UNIFIED FINITE ELEMENT MODEL FOR THE ANALYSIS OF …...The finite element method in Cartesian coordinates is formulated using different types of one, two and three dimensional isoparametric

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International Journal of Civil Engineering and Technology (IJCIET)

Volume 10, Issue 11, November 2019, pp. 385-397, Article ID: IJCIET_10_11_039

Available online at http://www.iaeme.com/ijciet/issues.asp?JType=IJCIET&VType=10&IType=11

ISSN Print: 0976-6308 and ISSN Online: 0976-6316

© IAEME Publication

UNIFIED FINITE ELEMENT MODEL FOR THE

ANALYSIS OF BEAMS ON ELASTIC

FOUNDATION

M. E. Onyia

Department of Civil Engineering, University of Nigeria, Nsukka. Nigeria.

E. O. Rowland-Lato

Department of Civil Engineering, University of Port Harcourt, Choba. Nigeria.

ABSTRACT

This paper presents a unified beam finite element for the analysis of shear

deformable beam on elastic foundation. The beam stiffness and foundation matrices

are obtained by introducing an analytical bending-shear rotation interaction factor

which enables the decoupling of the bending and shear curvatures. Cubic and

quadratic polynomials are used as shape functions for bending and shear deformation

respectively. The resulting element is free from shear locking, a major drawback of

previous finite element models, The results obtained using this element are seen to be

in very good agreement with classical beam theories for combined bending and shear

deformation. It is concluded that the effect of shear deformation on deflection of

beams on elastic foundation is significant for span-to-depth ratios of 5 or less, and

should therefore be accounted for in the design of such beams.

Keywords: Timoshenko Beam, Shear-locking, Winkler Foundation

Cite this Article: M. E. Onyia and E. O. Rowland-Lato, Unified Finite Element

Model for the Analysis of Beams on Elastic Foundation. International Journal of Civil

Engineering and Technology 10(11), 2019, pp. 385-397.

http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=10&IType=11

1. INTRODUCTION

In practical structures, beam members can take up a great variety of loads such as biaxial

bending, transverse shears, axial forces and torsion. Such complicated actions are typical of

spatial beams, which are used in three-dimensional frameworks and are subject to forces

applied along arbitrary directions in space. In civil engineering, there are numerous studies

dealing with problems related to soil-structure interaction, such as railroad tracks, highway

pavements, strip foundations and ground beams, in which the structure is modelled by means

of a beam on elastic foundation.

There are various types of foundation models such as Winkler, Pasternak, Vlasov, etc .A

well-known and widely used soil-structure model is the one devised by Winkler. According to

the Winkler model, the beam-supporting soil is modelled as a series of closely-spaced,

M. E. Onyia and E. O. Rowland-Lato


mutually independent, linear elastic vertical springs which provide resistance in direct

proportion to the deflection of the beam. In the Winkler model, the properties of the soil are

described only by the parameter k, which represents the stiffness of the vertical springs

(Avramidis et. al, 2006).

Various investigators have used approximate numerical methods such as finite difference

and finite element methods in the analysis of beams on elastic foundation. Al-Musawi (2005)

studied the linear elastic behavior of beams resting on elastic foundations with both

compressional and tangential resistances. The finite element method in Cartesian coordinates

is formulated using different types of one, two and three dimensional isoparametric elements

to compare and check the accuracy of the solutions. Al-Shraify (2005) presented a three-

dimensional nonlinear finite element model suitable for the analysis of reinforced concrete

members. Concrete was modelled using 20-node isoparametric quadratic elements, while the

reinforcing bars were modelled as axial members embedded within the concrete elements.

The nonlinear equations of equilibrium were solved using an incremental iterative technique

based on the modified Raphson method. Dinev (2012) proposed a new method of obtaining a

closed-form analytical solution of the problem of bending of a beam on an elastic foundation

using singularity functions. The basic equations were obtained by a variational formulation

based on the minimum of the total potential energy functional.

Finite element solution of beams has predominantly been based on the Euler-Bernoulli

beam theory (EBT) which neglects the existence of through-thickness shear strains variation

to justify the plane section hypothesis. On the other hand this theory is not applicable for

moderately short and thick beams. With the increase in the thickness of the beam, the shear

deformation effect becomes significant, and the error of response increases if the Euler-

Bernoulli theory is used (Antes, 2003). Correspondingly, the effect of shear deformation is

formulated in Timoshenko beam theory (TBT). However the finite elements derived from the

TBT have tended to be unsatisfactory as they exhibit shear locking due to a number of

possible causes enumerated by Carpenters (1986) and Prathap (1982, 1987).

In this paper, an analytical bending-shear rotation interaction factor is proposed, the

introduction of which enables the decoupling of the bending and shear curvatures in the

Euler-Bernoulli beam governing equations. This factor is derived from bending and shear

strain energy considerations in a loaded beam. The formulation allows for the approximation

of the decoupled displacement variables, namely the transverse displacement and shear

rotation, using cubic and quadratic polynomials respectively. This leads to the emergence of a

locking-free Timoshenko beam stiffness matrix and consistent load vector in the finite

element formulation.

2. FORMULATION OF THE TIMOSHENKO BEAM FINITE ELEMENT

Consider a straight beam with constant cross-section continuously supported on a deformable

elastic foundation which is modeled by linear springs with stiffness, K (See Fig. 1).

The beam material and the foundation medium are assumed to be linearly elastic,

homogeneous, isotropic and continuous.

Unified Finite Element Model for the Analysis of Beams on Elastic Foundation


Figure 1 Beam element resting on elastic foundation

The beam deflection w is divided into two components; that due to flexure wb and that due

to transverse shear ws. Consequently, the angle of rotation is divided into its constitutive

parts, that due to bending b and slope due to shear s, (Fig. 2).

Figure 2 Kinematics of a beam undergoing both bending and shear rotations

To ensure continuous interaction between the bending and shear parts, and to avoid the

use of partial derivatives, the following relationship for the total cross sectional rotation is

proposed (Onyia and Rowland-lato, 2018):

)()1()()( xxx sb (1)

is the bending-shear interaction factor and is expressed as the ratio of the bending

strain energy bU to total strain energy of a simply-supported beam under load.

That is:

1

1

sb

b

UU

U (2)

where = b

s

U

U

(3)

sU = shear strain energy

For a simply supported beam under a central point load P (Fig. 3), the bending moment at

a section, distance x from a support, is given by:



2)(

PxxM , x<L/2 and

22)(

LxP

PxxM , x>L/2

Figure 3 A simply supported beam under a point load P at the center

Considering the midspan moment, the bending strain energy is

L

b dxEI

xMU

0

2

2

)(

(4)

2)(

PxxM

Substituting for )(xM in Equation (4) and performing the integration gives

EI

LPU b

96

32

(5)

The shear force at any section, distance x from a support, is:

2)(

PxQ

The shear strain energy is

dxkAG

xQU

L

s 0

2

2

))(( (6)

Substituting for )(xQ in Equation (6) gives the shear strain energy as:

kAG

LPU s

8

2

(7)

kAGL

EI

U

U

b

s

2

12

(8)

where E= Young’s modulus

G= shear modulus

A= cross-sectional area

k = shear coefficient depending on the shape of cross-section .

Akobo (1984) and Edem (2006) both proposed that the bending-shear interaction factor be based on the value of for midspan point load (i.e.

Equation 8).



2.1. Interpolation Functions

The interpolation function for flexural deformation (wb) is based on the Hermite cubic

polynomial:

3

4

2

321 xaxaxaaxwb (9)

The slope, 2

432 32 xaxaadx

dwb

b (10)

The generalized nodal displacements for the Bernoulli beam are defined as bw and b .

From Equations (9) and (10):

11)0( awxw bb

34

23212)( LaLaLaawLxw bb (11)

21)0( ax bb

2

4322 32 LaLaaLx bb

Putting Equations (11) in matrix form:

4

3

2

1

2

32

2

2

1

1

3210

1

0010

0001

a

a

a

a

LL

LLLw

w

b

b

b

b

(12)

Solving by matrix inversion:

2

2

1

1

2323

22

4

3

2

1

/1/2/1/2

/1/3/2/3

0010

0001

b

b

b

b

w

w

LLLL

LLLL

a

a

a

a

(13)

From Equation (9):

34

2321 xaxaxaaxwb

Substituting for a1, a2, a3, a4 from Equations (13):

ii

ib uxxw

4

1

(14)

where ’s are given as

(15)

and iu denotes the column displacement vector T

bbbb ww 2211 ,,,

Using a quadratic polynomial to approximate the shear deformation, ws(x): 2

3211)( xbxbbxws (16)

i

, , ,



The slope, xbbdx

dws

s 32 21 (17)

The generalized nodal displacements for the shear beam are defined by sw and s .


11 1)0( bwxw ss

23212 1)( LbLbbwLxw ss

21 1)0( bx ss

LbbLx ss 322 21

In matrix form:

3

2

1

2

2

2

1

1 1

2101

11

0101

0011

b

b

b

L

LLw

w

s

s

s

s

Solving by matrix inversion:

b

2

2

1

1

2

2

1

2/102/10

2/1/12/1/1

2/1/12/1/)1(

2/1/12/1/11

s

s

s

s

w

w

LL

LL

LLL

LL

b

b

b

(18)

From Equation (16), 23211)( xbxbbxws

Substituting for b1, b2, b3 from Equation (18):

i

i

is uxxw

4

1

(19)

and si ' are given as

2

43

2

212

and ,2

,1L

x

L

xL

L

x

L

x

L

xL

L

x (20)

and iu denotes the column displacement vectors T

ssss ww 2211 ,,,

2.2. Formulation of Timoshenko Beam Finite Element Stiffness Matrix

The strain energy of the Euler-Bernoulli beam is given by

L

dxEI

xMU

0

2

2

)(

where M(x) is the bending moment.

But curvature is,

EI

xMx

L

dxxEIU

0

2

2

1

Also curvature is,



xdx

xdx

where θ(x) is the slope.

L

dxxEIU

0

2

2

1

From Equation (1), the total cross-sectional rotation is

)()1()()( xxx sb

L

sbsb dxEI

U

0

22 12

,

(21)

dx

d

dx

d ss

bb

,


ii

ib uxx

4

1

for the flexural beam

i

i

is uxx

4

1

for the shear beam

where ’s and and si ' are given by Equations (15) and (20) respectively.

From Castigliano’s theorem, the stiffness matrix is given by

sb

ji

Uuu

K ,

dxuxEIuu

dxuxEIuu

Kei i

k

i

L

xjii

k

i

L

xji

24

10

24

10

)(12

1)(

2

1.

where and 2

2

dx

d i

i

dxxxEIdxxxEIK

L

x

ji

L

x

ji

00

)()(1)()(

(22)

Substituting for the interpolation functions, the assembled unified beam element stiffness

matrix [K] is

22

22

3

)4(6)2(6

612612

)2(6)4(6

612612

LLLL

LL

LLLL

LL

L

EIK

(23)

1

i

2

2

dx

d i

i



For the foundation, the total energy for the extension of the neutral axis is

L

wf dxwKU

0

2

2

1

Kw is the modulus of subgrade reaction.


ii

ib uxxw

4

1

i

i

is uxxw

4

1

dxuxKdxuxKU i

k

i

L

x

wi

k

i

L

x

wsbf

24

10

24

10

)(12

1)(

2

1,

The foundation matrix is then

sbf

jif U

uuK ,

dxuxKuu

dxuxKuu

i

k

i

L

x

wji

i

k

i

L

x

wji

24

10

24

10

)(12

1)(

2

1

dxxxKdxxxKKei

L

x

jiw

L

x

jiwf

00

)()(1)()(..

(24)

Substituting for the interpolation functions, the unified foundation matrix is

22

22w

f

L78L3544 L7Φ6L35Φ 26

L3544280312L35Φ26140Φ108

L76L3526 L78L35Φ44

L35Φ 26 140Φ108L35Φ44280 312

840

LβKK

(25)

Kw is the modulus of subgrade reaction.

The governing equation for beam on Winkler foundation is

extf FUKK

(26)

where K is the Structure stiffness matrix

Kf is the foundation matrix

U is the vector of the structure nodal displacements

Fext is the vector of nodal external forces

extT

f FwwKKei 2211 ,,,..

The total energy in the unified beam element on elastic foundation loaded by normal load

q is given by:



LL

w

L

dxqwdxwKdxEI

qwU

0

0

0

2

0

2

2

1

2,,

(27)

where sb )1(

sb www )1(

L

dxqwF

0

0 is called the consistent load vector

2.3. Consistent Load Vector

For distributed loading on the beam, the consistent load vector is:

dxxqf

i

i

L

)(

4

0

(28)

The column vector f is then given by

dxqfT

L

4321

0

where ’s are given in Equation (15).

Substituting for the φi’s in Equation (28) and integrating for a uniformly distributed load,

then

TLLqL

f 6612

(29)

For concentrated load, P, at an arbitrary position on the beam, the consistent load vector is

given by

dxxPf

i

i )(

4

TPf 4321

(30)

Substituting for the φi’s, then

TLLPf82

182

1 (31)

3. RESULTS AND DISCUSSION

Consider a simply-supported beam on elastic foundation under uniformly distributed load.

Using Equation (26) and substituting for the unified structure stiffness matrix [K], the

foundation matrix [Kf] and the consistent load vector {f}, we have

L

LqL

w

w

LLLL

L

LLL

LL

Lk

LLLL

LL

LLLL

LL

L

EI

w

6

6

12

)78()3544( )76()3526(

)3544()280312()L3526()140081(

)76()L3526( )78()3544(

)3526()140081( )3544()280312(

840

4626

612612

2646

612612

2

2

1

1

22

22

22

22

3

(32)

i



Applying the boundary conditions w1 = w2 = 0:

120124

3

21

EI

qL (33)

Where is the foundation factor and is expressed as:

EI

Lkw

4

The maximum deflection wc at midspan is given as

180640

3526

1008021205

1201384

4

EI

qL

wc (34)

In terms of non-dimensional deflection

(35)

Chen et. Al (2004) solved the above simply-supported beam on elastic foundation whose

assumed properties were taken as:

Elastic modulus E = 1, Poisson’s ratio = 0.3, shear correction factor k =5/6

From Equation (8),

kAGL

EI2

12

For

12

3bdI , bdA ,

)1(2 v

EG

Then

l

d

k

v)1(2

(36)

Table 1 presents a comparison of the results obtained using the unified beam finite

element (UBE) solution and the analytical solution by Chen et. Al (2004) for different values

of L/d.

180640

3526

1008021205

1201384

1

4

qL

EIww c



Table 1 Comparison of Solutions for Non-dimensional Deflection of Simply-supported Beam on

Elastic Foundation under Uniformly Distributed Load

L/

d Non-dimensional Deflection 4qL

EIww c

=0 =10 =100

Analytica

l

UBE %

Diff

Analytica

l

UBE %

Diff

Analytica

l

UBE %

Diff

5 -0.01420 -

0.01335

5.99 -0.01277 -

0.01230

3.68 -0.00668 -

0.00714

-6.89

15 -0.01315 -

0.01305

0.76 -0.01191 -

0.01204

-

1.09

-0.00643 -

0.00706

-9.80

12

0

-0.01302 -

0.01302

0 -0.01181 -

0.01201

-

1.69

-0.00640 -

0.00705

-

10.16

Legend:

= Foundation Parameter

A plot of the results shown in Table 1 is presented in Figure 4.

Figure 4 A Plot of Non-dimensional Midspan Deflection ( w ) versus Span-to-depth (L/d) Ratio

Also the effect of the span-to-depth ratio and the foundation parameter Ω on the midspan

deflection of the beam on elastic foundation is investigated using the unified beam finite

element (UBE) model and the results shown in Table 2.

Table 2 Non-dimensional Midspan Deflection of Simply-supported Beam on Elastic Foundation

under UDL for Various Values of L/d Ratio and Foundation Parameter Ω

L/d Non-dimensional Midspan Deflection

4qL

EIww c

( x 102)

Ω=0 Ω=10 Ω=100 Ω=250 Ω=500

1 -2.11 -1.91 -0.94 -0.40 -0.13

2 -1.51 -1.38 -0.77 -0.41 -0.20

5 -1.33 -1.23 -0.72 -0.41 -0.23

10 -1.31 -1.21 -0.71 -0.41 -0.24

20 -1.30 -1.20 -0.71 -0.41 -0.24

100 -1.30 -1.20 -0.70 -0.41 -0.24

EI

Lkw

4



A Plot of the results shown in Table 2 is presented in Figure 5.

Figure 5 Plot of Maximum Midspan Deflection ( w ) versus Foundation Parameter (Ω)

The results in Table 1 show that the average error is less than 5%, indicating the results

obtained with the proposed numerical model have a very good agreement with the exact

solutions both for slender and deep beams. The accuracy of the proposed unified beam

element (UBE) model increases as the beam slenderness increases and decreases with an

increase in the foundation parameter . In general, the UBE model gives upper-bound

solutions for slender beams and high values of the foundation parameter.

The results in Table 2 show that the beam displacement decreases as the span-to-depth

ratio increases from 0 to 5; however, for L/d greater than 5, the displacement is relatively

constant for all values of the foundation parameter. This implies that shear deformation may

safely be ignored in the deflection design of beams whose L/d ratio is greater than 5, but not

for those with L/d is less than or equal to 5 for all values of the foundation parameter.

The effect of modulus of subgrade is expressed in terms of the dimensionless foundation

parameter (Ω) of the beam element. Table 2 shows that the beam displacement decreases as

the foundation parameter increases for all span-to-depth ratios.

4. CONCLUSION

A Unified beam finite element (UBE) free from shear locking has been developed to analyse

shear deformable beams on elastic foundation. The beam stiffness and foundation matrices

were obtained by introduction an analytical bending-shear rotation interaction factor which

enables the decoupling of the bending and shear curvatures. Cubic and quadratic polynomials

were used as shape functions to approximate flexural and shear deformations respectively.

The results obtained using this element in the analysis of Timoshenko beam on Winkler

foundationare are seen to be in very good agreement with classical beam theories for

combined bending and shear deformation. It is also observed that the accuracy of the

proposed unified beam element (UBE) model increases as the beam slenderness increases and

decreases with an increase in the foundation parameter . It is concluded that the effect of

shear deformation on deflection of beams on elastic foundation is significant for span-to-

depth ratios of 5 or less and should therefore be accounted for in the design of such beams.



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Factors in C0 Bending Elements,” Computer and Structure, 22, 39.

[3] Chen, W.Q., Lu, C.F. and Bian, Z.G. (2004). “A Mixed Method for Bending and Free

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Documents

UNIFIED FINITE ELEMENT MODEL FOR THE ANALYSIS OF …...The finite element method in Cartesian coordinates is formulated using different types of one, two and three dimensional isoparametric