Quasi-3D solutions for the vibration of solid and hollow slender structures with general boundary conditions

Computers and Structures xxx (2018) xxx–xxx

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Computers and Structures journal homepage: www.elsevier.com/locate/compstruc

Quasi-3D solutions for the vibration of solid and hollow slender structures with general boundary conditions Yukun Chen, Tiangui Ye ⇑, Guoyong Jin College of Power and Energy Engineering, Harbin Engineering University, Harbin 150001, PR China

a r t i c l e

i n f o

Article history: Received 17 March 2018 Accepted 4 October 2018 Available online xxxx Keywords: Quasi-3D solution Free vibration analysis Slender structures Improved Fourier series Carrera unified formulation

a b s t r a c t In this study, a uniform quasi-three-dimensional solution is presented for the free vibration analysis of solid and hollow slender structures with general boundary conditions. The boundary conditions are simulated by penalty method, which releases the limitation of geometrical boundary restraints on the selection of admissible functions. A combination of the improved Fourier series method (IFSM) and Taylor expansion (TE) is introduced to construct the displacement components of the structures. The nuclear matrices of the stiffness and mass matrices are derived using the energy variational principle. Subsequently, an assembly procedure based on the Carrera unified formulation (CUF) is proposed for the global matrices of higher-order configurations. In the current study, the displacement functions are not dependent on the specific shape of the section, and the proposed method is thus suitable for any uniform slender structure. Herein, different types of standard solid and hollow slender structures are studied. The efficiency and accuracy of the present method are illustrated by comparing the results with the reference data and solutions of the commercial codes. Finally, the effects of the geometrical dimensions and boundary conditions on the vibration characteristics are investigated. Ó 2018 Published by Elsevier Ltd.

1. Introduction Slender structures, such as open-section, box, and hollow cylindrical beams, are extensively used as basic structural components in applications in mechanical, ocean, and aerospace engineering. The dynamic characteristics of slender structures are very attractive in the engineering design phases to ensure a stable operation and low-noise radiation. Physically, each slender structure is a three-dimensional (3-D) body, and its vibration characteristics can be predicted directly using the methods derived from the linear elasticity theory. Nevertheless, the solution of the 3-D equations for slender structures is highly complex and may require significant internal storage and computer time. Beam theories have been applied in the dynamic analysis of slender structures for more than one hundred years. Accordingly, a 3-D slender structure can be simplified as a one-dimensional (1-D) model with suitable assumptions. The 1-D models are certainly simpler and require fewer computations than the higher-dimensional models, and are thus particularly suitable for large-scale problems. Over the last few decades, numerous beam theories have been proposed based on different assumptions, such ⇑ Corresponding author. E-mail addresses: [email protected] (Y. Chen), [email protected], [email protected] (T. Ye).

as classical beam theories, the generalized beam theory, and the higher-order beam theory. The conventional beam theories, including the Euler and Timoshenko beam theories, have been extensively used in modeling the static and dynamic behavior of beams. The Euler beam theory (EBT) is proposed on the assumption that the beam section is perpendicular to the neutral surface before and after deformation. When short beams are considered, the results have large errors. Subsequently, a refined theory known as the Timoshenko beam theory (TBT) was proposed in 1921 [1]. The TBT considers a uniform value to represent the shear distribution over the beam section, where a relevant shear correction factor is introduced to remedy the discrepancy between the actual stress state and assumed uniform value, which can be observed in the works of Timoshenko and Goodier [2] and Sokolnikoff [3]. Furthermore, Kaneko [4] reviewed the studies on the shear correction factor applied in circular and rectangular cross-section beams. Han et al. [5] compared the Euler, Rayleigh, shear, and Timoshenko theories that deal with a transversely non-slender beam, and concluded that both the shear and Timoshenko models could be applied to the beams with a small slenderness ratio. However, in regard to more complex structures, such as thinwalled, thick-walled, and open-section beams, the assumed uniform stress state of the TBT is unable to determine the real stress distribution of the structures. Various authors focused on the use

https://doi.org/10.1016/j.compstruc.2018.10.001 0045-7949/Ó 2018 Published by Elsevier Ltd.

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of shear correction factors to enhance the static and dynamic responses of the classical beam theories in the application of numerous complex structures [6–9]. Gruttmann et al. [10–12] evaluated the shear correction factors for numerous structures such as arbitrary shaped cross-sections, thin-walled, and bridgelike structures. Even then, in a review study by Dong et al. [13], it was illustrated that the development of a fully accepted formulation for the shear correction factors was difficult. El Fatmi [14–17] introduced a warping function for the crosssectional displacement of a beam, and indicated that the modified displacement function could be used to describe the stress over the beam section more efficiently. Ladevèze et al. [18,19] simplified the 3D elasticity equations into a beam-like formulation on the basis of the de Saint–Venant solution. Another improved method based on the classical theory is the generalized beam theory, which was initially proposed by Schardt [20,21] and was subsequently extended by Silvetre [22,23] and Bebiano [24]. Carrera et al. [25] proposed a modified 1-D model with a quasi3-D accuracy on the basis of the Carrera unified formulation (CUF), which was developed for the investigation of plates and shells over a decade [26,27]. The CUF is a hierarchical formulation that offers a procedure to obtain refined structural models. In this formulation, many interpolation functions, such as Taylor and Lagrange polynomials, are applied for the description of structural deformation, and the order of the interpolation functions can be regarded as a predefined free-parameter so that the higher-order theories can be derived without any need of additional procedures. Correspondingly, 1-D-CUF models were successfully used to solve complex problems [28–32]. In most of the published works, the finite element method (FEM) has been employed to discretize the 1-D-CUF models [28,29,31,33]. However, the inconsistencies in the interpolation method used in the FEM inevitably leads to shear locking, which is exacerbated as the height-to-length ratio decreases [34]. The shear locking problem remains to be one of the important research issues in FEM applications. This study explores an efficient solution method for a 1-D-CUF model by utilizing the improved Fourier series method (IFSM), which was originally proposed by Li and dealt with beams and plates [35,36]. For this method, the displacement functions are usually expressed as a combination of the complete Fourier cosine series and various additive terms, which are presented to overcome the limitation of the standard Fourier approach. Recently, the IFSM was employed to solve the dynamic problems of different structures [37–41]. The main advantages of this technique are that each displacement function can be simulated adequately and smoothly throughout the entire solution domain and the corresponding coefficient matrix can be inverted easily. In this work, the CUF and IFSM are combined to provide quasi-3-D solutions for the free vibration analysis of both solid and hollow structures. The study is organized as follows: first, the displacement components of the slender structures are presented by combining the Taylor expressions and improved Fourier series. Second, the Lagrangian energy function is established based on the strain, kinetic and boundary energy. Subsequently, the nuclear matrices of the stiffness and mass matrices are derived using the energy variational principle. Next, an assembly procedure for the global matrices of any desired higher-order configuration is introduced. On these bases, typical solid and hollow slender structures are considered to illustrate the efficiency and accuracy of the proposed method. Moreover, the vibration characteristics of slender structures subjected to classical and elastic restraints are then predicted. Furthermore, the effects of the boundary conditions and geometrical dimensions on the vibration characteristics are studied accordingly.

2. Theoretical formulations 2.1. Model description The two typical slender structures that are considered in the present work are shown in Fig. 1. Fig. 1(a) shows the dimensions of a rectangular box beam. A rectangular box beam with any hollow ratio can be obtained by setting the inner lengths h2 from 0 to h1, and b2 from 0 to b1. For example, a solid beam can be obtained by setting the inner lengths h2 and b2 to 0, whereas a thin-walled structure can be obtained by setting the inner edge length h2 almost to h1 and b2 to b1. Similarly, the hollow cylindrical beam presented in Fig. 1(b) can be described by the length L2, outer radius R1, and inner radius R2. The Cartesian coordinate system (x, y, and z) is fixed at the structure. The x- and z- axes define the cross-sectional directions, and the y-axis corresponds to the axial direction. 2.2. Displacement functions The hollow structures are simplified as 1D models by separating the displacement field into two function expansions: the crosssectional and axial basic terms. The displacement variables are fitted over the cross-section using the Taylor series expansion. Table 1 presents the compact form of the Taylor expression series, and the N-order expression terms are presented in the corresponding row of the table. It is important to note that the Taylor expression series can be rewritten in a unified form, and the order of the Taylor expressions is considered as a free parameter that controls the accuracy of the method. The displacement functions can be written in the following form

Uðx; y; zÞ ¼ u1 þ xu2 þ zu3 þ    þ zN uðNþ1ÞðNþ2Þ=2 ¼

C X F s us

s¼1

Vðx; y; zÞ ¼ v 1 þ xv 2 þ zv 3 þ    þ zN v ðNþ1ÞðNþ2Þ=2 ¼

C X Fsvs

ð1Þ

s¼1

Wðx; y; zÞ ¼ w1 þ xw2 þ zw3 þ    þ zN wðNþ1ðNþ2Þ=2 ¼

C X F s ws

s¼1

where Fs indicates the number s term of the Taylor expression, and variables (us, vs, ws) denote the axial displacements of the structure. Additionally, C represents the sum of the Taylor expansion terms. The variables are expressed as the improved Fourier cosine series along the beam axis. An example is provided to determine the number and form of the additive terms. Consider a function f(y) defined in a closed interval [0, L]. The function can be expanded according to the Fourier cosine series

f ðy Þ ¼

1 X mp am cos y; L m¼0

06y6L

ð2Þ

The displacement components of the structures merely require first-order derivatives owing to the direct derivation of the energy function presented in the following section. In according to a formal differentiation, the first-order derivative of f(y) can be written as 1 X am mp mp 0 f ðyÞ ¼  sin y L L m¼0

ð3Þ

From Eq. (3), it is clear that the first-order derivative of the Fourier cosine series is equal to 0 at the two ends of the closed interval. This only satisfies the simply supported condition. To overcome the limitation, two sine terms are introduced

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Y. Chen et al. / Computers and Structures xxx (2018) xxx–xxx

z

z

x y

x

L1 z h1 h2

o

y

L2

z R2

x

x o

b2 b1

R1

(a)

(b)

Fig. 1. Slender structures with hollow cross-sections: (a) Rectangular box beam, and (b) hollow cylindrical beam.

2.3. Energy expressions

Table 1 Taylor expansion series. N

C

Fs

0 1 2 3 ... N

1 3 6 10 ... (N + 1)(N + 2)/2

F1 = 1 F2 = x F3 = z F4 = x2 F5 = xz F6 = z2 F7 = x3 F8 = zx2 F9 = xz2 F10 = z3 ... F ðN2 þNþ2Þ=2 ¼ xN . . . F ðNþ1ÞðNþ2Þ=2 ¼ zN

   p  2p p1 ðyÞ ¼ sin  y ; p2 ðyÞ ¼ sin  y L L

ð4Þ

Therefore, each variable in Eq. (1) can be expressed in a refined form of the trigonometric series as follows

us ðyÞ ¼

M X

asm Wm ðyÞ; v s ðyÞ ¼

m¼2

M X

M X

bsm Wm ðyÞ; ws ðyÞ ¼

m¼2

csm Wm ðyÞ

m¼2

ð5Þ where aim, bim, and cim, denote the expansion coefficients, and M represents the truncation index, and



Wm ðyÞ ¼

cos km y m P 0 sin km

ym < 0

km ¼ mp=L

ð6Þ

The aim of this part of the study is to determine the global energy of the structure, which mainly contains the strain, boundary, and kinetic energy. First, we can derive the strain energy based on the assumption that the material is perfectly elastic. Starting from the strain displacement relations, we group the strain components as follows

u ¼ fU

ep ¼ Dp u  en ¼ Dn u ¼ DnX þ Dny u ep ¼ f exy eyy ezy gT ; en ¼ f exx exz ezz gT

M X C M X C X X asm F s Wm ðyÞ; Vðx; y; z; tÞ ¼ bsm F s Wm ðyÞ m¼2 s¼1

Wðx; y; zÞ ¼

0

ð7Þ A harmonic solution is utilized for the purpose of free vibration analysis

U ðx; y; z; tÞ ¼ U ðx; y; zÞeixt ; V ðx; y; z; t Þ ¼ V ðx; y; zÞeixt W ðx; y; z; tÞ ¼ W ðx; y; zÞeixt

0

1

0

C 0C A;

@ @y @ @z

Dn ¼

@ @y

@ @x B@ @ @z

0

0

0

0

0

1

@ C @x A @ @z

ð10Þ

rp ¼ Cpp ep þ Cpn en rn ¼ Cnp ep þ Cnn en 0

Cpp

C 44 B ¼@ 0 0 0 0 B ¼ @ C 21 0

ð11Þ

0 1 C 11 0 C 13 C B C C 22 0 A; Cnn ¼ @ 0 C 66 0 A C 31 0 C 33 0 Cc 1 0 1 0 0 0 C 12 0 C B C 0 C 23 A; Cnp ¼ @ 0 0 0 A 0 C 32 0 0 0 a

b

1

ð12Þ

where Cij (i, j = 1–6) represent the stiffness coefficients that can be expressed as

Eð1  tÞ ð1 þ tÞð1  2tÞ tE ¼ ð1 þ tÞð1  2tÞ E ¼ 2ð1 þ tÞ

C 11 ¼ C 22 ¼ C 33 ¼ ð8Þ

where x denotes the natural frequency of the structure, and pffiffiffiffiffiffiffi 1.

i¼

@ @x

where ep denotes the strain terms pointing to the y ordinate, and en indicates the strain terms in the cross-section. The corresponding stresses are derived basing on Hooke’s law

Cpn

m¼2 s¼1

@ @y

B Dp ¼ B @0 0

m¼2 s¼1

M X C X csm F s Wm ðyÞ

ð9Þ

with

Substituting Eq. (5) into Eq. (1) and merging similar terms, we rewrite the displacement components as follows

Uðx; y; zÞ ¼

W gT

V

C 12 ¼ C 13 ¼ C 23 C 44 ¼ C 55 ¼ C 66

ð13Þ

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where E stands for the Young’s modulus and t is the Poisson’s ratio. According to the 3-D theory of elasticity, the strain energy of the structure can be written as

Vs ¼

1 2

Z Z  l

A

1 2

Z h  A



ð14Þ

 i uT k0 u jy¼0 þ uT kL u jy¼L dA

ð15Þ

8 M C   X X mnij > < K  x2 Mmnij qjn ¼ 0 n¼2 j¼1 > : ð1 6 i 6 C; 2 6 m 6 M Þ

0

B k0 ¼ @ 0

0

0

kuL

C B 0 A; kL ¼ @ 0 kw0 0

kv 0

0

1

0

0 kv L 0

0

Z K mnij ¼ C 11 xx

1

Z

A

Z Z l

A

0T

0

qu u dAdy;

 T @U @U @W u ¼ @t @t @t 0

ð17Þ

Z

A

0

L

Z

A

¼ C 12 K mnij xy

Wm Wn dy

Wm W0n dy þ C 44

F i;x F j dA A

0

Z ¼ C 13 K mnij xz

L

Z

L

0

A

Z

F i F j dA

A

Z

Z

L

W0m Wn dy

F i F j;x dA A

0

Z

Wm Wn dy þ C 66

F i;x F j;z dA

W0m W0n dy

0

þ ku0 Wm jy¼0 Wn jy¼0 þ kuL Wm jy¼L Wn jy¼L Z

L

F i F j dA

0

Z

where ku0, kv0, and kw0, represent the stiffness coefficients of the springs distributed at the end of y = 0, and kuL, kvL, and kwL stand for those of y = L end. It should be noted that the springs are uniformly distributed over the entire boundary area. On these bases, the classical restraints, elastic supports, and their combination can be simply achieved by setting the linear springs at the corresponding values. For example, a clamped end and a free end can be realized by setting the spring stiffness coefficients tend to infinity and zero. More details about the determination of the spring stiffness coefficients can be seen in the work provided by Ye et al. [38]. The spring stiffness coefficients for the classical boundary conditions are listed in Table 2. The kinetic energy of the structure can be expressed in the following form

1 Tp ¼ 2

0

Z

Wm Wn dy þ C 44

F i;z F j;z dA



ð16Þ

L

F i;x F j;x dA

þ C 66

C 0 A kwL

ð22Þ

where Kmnij is the nuclear matrix of the global stiffness matrix, and Mmnij represents the mass nuclear matrix. The nucleus matrices are 3  3 arrays that neither depend on the order of the polynomials nor on the improved Fourier series. The nine components of the nuclear stiffness matrix are given by

Z

ku0

ð21Þ

1 6 i 6 C; 2 6 m 6 M

with

0



qim ¼ aim ; bim ; cim

Then



eTn rn þ eTp rp dAdy

In this work, the penalty method is used to handle the general boundary conditions. Three groups of penalty factors are introduced at each boundary of the structures. The penalty factors can be physically viewed as spring stiffness coefficients that are introduced to simulate the mechanical boundary conditions [42,43]. Thus, the potential energy stored in boundary restraints can be expressed as

Vb ¼

@X ¼0 @qim

Z

L

F i;z F j;x dA 0

A

Wm Wn dy ð23Þ

Z

Z ¼ C 12 K mnij yx

A

Z K mnij yy ¼ C 22

L

W0m Wn dy þ C 44

F i F j;x dA 0 L

Z

W0m W0n dy þ C 44

F i F j dA 0

A

Z

þ C 55

Z

L

F i;z F j;z dA

Z

Z

L

Wm W0n dy

F i;x F j dA 0

A

Z

Z

L

Wm Wn dy

F i;x F j;x dA 0

A

Wm Wn dy

 Z F i F j dA þ kv 0 Wm jy¼0 Wn jy¼0 þ kv L Wm jy¼L Wn jy¼L A

0

Z

Z K mnij yz

¼ C 23

L

0 m

W Wn dy þ C 55

F i F j;z dA 0

A

A

Z

Z

L

Wm W0n dy

F i;z F j dA 0

A

ð24Þ

where q is the density of the structure.

Z

Z

2.4. Solution procedure

K mnij ¼ C 13 zx

X ¼ Vs þ Vb  Tp

ð18Þ

According to the CUF, the vectors that contain the unknown coefficients can be expressed as T

qim ¼ faim ; bim ; cim g

ð19Þ

If a higher-order of polynomial N is considered (number of Taylor expansion C = (N + 1) (N + 2)/2), the unknowns will be T

qm ¼ fa1m ; b1m ; c1m ; a2m ; b2m ; c2m ; . . . ; aCm ; bCm ; cCm g

A

Z

The Lagrangian energy function can be written as

ð20Þ

Minimizing the Lagrangian energy function with respect to all the unknown coefficients yield

K mnij ¼ C 23 zy

F i;z F j dA A

0

Z

Wm W0n dy þ C 55 L

A

0

Z

Z

Boundary conditions

y=0

y=L

Clamped-Clamped (C-C) Simply supported (S-S) Free-Free (F-F)

ku0 = kv0 = kw0 = 1012 kv0 = 0, ku0 = kw0 = 1012 ku0 = kv0 = kw0 = 0

kuL = kvL = kwL = 1012 kvL = 0, kuL = kwL = 1012 kuL = kvL = kwL = 0

L

0

A

Z

Z

L

F i;x F j;z dA A

Z

Wm Wn dy þ C 55

F i;x F j;x dA

Z

0 L

F i F j;z dA A

0

Z

Z F i F j dA A

Wm Wn dy

þ kw0 Wm jy¼0 Wn jy¼0 þ kwL Wm jy¼L Wn jy¼L

0

Z

L

Wm Wn dy

W0m Wn dy W0m W0n dy

F i F j dA

A

ð25Þ where Fi,x and Fi,z indicate that Fi is differentiable with respect to x and z, respectively, and W0 n represents the first derivative of Wn.The nine components of the mass nuclear matrix are

Z

Z mnij M mnij ¼ Mmnij ¼q xx yy ¼ M zz

Table 2 Classical boundary conditions with respect to the end-supported structures.

Wm Wn dy þ C 66

F i;z F j;z dA

þ C 66 

0 L

Z

Z ¼ C 33 K mnij zz

L

F i;z F j;x dA

mnij ¼ M mnij ¼ M mnij xy ¼ M xz yx

F i F j dA

A Mmnij yz

¼

0 M mnij zx

L

Wm Wn dy

ð26Þ

¼ Mmnij ¼0 zy

The assembly procedure for the global stiffness matrix is achieved by the using the four indices m, n, i, and j, where the former two indices are related to the Fourier expansions Wm and Wn, and the latter two are related to the Taylor expansions, Fi and Fj. As

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Fig. 2. Assembly procedure for the global stiffness matrix.

Table 3 Convergence of the first five bending frequencies (Hz) for the S–S square solid beams (N = 1). Solutions

h/L = 0.1 n=1

a

h/L = 0.01 n=2

n=3

n=4

n=5

n=1

n=2

n=3

n=4

n=5

M 5 10 15 20 25 30

DOFs 72 117 162 207 252 297

Present method 117.746 452.084 117.746 452.084 117.746 452.084 117.746 452.084 117.746 452.084 117.746 452.084

962.14 958.35 958.35 958.34 958.34 958.34

1595.86 1587.86 1587.86 1587.84 1587.83 1587.83

2554.83 2300.99 2300.99 2300.83 2300.81 2300.80

1.195 1.195 1.195 1.195 1.195 1.195

4.777 4.777 4.777 4.777 4.777 4.777

10.952 10.762 10.744 10.741 10.740 10.740

20.502 19.116 19.084 19.076 19.075 19.074

38.253 30.010 29.799 29.776 29.767 29.764

No. & Elem. 10 B2 20 B2 40 B2 10 B3 20 B3 40 B3

DOFs 99 189 369 189 369 729

CUF-FEM [25] 119.139 473.328 118.058 457.071 117.790 453.138 117.704 451.971 117.701 451.846 –a 451.838

1056.5 981.05 963.39 958.91 957.68 957.61

1866.07 1650.79 1602.06 1592.39 1586.64 1586.26

2905.29 2434.78 2331.25 2317.45 2299.29 2298.06

1.209 1.198 1.195 1.194 – –

5.020 4.835 4.790 4.777 4.778 4.775

12.039 11.041 10.811 10.753 10.738 10.737

23.491 20.043 19.304 19.159 19.073 19.068

41.674 32.180 30.332 30.090 29.775 29.754

Not provided by the reference.

Table 4 Convergence of the first five bending frequencies (Hz) for the S-S thin-walled cylindrical beams (N = 4). M

5 10 15 20 25 30 35 40 Ref [25]

D/L = 0.1

D/L = 0.01

n=1

n=2

n=3

n=4

n=5

n=1

n=2

n=3

n=4

n=5

14.022 14.022 14.022 14.022 14.022 14.022 14.022 14.022 14.019

51.505 51.505 51.505 51.505 51.505 51.505 51.505 51.505 51.470

114.820 103.502 103.496 103.496 103.495 103.495 103.495 103.495 103.369

169.510 162.679 162.661 162.659 162.658 162.658 162.658 162.658 162.382

323.515 224.496 224.408 224.399 224.397 224.397 224.396 224.396 223.927

0.145 0.145 0.145 0.145 0.145 0.145 0.145 0.145 0.145

0.579 0.579 0.579 0.579 0.579 0.579 0.579 0.579 0.579

1.324 1.303 1.300 1.300 1.300 1.300 1.300 1.300 1.300

2.485 2.310 2.307 2.306 2.306 2.306 2.306 2.306 2.306

4.743 3.620 3.595 3.593 3.592 3.592 3.592 3.592 3.592

depicted in Fig. 2, the global stiffness matrix K is obtained from the circulation of the sub-matrices Kmn by varying indices m and n, while the sub-matrix Kmn is obtained with the circulation of nuclear matrices Kmnij by varying the indices i and j. The global

mass matrix M can be obtained in a similar manner. It is of significance to suggest that the global matrices of each higherorder configuration can be obtained by circulating the same nuclear matrix.

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(a)

(b)

Fig. 3. Bending mode shapes of the S-S square solid beam (h/L = 0.1, N = 1): (a) 117.746 Hz, and (b) 452.084 Hz.

(a)

(b)

Fig. 4. Bending mode shapes of the S-S thin-walled cylindrical beam (D/L = 0.1, N = 4): (a) 14.022 Hz, and (b) 51.505 Hz.

Table 5 The first three bending, torsional, shell-like, and the two longitudinal frequencies (Hz) of the square box beam (h2/h1 = 0.8, C-C, L = 20 m) predicted with different N-order polynomials. Mode types

ANSYS N=2

N=3

N=4

N=5

N=6

N=7

N=8

DOFs Bending

387 37.306 90.966 157.827

774 32.129 80.251 141.801

1290 30.369 73.112 125.928

1935 30.267 72.768 125.074

2709 29.995 71.748 122.869

3612 29.981 71.643 122.417

4644 29.841 71.067 121.018

5805 29.826 70.923 120.349

36,663 29.523 69.448 115.79

Torsional

80.788 161.576 242.365 133.255 265.807

80.788 161.576 242.365 132.885 264.632

80.788 161.576 242.365 132.491 263.799

73.91 147.871 221.931 132.409 263.629

73.824 147.628 221.392 132.359 263.509

72.97 145.912 218.797 132.353 263.498

72.935 145.759 218.358 132.334 263.413

72.816 145.489 217.865 132.330 263.406

71.696 143.04 213.63 132.22 262.60

–a – –

– – –

– – –

– – –

189.418 207.429 234.313

184.993 190.907 200.646

156.814 164.214 176.136

156.329 162.335 172.112

152.36 156.28 162.85

Longitudinal Shell-like

a

Polynomials N=1

Not provided by the theory.

A total number of 3  (M + 3)  (N + 1)  (N + 2)/2 linear equations related to the corresponding coefficients can be achieved and arranged in a matrix form as



K  x2 M q ¼ 0

ð27Þ

where q indicates the vector containing all the unknown Fourier coefficients, and K and M represent the global stiffness matrix and mass matrix, respectively. The vibration characteristics, including the natural frequencies and mode shapes, of the structures are calculated from the solution

of the characteristic equations. After the natural frequencies of the structure are determined, the mode shapes can be obtained by substituting the corresponding coefficients q into the displacement functions.

3. Numerical results and discussion Various examples are performed to illustrate the capacity and efficiency of the present solution. These examples include the free vibration analysis of square box beams and hollow cylinder beams

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Fig. 5. Relative errors in the present method and ANSYS of the square box beam (h2/h1 = 0.8, C-C).

with different hollow ratios and boundary conditions. The hollow ratios of the square box beam and hollow cylindrical beam are defined as h2/h1 and R2/R1, respectively. It should be noted that all the examples can be obtained by varying the inner hole of the same beams, as shown in Fig. 1. Thus, the variation from one case to another is simply owing to the adjustment of the predefined geometrical dimensions without modifications of the main procedure. In summary, the analysis is executed as follows: first, the convergence and validity of the present method are estimated with two cases involving the free vibration analysis of simply supported beams. The dynamic characteristics of the clamped hollow structures are then predicted for different orders of polynomials, based on which the principle for the selection of the cross-sectional polynomials is proposed. Finally, the hollow structures subjected to classical and elastic boundary restraints are investigated, and the effects of the boundary conditions on vibration characteristics are studied accordingly. In the following discussion, the material for the structures is assumed to be aluminum having the following properties: Young’s modulus E = 75 GPa, mass density q = 2700 kg/m3, and Poisson’s ratio t = 0.33. 3.1. Convergence studies and verification It is known that an excellent result can be obtained by choosing sufficient terms in the improved Fourier series solution. However, only a few terms could be considered in the actual calculations

Fig. 6. Relative errors in the present method and ANSYS for the hollow cylindrical beam (h2/h1 = 0.8, C-C).

owing to computational efficiency limitations. Two cases regarding the vibration analysis of simply supported beams with different cross-sections are presented to examine the convergence of the present solution method and determine the truncation number of the improved Fourier series. First, the square solid beam with h1 = 0.2 m, subjected to a simple constraint (SS), is investigated using a first-order polynomial. The first five bending frequencies of the square solid beam with different height-to-length ratios (h1/L1 = 0.1 and 0.01) are listed in Table 3. The results of the short beam (h1/L1 = 0.1) are shown in the left of the table, whereas the results of the slender beam (h1/L1 = 0.01) are listed in the right side of the table. Both the short and slender beams are predicted with different truncation schemes. It is significant that the present method has a satisfactory convergence, and it converges well even when only a few terms of the Fourier series are employed. The difference in the fifth bending frequency between the truncation configurations M = 10 and M = 30 is less than 0.01%. Furthermore, the present results are compared with CUF–FEM solutions [25] using the same order of polynomial, N = 1. Linear (B2) and quadratic (B3) finite elements along the beam axis were used in the CUF–FEM solutions. The numbers of degrees-of-freedom (DOFs) for the CUF–FEM and present method are also listed in Table 3. In comparison to the CUF–FEM, the present method converges rapidly to achieve the same level of accuracy owing to the high-order smoothness of the Fourier series.

Table 6 The first three bending, torsional, shell-like, and the two longitudinal frequencies (Hz) of the hollow cylindrical beam (R2/R1 = 0.8, C-C, L = 20 m) for different N-order polynomials. Mode types

a

Polynomials

ANSYS

N=1

N=2

N=3

N=4

N=5

N=6

N=7

N=8

DOFs Bending

387 33.228 82.872 146.313

774 28.327 72.182 129.751

1290 27.081 66.833 117.079

1935 27.05 66.723 116.793

2709 26.963 66.357 115.944

3612 26.963 66.356 115.941

4644 26.959 66.34 115.903

5805 26.959 66.34 115.903

36,600 26.847 66.018 115.34

Torsional

80.788 161.576 242.365

80.788 161.576 242.365

80.788 161.576 242.365

80.788 161.576 242.365

80.788 161.576 242.365

80.788 161.576 242.365

80.788 161.576 242.365

80.788 161.576 242.365

80.796 161.59 242.39

Longitudinal

133.067 265.605

132.721 265.070

132.535 265.066

132.533 265.036

132.518 265.034

132.517 265.032

132.516 265.033

132.516 265.033

132.03 264.05

Shell-like

–a – –

– – –

233.558 – 272.504

232.293 – 271.628

200.757 205.401 214.814

200.376 203.86 211.312

175.862 179.785 188.148

175.823 179.630 187.797

165.87 169.73 178.06

Not provided by the theory.

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(a) First bending mode, f = 29.826 Hz

(c) First longitudinal mode, f = 132.330 Hz

(b) First torsional mode, f = 72.816 Hz

(d) First shell-like mode, f = 156.329 Hz

Fig. 7. Mode shapes of the square box beam (h2/h1 = 0.8, C-C, L = 20 m) obtained with the eighth-order polynomials.

The next case is the estimation of the vibration characteristics of a thin-walled cylindrical beam with an SS boundary. The outer radius R1 is equal to 1 m and the hollow ratio R2/R1 is 0.98. The diameter-to-length ratio D/L is 0.1 or 0.01 (diameter D = 2R1). Fourth-order polynomials are employed in this case. The convergence and comparison of the first five bending frequencies are presented in Table 4. The natural frequencies obtained with the different types of truncation schemes are also compared with the CUFFEM solutions [25] in which the fourth-order Taylor expansions and 40 elements along the beam axis were used. As shown in Table 4, the present method has a good convergence, and the present results match well with the reference data. It is concluded that 40 terms of the Fourier series expression are sufficient to capture the axial displacement, and these terms will be employed in the following discussion. Moreover, the first two bending mode shapes of the square solid beam and thin-walled cylindrical beam are reported in Figs. 3 and 4 for illustrative purposes. 3.2. Higher-order configurations for hollow structures To better understand the ability of the present method to accurately predict the vibration characteristics of the uniform slender structures, a square box beam and a hollow cylindrical beam are considered in this subsection. The dynamic characteristics of the beams are investigated using higher-order configurations. The latter can be achieved by utilizing different N-orders of the polynomials over the beam sections. Moreover, the beams

are set to be clamped at both ends and their lengths are equal to 20 m. For the square box beam, the edge length of the outer square h1 is 2 m and the hollow ratio h2/h1 is 0.8. The results are compared with the 3-D solutions calculated by commercial FEM software in which the mesh size is chosen according to preliminary convergence analyses. Table 5 presents the first three bending, torsional, shell-like, and two longitudinal frequencies for the square box beam. The ‘‘shell-like” mode is presented by Carrera et al. [44] to describe the mode that mainly performs the cross-sectional deformation. The relative errors between the present results and those of the commercial FEM software are depicted in Fig. 5. The relative errors are defined as



f N  f FEM



di ¼ i FEMi  100%

f

ð28Þ

i

FEM where fN respectively represent the natural frequencies i and fi obtained by the present method and commercial FEM software. As it can be observed in Fig. 5, the differences in the longitudinal, bending, and torsional frequencies, obtained by the present method and ANSYS decrease rapidly by increasing the order of adopted polynomial N. Interestingly, there is a large error decrease from 12.5% to 3.0% between the third- and fourth-order polynomials for calculating the torsional modes. This is probably owing to the non-uniform warping deformation, which cannot be fully detected with lower-order polynomials. Furthermore, the

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Y. Chen et al. / Computers and Structures xxx (2018) xxx–xxx

(a) First bending mode, f = 26.959 Hz

(c) First longitudinal mode, f = 132.516 Hz

9

(b) First torsional mode, f = 80.788 Hz

(d) First shell-like mode, f = 175.823 Hz

Fig. 8. Mode shapes of the hollow cylindrical beam (R2/R1 = 0.8, C-C, L = 20 m) obtained with the eighth-order polynomials.

shell-like mode is first detected with the fifth-order polynomials but it results in a 25% relative error compared to the ANSYS code. Satisfyingly, the differences of the shell-like modes decrease dramatically with the increase in the order of the polynomial, N. The DOFs of the present method and ANSYS are also reported in Table 5. We note that the present method is substantially faster owing to the considerably fewer DOFs (5805) used in comparison with 36,663 in the case of the commercial FEM software. For the hollow cylindrical beam, the outer radius R1 is 1 m and the hollow ratio R2/R1 is equal to 0.8. The present results are compared with those obtained using the ANSYS software based on SOLID185 element discretization. Moreover, the first three bending, torsional, shell-like, and the two longitudinal frequencies calculated with different order of polynomials are listed in Table 6, and the relative errors between the present modes and those obtained with ANSYS are shown in Fig. 6. The present method matches well the commercial code when higher-order polynomials, such as the eighth- or ninth-order, are employed. As before, the computational efficiency is improved rapidly because the DOFs of the model can be reduced from 36,600 to 5805 using the eighthorder polynomials. In comparison to the square box beam, the same phenomenon can be observed such that the bending and longitudinal modes only vary in the first three polynomials, and the shell-like modes cannot be detected with lower-order polynomials. Alternatively, the torsional frequencies can be predicted accurately using the lower-order polynomials. Finally, some distinguishable mode shapes are shown in Figs. 7 and 8. Next, the vibration characteristics of the hollow structures with different hollow ratios are estimated. The comparisons of the

frequencies of the first bending, torsional, and longitudinal modes between the present results and ANSYS data are made in terms of the percentage difference as a function of the hollow ratio (h2/h1 or R2/R1). For clarity, only the values predicted with a polynomial that has an odd order number, are shown in Fig. 9. The first bending modes of the hollow structures are depicted in Fig. 9(a) and (b). As shown in Fig. 9(a), the percentage differences between the results are nearly proportional to the hollow ratio. Fig. 9(b) shows that the bending mode of the hollow cylindrical beam can be predicted by the higher-order polynomials (N > 3), within 1% accuracy. The first torsional modes of the hollow structures are shown in Fig. 9 (c) and (d). In Fig. 9(c), the percentage differences do not vary markedly when the hollow ratio is less than 0.5. As the hollow ratio increases from 0.5 to 0.9, the percentage differences increase gradually. In Fig. 9(d), the torsional mode of the hollow cylindrical beam can be calculated exactly with any N-order polynomials. Fig. 9(e) and (f) display the first longitudinal modes of the hollow structures, where the percentage differences increase gradually with the hollow ratio. In general, with the increase in the hollow ratio, the percentage differences of the natural frequencies between the present results and those obtained by ANSYS exhibit an increasing tendency. For this issue, higher-order polynomials can be employed to improve the computational accuracy of the present method that deals with a hollow structure with a large hollow ratio. 3.3. Hollow structures with general boundary conditions The free vibration analysis of the hollow structures, which are subjected to classical and elastic boundary restraints, are discussed

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Y. Chen et al. / Computers and Structures xxx (2018) xxx–xxx

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 9. Frequency differences of hollow structures (C-C) with different hollow ratios: (a, b) bending modes, (c, d) torsional modes, and (e, f) longitudinal modes.

Table 7 Natural frequencies (Hz) of the square box beam with classical boundary conditions (N = 8, h1 = 2 m, h2/h1 = 0.8, and L1 = 20 m). Modes

1 2 1 2 1 2

Bending Bending Shell-like Shell-like Torsional Torsional

F-F

C-F

C-C

S-S

Present

ANSYS

Error

Present

ANSYS

Error

Present

ANSYS

Error

Present

ANSYS

Error

33.555 79.309 92.560 94.079 71.698 142.426

32.226 77.873 93.092 96.339 71.715 143.039

4.12% 1.84% 0.57% 2.35% 0.02% 0.43%

31.242 74.615 92.233 94.418 35.932 107.34

30.117 73.465 93.223 95.874 35.913 107.55

3.74% 1.57% 1.06% 1.52% 0.05% 0.20%

30.520 70.269 94.838 106.416 71.807 142.656

29.489 69.349 95.497 107.17 71.844 143.31

3.50% 1.33% 0.69% 0.70% 0.05% 0.46%

15.362 54.279 93.190 99.167 71.698 142.428

14.683 52.656 93.854 100.349 71.715 143.043

4.62% 3.08% 0.71% 1.18% 0.02% 0.43%

in this subsection. Different boundary conditions are achieved by setting the stiffness coefficients of the springs at appropriate values. For the sake of brevity, three spring parameters, nx, ny, and

nz, which are defined as the ratios of the corresponding spring stiffness coefficient to the Young’s modulus, are considered here (i.e., nx = log10(kxl/E), ny = log10(kyl/E), and nz = log10(kzl/E)). The

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Y. Chen et al. / Computers and Structures xxx (2018) xxx–xxx Table 8 Natural frequencies (Hz) of the hollow cylindrical beam with classical boundary conditions (N = 5, R1 = 1 m, R2/R1 = 0.98, and L2 = 20 m). Modes

F-F Present

DSM [45]

Error

Present

DSM [45]

Error

Present

DSM [45]

Error

Present

DSM [45]

Error

1 2 1 2 1 2

30.932 77.041 17.713 17.801 80.788 161.576

30.932 77.041 17.709 17.777 80.788 161.576

0.00% 0.00% 0.02% 0.14% 0.00% 0.00%

5.089 29.159 17.819 20.604 40.394 121.182

5.065 29.088 17.805 20.580 40.394 121.181

0.48% 0.24% 0.08% 0.12% 0.00% 0.00%

28.714 69.382 20.505 32.333 80.788 161.576

28.576 69.110 20.484 32.222 80.786 161.573

0.48% 0.39% 0.10% 0.34% 0.00% 0.00%

14.021 51.504 18.399 25.456 80.788 161.576

14.022 51.503 18.405 25.460 80.786 161.573

0.00% 0.00% 0.03% 0.02% 0.00% 0.00%

Bending Bending Shell-like Shell-like Torsional Torsional

C-F

(a)

C-C

S-S

(b)

(c)

(d)

(e)

(f)

Fig. 10. Distinguished modes of the hollow structures with varying elastic boundary restraints (N = 5 for hollow cylindrical beam and N = 8 for square box beam): (a, b) first shell-like mode, (c, d) first bending mode, and (e, f) first torsional mode.

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Y. Chen et al. / Computers and Structures xxx (2018) xxx–xxx

clamped at one end (y = 0), and elastically restrained at the other end (y = L). Four elastic boundary conditions are considered in the calculation. They are the axial elastic condition (nx = nz = 5, ny = T), single radial elastic condition (ny = nz = 5, nx = T), two radial elastic condition (ny = 5, nx = nz = T), and axialradial elastic condition (nx = 5, ny = nz = T), where T is an intermediate variable that varies from 5 to 5. The variations in the natural frequencies of the beams with respect to the spring parameter nx, ny, and nz are depicted in Fig. 10. It can be observed that the natural frequencies of the hollow structures increase with the reinforcement of the restraints. When the spring parameters are less than 5, the natural frequencies remain constant. When the spring parameters increase between 5 and 3, the natural frequencies increase rapidly except in the case of several distinct situations. Once the spring parameters are larger than three, the natural frequencies become constant again. Regarding the shell-like modes shown in Fig. 10(a) and (b), the frequency values remain constant as a function of the single radial spring parameter nx, but increase with the growth of the two radial spring parameters nx and nz. For the bending modes shown in Fig. 10(c) and (d), the frequency values increase with the increase of the spring parameters within a certain range in all the four cases. From Fig. 10(e) and (f), it can be observed that the torsional mode increases with the radial spring parameters nx and nz. Finally, some mode shapes of the hollow cylindrical beam with varying elastic boundary restraints are shown in Figs. 11–13.

dimensions of the square box beam are: h1 = 2 m, h2/h1 = 0.8, and L1 = 20 m. Similarly, the dimensions of the hollow cylindrical beam are: R1 = 1 m, R2/R1 = 0.98, and L2 = 20 m. The eighth-order polynomials are used for the square box beam, whereas the fifth-order polynomials are used for the hollow cylindrical beam. First, the appropriateness of defining the end-conditions in terms of the spring parameters (nx, ny, and nz) is proved by the evaluation of the beams with different classical boundary conditions, namely, free-free (FF), clamped-free (CF), clamped-clamped (CC), and simply supported (SS). Tables 7 and 8 list the natural frequencies of the first two bending, shell-like, and torsional frequencies of the hollow structures. In Table 7, the results of the square box beam are compared with those obtained by the ANSYS software using the SOLID185 element configuration. The comparisons for the hollow cylindrical beam with different boundary conditions are listed in Table 8. The results are compared with those solutions obtained by the dynamic stiffness method (DSM) provided by Pagani et al. [45]. Moreover, the relative errors between the present modes and CUF–DSM solutions are reported in Table 8, and the relative DSM errors are computed according to di = |(fN )/fDSM |  100. As i  fi i illustrated in Tables 7 and 8, the present results are in excellent agreement with the reference data or commercial code. Thus, it is noted that defining the boundary conditions in terms of the spring parameters is advisable and efficacious. Subsequently, the free vibration of the hollow structures with elastic restraints is investigated. The hollow structures are

(a)

(b)

(c)

Fig. 11. First bending mode shape variation with the increase in the spring parameter (N = 5), ny = nz = 5, nx = T: (a) T = –5, (b) T = –2, and (c) T = 5.

(a)

(b)

(c)

Fig. 12. First shell-like mode shape variation with the increase in the spring parameter (N = 5), nx = nz = 5, ny = T: (a) T = 5, (b) T = 0, and (c) T = 5.

(a)

(b)

(c)

Fig. 13. First torsional mode shape variation with the increase in the spring parameter (N = 5), ny = 5, nx = nz = T: (a) T = 5, (b) T = 1, and (c) T = 5.

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4. Conclusions This study introduced a unified quasi-3D solution for the free vibration analysis of both solid and hollow slender structures with general boundary conditions. The boundary conditions were implemented using the penalty method, in which case the selection of admissible functions was no longer dependent on the geometrical boundary restraints. The displacement components of the structures were presented by the combination of the Taylor expressions and improved Fourier series. The nuclear matrices of the stiffness and mass matrices were derived using the energy variational principle. An assembly procedure based on the Carrera unified formulation (CUF) was then proposed for the global matrices of the higher-order configurations. In this study, the displacement functions were not dependent on the specific shape of the section, and the proposed method was thus suitable for any uniform slender structure. Furthermore, the proposed method offered a unified analysis operation under which the boundary constraints were considered as predefined parameters. This simplified the boundary condition analysis in such a way that was still efficient in the application of hollow slender structures in engineering. Different types of typical solid and hollow slender structures were studied. The efficiency and accuracy of the present method were illustrated by comparing the present results with the reference data and commercial code. It was found that the proposed method needed much fewer DOFs to achieve the same level of accuracy as the commercial code. The calculation error in the specific configuration increased as the hollow ratio increased. Thus, a higher-order configuration should be employed to ensure the computational accuracy when dealing with a large hollow ratio structure. When the elastic boundary condition was considered, the natural frequencies increased rapidly as the stiffness parameters increased within a certain range, but all of them remained nearly constant at the two ends of the range.

Conflict of interest statement The authors declare that there is no conflict of interest regarding the publication of this paper. Acknowledgement The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Nos. 51822902, 51709066, and 51775125) and the Fundamental Research Funds for the Central Universities of China (No. HEUCFG201713). References [1] Timoshenko PSPX. On the transverse vibrations of bars of uniform crosssection. Philos Mag 1922;253:125–31. [2] Timoshenko SP, Goodier JN. Theory of elasticity. 3rd ed. New York: McGrawHill; 1970. [3] Sokolnikoff IS. Mathematical theory of elasticity. New York: McGraw-Hill; 1956. [4] Kaneko T. On Timoshenko’s correction for shear in vibrating beams. JPhD 1975;8:1927–36. [5] Han SM, Benaroya H, Wei T. Dynamics of transversely vibrating beams using four engineering theories. J Sound Vib 1999;225:935–88. [6] Cowper GR. The shear coefficient in Timoshenko’s beam theory. J Sound Vib 1966;87:621–35. [7] Murty AVK. On the shear deformation theory for dynamic analysis of beams. J Sound Vib 1985;101:1–12. [8] Pai PF, Schulz MJ. Shear correction factors and an energy-consistent beam theory. Int J Solids Struct 1999;36:1523–40. [9] Mech I, Tounsi A, Benattaa MA, Bediaa EAA. Deformation of short composite beam using refined theories. J Math Anal Appl 2008;346:468–79. [10] Gruttmann F, Wagner W. Shear correction factors in Timoshenko’s beam theory for arbitrary shaped cross-sections. Comput Mech 2001;27:199–207.

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