The dynamics of a semi-infinite Euler–Bernoulli beam on unilateral elastic springs is investigated. The mechanical model is governed by a moving-boundary hyperbolic problem, which cannot be solved in closed form. Therefore, we look for approximated solutions following two different approaches. From the one side, approximate analytical solutions are obtained by means of perturbation techniques. On the other side, numerical solutions are determined by a self-made finite element algorithm. The analytical and numerical solutions are compared with each other, and the effects of the problem nonlinearity on the beam motion are analyzed. In particular, the superharmonics oscillations and the resonances are investigated in depth.

Issue Section:
Research Papers
Keywords:
beams (structures), elasticity, springs (mechanical), hyperbolic equations, perturbation techniques, finite element analysis, oscillations, dynamics, soil, beams on tensionless foundations, finite elements in an infinite domain, asymptotic developments, cutoff frequencies
Topics:
Displacement, Oscillations, Resonance, Simulation, Soil, Springs, Finite element analysis, Boundary-value problems
1.
Tsai
,
N. C.
, and
Westmann
,
R. E.
, 1967, “
Beams of Tensionless Foundation
,”
J. Engrg. Mech. Div.
0044-7951,
93
, pp.
1
–
12
.
2.
Ascione
,
L.
, and
Grimaldi
,
A.
, 1984, “
Unilateral Contact Between a Plate and an Elastic Foundation
,”
Meccanica
0025-6455,
19
, pp.
223
–
233
.
Crossref
Search ADS
 
3.
Zhang
,
Y.
, and
Murphy
,
K. D.
, 2004, “
Response of a Finite Beam in Contact with a Tensionless Foundation under Symmetric and Asymmetric Loading
,”
Int. J. Solids Struct.
0020-7683,
41
(
24–25
), pp.
6745
–
6758
.
4.
de Holanda
,
A. S.
, and
Gonçalves
,
P. B.
, 2003, “
Postbuckling Analysis of Plates Resting on a Tensionless Elastic Foundation
,”
J. Eng. Mech.
0733-9399,
129
(
4
), pp.
438
–
448
.
Crossref
Search ADS
 
5.
Lin
,
L.
, and
Adams
,
G. G.
, 1987, “
Beam on Tensionless Elastic Foundation
,”
J. Eng. Mech.
0733-9399,
113
, pp.
542
–
553
.
Crossref
Search ADS
 
6.
Güler
,
K.
, 2004, “
Circular Elastic Plate Resting on Tensionless Pasternak Foundation
,”
J. Eng. Mech.
0733-9399,
130
(
10
), pp.
1251
–
1254
.
Crossref
Search ADS
 
7.
Khathlan1
,
A. A.
, 1994, “
Large-Deformation Analysis of Plates on Unilateral Elastic Foundation
,”
J. Eng. Mech.
0733-9399,
120
(
8
), pp.
1820
–
1827
.
Crossref
Search ADS
 
8.
Ioakimidis
,
N. I.
, 1988, “
Beams on Tensionless Elastic Foundation: Approximate Quantifier Elimination with Chebyshev Series
,”
Int. J. Numer. Methods Eng.
0029-5981,
39
(
4
), pp.
663
–
686
.
Crossref
Search ADS
 
9.
Silva
,
A. R. D.
,
Silveira
,
R. A. M.
, and
Gonçalves
,
P. B.
, 2001, “
Numerical Methods for Analysis of Plates on Tensionless Elastic Foundations
,”
Int. J. Solids Struct.
0020-7683,
38
, pp.
2083
–
2100
.
Crossref
Search ADS
 
10.
Weitsman
,
Y.
, 1970, “
On Foundation that Reacts in Compression Only
,”
ASME J. Appl. Mech.
0021-8936,
37
(
4
), pp.
1019
–
1030
.
Crossref
Search ADS
 
11.
Celep
,
Z.
,
Malaika
,
A.
, and
Abu-Hussein
,
M.
, 1989, “
Forced Vibrations of a Beam on a Tensionless Foundation
,”
J. Sound Vib.
0022-460X,
128
(
2
), pp.
235
–
246
.
Crossref
Search ADS
 
12.
Choros
,
J.
, and
Adams
,
G. G.
, 1979, “
A Steadily Moving Load on an Elastic Beam Resting on a Tensionless Winkler Foundation
,”
ASME J. Appl. Mech.
0021-8936,
46
(
1
), pp.
175
–
180
.
Crossref
Search ADS
 
13.
Cokşun
,
I.
, 2003, “
The Response of a Finite Beam on a Tensionless Pasternak Foundation Subjected to a Harmonic Load
,”
Eur. J. Mech. A/Solids
0997-7538,
22
(
1
), pp.
151
–
161
.
14.
Cokşun
,
I.
, 2000, “
Non-Linear Vibrations of a Beam Resting of a Tensionless Winkler Foundation
,”
J. Sound Vib.
0022-460X,
236
(
3
), pp.
401
–
411
.
15.
Celep
,
Z.
, 1990, “
In-Plane Vibrations of Circular Rings on a Tensionless Foundation
,”
J. Sound Vib.
0022-460X,
143
(
3
), pp.
461
–
471
.
Crossref
Search ADS
 
16.
Nayfeh
,
A. H.
, 1981,
Introduction to Perturbation Methods
,
Wiley
, New York.
17.
Kevorkian
,
J.
, and
Cole
,
J. D.
, 1981,
Perturbation Methods in Applied Mathematics
,
Springer
, Berlin.
18.
Callegari
,
M.
,
Lancioni
,
G.
, and
Lenci
,
S.
, 2005, “
A Numerical Model for the Nonlinear Dynamics of a Semi-infinite Beam on Unilateral Winkler Soil
,”
Proceedings AIMETA05
, Firenze, September 11–15 (CD-ROM).
19.
Andersen
,
L.
,
Nielsen
,
S. R. K.
, and
Kirkegaard
,
P. H.
, 2001, “
Finite Element Modelling of Infinite Euler Beams on Kelvin Foundation Exposed to Moving Loads in Convected Coordinates
,”
J. Sound Vib.
0022-460X,
241
(
4
), pp.
587
–
604
.
Crossref
Search ADS
 
20.
Liu
,
T.
, and
Li
,
Q.
, 2003, “
Transient Elastic Wave Propagation in an Infinite Timoshenko Beam on Viscoelastic Foundation
,”
Int. J. Solids Struct.
0020-7683,
40
(
2
), pp.
3211
–
3228
.
21.
Graff
,
K. F.
, 1975,
Wave Motion in Elastic Solids
,
Oxford University Press
, London, UK.
22.
Achenbach
,
J. D.
, 1973,
Wave Propagation in Elastic Solids
,
North–Holland
, Amsterdam, The Netherlands.
23.
McGhie
,
R. D.
, 1990, “
Flexural Wave Motion in Infinite Beam
,”
J. Eng. Mech.
0733-9399,
116
(
3
), pp.
531
–
548
.
Crossref
Search ADS
 
24.
Hughes
,
T. J. R.
, 1987,
The Finite Element Method; Linear Static and Dynamic Finite Element Analysis
,
Prentice–Hall
, Englewood Cliffs, N.J.
25.
Callegari
,
M.
,
Carini
,
C. B.
,
Lenci
,
S.
,
Torselletti
,
E.
, and
Vitali
,
L.
, 2003, “
Dynamic Models of Marine Pipelines for Installation in Deep and Ultra-deep Waters: Analytical and Numerical Approaches
,”
Proceedings AIMETA03
, Ferrara, Italy, September 9–12 (CD-ROM).
26.
Lenci
,
S.
, and
Callegari
,
M.
, 2005, “
Simple Analytical Models for the J-lay Problem
,”
Acta Mech.
0001-5970,
178
(
1–2
), pp.
23
–
39
.
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