Abstract—The basic idea of the present study is proposed by Jang
et al. (2012), and the aim is to introduce the nonlinear procedure for
the numerical identification of static deflection of an infinite beam on a
nonlinear elastic foundation using one-way spring model. Jang’s
method involves Green’s function technique and an iterative method
using the pseudo spring coefficient.
Keywords—Jang’s method; One-way spring model; Nonlinear
elastic foundation; Green’s function;
I. INTRODUCTION
N the field of structural engineering, the accurate modeling of
an infinite beam on a nonlinear elastic foundation is crucial
role on the practical engineering design application. Especially,
in naval architecture, ships and ship-shaped offshore structures
usually consist of various curved beam and plate components.
So there have been many theoretical and experimental
researches concerning the accurate modeling for the
manufacturing.
Especially, among the large number of studies, some
researchers studied the closed form solutions using the Green’s
function techniques for the static and dynamic response of a
uniform beam which is resting on a linear elastic foundation.
[1-7] Lee et al. [8-10] and Kuo and Lee developed the exact and
semi-exact analysis of a non-uniform beam on a nonlinear
elastic foundation [11]. Beaufait and Hoadley approximated the
stress-strain relationship to be hyperbolic and modeled it as
bilinear curve to handle the nonlinearity [12]. Soldatos and
Selvadurai also applied the hyperbolic type elastic foundation to
identify the finite or infinite beam [13].
Recently, Jang et al. proposed Jang’s method for the
nonlinear deflection of an infinite beam on a nonlinear elastic
foundation [14]. He also advanced the studies on the large
deflection and variable cross section of an infinite beam on
nonlinear elastic foundation [15-17]. He also studied the
existence and uniqueness of the nonlinear deflections of an
Jinsoo Park is a doctoral degree student in Pusan National University,
supervised by Prof. T.S. Jang. 2, Busandaehak-ro, 63beon-gil, Geumjeong-gu,
Busan, 609-735, Republic of Korea (e-mail: [email protected]).
T.S. Jang, is a professor in Dept. Naval Architecture and Ocean Engineering,
Pusan National University. 2, Busandaehak-ro, 63beon-gil, Geumjeong-gu,
Busan, 609-735, Republic of Korea (corresponding author’s phone: +82 51
510 2789; e-mail: taek@ pusan.ac.kr).
infinite beam resting on a nonlinear elastic foundation using
Jang’s method [18]. From the research, Park applied the Jang’s
method using the realistic nonlinear elastic foundation, one-way
spring model [19]. The applied nonlinear elastic foundation is
active only when the beam is pressing against the foundation.
Finally, in this paper, we numerically identify the nonlinear
deflection of an infinite beam on a nonlinear elastic foundation
using one-way spring model.
II. SYSTEM
A. Jang’s method
The governing equation of the Euler-Bernoulli’s beam on a
nonlinear elastic foundation is as follows [15, 19]:
4
4 p p
d uEI k u f u w x k u
dx (1)
Where the nonlinear spring force f depends on the
deflection and denotes as follows [19]:
for 0,
for 0,0,
uk u N uf u
u (2)
In (1) and (2), E , I , k , pk , N u and w x are Young’s
modulus, the mass moment of inertia, a linear spring coefficient,
a pseudo spring coefficient, a nonlinear part of spring force and
external load, respectively. Figure 1 depicts the graphical
illustration of the system, and Figure 2 shows f u in (2).
[Park]
The iterative solution of the system is as follows [15, 19]:
1 , ;
, ;
n p
p n
u x G x k w d
G x k K u d
, 1,2,...n (3)
Where
pK u k u f u . (4)
The Green’s function with pseudo linear spring coefficient in
(3) can be expressed as
/ 2, ; sin
2 42
x
p
xG x k e
k,
4 / pk EI (5)
A Numerical Experiments of an Infinite beam
on a Nonlinear Elastic Foundation using
Jang’s Iterative Method
Jinsoo Park, and T.S. Jang
I
International Conference Recent treads in Engineering & Technology (ICRET’2014) Feb 13-14, 2014 Batam (Indonesia)
http://dx.doi.org/10.15242/IIE.E0214528 75
Finally, (3) can be expressed as a discretized form as
1
1
, ;
, ;
N j p j
n j
jp n j
G x k wu x W
G x k K u,
0, 1, 2, ...j , (6)
Where jW denotes the weights for the integration rule, N is
the total number of subinterval ,R R . R satisfies the
boundary condition of (1) [15, 19]:
u , xu ,
xxu , and xxxu 0 as | |x (7)
Fig. 1 An infinite beam on a nonlinear elastic foundation: one-way
spring model [19]
Fig. 2 One-way spring model [19]
III. NUMERICAL RESULTS
A. Nonlinear spring model 1
The first numerical experiment assume the nonlinear part of
spring force N u in (2) is [19]
3N u u , (8)
So the spring force in (2) is derived as
3 for 0,
for 00,
uk u uf u
u
. (9)
The principal properties are listed in Table I.
First of all, to determine the validity of the iterative method,
we assume the exact deflection of an infinite beam to be
2
sin xu x x e [15, 19] while the external load is derived as
TABLE I
PRINCIPAL PROPERTIES [20]
Symbol Properties Value
EI Flexural rigidity 2500kNm
pk Pseudo linear spring
coeff.
2250 /kN m
k Linear spring coeff. 2500 /kN m
Nonlinear spring coeff. 4250 /kN m
2 23 2 456 32 cos 24 72 16 sinx xw x EI x x e x EI x x e x
2 23 3 sin sinx xke x e x (10)
when 0u , and
2 23 2 456 32 cos 24 72 16 sinx xw x EI x x e x EI x x e x
(11)
when 0u . Figure 3 depicts w x in (11) .
Fig. 3 Applied external load according to the exact solution:
2
sin xu x x e
Figs. 4 and 5 demonstrate the converged solution and its
convergence behaviors, respectively. And Figure 6 shows the
errors of the solutions for the iteration number n [15, 19],
2
2
exact n
exact
u uError n
u
, (12)
where 2 is
2L norm.
Fig. 4 Converged solution compared to the exact one
International Conference Recent treads in Engineering & Technology (ICRET’2014) Feb 13-14, 2014 Batam (Indonesia)
http://dx.doi.org/10.15242/IIE.E0214528 76
B. Nonlinear spring model 2
The nonlinear spring force f u in (2) is assumed to be,
2 for 0,
for 00,
uk u uf u
u
r (13)
The principal properties of the second numerical experiment
are listed in Table I.
Fig. 5 Convergence behaviors of the solutions
Fig. 6 L-2 errors in (12)
Using the exact solution 2xu e , the external load is derived
as follows:
2 2 2
2
2 4 2
2 4
12 48 16 0
012 48 16
x x x
x
EI x x e ke e u xw x
u xEI x x e
(14)
Which is demonstrated in Figure 7. Figs. 8-10 show the
converged solution compared to the exact one convergence and
the behavior of the solution and L-2 errors, respectively.
IV. CONCLUSION
In this study, numerical experiments are carried out on the
identification of static deflection of an infinite beam on a fully
nonlinear elastic foundation using one-way spring model. To
find the highly nonlinear solutions, the Jang’s method, which
involves Green’s function technique and uses the pseudo spring
constant, is applied. Finally, the applied iterative method is
relatively simple but yields accurate solutions with relatively
fast convergence rate.
Fig. 7 Applied load from the exact solution: 2xu x e
Fig. 8 The converged solution: 100n
Fig. 9 Convergence behaviors of the solutions
Fig. 10 L-2 errors of 2nd numerical experiment
International Conference Recent treads in Engineering & Technology (ICRET’2014) Feb 13-14, 2014 Batam (Indonesia)
http://dx.doi.org/10.15242/IIE.E0214528 77
ACKNOWLEDGMENT
The research is supported by Basic Science Research
Program through the National Research Foundation of Korea
(NRF) funded by the Ministry of Education (Grant No.
2011-0010090).
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International Conference Recent treads in Engineering & Technology (ICRET’2014) Feb 13-14, 2014 Batam (Indonesia)
http://dx.doi.org/10.15242/IIE.E0214528 78