A p-version, first order shear deformation, finite element for geometrically non-linear vibration of...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2004; 61:2696–2715Published online 29 October 2004 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1180

A p-version, first order shear deformation, finite element forgeometrically non-linear vibration of curved beams

Pedro Ribeiro∗,†

IDMEC/DEMEGI, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias,4200-465 Porto, Portugal

SUMMARY

A p-version, hierarchical finite element for curved, moderately thick, elastic and isotropic beams isintroduced. The convergence properties of the element are analysed and some results are compared withresults published elsewhere or calculated using a commercial finite element package. It is verified that,with the proposed element, shear locking does not affect the computation of the natural frequencies andthat low dimensional, accurate models are obtainable. Geometrically non-linear vibrations due to finitedeformations, which occur for harmonic excitations with frequencies close to the first three naturalfrequencies of vibration, are investigated using Newmark’s method. The influence of the thickness,longitudinal inertia and curvature radius on the dynamic behaviour of curved beams are studied.Copyright � 2004 John Wiley & Sons, Ltd.

KEY WORDS: curved beams; p-version; hierarchical finite element method; geometrically non-linearvibrations

1. INTRODUCTION

Often, the finite element method is applied to study arches or curved beams. One of the firstworks on the linear vibration of arches by the FEM was presented by Petyt and Fleischer[1], who applied three finite element models to study the radial vibrations of curved beams,with different subtended angles and boundary conditions. More recently, in Reference [2],a formulation was proposed to alleviate the problem of shear and membrane locking andthree examples—none in dynamics—were presented. Eisenberger and Efraim [3] introduced adynamic stiffness matrix for a circular beam, and calculated the natural frequencies of vibration.In Reference [4] a plane arch finite element was presented and the linear free vibrations ofarches analysed. Different curvature radius and boundary conditions were considered, and the

∗Correspondence to: P. Ribeiro, IDMEC/DEMEGI, Faculdade de Engenharia, Universidade do Porto, RuaDr. Roberto Frias, 4200-465 Porto, Portugal.

†E-mail: [email protected]

Contract/grant sponsor: FEDER; contract/grant number: POCTI/1999/EME/32641Contract/grant sponsor: Portuguese Science and Technology Foundation

Received 13 March 2003Revised 18 April 2004

Copyright � 2004 John Wiley & Sons, Ltd. Accepted 16 April 2004

A P-VERSION, FINITE ELEMENT FOR CURVED BEAMS 2697

influence of tangential and rotatory inertia was discussed. Reference [5] is dedicated to thederivation of a shear-flexible curved beam element formulation.

Some publications also exist concerning non-linear analysis of curved beams. For example,Bi and Dai [6] studied the dynamic behaviour of a shallow arch under periodic excitations. Atwo mode expansion was followed, and the ensuing two non-linear equations of motion wereintegrated numerically. In Reference [7], non-linear steady state vibration of beams, frames andshallow arches are analysed by the finite element and incremental harmonic balance methods.A h-version straight beam element is adopted and fundamental, super-harmonic, sub-harmonic,and combination resonances are computed. An interesting analysis of internal resonances inundamped, free vibrations of curved Timoshenko beams is presented in Reference [8], wherethe Galerkin method is applied both in the space and time domains, in order to discretizethe non-linear differential equations of motion. This approach is somewhat similar to use thefinite element and the harmonic balance methods. A consistent finite element theory for curvedand twisted beams is proposed in Reference [9]. It is a general theory, which accounts fortranslational and rotational displacements of the beam’s cross-sections, warping and distortion.

In the p-version of the FEM, the accuracy of the approximation is improved by increasingthe number of shape functions over the elements, keeping the mesh constant. Several worksdemonstrate that, in general, the p-version of the FEM gives accurate results with fewerdegrees of freedom than the h-version [10–15]. If the set of functions corresponding to anapproximation of lower order p, constitutes a subset of the set of functions corresponding tothe approximation of order p + 1, then the p-version of the FEM is called ‘hierarchical finiteelement method’ (HFEM), which has additional benefits [10].

In the following pages, a p-version, hierarchical finite element for moderately thick,isotropic, linear elastic, curved beams is derived, linear and geometrically non-linear vibra-tions are studied. First order shear deformation theory is followed and the rotatory inertia isconsidered. The convergence properties of the element are investigated and, in order to validateand to demonstrate its advantages, results are compared with published ones and with valuescalculated using a commercial FEM package. The non-linear equations of motion are solved inthe time domain by Newmark’s method. The influence on the beams’ linear and non-linear dy-namics of several parameters like the thickness and the ratio of curvature are studied. Motionsinvolving the first and higher order modes are analysed.

2. CURVED BEAM P -VERSION HIERARCHICAL FINITE ELEMENT

In this section, a hierarchical finite element for elastic curved beams, with uniform thicknessh is presented. The undeformed beam’s geometry is defined from a reference straight line,introducing an initial displacement wi (Figure 1). In a rectangular co-ordinate system, Oxz, thequadratic centroidal axis of the beam is expressed as

wi(x) = −1

2

(x2

R

)(1)

where R is the principal radius of curvature.Beams where the curvature radius is significantly greater than the length are considered. In

this case, Equation (1) is valid for circular beams and for beams with parabolic shapes. In

Copyright � 2004 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 61:2696–2715

2698 P. RIBEIRO

β R

Owi

x

C

z

Figure 1. Curved beam.

fact, for a circular arch the following relation holds:

wi = R cos � − R (2)

But for R?x and −�/2 < � < �/2

� ∼= arcsin( x

R

)⇒ cos � ∼=

√1 −

( x

R

)2

Developing the latter in Taylor series around x = 0, one obtains√1 −

( x

R

)2 = 1 − 1

2R2 x2 + h.o.t.

Because R?x, higher order terms can be neglected, and Equation (2) becomes similar toEquation (1).

In general, a small number of p elements is enough to analyse a structure, reducing possibleproblems due to inter-element discontinuities. Actually, in this paper, because simple beams areanalysed, only one finite element is employed. Timoshenko’s model [16], where it is assumedthat the transverse shear deformation is constant along the cross section, is followed. Therefore,the displacement components along the x direction at any point of the beam—denoted by u—are functions of the centroidal axis longitudinal translations u0, and of the independent rotationof the normal to the centroidal axis about the y axis, which is denoted by �0. The superscript‘0’ represents the centroidal axis and the rotations follow the right-hand rule. Hence, thecomponents of the displacement are:

u(x, z, t) = u0(x, t) + z�0(x, t) (3)

w(x, z, t) = w0(x, t) + wi(x) (4)

where w is the displacement along the z direction and w0 sets the initial configuration.



The longitudinal and shear strains are

�x = �0x − z�x (5)

�zx = �0zx (6)

when R is significantly greater than L, the longitudinal strain at z = 0, �0x , can be approximately

defined by [17]

�0x = u0

,x + w0

R+ 1

2(w0

,x)2 (7)

where the comma represents the partial derivative and �x is the curvature, given by

�x = −��0

�x(8)

For large R, the transverse shear strain may be computed as

�0zx = w0

,x + �0 (9)

For each element, the middle surface displacements are expressed in the form

u0

w0

�0

=

{Nu}T 0 0

0 {Nw}T 0

0 0 {N�}T

qu

qw

q�

(10)

where {qu}is the vector of generalized longitudinal displacements, {qw} is the vector of trans-verse displacements, and {q�} is the vector of generalized rotations about the y axis. Thecomplete matrix of shape functions is constituted by the row vectors of longitudinal, transverseand rotational shape functions, which are, respectively,

{Nu}T = {g1(�), g2(�), · · · gpi(�)} (11)

{Nw}T = {f1(�), f2(�), · · · fpo(�)} (12)

{N�}T = {�1(�), �2(�), · · · �p�(�)} (13)

po, pi and p� are the numbers of respective displacement shape functions employed. � is thelocal co-ordinate.

The constitutive equations for linear elastic isotropic materials are{�x

xz

}=

[E 0

0 G

]{�x

�xz

}(14)

where E is the Young modulus and G is the shear modulus of elasticity, which is given byE/(2(1 + �)). The letter � represents the ratio of Poisson.


2700 P. RIBEIRO

The longitudinal strain component can be written as

�x = [1 z]({

�po

�bo

}+

{�I

0

}+

{�pL

0

})(15)

where the linear longitudinal and bending strains, {�p0 } and {�b0}, and the geometrically non-linearlongitudinal strain, {�pL}, are defined as

�po = u0,x, �bo = ��0

�x, �pL = (w0

,x)2

2(16)

The contribution of the initial deformation to the longitudinal strain is

�I = w0

R(17)

The former stress–strain relationships are given in terms of shape functions and generalizeddisplacements in Appendix A.

The stress and couple resultants per unit length, Tx and Mx , respectively, are defined by

Tx =∫ h/2

−h/2�x dz (18)

Mx =∫ h/2

−h/2�xz dz (19)

The shear stress resultant is

Qx =∫ h/2

−h/2xz dz (20)

Substituting Equation (14) into Equations (18)–(20), one arrives at the following constitutiverelations for the curved beam:{

T

M

}=

[A 0

0 D

]({�po

�bo

}+

{�I

0

}+

{�pL

0

})(21)

Qx = Eh

2(1 + �)�xz (22)

with

A, D =∫

z

E(1, z2) dz (23)

The equations of motion are derived by equating the sum of the virtual work of the inertiaforces and of the elastic restoring forces to zero. The virtual work of the internal forces is



given by

Win = −b

∫L

�p0

�b0

T

+

�pL

0

T

+

�I

0

T

×A 0

0 D

�p0

�b0

+

�pL

0

+

�I

0

dx

−b

∫L

�zx

Eh

2(1 + �)�zx dx (24)

The virtual work of the inertia forces is

Wj = −b

∫ h/2

−h/2

∫L

�( uu + ww) dx dz (25)

From these expressions, one derives the stiffness and mass matrices.The virtual work of the external forces is given by

Wex =∫

L

[P j

w(t) (x − xj ) + P dw(x, t)

] w(x, t) + [

Pju (t) (x − xj ) + P d

u (x, t)] u(x, t) dx

+∫

L

[Mj(t) (x − xj ) + Md(x, t)

] �(x, t) dx = { qu qw q�}T

F Eu

F Ew

ME

where {F Eu , F E

w, ME} represents the vector of generalized external applied forces and (x −xj ) represents a spatial Dirac delta function. P j (t) and Mj(t) are concentrated forces ormoments acting at point x = xj , P d(x, t) and Md(x, t) represent distributed forces or moments.The subscripts u and w distinguish longitudinal from transverse forces.

A simplified model is considered for damping: proportional to the linear stiffness, witha damping coefficient �. Thus, we have the following time domain equations of motion ingeneralized coordinates:

[Mp] 0 0

0 [Mb] 0

0 0 [MR]

qu

qw

q�

+

[Kp1 ] [Kpc

1 ] 0

[Kcp1 ] [Kcc

1 ] + [K�111 ] [K�12

1 ]0 [K�21

1 ] [Kb1 ] + [K�22

1 ]

qu

qw

q�


2702 P. RIBEIRO

+ �

[Kp1 ] [Kpc

1 ] 0

[Kcp1 ] [Kcc

1 ] + [K�111 ] [K�12

1 ]0 [K�21

1 ] [Kb1 ] + [K�22

1 ]

qu

qw

q�

+

0 [K2] 0

[K3] [K4] + [Kc2 ] + [Kc

3 ] 0

0 0 0

qu

qw

q�

=

F Eu

F Ew

ME

(26)

Matrices of the type [M] and [K1] are constant, matrices of the types [K2] and [K3] dependlinearly on the solution, and matrix [K4] depends quadratically on the solution. The superscriptsp, b, R, c, � represent, respectively, longitudinal, bending, rotational, initial curvature and sheareffects. Some of these matrices are equal to ones given in Reference [14] for a straight thinbeam and the other matrices are given in Appendix B.

Each of the displacement components—u, w and �—is associated with a particular set ofshape functions. The transverse displacement is expressed as a function of Legendre polynomialsin the Rodrigues’ form plus the four Hermite cubics [4]. Only one of the first four shapefunctions—the Hermite cubics—has either displacement or rotation different from zero at theend of each element. All higher order hierarchical functions—the Legendre polynomials—havezero amplitudes and slopes at � = −1 and � = 1. A set of polynomials called the g set willbe applied in conjunction with linear functions for the longitudinal displacements and for therotations. The slope of these g functions is not zero at the boundaries, but their value is.

3. NUMERICAL APPLICATIONS

3.1. Linear vibration

In order to carry out a convergence study and confirm the accuracy of the p-version hierarchicalfinite element here introduced, the linear free vibrations of straight and curved beams areinvestigated.

The shear correction factor, , depends on the beam’s cross section, and different values havebeen suggested [16, 18, 19]. Kaneko [19] concluded that the value implied by Timoshenko [16]come closest to experimental results. For rectangular beams, this value is = (5+5�)/(6+5�).Hutchinson [18] suggested a more elaborated formula, where is a function of the aspectratio. However, in the particular case of rectangular beams, the same author arrived at theconclusion that ‘the experimental results neither confirm nor negate the dependence of the newshear coefficient on the aspect ratio’. Therefore, in this work the value = (5 + 5�)/(6 + 5�)will be used, except when comparing results with the ones of other authors.

In Table I the first four natural frequencies of a straight beam are studied and values arecompared with results from Timoshenko’s solution, given in Reference [20], and with resultsfrom Archer [21], Dong and Wolf [22] and Reddy [20]. The values of the present approachare close to the ones of other authors, and taking the 50 DOF model as a reference, the errorof the model with 30 DOF is lower than 0.114%.



Table I. First four frequencies (rad/s) of a simply supported beam (L = 200 in,h = 6

√5 in, b = 7

√5/2 in, � = 0.25389 × 10−3 lb s2/in4, E = 10 × 106 psi,G = 3.75 × 106 psi,� = 0.33, = 5/6).

Mode number

Number of shape functions/references 1 2 3 4 DOF

po = 7, pi = 7, p� = 7 188.201 736.320 1625.62 2824.72 21po = 8, pi = 8, p� = 8 188.201 736.320 1601.27 2823.20 24po = 10, pi = 10, p� = 10 188.201 736.307 1600.91 2728.52 30po = 15, pi = 15, p� = 15 188.201 736.307 1600.91 2725.40 50po = 15, pi = 15, p� = 15 191.534 736.978 1603.94 2733.70 50 = (5 + 5�)/(6 + 5�)

Timoshenko solution [20] 188.203 736.313 1600.92 2725.43Archer [21], 20 elements 188.205 736.451 1602.33 2732.77Dong and Wolf [22], 20 elements 188.209 736.423 1601.52 2727.43Reddy [20], 12 elements 188.198 736.152 1599.48 2719.37

Table II. First four frequencies (rad/s) of a clamped–clamped curved beam (L = 0.5 m, h = 0.002 m,b = 0.02 m, R = 2.5 m, � = 2778 Kg/m3, E = 7 × 1010 Pa, � = 0.34, = 5/6).

Mode number

Number of shape functions 1 2 3 4 DOF

po = 7, pi = 7, p� = 7 713.912 1199.29 1962.69 2334.32 21po = 8, pi = 8, p� = 8 713.909 1197.41 1953.34 2334.29 24po = 9, pi = 9, p� = 12 713.902 1197.39 1953.24 2314.71 30po = 10, pi = 10, p� = 13 713.901 1197.38 1953.09 2314.49 33po = 12, pi = 12, p� = 15 713.900 1197.37 1953.09 2314.25 39po = 16, pi = 16, p� = 18 713.900 1197.37 1953.09 2314.25 50Commercial FEM 714.806 1197.82 1953.64 2319.16

In Tables II and III the values of the natural frequencies of two curved bars are comparedwith the ones calculated using a commercial FEM, and it is shown that with the proposedhierarchical finite element accurate results are achieved with a small number of degrees offreedom. Not surprisingly, the value taken for influences the natural frequency.

The linear natural frequencies for a curved beam given in Reference [3] are computed aswell. The beam’s adimensional properties are L/r = 23.56; R/r = 15, where r is the radius ofgyration (r = √

I/A), and G/E = 0.3. The goal of this comparison is to somehow verify theproposed model using the analytical solutions of Reference [3], and the solutions of Reference[23], quoted in Reference [3]. It must be pointed out that this beam does not respect therelation R > L, and it is interesting to realize that, in spite of this, the values computed arefairly close to the ones given in those references (Table IV).


2704 P. RIBEIRO

Table III. First four frequencies (rad/s) of a clamped–clamped curved beam (L = 0.5 m, h = 0.002 m,b = 0.02 m, R = 5 m, � = 2778 Kg/m3, E = 7 × 1010 Pa, � = 0.34, = 5/6).

Mode number

Number of shape functions 1 2 3 4 DOF

po = 7, pi = 7, p� = 8 714.556 832.771 1462.88 2331.08 22po = 8, pi = 8, p� = 8 714.556 832.771 1462.88 2331.08 24po = 9, pi = 9, p� = 12 714.550 832.783 1469.83 2315.34 30po = 10, pi = 10, p� = 13 714.549 832.783 1469.82 2315.07 33po = 12, pi = 12, p� = 15 714.548 832.783 1469.81 2315.00 39po = 16, pi = 16, p� = 18 714.547 832.782 1469.81 2314.99 50Commercial FEM 717.221 832.587 1476.01 2324.84

Table IV. First four non-dimensional frequencies (�L2√�A/(EI)) of a clamped–clamped curved beam.

Mode number

References or number of shape functions 1 2 3 4 DOF

po = 5, pi = 7, p� = 7 35.82 43.13 80.41 82.57 19po = 16, pi = 16, p� = 18 35.82 42.46 80.36 81.27 50Reference [3] 36.7030706 42.26350131 82.23297187 84.49146227 —Reference [23] 36.81 42.44 82.5 84.3 —

Curved beam elements based on first order shear deformation theory often suffer from shearlocking. To alleviate this problem selective/reduced integration schemes, penalty parametermethods and mixed formulations have been used [2]. In order to demonstrate that the elementproposed in this work is not prone to locking phenomena, the natural frequencies of very thinarches were computed, and results compared with the ones of thin beam theory (Table V).The latter simply results from assuming that the shear deformation, given by Equation (9), iszero.

According to Reference [4] it is important to consider the tangential inertia in the analysis ofarches, even in the case of thin ones. In order to verify this, the linear natural frequencies werecomputed with and without considering the tangential inertia (Table V). The model withouttangential inertia simply results from equating matrix [Mp] to a null matrix and carrying outa condensation of Equation (26), thus reducing the number of unknowns. It is apparent thatfor the lower order modes of CC beams, the longitudinal inertia does not affect significantlythe linear natural frequencies; however, it becomes more important when higher order modesare studied. In any case, in the ensuing analyses the longitudinal inertia is always considered.

3.2. Geometrically non-linear vibration

In the following, several time domain results on non-linear vibrations are presented. These wereobtained by Newmark’s method with Newmark’s parameters [24]. Ten shape functions were



Tabl

eV

.L

inea

rfr

eque

ncie

sof

CC

beam

s(r

ad/s)

com

pute

dby

the

p-v

ersi

on,

HFE

M.

b=

0.02

m,

L=

1m

,R

=5

m,

=

5/6.

L/h

=10

000

L/h

=10

00L

/h

=10

0L

/h

=50

L/h

=10

FS

DT

FS

DT

FS

DT

FS

DT

FS

DT

FS

DT

FS

DT

FS

DT

FS

DT

FS

DT

Mod

eB

-ET

IN

oT

IB

-ET

IN

oT

IB

-ET

IN

oT

IB

-ET

IN

oT

IB

-ET

IN

oT

I

18.

9286

8.92

898.

9398

89.2

6789

.268

89.3

7687

1.98

871.

7687

1.93

1052

.210

51.1

1051

.233

47.1

3152

.331

52.8

216

.096

16.0

9616

.102

160.

3616

0.35

160.

4189

2.61

891.

1889

2.26

1785

.217

73.9

1776

.089

21.4

7781

.277

93.7

331

.756

32.0

5832

.076

298.

6229

9.42

299.

5617

99.7

1795

.017

95.4

3523

.434

85.2

3485

.915

793

1412

414

128

448

.579

48.5

7948

.587

432.

4843

2.45

432.

5228

97.5

2901

.429

02.7

5791

.457

26.2

5729

.017

519

1578

721

511

FSD

T:

Firs

tor

der

shea

rde

form

atio

nH

FEM

mod

el:

po

=8,

pi=

8p

�=

8,n

=24

.B

-E:

Ber

noul

li–E

uler

HFE

Mm

odel

:p

o=

8,pi

=8,

n=

16.

TI:

Tang

entia

lin

ertia

.


2706 P. RIBEIRO

-0.05

-0.03

-0.01

0.01

0.03

0.05

0.94 0.96 0.98 1

-0.2

-0.1

0

0.1

0.2

0.94 0.96 0.98 1

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.94 0.96 0.98 1

t

w/h

w/h

w/h

t (a)

(c)

(b)

t

Figure 2. Time histories at the middle of the beam, when excitation frequency � = 0.9�l1 and:(a) F = 5 × 102 N; (b) F = 5 × 103 N: and (c) F = 5 × 103 N.

used to approximate each generalized displacement (po = pi = p� = 10). As we just saw, withthis number of shape functions very accurate models are obtained. Nevertheless, the numberof degrees of freedom is sufficiently small to allow one to carry out the computations in arelatively short time.

Figures 2–6 show the response of arches to a transverse harmonic point excitation withfrequencies �/�l1 = 0.9, 1 and 1.1, amplitudes 500, 2000 and 5000 N, applied at mid span.The beams width, length, curvature radius and thickness are b = 0.02m, L = 1m, R = L/0.4mand h = L/50m. The material is aluminium, with properties � = 2778Kg/m3, E = 7×1010 Pa,� = 0.34, the loss factor is � = 1.0×10−5 and the shear correction factor is = (5+5�)/(6+5�).In most figures, only the steady state response is displayed, i.e. transients have died out.

The responses shown in Figure 2 are always periodic, with the first harmonic predominant.In the Figures 3 and 4, which correspond to an excitation with a frequency equal to the firstlinear natural frequency and to 1.1 �l1, the arch behaves in a very unstable manner when theexcitation amplitude is large. When the response is not periodic, as occurs in Figures 3(c) and4(c), the displacement is much more pronounced inwards (negative transverse displacements)then outwards.

The previous analysis is confirmed by the phase plots shown in Figure 5, where it can beverified that for smaller amplitudes of excitation the displacement is almost symmetric withrespect to the curved beam’s middle axis and that the phase plots deviate slightly from anellipse. However, as shown in Figure 6, for larger excitation amplitudes, the response is, forthe lower excitation frequency, visibly not harmonic, and for larger excitation frequencies, notat all periodic. Moreover, the displacements become quite larger in the negative than in thepositive vertical direction. The results indicate that unstable motion appears. It should be notedthat for very large amplitudes of vibration it will be necessary to verify the validity of theelastic constitutive law assumed here.



-0.05

-0.03

-0.01

0.01

0.03

0.05

0.94 0.96 0.98 1

-0.2

-0.1

0

0.1

0.2

0.94 0.96 0.98 1

-4

-3

-2

-1

0

1

0 0.02 0.04 0.06 0.08 0.1

t

w/h

w/h

w/h

t

t(c)

(a) (b)

Figure 3. Time histories at the middle of the beam, when excitation frequency � = �l1 and:(a) F = 5 × 102 N; (b) F = 2 × 103 N: and (c) F = 5 × 103 N.

-0.08-0.06-0.04-0.02

00.020.040.060.08

0.94 0.96 0.98 1

-4

-3

-2

-1

0

1

2

0 0.01 0.02 0.03

t

t t

w/h

w/h

w/h

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.54 0.56 0.58

(a) (b)

(c)

Figure 4. Time histories at the middle of the beam, when excitation frequency � = 1.1�l1 and:(a) F = 5 × 102 N; (b) F = 2 × 103 N and: (c) F = 5 × 103 N.

Figure 7 shows the time domain responses and the phase plots, for beams with differentcurvature ratios. The frequency of excitation is always the first linear natural frequency of thecurved beam under analysis, and the amplitude of excitation is 5000 N. The growth in the


2708 P. RIBEIRO

-2.5

-1.5

-0.5

0.5

1.5

2.5

-0.065 -0.015 0.0350.065

-12

-8

-4

0

4

8

12

-0.3 -0.1 0.1 0.3

w/h

w w

w/h(a) (b)

Figure 5. Phase plots at the middle of the beam; when F = 5 × 102 N (a): and F = 2 × 103 N (b).The different frequencies are represented as: −� = 0.9�l1, (◦) � = �l1 and (�)� = 1.1�l1.

-20

-10

0

10

20

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

-100

-5 0

0

50

100

-4 -3 -2 -1 0 1 2

-100

-50

0

50

100

-4 -2 0 2

w(b)

w(a)

w(c)

w

ww

Figure 6. Phase plots at the middle of the beam, when F = 5 × 103 N and:(a) � = 0.9�l1; (b) � = �l1; and (c) � = 1.1�l1.

curvature radius eventually causes the beam to buckle, explaining the highly asymmetric (withrespect to the x axis) dynamic behaviour of the deeper arch.

In Figures 8–11 the time domain responses and the phase plots are displayed for beams withdifferent thicknesses, when R = L/0.2 and L/0.4. The frequency of excitation is always thefirst linear natural frequency and the amplitude of excitation is 2000 N. Although the responseto the harmonic excitation is always periodic, the complexity of the time and phase plotsindicates that several harmonics are present in the motion of the thinner beam (L/h = 100).



-1.5

-1

-0.5

0

0.5

1

1.5

0.92 0.94 0.96 0.98 1

-3

-2

-1

0

1

0.42 0.44 0.46 0.48 0.5

-5

-4

-3

-2

-1

0

1

2

0.92 0.94 0.96 0.98 1

-4.5

-3.5

-2.5

-1.5

-0.5

0.5

1.5

0.00 0.02 0.04 0.06 0.08 0.10

w/h

w/h

w/h

w/h

w/h

tt

t t(a) (b)

(c) (d)

Figure 7. Time histories at the middle of the beam, when � = �l1, F = 5×103 N and: (a) L/R = 0.1;(b) L/R = 0.2; (c) L/R = 0.3; and (d) L/R = 0.4.

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.9 0.92 0.94 0.96 0.98 1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.9 0.92 0.94 0.96 0.98 1

-8

-6

-4

-2

0

2

4

0 0.05 0.1 0.15 0.2 0.25

w/h

w/h

w/h

t

tt(a) (b)

(c)

Figure 8. Time histories at the middle of a beam (b = 0.02, L = 1, R = L/0.2), when � = �l1,F = 2 × 103 N, and: (a) h = L/20; (b) h = L/50; and (c) h = L/100.


2710 P. RIBEIRO

-2.5

-1.5

-0.5

0.5

1.5

2.5

-0.03 -0.02 -0.01 0 0.01 0.02 0

-15

-10

-5

0

5

10

15

-0.6 -0.4 -0.2 0 0.2 0.4

-70

-50

-30

-10

10

30

50

-8 -6 -4 -2 0 2 4

w/h

w/hw/h

(a) (b)

(c)

w

ww

Figure 9. Phase plots of point at the middle of the beam (b = 0.02, L = 1, R = L/0.2), when � = �l1,F = 2 × 103 N, and: (a) h = L/20; (b) h = L/50; and (c) h = L/100.

In Figures 12, 13 one can see the time domain responses and the phase plots, for a beamexcited at a quarter of its length (x = L/4), with frequencies equal to the first three linearnatural frequencies. The beam geometric properties are b = 0.02, L = 1, h = L/50, R = L/0.2,the loss factor is � = 1.0 × 10−5, and the excitation amplitude is F = 5 × 103 N. The non-linear response to harmonic excitations with frequencies equal to the first and third linearnatural frequencies is not symmetric with respect to the vertical, z, axis, as occurs with thelinear mode. Moreover, the response to a force with a frequency equal to the second linearnatural frequency is not antisymmetic with respect to the y axis, as occurs with the linearmode.

4. CONCLUSIONS

A p-version, hierarchical finite element, for linear elastic, isotropic and curved beams was pre-sented. First order shear deformation, or Timoshenko’s, theory was followed and the geometricalnon-linearity due to finite deformations was considered. The accuracy and the reliability of theelement were demonstrated by comparing numerical results with published ones and with val-ues obtained using a commercial FEM package. The element proposed requires less degrees of



-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.075 0.08 0.085 0.09 0.095

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.499 0.504 0.509 0.514

-12

-10

-8

-6

-4

-2

0

2

4

0.9 0.92 0.94 0.96 0.98 1

w/h w/h

w/h

tt

t

(a) (b)

(c)

Figure 10. Time histories at the middle of a beam (b = 0.02, L = 1, R = L/0.4), when � = �l1,F = 2 × 103 N, and: (a) h = L/20; (b) h = L/50; and (c) h = L/100.

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-0.015 -0.005 0.005 0.015

-8

-6

-4

-2

0

2

4

6

8

-0.25 -0.15 -0.05 0.05 0.15

-100

-50

0

50

100

-12 -8 -4 0 4

w/h w/h

w/h

(a) (b)

(c)

w

ww

Figure 11. Phase plots of point at the middle of the beam (b = 0.02, L = 1, R = L/0.4), when� = �l1, F = 2 × 103 N, and: (a) h = L/20; (b) h = L/50; and (c) h = L/100.


2712 P. RIBEIRO

-1.5

-1

-0.5

0

0.5

0.95 0.97 0.99

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.965 0.975 0.985 0.995

-0.25

-0.15

-0.05

0.05

0.15

0.25

0.88 0.885 0.89 0.895 0.9

w/h

w/h

w/h

t

t t (a) (b)

(c)

Figure 12. Time histories at x = L/4 (−) and x = −L/4 (�): (a) � = �l1;(b) � = �l2; and (c) � = �l3.

-25

-15

-5

5

15

25

-1.5 -1 -0.5 0 0.5

-20

-15

-10

-5

0

5

10

15

20

-0.6 -0.4 -0.2 0 0.2 0.4

-15

-10

-5

0

5

10

15

-0.25 -0.15 -0.05 0.05 0.15

a

w/h

w/hw/h

(a) (b)

(c)

Figure 13. Phase plots at x = L/4 (−) and x = −L/4 (�): (a) � = �l1;(b) � = �l2; and (c) � = �l3.



freedom for accuracy than h-version finite elements. Moreover, it does not suffer from lockingphenomena, allowing one to study thick and thin beam structures.

The effects of the beams’ curvatures, thickness and vibration amplitude on the dynamicsresponse were analysed for frequencies close to the fundamental frequency. Both non-harmonicand non-periodic responses to harmonic excitations were found, particularly for thinner or morecurved beams. Assymmetric (with respect to the longitudinal axis) and unstable motions dueto buckling like phenomena were also observed.

Non-linear responses around the first and higher natural frequencies were investigated ap-plying a force at a quarter of the beam. Unlike the natural modes of vibration, the responsearound the second natural frequency was far from antisymmetric, and the response around thefirst and third mode was not symmetric.

APPENDIX A: STRAIN–DISPLACEMENT RELATIONS

{�p0 } = {Nu,x}T{qu} (A1)

{�bo} = {N�,x}T{q�} (A2)

{�pL} = 12 {qw}T{Nw

,x}{Nw,x}T{qw} (A3)

(�I ) = 1

R{Nw}T(qw) (A4)

{�zx} = [{Nw,x}T {N�}T] {qw

q�

}(A5)

APPENDIX B: STIFFNESS AND MASS MATRICES

The longitudinal stiffness matrix [Kp1 ] is equal to the straight beam one, given in Refer-

ences [13, 14]. The bending matrix [Kb1 ], and the matrices due to the beam’s initial curvature([Kcp

1 ], [Kpc1 ] and [Kcc

1 ]) are

[Kb1 ] = bD

∫L

{N�,x}{N�

,x}T dx (B1)

[Kpc1 ] = bA

R

∫L

{Nu,x}{Nw}T dx (B2)

[Kcp1 ] = [Kpc

1 ]T (B3)

[Kcc1 ] = bA

R2

∫L

{Nw}{Nw}T dx (B4)


2714 P. RIBEIRO

The linear shear stiffness matrix [K�1] is

[K�1] = bGh

∫L

{Nw

,x}{Nw,x}T {Nw

,x}{N�}T

{N�}{Nw,x}T {N�}{N�}T

dx (B5)

where the shear modulus is G = E/2(1 + �).There are three non-linear stiffness matrices equal to the thin straight beam ones, which are

given in Reference [14]. These are matrix [K2{qw}], which depends linearly on the generalizedtransverse displacements {qw}, matrix [K4{qw}], which is a quadratic function of {qw}, andmatrix [K3], which is equal to 2[K2]T.

Two non-linear stiffness matrices that do not exist in straight beams also appear. These arematrices [Kc

2 ] and [Kc3 ]. [Kc

2 ] is given by

[Kc2 ] = bA

2R

∫L

{Nw}{Nw,x}Tw0

,x dL (B6)

and matrix [Kc3 ] is equal to 2[Kc

2 ]T.The consistent mass matrix is

[M] = �h

b

∫L

{Nu}{Nu}T dx 0 0

0 b

∫L

{Nw}{Nw}T dx 0

0 0 b h2

12

∫�{N�}{N�}T dx

(B7)

and may be written as

[M] =

[Mp] 0 0

0 [Mb] 0

0 0 [MR]

(B8)

[Mp] and [Mb] are the longitudinal and transverse inertia matrices, and [MR] is due to therotatory inertia.

ACKNOWLEDGEMENTS

The support of this work by the Portuguese Science and Technology Foundation and FEDER, underproject POCTI/1999/EME/32641, is gratefully acknowledged.

REFERENCES

1. Petyt M, Fleischer CC. Free vibration of a curved beam. Journal of Sound and Vibration 1971; 18:17–30.2. Lee SS, Koo JS, Choi JM. Development of a new curved beam element with shear effect. Engineering

Computations 1996; 13:9–25.



3. Eisenberger M, Efraim E. In-plane vibrations of shear deformable curved beams. International Journal forNumerical Methods in Engineering 2001; 52:1221–1234.

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5. Raveendranath P, Singh G, Rao GV. A three-noded shear-flexible curved beam element based on coupleddisplacement field interpolations. International Journal for Numerical Methods in Engineering 2001; 51:85–101.

6. Bi Q, Dai HH. Analysis of non-linear dynamics and bifurcations of a shallow arch subjected to periodicexcitation with internal resonance. Journal of Sound and Vibration 2000; 233:557–571.

7. Chen SH, Cheung YK, Xing HX. Nonlinear vibration of plane structures by finite element and incrementalharmonic balance method. Nonlinear Dynamics 2001; 26:87–104.

8. Luczko J. Bifurcations and internal resonances in space-curved rods. Computer Methods in Applied Mechanicsand Engineering 2002; 191:3271–3296.

9. Petrov E, Géradin M. Finite element theory for curved and twisted beams based on exact solutions forthree-dimensional solids. Part 1: beam concept and geometrically exact nonlinear formulation. ComputerMethods in Applied Mechanics and Engineering 1998; 165:43–92.

10. Meirovitch L, Baruh H. On the inclusion principle for the hierarchical finite element method. InternationalJournal for Numerical Methods in Engineering 1983; 19:281–291.

11. Bardell NS. The application of symbolic computing to the hierarchical finite element method. InternationalJournal for Numerical Methods in Engineering 1989; 28:1181–1204.

12. Han W, Petyt M. Geometrically nonlinear vibration analysis of thin, rectangular plates using the hierarchicalfinite element method—I: the fundamental mode of isotropic plates. Computers and Structures 1997; 63:295–308.

13. Ribeiro P, Petyt M. Non-linear vibration of beams with internal resonance by the hierarchical finite elementmethod. Journal of Sound and Vibration 1999; 224:591–624.

14. Ribeiro P. The second harmonic and the validity of Duffing’s equation for vibration of beams with largedisplacements. Computers and Structures 2001; 79:107–117.

15. Szabó BA, Babuska I. Finite Element Analysis. John Wiley and Sons: New York, 1991.16. Timoshenko SP. On the transverse vibrations of bars of uniform cross section. Philosophy Magazine 1922;

43:125–131.17. Narita Y, Leissa A. Vibrations of completely free shallow shells of curvilinear planform. Journal of Applied

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