a numerical experiments of an infinite beam on a nonlinear...

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



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



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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