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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,
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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.
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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:
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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).
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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
, , ,
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The slope, xbbdx
dws
s 32 21 (17)
The generalized nodal displacements for the shear beam are defined by sw and s .
From Equations (16) and (17):
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,
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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
,
From Equations (14) and (19):
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
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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.
From Equations (14) and (19):
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:
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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
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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
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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
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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.
Unified Finite Element Model for the Analysis of Beams on Elastic Foundation
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