when Q is located on the boundary Γ it is denoted as q), when P ~ SCj , φP , ΦSS , Φ ~ TS , j ¼ Y; Z) and to the fundamental solution [53] applied to the warping function ðiÞ (i ¼ ΦCj , Φ S GðP; Q Þ ¼

1 ln r ðP; Q Þ 2π

(33)

and after converting the remaining domain integrals into line ones along the boundary Γ [53], yields Z Z h i  1  1 P P P Λ3 ðr Þi  Λ4 ðr Þi;n dsq  Λ1 ðr ÞΦ Ci ðqÞ þ Λ2 ðr ÞΦ Ci;n ðqÞ dsq ; i ¼ Y; Z U ϵðP ÞΦ Ci ðP Þ ¼ 2π Γ 2π Γ ~ Ci ðP Þ ¼  ϵðP ÞΦ S

1 IΦ 2π i

Z

  1 Λ5 ðr Þi Λ6 ðr Þi;n dsq þ 2π Γ

Z  Γ

Z  i 1 h ~ PCi ðqÞ þΛ2 ðr ÞΦ ~ PCi;n ðqÞ dsq ; Λ3 ðr ÞφPCi  Λ4 ðr ÞφPCi;n dsq  Λ1 ðr ÞΦ 2π Γ

(34a) i ¼ Y; Z (34b)

Z i 1 h Λ1 ðr ÞφPS ðqÞ þ Λ2 ðr ÞφPS;n ðqÞ dsq ϵðP ÞφPS ðP Þ ¼  2π Γ S

ϵðP ÞΦ S ðP Þ ¼

1 2π

Z  Γ

 1 Λ3 ðr ÞφPS  Λ4 ðr ÞφPS;n dsq  2π

U

Z h Γ

i S S Λ1 ðr ÞΦ S ðqÞ þ Λ2 ðr ÞΦ S;n ðqÞ dsq

(34c)

(34d)

Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

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Table 3 Discrepancies (%) of natural frequencies between GWBT- Solid FEM [55] and TBT [55]- Solid FEM [55] of the simply supported beam of example 1, for various lengths. l ¼ 2.5 m

Natural frequency type

1st v 2nd v 3rd v 4th v 1st w 2nd w 3rd w 4th w 1st θx 2nd θx 3rd θx 4th θx

l ¼ 3.5 m

l ¼4.5 m

l ¼ 5.5 m

GWBT

TBT

GWBT

TBT

GWBT

TBT

GWBT

TBT

 0.14  0.10  0.06  0.02  0.01  0.05  0.05 0.00  0.10  0.16  0.15  0.15

12.98 11.04 8.84 6.94 7.34 3.19 2.01 1.83 9.81 4.63  4.64  16.19

 0.27  0.24  0.21  0.18 0.00  0.04  0.07  0.07  0.08  0.24  0.25  0.24

13.36 12.25 10.76 9.21 9.53 4.97 2.93 2.13 9.64 8.35 3.54  3.07

 0.43  0.41  0.38  0.36 0.01  0.04  0.07  0.08  0.06  0.33  0.38  0.38

13.53 12.83 11.82 10.65 10.88 6.63 4.11 2.84 8.79 9.61 7.04 2.94

 0.62  0.61  0.59  0.57 0.00  0.03  0.07  0.09  0.04  0.42  0.52  0.53

13.62 13.14 12.43 11.56 11.71 8.01 5.30 3.70 7.79 9.93 8.72 6.10

1.00 GWBT Solid FEM

0.80

TBT

0.60

0.40

0.20

0.00 0.0

0.5

1.0

1.5

2.0

2.5

x (m) Fig. 5. 1st torsional modeshape of the clamped beam of length l ¼2.5 m of example 1 as obtained from Solid FEM [55] (a) in comparison with those obtained from GWBT and TBT [55] (b).

~ S ðP Þ ¼  ϵðP ÞΦ T

1 I St 2π I φP φP S

S

Z  Γ

Z  Z h   i 1 1 ~ TS ðqÞ þ Λ2 ðr ÞΦ ~ TS;n ðqÞ dsq Λ5 ðr ÞφPS Λ6 ðr ÞφPS;n dsq þ Λ3 ðr ÞφSS Λ4 ðr ÞφSS;n dsq  Λ1 ðr ÞΦ 2π ΓΩm 2π ΓΩm (34e)

where I ΦY ¼ APY =I ZZ , I ΦZ ¼ APZ =I YY . The parameter εðP Þ ¼ 1, 1=2 or 0 depending on whether the point P is in the domain Ωm , on h i1=2 and subscript q indicates that the boundary Γ (P  p) or outside Ω, respectively, r ¼ r ðP; Q Þ ¼ jQ  P j ¼ ðξ  xÞ2 þ ðη  yÞ2 point q varies along the boundaries of the cross section during integration and differentiation while P (or p) is retained constant. Furthermore, kernels Λi ðr Þ (i ¼ 1; …; 6) are given as r ;n Λ2 ðr Þ ¼ ln r r

1 1 1 ln r  rr;n Λ4 ðr Þ ¼ ðln r 1Þr 2 Λ3 ðr Þ ¼ 2 2 4 Λ1 ðr Þ ¼ 

Λ 5 ðr Þ ¼



1 5 ln r  r3 r ;n 16 4

Λ6 ðr Þ ¼



1 3 ln r  r4 64 2

(35a) (35b)

(35c)

The integral representations of the solution derivatives with respect to y; z in every point P of the domain Ω can be given   P ~ SCj , φP , by differentiating accordingly the integral representations (33). The unknown quantities ðiðP ÞÞ and i;n ðP Þ (i ¼ ΦCj , Φ S T S ~ S , j ¼ Y; Z) on the boundary can be evaluated from the solution of a boundary integral equation on the boundary Γ, ΦS , Φ which is derived from Eq. (34) considering a point p and a point q lying on the boundary Γ of the region Ω (εðP Þ  εðpÞ ¼ 1=2) [5]. In order to approximate the boundary integrals appearing in these boundary integral equations, the boundary Γ is   P ~ SCj , φP , ΦSS , Φ ~ TS , j ¼ Y; Z) are assumed to vary according to divided into N boundary elements, on which ðiðP ÞÞ, i;n ðP Þ (i ¼ ΦCj , Φ S Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

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15

1400 1200

f (Hz)

1000

GWBT TBT Solid FEM 1st v

800

2nd v 3rd v 4th v

600 400 200 0 2.5

3.5

4.5

5.5

l (m)

1200

f (Hz)

1000 GWBT TBT Solid FEM 1st w nd 2 w

800 600

3rd w 4th w

400 200 0 2.5

3.5

4.5

5.5

l (m)

1400 1200 GWBT TBT Solid FEM

f (Hz)

1000

1st θ x

800

2nd θ x 3rd θx

600

4th θ x

400 200 0 2.5

3.5

4.5

5.5

l (m) Fig. 6. Comparison of the first four bending natural frequencies with deflections along the y (a) and the z (b) axis and the first four torsional natural frequencies (c) among GWBT, TBT [55] and Solid FEM [55] solutions of the clamped beam of example 1, for various lengths.

Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

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Fig. 7. Comparison of the first four torsional natural frequencies of the cantilever beam of example 1 (a) and of the first four bending natural frequencies with deflections along the y axis of the simply supported beam (b) of example 1, for various lengths, among GWBT, TBT [55] and Solid FEM [55] solutions.

a certain law (constant, linear, parabolic etc.). Thus, using constant boundary elements and the collocation technique, the following system of linear algebraic equations is obtained for each subdomain as    H ^i ¼ f  G ^i;n ;

~ Cj ; φPS ; ΦS ; Φ ~ S ; j ¼ Y; ZÞ ði ¼ ΦCj ; Φ P

S

S

T

(36)

where H, G known N N coefficient matrices originating from the integration of the kernels (see Eq. (34)) over the boundary curve. f is a N 1 known vector containing the values of the first integrals in the right hand side of Eq. (34). It is worth here noting that in order to avoid singular behavior during the inversion of the coefficient matrix H due to the fact that a Neumann type boundary value problem is solved, the regularization technique presented in [24] is employed. P ~ SCj , φP , ΦSS , Φ ~ TS , j ¼ Y; Z) and its derivatives can be evaluated on any arbitrary Subsequently the warping function i (i ¼ ΦCj , Φ S point of the cross sectional domain applying integral representations (34) and their derivatives and employing the already developed boundary element mesh. In order to avoid near singular behavior when an internal point approaches the boundary, the element subdivision technique is employed [53]. In order to maintain the pure boundary character of the formulation all the geometric constants defined in Eq. (Α.1) are evaluated by converting the domain integrals into line ones along the boundary using integration by parts, the Gauss theorem and the Green identity. The expressions of these constants are given in the Appendix A of [46].

Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

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1200 1000 GWBT TBT Solid FEM 1st v

f (Hz)

800 600

2nd v 3rd v th 4 v

400 200 0 2.5

3.5

4.5

5.5

l (m) 1800 1600 1400

f (Hz)

1200

GWBT TBT Solid FEM 1st w

1000

2nd w 3rd w 4th w

800 600 400 200 0

2.5

3.5

4.5

5.5

l (m) Fig. 8. Comparison of the first four bending natural frequencies with deflections along the y (a) and the z axis (b) among GWBT, TBT [55] and Solid FEM [55] solutions, of the clamped beam of example 2, for various lengths.

Table 4 Discrepancies (%) of natural frequencies between GWBT- Solid FEM [55] and TBT [55]- Solid FEM [55] models of the clamped beam of example 2, for various lengths. Natural frequency type

1st v 2nd v 3rd v 4th v 1st w 2nd w 3rd w 4th w 1st θx 2nd θx 3rd θx 4th θx

l ¼ 2.5 m

l ¼3.5 m

l ¼ 4.5 m

l ¼ 5.5 m

GWBT

TBT

GWBT

TBT

GWBT

TBT

GWBT

TBT

0.97 1.82 2.37 2.85 0.22 0.36 0.38 0.34 0.30 0.28 0.24 0.19

8.17 7.25 7.35 7.56 20.02 20.57 21.22 21.73 1.36 0.97 0.30  0.70

0.45 0.96 1.36 1.69 0.09 0.18 0.20 0.17 0.23 0.20 0.16 0.10

9.43 7.67 7.08 6.86 18.24 19.43 20.08 20.67 1.04 0.88 0.60 0.16

0.21 0.53 0.81  0.45 0.02 0.07 0.09 0.06 0.19 0.16 0.12 0.05

10.51 8.55 7.49 5.50 17.03 18.37 19.15 19.77 0.83 0.76 0.61 0.38

0.07 0.28 0.48 0.64  0.04  0.01 0.00  0.02 0.16 0.14 0.09 0.03

11.30 9.44 24.13 7.30 16.20 17.48 18.34 19.00 0.70 0.65 0.57 0.43

Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

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Table 5 Discrepancies (%) of natural frequencies between GWBT- Solid FEM [55] and TBT [55]- Solid FEM [55] models of the cantilever beam of example 2, for various lengths. Natural frequency type

1st v 2nd v 3rd v 4th v 1st w 2nd w 3rd w 4th w 1st θx 2nd θx 3rd θx 4th θx

l ¼2.5 m

l ¼ 3.5 m

l ¼ 5.5 m

l¼ 4.5 m

GWBT

TBT

GWBT

TBT

GWBT

TBT

GWBT

TBT

0.05 0.67 1.20 1.64 0.00 0.12 0.16 0.15 0.19 0.25 0.29 0.29

12.22 7.78 6.47 5.74 14.48 15.97 17.43 18.18 0.77 0.64 0.28  0.39

 0.03 0.26 0.57 0.86  0.04 0.02 0.05 0.04 0.15 0.16 0.17 0.16

12.91 9.55 7.51 6.27 14.13 15.39 16.64 17.70 0.57 0.52 0.39 0.12

 0.08 0.07 0.26 0.45  0.07  0.04  0.03  0.04 0.13 0.13 0.11 0.09

13.24 10.76 8.68 7.13 14.00 39.19 15.98 17.01 0.46 0.44 0.37 0.24

 0.12  0.03 0.08 0.20  0.10  0.09  0.09  0.11 0.12  7.81 0.08 0.05

13.41 11.57 9.68 8.06 13.93 14.59 15.48 16.41 20.60  7.52 0.33 0.26

Table 6 Discrepancies (%) of natural frequencies between GWBT- Solid FEM [55] and TBT [55]- Solid FEM [55] models of the simply supported beam of example 2, for various lengths. Natural frequency type

1st v 2nd v 3rd v 4th v 1st w 2nd w 3rd w 4th w 1st θx 2nd θx 3rd θx 4th θx

l ¼2.5 m

l ¼ 3.5 m

l ¼ 4.5 m

l ¼ 5.5 m

GWBT

TBT

GWBT

TBT

GWBT

TBT

GWBT

TBT

 0.01 0.11 0.43 0.95  0.02  0.03  0.06  0.11 0.06 0.03  0.01  0.07

10.69 7.02 5.71 5.78 15.09 17.76 19.83 21.15 0.07  0.04  0.45  1.28

 0.05  0.01 0.08 0.28  0.04  0.05  0.09  0.14 0.06 0.03  0.02  0.08

11.98 8.84 6.70 5.78 14.47 16.23 18.10 19.58 0.07 0.04  0.10  0.40

 0.01 0.16 0.31 0.44  0.05  0.03  0.02  0.05 0.44 0.38 0.29 0.18

12.70 10.40 8.30 6.94 14.22 15.42 16.96 18.38 0.45 0.41 0.30 0.12

 0.12  0.12  0.11  0.11  0.10  0.11  0.15  0.20 0.06 0.03  0.02  0.09

12.99 11.12 9.10 7.51 14.07 14.88 16.05 17.29 0.06 0.06 0.02  0.06

1.00 GWBT

Solid FEM

0.80

TBT

0.60

0.40

0.20

0.00 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

x (m) Fig. 9. 1st bending modeshape with deflections along the y axis of the clamped beam of length l ¼4.5 m of example 2 as obtained from Solid FEM [55] (a) in comparison with those obtained from GWBT and TBT [55] (b) models.

Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

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19

1400 1200

f (Hz)

1000

GWBT TBT Solid FEM 1st w

800

2nd w 3rd w 4th w

600 400 200 0 2.5

3.5

l (m)

4.5

5.5

1600 1400

f (Hz)

1200

GWBT TBT Solid FEM 1st w

1000

2nd w 3rd w 4th w

800 600 400 200 0

2.5

3.5

l (m)

4.5

5.5

Fig. 10. Comparison of the first four bending natural frequencies with deflections along the z axis among GWBT, TBT [55] and Solid FEM [55] solutions of the cantilever (a) and simply supported (b) beam of example 2, for various lengths.

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. A commercial FEM package with solid and beam modeling capabilities [55] is employed to compare and verify the results of the proposed method. 4.1. Example 1 In the first example, the free vibration characteristics of four HEB500 cross section (Fig. 4a) steel (E¼210 GPa, G¼80.8 GPa, v ¼ 0:3, ρ ¼ 7:85 kN s2 =m4 ) beams of lengths l ¼2.5 m, 3.5 m, 4.5 m, 5.5 m for three types of boundary conditions (clamped at both ends (Fig. 4c), cantilever (Fig. 4d), simply supported with fixed torsional rotation (Fig. 4e)) have been analyzed. More specifically, the generalized eigenvalue problem of Eq. (30) has been solved to obtain the natural frequencies for the Generalized Warping Beam Theory (GWBT). For comparison reasons Timoshenko Beam Theory taking into account primary torsional warping (TBT [55]) and Solid (3-D hexahedral elements extruded by quadrilateral surfaces) FEM [55] models have been analyzed. In the Solid models, rigid diaphragms at every set of nodes having the same longitudinal coordinate have been employed, pointing out that the diaphragms are designed so as each set of nodes has the same translation and rotation in the cross section plane, corresponding to the assumption employed in the proposed method that the cross section maintains its shape at the transverse directions during deformation (no distortion effects). In GWBT solutions 88 elements have been used, while in TBT 100 elements per meter of length and 24,300 solid elements in Solid FEM model. There are three types of natural frequencies under examination: bending natural frequencies with deflections along the y axis (v natural frequencies), along the z axis (w natural frequencies) and torsional natural frequencies (θx natural frequencies). Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

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Fig. 11. Cross section (a), boundary conditions and loading (b) and load cases (c) of the beam of example 3.

In Tables 1–3 the natural frequencies as obtained from GWBT and TBT are compared with Solid FEM solutions for the clamped, cantilever and simply supported boundary condition, respectively. Fig. 5a and b illustrates the 1st torsional modeshape of the bar of length l¼2.5 m as obtained from Solid FEM [55] and compared with those from GWBT and TBT [55], respectively. Fig. 6a–c illustrates the first four bending natural frequencies with deflections along the y axis, the z axis and torsional natural frequencies, respectively as obtained from GWBT, TBT [55] and Solid FEM [55] solutions for the clamped boundary condition. Fig. 7a and b shows the first four torsional natural frequencies for the cantilever boundary condition and the first four bending natural frequencies with deflections along the y axis for the simply supported boundary condition, respectively, as calculated by GWBT, TBT [55] and Solid FEM [55] solutions. From the results of Tables 1–3, it is concluded that TBT cannot predict the natural frequencies as accurate as GWBT can. In more detail, the discrepancies of GWBT with respect to Solid solutions is up to 2.00% (absolute value) for any of the examined boundary condition or length. On the contrast the discrepancies of TBT with respect to Solid FEM solutions is up to 54.61% for Tables 1, 52.49% for Table 2 and 16.91% for Table 3 (absolute values). Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

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x 10

21

−6

0 −2 −4 −6 w (m)

−8

GWBT TBT E/BBT Solid FEM

−10 −12 −14 −16 −18 −20 0.00

x 10

0.01

0.02 t (sec)

0.03

0.04

−4

1

w (m)

0.5 GWBT TBT E/BBT Solid FEM

0

−0.5

−1

−1.5 0.00

0.01

0.02 t (sec)

0.03

0.04

Fig. 12. Time histories of the displacement w at x ¼ l=2 of the beam of example 3. Load Cases I (a), II (b).

4.2. Example 2 In the second example, the free vibration characteristics of four RHS_500x300 20 cross section (Fig. 4b) steel (E ¼210 GPa, G¼80.8 GPa, v ¼ 0:3, ρ ¼ 7:85 kNs2 =m4 ) beams of lengths l¼2.5 m, 3.5 m, 4.5 m, 5.5 m, for three types of boundary conditions (clamped at both ends (Fig. 4c), cantilever (Fig. 4d), simply supported with fixed torsional rotation (Fig. 4e)) have been obtained through the three methods outlined in the previous example. As in the first example, there are three types of natural frequencies under examination. (Fig. 8) In Tables 4–6 the natural frequencies as obtained from GWBT and TBT [55] are compared with Solid FEM [55] solutions for the clamped, cantilever and simply supported boundary condition, respectively. Fig. 9a and b illustrates the 1st bending modeshape with deflections along the y axis of bar of length l¼4.5 m for the clamped boundary condition as obtained from Solid FEM [55] and comparison among those obtained from GWBT, TBT taking into account primary torsional warping and Solid FEM, respectively. Fig. 8a, and b illustrates the first four bending natural frequencies with deflections along the y axis and the z axis, respectively, as obtained from GWBT, TBT [55] and Solid FEM [55] solutions for the clamped boundary condition. Fig. 10a and b shows the first four bending natural frequencies with deflections along the z axis for the cantilever and the simply supported boundary condition respectively, as calculated by GWBT, TBT [55] and Solid FEM [55] solutions. From the results of Tables 4–6, it is concluded that even in closed shaped cross sections TBT [55] cannot predict the bending natural frequencies as accurate as GWBT can. In more detail the discrepancies of GWBT with respect to Solid FEM [55] solutions is up to 2.85% (absolute value) for any of the examined boundary condition or length. On the contrast the discrepancies of TBT [55] with respect to Solid FEM [55] solutions is up to 24.13% for Table 4, 18.18% for Table 5 and 21.15% for Table 6 (absolute values). Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

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22

1000 900 800

σ xx (kPa)

700 GWBT TBT E/BBT Solid FEM

600 500 400 300 200 100 0 0.00

0.01

0.02 t (sec)

0.03

0.04

4000

σxx (kPa)

2000 GWBT TBT E/BBT Solid FEM

0

−2000 −4000 −6000 0.00

0.01

0.02

0.03

0.04

t (sec) Fig. 13. Time histories of the normal stress σ xx at the point A of the end 1 of the beam of example 3. Load Cases I (a), II (b).

4.3. Example 3 In the third example, the forced vibrations of a double cell box shaped cross section (Fig. 11a) steel (E ¼210 GPa, G¼80.8 GPa, v ¼ 0:3, ρ ¼ 7:85 kN s2 =m4 ) beam of length l ¼5 m, clamped at both ends (Fig. 11b) have been examined. The beam is subjected to a uniformly distributed dynamic load pz ðt Þ (Fig. 11b and c). The load is eccentrically placed so it produces a torsional load mðt Þ ¼ a U pz ðt Þ where a ¼0.5 m is the half of the cross section width (Fig. 11a). Two load cases have been examined (Fig. 11c). The time history of loading is constant for load case I and sinusoidal for load case II. The frequency of load case II is 735.91 rad/s that is very close (805.68 rad/s) to the first natural frequency with deflections along the z axis, as calculated by GWBT. More specifically, the initial boundary value problem of Eqs. (8), (10) and (11) has been solved to obtain the response for the Generalized Warping Beam Theory (GWBT). For comparison reasons Euler–Bernoulli Beam Theory (E/BBT [55]), Timoshenko Beam Theory (TBT [55]) and Solid (3-D hexahedral elements extruded by quadrilateral surfaces) FEM (Solid FEM [55]) models have been used. In the Solid model, rigid diaphragms at every set of nodes spaced 0.5 m with the same properties as in the first example have been employed. In GWBT solution 22 elements have been used as in TBT [55] and E/BBT [55], while 10,800 solid elements have been employed for the Solid FEM [55] model. Fig. 12a and b illustrates the time histories of the displacement w at x ¼ l=2 for Load Cases I and II, respectively. Fig. 13a and b shows the time histories of the normal stress σ xx at the point A (see Fig. 11a) of the end 1 cross section of the beam for Load Cases I and II, respectively. From these figures, it is deduced that the results of the Solid FEM [55] solution are much closer to the proposed methodology (GWBT) than the TBT [55], so the effect of Generalized Warping is pronounced. The results of E/BBT [55] indicate that it is nessesesary to take into account shear deformations in the analysis. Moreover in the load case II (resonance) it is observed that in E/BBT [55] does not occur resonance as in other cases of analysis.

Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

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23

Fig. 14. Cross section (a), boundary conditions, loading (b) and load cases (c) of the beam of example 4.

4.4. Example 4 As a final example, the forced vibrations of a box shaped cross section (Fig. 14a) steel (E ¼21 GPa, G¼8.08 GPa, v ¼ 0:3, ρ ¼ 2:5 kN s2 =m4 ) beam of length l ¼30 m, clamped at end 1 and simply supported with fixed torsional rotation at end 2 (Fig. 14b) have been examined. The beam is subjected to a uniformly distributed dynamic load pz ðt Þ (Fig. 14b,c) along the axis of symmetry Z. Three load cases have been examined (Fig. 14c). The time history of loading is constant for load case I and sinusoidal for load cases II and III. The frequency of load case III is 51.48 rad/s being very close to the first natural frequency (56.44 rad/s), that is not valid for load case II with frequency 30.00 rad/s. More specifically, the initial boundary value problem of Eqs. (8), (10) and (11) has been solved to obtain the response of Generalized Warping Beam Theory (GWBT). For comparison reasons Timoshenko Beam Theory taking into account primary torsional warping (TBT [55]) and Solid (3-D hexahedral elements extruded by quadrilateral surfaces) FEM [55] models have Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

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24

−0.00020

w (m)

−0.00060 GWBT TBT Solid FEM

−0.00100

−0.00140

−0.00180 0.0

0.1

0.2

0.3 t (sec)

0.4

0.5

0.6

σxx (kPa)

600

400

GWBT TBT Solid FEM

200

0 0.0

0.1

0.2

0.3 t (sec)

0.4

0.5

0.6

Fig. 15. Time history of the displacement w at x ¼16.94 m (a) and normal stress σ xx at the point A at x ¼1.5 m (b) of the beam of example 4, for Load Case I.

Fig. 16. Distribution of normal stresses σ xx at x¼ 1.5 m for Load Case I of example 4, according to Solid FEM [55] (a), GWBT (b) and TBT [55] (c) solutions.

been analyzed. In the Solid model, rigid diaphragms at every set of nodes with the same properties as in the first example have been used. In GWBT solution 62 elements have been employed as in TBT [55], while 16,400 solid elements have been employed for the Solid FEM [55] model. Figs. 15a, 17a and 18a illustrate the time history of the displacement w at x¼16.94 m (position of maximum displacement) for Load Cases I, II and III, respectively. Figs. 15b, 17b and 18b illustrate the time history of normal stress σ xx at the point A (Fig. 14a) at x ¼1.5 m of the beam for Load Cases I, II and III, respectively. From these figures, it is deduced that the Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

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0.0020

0.0012

0.0004

w (m)

GWBT TBT Solid FEM

−0.0004

−0.0012

−0.0020

0.0

0.1

0.2

0.3 t (sec)

0.4

0.5

0.6

σ xx (kPa)

400

0

GWBT TBT Solid FEM

−400

−800

0.0

0.1

0.2

0.3

0.4

0.5

0.6

t (sec) Fig. 17. Time history of the displacement w at x¼ 16.94 m (a) and normal stress σ xx at point A at x ¼1.5 m (b) of the beam of example 4, for Load Case II.

results of the Solid FEM solution are closer to the proposed method (GWBT) than the TBT, so the effect of Generalized Warping is pronounced. Fig. 16a–c shows the distribution of normal stresses σ xx at x ¼1.5 m for Load Case I according to Solid FEM, GWBT and Timoshenko solutions, respectively, at time instances that appear the maximum stress at point A for each solution. The discrepancies of the distribution and the extreme values of GWBT and TBT with respect to Solid FEM solutions indicate that GWBT is much more appropriate for the analysis than the TBT.

5. Conclusions In this paper a boundary element solution is developed for the nonuniform warping dynamic analysis of arbitrary cross section beams including shear lag effects due to both flexure and torsion. The beam is subjected to arbitrarily distributed or concentrated external loading, and its ends are supported by the most general boundary conditions. The formulation based on displacements stems from an improved stress field and is capable of yielding more accurate stress component values. The main conclusions that can be drawn from this investigation are as follows:

 The developed procedure retains most of the advantages of a BEM solution over a FEM approach, although it requires     

longitudinal domain discretization. The accuracy and the validity of the proposed method are noteworthy. TBT cannot predict the bending natural frequencies as accurate as GWBT can. Employing GWBT, normal stresses are evaluated with high accuracy compared with solid FEM models. The developed formulation is capable of giving accurate results without increasing the number of the involved kinematical unknowns. In the treated examples, the influence of flexural and torsional shear lag effects was apparent, modifying significantly the distribution of normal stress. Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

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26

0.010 0.006

w (m)

0.002 GWBT TBT Solid FEM

−0.002 −0.006 −0.010 −0.014 0.0

0.1

0.2

0.3 t (sec)

0.4

0.5

0.6

σxx (kPa)

2000

GWBT TBT Solid FEM

0

−2000

−4000 0.0

0.1

0.2

0.3

0.4

0.5

0.6

t (sec) Fig. 18. Time history of the displacement w at x ¼16.94 m (a) and normal stress σ xx at point A of the end 1 cross section (b) of the beam of example 4, for Load Case III.

Acknowledgments This research (for the first two authors) has been co-financed by the European Union (European Social Fund-ESF) and Greek National Funds through the Operational Program "Education and Lifelong Learning" of the National Strategic Reference Framework (NSRF)-Research Funding Program: THALES: Reinforcement of the interdisciplinary and/or interinstitutional research and innovation.

Appendix. : Geometric constants The geometric constants of the beam can be obtained by the following definitions Z A¼ dΩ

(A.1a)

Ω

Z SY ¼

Z dΩ

(A.1b)

Y dΩ

(A.1c)

Z 2 dΩ

(A.1d)

Ω

Z SZ ¼

Ω

Z I YY ¼

Ω

Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i

I.C. Dikaros et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Z I ZZ ¼

27

Y 2 dΩ

(A.1e)

YZ dΩ

(A.1f)

Ω

Z I YZ ¼

Ω

Z Si ¼

Ω

i ¼ φPS ; φSS ; φPCY ; φPCZ

ðiÞ dΩ;

(A.1g)

Z I ij ¼

Ω

i; j ¼ y; z; φPS ; φSS ; φPCY ; φPCZ

ðiÞðjÞ dΩ;

Z Dij ¼

Ω

i; j ¼ ΦSS ; ΦΤS ; ΦPCY ; ΦSCY ; ΦPCZ ; ΦSCZ

∇ðiÞ U ∇ðjÞ dΩ; I Pt ¼

Z  Ω

 y2 þz2 zφPS;y þyφPS;z dΩ Z

Ip ¼

Ω

 2  y þ z2 dΩ

(A.1h)

(A.1i)

(A.1j)

(A.1k)

while it holds that APY  DΦP ΦP

(A.2a)

ASY  DΦS ΦS

(A.2b)

APZ  DΦP ΦP

(A.2c)

ASZ  DΦS ΦS

(A.2d)

I Sx  DΦSx ΦSx

(A.2e)

I Tx  DΦTx ΦTx

(A.2f)

Z

Z

Y

Y

Z

Z

Y

Y

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Please cite this article as: I.C. Dikaros, et al., Generalized warping effect in the dynamic analysis of beams of arbitrary cross section, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.01.022i