Linear and nonlinear vibration of non-uniform beams on two-parameter foundations using p-elements

8/9/2019 Linear and nonlinear vibration of non-uniform beams on two-parameter foundations using p-elements

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Linear and nonlinear vibration of non-uniform beams on two-parameter

foundations using p-elements

B. Zhu a,b,*, A.Y.T. Leung b

a Department of Civil Engineering, Zhejiang University, 388#, Yuhangtang Road, Hangzhou, Zhejiang 310058, Chinab Department of Building and Construction, City University of Hong Kong, China

a r t i c l e i n f o

Article history:Received 28 August 2008

Received in revised form 13 December 2008

Accepted 15 December 2008

Available online 20 January 2009

Keywords:

Beams

Foundations

Vibration

Nonlinear

p-Element

a b s t r a c t

A hierarchical finite element is presented for the geometrically nonlinear free and forced vibration of anon-uniform Timoshenko beam resting on a two-parameter foundation. Legendre orthogonal polynomi-

als are used as enriching shape functions to avoid the shear-locking problem. With the enriching degrees

of freedom, the accuracy of the computed results and the computational efficiency are greatly improved.

The arc-length iterative method is used to solve the nonlinear eigenvalue equation. The computed

results of linear and nonlinear vibration analyses show that the convergence of the proposed element

is very fast with respect to the number of Legendre orthogonal polynomials used. Since the elastic foun-

dation and the axial load applied at both ends of the beam affect the ratios of linear frequencies asso-

ciated with the internal resonance, they influence the nonlinear vibration characteristics of the beam.

The axial tensile stress of the beam in nonlinear vibration is investigated in this paper, and attention

should be paid to the geometrically nonlinear vibration resulting in considerably large axial tensile

stress in the beam.

Crown Copyright 2008 Published by Elsevier Ltd. All rights reserved.

1. Introduction

The dynamic infinite element[1,2], the exact high-order accu-

rate artificial boundaries [3,4] and some other numerical tech-

niques simulating the infinite media were well developed for the

vibration analysis of soilstructure interaction. Some soilstruc-

ture interaction problems such as the pile foundation and the min-

ing panel can also be approximately modeled by means of a beam

resting on an elastic foundation [57]. Nonlinear vibration of these

structures will be induced when subjected to dynamic forces of

earthquake, machine oscillation, etc. Compared with those meth-

ods dealing with the infinite media of the soilstructure interac-

tion, the use of two-parameter foundations is a more rough

approximation of representing the infinite medium on which the

beam is rested But the finite element method (FEM) based on the

two-parameter foundation approximation can be easily used to

analyze the soil-beam interaction problems, and this method has

also become one of the most popular methods for these problems

[810]. If the beam is with immovable supports, the equation of

transverse vibration is nonlinear due to the significant influence

of axial force, and the nonlinear vibration characteristics of the

beam are much different from the linear ones [11,12].

Since the nonlinear stiffness matrices are required to be recon-

structed during iterations, the computational complexity increases

considerably with the increase of the number of degrees of free-

dom (DOFs). Therefore, using less number of DOFs for the same

accuracy is very desirable for a nonlinear vibration analysis. In

general, the p-version elements converge more rapidly than the

h-version elements in a finite element analysis [1114]. One whole

non-uniform beam can be modeled by just one element while sat-

isfying the accuracy requirement using the p-version element.

Both the properties make the p-version elements more popular

for the nonlinear vibration analyses of beams [11,12,15,16]. So

far, the studies of nonlinear vibration analyses are limited to the

uniform beams, most of which are based on the EulerBernoulli

beam theory.

In this paper a hierarchical finite element (HFE) for nonlinear

free and forced vibration analyses of non-uniform Timoshenko

beams resting on two-parameter foundations is presented. An ini-

tial axial load is applied at both ends of the beam. The element

takes into account the effects of transverse shear deformation

and rotatory inertia. The nonlinear eigenvalue equation of the

beam is obtained by applying the harmonic balance method

(HBM) and solved by the arc-length iterative method. Convergence

of the present hierarchical element for linear and nonlinear vibra-

tion is studied and the effects of the foundation parameters, the

initial axial load and the variation of the section of the beam to

the nonlinear free vibration are investigated.

0266-352X/$ - see front matter Crown Copyright 2008 Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.compgeo.2008.12.006

* Corresponding author. Address: Department of Civil Engineering, Zhejiang

University, 388#, Yuhangtang Road, Hangzhou, Zhejiang 310058, China.

E-mail address:[email protected](B. Zhu).

Computers and Geotechnics 36 (2009) 743750

Contents lists available at ScienceDirect

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2. Formulation

2.1. Finite element formulation

A p-version element of a non-uniform Timoshenko beam rest-

ing on a two-parameter elastic foundation is depicted in Fig. 1.

The elastic foundation model is characterized by the Winkler foun-

dation modulus k and the shear foundation modulus kG. For the

sake of simplicity, it is assumed that the cross-section of the beam

is rectangular, the beams width and depth, the foundation param-

etersk and kG are functions ofx given by

bx Xi0

bixi; tx

Xi0

tixi; kx

Xi0

kixi;

kGx Xi0

kGixi 1

It is noted that there are additional internal degrees of freedom

(DOFs) yi and hi (3 6 i 6 p 2) as shown inFig. 1. The C0 shapefunctionsN for transverse displacementw and rotation h are

Nin fin i 1; 2;. . . 2

where n=x/l. The first two terms are traditionally used for C 0 2-

node linear element and the additional enriching shape functions

using Legendre orthogonal polynomials fi(n) (36 i 6 p 2) lead tozero displacements at both nodes[7]. The geometrically nonlinear

axial strain, the flexural strain and the shear strain are given by

eou

ox

1

2

ow

ox

23a

joh

ox 3b

/ owox

h 3c

whereu is the axial displacement,w is the transverse displacement

andh is the rotation. Neglecting the longitudinal inertia forces and

the longitudinal displacements, one can interpolatew andh as

w

h

N 0

0 N

qw

qh

Nw

Nh

q 4

whereq is the vector of generalized degrees of freedom; qw andqhare vectors of generalized transverse displacements and rotations,

respectively. Then the flexural strain and the shear strain in Eq.

(3)are given by

joh

ox

oNh

ox q Bhq 5a

/ow

ox h

oNw

ox Nh

q Bw Nhq Bsq 5b

Taking into account the effects of the transverse shear deforma-

tion, the elastic foundation and the rotatory inertia, the Hamilton

principle yields the nonlinear motion equation

x2Ml Kl K4 0

0 0 q Q 6

whereQis the vector of generalized external excitation force. The

linear stiffness and mass matrices for transverse vibration Kl and

Ml are given by[17]

Kl

Z 10

l BhTEIBh Bs

Tk

0GABs Nw

TkNw

h Bw

TkGPBwi

dnKw Kwh

Khw Kh

7a

Ml

Z 10

l NwTqANw NhTqINhdn diagMw;Mh 7b

whereKhw KTwh;Eis the Youngs modulus;q is the mass density;A(x) is the cross-sectional area; I(x) is the second moment of area; k0

is the shear correction factor; G is the shear modulus; and Pis the

initial axial force applied at both ends of the beam.K4is a quadratic

function ofqw and can be expressed as[15,16]

K41

2

Z 10

ElAxN;x

TN

;xqwqTwN;x

TN

;x

h idn 8

where N,x is the first-order differential of N with respect to x. It

should be noted that the linear matrices in Eq. (7) and the nonlinear

matrix in Eq.(8)can be obtained with analytic integrations.

2.2. Arc-length iteration for free and forced nonlinear vibration

The external excitation force considered per unit length is given

by Q0(x,t), and then the vector of generalized external force has theform

Nomenclature

yi degrees of freedom of transverse displacementshi degrees of freedom of rotationsNi element shape functionsP number of enriching shape functionsE Youngs modulus

G shear modulusH density of the beamk0 shear correction factorm Poissons ratiob width of the beamt depth of the beamL length of the beamA cross-sectional areaI second moment of area

rg radius of gyrationx natural frequencyc frequency parameterPr non-dimensional parameter of initial axial loadk non-dimensional parameter of Winkler foundation

moduluskG non-dimensional parameter of shear foundation modu-

lusqw vector of generalized transverse displacementsqh vector of generalized rotationsq vector of generalized total DOFsKl; Ml linear stiffness and mass matricesK4 nonlinear stiffness matrixQ vector of generalized external excitation force

1y1

2y2y i

l

k, kG

y

P P xi

Fig. 1. A p-version element of non-uniform Timoshenko beam resting on elasticfoundation.

744 B. Zhu, A.Y.T. Leung/ Computers and Geotechnics 36 (2009) 743750


3/8


4/8

The expressions of the nonlinear matrix eK4 can be found in Ref.[18]. In this work arc-length iterative method is used to solve the

nonlinear eigenvalue problem[16,19]. The first point in backbone

curve and the frequency response function (FRF) curve is obtained

by the Newton method. Then, the subsequent points are obtained

by the arc-length method. When the residual force vector F is suf-

ficiently small, the solutions of the next equilibrium statem+1qandm+1x2 is obtained. During each increment, the following constrainequation controls each iteration:

q mqT

q mq DqTDq s2 13

where Dq is the total increment ofq. Inserting the obtained solu-tions ofq into Eq. (4), one finds the transverse displacement of a

certain harmonic

wi Nqwci 14

3. Numerical results

In this paper, uniform and non-uniform Timoshenko beams are

analyzed. The slenderness ratio of the beam is L/rg, in which Lis the

length of the beam and rgffiffiffiffiffiffiffiffiffiffiffiffi

I0=A0p

is the radius of gyration with

A0= b0t0andI0 b0t30=12. The following non-dimensional parame-ters are defined: c x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqA0L

2=EI0q

, Pr= PL2/(EI0), k kL4=EI0;

kG

kGL2=EI0

.

3.1. Linear vibration analyses

With only onep-element, the computed first three natural fre-

quencies of a uniform hingedhinged beam fully supported on a

constant two-parameter foundation are listed in Table 1. The fol-

lowing parameters are used in the computation: L= 25 m, G/

E= 3/8,m = 1/3,k0

= 2/3,b0 1=ffiffiffiffiffiffi

12p

andt0ffiffiffiffiffiffi

12p

. The results are

Table 4

Results of nonlinear free vibration for a thin hingedhinged beamk0; kG0; Pr0; L=rg1000.

Exact Ref.[19] Ref.[18] Present

a/rg x/xl x/xl x/xl One harmonic Two harmonics Three harmonics

a/rg x/xl a/rg x/xl a/rg x/xl

1.0000 1.0892 1.0897 1.0865 1.0034 1.2711 0.9973 1.0977 1.0006 1.0983

2.0000 1.3177 1.3229 1.3331 1.9989 1.8160 1.9991 1.3354 1.9990 1.3353

3.0000 1.6256 1.6400 1.6422 3.0093 2.4382 2.9991 1.6359 3.0210 1.6431

1 2 3 4 5 6 7 80.0

0.2

0.4

0.6

0.8

1.0

p=6

p=7

p=8

L/rg=100

k'=2/3

=0.25

=0.6 4

G= 2

Pr=0.6 2

|w1

+w3

+w5

|/r

g

/ l1

Fig. 4. Convergence study for a uniform hingedhinged beam.

1.00 1.04 1.08 1.12 1.16 1.20-3

-2

-1

0

1

2

3

4

=0.006 4, G=0.01

2

=0.060 4, G=0.10

2

=0.120 4, G=0.20

2

=0.180 4, G=0.30

2

Pr=0.006

2

L/rG=100

A

mplituderatiow1

/r

g

Frequency ratio / l1

1.00 1.04 1.08 1.12 1.16 1.20-0.75

-0.50

-0.25

0.00

0.25

=0.0064,

G=0.01

2

=0.0604,

G=0.10

2

=0.1204,

G=0.20

2

=0.1804,

G=0.30

2

Pr=0.006

2

L/rG=100

Amplitudera

tiow3

/r

g


1.00 1.04 1.08 1.12 1.16 1.200.0

0.5

1.0

1.5

2.0

2.5

3.0

=0.0064,

G=0.01

2

=0.0604

, G=0.102

=0.1204,

G=0.20

2

=0.1804,

G=0.30

2

Pr=0.006

2

L/rG=100

Amplituderatio|w1+w3+w5|/r

g


Fig. 5. Backbone curves of a uniform clampedhinged beam.



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compared with the existing solutions in Refs. [2022]. Ref. [22]

introduced the Hermite C1 shape functions to interpolate the trans-

verse displacement and the rotation, respectively. It can be ob-

served that the convergence of the proposed element is the

fastest. The present solutions with 18 DOFs are in excellent agree-

ment with the exact solutions.

The second example is a symmetric rectangular cross-sectional

beam with linearly varying width and depth (seeFig. 2). Both ends

of the beams are hinged. Data in Refs. [22,23]are adopted here asfollows, L = 100 m, E= 3 106 N/m2, m = 0.3,k 0 = 0.85,q = 1 kg/m3.

The beam is analyzed by twop-version beam elements. For the left

element, b0= 12, b1= 3/25, t0= 10 and t1= 1/10. Comparison be-

tween the solutions with two p-elements and those in literatures

[22,23]is carried out inTable 2. One can observe that the present

element converges more rapidly than that given in Ref. [22].

Unless otherwise stated in the rest of this work, the computa-

tional parameters are: Youngs modulus E= 2.1295 1011 N/m2,the shear correction factor k0= 2/3, Poissons ratio m= 0.25, themass densityq= 7829 kg/m3, the length of beamL= 1 m, the axialload Pr= 0.6p

2, foundation parameters k0 0:6p4 and kG0 p2,number of additional hierarchical shape functionsp = 8. The linear

and nonlinear vibration is analyzed using only one p-element with

three harmonics.

Three square cross-sectional hingedhinged beams resting on

different foundations are analyzed. An axial load Pr= 0.6p2 is ap-

plied at both ends of the beam. The width and depth of the first

beam are constant. For the second beam, the width and depth

are linearly varying, andb1= t1= b0,k1 k0,kG1 kG0. For the thirdbeam, the width and depth are quadratic functions of x, and

b1= t1= 0,b2= t2= b0, k1

kG1

0; k2

k0, kG2

kG0. The common

parameters b0 ffiffiffiffiffiffi12p =10,t0 ffiffiffiffiffiffi12p =10 are used in the analyses inall three cases. With the number of additional hierarchical shape

functionsp= 8, the computed results of these three beams resting

on different foundations are listed in Table 3. For the uniform

beam, the computed first three frequencies by the present element

with 18 DOFs are more accurate than those of Ref. [17] with 32

DOFs. A finite element considering the coupling relationship be-

tween the transverse displacement and the rotation is used in

1.00 1.04 1.08 1.12 1.16 1.20-3

-2

-1

0

1

2

3

4

Pr=0.006 2

Pr=0.012 2

Pr=0.024 2

Pr=0.036 2

=0.006 4, G=0.01 2

L/rG=100

Amplitude

ratiow1

/r

g


1.00 1.04 1.08 1.12 1.16 1.200.0

0.2

0.4

0.6

0.8

1.0

Pr=0.006 2

Pr=0.012 2

Pr=0.024 2

Pr=0.036 2

=0.0064,

G=0.01

2

L/rG=100

Amplituderatiow

5

/r

g


1.00 1.04 1.08 1.12 1.16 1.20

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Pr=0.006 2

Pr=0.012 2

Pr=0.024 2

Pr=0.036 2

=0.006 4, G=0.01 2

L/rG=100

Amplituderatio|w1+w3+w5|/r

g


Fig. 6. Backbone curves of a uniform clampedclamped beam.

c

b0

b0t0

t0

b

5b0

5t0

2t0

t0b0

t0

a

2b0

b0L

Fig. 7. A uniform beam and two non-uniform beams.

1.0 0 1.05 1.1 0 1.15 1.2 0 1.25 1.300

1

2

3

4

Uniform beam

Non-uniform beam 1

Non-uniform beam 2

L/rg=100

k'=2/3

=0.25

0=0.6 4

G0

= 2

Pr=0.6 2

|w1

+w3

+w5

|/rg

/l1

Fig. 8. Backbone curves of three beams.

B. Zhu, A.Y.T. Leung/ Computers and Geotechnics 36 (2009) 743750 747


6/8

Ref.[17]. The convergence of this element is fast, but the assump-

tion that the shear strain over the element is constant is not suit-

able for non-uniform beams.

The axial load reduces the stiffness and also the natural fre-

quencies of the Timoshenko beam resting on elastic foundation.

Fig. 3plots the frequency parametercof a uniform hingedhinged

beam with slenderness ratio L/rg= 100 as a function of the axial

load parameterPr

. The critical axial loads

Prmake the frequencies

of the beam vanish.

3.2. Nonlinear vibration analyses

A thin uniform EulerBernoulli beam with hingedhinged ends

and the parameters k 0; kG 0,Pr= 0,L/rg= 1000 is analyzed byone p-element with one, two and three odd harmonics, respec-

tively. In Table 4, a comparison of the present results with the exact

solutions and those previously publishedin Refs. [18,19] is given. In

the table, xl is the fundamental linear natural frequency;a= |P

wi|/

rgwith wi are the vibration amplitudes of the first, third or fifth har-

monics in the middle of the beam. The results in Ref. [19]are ob-

tained by one hierarchical element and in Ref. [18]are found by

six finite elements. Good agreement between the present solutions

with three harmonics and the existing results is observed.

To study the convergence of the present hierarchical element

for nonlinear vibration analyses, the free vibration of a uniform

hingedhinged beam withL/rg= 100 resting on an elastic founda-

tion withk 0:6p4, kG p2 is analyzed using one p-element withdifferent numbers of hierarchical terms. An axial loadPr= 0.6p

2 is

applied at both ends of the beam. The backbone curves describing

the relation between amplitude of vibration and the frequency are

plotted inFig. 4. The parameterxl1 is the fundamental linear nat-ural frequency, and wi represents the amplitudes of vibration dis-

placements at x

= 0.25L

. Hardening spring effect is observed in

the plot. For the first three nonlinear modes, the solutions of the

present element withp = 8 can achieve excellent convergence.

An internal resonance of order three exists as the ratio of the

second and the first linear natural frequencies is about three for

the clampedhinged beam, and a fifth order internal resonance is

seen in the clampedclamped beam[18]. If the first frequency in-

creases, a characteristic loop will be observed on the backbone

curves. To study the effects of the elastic foundations and the axial

loads on the nonlinear free vibration of the Timoshenko beam, the

response curves around the fundamental linear frequency of a

clampedhinged beam and a clampedclamped beam with differ-

ent elastic foundations and axial loads are shown in Figs. 5 and 6,

respectively. In the plotswi represents the amplitudes of vibration

displacements at x = 0.5L. Since the ratio of the second linear fre-

quency to the fundamental is decreased with an increase of elastic

foundation parameters, the characteristic loop on the backbone

curve is closer to the fundamental linear frequency if the parame-

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

/l1=1.0000

/l1=1.1003

/l1=1.2060

/l1=1.3020

L/rg=100

k'=2/3

=0.25

0=0.64

G0

= 2

Pr=0.62

w1

/w1max

x/L

0.0 0.2 0.4 0.6 0.8 1.0-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

/l1=1.0002

/l1=1.1003

/l1=1.2060

/l1=1.3020

L/rg=100

k'=2/3

=0.25

0=0.64

G0

= 2

Pr=0.6 2

w3

/w3max

x/L

Fig. 9. Mode shapes of a non-uniform hingedhinged beam.

0.95 1.00 1.05 1.100.0

0.2

0.4

0.6

0.8

FRF curve, Q0=289 N/m

2


2


2


2

Backbone curve

L/rg=100

k'=2/3

=0.25

0=0.6

4

G0

=2

Pr=0.6

2

w1

/r

g

/ l1

0.975 1.000 1.025 1.0500.000

0.001

0.002

0.003

0.004


2


2


2


2

Backbone curve

L/rg=100

k'=2/3

=0.25

0=0.6 4

G0

= 2

Pr=0.6 2

w3

/r

g

/ l1

Fig. 10. Frequency response function curves of a non-uniform hingedhingedbeam.



7/8

ters of the elastic foundations are increased. The ratio of the second

linear frequency to the fundamental is less than 3.0 for the case

with k 0:180p4, kG 0:30p2 and the characteristic loop on thebackbone curve vanishes. On the other hand, the ratioxl2/xl1is in-creased and the characteristic loops move toward the right with

increasing axial loads applied at both ends of the clamped

clamped beam.

A uniform and two non-uniform beams with linearly varying

width and depth are shown in Fig. 7. Their slenderness ratio L/

rg= 100. The ends of the beams are hingedhinged. The backbone

curves of the first mode in the middle of beams are presented in

Fig. 8. InFig. 9a and b the mode shapes of the first harmonic and

the third harmonic associated with different maximum amplitudes

of vibration displacements are shown, respectively. The effect of

the third harmonic is larger than that of the first, and the mode

shapes deviate a little to the end of the beam with smaller cross-

sectional area. For the non-uniform beam 1 subjected to different

uniform transverse loads, the frequency response function (FRF)

curves around the first mode are shown inFig. 10. It can be seen

that the response of the third harmonic is very small for this

hingedhinged beam resting on the elastic foundations.

As shown inFig. 11, a pile with an axial loadPapplied at its top

and tip is embedded in three layers of soil. The pile properties are:

Youngs modulus E= 3 1010 N/m2, the shear correction factork0= 0.85, Possions ratio m = 0.20, the mass density q= 2500 kg/m3, the diameter of cross-section D = 0.6 m. The axial load param-

eter Pr= 0.6p2. The foundation parameters of the three layers of

soil kL1 6:0p4; kGL1 0:1p2; kL2 1:2p4; kGL2 0:02p2; kL3 300p

4; kGL3 50p2. With a bearing platform on the pile top and thehard soil surrounding the pile tip, the pile top and tip can be as-

sumed as fixed and immovable, respectively.

The backbone curves around the first modes of the pile with

clampedclamped and clampedhinged ends are plotted in

Fig. 12. There is little difference between the backbone curves of

the pile with different boundary conditions. The reason is that

the hard soil surrounding the pile tip constrains its rotation. With

varying parameters of the second layer of soil and other parame-ters fixed, the values of the axial stress Nt P=A around the firstmode of the clampedhinged pile are shown inFig. 13, where Ntis the average axial force arising from the geometric nonlinear

vibration and is defined in[18]as

Nt 1

2

Z 10

EAXi

qTwiN0xNT0x

Xi

qwidn 15

Please note that there is a characteristic loop on the curve for

the case with the smallest foundation parameters kL2 0:6p4,

kL2, kGL2

y

P

x

P

kL1, kGL1

kL3, kGL3

3m

14.5m

2.5

m

L

Fig. 11. A pile embedded in three layers of soil.

30 35 40 45 500

1

2

3

4

l1l1

Clamped-Hinged Beam

Clamped-Clamped Beam

w1

/r

g

Natural Frequency (rad/s)

30 35 40 45 50-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

Clamped-Hinged Beam

Clamped-Clamped Beam

w3

/r

g

Natural Frequency (rad/s)

Fig. 12. Comparison of backbone curves of the pile with different boundary

conditions.

30 35 40 45 50 55 60-10

-5

0

5

10

L2

=0.6 4; GL2

=0.01 2

L2

=1.2 4; GL2

=0.02 2

L2

=2.5 4; GL2

=0.05 2

L2

=6.0 4; GL2

=0.12 2

Axialstress(MPa)

Natural Frequency(rad/s)

Fig. 13. Axial stress of the clampedhinged pile.

B. Zhu, A.Y.T. Leung/ Computers and Geotechnics 36 (2009) 743750 749


8/8

kGL2 0:01p2. The increasing stress is dangerous if the pile was notdesigned for resisting the axial tensile stress. For the clamped

hinged pile subjected to different uniform transverse loads, the

FRF curves of the first harmonic around the first mode shown in

Fig. 14is indicated by different line types and the backbone curve

by a dash line. The response of all free and forced vibration is cal-

culated atx = 0.5L.

4. Conclusions

In this paper, a hierarchical finite element is presented for geo-

metrically nonlinear free and forced vibration of the non-uniform

beam resting on two-parameter foundations. For linear vibration

analyses, only one present p-element can predict very accurate

solutions of both uniform and non-uniform beams. Compared withthe traditionalh-version elements, its convergence is much faster

with respect to the number of Legendre orthogonal polynomials

used. Nonlinear vibration of non-uniform beams resting on two-

parameter foundations can be analyzed by just one proposedp-ele-

ment with different harmonics without difficulty, and excellent

convergence and accuracy are also observed in the analyses.

The elastic foundation parameters can heavily influence both of

linear frequencies and nonlinear vibration characteristics of the

beam. Due to the nonlinear vibration, large axial tensile stress is in-

duced in the beam resting on the two-parameter foundation and

subjected to transverse loadings, which should be paid attention

to in the engineering.

Acknowledgement

The research is supported by the National Natural Science Foun-

dation of China (Grant No. 50608062, 50809060 and 50708095).

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0.90 0.95 1.00 1.05 1.100.0

0.4

0.8

1.2

1.6

2.0

FRF curve, Q0=100 N/m2



Backbone curve

w1/

rg

/l1

Fig. 14. FRF curves of the clampedhinged pile.


Linear and nonlinear vibration of non-uniform beams on two-parameter foundations using p-elements

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