KrigingBased Timoshenko Beam Element for Static and Free Vibration Analyses
ABSTRACT An enhancement of the finite element method using Kriging interpolation (KFEM) has been recently proposed and applied to solve one and two dimensional linear elasticity problems. The key advantage of this innovative method is that the polynomial refinement can be performed without adding nodes or changing the element connectivity. This paper presents the development of the KFEM for static and free vibration analyses of Timoshenko beams. The transverse displacement and the rotation of the beam are independently approximated using Kriging interpolation. For each element, the interpolation function is constructed from a set of nodes within a prescribed domain of influence comprising the element and its several layers of neighbouring elements. In an attempt to eliminate the shear locking, the selectivereduced integration technique is utilized. The developed beam element is tested to several static and free vibration problems. The results demonstrate the excellent performance of the developed element.

Article: Moving Kriging reconstruction for high order finite volume computations of compressible Flows
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ABSTRACT: This paper describes the development of a highorder finite volume method for the solution of compressible viscous flows on unstructured meshes. The novelty of this approach is based on the use of moving Kriging shape functions for the computation of the derivatives in the numerical flux reconstruction step at the cell faces. For each cell, the successive derivatives of the flow variables are deduced from the interpolation function constructed from a compact stencil support for both Gaussian and quartic spline correlation models. A particular attention is paid for the study of the influence of the correlation parameter onto the accuracy of the numerical scheme. The effect of the size of the moving Kriging stencil is also investigated. Robustness and convergence properties are studied for various inviscid and viscous flows. Results reveal that the moving Kriging shape function can be considered as an interesting alternative for the development of highorder methodology for complex geometries.Computer Methods in Applied Mechanics and Engineering 01/2013; 253:463478. · 2.63 Impact Factor
Page 1
ISSN 14109530 print / ISSN 1979570X online
Krigingbased Timoshenko Beam Element for Static and Free
Vibration Analyses
42
Civil Engineering Dimension, Vol. 13, No. 1, March 2011, 4249
Wong, F.T.1 and Syamsoeyadi, H.2
Abstract: An enhancement of the finite element method using Kriging interpolation (KFEM)
has been recently proposed and applied to solve one and two dimensional linear elasticity
problems. The key advantage of this innovative method is that the polynomial refinement can be
performed without adding nodes or changing the element connectivity. This paper presents the
development of the KFEM for static and free vibration analyses of Timoshenko beams. The
transverse displacement and the rotation of the beam are independently approximated using
Kriging interpolation. For each element, the interpolation function is constructed from a set of
nodes within a prescribed domain of influence comprising the element and its several layers of
neighbouring elements. In an attempt to eliminate the shear locking, the selectivereduced
integration technique is utilized. The developed beam element is tested to several static and free
vibration problems. The results demonstrate the excellent performance of the developed element.
Keywords: Finite element, kriging, Timoshenko beam, shear locking, selectivereduced
integration.
Introduction
In an attempt to improve the elementfree Galerkin
method with Kriging interpolation [1], Plengkhom
and KanokNukulchai [2] presented a new class of
FEM by introducing Kriging shape functions in the
conventional FEM. In this method, Kriging
interpolation (KI) is constructed for each element
using a set of nodes in a domain of influencing nodes
(DOI) composed of several layers of elements (the
DOI is in the form of polygon for 2D problems).
Combining the KI of all elements, the global field
variable is thus approximated by piecewise KI. For
evaluating the integration in the Galerkin weak
form, the elements are employed as integration cells.
The method subsequently referred to as Kriging
based FEM (KFEM) [3].
The advantages of the KFEM are: (1). Highly
accurate field variables and their gradients can be
obtained even using the simplest form of elements
(triangles in the 2D domain and tetrahedrons in the
3D domain). (2). The polynomial refinement can be
achieved without any change to the element or mesh
structure. (3). Unlike the moving Kriging element
free Galerkin method [1], the formulation and coding of
the KFEM are very similar to the conventional FEM
1 Department of Civil Engineering, Petra Christian University,
Surabaya, INDONESIA
Email: wftjong@petra.ac.id
2 PT. Hakadikon Pratama, Surabaya, INDONESIA
Note: Discussion is expected before June, 1st 2011, and will be
published in the “Civil Engineering Dimension” volume 13, number
2, September 2011.
Received 9 April 2010; revised 6 November; accepted 28 November
2010.
so that any existing generalpurpose FE program
can be easily extended to incorporate the enhanced
method. Thus, the KFEM has a high chance to be
accepted in real engineering practices.
In the pioneering work [2], the KFEM was developed
for static analyses of 1D bar and 2D plane
stress/planestrain solids. Subsequently, it was
developed for analyses of ReissnerMindlin plates [3,
4] and improved through the use of adaptive
correlation parameters and by introducing the
quartic spline correlation function. A drawback of
the KFEM is that the interpolation function is
discontinuous at the interelement boundaries
(except in 1D problems). In spite of this discontinuity,
using appropriate choice of shape function parameters,
the KFEM passes weak patch tests and therefore
the convergence is guaranteed [5]. The basic concepts
and advances of the KFEM have been presented in
several papers [68]. The current development is the
extension and application of the KFEM to different
problems in engineering, such as general plate and
shell structures [9, 10] and multiscale mechanics
[11].
Despite many attractive features of the KFEM, in
the application to sheardeformable plates and
shells, the drawback of transverse shear locking and
membrane locking presents in the KFEM [3, 4, 9,
10]. The use of high order basis (cubic and quartic)
in KI can alleviate the shear and membrane
lockings, but there is no guarantee to eliminate the
lockings completely. Until the writing of this paper,
to the authors’ knowledge, an effective method to
completely eliminate the lockings in the KFEM is
not yet invented.
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Wong, F.T.et al / Krigingbased Timoshenko Beam Element for Static and Free Vibration Analyses / CED, Vol. 13, No. 1, March 2011, pp. 42–49
43
In attempt to invent the method to eliminate the
drawback of shear locking in sheardeformable plate
and shell problems, it is instructive to study the K
FEM in the simpler context of the Timoshenko beam
since this problem can be considered as a 1D
degeneration from the ReissnerMindlin plate. It is
the aim of this paper to present the development and
testing of the KFEM for analyses of static and free
vibration of Timoshenko beams. The developed
element is tested to several beam problems and its
performance is compared with high performance
Timoshenko beam element developed by Friedman
and Kosmatka [12].
Kriging Interpolation in the KFEM
Named after Danie G. Krige, a South African mining
engineer, Kriging is a wellknown geostatistical
technique for spatial data interpolation in geology
and mining [13, 14]. Using this interpolation, every
unknown value at a point can be interpolated from
known values at scattered points in its specified
neighborhood. Here the concepts of the KI in the
context of KFEM are briefly reviewed.
Consider a continuous field variable u(x) defined in a
domain Ω. The domain is represented by a set of
properly scattered nodes xi, i=1, 2, …, N, where N is
the total number of nodes in the whole domain.
Given N field values u(x1), …, u(xN), the problem of
interest is to obtain an estimated value of u at a
point x0 ∈ Ω.
The Kriging estimated value uh(x0) is a linear
combination of u(x1), …, u(xn), i.e.
n
ii
i
uu
=
xx
where λi’s are the unknown Kriging weights and n is
the number of nodes surrounding point x0, inside and
on the boundary of a DOI Ωx0 ⊆ Ω.. Considering each
function values u(x1), …, u(xn) as the realizations of
random variables U(x1), …, U(xn), Eq. (1) can be
written as
n
ii
i
UU
=
xx
The Kriging weights are determined by requiring the
estimator Uh(x0) unbiased, i.e.
E()()0
UU
⎡⎤
−=
⎣⎦
where E[•] is the expected value operator, and by
minimizing the variance of estimation error,
var[Uh(x0)–U(x0)]. Using the method of Lagrange [13,
14] for the constraint optimization problem, the
requirements of minimum variance and unbiased
estimator lead to the following Kriging equation
system (see Wong [10] for the complete derivation):
Rλ + Pµ = r(x0)
h
0
1
()( )
λ=∑
(1)
h
0
1
()( )
λ=∑
(2)
h
00
xx
(3)
PTλ = p(x0)
in which
C
⎡
⎢
=⎢
⎢
⎣
p
⎡
⎢
=⎢
⎢
⎣
(4)
111
1
()...
...
()
...
(
h
...
(
h
) ...)
n
nnn
x
C
CC
⎤
⎥
⎥
⎥
⎦
⎤
⎥
⎥
⎥
⎦
hh
R
;
111
1
( )
...
(
x
...
...
( )
...
(
m
p
) ...)
m
nn
p
p
x
P
x
;
λ = [λ1 … λn]T ; µ = [µ1 … µm]T
R(x0) = [C(h10) C(h20) ... C(hn0)]T ;
p(x0) = [p1(x0) ... pm(x0)]T
R is an n x n matrix of covariance of U(x) at nodes x1,
…, xn; P is an n x m matrix of polynomial values at
the nodes; λ is an n x l vector of Kriging weights; µ is
an m x l vector of Lagrange multipliers; r(x0) is an n
x l vector of covariance between the nodes and the
node of interest, x0; and p(x0) is an m x l vector of
polynomial basis at x0. While C(hij) = cov[U(xi), U(xj)]
is the covariance between U(x) at node xi and U(x) at
node xj. The unknown Kriging weights λ and
Lagrange multipliers µ are obtained by solving the
Kriging equations, Eqs. (4).
The expression for the estimated value uh, Eq. (1),
can be rewritten in matrix form
uh (x0) = λTd
where d = [u(x1) … u(xn)]T is an n x l vector of nodal
values. Since the point x0 is an arbitrary point in the
DOI, the symbol x0 can be replaced by symbol x.
Thus, using the usual finite element symbol, Eq. (5)
can be expressed as
=∑
xN x d
in which N(x)= λT(x) is the matrix of shape functions.
In order to construct Kriging shape functions in Eq.
(6), a polynomial basis function and a correlation
function should be chosen. Basis functions ranging
from polynomial of the degree one up to four have
been utilized in the past works on the KFEM [211].
In the problems of shear deformable plates and
shells, it is necessary to use cubic or quartic
polynomial basis in order to alleviate the shear and
membrane lockings [4, 9, 10].
The correlation function ρ (h) is defined as:
ρ(h) = C(h)/σ 2
where h is a vector separating two points x and x+h
and σ2 is the variance of the random function U(x).
In the KFEM, factor σ2 has no effect on the final
results and it was taken equal to 1 in this study.
(5)
h
1
( )( )( )
x
n
ii
i
uNu
=
=
(6)
(7)
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Wong, F.T.et al / Krigingbased Timoshenko Beam Element for Static and Free Vibration Analyses / CED, Vol. 13, No. 1, March 2011, pp. 42–49
44
Following the previous works [310], Gaussian and
Quartic Spline correlation function (QS) were
chosen. Gaussian correlation function is defined as
ρ (h) = ρ (h) = exp((θ h/d)2)
and QS is defined as
(8)
234
1 6(
⎧ −
=⎨
⎩
/ )
h d
θ
8(/ )
h d
θ
3( / ) for 0
h d
θ
/
>
1
( )
h
( )
h
0 for /1
h d
h d
θ
ρρ
θ
+−≤≤
=
(9)
In these equations, θ>0 is the correlation parameter,
h = h, i.e. the Euclidean distance between points x
and x+h, and d is a scale factor to normalize the
distance. Factor d was taken to be the largest
distance between any pair of nodes in the DOI.
For each aforementioned correlation function,
appropriate values for the correlation parameter
should be chosen so that the KFEM will give
reasonable results. Plengkhom dan Kanok
Nukulchai [2] proposed a rule of thumb for choosing
the parameter as follows: Correlation parameter, θ,
should be selected so that it satisfies the lower bound
condition,
∑
10
1
11 10
n
iN
=
a
i
−+
− ≤ ×
(10a)
and also satisfies the upper bound condition,
det(R) < 1 x 10b (10b)
where a is the order of basis function and b is the
dimension of problem (1, 2, or 3).
Variational Form of Timoshenko Beam
Governing Equations
The basic assumption of Timoshenko beam theory is
that a plane normal to the beam axis in the
undeformed state remains plane in the deformed
state but it does not necessarily in normal direction
to the neutral axis [16]. The theory accounts for both
the transverse shearing strain and the rotary inertia
in a dynamic analysis. This section presents the
variational formulation of Timoshenko beam following
that given by Friedman and Kosmatka [12].
Fig. 1 shows the coordinate system used in the
following formulation. The displacement components
in x and z directions can be respectively written as a
function of coordinate x and time t as
u =  zψ (x,t)
w = w(x,t)
where u is the axial displacement of a material point
at coordinate (x, z), w is the transverse displacement
(deflection) of the neutral axis and ψ is the rotation
of the crosssection.
(11a)
(11b)
Figure 1. Coordinate system, deflection and rotation of the
beam
Using the smallstrain and displacement equations
for general solids, Eqs. (11a) and (11b) give the
nonzero strain components:
εxx =  zψ,x
γxz = w,x  ψ
where εxx is the normal strain in x direction and γxz is
the transverse shearing strain. The commas denote
the first partial derivatives with respect to the
variable next to it (i.e. x).
The variational equation of motion of the beam can
be derived using Hamilton’s principle [12], viz.
()
∫
1
t
where δU, δT, and δWe, are the variations of the
strain energy, the kinetic energy, and the work of
external forces, respectively. The strain energy is
given as
1
2
in which L and A are the length and the cross
sectional area of the beam respectively, and
E
kG
γ
⎩⎭
is the vector of normal and shearing stresses, and
ε
γ
⎩⎭
is the vector of normal and shearing strains. In Eq.
(15a), E and G are Young’s and the shear moduli of
the beam material respectively, and k is a shear
correction factor that is dependent upon the cross
section geometry. Substituting Eqs. (15a) and (15b)
into Eq. (14), considering the strains, Eq. (12a) and
Eq. (12b), and integrating over the crosssection, Eq.
(14) yields
()
0
22
∫∫
where I is the moment of inertia of the crosssection.
(12a)
(12b)
=−−=∏
2
0
t
edtWTU
δδδδ
(13)
T
0
L
A
UdAdx
=∫ ∫σ ε
(14)
xx
xz
ε
⎧⎫
⎬
=⎨
σ
(15a)
xx
xz
⎧⎫
⎬
=⎨
ε
(15b)
()
22
0
11
, ,
LL
xx
UEIdxkGA wdx
ψψ−=+
(16)
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Wong, F.T.et al / Krigingbased Timoshenko Beam Element for Static and Free Vibration Analyses / CED, Vol. 13, No. 1, March 2011, pp. 42–49
45
The kinetic energy of the beam is given as
(
0
2
∫ ∫
where ρ is mass density (per unit volume) of the
beam material. The dots signify the first partial
derivatives with respect to the time variable t.
Substituting Eqs. (11a) and (11b) and integrating
over the crosssection, Eq. (17) can be expressed as
11
22
∫∫
Finally, the work of external forces is given as.
LL
W w q dxm dx
ψ=+
∫∫
in which q and m are the distributed forces and
moments along the length of the beam.
Substituting Eq. (16), Eq. (18) and Eq. (19) into Eq.
(13), applying the variational operations, and applying
Hamilton’s principles lead to the variational equation
(or the weak form) for the Timoshenko beam as
follows,
wAw dxI dx
δρ δψ ρ ψ+
∫∫
∫∫
∫∫
The double dots signify the second derivatives with
respect to time t.
The bending moment and shear force along the
beam can be calculated from the deflection w and
rotation ψ as follows:
,x
M EI ψ=
()
,x
Q kGA w
ψ=−
Formulation of Krigingbased Timoshenko
Beam Element
Suppose a beam is divided into a number of finite
elements. For each element, KI is constructed based
upon a set of nodes in a DOI including the element
itself and a predetermined number of neighboring
elements (see Wong and KanokNukulchai [8] for
detailed explanations). Consider now an element
with its DOI including n nodes. The displacement
components over the element are approximated by
KI as follows:
w = Nw d
ψ = Nψ d
where
Nw = {N1 0 N2 0 ... Nn 0}
Nψ = {0 N1 0 N2 ... 0 Nn}
d = {w1 ψ1 w2 ψ2 ... wn ψn}T
)
22
1
L
A
Tu
&
w
&
dA dx
ρ=+
(17)
22
00
LL
T Aw dx
ρ
&
Idx
ρ ψ=+
&
(18)
00
(19)
()()
00
00
00
,, ,,
LL
LL
xxxx
LL
EIdxw kGA wdx
wq dx
δ
m dx
δψψδ δψ−ψ−
δψ
+
+
=+
&&
&&
(20)
(21)
(22)
(23a)
(23b)
(23c)
(23d)
(23e)
and the shape functions (N1, N2, …., Nn) are Kriging
shape functions, which are obtained by solving
Kriging equations, Eqs. (4).
Entering the approximated displacement functions
Eqs. (23a) and (23b) into the variational equation,
Eq. (20), leads to the discrete equation of motion for
Timoshenko beam element, i.e.
( )( ) ( ) tftdktdm
=+
& &
where
LL
Adx
ψ
ρ=+
∫∫
is the element consistent mass matrix,d&& is the
element nodal acceleration vector,
L
L
T
x ,
ψψ
−
is the element stiffness matrix, and
LL
qdxmdx
ψ
=+
∫∫
is the element equivalent nodal force vector. Note
that the size of the matrices m and k, d, and f
depends on the number of nodes in the DOI, n. For
static problems, Eq. (24a) simply reduces to
kd = f
The integrations in Eqs. (24b), (24c), and (24d) are
evaluated using Gauss quadrature. It is well known
in the conventional FEM that if the stiffness matrix,
Eq. (24c), is evaluated using full integration, then the
element becomes too stiff for thin beams (shear
locking phenomenon, see e.g. Hughes et al. [15] and
Reddy [16]). One of the techniques to overcome the
shear locking is the selectivereduced integration
(SRI), in which the stiffness matrix is evaluated
using full integration for the bending term (the first
term in the righthand side of Eq. (24c)) and using
reduced integration for the shearing term (the
second term in the righthand side of Eq. (24c)). The
effectiveness of the SRI technique to eliminate the
shear locking in the Krigingbased Timoshenko
beam is investigated in this study through a series of
numerical tests.
Using the finite element assembly procedure, one
can obtain the global discretized equation for the
beam vibration as follows:
( )( ) ( ) tFtDKtDM
=+
& &
where M and K are the global mass and stiffness
matrices, respectively, D is the global nodal
displacement vector,
D& & is the global nodal
acceleration vector, and F is the global nodal force
vector. The equations for static and undamped free
vibration problems can be respectively written as
K D = F
MD& & (t) + K D(t) = 0
(24a)
TT
00
I dx
ψ
ρ
ww
mNNNN
(24b)
dx)NN( kGA
)NNdx EINNk
x ,w
0
T
x ,w
0
x ,
ψ
ψ
−+=
∫∫
(24c)
00
w
fNN
(24d)
(25)
(26)
(27)
(28)
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Wong, F.T.et al / Krigingbased Timoshenko Beam Element for Static and Free Vibration Analyses / CED, Vol. 13, No. 1, March 2011, pp. 42–49
46
The free vibration of a structure with the natural
frequency ω can be written as
( )()
ϕ+=
ωt sin
0
where D0 is the amplitude of the vibration and φ is
the phase angle. Entering Eq. (29) into Eq. (28)
leads to the eigen equation
(K ― ω2M)D0 = 0
where D0 is a vector of eigenvalues, which is the
mode shape of vibration, corresponding to an
eigenvector ω 2.
Numerical Results
The performance of the developed Krigingbased
Timoshenko beam element was tested using a
number of different static and freevibration
problems. In the following, for the sake of brevity,
only the results for the beam element with the K
FEM options of the cubic basis, threelayer DOI (Fig.
2), and Gaussian correlation function (P33G) are
presented (see Syamsoeyadi [17] for the compre
hensive results). The issues considered include the
shear locking phenomenon, the accuracy and the
convergence of the displacements, bending moments,
shear forces, and natural frequencies.
Firstly, the appropriate range for the correlation
parameter θ and the efficient number of Gaussian
sampling points for evaluating the integrations in
Eqs. (24b), (24c) and (24d) were determined by
conducting a series of numerical tests. The tests for
finding the lower bound and the upper bound of θ
satisfying Eqs. (10a) and (10b) revealed that for the
beam element with Kriging option P33G, the lower
and upper bound values for θ are 104 and 1.9,
respectively. In the subsequent analyses the
parameter θ was taken to be equal to 1, which was
nearly the midvalue between the lower and upper
bounds.
To obtain an accurate yet efficient number of the
integration sampling points, a cantilever beam
subjected to triangulardistributed load as shown in
Fig. 3 was analyzed using different number of
sampling points for evaluating the stiffness matrices,
Nsamp. Subsequently, it was analyzed with different
number of sampling points for evaluating the force
vector, Nbody,. The freeend deflection of the beam
was observed and compared to the exact solution
[12], viz.
DtD
(29)
(30)
Figure 2. Threelayer DOI of a typical 1D element
4
0
EI
t
5
1
12
q L
w
φ
⎛
⎜
⎝
⎞
⎟
⎠
⎞
⎟
⎠
=+
(31a)
2
1(12 11 )
5
Where qo, L, E, I, φ, v, and h are respectively the
value of the triangular load at the clamped end, the
length, the modulus of elasticity, the moment of
inertia, the ratio of bending stiffness to shearing
stiffness, Poisson’s ratio and the thickness of the
beam.
The results were presented in Tables 1 and 2. The
tables show that at least two sampling points are
needed to yield accurate integrations. In the
subsequent analyses Nsamp was taken to be equal to 3
while Nbody was taken to be equal to 2. In the case
where the SRI technique was employed, the number
of sampling points for the bending term was 3 while
that for the shearing term was 1.
Shear Locking
Shear locking is a phenomenon where the beam
element is excessively stiff for the range of very small
thickness (or the lengthtothickness ratio, L/h, is
very large). To observe this phenomenon, a clamped
clamped beam with L = 10 m, E = 2000 kN/m2, k =
0.84967, v = 0.3, subjected to uniformlydistributed
load, q=1 kN/m, was analyzed using eight elements.
The height of the beam was varied from thick, L/h=5,
up to extremely thin, L/h=10000.
h
L
φν⎛
=+
⎜
⎝
(31b)
Figure 3. Cantilever beam (E=2000 kN/m2, k=0.84967,
v=0.3) subjected to unit triangular distributed load (kN/m),
divided into eight equal finite elements
Table 1. Relative displacement errors for different number
of sampling points on stiffness matrix (in this case Nbody = 2)
Nsamp
1
2
3
4
Relative error (%)
2.23034
0.96674
0.96674
0.96674
Table 2. Relative displacement errors for different number
of sampling points on equivalent nodal force vector (in this
case Nsamp = 2)
Nbody
1
2
3
4
Relative error (%)
0.28165
0.96674
0.96674
0.96674
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Wong, F.T.et al / Krigingbased Timoshenko Beam Element for Static and Free Vibration Analyses / CED, Vol. 13, No. 1, March 2011, pp. 42–49
47
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1 10 1001000 10000
L/h
w/wEuler
P33G
P33G (SRI)
Figure 4. Normalized deflection at the midspan of the
beam for different lengthtothickness ratios
The deflection at the midspan was observed and
normalized to the EulerBernoulli beam solution, i.e.
qL4
wEuler = 384EI
Theoretically, as the beam becomes very thin, the
solution of Timoshenko beam theory will converge to
the solution of EulerBernoulli beam theory. The
results of the analyses were presented in Fig. 4 for
the cases of using full integration, P33G, and
selectivereduced integration, P33G (SRI). The
figure shows that the element with full integration
suffers shear locking while the element with SRI
indicates no locking. Thus, the SRI technique is
effective to eliminate the shear locking. This finding
confirms the conclusion in the previous study [18].
Note that the SRI technique is also applicable for
Krigingbased rectangular ReissnerMindlin plate
elements [18] but it is not applicable for triangular
elements [10].
Accuracy and Convergence
To study the accuracy and convergence of the static
solutions of the beam elements using P33G and P3
3G SRI, the beam subjected to triangular
distributed load (Fig. 3) was analyzed with the
parameter E=1000 kN/m2, I=0.020833 m4, A=1 m2,
G=384.61 kN/m2, k=0.84967, v=0.3, L=4 m, L/h=8.
The beam was divided into 2, 4, 8, and 16 elements.
The deflections at the free end, the bending moments
and the shearing forces at the clamped end were
observed. The results were normalized with respect
to their respective exact solutions and presented in
Tables 35 together with the results of the
superconvergent beam element of Friedman and
Kosmatka (F&K) [12]. The tables show that both P3
3G and P33G (SRI) can produce very accurate
displacements and reasonably accurate moments.
The element with P33G gives slightly better results
than the element with SRI. The shearing forces
directly calculated from the shearing strains (Eq. 22),
however, cannot produce accurate results. The use of
SRI worsen the shearing forces. It is worthy to note
(32)
that this fact is also true for the traditional
Timoshenko beam element [19]. It is interesting to
notice that the results using full integration for the
displacement and moment, converge from below,
while those using SRI converge from above. This is
because the use of SRI makes the stiffness matrix
becomes ‘less stiff’ than the actual.
Free Vibration
To investigate the accuracy and convergence of the
natural frequency solutions from the beam element
with P33G and P33G (SRI), free vibrations of a
thick and a very thin simplysupported beams were
considered. The beam parameters are L=10 m, the
width b=1 m, E= 2x109 N/m2, ν=0.3, ρ=10 kg/m3,
L/h=5 for the thick beam and L/h=1000 for the very
thin beam. The beams were divided into 4, 8, 16, and
32 elements. The resulting natural frquencies were
expressed in the form of nondimensional frequency
parameter, i.e.
m
L
ii
where λi is the nondimensional frequency parameter,
ωi is the angular natural frequency of the vibration
modei and m is the mass per unit length. For the
thick beam, the convergent solutions of the
pseudospectral method [20] were taken to be the
reference solutions to assess the accuracy. Whilst for
the very thin beam, the exact solutions of the Euler
Bernoulli beam [18] were used as the reference
solutions. The normalized results of the analyses
using full integration and SRI were presented in
Tables 6 and 7 together with the reference solutions.
Table 3. Deflections at the free end normalized to the
deflection of exact solution
EI
λ
2
ω=
(33)
Number of
elements
4
8
16
P33G P33G (SRI) F&K
0.9998
0.9999
0.9999
1.0042
1.0005
1.0000
1
1
1
Table 4. Bending moments at the clamped end normalized
to the bending moment of exact solution
Number of
elements
4
8
16
P33G P33G (SRI) F&K
0.9350
0.9924
0.9993
1.0778
1.0441
1.0217
0.96
1
1
Table 5. Shearing forces at the clamped end normalized to
the shearing force of exact solution
Number of
elements
4
8
16
P33G P33G (SRI) F&K
1.6338
1.1146
1.0175
4.1432
1.8706
1.4110
0.8
0.9
1
Page 7
Wong, F.T.et al / Krigingbased Timoshenko Beam Element for Static and Free Vibration Analyses / CED, Vol. 13, No. 1, March 2011, pp. 42–49
48
The tables demonstrate that for the case of thick
beam the performance of the elements is excellent,
both with full and reduced integrations. For very
thin beams, a very fine mesh is needed to achieve
good accuracy for high mode frequencies. The
element with SRI outperforms the element with full
integration for the case of thin beam.
Conclusions
Krigingbased Timoshenko beam elements have
been developed and tested in the static and free
vibration problems. The test results show that the
shear locking can be eliminated using the SRI
technique. The elements, both with full and reduced
integrations, can produce accurate displacement,
bending moment, and natural frequencies. For thick
and not very thin beams, the element with full
integration gives better overall results while for very
thin beams, the element with SRI is better. The
shearing forces directly calculated from the shearing
strains, however, are not accurate. While the SRI
technique works well for eliminating the shear
locking, the use of this technique worsen the
shearing force results. Alternative methods to
eliminate the shear locking needs to be studied in
the future research.
References
1. Gu, L., Moving Kriging Interpolation and Element
Free Galerkin Method, International Journal for
Numerical Methods in Engineering, John Wiley
and Sons, 56 (1), 2003, pp. 111.
Table 6. Normalized frequency parameter for the thick beam (L/h=5)
λ/λ*
4 element
Full
1.0080
1.0697
1.1472
1.4321
1.2654
1.1991
1.3375
1.3814







8 element
Full
1.0047
1.0144
1.0263
1.0436
1.0682
1.0945
1.0612
1.0562
1.0916
1.0830
1.0859
1.1166
1.1747
1.2066
1.4033
16 element
Full
1.0047
1.0139
1.0224
1.0294
1.0352
1.0402
1.0612
1.0562
1.0452
1.0467
1.0506
1.0380
1.0570
1.0310
1.0644
32 element
Full
1.0047
1.0139
1.0224
1.0293
1.0347
1.0390
1.0612
1.0562
1.0425
1.0467
1.0453
1.0380
1.0476
1.0310
1.0495
Mode
Shape
λ*
SRI
0.9949
0.9485
1.1091
1.4186
1.2406
1.1673
1.2409
1.3005







SRI
1.0041
1.0086
1.0050
0.9973
1.0079
1.0666
1.0649
1.0556
1.0915
1.1217
1.1063
1.1104
1.1640
1.1752
1.3669
SRI
1.0046
1.0134
1.0211
1.0264
1.0292
1.0295
1.0603
1.0480
1.0355
1.0421
1.0333
1.0339
1.0378
1.0279
1.0489
SRI
1.0047
1.0138
1.0222
1.0289
1.0342
1.0382
1.0608
1.0555
1.0413
1.0460
1.0436
1.0373
1.0453
1.0304
1.0465
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
3.0453
5.6716
7.8395
9.6571
11.2220
12.6022
13.0323
13.4443
13.8433
14.4378
14.9766
15.6676
16.0241
16.9584
17.0019
Note: * Convergent solution from the the pseudospectral method [20]
Table 7. Normalized frequency parameter for the very thin beam (L/h=1000)
λ/λ*
4 element
Full
1.0535
1.1277
8.0782
220.1137
176.0909
146.7428
125.7805
110.0582







8 element
Full
1.0263
1.1234
1.3326
2.4924
4.2911
6.5564
8.9450
110.0568
97.8283
88.0457
80.0417
73.3720
67.7285
62.8912
58.7001
16 element
Full
1.0004
1.0059
1.0227
1.0707
1.1270
1.2379
1.5666
2.0798
2.6893
3.3680
4.1110
4.9077
5.7165
6.4285
6.8404
32 element
Full
1.0000
1.0001
1.0005
1.0016
1.0035
1.0063
1.0102
1.0156
1.0254
1.0441
1.0758
1.1235
1.1964
1.3053
1.4512
Mode
Shape
λ*
SRI
0.9903
0.9314
1.4430
18.3873
126.4865
147.1230
128.3333
113.7652







SRI
0.9995
0.9947
0.9822
0.9720
1.0059
1.1981
2.3078
12.8434
51.8049
75.5844
77.8939
73.2515
67.7595
62.9363
60.0300
SRI
1.0000
0.9997
0.9989
0.9973
0.9948
0.9920
0.9905
0.9936
1.0069
1.0381
1.0982
1.2059
1.4031
1.8181
3.0031
SRI
1.0000
1.0000
0.9999
0.9998
0.9997
0.9994
0.9991
0.9986
0.9981
0.9976
0.9972
0.9971
0.9975
0.9987
1.0012
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
3.1416
6.2832
9.4248
12.5664
15.7080
18.8496
21.9911
25.1327
28.2743
31.4159
34.5575
37.6991
40.8407
43.9823
47.1239
Note: * Exact solution of EulerBernoulli beam theory
Page 8
Wong, F.T.et al / Krigingbased Timoshenko Beam Element for Static and Free Vibration Analyses / CED, Vol. 13, No. 1, March 2011, pp. 42–49
49
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