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An accurate curved beam element based on trigonometrical mixed polynomial function Article  in  International Journal of Structural Stability and Dynamics · May 2013 DOI: 10.1142/S0219455412500848

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International Journal of Structural Stability and Dynamics Vol. 13, No. 4 (2013) 1250084 (19 pages) # .c World Scienti¯c Publishing Company DOI: 10.1142/S0219455412500848

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AN ACCURATE CURVED BEAM ELEMENT BASED ON TRIGONOMETRICAL MIXED POLYNOMIAL FUNCTION

Y. Q. TANG, Z. H. ZHOU and S. L. CHAN* Department of Civil and Structural Engineering Hong Kong Polytechnic University, Hong Kong, P. R. China *[email protected] Received 11 May 2012 Accepted 24 August 2012 Published 2 April 2013 A displacement-based, novel curved beam element is proposed for e±cient and reliable analysis of frames composed of curved members. The accuracy of the proposed element is not controlled by the subtended angle of the element with the angle up to
. In contrast to the conventional method, the interpolation function for displacement is based on the in¯nitesimal straight beam sections extracted from the curved element. Consequently, the strain energy of the curved beam element can be integrated by the in¯nitesimal sections along the element length. The relationship between the displacements and the corresponding strains in the straight beam is simpler than that in the curvilinear co-ordinate description widely adopted by many researchers in their element derivations. This technique is formulated to avoid couplings between the tangential and radial displacement variables in the strain ¯eld and its successful utilization is also demonstrated herein. Furthermore, the relation between displacements and strains of the in¯nitesimal straight beam section is equivalent to that of the curved beam in the curvilinear co-ordinate description. Finally, the analysis results of several bench marked examples by the proposed curved beam element are presented. The results show the high accuracy and e±ciency of the proposed element against the classical curved beam element. Keywords: Curved beam element; ¯nite element method; displacement-based; in¯nitesimal straight beam section.

1. Introduction During the past few decades, much research work has been conducted on the utilization of ¯nite element method (FEM) for solving curved beam problems. The coupling e®ect between the bending and the stretching complicates the analysis of curved beams. Although many curved beam elements have been formulated on the basis of energy methods such as the Castigliano's theorem,1 a versatile approach

* Corresponding

author.

1250084-1

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Y. Q. Tang, Z. H. Zhou & S. L. Chan

using the ¯nite element (FE) formulation appears to be unavailable and should be further investigated in order to make the analysis reliable and accurate. The earliest attempt of FE method developed from a thin curved beam element is based on the Kirchho® theory with C0 continuous tangential displacement and C1 continuous normal displacement (the cubic-linear or CL element), and shear °exible curved element based on the Mindlin theory with independent C0 continuous displacements and rotation. However, these curved beam elements and some low-order elements were de¯cient when applied to practical problems because the models need many elements to obtain accurate solutions. This defect is due to an element locking phenomenon which is an over-sti® behavior of a curved beam that resulted from large axial to bending sti®ness. In order to remedy the locking phenomenon, higher-order interpolations or mixed polynomial
trigonometric displacement ¯elds have been proposed to formulate the curved beam element. The analytical models need fewer elements to obtain an accurate solution, but they require more nodes per element or more degrees of freedom per node.2
7 Some researchers use the method of reduced or selective integration of shear and membrane energy terms to alleviate the sti® locking results from spurious constraints terms contained in the strain energy.8
11 To this end, Prathap8 conducted an extensive study on the CL curved beam element and he attributed the failure of the CL element in providing accurate results for very thin and deep arch problems to membrane locking. He also showed how the element could be re-formulated to furnish acceptable results with using one-point Gaussian integration for the membrane energy, which is named as the CL1 element. The ¯eld consistency approach proposed by Prathap and co-workers identi¯ed the spurious constraints of the inconsistent strain ¯eld with these constraints eliminated in the ¯nal formulation.12,13 This approach ensures a variationally correct and orthogonally consistent strain ¯eld. Besides these traditionally common displacement-based ¯nite element methods reported in previous papers, other types of ¯nite elements have been proposed. The hybrid/mixed elements with equilibrium equations and constitutive relations are imposed to the variation of the related energy function.14
18 The coupled displacement ¯eld element with a cubic polynomial ¯eld for radial displacement is ¯rst assumed. Then the displacement ¯eld for tangential displacement and section rotation are derived using equilibrium equations and constitutive relations. The displacement ¯eld then properly couples polynomial coe±cients which produce consistently vanishing membrane and shear strain components in their respective constrained physical limits.19 The curvature-based beam element represents the bending energy directly and obtains the displacement and other energy components with the equilibrium equations and constitutive relations free from locking and this approach is reported to be e±cient.19
22 All these elements have one common feature, i.e., they need the aid from equilibrium equations to obtain an accurate solution. However, these elements require more complex derivations with complicated expressions for the sti®ness matrices. 1250084-2

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An Accurate Curved Beam Element Based on Trigonometrical Mixed Polynomial Function

In this paper, a new two-node curved beam element derived by the displacementbased method is proposed. Unlike other previous studies, the relationship between strains and displacements in the curvilinear co-ordinate system is not used here. Instead, the in¯nitesimal straight beam section of the curved beam element is studied and the relationship between displacements and strains in the straight beam is established. Actually, the relationship here is equivalent to the one in the curvilinear co-ordinate description, but the corresponding expression is simpler and the coupling of the tangential displacement and the radial displacement is avoided in the energy expression. Another di®erence is that the curved beam element model without rigidbody movement is used in the derivation. To simplify the formulation, the membrane and °exible deformations are considered separately, making use of the property that the membrane strain is much smaller than the bending strain. The dominant bending sti®ness is derived through the displacement-based method ¯rst and the e®ect of the membrane sti®ness derived from the energy method is subsequently added to modify the sti®ness matrix. The equilibrium equations are not needed here. Finally, ¯ve examples are solved and the results are compared with other methods. The results show that the proposed new element has excellent performance when using only one element per member in analysis. 2. Assumptions The present formulation is based on the following assumptions: (a) (b) (c) (d)

The curved element is thin, prismatic and elastic. The applied loads are conservative. Small strain and small displacement are assumed Shear and warping deformations are neglected.

Figure 1 describes the pinned curved beam element of radius R and subtended angle  based on classical thin beam theory. The tangential displacement, radial displacement and the cross-section rotation are de¯ned as u; v and , respectively.

Fig. 1. Curved beam element modal.

1250084-3

Y. Q. Tang, Z. H. Zhou & S. L. Chan

The curved beam element model without rigid-body movement has two nodes and three degrees of freedom. P is the horizontal pull along axial x-direction, the corresponding displacement is e, and Mi is the bending moment at node i and the corresponding rotation is i (i ¼ 1, 2).

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3. Formulation of Displacement Function The in¯nitesimal straight beam section of the curved beam element is shown in Fig. 2. The whole curved beam can be regarded as a structure made of in¯nitesimal straight beam sections which obey the law of the Euler
Bernoulli beam. The in¯nitesimal straight section ds could generate the displacements du 0 and dv 0 due to membrane force and bending moment respectively. Special attention should be paid to the displacement components of u 0 and v 0 , which are the integrals of du 0 and dv 0 respectively and they are di®erent from the u and v in the curvilinear co-ordinate system which contains the rigid-body movement relative to the in¯nitesimal straight beam section. The membrane strain " 0 and the bending strain  0 of the in¯nitesimal straight beam section are described by the strain
displacement relations as du 0 ; ds dv 0 0 ¼ ; ds

"0 ¼

0 ¼

d 2v 0 : ds 2

ð1aÞ ð1bÞ ð1cÞ

Since the axial deformation is much smaller than the bending deformation, they can be considered separately. Assuming ¯rst that there is no axial deformation in the curved beam element, the displacement interpolation for v 0 is given by v 0 ¼ a0 þ a1  þ a2  2 þ a3 sinðÞ þ a4 cosðÞ;

Fig. 2. In¯nitesimal section of the curved beam element.

1250084-4

ð2Þ

An Accurate Curved Beam Element Based on Trigonometrical Mixed Polynomial Function

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in which  ¼ ’= and a0 to a4 are the coe±cients obtained from the boundary conditions and we have ds ¼ Rd. The incremental displacement of the in¯nitesimal straight beam section in the Cartesian co-ordinate system can be written as dux and duy which are along respectively the x-axis and y-axis. The transformation relationship between dux , duy and du 0 and dv 0 is shown in the following equations:
  
 dux cosðÞ sinðÞ du 0 ¼ ð3Þ duy
sinðÞ cosðÞ dv 0 The displacements ux and uy at an arbitrary point of the curved beam element are the integrals of dux and duy respectively. We have du 0 ¼ 0 from the assumption that there is no membrane deformation in the curved beam element. Z  : ux ¼ v 0 sinðÞd; ð4aÞ
1=2

Z uy ¼


1=2

:

v 0 cosðÞd;

ð4bÞ

where the dot represents a di®erentiation with respective to the natural co-ordinate. The displacements u and v in the curvilinear co-ordinate system at an arbitrary point of the curved beam element have the following transform relationship with ux and uy :
  
 cosðÞ
sinðÞ ux u ¼ : ð5Þ uy v sinðÞ cosðÞ In the curvilinear co-ordinate system, the strain
displacement relationships are given by

¼

"¼

du v þ ; Rd R

ð6aÞ

¼

dv u
; Rd R

ð6bÞ

d 2v R 2  2 d 2




du R 2 d

:

ð6cÞ

It can be proven that " ¼ " 0 ;  ¼  0 , and  ¼  0 from the foregoing equations. Therefore the method of using the in¯nitesimal straight beam sections to simulate the curved beam element is possible and the boundary conditions derived from the in¯nitesimal straight beam section can be used. ";  and  are used to replace " 0 ;  0 and  0 as follows. The boundary condition can be obtained from Fig. 1. 1 At  ¼
; 2 1250084-5

v 0 ¼ 0;

ð7aÞ

Y. Q. Tang, Z. H. Zhou & S. L. Chan

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1 dv 0 At  ¼
; ¼ 1 ; 2 Rd 1 dv 0 ¼ 2 ; At  ¼ ; 2 Rd 1 At  ¼ ; ux ¼ e; 2 1 At  ¼ ; uy ¼ 0: 2

ð7bÞ ð7cÞ ð7dÞ ð7eÞ

The coe±cients a0 to a4 of the function v 0 can be eliminated from Eqs. (7a)
(7e). And the displacement interpolative function can be written as v 0 ¼ fN1 N2 N3 g  fe R1 R2 g T ; where N1 N2 and N3 are the shape functions given by       4 cos N1 ¼
4 cosðÞ þ ð1
4 2 Þ sin ; 2C 2 2 1 ðA þ A1  þ A2  2 þ A3 sinðÞ þ A4 cosðÞÞ; N2 ¼ 16ð
sinðÞÞC 0 N3 ¼

ð8Þ

ð9aÞ ð9bÞ

1 ðB þ B1  þ B2  2 þ B3 sinðÞ þ B4 cosðÞÞ 16ð
sinðÞÞC 0

ð9cÞ

and C ¼  2 þ  sinðÞ þ 4 cosðÞ
4; A0 ¼ 80
45 2 þ 6 4 þ 16ð
8 þ  2 Þ cosðÞ þ 3ð16
 2 Þ cosð2Þ þ 4ð4 þ  2 Þ sinðÞ þ 24 sinð2Þ; A1 ¼
28 2 þ 8 4 þ 32 2 cosðÞ
4 2 cosð2Þ þ 16að
2 þ  2 Þ sinðÞ þ 16 sinð2Þ; A2 ¼
4 2
8 4 þ 4 2 cosð2Þ þ 16 3 sinðÞ;  A3 ¼
32 sin ð
4 þ  2 þ 4 cosðÞ þ  sinðÞÞ; 2         3 3 A4 ¼
16
2ð1 þ  2 Þ cos þ 2 cos þ  5 sin þ sin ; 2 2 2 2 B0 ¼ 112
15 2 þ 2 4 þ 16ð
8 þ 3 2 Þ cosðÞ þ ð16
 2 Þ cosð2Þ þ 4að
16 þ 3 2 Þ sinðÞ þ 8 sinð2Þ; B1 ¼ A1 ; B2 ¼
A2 ; B3 ¼ A3 ; B4 ¼
A4 : When the subtended angle  ¼
2 , the shape functions are shown in Fig. 3. 4. Sti®ness Matrix The total potential energy function of the curved beam element (which is made up of in¯nitesimal straight beam sections and does not have the membrane deformation) 1250084-6

An Accurate Curved Beam Element Based on Trigonometrical Mixed Polynomial Function

N1 • 0.5

0.5

ξ

• 0.5 • 1.

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(a)

N2 N3

0.1 0.05

• 0.5

0.5

ξ

• 0.05 • 0.1

(b) Fig. 3. Shape functions.

can be expressed as 1
¼ EI 2

Z


1=2

 2 d
W ;

ð10Þ

1=2

in which EI is the °exural rigidity and W the external work. Apart from the conventional displacement
strain relations in a curvilinear coordinate description, the bending strain can be derived from the Eq. (1c) which only contains the assumed radial displacement interpolation function v 0 . So the coupling of the displacements is avoided in the total potential energy function. In the curvilinear co-ordinate system, both the displacement interpolation functions are needed even with the inextensibility condition. The external work is given by Z
1=2 T W ¼ fP M1 M2 g  fe 1 2 g þ fpu pv m g  fu v g T Rd; 1=2

ð11Þ where pu ; pv and m are the distributed tangential, radial and moment loads along the circular arc respectively. Note that, the displacements u and v are obtained from the Eq. (5). 1250084-7

Y. Q. Tang, Z. H. Zhou & S. L. Chan

By making use of Eqs. (1c), (8) to (11) and the principle of minimum potential energy 
¼ 0, the relation between forces and displacements can be obtained as 8 9 8 e9 < e =

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Pe ¼

dW ; de

The sti®ness matrix ½ke  is given by

M 1e ¼

dW ; d1 2

M 2e ¼

dW : d2

ð13Þ

3 k12 k13 EI 4 ½ke  ¼ k22 k23 5; 2R 3 ð
sinðÞÞC sym k33 k11

ð14Þ

where k11 ¼ 4ð
sinðÞÞ;      k12 ¼
4R  cos
2 sin ð
sinðÞÞ; 2 2 k13 ¼
k12 ; k22 ¼
R 2 ð9
6 2
16 cosðÞ þ 7 cosð2Þ þ 8 sinðÞ þ 2 sinð2ÞÞ; k23 ¼
R 2 ð15 þ 2 2 þ 4ð
4 þ  2 Þ cosðÞ þ cosð2Þ
12 sinðÞÞ; k33 ¼ k22 : The sti®ness matrix [ke ] could be proven to be the inverse of the °exibility matrix derived from the °exibility method, where the end de°ection components of the element can be expressed as [fe ] fP M1 M2 g T and [fe ] is the °exibility matrix in the Eq. (15).23 Hence the proposed sti®ness matrix is the exact expression of the curved beam element with the assumption of negligible membrane strain. ½fe  ¼

R 8EIsin 2 ðÞ 3 2 2 2 4R sin ðÞð2 þ 4cos 2 ðÞ 8Rsin 2 ðÞð cosðÞ
8Rsin 2 ðÞð cosðÞ 7 6
3 sinð2ÞÞ
sinðÞÞ
sinðÞÞ 7 6 7 6 7 6 2 þ 4sin 2 ðÞ 2
4sin 2 ðÞ 7 6 7; 6 6 7
sinð2Þ
sinð2Þ 7 6 7 6 7 6 2 þ 4sin 2 ðÞ 5 4
sinð2Þ

sym

ð15Þ where  ¼ 2 . 1250084-8

An Accurate Curved Beam Element Based on Trigonometrical Mixed Polynomial Function

For simplicity, only the displacement produced by bending is considered in the above formulae, while the e®ects on axial deformation of membrane strain may be considered as follows. From the condition of static equilibrium, the axial force is given by 0 1 R 1=2 M1 þ M2 þ
1=2 m Rd 1 Z 1=2 NðÞ ¼ @ þ ðp cosðÞ
pu sinðÞÞRdA 2R sinða=2Þ 2
1=2 v Z  ðpu cosð
Þ þ pv sinð
ÞÞRd: sinðÞ þ P cosðÞ þ Int. J. Str. Stab. Dyn. 2013.13. Downloaded from www.worldscientific.com by HONG KONG POLYTECHNIC UNIVERSITY on 05/19/16. For personal use only.


1=2

ð16Þ Then the horizontal displacement due to axial deformation can be derived by the Castigliano's theorem as !, Z 1=2 NðÞ 2 eA ¼ d Rd dP
1=2 2EA : ð17Þ  þ sinðÞ ¼ PR þ eb ðpu ; pv Þ 2EA By making use of the symmetry of the curved beam element, the horizontal displacement contains only the three load variables P ; pu ; pv , and the term eb is due to distributed loads pu ; pv . When the membrane strain is considered, the relationship between forces and displacements can be modi¯ed as 8 9 8 e9 < e =

I ð 2 þ  sinðÞÞ; AR 2

K11 ¼ k11 ; K22 K23

K12 ¼ k12 ; K13 ¼ k13 ; I ¼ k22 þ ð
1 þ 4 2
2 2 cosðÞ þ cosð2Þ þ 2 sinðÞ
 sinð2ÞÞ; A I ¼ k23 þ ð1
2 2 cosðÞ
cosð2Þ þ 2 sinðÞ
 sinð2ÞÞ; A

K33 ¼ K22 : 1250084-9

ð19Þ

Y. Q. Tang, Z. H. Zhou & S. L. Chan

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The equivalent nodal load vector in the right term is 8 e9 8 9 8 e 9

> > = = < p > < <ep= e M 1p ¼ ½Ke ½ke 
1 dWp =d1 þ ½Ke  0 ; > > > : ; ; ; : e> : 0 dWp =d2 M 2p

ð21Þ

where Wp is the external work of the distributed load. In order to use the proposed curved beam element in the three-dimensional spaces allowing for rigid-body movements, the sti®ness matrix can be determined as ½KT  ¼ ½L½T  T ½ke ½T ½L T ;

ð22Þ

where [T ] is the internal to external transformation matrix relating the three independent internal forces and displacements to the external six forces and displacements, [L] is the local to global transformation matrix. The matrices [T ] and [L] are the same as those used in PEP element,24 i.e., 2 3
1 0 0 1 0 0 6 7 ½T  ¼ 4 0 1=l 1 0
1=l 0 5; ð23aÞ 0 1=l 0 0
1=l 1  0  L 0 ½L ¼ ; ð23bÞ 0 L0 2 3 X2
X1
ðY2
Y1 Þ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 where ½L 0  ¼ 4 Y2
Y1 X2
X1 0 5 and l ¼ ðX2
X1 Þ 2 þ ðY2
Y1 Þ 2 . l 0 0 1 The curved beam element in the global plane space is shown in the Fig. 4. The ¯nal equivalent nodal load vector of the distribute loads is given by 8 e9 < Pp = e T Me fF p g61 ¼ ½L½T  : ð24Þ : 1p e ; M 2p The consideration of the membrane and bending deformations separately is based on the fact that the membrane strain is much smaller than the bending strain. Also, the tangential displacement of the in¯nitesimal straight beam can be assumed as a 1250084-10

An Accurate Curved Beam Element Based on Trigonometrical Mixed Polynomial Function

( X 2 , Y2 )

2

( X1 , Y1 )

l

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1

Fig. 4. Curved beam element in the global coordinate system.

displacement interpolation function initially and the coe±cients can be obtained by the Ritz method, because the boundary conditions are not su±cient for determining all the coe±cients directly. 5. Numerical Examples In order to evaluate the performance of the proposed curved beam element, a series of typical and widely studied problems are selected and analyzed. The results obtained in this study are compared with the exact solutions solved by Castigliano's theorem and the results obtained by other ¯nite element approaches require signi¯cantly more computational e®ort. 5.1. Example 1: An arch with various subtended angles under horizontal pull A horizontal concentrated load P at end of the arch with subtended angle  is shown in the Fig. 5. The displacement and rotation of the node 2 under load P are compared with the exact solutions and summarized in Table 1. The exact solutions solved by Castigliano's theorem of the example are given by e ¼

PR PR 3 ð þ sinðÞÞ þ ð2 þ  cosðÞ
3 sinðÞÞ; 2EA 2EI

ð25aÞ

 1 ¼

    PR 2   cos
2 sin ; 2 2 2EI

ð25bÞ

 2 ¼
 1 :

ð25cÞ

It can be shown that the curved beam element converges well to the exact solutions of arbitrarily subtended angle under the horizontal pulling force using one element only. Actually, the results of the element proposed are the exact solutions in this example. Conventionally, when the straight beam elements are used in analysis of arch, a number of cubic straight beam elements are used to model the arch in Example 1 1250084-11

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Y. Q. Tang, Z. H. Zhou & S. L. Chan

P=1kN R=1m E=2.05e8kN/m2 circular hollow section: 48.3mmx5.0mm Fig. 5. An arch with various subtended angles under horizontal pull.

Table 1. Comparison of the ¯nite element solutions with the exact solutions. Angle () Degree

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180

One element proposed

Comparison

eðmÞ

2 ðradÞ

e=e 

2 = 2

1.28928e
6 3.77491e
6 1.34459e
5 4.55845e
5 1.28695e
4 3.07608e
4 6.45547e
4 1.22499e
3 2.14720e
3 3.53042e
3 5.50670e
3 8.21745e
3 1.18079e
2 1.64206e
2 2.21884e
2 2.92264e
2 3.76270e
2 4.74478e
2

6.68472e
6 5.33556e
5 1.79390e
4 4.22954e
4 8.20414e
4 1.40577e
3 2.21011e
3 3.26112e
3 4.58259e
3 6.19401e
3 8.11020e
3 1.03410e
2 1.28908e
2 1.57589e
2 1.89388e
2 2.24182e
2 2.61794e
2 3.01990e
2

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

when the subtended angle  ¼
=2 and the convergence of the horizontal displacement with increasing number of elements is shown in Table 2 with the exact solution given in Table 1 as e ¼ 2:14720e
3m. From Table 2, it can be seen that the exact solution can be approached only when using a large number of straight beam elements. 1250084-12

An Accurate Curved Beam Element Based on Trigonometrical Mixed Polynomial Function Table 2. Convergence with increase in number of straight beam elements. No. of straight beam elements

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5 10 20 50 100

eðmÞ

e=e 

2.0000e
3 2.1098e
3 2.1378e
3 2.1457e
3 2.1468e
3

0.93145 0.98258 0.99562 0.99930 0.99981

5.2. Example 2: An arch with various subtended angles under a concentrated bending moment The unsymmetrical load is applied to the arch to examine the proposed curved beam element. A concentrated moment M at the node 2 with sliding support is shown in Fig. 6. The displacement and rotation of the node 2 under the concentrated moment M are compared with the exact solutions and summarized in Table 3. The exact solutions solved by Castigliano's theorem of the example are given by     MR 2  e ¼ 2 sin
 cos ; ð26aÞ 2 2 2EI    sinðÞ R cosðÞ R sinðÞ
þ
; ð26bÞ  1 ¼ M 8EAR 8EAR 8EI 8EI   2  sinðÞ R R cosðÞ R sinðÞ 
þ

 2 ¼ MCsc : ð26cÞ 2 8EAR 8EAR 4EI 8EI 8EI From the example, it can be seen that the curved element has excellent performance in calculation of horizontal displacements and has a small di®erence in rotation under the concentrated moment even in a deep arch (40  <  < 180  Þ.3

M=1kNm R=1m E=2.05e8kN/m2 circle hollow section: 48.3mmx5.0mm Fig. 6. An arch with various subtended angles under a concentrate moment.

1250084-13

Y. Q. Tang, Z. H. Zhou & S. L. Chan Table 3. Comparison of the ¯nite element solutions with the exact solutions.

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Angle(Þ Degree

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180

One element proposed

Compared with exact solutions

eðmÞ

1 ðradÞ

2 ðradÞ

e=e 

1 = 1

2 = 2

6.68472e
6 5.33556e
5 1.79390e
4 4.22854e
4 8.20414e
4 1.40577e
3 2.21011e
3 3.26112e
3 4.58259e
3 6.19401e
3 8.11020e
3 1.03410e
2 1.28908e
2 1.57589e
2 1.89388e
2 2.24182e
2 2.61794e
2 3.01990e
2


8.78008e
4
1.75332e
3
2.62320e
3
3.48477e
3
4.33497e
3
5.17048e
3
5.98762e
3
6.78229e
3
7.54976e
3
8.28459e
3
8.98039e
3
9.62958e
3
1.02231e
2
1.07497e
2
1.11960e
2
1.15449e
2
1.17751e
2
1.18591e
2

1.75735e
3 3.51740e
3 5.28289e
3 7.05668e
3 8.84184e
3 1.06417e
2 1.24599e-2 1.43006e
2 1.61685e
2 1.80690e
2 2.00086e
2 2.19948e
2 2.40367e
2 2.61453e
2 2.83344e
2 3.06209e
2 3.30261e
2 3.55774e
2

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

1.00012 1.00012 1.00012 1.00012 1.00012 1.00013 1.00013 1.00013 1.00014 1.00014 1.00015 1.00015 1.00016 1.00017 1.00018 1.00020 1.00021 1.00024

0.99994 0.99994 0.99994 0.99994 0.99994 0.99994 0.99994 0.99994 0.99994 0.99994 0.99993 0.99993 0.99993 0.99993 0.99993 0.99993 0.99992 0.99992

Exact solution of the horizontal displacement is obtained by the proposed element in this example, and the di®erence of rotation is due to the ignorance of the e®ect of membrane deformation to the rotation. Nevertheless, this e®ect is small because the membrane sti®ness is much larger than the bending sti®ness. 5.3. Example 3: An arch with various subtended angles under horizontally distributed load A uniform load q distributed along the horizontal direction on the arch with two ends pinned is shown in Fig. 7, so we have the tangential distributed load pu ¼ q  cosðÞ sinðÞ and the radial distributed load pv ¼
q cos ðÞ 2 , and the equivalent nodal load of the uniform load q can be obtained through Eqs. (11) to (21). The horizontal resistant force of the arch and the rotations of the ends solved by the proposed ¯nite element are compared with the exact solutions shown in the Table 4. The exact solutions solved by Castigliano's theorem of the example are given by qR 4I
AR 2 H  ¼ pffiffiffi  ; 3 2 Ið2 þ
Þ þ 2Að
3 þ
ÞR 2  1 ¼
 2 ¼


qR 4 Ið
22 þ
Þ þ Að22
7
ÞR 2  ; 24EI Ið2 þ
Þ þ 2Að
3 þ
ÞR 2

ð27aÞ ð27bÞ

in which A is the cross-sectional area and I is the inertia moment of cross-section. The proposed curved beam element has a good performance and the solutions are close to the exact. For the case when no unsymmetrical moment acts on the arch, the proposed curved element obtains the exact solutions. 1250084-14

An Accurate Curved Beam Element Based on Trigonometrical Mixed Polynomial Function

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UNIFORM LOAD q

q=1kN/m R=1m E=2.05e8kN/m2 circle hollow section: 48.3mmx5.0mm Fig. 7. An arch with two ends pinned under the uniform load.

Table 4. Comparison of the ¯nite element solutions with the exact solutions. One element proposed H=kN 0.827964

Compared with exact solutions

1 =rad

H=H 

1 = 1


1.93461e
5

1.00000

1.00000

5.4. Example 4: A quarter circular cantilever ring The cantilever ring shown in Fig. 8 is a problem that has been investigated by many researchers. The quarter ring is under a radially concentrated load P at free end. The sti®ness matrix [Ke ] has been used here to compute the displacements and rotation of the free end over a wide range of slenderness ratios from R=h ¼ 4 to 1,000. The slenderness ratios are used to adjust the proportion between the membrane and bending sti®ness, and the membrane sti®ness increases with the decrease of the slenderness ratio. The exact solutions can be derived by Castigliano's theorem and they are given by PR PR 3 þ ; 2EA 2EI P
R P
R 3
; v 2 ¼
4EA 4EI PR 2  2 ¼
: EI

u 2 ¼


1250084-15

ð28aÞ ð28bÞ ð28cÞ

Y. Q. Tang, Z. H. Zhou & S. L. Chan

Int. J. Str. Stab. Dyn. 2013.13. Downloaded from www.worldscientific.com by HONG KONG POLYTECHNIC UNIVERSITY on 05/19/16. For personal use only.

R=10 Unit width E=10.5e6 P=1

Fig. 8. A quarter circle cantilever ring. Table 5. Comparison of the ¯nite element solutions with the exact solutions. Slenderness ratio (R/h) 4 10 20 50 100 200 500 1000

One element proposed

Compared with exact solutions

u2

v2

2

u2 =u 2

v2 =v 2

2 = 2

3.63810e
5 5.70952e
4 4.57045e
3 7.14262e
2 5.71424e
1 4.57142 71.4285 571.429


5.77455e
5
8.98346e
4
7.18288e
3
1.12203e
1
8.97605e
1
7.18080
112.200
897.598


7.31429e
6
1.14286e
4
9.14286e
4
1.42857e
2
1.14286e
2
9.14286e
1
14.2857
114.286

0.99851 0.99976 0.99994 1.00000 1.00000 1.00000 1.00000 1.00000

0.99906 0.99985 0.99996 1.00000 1.00000 1.00000 1.00000 1.00000

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

The results of curved beam ¯nite element are still close to the exact solutions even in a thick arch (R=h < 40).3 It can be seen that the di®erence increases when the slenderness ratio decreases, because the results are more a®ected by the membrane sti®ness with the slenderness ratio decreases and the proposed curved beam element simpli¯es the e®ect of membrane deformation. 5.5. Example 5: A thin pinched ring A typical pinched ring is shown in Fig. 9 and there are two identical and opposite forces P through the diameter at top and bottom of the ring respectively. A quadrant of the ring is modeled to check the proposed curved beam element with symmetrical boundary conditions. The exact solutions can be derived by Castigliano's theorem and given by ð

4ÞPR 3 PR ;
4EA 4
EI ð
2
8ÞPR 3
PR V 2 ¼ : þ 8EA 8
EI U 1 ¼


1250084-16

ð29aÞ ð29bÞ

An Accurate Curved Beam Element Based on Trigonometrical Mixed Polynomial Function

Fig. 9. A thin pinched ring.

1

0.95

V2 /V2*

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R=4.953 h=0.094 Unit width E=10.5e6 P=1

0.9

CL1 Element

0.85

Present Element

0.8 1

2

3

4 5 6 7 8 No. of Elements in Quadrant

9

10

Fig. 10. Convergence of radial displacement under point load of the pinched ring.

The curved beam element proposed herein has a high accuracy in this example. The comparison of a classical curved beam element with the exact solution is shown in Fig. 10. In order to obtain a better representation of the comparison, a nondimensional axis is chosen. One of the classical beam elements, named as CL1 element by Prathap, is a basic curved element with one-point Gaussian integration for the membrane energy.8 Both the present element and CL1 element are displacement-based elements, but the present element has a much higher accuracy when using one element. Although the CL element and the low-order element have been studied extensively by many 1250084-17

Y. Q. Tang, Z. H. Zhou & S. L. Chan Table 6. Comparison of the ¯nite element solutions with the exact solutions.

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One element proposed

Compared with exact solutions

U1

V2

U1 =U 1

V2 =V 2


1.14192e
2


1.24388e
2

0.99997

0.99997

researchers, their displacement functions are not suitable for the curved beam element. The exact displacement functions of the °exible curved beam are proposed in Eq. (5). They are complex mixed polynomial
trigonometric functions that these functions are impossible to obtain a good solution by the cubic
linear or other loworder displacement ¯elds. 6. Concluding Remarks Presented herein is a new ¯nite curved beam element using the displacement and strain ¯elds of the in¯nitesimal straight beam section. The ¯ve examples considered show that this curved beam element has a high accuracy when using a single element per member. This concept of using the strain energy of the in¯nitesimal straight beam section has been used in the energy method with Castigliano's theorem when the curved beam is thin, and the result is regarded as the exact solution. Surprisingly, few researchers use this concept to establish the displacement and strain ¯eld of a curved beam element. Interestingly, the displacement of the in¯nitesimal straight beam section herein is di®erent from the displacement of an arbitrary point of the curved beam in the curvature coordinate system, but they lead to the same strain expression and strain energy. Through this method, the exact displacement function without membrane strain has been established. The membrane deformation is simpli¯ed in the proposed curved beam element which has an excellent performance in the ¯ve examples when the membrane strain is much smaller than the bending strain which is a common characteristic of arches. Acknowledgment The authors acknowledge the ¯nancial support by the Research Grant Council of the Hong Kong SAR Government on the projects \Collapse Analysis of Steel Tower Cranes and Tower Structures (PolyU 5119/10E)" and \Stability and second-order analysis and design of re-used and new sca®olding systems (PolyU 5116/11E)". References 1. S. P. Timoshenko, Strength of Materials (D. Van Nostrand Company, Inc., 1946). 2. D. G. Ashwell and R. H. Gallagher, Finite Elements for Thin Shells and Curved Members (John Wiley & Sons, London, 1976). 1250084-18

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An Accurate Curved Beam Element Based on Trigonometrical Mixed Polynomial Function

3. D. Dawe, Some High-Order Elements for Arches and Shells, Finite Elements for Thin Shells and Curved Members (John Wiley, London, 1976), pp. 131
153. 4. D. Dawe, Numerical studies using circular arch ¯nite elements, Comput. Struct. 4(4) (1974) 729
740. 5. D. Dawe, Curved ¯nite elements for the analysis of shallow and deep arches, Comput. Struct. 4(3) (1974) 559
580. 6. H. Meck, An accurate polynomial displacement function for unite ring elements, Comput. Struct. 11(4) (1980) 265
269. 7. D. Ashwell, A. Sabir and T. Roberts, Further studies in the application of curved ¯nite elements to circular arches, Int. J. Mech. Sci. 13(6) (1971) 507
517. 8. G. Prathap, The curved beam/deep arch/¯nite ring element revisited, Int. J. Num. Meth. Eng. 21(3) (1985) 389
407. 9. G. Prathap and G. Bhashyam, Reduced integration and the shear °exible beam element, Int. J. Num. Meth. Eng. 18(2) (1982) 195
210. 10. H. Stolarski and T. Belytschko, Membrane locking and reduced integration for curved elements, J. Appl. Mech. 49 (1982) 172. 11. H. Stolarski and T. Belytschko, Shear and membrane locking in curved C0 elements, Comput. Meth. Appl. Mech. Eng. 41(3) (1983) 279
296. 12. C. R. Babu and G. Prathap, A linear thick curved beam element, Int. J. Num. Meth. Eng. 23(7) (1986) 1313
1328. 13. G. Prathap and C. R. Babu, An isoparametric quadratic thick curved beam element, Int. J. Num. Meth. Eng. 23(9) (1986) 1583
1600. 14. A. Saleeb and T. Chang, On the hybrid-mixed formulation of C0 curved beam elements, Comput. Meth. Appl. Mech Eng. 60(1) (1987) 95
121. 15. B. Reddy and M. Volpi, Mixed ¯nite element methods for the circular arch problem, Comp. Meth. Appl. Mech. Eng. 97(1) (1992) 125
145. 16. H. Dor¯ and H. Busby, An e®ective curved composite beam ¯nite element based on the hybrid-mixed formulation, Comput. Struct. 53(1) (1994) 43
52. 17. J. G. Kim and Y. Y. Kim, A new higher-order hybrid-mixed curved beam element, Int. J. Num. Meth. Eng. 43(5) (1998) 925
940. 18. C. Zhang and S. Di, New accurate two-noded shear-°exible curved beam elements, Comput. Mech. 30(2) (2003) 81
87. 19. P. Raveendranath, G. Singh and B. Pradhan, A twonoded locking
free shear °exible curved beam element, Int. J. Num. Meth. Eng. 44(2) (1999) 265
280. 20. P. G. Lee and H. C. Sin, Locking-free curved beam element based on curvature, Int. J. Num. Meth. Eng. 37(6) (1994) 989
1007. 21. S. Y. Yang and H. C. Sin, Curvature-based beam elements for the analysis of Timoshenko and shear-deformable curved beams, J. Sound Vib. 187(4) (1995) 569
584. 22. H. Sa®ari and R. Tabatabaei, An accurate fourier curvature function for ¯nite ring elements, Int. J. Appl. Math. Mech. 2(2) (2006) 75
93. 23. W. Jenkins, Matrix and Digital Computer Methods in Structural Analysis (McGraw-Hill, London, 1969). 24. S. L. Chan and Z. H. Zhou, Pointwise equilibrating polynomial element for nonlinear analysis of frames, J. Struct. Eng. 120 (1994) 1703.

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