IJNME-Arch-Free.pdf

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2003; 58:1907–1936 (DOI: 10.1002/nme.837)

Free vibration analysis of arches using curved beam elements

Jong-Shyong Wu∗;† and Lieh-Kwang Chiang

Department of Naval Architecture and Marine Engineering; National Cheng-Kung University;Tainan 701; Taiwan

SUMMARY

The natural frequencies and mode shapes for the radial (in-plane) bending vibrations of the uniformcircular arches were investigated by means of the �nite arch (curved beam) elements. Instead of thecomplicated explicit shape functions of the arch element given by the existing literature, the simpleimplicit shape functions associated with the tangential, radial (or normal) and rotational displacementsof the arch element were derived and presented in matrix form. Based on the relationship betweenthe nodal forces and the nodal displacements of a two-node six-degree-of-freedom arch element, theelemental sti�ness matrix was derived, and based on the equation of kinetic energy and the implicit shapefunctions of an arch element the elemental consistent mass matrix with rotary inertia e�ect consideredwas obtained. Assembly of the foregoing elemental property matrices yields the overall sti�ness andmass matrices of the complete curved beam. The standard techniques were used to determine the naturalfrequencies and mode shapes for the curved beam with various boundary conditions and subtendedangles. In addition to the typical circular arches with constant curvatures, a hybrid beam constructed byusing an arch segment connected with a straight beam segment at each of its two ends was also studied.For simplicity, a lumped mass model for the arch element was also presented. All numerical resultswere compared with the existing literature or those obtained from the �nite element method based onthe conventional straight beam element and good agreements were achieved. Copyright ? 2003 JohnWiley & Sons, Ltd.

KEY WORDS: arch (curved beam) element; straight beam element; sti�ness matrix; consistent massmatrix; lumped mass matrix

1. INTRODUCTION

As shown in Reference [1], while the application of �nite elements to the �at structuressuch as beams and plates is well established, the solution for the curved structures, suchas arches, rings and shells, is not yet completely understood. This situation has not changedmuch and so a lot of researchers are still interested in this area of study. Since this

∗Correspondence to: Jong-Shyong Wu, Department of Naval Architecture and Marine Engineering, NationalCheng-Kung University, Tainan 701, Taiwan.

†E-mail: [email protected]

Contract=grant sponsor: National Science Council; contract=grant number: NSC89-2611-E-006-049

Received 28 November 2001Revised 22 November 2002

Copyright ? 2003 John Wiley & Sons, Ltd. Accepted 3 February 2003

1908 J.-S. WU AND L.-K. CHIANG

paper focuses on the in-plane vibrations of the arches, most of the information regardingthe out-of-plane behaviour of the curved beams has been neglected here. Among the existingreports, most of them aim at the derivations of displacement functions (or shape functions)and sti�ness matrices of the arch (curved beam) elements [2–8]. Although these reports arevery useful for the static analysis of the arches, the mass matrix of the arch element is alsorequired for the dynamic analysis of arches. However, the information in this aspect is rare.To the author’s knowledge, References [1] and [9–13] are the few papers most concernedwith this.In References [1] and [9], the e�ect of rotary inertia was neglected and the sti�ness matrix

and consistent mass matrix of the arch element in ‘explicit’ forms were derived based on thetwo functions for the tangential and radial displacements. In Reference [10], by consideringthe e�ects of rotary inertia and warping torsion and neglecting the e�ect of shear deformation,both the sti�ness and mass matrices of the thin-walled (spatial) curved beam element werederived from the energy variation theory. In Reference [11], the ‘explicit’ shape functionsof Reference [8] were used to derive the sti�ness matrix and consistent mass matrix of thearch element by using the energy variation theory and the unit-displacement method, respec-tively. In the current paper, the e�ect of rotary inertia was considered and the sti�ness matrixand consistent mass matrix of the arch element in ‘implicit’ forms were derived based on thethree functions given by Reference [3] for the tangential displacement, the radial displacementand the rotational angle. Where the sti�ness matrix of the arch element was obtained fromthe force–displacement relations given by Reference [3] and the mass matrix was directlyderived from the equation of kinetic energy for the arch element. Besides, in References[9, 10, 12, 13] all the displacement functions are in terms of the circumferential (circular)co-ordinate z, but in this paper all the displacement functions are in terms of the angularco-ordinate � given in Reference [3] and was widely used by many researchers [6–8, 11].It is noted that if X denotes a physical parameter (such as force or displacement) then thedimension of @X=@� is the same as that of X , but this is not true for the dimensions of @X=@zand X . Comparing with the 18 tedious ‘explicit’ shape functions reported in the latest liter-ature [8], the shape functions presented in this paper are in ‘implicit’ matrix form [10] andhave the advantages of easy derivation, easy computer programming and ease of getting ridof mistakes.In addition to the consistent mass matrix, a lumped mass model for the arch element

was presented. Excellent agreements between the natural frequencies obtained from the con-sistent arch mass matrix, the lumped arch mass matrix, the straight beam element modeland the existing literature [11] verify the availability of the presented approaches. In or-der to incorporate with the arch element to perform the free vibration analysis of a hybridbeam composed of an arch segment and two straight beam segments, the modi�ed prop-erty matrices for the straight beam element were also derived and use of the arch elementstogether with the straight beam elements to the free vibration analysis of the hybrid beamwas tried.For the dynamic characteristics of a curved beam, in addition to the foregoing ‘in-plane’

vibration, the ‘out-of-plane’ vibration is also an important problem in practical application.Therefore, several researchers have devoted themselves to the study of the last problem. Ref-erences [12–14] were found to be the three pertinent papers. In References [12] and [13],the dynamic responses of a horizontally curved I-girder bridge due to a two-axledfour-wheeled vehicle were studied using the �nite element method, where the three

Copyright ? 2003 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2003; 58:1907–1936

FREE VIBRATION ANALYSIS OF ARCHES 1909

major structural components, the roadway slab, the I-girders and the diaphragms, were dis-cretized by using the annular plate elements, the thin-walled curved beam elements and thestraight prismatic beam elements, respectively. In Reference [14], the dynamic responses of ahorizontally curved beam subject to a moving load were solved with the classical analyticalmethod. The centrifugal force due to curvilinear motion of the moving load was considered inthe last three references. In Reference [15], the analytical solutions of the eigenvalue equationfor the buckling of the thin-walled curved members have been solved, however, extensionof the formulation to include the element mass matrix is required for the dynamic analy-sis of the thin-walled curved members. In Reference [16], for simplicity and generality ofthe presented theory, the similar idea of constructing the ‘implicit’ shape functions has alsobeen proposed and developed in the context of meshfree method for the analyses of theelastostatic, buckling, free vibration and forced vibration characteristics of the thin straightbeam.Since, in this paper, the derivation of sti�ness matrix for the arch element is to refer

to Reference [3] and the displacement functions of Reference [3] look like those of Refer-ence [13], a comparison between References [3] and [13] and the current paper was made.The main �ndings are: (i) The procedure for the derivation of sti�ness matrix in Reference[3] is similar to that of Reference [13]. (ii) The in-plane motion and out-of-plane motionof the three-dimensional (3D) curved beam are coupled in Reference [3], but uncoupled inReference [13]. (iii) The warping e�ect is neglected in Reference [3] and is considered inReference [13]. (iv) The displacement functions of Reference [3] are in terms of 12 unknownconstants and those of Reference [13] are in terms of 100 unknown constants. (v) Reference[3] adopted the angular co-ordinate � and Reference [13] adopted the circumferential (circu-lar) co-ordinate z. (vi) Reference [3] derived only the sti�ness matrix of a 3D curved beamelement and Reference [13] derived both the sti�ness matrix and consistent mass matrix of a3D curved beam element. Since the displacement functions of Reference [3] are much simplerthan those of Reference [13], this paper used the former to derive both the sti�ness matrixand consistent mass matrix of an in-plane curved beam element. The consistent mass matrixof this paper was derived from the equation of kinetic energy, which is di�erent from thatof Reference [13] derived from the principle of virtual work. In addition to the consistentmass matrix, the current paper tried to use the ‘lumped mass matrix’ to solve the problemand achieved satisfactory results, but this is not true for Reference [13]. Where the ‘lumpedmass matrix’ was not derived from the energy method.

2. DISPLACEMENT FUNCTIONS AND SHAPE FUNCTIONS

For the arch element shown in Figure 1, if the x-axis is the symmetric axis for the cross-sectionof the arch and the e�ect of shear deformation is ignored then the displacement functions forthe radial displacement ux, the circumferential displacement u� and the rotational angle y aregiven by [3]

ux =G1 +G2 cos �+G3 sin �+G4� sin �+G6� cos � (1a)

u� =G1C�+G2 sin �−G3 cos �+G4(sin �−� cos �)+ G5+G6(cos �+� sin �) (1b)



R1y

2y

y

z

y

x

z

1xu

xo

xu

θu

2xu

2uθ

θ

b

a1uθ

1θ

θ

2θ

�

�

�

12

Figure 1. The de�nition for the in-plane element displacements, ux, u� and y, for an arch element andthe associated reference local and global co-ordinate systems, xyz and xyz.

y =G1

(CR

)�+G4

(2R

)sin �+G5

(1R

)+G6

(2R

)cos � (1c)

where

C=1+ (Iy=AR2) (2)

In Equation (1), G1–G6 are the integration constants determined by the boundary conditionsof the arch element, while in Equation (2), A is the cross-sectional area, R is the averageradius of curvature of the arch element and Iy is the moment of inertia of the area A aboutthe y-axis given by [3]

Iy=∫A

x2

1− (x=R) dA (3)

Since the natural frequencies and mode shapes of the arches and a hybrid beam willalso be calculated with the conventional straight beam elements (in addition to the archelements) in this paper, both the local reference co-ordinate system, xyz, and the global ref-erence co-ordinate system, xyz, required for the straight beam elements are introduced inFigure 1.Writing Equation (1) in matrix form gives

{u}=[H ]{G} (4)



where

{u}= {ux u� y } (5a)

[H ] =

1 cos � sin � � sin � 0 � cos �

C� sin � − cos � sin �− � cos � 1 cos �+ � sin �

CR

� 0 02Rsin �

1R

2Rcos �

(5b)

{G}= {G1 G2 G3 G4 G5 G6 } (5c)

In Equations (4) and (5), the symbols [·] and {·} represent the rectangular (or square) matrixand the column vector, respectively.Applying to Equation (4) the boundary conditions for the arch element shown in Figure 1,

one obtains

{�}=[B]{G} (6)

where

{�}= {ux1 u�1 y1 ux2 u�2 y2}

(7)

[B]=

1 cos �1 sin �1 �1 sin �1 0 �1 cos �1

C�1 sin �1 − cos �1 sin �1 − �1 cos �1 1 cos �1 + �1 sin �1CR�1 0 0

2Rsin �1

1R

2Rcos �1

1 cos �2 sin �2 �2 sin �2 0 �2 cos �2

C�2 sin �2 − cos �2 sin �2 − �2 cos �2 1 cos �2 + �2 sin �2CR

�2 0 02Rsin �2

1R

2Rcos �2

(8)

From Equation (6) one obtains

{G}=[B]−1{�} (9)

The substitution of Equation (9) into Equation (4) yields

{u}=[N ]{�} (10)

where [N ] is a matrix of the shape functions de�ned by

[N (�)]= [H ][B]−1 =

N11(�) N12(�) · · · N16(�)

N21(�) N22(�) · · · N26(�)

N31(�) N32(�) · · · N36(�)

(11)



In theory, one may obtain a 6 × 6 square matrix for [B]−1 with all coe�cients beingin ‘explicit’ forms. However, to avoid manipulating the complicated matrix multiplicationand listing the tedious lengthy mathematical expressions in the subsequent derivations, the‘implicit’ form of [B]−1 will be used for the formulation of this paper and so will be therelated sti�ness and mass matrices of the arch element. Once the angular co-ordinates ofnode ©1 and node ©2 for the arch element, �1 and �2, are given, one may obtain the val-ues of matrix [B] and its inverse [B]−1 from Equation (8). To insert the values of [H ]and [B]−1 into Equation (11) the values of the shape function matrix [N (�)] will haveto be determined. It is evident that the implicit shape functions given by Equation (11)are much simpler than the explicit ‘exact’ shape functions given in Tables I–III of Ref-erence [8], particularly for the computer programming. Based on the numerical examplein Reference [8], the 18 curves for the shape functions Nmn (m=1–3; n=1–6) were ob-tained from Equation (11) and compared with the corresponding ones obtained from theexplicit ‘exact’ shape functions given in Tables I–III of Reference [8] and excellent agree-ments were achieved. However, it has been found that the coe�cient C(1)4 for the shapefunction N (1)

vu and that C(2)4 for N (2)vv in Table I of Reference [8] are wrong and the correct

expressions are

C(1)4 =− A′

2D2�2s0 +

12D′A′′s0�s+

2D1

c0s (a)

C(2)4 =− A′

2D2�2c0 +

2D1(1− c)c0 +

12(D′ −D′′c)A′′�s0 + c0 (b)

The above two expressions were provided by the second author of Reference [8] throughcorrespondence. It is noted that all the symbols appearing in Equations (a) and (b) are com-pletely the same as those in Reference [8] except that the right superscripts (1) and (2) forthe coe�cients C4 are newly added for convenience here. In theory, the �nite element methodis an ‘approximate’ method, thus the ‘exact’ shape functions reported in Reference [8] onlymean that they enable the behaviour of the arch to be calculated exactly for any mesh den-sity (i.e. any number of arch elements) and they are also called ‘natural’ shape functions[1, 2, 9].

3. STIFFNESS MATRIX FOR ARCH ELEMENT

Instead of the energy variation theory [1, 2, 6–11], this paper employs the force–displacementrelations to derive the sti�ness matrix of the arch element. From Reference [3] one obtainsthe following force–displacement relations:

Fx = F ′� (12a)

F� =EAR(u′� − ux)− My

R(12b)

My =EIyR2(ux + u′′x ) (12c)



Table I. The central displacements, ux, u� and y, for an arch with clamped–clamped (case nos. 1,2 and 3) and hinged–hinged (case nos. 4, 5 and 6) supports due to the concentrated central loads:

Fx=1 kN, F�=1 kN and My=1 kNm.

Present paper

Case Supporting and Central SB element CB elementno. loading conditions displacements∗ Reference [8] (n=20) (n=4)

1

kN1=xFux (m)u� (m) y (rad)

0:22050:00000:0000

0:220480:00000:0000

0:219690:00000:0000

2

kN1=θF ux (m)u� (m) y (rad)

0:00000:1136

−0:3683

0:00000:11289

−0:3642

0:00000:11176

−0:35918

3

kN1=yM

ux (m)u� (m) y (rad)

0:0000−0:09211:0375

0:0000−0:09101:0373

0:0000−0:083931:0357

4

kN1=xF ux (m)u� (m) y (rad)

0:25460:00000:0000

0:25360:00000:0000

0:25160:00000:0000

5

kN1=θFux (m)u� (m) y (rad)

0:00000:2770

−0:8122

0:00000:27346

−0:80496

0:00000:2687

−0:79559

6

kN1=yMux (m)u� (m) y (rad)

0:0000−0:20301:3383

0:0000−0:20121:3396

0:0000−0:19881:3390

∗Note: The actual magnitudes of ux, u� and y are the listed values times 10−6.

where the primes denote the derivatives with respect to the angular co-ordinate �, and since �is a non-dimensional parameter, the dimension of X ′ or X ′′ is the same as that of X (X =F�,u� or ux). From Equations (1) and (12) and the following relationship for the equilibrium of



Table II. (a) The lowest �ve natural frequencies of the simply supported (uxi=0; i=1; 2) curved beam,!i (i=1–5) (rad=s), with rotary inertia neglected.

Present paper

By SB elements By CB elementsMode Exactno. i solutions [9] 20 elements 40 elements 2 elements 4 elements 6 elements

1 0.349 0.472 0.472 0.349 0.349 0.3492 1.571 1.733 1.733 1.572 1.572 1.5723 3.612 3.790 3.789 3.725 3.615 3.6134 6.470 6.652 6.652 8.212 6.474 6.4745 10.144 10.325 10.324 14.307 10.274 10.162

(b) The lowest three natural-frequency ratios, !0ei =!dei , for the arch with shear deformation neglected obtained

from Reference [11] and present paper.

Natural-frequency ratios∗, !0ei =!dei

Thicknessratios, a=R Methods !0e1 =!de

1 !0e2 =!de2 !0e3 =!de

3

0.250 Reference [11]† 1.1716 1.0291 1.1397Present 1.1845 1.0360 1.1508

0.100 Reference [11] 1.0329 1.0417 1.0130Present 1.0357 1.0477 1.0257




∗!dei is the i-th natural frequency with all e�ects (including axial compressibility, rotatory inertia and sheardeformation) considered as given by Table IV of Reference [11].

†!0ei is the i-th natural frequency with shear deformation neglected as given by Table VI of Reference [11].

Table III. The lowest �ve natural frequencies of a clamped–clamped 180◦ circulararch, !i (i=1–5) (rad=s).

CB elements

SB elements Lumped mass Consistent massModeno. 20 elements 40 elements 20 elements 40 elements 20 elements 40 elements

1 1236.276 1233.466 1234.918 1234.933 1234.953 1234.939(−0:119%) (−0:000%)

2 2664.279 2658.408 2662.838 2662.957 2663.102 2662.990(−0:172%) (−0:001%)

3 4952.152 4940.879 4948.450 4949.230 4950.186 4949.452(−0:173%) (−0:004%)

4 6718.584 6706.799 6713.089 6714.779 6716.738 6715.114(−0:124%) (−0:005%)

5 8805.505 8785.752 8791.400 8794.968 8799.000 8795.531(−0:111%) (−0:006%)



nodal forces

{Fx1 F�1 My1}= −{Fx2 F�2 My2} (13)

one obtains

{F} = [D]{G} (14)

where

{F} = {Fx1 F�1 My1 Fx2 F�2 My2} (15)

[D] =EIyR2

0 0 0 − 2Rsin �1 0 − 2

Rcos �1

0 0 02Rcos �1 0 − 2

Rsin �1

−1 0 0 −2 cos �1 0 2 sin �1

0 0 02Rsin �2 0

2Rcos �2

0 0 0 − 2Rcos �2 0

2Rsin �2

1 0 0 2 cos �2 0 −2 sin �2

(16)

Introducing the values of [G] de�ned by Equation (9) into Equation (14) leads to

{F}=[D][B]−1{�}=[K]{�} (17)

where

[K] = [D][B]−1 (18)

which is the sti�ness matrix of the arch element in implicit form. Although the in-plane forces(and moments) for an arch element, given by Equation (12), are not functions of the rotationalangle y, they are actually closely related with y because the element displacements, ux andu�, are de�ned by the six integration constants, G1–G6, and these constants are functions of y as one may see from Equations (1) and (9).

4. MASS MATRIX FOR ARCH ELEMENT

The kinetic energy of the arch element is given by [9]

T =12

∫ �2

�1[�A(ux

2 + u�2) + �Iy y

2]R d� (19)

where the overdots denote the derivatives with respect to time t, � is the mass density ofthe arch material and Iy is the moment of inertia of the cross-sectional area de�ned by



Equation (3). It is noted that the third term on the right-hand side of Equation (19), �Iy y2,

represents the rotary inertia, which is not considered in References [1] and [9].For harmonic free vibrations, one has

{u}= { �u}ei!t (20)

where { �u} is the amplitude of {u}, ! is the natural frequency of the arch, t is time andi =

√−1.Substituting Equations (4) and (20) into Equation (19) yields

T = 12 !

2{�}T[M ]{�} (21)

where [M ] is the consistent mass matrix of the arch element given by

[M ]=�R([B]−1)T(∫ �2

�1[H ]T[�][H ] d�

)[B]−1 (22)

with

[�]=

A 0 0

0 A 0

0 0 Iy

(23)

To determine the consistent mass matrix of an arch element, [M ], using Equation (22), it isrequired to calculate the following integration:

[ �H ]=∫ �2

�1[H ]T[�][H ] d� (24)

and all the other numerical calculations are performed by the computer. The results for theintegration de�ned by Equation (24) are shown in Appendix A. Since [ �H ] is a 6 × 6 sym-metrical square matrix, one requires only to calculate the 21 coe�cients of the matrix. Thisis also much simpler than the 108 constants for the 18 shape functions shown in TablesI–III of Reference [8].For the purpose of comparison and simplicity, a lumped mass matrix for the arch element

is also presented in this paper:

[M ∗]= pm1 m1 J1 m2 m2 J2y (25)

with

m1 =m2 = 12 �AR� (26a)

J1 = J2 = 12 �IyR� (26b)



The symbol p·y in Equation (25) denotes a diagonal matrix and the notation � (= �2− �1) inEquation (26) denotes the subtended angle of the arch element (see Figure 1).

5. STIFFNESS AND MASS MATRICES FOR A STRAIGHT-BEAM ELEMENT

For a hybrid beam composed of arch segments and straight-beam segments, one may performthe free or forced vibration analysis using a combination of arch elements and straight-beamelements. In such a case, attention must be paid to the compatibility of deformations at thejoints where the arch (curved-beam) elements and straight-beam elements are connected. Ingeneral, the sti�ness and mass matrices for a straight-beam element were derived based onthe local co-ordinate system, xyz, with the x-axis coincident with the longitudinal axis ofthe beam element, as shown in Figure 2(a) [17]. However, in order to incorporate with theproperty matrices of the arch element derived in the previous subsections, one must changethe local co-ordinate system for the straight-beam element to that with the z-axis coincidentwith the longitudinal axis of the straight-beam element and with the x-axis coincident withthe radial direction of the arch element, as shown in Figure 2(b). In addition to exchang-ing the co-ordinate axes between the two local co-ordinate systems shown in Figures 2(a)and (b), the order of numbering for the nodal displacements must also be changed fromthe order for (u1; u2; u3; : : : ; u6) as shown in Figure 2(a) to the order for (ux1; u�1; y1; : : : ; y2)as shown in Figure 2(b). The transformation of the nodal displacements, ux1; u�1; y1; : : : and y2, with respect to the local co-ordinate system, xyz, will yield the nodal displacements,�u1; �u2; �u3; : : : and �u6, with respect to the global co-ordinate system, xyz, as shown in Figure3. By considering all the above-mentioned factors, the modi�ed sti�ness and mass matri-ces for a straight-beam element used to incorporate with the arch element derived in this

x

z

1xu

1θu

2xu

2θu

2y1y

l

1u

2u

3u

4u

5u

6u

x

z

(a)

(b)

l

��

1

12

2

Figure 2. The local co-ordinate system, xyz, for the in-plane straight-beam elements: (a) x-axis is alongthe beam length; and (b) z-axis is along the beam length.



γ

β

1u

2u

1xu

1θu

z

5u

4u

2xu

2θu26 yu =

z

xo

13 yu =

γβ

β

γ

x

β

2θ

1θ

α

�

� 1

2

Figure 3. The relationship between the nodal displacements for the curved-beam (arch) element(ux1; u�1; y1; : : : ; y2) and the straight-beam element ( �u1; �u2; �u3; : : : ; �u6).

paper are

[K] =

12EIy=‘3

0 EA=‘ Symmetric

6EIy=‘2 0 4EIy=‘

−12EIy=‘3 0 −6EIy=‘2 12EIy=‘3

0 −EA=‘ 0 0 EA=‘

6EIy=‘2 0 2EIy=‘ −6EIy=‘2 0 4EIy=‘

(27)

[M ] = �A‘

1335+6Iy5A‘2

013

Symmetric

11‘210

+Iy10A‘

0‘2

105+2Iy15A

970

− 6Iy5A‘2

013‘420

− Iy10A‘

1335+6Iy5A‘2

016

0 013

−13‘420

+Iy10A‘

0 − ‘2

140− Iy30A

−11‘210

− Iy10A‘

0‘2

105+2Iy15A

(28)



These two expressions were obtained from the sti�ness and mass matrices of the straight-beam element given by Reference [17] by changing the orders and signs of the associatedcoe�cients in the rows and columns of the element matrices, where ‘ is the length of thestraight-beam element.

6. DISPLACEMENTS FROM CURVED-BEAM AND STRAIGHT-BEAM ELEMENTS

The relationship between the nodal displacements for the curved-beam (CB) element, ux1; u�1, y1; : : : and y2, and those for the straight-beam (SB) element, �u1; �u2; �u3; : : : and �u6, are shownin Figure 3, where the CB (arch) element is represented by the solid lines and the SB elementby the dashed lines. Since all the sti�ness and mass matrices of the SB elements must betransformed to the ones with respect to the global co-ordinate system, xyz, and then areassembled to establish the overall sti�ness and mass matrices of the whole structure, thecomputer outputs for either the modal displacements obtained from the free vibration analysisor the actual displacements obtained from the forced vibration analysis are with respect to theglobal co-ordinate systems, xyz, when the problem is solved with the SB elements. This isthe reason why the nodal displacements with respect to the global (xyz) co-ordinate system,�ui (i=1–6), instead of those with respect to the local (xyz) co-ordinate system, ui (i=1–6),were shown in Figure 3. It is evident that all the directions of the nodal displacements forthe CB element are di�erent from those for the SB element except the rotational angles

�u3 = y1 (29a)

�u6 = y2 (29b)

Therefore, the following expressions should be used to transform the displacement componentsobtained from the CB elements, uxi and u�i (i=1; 2), into the vertical ones (in the �z-direction),d �zi (i=1; 2), if comparisons between the arch elements and the SB elements are based onthe vertical displacements:

d �z1 =−ux1 cos �+ u�1 sin � (for node ©1 ) (30a)

d �z2 =−ux2 cos �− u�2 sin � (for node ©2 ) (30b)

where � and � are the angles between the radii passing through nodes ©1 and ©2 and thevertical �z-axis, respectively, as shown in Figure 3. The positive values of d �z1 and d �z2 indicatethe vertical displacements in the upward (+�z) direction.Similarly, if the comparisons are based on the radial displacements, then one should use the

following relations to transform the nodal displacements of the SB element, �ui (i=1; 2; 4; 5),into the radial ones:

dr1 = �u1 sin � − �u2 cos � (for node ©1 ) (31a)

dr2 = �u4 sin �− �u5 cos � (for node ©2 ) (31b)



The positive values of dr1 and dr2 indicate the radial displacements pointing to the curvaturecentre �o of the arch (i.e. to the +uxi (i=1; 2) directions).In this paper, all the mode shapes were plotted based on the vertical modal displacements.

Hence Equations (30a) and (30b) were used for the transformations.

7. FREE VIBRATION ANALYSIS USING CB, SB AND CB–SB ELEMENTS

After the element sti�ness matrix [K] and the element mass matrix [M ] (or [M ∗]) are de-termined, one may assemble all the elemental property matrices to establish the equation ofmotion for free vibration of the complete system

[M ]{ �u}+ [K]{u} = 0 (32)

where {u} and { �u} denote the overall nodal displacement vector and overall nodal accelerationvector, respectively. Solving the corresponding eigenvalue equation

([K]−!2[M ]){u∗}=0 (33)

will yield the natural frequencies !i (i = 1; 2; : : :) and the associated mode shapes {u∗}i ofthe complete system.For a circular arch, it is well known that one may solve the problem with the monotype of

CB (arch) element or SB element. However, for a hybrid beam composed of arch segmentsand straight beam segments, one may solve the problem with the hybrid (CB–SB) elementsin addition to the monotype of elements. This is one of the solutions that this paper triesto seek.

8. NUMERICAL RESULTS AND DISCUSSIONS

To con�rm the reliability of the approaches presented, the shape functions, the sti�ness matrixand the natural frequencies obtained in this paper were compared with the existing literature.After that, the free vibration analysis of a 180◦ circular arch and a hybrid beam was performed.Unless particularly stated, all the numerical values in this paper were obtained based on theconsistent mass matrix with e�ect of rotary inertia considered.

8.1. Validation of shape functions

For convenience, we set

{�1 �2 �3 �4 �5 �6 }= {ux1 u�1 y1 ux2 u�2 y2} (34)

It is evident that if �1 = 1 and �i=0 (i=1–6; i �=1), then from Equations (10) and (11) oneobtains

ux(�) =N11(�) (35a)

u�(�) =N21(�) (35b)

y(�) =N31(�) (35c)



Similarly, if �2 = 1 and �i=0 (i=1–6; i �=2), then one has

ux(�) =N12(�) (36a)

u�(�) =N22(�) (36b)

y(�) =N32(�) (36c)

Finally, if �6 = 1 and �i=0 (i=1–6; i �=6), then one has

ux(�) =N16(�) (37a)

u�(�) =N26(�) (37b)

y(�) =N36(�) (37c)

Based on Equations (35)–(37) and the other similar relations, one obtains the 18 displacement(or shape) function curves as shown in Figures 4(a)–(f), where the curves with attachments(····•····ux; – – N – – u�; —�— y) were obtained in this paper based on Equations (5b), (8),(10) and (11), while the curves without attachments (·········ux; – – – – u�; — y) were obtainedfrom a computer program developed by the authors based on the mathematical expressionsfor the shape functions shown in Tables I–III of Reference [8]. It is noted that the correctedexpressions (a) and (b) as shown in Section 2 of this paper, were used. The abscissa for eachof Figures 4(a)–(f) denotes the angular co-ordinate � (radians) and the ordinate representsthe radial displacement ux (metres), circumferential displacement u� (metres) or rotationalangle y (radians). The excellent agreement between the curves obtained from this paper andthe corresponding ones based on Reference [8] veri�es the correctness of the derived shapefunctions de�ned by Equation (11). The given data for the arch studied in this subsection are[8]: radius R = 4m, total subtended angle ��=120◦, radial thickness of cross-section a=0:6m,axial thickness of cross-section b=0:4 m, Young’s modulus E=3 × 1010 N=m2, Poisson’sratio �=0:17, mass density �=2:53 × 103 kg=m3, membrane factor e=0:00684 and shearfactor d=0.

8.2. Validation of sti�ness matrix

For the arch discussed in the last subsection, if its two ends are �xed (see case nos. 1, 2and 3 in Table I) and its centre is subjected to a downward vertical force Fx=1kN (case 1),a rightward horizontal force F�=1 kN (case 2) or a clockwise moment about the horizontaly-axis My=1 kNm (case 3), then the radial displacements ux (m), the circumferential dis-placements u� (m) and the rotational angles y (rad) at the centre of the arch are shown incolumns 4–6 of Table I. The values of ux, u� and y listed in column 4 were obtained fromReference [8], and those listed in columns 5 and 6 were the results of this paper calculatedon the basis of the SB element and the CB element, respectively. It is evident that either theresults based on the SB element or those based on the CB element are very close to thoseof Reference [8]. However, the total number of �nite elements used for the SB element isn=20 and that for the CB element is n=4. This is because the results based on the SB



-0.5 -0.3 0.0 0.3 0.5

Angular coordinates, θ (rad)

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Dis

plac

emen

ts,u

x(m

),u θ(

m),

ψy(

rad)

Present

Ref.[8]

u N ( θ)

u ( θ

( θ

)=

θ)=

N ( θ)

( θ)= N ( θ)

N ( θ)

N ( θ)

N ( θ)

u ( θ)=

u ( θ)=

( θ)=

(Node 2)(Node 1)

Angular coordinates, θ (rad)(Node 2)(Node 1)


-0.5 -0.3 0.0 0.3 0.5

-1 .0

-0 .5

0.0

0.5

1.0

1.5

2.0

2.5

Dis

plac

emen

ts,u

x(m

),u θ(

m),

ψy(

rad)

Present

Ref.[8]

-0 .5 -0 .3 0 .0 0 .3 0.5

-0 .5

0.0

0.5

1.0

1. 5

Dis

plac

emen

ts,u

x(m),

u θ(m),

ψy(r

ad)

Present

Re f.[8]

ψ

ψ

u N ( θ)

u ( θ

( θ

)=

)=

N ( θ)

( θ)= N ( θ)

)N ( θ

N ( θ)

N ( θ)

u ( θ)=u ( θ)=

( θ)=

ψ

ψ

u N ( θ)

( θ)u ( θ

( θ

)=

)=

N

( θ)= N ( θ)

)N ( θ

N ( θ)

( θ)N

u ( θ)=u ( θ)=

( θ)=

ψ

ψ

(a)

(b)

(c)

Figure 4. The displacement (or shape) function curves obtained from the present paper (· · · ·• · · · ·ux;– – N – – u�; —�— y) and those based on Reference [8] (· · · · · · · · ·ux; – – – – u�; —— y) for

�i=0 (i=1–6) but: (a) �1 = 1; (b) �2 = 1; (c) �3 = 1; (d) �4 = 1; (e) �5 = 1; and (f) �6 = 1.



-0.5 -0.3 0.0 0.3 0.5

-0.5

0. 0

0. 5

1. 0

1. 5

2. 0

Present

Ref. [8]u (θ)=N (θ)

u (θ)=N (θ)

(θ)=N (θ)

u (θ)=N (θ)

u (θ)=N (θ)

(θ)=N (θ)

-0. 5 -0 .3 0.0 0.3 0.5

-1 .0

-0 .5

0.0

0.5

1.0

1.5

2.0

Present

Ref.[8]

u (θ)=N (θ)

u (θ)=N (θ)

(θ)=N (θ)

u (θ)=N (θ)

u (θ)=N (θ)

(θ)=N (θ)

-0.5 -0.3 0.0 0.3 0.5

-0.5

0.0

0.5

1.0

Present

Ref.[8]

u (θ)=N (θ)

u (θ)=N (θ)

(θ)=N (θ)

u (θ)=N (θ)

u (θ)=N (θ)

(θ)=N (θ)

Dis

plac

emen

ts,u

x(m

),u θ(

m),

ψy(

rad)

Dis

plac

emen

ts,u

x(m

),u θ(

m),

ψy(

rad)

Dis

plac

emen

ts,u

x(m

),u θ(

m),

ψy(

rad)




ψ

ψ

ψ

ψ

ψ

ψ

(d)

(e)

(f)

Figure 4. Continued.



element will be more closer to those of Reference [8] when the total number of elementsincreases, but the results based on the CB element are very close to those of Reference [8]once n¿4 and increase in the total number of the elements, n, hardly a�ects the results. Thismay be the reason why the shape function curves of this paper are almost coincident withthe ‘exact’ shape function curves of Reference [8] as shown in Figure 4. Of course, this willbe one of the advantages that the CB element is superior to the SB element. The situationfor case nos. 4, 5 and 6 are the same as case nos. 1, 2 and 3, the only di�erence is that thesupport condition is hinged–hinged instead of clamped–clamped. It is seen that the results ofthis paper are also in good agreement with those of Reference [8] for the hinged–hinged arch.

8.3. Validation of natural frequencies

The information regarding the free vibration analysis of arches is rare. References [1] and[9–11] have been found to be most concerned with the free vibration analyses of arches. Thedimensions and the material constants for the present example are [9]: radius of curvatureR=30 in, radial thickness of rib a=1:289× 10−2 in, axial thickness of rib b=1:008 in, cross-sectional area A = 0:013in2, total subtended angle of the complete arch ��=1rad =57:3◦, total(arc) length L=R ��=30 in, Young’s modulus E=107 lb=in2, mass density of arch material�=0:1 lb=in3.By neglecting the e�ect of rotary inertia, the lowest �ve natural frequencies !i (i=1–5)

(rad=s) of the arch are shown in Table II(a) for the simply supported conditions (with radialdisplacements ux1 = ux2 = 0). The natural frequencies listed in column 2 of Table II(a) are theexact solutions calculated from the frequency equation listed in the appendix of Reference[9] and those listed in columns 3–7 are the �nite element solutions of this paper. Among thelatter, those in columns 3 and 4 are obtained using the SB element and those in columns5, 6 and 7 using the CB element. It is seen that all the natural frequencies obtained fromthe CB element rapidly converge to the corresponding exact ones when the total number ofelements increases from n=2 to 6, but this in not true for the SB element even if n=40.From Table II(a) one also sees that the natural frequencies obtained from the �nite elementmethods (either based on SB or CB) are slightly larger than the exact solutions and convergemonotonically to the exact ones from above. This trend of �nite element solutions agrees withthat of Reference [9].Based on the following dimensions and material constants for the arch given by Reference

[11], the lowest three natural-frequency ratios of the arch, !0ei =!dei (i=1 to 3), obtained from

this paper (with total number of �nite elements, n=6) were compared with the correspondingones obtained from Reference [11] as shown in Table II(b): radius of curvature, R=1 m;total subtended angle of the complete arch, ��=2�=3rad; Young’s modulus, E=1N=m2; Maasdensity, �=1 kg=m3; axial thickness b and radial thickness a of cross-section, b= a=0:25 to0:01m (varied). In Table II(b), the symbol !de

i denotes the i-th natural frequency of the archby considering all the e�ects (including the axial compressibility, rotatory inertia and sheardeformation) given by Table IV of Reference [11], while the symbol !0ei denotes the i-thnatural frequency of the arch by neglecting the e�ect of shear deformation (and considering thee�ects of axial compressibility and rotatory inertia only) obtained from Table VI of Reference[11] and the present paper. It is seen that good agreement between the lowest three natural-frequency ratios, !0ei =!de

i (i=1 to 3), in the range of thickness ratios, a=R=0:25–0.01, wasachieved.



Table IV. The lowest six natural frequencies of the hinged–hinged hybridbeam, !i (i=1–6) (rad=s), with points P and Q free and pinned.

Mode no. SB–SB method SB–CB method

(a) Points P and Q free1 719.032 694.3382 2115.542 1893.9403 3945.599 3178.1004 5473.203 5602.4175 6802.270 7595.1796 8291.487 8778.209

(b) Points P and Q pinned (ux = u� = 0)1 2602.559 2605.2852 5059.209 5068.2243 8015.363 8008.6464 8850.068 8860.0305 10552.128 10557.9056 11849.540 11852.224

For a uniform arch, both the element sti�ness matrix and element mass matrix (eitherconsistent or lumped) are invariant if the subtended angle of the arch element, � (= �2− �1),is a constant as one may see from the output of the computer program. In addition, becausethe formulations for the arches are based on the polar co-ordinate system, transformation ofeach element property matrix (from local co-ordinate system into global one) is not requiredbefore assembling for the CB method. This is another advantage that the CB element issuperior to the SB element for the dynamic analysis of circular arches.

8.4. Free vibration analysis of a curved beam

The arch with radius of curvature R=30′′ and total subtended angle of the complete arch��=1 rad =57:3◦, illustrated in the last subsection, may be impractical because of its smallsize. Thus a more realistic clamped–clamped 180◦ circular arch is studied in this subsec-tion: radius of curvature R=0:5 m, radial thickness of rib a=0:06 m, axial thickness ofrib b=0:04m, cross-sectional area A=0:0024m2, total subtended angle of the complete arch��=� radians=180◦, total (arc) length L=R ��=1:5708m, Young’s modulus E=12×1010N=m2,mass density of arch material �=7:2× 103 kg=m3.From Table II one �nds that the accuracy of the CB element is much better than that of the

SB element. Thus the natural frequencies obtained from the CB element with consistent massmatrix as shown in the �nal column of Table III will be selected as the Reference frequenciesof comparisons. From Table III it is seen that, for the CB element, the lowest �ve naturalfrequencies obtained from the lumped-mass model are very close to the consistent-mass modelwith maximum percentage di�erence less than 0.006%. The lowest �ve natural frequenciesobtained based on the SB element are also very close to those based on the CB element withmaximum percentage di�erence less than 0.173%. Therefore, the lumped-mass model for theCB element is a simple and accurate model for the dynamic analysis of the arches. Of course,the conventional SB element is also available for the dynamic analysis of the arches if thesuitable size of beam element is used.



-1.0 -0.5 0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0

Non-dimensional coordinates, ξ = (θ-0.5π)/0.5π


0.0

0.5

1.0

Ver

tica

l mod

al d

ispl

acem

ents

, {u z }

1-~*

0.0

0.5

1.0

Ver

tica

l mod

al d

ispl

acem

ents

,{u z }

2-~*

(a)

(b)

SB

CB (Lumped mass)

CB (Consistent mass)

Static form

SB

CB (Lumped mass)


Static form

Figure 5. The lowest �ve mode shapes for the clamped–clamped 180◦ circular arch obtained from theSB element, · · · ·• · · · ·, the CB element with lumped-mass model, – – N – – and the CB element withconsistent-mass model, —�—: (a) 1st mode; (b) 2nd mode; (c) 3rd mode; (d) 4th mode; and (e) 5th

mode. The thick dashed lines (– – – –) represent the static con�gurations of the circular arch.

The lowest �ve mode shapes for the clamped–clamped 180◦ circular arch are shown inFigures 5(a)–(e), where the abscissa of each �gure denotes the non-dimensional co-ordinates=(� − 0:5�)=0:5� and the ordinate denotes the vertical modal displacements {u∗�z }i with re-spect to the static con�guration of the circular arch shown by the thick dashed line (– – – –),the mode shapes obtained from the SB element are represented by the curves · · · · • · · · ·,while those from the CB element with lumped-mass model and with consistent-mass modelare denoted by – – N – – and — � —. It is seen that the mode shapes obtained from alltechniques are very close to each other, particularly those based on the CB element with thelumped-mass and the consistent-mass models.From Figures 5(a) and (b) one sees that, for the �rst and second mode shapes of the

clamped–clamped 180◦ circular arch, their ‘crests’ are near the position at 1=4 (or 3=4) of the



-1.0 -0.5 0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0




0.0

0.5

1.0

0.0

0.5

1.0

0.0

0.5

1.0

Ver

tical

mod

al d

ispl

acem

ents

, {u z }

3-~*

Ver

tica

l m

odal

dis

plac

emen

ts,{

u z }4

-~*V

ertic

al m

odal

dis

plac

emen

ts,{

uz }

5-~*

(c)

(d)

(e)

SB

CB (Lumped mass)


Static form

SB

CB (Lumped mass)


Static form

SB

CB (Lumped mass)


Static form




a=6 cm

b=4 cm(B-B section)

z

R=0.5mo120

QO

P

DA

B

B

xo

o90 o90

maxx maxx

O

Figure 6. The dimensions for the hybrid beam studied in this paper.

arch length and their ‘nodes’ are near the middle of the arch. Therefore, when the circulararch is subjected to a load moving circumferentially, the maximum response will occur atthe position near 1=4 of the arch length. This is di�erent from a clamped–clamped ‘straight’beam subjected to a moving load, where the maximum response will occur at the middle ofthe straight beam.

8.5. Free vibration analysis of a hybrid beam

The hybrid beam studied in this subsection is shown in Figure 6. It is constructed by usinga 120◦ curved-beam segment connected with a straight-beam segment at each of its twoends. The two straight-beam segments are identical and along the tangential directions ofthe curved-beam segment at the two connecting points, P and Q, respectively. This assuresthe continuity of the hybrid beam at the connecting joints (P and Q). The dimensions forthe hybrid beam are shown in Figure 6 and the material properties are: Young’s modulusE=12× 1010 N=m2, mass density �=7:2× 103 kg=m3.The free vibration analysis of the hybrid beam was performed with two methods: the

SB–SB method and the SB–CB method, where the SB–SB method is a conventional �niteelement method with all beam elements (either straight or curved) being modelled by thestraight-beam (SB) elements. However, in the SB–CB method, the curved part of the hybridbeam is modelled by the curved-beam elements and the straight part of the hybrid beam ismodelled by the straight-beam elements. Because the co-ordinate system for the curved-beamelements is di�erent from that for the straight-beam elements, attention must be paid to thederivations of the sti�ness matrix and mass matrix for the straight-beam element as shownin Equations (27) and (28), respectively. In addition, all the element property matrices, foreither the curved-beam element or the straight-beam element, must be transformed into theones in terms of the common global co-ordinates before they are assembled.When the hybrid beam is hinged–hinged at the two ends (uxi= u�i=0, i=A;D), the low-

est six natural frequencies obtained from the above-mentioned two methods are shown in



Table V. The lowest six natural frequencies of the clamped–clamped hybridbeam, !i (i=1–6) (rad=s), with points P and Q free and pinned.

Mode no. SB–SB method SB–CB method

(a) Points P and Q free1 1220.042 1190.2232 2818.803 2699.9433 5042.719 3757.6784 6066.225 6190.5285 7701.025 8169.5286 8871.101 10698.151

(b) Points P and Q pinned (ux = u� = 0)1 2713.308 2716.1272 5172.868 5181.7053 8248.240 8244.3684 9835.386 9851.7825 14014.843 14024.3286 14347.456 14316.507

Table IV(a) for the case with points P and Q unconstrained and in Table IV(b) for points Pand Q pinned. The numbers of �nite elements for the left straight-beam segment (AP), middlecurved-beam segment (PQ) and right straight-beam segment (QD) are 10, 20 and 10, respec-tively. In other words, the total number of �nite elements for the whole hybrid beam is 40.Theoretically, the SB–SB method is available for most of the structures. Thus it is reasonableto use its results as the reference natural frequencies of comparisons. From Table IV(a) onesees that the natural frequencies obtained from the two methods diverge from each other tosome degree, but this is not true for Table IV(b). The last phenomenon may be due to thefact that the existence of the two pins at the two conjunctions, P and Q, signi�cantly reducesthe interference of the translational displacements between the curved-beam segment and thestraight-beam segments for Table IV(b).When the hybrid beam is in the clamped–clamped supporting conditions, the lowest six

natural frequencies obtained from the two methods are shown in Table V(a) for pointsP and Q free and in Table V(b) for P and Q pinned. It is evident that the above con-clusions drawn from the hinged–hinged hybrid beam are also available for Tables V(a)and (b).The lowest �ve mode shapes for the hinged–hinged hybrid beam with points P and Q

pinned are shown in Figures 7(a)–(e), where the abscissa of each �gure denotes the non-dimensional co-ordinates =(�− 0:5�)=0:5� and the ordinate denotes the vertical modal dis-placements {u∗�z }i with respect to the static con�guration of the hybrid beam shown by thethick dashed line (– – – –), the mode shapes obtained from the SB–SB method are repre-sented by the curves · · • · ·, while those obtained from the SB–CB method are denoted by—�—. Since the natural frequencies obtained from the two methods are very close to eachother, so are the corresponding mode shapes. The same conclusions may also be drawn fromFigures 8(a)–(e) for the lowest �ve mode shapes of the clamped–clamped hybrid beam withpoints P and Q pinned.



-1.0 -0.5 0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0




0.0

0.5

1.0

Ver

tica

l mod

al d

ispl

aV

erti

cal m

odal

dis

pla

cem

ents

, {u z

}1

-~*

0.0

0.5

1.0

cem

ents

, {u z

}2

-~*

0.0

0.5

1.0

Ver

tica

l mod

aldi

spla

cem

ents

,{u z

}3

-~*

(a)

(b)

(c)

SB-SB

SB-CB

Static form

SB-SB

SB-CB

Static form

SB-SB

SB-CB

Static form

Figure 7. The lowest �ve mode shapes for the hinged–hinged hybrid beam obtained from the SB–SB method, · · · · • · · · · and the SB–CB method, —�—: (a) 1st mode; (b) 2nd mode; (c)3rd mode; (d) 4th mode; and (e) 5th mode. The thick dashed lines (– – – –) represent the static

con�gurations of the hybrid beam. Points P and Q are pinned.



-1.0 -0.5 0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0



0.0

0.5

1.0

Ver

tica

l mod

al d

ispl

acem

ents

, {u

z }4

-~*

0.0

0.5

1.0

Ver

tica

l mod

al d

ispl

acem

ents

, {u

z }5

-~*

(d)

(e)

SB-SB

SB-CB

Static form

SB-SB

SB-CB

Static form


9. CONCLUSIONS

1. Although the shape functions presented in this paper are much simpler than the existingones, numerical results show that their accuracy is very close to that of the existing ones.

2. For a uniform arch with constant curvature, both the element sti�ness matrix and elementmass matrix (either consistent or lumped) are invariant if the subtended angle of the archelement is a constant. Furthermore, because the formulations are based on the polar co-ordinate system, transformation of each element matrix (from local co-ordinate systeminto global one) is not required before assembling. These are the advantages that thearch element is superior to the straight-beam element for the dynamic analysis of arches.Besides, for the same mesh density the accuracy of the arch element presented in thispaper is much better than that of the conventional straight beam element.

3. For an arch element, the lumped mass matrix is much simpler than the consistent massmatrix, but the accuracy between them is indistinguishable. Thus, replacing the compli-cated consistent mass matrix by the simple lumped mass matrix is also a good choicefor the dynamic analysis of arches.



-1.0 -0.5 0.0 0.5 1.0Non-dimensional coordinates, ξ = (θ-0.5π)/0.5π

0.0

0.5

1.0

Ver

tica

l mod

al d

ispl

acem

ents

, {u

z }1

-~*

(a)


0.0

0.5

1.0

Ver

tica

l mod

al d

ispl

acem

ents

, {u

z }2

-~*

(b)


0.0

0.5

1.0

Ver

tica

l mod

al d

ispl

acem

ents

, {u

z }3

-~*

(c)

SB-SB

SB-CB

Static form

SB-SB

SB-CB

Static form

SB-SB

SB-CB

Static form

Figure 8. The lowest �ve mode shapes for the clamped–clamped hybrid beam obtained from theSB–SB method, · · · · • · · · · and the SB–CB method, —�—: (a) 1st mode; (b) 2nd mode;(c) 3rd mode; (d) 4th mode; and (e) 5th mode. The thick dashed lines (– – – –) represent the

static con�gurations of the hybrid beam. Points P and Q are pinned.




0.0

0.5

1.0

Ver

tica

l mod

al d

ispl

acem

ents

, {u

z }4

-~*

(d)


0.0

0.5

1.0

Ver

tica

l mod

al d

ispl

acem

ents

, {u z

}5

-~*

(e)

SB-SB

SB-CB

Static form

SB-SB

SB-CB

Static form


4. For the hybrid beam illustrated in this paper, if the conjunctions between the curvedbeam segment and the straight beam segments are pinned so that the interference oftranslational displacements between the di�erent beam segments is signi�cantly reduced,then the natural frequencies and the associated mode shapes obtained from the SB–SBmethod are very close to those obtained from the SB–CB method, where SB and CBdenote straight beam and curved beam, respectively.

APPENDIX A. COEFFICIENTS OF MATRIX [ �H ]

Integration according to Equation (24) will give the coe�cients of the matrix [ �H ], �Hij (i; j=1–6). Because of symmetry, only the coe�cients in the left lower triangle are shown below:

�H11 =[A�+ C2

(A+

IyR2

)(�3

3

)]�2�1



�H 21 = A[(1 + C) sin �− C � cos �]�2�1

�H 22 = A[�]�2�1

�H 31 = A[−(1 + C) cos �− C� sin �]�2�1

�H 32 = 0

�H 33 = A[�]�2�1

�H 41 =[(

A+ 3AC +2CIyR2

)(sin �− � cos �)− AC�2 sin �

]�2�1

�H 42 =A2[�− sin � cos �]�2�1

�H 43 =A2[�2 − sin2 �]�2�1

�H 44 =[A�3

3+(A+

2IyR2

)(�− sin � cos �)− A� sin2 �

]�2�1

�H 51 =[(

C2

)(A+

IyR2

)�2]�2�1

�H 52 = A[− cos �]�2�1

�H 53 = A[− sin �]�2�1

�H 54 =[−(2A+

2IyR2

)cos �− A� sin �

]�2�1

�H 55 =[(

A+IyR2

)�]�2�1

�H 61 =[(

A+ 3AC +2CIyR2

)(cos �+ � sin �)− AC�2 cos �

]�2�1

�H 62 =A2[�2 + sin2 �]�2�1



�H 63 =A2[−�− sin � cos �]�2�1

�H 64 =12

[−A

(� sin 2�+

cos 2�2

)+(A+

4IyR2

)sin2 �

]�2�1

�H 65 =[(2A+

2IyR2

)sin �− A� cos �

]�2�1

�H 66 =[A�3

3+ A sin � (cos �+ � sin �) +

(2IyR2

)(�+ sin � cos �)

]�2�1

where

[f(�)]�2�1 =f(�2)− f(�1)

ACKNOWLEDGEMENTS

This paper is part of the project under NSC89-2611-E-006-049, National Science Council, Republicof China. The �nancial support of NSC is greatly acknowledged. The authors also thank very muchProfessor J. Rakowski, author of References [8] and [11], for providing the correct expressions givenin Equations (a) and (b) of this paper.

REFERENCES

1. Sabir AB, Ashwell DG. A comparison of curved beam �nite elements when used in vibration problems. Journalof Sound and Vibration 1971; 18(4):555–563.

2. Ashwell DG, Sabir AB, Roberts TM. Further studies in the application of curved �nite elements to circulararches. International Journal of Mechanical Sciences 1971; 13:507–517.

3. Lebeck AO, Knowlton JS. A �nite element for the three-dimensional deformation of a circular ring. InternationalJournal for Numerical Methods in Engineering 1985; 21:421–435.

4. Palaninathan R, Chandrasekharan PS. Curved beam element sti�ness matrix formulation. Computers andStructures 1985; 21(4):663–669.

5. Stolarski HK, Chiang YM. The mode-decomposition, C0 formulation of curved, two-dimensional structuralelements. International Journal for Numerical Methods in Engineering 1989; 28:145–154.

6. Ryu HS, Sin HC. Curved beam elements based on strained �elds. Communications in Numerical Methods inEngineering 1996; 12:767–773.

7. Litewka P, Rakowski J. An e�cient curved beam �nite element. International Journal for Numerical Methodsin Engineering 1997; 40:2629–2652.

8. Litewka P, Rakowski J. The exact thick arch �nite element. Computers and Structures 1998; 68:369–379.9. Petyt M, Fleischer CC. Free vibration of a curved beam. Journal of Sound and Vibration 1971; 18(1):17–30.10. Yoo CH, Fehrenbach JP. Natural frequencies of curved girders. Journal of the Engineering Mechanics Division

(ASCE) 1981; 107(EM2):339–354.11. Litewka P, Rakowski J. Free vibrations of shear-�exible and compressible arches by FEM. International Journal

for Numerical Methods in Engineering 2001; 52:273–286.12. Chaudhuri SK, Shore S. Dynamic analysis of horizontally curved I-girder bridges. Journal of the Structural

Division (ASCE) 1977; 103(ST8):1589–1604.13. Chaudhuri SK, Shore S. Thin walled curved beam �nite element. Journal of the Engineering Mechanics Division

(ASCE) 1977; 103(EM5):921–937.



14. Yang YB, Wu CM, Yau JD. Dynamic response of a horizontally curved beam subjected to vertical and horizontalmoving loads. Journal of Sound and Vibration 2001; 242(3):519–537.

15. Usami T, Koh SY. Large displacement theory of thin-walled curved members and its application tolateral-torsional buckling analysis of circular arches. International Journal of Solids and Structures 1980;16:71–95.

16. Gu YT, Liu GR. A local point interpolation method (LPIM) for static and dynamic analysis of thin beams.Computer Methods in Applied Mechanics and Engineering 2001; 190:5515–5528.

17. Przemieniecki JS. Theory of Matrix Structural Analysis. McGraw-Hill: New York, 1968.


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