Deflection of Non-Uniform Beams Resting on a

Non-Linear Elastic Foundation Using (GDQM)

Ramzy M. Abumandour1, Islam M. Eldesoky

1, Mohamed A. Safan

2, R. M. Rizk-Allah

1, and Fathi A.

Abdelmgeed1

1Basic Engineering Sciences Department, Faculty of Engineering, Menofia University, Egypt

2Department of Civil Engineering, Faculty of Engineering, Menoufia University, Egypt

Email: { ramzy_0000, eldesokyi, msafan2000, rizk_masoud, fathi_azeem}@yahoo.com

AbstractIn this study, a method of a new technique of

GDQM is presented for determining the deflection of a non-

uniform beam resting on a non-linear elastic foundation,

subjected to axial and transverse distributed force. The

nonlinear subgrade model which describes the foundation

includes the linear and nonlinear Winkler (normal)

parameters and the linear Pasternak (shear) foundation

parameter. The nonlinear 4th order differential equation of

beam is solved using a new technique. In construction of

numerical scheme a GDQM is used to transform the

differential equation into a set of nonlinear algebraic

equations. Then, these nonlinear algebraic equations solved

by Newtons method. Comparison the present results with

the previous solutions proves the accuracy of this

combination.

Index Termsnonlinear elastic foundation, deflection,

GDQM and newtons method

I. INTRODUCTION

The analysis of beams resting on elastic foundation

subjected to axial loading are very common in structural

systems under actual operating conditions [1] and

therefore an accurate and reliable method of analysis is

required especially when the properties of their cross-

section are variable. The applications for beams resting

on elastic foundation are one of the important topics in

many engineering fields due to its wide applications in

engineering, such as mainly: In civil engineering e.g. in

the design of structural components in buildings, aircrafts,

ships, buried pipes and concrete pavement slabs and

bridges and network of beams in the construction of floor

systems for ships, buildings, and bridges, submerged

floating tunnels, buried pipelines etc. In railway

engineering, e.g. the design of railroad tracks. In

mechanical engineering e.g. disc brake pad, shafts

supported on ball, roller, or journal bearings, vibrating

machines on elastic foundations.

The work has been done on this subject is limited only

to the linear response of uniform beams on linear elastic

foundation [2], uniform beams on nonlinear elastic

foundation [3]-[7] and non-uniform beams on nonlinear

elastic foundation [8]. Numerical solutions for beams

resting on elastic foundations using DQEM [9]. Some

Manuscript received April 7, 2016; revised August 10, 2016.

problems in structural analysis resting on fluid layer

using GDQM [10]. Free vibration of uniform and non-

uniform beams resting on fluid layer under axial force

using the GDQM [11].

In this paper, the numerical solution using a

combination of a GDQM and Newtons method for

solution the fourth order differential equation of beam

under appropriate boundary conditions is nonlinear

ordinary differential equations. Also, the effect of the

linear and nonlinear Winkler (normal) parameters and the

linear Pasternak (shear) foundation parameter is

presented.

II. FORMATION OF THE PROBLEM

A. Formulation of the Problem

For a general elastically end restrained non-uniform

Euler Bernoulli beam of finite length L, resting on a non-

linear-elastic foundation subjected to axial load and

transverse forces, the dynamic flexural deflection v (x, t)

satisfies the ordinary differential equation:

2 2 2 23

1 22 2 2 2A EI p k k

t x x x

2

3 2( ek F x t x L

x

(1)

Figure 1. Geometry of the beam resting on non-linear elastic foundation

where, v (x,t) is the flexural deflection, is the density of

the beam material, A is the area of the beam section, E(x)

is the Young's modulus, I(x) is the moment of inertia,

P(x,t) is the axial force, F(x,t) is the transverse force, k1 is

the linear Winkler (normal) foundation parameter, k2 is

the nonlinear Winkler (normal) foundation parameter and

k3 is the linear Pasternak (shear) foundation parameter.

Normalizing (1) then the non-dimensional governing

equation of a Bernoulli-Euler beam as:

International Journal of Structural and Civil Engineering Research Vol. 6, No. 1, February 2017

2017 Int. J. Struct. Civ. Eng. Res.doi: 10.18178/ijscer.6.1.52-56

52

, ) 0, 0

4 3 2 2

4 3 2 2

( ) ( ) ( ) ( ) ( )( ) 2

d W X dS X d W X d S X d W XS X

dX dX dX dX dX

23

1 2 3 2

( )( ) ( ) ( ) 0, 0 1.

d W XK W X K W X K F X X

dX

(2)

where the non-dimensional coefficients are;

42

11

0 0

, , , ,k LV x pL

W X P KL L EI EI

26 3

322 3

0 0

( ), F(X) .

k Lk L f x LK K

EI EI Eand

I

Equation (1) is a 4th

order ordinary differential

equation with inertia ratio S(X) = (1+1X)2

. In the case

of non-uniform beam will study one case of inertia ratio

S(X); 1 = 0.5 and 2 = 1.0. It requires 4 boundary

conditions, two at X = 0, and two at X = 1. In the present

work, the following three types of boundary conditions

are considered:

For ClampedClamped (CC),

(0)(0) 0dW

WdX

and

( )( ) 0

dW LW L

dX (3)

For SimplySimply (SS);

2

2

(0)(0 ) 0

d WW

dXand

2

2

( )( ) 0

dW LW L

dX (4)

For ClampedSimply (CS);

(0)(0) 0

dWW

dXand

2

2

( )( ) 0

dW LW L

dX (5)

B. Method of Solution

The coordinates of the grid points are chosen

according to Chebyshev-Gauss-Lobatto as:

1 1( ) 1 cos , 1,2, , .

2 1

iX i i N

N

(6)

Applying the GDQ method to (2) yields;

(1) (2)

, , ,

1 1 1

2( ) ( ) ( )N N N

i i j j i i j j i i j j

j j j

S X D W S X C W S X B W

3

, 1 2 3 ,

1 1

N N

i j j i i i j j

j j

P B W K W K W K B W

( ) 0, 1,2, , .iF X i N (7)

where Wi, i=1, 2, 3,, N, is the functional value at the

grid Xi, Bij, Cij and Dij is the weighting coefficient matrix

of the second, third and fourth order derivatives. S(2)

(Xi)

and S(1)

(Xi) are the second and first order derivatives of

S(X) at Xi.

Similarly, the derivatives on the boundary conditions

can be discretized by the GDQ method, as a result, the

numerical boundary conditions can be written as:

1 0.W (8-1)

( 0)

1,

1

0.N

n

k k

k

C W

(8-2)

0.NW (9-1)

( 1)

,

1

0.N

n

N k k

k

C W

(9-2)

where n0, n1 may be taken as either 1 or 2. By choosing

the value of n0 and n1, (8) and (9) can give the following

four sets of boundary conditions,

n0 = 1, n1 = 1 (CC) supported,

n0 = 1, n1 = 2 (CS) supported.

n0 = 2, n1 = 1 (SC) supported,

n0 = 2, n1 = 2 (SS) supported.

Equations (8-1) and (9-1) can be easily substituted into

the governing equation. This is not the case for (8-2) and

(9-2) which, can couple these two equations together to

give two solutions, W2 and WN-1, as: 2

2

3

11. .

N

k

k

W AXK WAXN

(10)

2

1

3

1. .

N

N k

k

W AXKN WAXN

(11)

where; ( 0) ( 1) ( 0) ( 1)

1, , 1 1, 1 ,1 ,n n n n

k N N N N kC C C CAXK ( 0) ( 1) ( 0) ( 1)

1,2 , 1, ,2 ,n n n n

N k k NC C C andCAXKN

( 1) ( 0) ( 0) ( 1)

,2 1, 1 1,2 , 1.n n n n

N N N NC C C CAXN

According to (10) and (11), W2 and WN-1 are expressed

in terms of W3, W4,,WN-2. Substituting (8-1), (9-1), (10)

and (11) into (7) gives:

2 2 2(1) (2)

, , ,

3 3 3

2( ) ( ) ( )N N N

i i j j i i j j i i j j

j j j

S X D W S X C W S X B W

2 23

, 1 2 3 ,

3 3

1,N N

i j j i i i j j

j j

P B W K W K W K B W

3,4, , 2.i N (12)

TABLE I. DEFLECTION OF NON-UNIFORM SIMPLYSIMPLY BEAM RESTING ON A NONLINEAR ELASTIC FOUNDATION UNDER UNIFORMLY

DISTRIBUTED FORCE (K1 = 1.0, K2 = 1.0, K3 = 1.0, P=1.0, F=1.0).

X W(GDQM) W (exact) W (C-C)

Uniform (CC) Uniform (CC) 1 = 0.5, 2 = 1.0

0 0 0 0

0.012536 -6.38E-06 -6.38E-06 -5.84E-06

0.049516 -9.23E-05 -9.23E-05 -8.35E-05

0.109084 -0.000393538 -0.000393538 -0.00035

0.188255 -0.000973019 -0.000973019 -0.00085

0.283058 -0.001715963 -0.001715963 -0.00146

0.38874 -0.002352658 -0.002352658 -0.00195

0.5 -0.002604167 -0.002604167 -0.0021

0.61126 -0.002352658 -0.002352658 -0.00184