Finite-Element Formulation for the Linear Steady

Finite-Element Formulation for the Linear Steady-State Response of Asymmetric Thin-Walled Members under Harmonic Forces

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Mohammed Ali Hjaji1 and Magdi Mohareb, M.ASCE2 Abstract: A closed-form solution and finite-element formulation are developed for the dynamic analysis of thin-walled members with asymmetric open sections subjected to harmonic forces. The dynamic equations of motion and associated boundary conditions are derived from Hamilton’s principle. The formulation is based on a generalized Vlasov-Timoshenko beam theory and accounts for the effects of shear deformation caused by bending and warping and translational and rotary inertia effects. It also captures the effects of flexural-torsional coupling caused by cross-sectional asymmetry. From this a general closed-form solution is obtained. A family of shape functions is then developed based on the exact solution of the coupled field equations and is used to formulate a beam finite element. The new element has two nodes with six degrees of freedom per node and successfully captures the coupled bending-torsional static and steady-state responses of asymmetric thinwalled members under harmonic forces. Results based on the closed-form solution and finite-element formulation are assessed and validated against other well-established finite-element solutions. DOI: 10.1061/(ASCE)EM.1943-7889.0000849. © 2014 American Society of Civil Engineers. Author keywords: Thin-walled members; Shear deformation; Warping; Harmonic loads; Finite element.

Introduction and Scope Thin-walled members are widely used in the design of many structural components in aerospace structures, steel building construction, steel bridges, ship and marine structural frames, and so forth. In such applications, thin-walled members are subjected to harmonic excitations caused by machinery, aerodynamic forces, traffic loads, and wave motion. Also, harmonic forces can be induced from unbalance in rotating machinery and propellants and reciprocating machines. Under harmonic forces, the steady-state component of the response of the structural member is sustained for a long time and is therefore of particular importance in fatigue design. In contrast, the transient component of the response, which is induced only at the beginning of the excitation, tends to dampen out quickly and is of little or no importance in fatigue design. Within this context, the present study aims at developing an accurate and efficient solution, which captures the steady-state response component of thin-walled members subjected to general harmonic forces.

Literature Review General The classical thin-walled beam theory developed by Vlasov (1961) is widely used for the analysis members with open cross sections. The 1 Lecturer, Dept. of Mechanical and Industrial Engineering, Univ. of Tripoli, Tripoli, Libya. 2 Professor, Civil Engineering Dept., Univ. of Ottawa, Ottawa, ON, Canada K1N 6N5 (corresponding author). E-mail: [email protected] Note. This manuscript was submitted on March 26, 2014; approved on July 15, 2014; published online on August 11, 2014. Discussion period open until January 11, 2015; separate discussions must be submitted for individual papers. This paper is part of the Journal of Engineering Mechanics, © ASCE, ISSN 0733-9399/04014126(16)/$25.00.

© ASCE

theory is based on two kinematics assumptions: (1) a beam cross section remains rigid (undeformed) in its own plane; and (2) the transverse shear deformations within the section midsurface are considered negligible. The second assumption signifies that within the Vlasov theory, warping deformations are captured, whereas shear deformations are neglected. Given the vast literature on the subject, the present literature survey focuses on the dynamic analysis of thinwalled members with asymmetric cross sections, that is, members with doubly symmetric and monosymmetric cross sections are not included in the present survey. The interested reader is referred to previous work in Hjaji and Mohareb (2011a, b) and Hjaji (2013) for a review on doubly symmetric and monosymmetric sections. Review of Analytical Solutions Formulations Excluding Shear Deformation Effects The coupled flexural torsional dynamic stiffness matrix for thinwalled beams was numerically developed by Friberg (1985) and Leung (1991), who incorporated coupling effects caused by axial force. In a subsequent study, Leung (1992) developed a dynamic stiffness matrix solution, which incorporated coupling effects caused by axial compressive force and strong-axis moments. Frequencydependent shape functions based on the exact solutions of the governing differential equations were obtained and then used to formulate the dynamic stiffness matrix. Using the principle of virtual work, Chen and Tamma (1994) used the FEM in conjunction with an implicit-starting unconditionally stable numeric integration for the vibration analysis of thin-walled members subjected to constant axial force. Using finite-element analysis, Tanaka and Bercin (1997) investigated the coupled flexural-torsional free-vibration analysis of thin-walled beams of asymmetric open cross section. Based on D’Alembert’s principle, Tanaka and Bercin (1999) formulated the governing differential equations of motion for triply coupled vibrations based on nonorthogonal coordinates in which the resulting governing field equations included the product of the inertia term. The resulting equations were solved in an exact sense by

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Arpaci and Bozdag (2002). Arpaci et al. (2003) extended their work to include the effect of rotary inertia and developed an analytical solution for predicting the undamped natural frequencies. Di Egidio et al. (2003) developed a dynamic thin-walled beam theory, which neglects shear deformation effects but captures nonlinear warping effects. Kim et al. (2003b) conducted a free-vibration analysis for the flexural-torsional behavior of thin-walled beams with asymmetric sections under eccentric axial loads. The displacement functions were obtained, and the exact dynamic stiffness matrices were evaluated using force-deformation relationships. Vörös (2004) analyzed the free-coupled vibration of thin-walled beams with asymmetric open cross sections in which the warping deformation effect is incorporated. Using D’Alembert’s principle, Jun et al. (2004a) formulated the equations of motions and the dynamic transfer matrix to determine the natural frequencies and mode shapes of axially loaded thin-walled beams. Using Hamilton’s principle, Mohri et al. (2004) formulated the governing equations for prebuckling and postbuckling dynamic behaviors in thin-walled composite members under compressive forces and transverse forces. Their solution captures nonlinear warping and bending-torsion coupling. Based on Galerkin’s approach, the governing partial differential equations were reduced to a nonlinear ordinary coupled differential equation system only in time. Using the principle of virtual work, Prokic (2006) derived a system of equations for triply coupled free vibrations of thin-walled beams. Closed-form solutions were obtained for the natural frequencies of simply supported thin walls of asymmetric cross sections. Zhu and Leung (2006) formulated the dynamic stiffness matrix for open thin-walled beams by using frequencydependent shape functions, which exactly satisfy the governing differential equations for free vibration. Based on Hamilton’s principle, Vo et al. (2010) studied the coupled flexural-torsional free vibration for thin-walled open composite beams with arbitrary layups subjected to constant axial compressive force. The theory was based on classical lamination theory and accounts for material anisotropy couplings. Most of these studies were based on the Vlasov beam theory and account for warping, rotary inertia, and axial force effects, except for the studies by Chen and Tamma (1994), Tanaka and Bercin (1997, 1999), Arpaci et al. (2003), Kim et al. (2003b), Vörös (2004), Prokic (2006), Vo et al. (2010), who omitted the axial force effects, and Jun et al. (2004b), who omitted the rotary inertia effects. Using D’Alembert’s principle and the principle of virtual work, Chen and Hsiao (2007) formulated the governing equations for the coupled axial-torsional vibration response of the thin-walled beam of Z cross sections. The coupled axial-torsional natural frequencies were obtained by solving the equations of motion. Kim et al. (2007) developed the static and dynamic stiffness matrices for coupled flexural-torsional stability and free-vibration analyses for thinwalled beam subjected to linearly varying axial force. Their solution was based on Vlasov’s kinematics, and the static and dynamic stiffness matrices were derived based on power series. In their equations of motion and force-deformation relationships, they retained secondorder terms of the rotations. Using the principle of virtual work, Vörös (2008, 2009) formulated the governing field equations for the bending-torsional coupled vibration of thin-walled beams subjected to longitudinal loads. Vörös’ solutions account for large rotations. Using D’Alembert’s principle, Altintas (2010) derived the governing equations of motion for axially loaded thin-walled members that investigated the effects of material properties and axial force level on the natural frequencies using the finite-difference method. The aforementioned studies have the commonality of neglecting shear deformation effects. Recently, Gonçalves et al. (2010) developed a generalized beam theory (GBT) solution, which captures shear deformation effects in thin-walled beams. The treatment allows the © ASCE

isolation of the shear deformation effect. Within the GBT solution methodology, Ranzi and Luongo (2011) developed an efficient technique in which they adopted the natural modes of the cross section to express the displacements within the plane of the cross section. Formulations Including Shear Deformation Effects Because shear deformation has a significant effect on problems where higher modes of vibrations are required or where a member is subjected to harmonic forces with high frequencies, modified versions of the Vlasov theory were developed by many researchers to capture transverse shear deformation effects. Laudiero and Savoia (1991) studied the flexural-torsional vibrations of thin-walled beams with open and closed cross sections. Secondary warping and shear lag effects were incorporated in their formulation. Tanaka and Bercin (1997) studied the coupled flexural-torsional free vibrations of thinwalled members of asymmetric open C-sections. Kollár (2001) developed a theory of free-vibration analysis of thin-walled open section composite beams and provided closed-form solutions for the coupled flexural-torsional natural frequencies for simply supported beams. Cortínez and Piovan (2002) developed an analytical solution for the free-vibration analysis of composite thin-walled beams. Kim et al. (2003a) formulated the exact dynamic and static stiffness matrices for the free-vibration and stability analysis of thin-walled sheardeformable beams. Also, they incorporated flexural-torsional coupling effects caused by the asymmetry of the cross sections. In a subsequent study, Kim and Kim (2005) adopted the theory in Kim et al. (2003a) to formulate the dynamic stiffness matrix element for the flexuraltorsional free vibration of asymmetric shear-deformable thin-walled beams. By applying the Hellinger-Reissner variational principle, the governing equations of motion were derived for the coupled vibration response and force-deformation relationships. Using the virtual work principle, Prokic (2006) derived the differential equations for the coupled vibrations for thin-walled beams capturing shear deformation effects caused by bending. The closed-form solution for the natural frequencies was derived for the case of simply supported beams. Vo and Lee (2009a, b) presented a general analytical solution based on the shear-deformable beam theory for the study of flexural-torsional buckling and vibration analysis of open thin-walled composite beams. Ambrosini (2009) developed a general theory for coupled flexuraltorsional free vibrations for thin-walled beams of open cross sections. de Borbón and Ambrosini (2010) extended the theory for coupled flexural-torsional vibrations of thin-walled beams while incorporating the effect of the axial forces. In Ambrosini (2010), an experimental study for the free vibration of thin-walled beams with asymmetric open cross sections was conducted, and the results were used to assess the accuracy of various theoretical solutions. These studies account for the effects of shear deformation, warping, and rotary inertia. Finite-Element Formulations In general, finite elements are based on three types of shape functions: (1) approximate polynomial interpolation functions, (2) shape functions based on the exact solution of the static equilibrium equations, and (3) shape functions based on the exact solution of the dynamic equations of motion. Formulations based on the approximate shape functions are most common and are included in the work of Chen and Tamma (1994), Hashemi and Richard (2000a, b), Lee and Kim (2002a, b), Kim and Kim (2005), Vörös (2008, 2009), Vo and Lee (2009a, b, 2010, 2011), and Vo et al. (2009, 2010, 2011). Solutions based on the exact solution for static equilibrium equations include the work of Mei (1970) and Hu et al. (1996). They have the advantage of avoiding locking problems, which could arise in some of the solutions based on polynomial interpolation. Finite-element

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solutions based on the exact solution of the dynamic equations of motion include the work of Hjaji and Mohareb (2011b) and Hjaji (2013). They offer the following two advantages: (1) they eliminate discretization errors arising in conventional interpolation schemes and converge to the solution using a minimal number of degrees of freedom; and (2) they lead to elements that are free from shear locking. However, the solutions based on this approach are applicable only for doubly symmetric sections (Hjaji and Mohareb 2011b) and monosymmetric sections (Hjaji 2013); therefore, they are unable to capture the triply coupled response in asymmetric sections. Within this context, this paper aims to develop a finiteelement formulation for dynamic steady-state analysis of thin-walled open members with asymmetric open sections. The formulation sought is based on exact shape functions and captures shear deformation effects caused by bending and warping, translational and rotary inertias, and bending-torsional coupling effects caused by the asymmetry of the cross section.

Assumptions The main assumptions of the formulation are as follows: 1. The solution is applicable to members of general open thinwalled cross sections. 2. The solution is restricted to prismatic members. 3. The material remains linearly elastic throughout deformation. 4. Strains and rotations are small. Under the small rotation assumption, the coupling between the axial deformation in the member and its torsional flexural behavior is neglected. 5. The cross section of the member remains rigid (undistorted) in its own plane throughout deformation, in line with the first Vlasov assumption (Vlasov 1961). 6. Damping is neglected in the formulation. For typical damping ratios of 2–3% in steel members, the omission of damping leads to an overestimation of the resulting displacement amplitudes, which is rather small when the exciting frequency is offset from the natural frequencies of the member. 7. For the flexural response, a cross section originally planar remains planar throughout deformation, but it does not remain perpendicular to the cross section after deformation, that is, the transverse shear deformations of the middle surface of the cross section are incorporated in the assumed kinematics (in a manner similar to the Timoshenko beam theory). 8. For the torsional response, the section is assumed to undergo warping in a manner analogous to the Vlasov beam. 9. The Timoshenko assumption in bending deformation (Item 7) is further generalized to warping deformation (i.e., shear strains induced by warping at the middle surface are assumed as nonzero and are characterized by a generalized displacement function multiplied by the sectorial coordinate). The assumed kinematics in Items 7 and 8 can be conceived as a combination of the Vlasov and Timoshenko theories. Similar kinematics are assumed within different analysis types in Laudiero and Savoia (1991), Back and Will (1998), Cortínez and Piovan (2002), Kim et al. (2003a), Machado and Cortínez (2007), Vo and Lee (2010), and Wu and Mohareb (2011).

Fig. 1. (a) Local coordinate system and displacement components of a point on the cross-sectional contour; (b) tangential and normal displacements

section centroid C. The second is the local coordinate system, (s, n, Z), where the s-axis is tangent to the midsurface and is directed along the contour of the cross section at a generic point, p, on the contour, and the n-axis is normal to the s-coordinate and Z-axis. According to the first Vlasov assumption, the section is assumed undeformable in its own plane. As a result, the in-plane displacements, up ðz, s, tÞ and vp ðz, s, tÞ, both oriented along the global coordinate system of point p½xðsÞ, yðsÞ on the contour of the cross section are related to the displacement components, uðz, tÞ and vðz, tÞ, at the shear center, Sc , through up ðz, s, tÞ ¼ uðz, tÞ 2 ðy 2 ys Þuz ðz, tÞ

(1)

vp ðz, s, tÞ ¼ vðz, tÞ þ ðx 2 xs Þuz ðz, tÞ

(2)

where uz ðz, tÞ 5 twist angle of the cross section; xðsÞ and yðsÞ 5 coordinates of point p; s 5 curvilinear coordinate; and xs and ys 5 coordinates of the shear center along the X-direction and Y-direction, respectively. Based on Assumptions 7 and 8, the longitudinal displacement, wp ðz, s, tÞ, of point p is obtained by (Back and Will 1998) wp ðz, s, tÞ ¼ wðz, tÞ þ yðsÞux ðz, tÞ 2 xðsÞuy ðz, tÞ þ vðsÞcðz, tÞ (3) where wðz, tÞ 5 average longitudinal displacement along the Z-direction; ux ðz, tÞ and uy ðz, tÞ 5 bending rotations of the cross section about the X-axis and Y-axis, respectively; cðz, tÞ 5 function that characterizes the magnitude of the warping deformation; and vðsÞ 5 warping function of the cross section. The tangential displacement, jðz, s, tÞ, of point pðx, yÞ on the midsurface can be expressed in terms of the shear center displacements, uðz, tÞ and vðz, tÞ, and the rotation of the cross section, uz ðz, tÞ, as jðz, s, tÞ ¼ uðz, tÞ

d d xðsÞ þ vðz, tÞ yðsÞ þ hðsÞuz ðz, tÞ ds ds

(4)

_ _ _ _ where hðsÞ 5 xðsÞsin a 2 yðsÞ cos a , sin a 5 dyðsÞ=ds, cos a 5 dxðsÞ _ =ds, and aðsÞ 5 angle between the tangent of point p and the X-axis (Fig. 1); and hðsÞ 5 perpendicular distance from the shear center to the tangent to the contour at point p.

Displacement Fields Fig. 1 shows the midsurface (contour) of a general thin-walled cross section. Two sets of right-handed coordinate systems are shown. The first is the orthogonal Cartesian coordinate system, (X, Y, Z), in which the Z-axis is parallel to the longitudinal axis of the member and the X-axis and Y-axis are the principal axes passing through © ASCE

Strain-Displacement Relationships Based on the small-strain assumption, the longitudinal strain, ɛ zz , and shear strain, gzs , for point pðx, yÞ are related to the displacement fields through

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ɛ zz

∂wp , ∂z

gzs

∂j ∂wp þ ∂z ∂s

(5)

arbitrary time limits, t1 and t2 . The expressions of the variations dT p , dU p , and dW p are (e.g., Librescu and Song 2006)

Variational Formulation

dT p ¼

The variational form of the Hamilton’s principle is expressed as ðt2 dP ¼

p

(7)

0 A

dðT 2 U Þdt þ dW p dt ¼ 0

(6) p

p

ðℓ ð

dU ¼

t1 p

where T 5 kinetic energy; U 5 internal strain energy; W 5 work done by external forces; and the integration is performed between the Downloaded from ascelibrary.org by Ottawa, University Of on 07/07/15. Copyright ASCE. For personal use only; all rights reserved.

r u_ p du_ p þ v_p dv_ p þ w_ p dw_ p dAdz

ðt2

p

t1 p

ðℓ ð

ðℓ ð

Ggzs dgzs dAdz þ GJu9du z 9dz z

Eɛ zz dɛ zz dAdz þ 0 A

ðℓ

0 A

0

(8)

ℓ ℓ dW p ¼ ½Nz ðze , tÞdwðze , tÞℓ0 þ ½Vx ðze , tÞduðze , tÞℓ0 þ Vy ðze , tÞdvðze , tÞ 0 þ ½Mx ðze , tÞdux ðze , tÞℓ0 þ My ðze , tÞduy ðze , tÞ 0 þ ½Mz ðze , tÞduz ðze , tÞℓ0

þ ½Mw ðze , tÞdcðze , tÞℓ0

ðℓ h þ qx ðz, tÞduðz, tÞ þ qy ðz, tÞdvðz, tÞ þ qz ðz, tÞdwðz, tÞ þ mx ðz, tÞdux ðz, tÞ 0

i þ my ðz, tÞduy ðz, tÞ þ mz ðz, tÞduz ðz, tÞ þ mw ðz, tÞdcðz, tÞ dz

(9)

where E 5 Young’s modulus; G 5 shear modulus; r 5 density of member material; J 5 Saint-Venant torsional constant; and A 5 crosssectional area. All primes denote derivatives with respect to coordinate z, whereas dots denote derivatives with respect to time. In Eq. (9), qz ðz, tÞ, qy ðz, tÞ, and qx ðz, tÞ are the distributed longitudinal, transverse, and lateral harmonic forces, respectively; mz ðz, tÞ is the twisting harmonic moment; mx ðz, tÞ and my ðz, tÞ are the biaxial harmonic bending moments along the X-direction and Y-direction; mw ðz, tÞ is the harmonic bimoment; symbol Nz ðze , tÞ is the concentrated longitudinal force at both ends (ze 5 0, ℓ); Vx ðze , tÞ and Vy ðze , tÞ are the transverse forces; Mz ðze , tÞ is the twisting moment; Mx ðze , tÞ and My ðze , tÞ are the bending end moments; and Mw ðze , tÞ is the end bimoment, all applied at the beam ends. All applied generalized forces are assumed to have the same sign convention as those of the end-displacement components (Fig. 1).

Expressions for Applied Forces The member is assumed to be subjected to general applied harmonic forces within the member qx ðz, tÞ, qy ðz, tÞ, qz ðz, tÞ, mx ðz, tÞ, my ðz, tÞ, mz ðz, tÞ, mw ðz, tÞ ¼ qx ðzÞ, qy ðzÞ, qz ðzÞ, mx ðzÞ, my ðzÞ, mz ðzÞ, mw ðzÞ eiVt

(10)

and have generalized harmonic forces at both ends Nz ðze , tÞ, Vx ðze , tÞ, Vy ðze , tÞ, Mx ðze , tÞ, My ðze , tÞ, Mz ðze , tÞ, Mw ðze , tÞ ¼ N z ðze Þ, V x ðze Þ, V y ðze Þ, M x ðze Þ, M y ðze Þ, M z ðze Þ, M w ðze Þ eiVt , for ze ¼ 0, ℓ

(11)

pffiffiffiffiffiffiffi in which V 5 circular exciting frequency of the applied forces; and i 5 21 is the imaginary constant.

Steady-State Displacement Functions Under the given harmonic forces, displacements corresponding to the steady-state component of the response are assumed to take the form

Uðz, tÞ 17 ¼ U 1 ðzÞ 11 U S ðzÞ 16 17 eiVt ¼ w ðzÞ uðzÞ vðzÞ ux ðzÞ

uy ðzÞ uz ðzÞ cðzÞ 17 eiVt

(12)

where wðzÞ, uðzÞ, vðzÞ, ux ðzÞ, uy ðzÞ, uz ðzÞ, cðzÞ 5 amplitude space functions for longitudinal translation, bending translations, bending rotations, torsional rotation, and warping deformation, respectively. Because the present formulation is intended to capture only the steady-state response of the system, the displacement fields postulated in Eq. (12) neglect the transient component of the response. © ASCE

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rAV2 þ EA D2 wðzÞ ¼ qz ðzÞ

Governing Field Equations


From Eqs. (1)–(4), by substituting into the strain-displacement relationships in Eq. (5), the new expressions are then substituted into the energy, Eqs. (7)–(9). The resulting energy equations are substituted into Hamilton’s principle (Eq. 6). By enforcing the orthogonality Ð conditions A hx, y, xy, xv, yv, vi137 dA 5 h0i137 ; performing integration by parts with respect to t; evoking the stationary condition of the Hamilton’s functional; noting that all the variations of the coefficients at the time limits, t1 and t2 , are zero, that is, dwðz, tÞjtt21 5 0, duðz,tÞjtt21 50, dvðz,tÞjtt21 50, dux ðz,tÞjtt21 50, duy ðz,tÞjtt21 50, duz ðz,tÞjtt21 50, and dcðz,tÞjtt21 50; and substituting from the harmonic expressions in Eqs. (10)–(12), one recovers the field equations

2

ZS

66

6 6 6 6 6 6 ¼6 6 6 6 6 4

2 rAV2 þ GDxx D2

2GDxy D2

2 rAV2 þ GDyy D2

66

U S ðzÞ 61 ¼ QS ðzÞ 61

(14)

in which

QS ðzÞ 16 ¼ qx ðzÞ qy ðzÞ mx ðzÞ my ðzÞ mz ðzÞ mw ðzÞ 16 U S ðzÞ 16 ¼ uðzÞ vðzÞ ux ðzÞ uy ðzÞ uz ðzÞ cðzÞ 16

GDxy D

2 rAV2 ys þ GDhx D2 rAV2 xs 2 GDhy D2

2GDhy D

GDxy

2GDhy D

2GDhy

rIyy V2 2 GDxx þ EIyy D2

GDhx D

2GDxy D

GDxx D

2GDyy D rIxx V2 2 GDyy þ EIxx D2

symm:

ZS

(13)

2 rAV2 ro2 þ GðDvv þ JÞD2

3

2GDhx D

GDhx

2GDvv D rCw V2 2 GDvv þ ECw D2

7 7 7 7 7 7 7 7 7 7 7 5

Ð where ro2 5 ð1=AÞ A ðh2 1 r 2 ÞdA 5 x2s 1 y2s 1 ðIxx 1 Iyy Þ=A 5 polar radius of gyration about the shear center; D 5 differential operator, that is, D [ d=dz; and D2 [ d2 =dz2 . In Eqs. (13) and (14), the following cross-sectional properties were defined: 2 2 2 # ð" dx dy dx dy dv dx dv dy dv 2 2 2 Ixx , Iyy , Cw , Dxx , Dyy , Dxy , Dhx , Dhy , Dvv ¼ y , x , v , dA (15) , , , , , ds ds ds ds ds ds ds ds ds A

Eq. (13) provides the governing equation for longitudinal deformation of the member, which is uncoupled from the remaining field equations and was solved independently (Hjaji and Mohareb 2011a). Eq. (14) governs the solution for the coupled system biaxial bending-torsional response of the member. Therefore, the present study only focuses on the solution of the coupled system of equations presented in Eq. (14). The aforementioned equations can be obtained from the work by Laudiero and Savoia (1991) after neglecting the shear lag effects and secondary warping terms. The present study differs from that of Laudiero and Savoia (1991) in three respects. First, Laudiero and Savoia developed an approximate solution based on trigonometric series expansions for the field equations for simply supported members. In contrast, the present treatment solves the coupled differential equations for general closed-form solutions. Second, whereas Laudiero and Savoia investigated the free-vibration analysis of thin-walled members, the current solution provides the steady-state response under general harmonic forces. Finally, in the current study, the closed-form solutions derived are used to formulate exact shape functions and develop a superconvergent finite element. This contrasts with the solution in Laudiero and Savoia (1991), who focused on providing an analytical solution to specific boundary conditions.

Closed-Form Solution of Coupled Field Equations The homogeneous solution of the governing coupled field equations, ½ Z S 636 fU S ðzÞg631 5 fQS ðzÞg631 , is obtained by setting the loading terms in the field equations to zero, that is, fQS ðzÞg631 5 0. The solution of the space displacement functions, fU S ðzÞg631 , is then assumed to take the following exponential form: 6 P U S ðzÞ 16 ¼ uðzÞ vðzÞ ux ðzÞ uy ðzÞ uz ðzÞ cðzÞ 16 ¼ ha1 j a2 j a3 j a4 j a5 j a6 ii,16 emi z

(16)

i51

From the space displacements postulated in Eq. (16), by substituting into Eq. (14), one obtains the quadratic eigenvalue problem

m2i M 66 þ mi C 66 þ K 66 fagi,61 ¼ 0

(17)

where fagi 5 eigenvectors corresponding to eigenvalues mi ; and matrices ½M 636 , ½ C 636 , and ½ K 636 are defined by © ASCE

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2

M

66

6 6 6 ¼6 6 6 4

2GDxx


6 6 6 C 66 ¼ 6 6 6 4

K 66

6 6 6 6 ¼6 6 6 4

2rAV2

0 2rAV2

0 0 EIxx

2GDhx 2GDhy 0 0 2GðDvv þ JÞ

0 0 0 EIyy

symm:

2

2

2GDxy 2GDyy

0

2GDxy 2GDyy 0

0 0

GDxx GDxy 0 0

0 0 2GDhy GDhx 0

symm:

0 0 rIxx V2 2 GDyy

3 0 0 7 7 0 7 7 0 7 7 0 5 ECw

3 2GDhx 2GDhy 7 7 7 0 7 7 0 7 2GDvv 5 0 2rAV2 ys rAV2 xs 0 0 2rAV2 ro2

0 0 GDxy rIyy V2 2 GDxx

symm:

(18a)

(18b)

3 0 7 0 7 7 2GDhy 7 7 2GDhx 7 7 5 0 2 rCw V 2 GDvv

(18c)

The quadratic 6 3 6 eigenvalue problem in Eq. (17) is transformed into an equivalent 12 3 12 unsymmetric linear right-handed eigenvalue problem (Saad 1992) as " # " #! ( ) fagi,61 ½In 66 ½In 66 ½066 ½066 mi 2 ¼ f0g121 (19) mi fagi,61 2K 66 2C 66 ½066 M 66 1212

121

which is then solved for the 12 eigenpairs mi and fagi,631 . For an asymmetric section, it is observed that all 12 roots are nonzero and distinct. Therefore, the homogeneous solution of the system of Eq. (19) takes the form

U S ðzÞ 61 ¼ A 612 EðzÞ 1212 fCg121 (20) where ½ A6312 5 fag1 fag2 fag3 fag4 . . . fag12 5 matrix of right eigenvectors; ½ EðzÞ12312 5 diag em1 z em2 z em3 z em4 z . . . em12 z 5 diagonal matrix of exponential functions emi z (i 5 1, 2, 3, . . . , 12); and fCg1231 5 vector of unknown integration constants to be determined from the boundary conditions of the problem. It is emphasized that Eq. (20) is valid when all 12 roots are distinct nonzero roots, that is, mi mj for all i j, which is the case for asymmetric sections. When the section has a single axis or two axes of symmetry, some of the roots vanish and/or become repeated. For instance, in a doubly symmetric wide flange section, sectional properties Dhx , Dhy , and Dxy vanish and the 6 3 6 coupled system of equations given in Eq. 14 simplifies to three separate 2 3 2 coupled systems of equations. The first coupled system relates the displacement fields, uðzÞ and uy ðzÞ, which describe the flexural behavior about the y-axis. The second system relates vðzÞ and ux ðzÞ and describes the flexural behavior about the x-axis, whereas the last system relates uz ðzÞ and cðzÞ and describes the twist/warping response. Each of the three systems is then solved independently (Hjaji and Mohareb 2011a).

Finite-Element Formulation This section formulates a two-noded finite element with six degrees of freedom per node. The element is based on a family of shape functions, which exactly satisfy the equilibrium of motions.

Exact Shape Functions The vector of unknown integration constants fCg1231 is expressed in terms of the nodal displacements by enforcing the conditions uð0Þ 5 u1 , vð0Þ 5 v1 , ux ð0Þ 5 ux1 , uy ð0Þ 5 uy1 , uz ð0Þ 5 uz1 , and cð0Þ 5 c1 and uðℓÞ 5 u2 , vðℓÞ 5 v2 , ux ðℓÞ 5 ux2 , uy ðℓÞ 5 uy2 , uz ðℓÞ 5 uz2 , and T cðℓÞ 5 c2 . The displacement field functions hU S ðzÞi136 are expressed in terms of nodal displacements U N ðzÞ ¼ u1 v1 ux uy uz c1 u2 v2 ux uy uz c2 112

1

1

1

2

2

2

112

yielding © ASCE

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(

) U S ð0Þ 61

U S ðℓÞ 61 # " A 612 Eð0Þ 1212 ¼ fCg121 A 612 EðℓÞ 1212

Discretized Equilibrium Equations

U N ðzÞ 121 ¼

¼ ½R1212 fCg121

(21)


From Eq. (21), by substituting into Eq. (20), one obtains

U S ðzÞ 61 ¼ A 612 EðzÞ 1212 ½R21 1212 U N ðzÞ 121

¼ ½HðzÞ612 U N ðzÞ 121 (22) where ½HðzÞ6312 5 ½ A 6312 ½ EðzÞ12312 ½R21 12312 5 matrix of 72 shape functions. The interpolation shape functions provided in Eq. (22) exactly satisfy the homogeneous solution of the field coupled equations and are dependent on the member span, cross-sectional geometry, and exciting frequency of the applied forces.

Xa 2

Xb

66

6 6 6 ¼6 6 6 4

½Hd ðzÞT126 ¼

hn

H1,9 jðzÞ

o 121

XS

66

Eqs. (1)–(4) are substituted into Eq. (5) to express the strain relationships in terms of displacement functions. The resulting expressions are substituted into Eqs. (10)–(12) to express the variation of the energy expressions given in Eqs. (7)–(9). When the resulting energy expressions and Eq. (22) are substituted into Hamilton’s principle [Eq. (6)] and the orthogonality conÐ ditions, A ðx, y, xy, xv, yv, vÞdA 5 ð0, 0, 0, 0, 0, 0Þ, are evoked, one recovers

½Ke 1212 2 V2 ½Me 1212 1212 U N 121 ¼ fFe g121 (23) Ðℓ where ½Ke 12312 5 0 f½H9ðzÞT1236 ½ X a 636 ½H9ðzÞ6312 1 ½Hd ðzÞT1236 ½X b 636 ½Hd ðzÞ6312 gdz 5 element stiffness matrix; ½Me 12312 Ðℓ 5 0 f½HðzÞT1236 ½X S 636 ½HðzÞ6312 1 ½Hr ðzÞ12312 gdz 5 element mass matrix; and the element load vector of the applied harÐℓ monic forces is given by fFe ðzÞg1231 5 0 ½HðzÞT1236 fPðzÞg631 dz, where

h i ¼ diag 0 0 EIxx EIyy GJ ECw

GDxx

GDxy GDyy

GDxy GDyy GDyy

2GDxx 2GDxy 2GDxy GDxx

symm:

66

GDhx GDhy GDhy 2GDhx GDww

66

3 GDhx GDhy 7 7 GDhy 7 7 2GDhx 7 7 GDww 5 GDww 66

¼ diag rA rA rIxx rIyy rAro2 rCw

n o H2,9 jðzÞ

121

n o

H3, j ðzÞ 121 H4, j ðzÞ 121 H5,9 jðzÞ

121

i H6, j ðzÞ 121

2

½Hr 1212

3 ys rA H5, j ðzÞ 112 6 7

7 H5, j ðzÞ 112 ¼ H1, j ðzÞ H2, j ðzÞ H5, j ðzÞ 123 6 4 2xs rA 5 rA ys H1, j ðzÞ 112 2 xs H2, j ðzÞ 112

Examples and Discussion In this section, examples for thin-walled open beams of asymmetric cross sections subjected to general harmonic forces and various boundary conditions are presented to assess the validity, accuracy, and applicability of the present finite-element formulation. Provided are comparisons with results based on other established solutions for steady-state dynamic response. The material properties used in all examples are E 5 200 GPa, G 5 77 GPa, and r 5 7,850 kg=m3 . Three solutions are provided for comparison. They are as follows: (1) a solution based on the Vlasov beam © ASCE

theory, which neglects shear deformation and distortional effects; (2) a solution based on the ABAQUS 6.11 two-noded B31OS beam element with seven degrees of freedom per node (i.e., three translations, three rotations, warping deformation), which accounts for shear deformation only for bending but ignores shear deformation caused by warping deformation and distortional effects; and (3) a solution based on the ABAQUS S4R shell element solution (four-noded doubly curved shell element with six degrees of freedom per node, i.e., three translations and three rotations), which captures shear deformation and distortional effects of the cross section.

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Example 1: Long Cantilever under Uniformly Distributed Twisting Moments A 4.0-m-span cantilever with a thin-walled asymmetric channel section of unequal flanges as shown in Fig. 2 is subjected to uniformly distributed twisting moment mz ðz, tÞ 5 0:40eiVt kNm=m. The channel section has an upper flange width of bu 5 80 mm, lower flange width of bℓ 5 40 mm, and midsurface height of H 5 100 mm. The flange thicknesses are tf 5 10 mm, and the web thickness is tw 5 8 mm. It is required to (1) extract the natural frequencies from a steady-state dynamic analysis, and (2) determine the steady-state response for an exciting frequency, V 5 1:356 v1 . (As observed under the results of Item 1, the first natural frequency is v1 5 3:680 Hz.) The sectional properties for the C-section with respect to the XCY principal coordinate system through the centroid are A 5 0:20 3 104 mm2 , Ixx 5 3:723 3 106 mm4 , Iyy 5 0:878 3 106 mm4 , J 5 0:5707 3 105 mm4 , and Cw 5 0:8608 3 109 mm6 . The principal coordinates are inclined through angle b 5 17:14° (Fig. 2). The centroidal coordinates in the global coordinate system are ðCx , Cy Þ 5 ð20, 60 mmÞ, whereas the coordinates of the shear center, Sc , along the principal directions are ðXs , Ys Þ 5 ð242:83, 10:29 mmÞ. In the ABAQUS shell model, the cantilever is subdivided into 200 S4R elements along the longitudinal axis of the member, eight elements along the top flange, four elements along the bottom flanges, and 10 elements along the web height (i.e., ABAQUS shell model consists of 4,400 S4R elements). In the case of the ABAQUS beam model, a total of 200 B31OS beam elements were needed to achieve convergence, whereas the finite-element solution developed in the present study is based on exact shape functions and is conducted using a single finite element with two nodes and six degrees of freedom per node. It was observed to give results exactly matching those based on the closed-form solution up to five significant digits. Extracting Natural Frequencies Multiple steady-state response analyses under distributed twisting moments mz ðz, tÞ 5 0:40eiVt kNm=m are performed for an exciting frequency, V, varying from nearly zero to 100 Hz. The natural frequencies are then extracted from the peaks of the displacementfrequency relationships. The lateral displacement, uA , transverse displacement, vA , bending rotations, uy and ux , twist angle, uz , and warping deformation, c, at the cantilever tip against the exciting frequency, V, are depicted in Figs. 3(a, c, e, g, i, and k), respectively. The solutions, based on the Vlasov beam theory, ABAQUS B31OS, and ABAQUS shell solutions are overlaid on the same diagrams for comparison. Each peak indicates resonance and identifies natural frequencies of the beam. Therefore, the first eight natural frequencies extracted from the peaks of Figs. 3(a, c, e, g, h, i, and k) are provided in Table 1. In addition, the first eight modes for lateral and transverse displacements, related bending rotations, twist angles, and warping deformation are shown in Figs. 3(b, d, f, h, j, and l), respectively.

In the shell finite-element analysis model, the bending rotation angles, ux and uy , and the warping deformation, c, for a given section, zi , are determined from the ABAQUS shell element model by extracting the longitudinal displacements at section zi at the four corner points i 5 1, 2, 3, 4 of the cross sections. Eq. (3) is then used to express the longitudinal displacements leading to four equations of the form w ðz, si Þ 5 w ðzÞ 1 yðsi Þux ðzÞ 2 xðsi Þuy ðzÞ 1 vðsi ÞcðzÞ, where xi , yi for i 5 1, 2, 3, 4 are the coordinates of the four corner points along the principal directions, ðX, YÞ, and vi for i 5 1, 2, 3, 4 are the corresponding warping function values. This leads to the system of equations 9 9 8 3 8 2 1 yðs1 Þ 2xðs1 Þ vðs1 Þ wðz, s1 Þ > wðzÞ > > > > > > > > > > = < wðz, s Þ = 6 1 yðs Þ 2xðs Þ vðs Þ 7 < u ðzÞ > 2 2 2 2 7 x 6 ¼6 7 > 4 1 yðs3 Þ 2xðs3 Þ vðs3 Þ 5 > wðz, s3 Þ > uy ðzÞ > > > > > > > > > : ; ; : wðz, s4 Þ 41 1 yðs4 Þ 2xðs4 Þ vðs4 Þ 44 cðzÞ 41 (24) which is then solved for the vector of generalized displacements hwðzÞ ux ðzÞ uy ðzÞ cðzÞi134 . The resulting warping values for cðzÞ are depicted in Figs. 3(k and l). Table 1 provides the first eight natural frequencies as extracted from the steady-state response analyses based on all four solutions. As a general observation, the present solution predicts frequencies in excellent agreement with those based on the Vlasov closed-form and ABAQUS beam and shell solutions. This is particularly the case for the lower eigenfrequencies. The agreement is observed to slightly deteriorate for higher frequency modes but still remain close. For the eighth eigenfrequency prediction, all four solutions agree within a few percentage points. Frequencies predicted by the Vlasov solution are the highest (because the solution neglects shear deformation and distortion effects and provides the stiffest representation of the structure), whereas the ABAQUS shell solution has the lowest values (because it incorporates shear deformation and distortional effects and provides the most flexible and most accurate representation of the structure). The eigenfrequencies predicted by the ABAQUS shell model differ from those based on the present finiteelement solution by 2.32%, from the B31OS solution by 2.58%, and from the Vlasov solution by 3.09%. In other words, the present theory provides the best agreement with the shell solution. From a computational viewpoint, the present solution provides by far the most efficient solution. Using a single finite element with 12 degrees of freedom gives results in excellent agreement with the ABAQUS shell model with 27,700 degrees of freedom and the ABAQUS B31OS model with 1,400 degrees of freedom. Steady-State Dynamic Response The steady-state results for an exciting frequency, V 5 1:356v1 4:990 Hz, are illustrated in Figs. 4(a–f). The steady-state amplitudes for the lateral displacement, uA ðzÞ, transverse displacement, vA ðzÞ, bending rotations, ux ðzÞ and uy ðzÞ, angle of twist, uz ðzÞ, and warping deformation, cðzÞ, based on the present formulations are in excellent agreement with the results obtained by the Vlasov beam theory and ABAQUS B31OS and ABAQUS shell models. This signifies that shear deformation and distortional effects play a rather minor role in the steady-state dynamic response for such a long-span cantilever. Example 2: Shear Deformation Effect

Fig. 2. Cantilever asymmetric C-section under uniformly distributed twisting moment

© ASCE

To illustrate the shear deformation effects, a short 1.0-m-span cantilever is considered. The cross section is taken as identical to

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Fig. 3. Natural frequencies and vibration modes for cantilever with asymmetric C-section under harmonic torsion

Example 1. The member is subjected to a uniformly distributed harmonic twisting moment mz ðz, tÞ 5 8:0eiVt kNm=m. It is required to conduct a steady-state dynamic analysis and extract the natural frequencies of the beam and determine the steady-state dynamic response of the cantilever under the given load with exciting frequency V 5 0:84v1 5 48:05 Hz (the first natural frequency of the cantilever is 57.20 Hz). The Vlasov closed-form solution, ABAQUS B31OS beam solution, and S4R shell element models are provided for comparison. In the ABAQUS shell solution, the cantilever is subdivided into 100 elements in the longitudinal direction, and as in Example 1, there are eight and four elements along the width of top and bottom flanges and 10 elements through the web height. Therefore, the ABAQUS shell model consists of 2,200 S4R shell elements. For the B31OS solution model, a total of 100 beam elements are used. © ASCE

Shear Deformation Effects on Natural Frequencies The steady-state response is illustrated in Figs. 5(a and b) over an exciting frequency range from nearly zero to 700 Hz. To illustrate the influence of shear deformation on the natural frequencies, plots representing the amplitudes of the lateral displacement, uA , and angle of twist, uz , at the cantilever tip versus the exciting frequency are provided in Figs. 5(a and b). These curves exhibit six peaks, which correspond to the first six natural frequencies of the member. For the lower frequencies, all four solutions closely predict the location of peaks associated with the first four natural frequencies. For higher frequencies, deviations in the locations of the peaks are observed between all four solutions. The ABAQUS shell model predicts the lowest values of the eigenfrequencies, whereas the Vlasov beam theory predicts the highest values. The present solution yields slightly higher values than those based on the shell model.

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Fig. 3. (Continued.)

Table 1. Natural Frequencies (Hz) for a Long Cantilever with an Asymmetric Section Difference (%) Mode S4R (27,700 Present (12 B31OS (1,400 Vlasov number degrees of freedom) degrees of freedom) degrees of freedom) (closed form) ðS4R 2 presentÞ=S4R ðS4R 2 B31OSÞ=S4R ðS4R 2 VlasovÞ=S4R 1 2 3 4 5 6 7 8

© ASCE

3.679 7.251 22.05 23.45 39.82 62.59 72.83 92.91

3.680 7.265 22.35 23.75 40.37 62.82 73.86 95.07

3.686 7.269 22.46 23.78 40.45 63.11 74.03 95.31

3.687 7.280 22.48 23.81 40.68 63.32 74.32 95.78

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20:03 20:19 21:36 21:28 21:38 20:29 21:41 22:32

20:19 20:25 21:86 21:41 21:58 20:75 21:65 22:58

20:22 20:40 21:95 21:54 22:16 21:09 22:05 23:09

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Fig. 4. Steady-state responses for cantilever asymmetric C-section under distributed harmonic torsion

This suggests that for the present beam, the effects of distortion are minor, whereas the shear deformation effects gain significance for higher frequencies. A similar observation was reported in the work of Tanaka and Bercin (1997). Table 2 provides a comparison of the first six natural frequencies. The results show very good agreement between all four solutions. For higher frequencies, the present formulation (capturing the shear deformation effects) overpredicts the frequencies by less than 2.0%, whereas the Vlasov beam theory, which does not capture the shear deformation effects, overpredicts the corresponding frequencies by 5.28–8.44% compared with the ABAQUS shell element model. Shear deformation effects are prominent in short-span beams and for higher eigenfrequencies. © ASCE

Displacement Response and Effect of Shear Deformations The steady-state displacement amplitudes at the cantilever tip, which are subjected to exciting frequency V 5 0:84 v1 5 48:05 Hz are provided in Table 3. The displacements and rotations predicted by the present study are in excellent agreement with those of the ABAQUS S4R shell element model. For steady-state dynamic loading (Table 3), the warping deformation, c, at the cantilever tip as predicted by the ABAQUS shell element model is observed to depart from the present solution by 17.94%, by 19.07% from the B31OS solution, and by 20.20% from the Vlasov solution. Again, the difference is attributed to the localized distortion effects at the cantilever tip, which are captured only in the shell model.

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The amplitude responses for the lateral displacement, uA ðzÞ, transverse displacement, vA ðzÞ, angles of rotations, ux ðzÞ and uy ðzÞ, angle of twist, uz ðzÞ, and warping deformation, cðzÞ, are plotted in Figs. 6(a–f), respectively, under the exciting frequency, V 5 0:84 v1 5 48:05 Hz. Results based on the present solution are observed to be in excellent agreement with those of the ABAQUS shell element solution. Because of the omission of the shear deformation fully in the Vlasov beam theory and partially in the ABAQUS beam model (because B31OS does not capture shear deformation caused by warping), the predicted results obtained by these two solutions show some deviation from the ABAQUS shell model results. This leads to the conclusion that the shear deformation effects become important for short beams under harmonic forces. In contrast, the present solution provides very good agreement with shell model results. Example 3: Three-Span Continuous Beam A three-span continuous beam with an asymmetric J-section (Fig. 7) is subjected to three harmonic forces: uniformly distributed transverse force qy ðz, tÞ 5 12:0eiVt kN=m, concentrated twisting moment Mz 5 8:0eiVt kNm, and concentrated transverse force Py 5 24:0eiVt kN. The section dimensions are bu 5 160 mm, bℓ 5 80 mm, tf 5 20 mm, tw 5 15 mm, and H 5 200 mm. The coordinates of centroid C are Cx 5 8:205 mm and Cy 5 120:5 mm. The coordinates of shear center Sc along the principal coordinates are ðXs , Ys Þ 5 ð223:89, 42:24 mmÞ, the orientation of the principal direction is b 5 9:46°, and the cross-sectional properties are A 5 0:78 3 104 mm2 , Ixx 5 56:16 3 106 mm4 , Iyy 5 8:489 3 106 mm4 , J 5 0:8650 3 106 mm4 , Cw 5 57:00 3 109 mm6 , Dxx 5 47:51 3 104 mm2 , Dyy 5 30:49 3 104 mm2 , Dhx 5 23:705 3 104 mm3 , Dhy 5 1:296 3 104 mm3 , Dxy 5 22:918 3 104 mm2 , and Dvv 5 47:14 3 106 mm4 . It is required to assess the accuracy and efficiency of the present finite-element formulation solution in determining the steady-state dynamic response under an exciting frequency, V 5 1:74v1 5 19:77 Hz, where v1 , the first natural frequency of the three-span beam, is v1 5 11:36 Hz.

Fig. 5. Deformation versus exciting frequency analysis of short cantilever asymmetric C-section: (a) lateral displacement; (b) angle of twist

Table 2. Natural Frequencies (Hz) for Short Cantilever with an Asymmetric Section Difference (%) Mode S4R (13,900 Present (12 B31OS (700 Vlasov number degrees of freedom) degrees of freedom) degrees of freedom) (closed form) ðS4R 2 presentÞ=S4R ðS4R 2 B31OSÞ=S4R ðS4R 2 VlasovÞ=S4R 1 2 3 4 5 6

57.20 72.32 172.9 264.1 358.2 550.7

57.34 73.76 174.8 265.9 361.8 559.4

57.96 74.23 175.5 276.6 368.8 592.3

20:24 21:99 21:10 20:68 21:01 21:58

58.18 74.60 180.3 278.7 377.1 597.2

21:33 22:64 21:50 24:73 22:96 27:55

21:71 23:15 24:28 25:53 25:28 28:44

Table 3. Steady-State Response of Short Cantilever Asymmetric C-Section under Distributed Harmonic Torsion Difference (%) Variable uA (mm) vA (mm) ux (1023 rad) uy (1023 rad) uz (1023 rad) c (1026 rad=mm)

© ASCE

S4R (27,700 Present (12 B31OS (1,400 Vlasov ðS4R 2 presentÞ= ðS4R 2 B31OSÞ= ðS4R 2 VlasovÞ= degrees of freedom) degrees of freedom) degrees of freedom) (closed form) S4R S4R S4R 4.315 24.73 214.84 39.7 1,121 2755.8

4.170 23.47 214.17 38.91 1,076 2620.2

3.111 22.48 213.59 36.79 1,043 2611.7

2.728 21.87 213.23 35.32 1,020 2603.1

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3.36 5.10 4.51 1.99 4.01 17.94

27.90 9.10 8.42 7.33 6.96 19.07

36.78 11.56 10.85 11.03 9.01 20.20

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Fig. 6. Steady-state response for cantilever asymmetric C-section under distributed harmonic torsion

Three solutions are provided for the problem. The first solution is based on the ABAQUS shell model where the three-span beam is subdivided into 400 shell S4R elements along the longitudinal direction, 10 elements along height of the web, and eight and four elements along the width of the upper and lower flanges, respectively. The model consists of 8,800 shell S4R elements with six degrees of freedom per node, which leads to approximately 55,000 degrees of freedom. The second solution is based on the ABAQUS finite-beam model of 200 beam B31OS elements in which a total of 1,400 degrees of freedom were needed to attain the required accuracy. The third solution is based on the present formulation. The member is subdivided into only five beam elements along the member span, i.e., the model has only 30 degrees of freedom. Displacement Response The displacement functions, uðzÞ, vðzÞ, ux ðzÞ, uy ðzÞ, uz ðzÞ and cðzÞ, are provided in Figs. 8(a–f) for the steady-state dynamic case (V 5 1:74v1 5 19:77 Hz), respectively. The figures show excellent © ASCE

Fig. 7. Three-span continuous beam with asymmetric C-section under harmonic forces

agreement between the nodal displacement functions predicted by the present finite-element model (using 30 degrees of freedom) and the ABAQUS finite-element models, the shell model (using 55,000 degrees of freedom) and the beam model (using 1,400 degrees of freedom). The computational effort in the present solution is several

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Fig. 8. Steady-state dynamic analysis of three-span continuous beam with J-shaped cross section under harmonic forces

orders of magnitudes less than that of other solutions. This is a natural outcome of the fact that the present finite element is based on the shape functions, which exactly satisfy the homogeneous form of the equations of motion, which in turn eliminates discretization errors encountered in finite-element formulations.

Summary and Conclusions 1. The equations of motion and related boundary conditions are derived for thin-walled members with asymmetric open cross sections under general harmonic forces. The formulation captures shear deformation effects caused by bending and warping, translational and rotary inertia effects, and bending-torsional coupling effects caused by nonsymmetry of the cross section. 2. Closed-form solutions are formulated for the resulting coupled system of equations. © ASCE

3. A family of shape functions is derived based on the exact homogeneous solution of the governing equations. 4. A superconvergent finite element is then formulated based on the exact shape functions. 5. The new element involves no discretization errors and generally provides excellent results with a minimal number of degrees of freedom. 6. The present solution is capable of efficiently capturing the steadystate responses of members under harmonic forces. It is also able to predict the natural frequencies and mode shapes of the system. 7. The solution successfully captures the coupled bendingtorsional response of asymmetric cross sections under harmonic forces. 8. Comparisons with the Vlasov beam theory solution demonstrate the importance of shear deformation effects on shortspan thin-walled members under higher exciting frequencies.

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s 5 curvilinear coordinate along midsurface of the section; T p 5 kinetic energy; t 5 time; t1 , t2 5 time intervals; U p 5 internal strain energy; fU N g1231 5 vector of nodal displacements; fU S ðzÞg631 5 vector of space displacement functions; u, v 5 displacements of shear center along the principal X-axis and Y-axis; up , vp , wp 5 displacements of point p on midsurface of section along the X-axis, Y-axis, and Z-axis; Vj ðz, tÞ 5 shear forces along the x-axis and y-axis (for j 5 x, y); W p 5 work done by applied forces; w 5 average longitudinal displacement along the Z-axis; X, Y, Z 5 principal coordinate system; Xs , Ys 5 coordinates of shear center along the principal X-axis and Y-axis; x, y, z 5 Cartesian coordinate system; xðsÞ, yðsÞ 5 coordinate of arbitrary point on midsurface of the section along the X-axis and Y-axis; _ aðsÞ 5 angle between the tangent to the cross section and principal X-axis; gzs 5 shear strain at midsurface of cross section; ɛ zz 5 longitudinal normal strain; h, j 5 tangential and normal displacements of point p along the n-direction and s-direction; ux , uy , uz 5 rotation angles around the X-axis, Y-axis, and Z-axis, respectively; r 5 density of material; cðz, tÞ 5 warping deformation function; V 5 exciting frequency; and vðsÞ 5 warping function of cross section.

9. Comparisons with shell element solutions show that distortional effects are more pronounced in cantilevers with shorter spans than longer spans.


Notation The following symbols are used in this paper: A 5 cross-sectional area; ½A6312 5 matrix containing the right eigenvectors; bt , bb 5 widths of top and bottom flanges; C 5 centroid of cross section; Cw 5 warping constant; fCg1231 5 vector of unknown integration constants; Dxx , Dyy , Dxy , Dhx , Dhy , Dvv 5 section properties defined by Eq. (16); E 5 modulus of elasticity; ½EðzÞ12312 5 diagonal matrix of exponential functions; fFe g1231 5 element load vector of applied forces; G 5 shear modulus; H 5 height of beam cross section from the flanges midsurface; ½HðzÞ6312 5 matrix of exact shape functions; hðsÞ 5 normal distance between shear center and tangent to midsurface; Ixx , Iyy 5 moment of inertias of the cross section about the principal X-axis and Y-axis, respectively; J 5 torsional constant; ½Ke 12312 5 element stiffness matrix; ℓ 5 length of member; ½Me 12312 5 element mass matrix; Mj ðz, tÞ 5 concentrated harmonic moment about the jth direction (for j 5 x, y, z); Mw ðz, tÞ 5 concentrated bimoment; mj ðz, tÞ 5 distributed harmonic moments about the jth direction (for j 5 x, y, z); mw ðz, tÞ 5 distributed harmonic bimoment; Nz 5 concentrated end forces along longitudinal axis; n, s, z 5 local curvilinear coordinate system; qj ðz, tÞ 5 distributed harmonic forces along x-direction, y-direction, and z-direction (for j 5 x, y, z); ro 5 polar radius of gyration; rðsÞ 5 distance between shear center to the normal to midsurface; Sc 5 shear center of cross section; © ASCE

References ABAQUS 6.11 [Computer software]. Providence, RI, Dassault Systèmes. Altintas, G. (2010). “Effect of material properties on vibrations of nonsymmetrical axially loaded thin-walled Euler-Bernoulli beams.” Math. Comput. Appl., 15(1), 96–107. Ambrosini, D. (2009). “On free vibration of nonsymmetrical thin-walled beams.” Thin Walled Struct., 47(6–7), 629–636. Ambrosini, D. (2010). “Experimental validation of free vibrations from nonsymmetrical thin walled beams.” Eng. Struct., 32(5), 1324–1332.

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J. Eng. Mech. 2015.141.

J. Eng. Mech.