# Kriging-Based Timoshenko Beam Element for Static and Free Vibration Analyses

**ABSTRACT** An enhancement of the finite element method using Kriging interpolation (K-FEM) 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 K-FEM 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 selective-reduced 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.

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**ABSTRACT:**Recently, Kriging-based finite element method (K-FEM) has been developed for analysis of Reissner-Mindlin plates. This method provides sufficient flexibility in customizing the interpolation function for desired smoothness and accuracy. In the application to thin plates, however, the well-known finite element drawback of transverse shear locking still remains in the K-FEM, particularly when low order basis function is used. In this study, the concept of assumed natural transverse shear strain is introduced to alleviate the shear locking. The positions of the shear strain sampling points and the assumed natural strain fields are determined approximately by assuming the deflection field is linear within the triangular integration cells. Numerical tests on a hard simply supported square plate and on a clamped circular plate are carried out to assess the effectiveness of the present method. The tests show that the shear locking can be considerably reduced but it still cannot be completely eliminated. In the application to thick plates, however, the solutions of the present method are less accurate compared to those of the standard K-FEM.01/2006; - International Journal of Computational Methods 01/2009; 6(1):1-27. · 0.48 Impact Factor
- SourceAvailable from: Worsak Kanok-Nukulchai[show abstract] [hide abstract]

**ABSTRACT:**An enhancement of the finite element method with Kriging shape functions (K-FEM) was recently proposed. In this method, the field variables of a boundary value problem are approximated using ‘element-by-element’ piecewise Kriging interpolation (el-KI). 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. This paper presents a numerical study on the accuracy and convergence of the el-KI in function fitting problems. Several examples of functions in two-dimensional space are employed in this study. The results show that very accurate function fittings and excellent convergence can be attained by the el-KI.Civil Engineering Dimension. 01/2009;

Page 1

ISSN 1410-9530 print / ISSN 1979-570X online

Kriging-based Timoshenko Beam Element for Static and Free

Vibration Analyses

42

Civil Engineering Dimension, Vol. 13, No. 1, March 2011, 42-49

Wong, F.T.1 and Syamsoeyadi, H.2

Abstract: An enhancement of the finite element method using Kriging interpolation (K-FEM)

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 K-FEM 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 selective-reduced

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, selective-reduced

integration.

Introduction

In an attempt to improve the element-free Galerkin

method with Kriging interpolation [1], Plengkhom

and Kanok-Nukulchai [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 (K-FEM) [3].

The advantages of the K-FEM 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 K-FEM 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 general-purpose FE program

can be easily extended to incorporate the enhanced

method. Thus, the K-FEM has a high chance to be

accepted in real engineering practices.

In the pioneering work [2], the K-FEM was developed

for static analyses of 1D bar and 2D plane-

stress/plane-strain solids. Subsequently, it was

developed for analyses of Reissner-Mindlin plates [3,

4] and improved through the use of adaptive

correlation parameters and by introducing the

quartic spline correlation function. A drawback of

the K-FEM is that the interpolation function is

discontinuous at the inter-element boundaries

(except in 1D problems). In spite of this discontinuity,

using appropriate choice of shape function parameters,

the K-FEM passes weak patch tests and therefore

the convergence is guaranteed [5]. The basic concepts

and advances of the K-FEM have been presented in

several papers [6-8]. The current development is the

extension and application of the K-FEM to different

problems in engineering, such as general plate and

shell structures [9, 10] and multi-scale mechanics

[11].

Despite many attractive features of the K-FEM, in

the application to shear-deformable plates and

shells, the drawback of transverse shear locking and

membrane locking presents in the K-FEM [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 K-FEM is

not yet invented.

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Wong, F.T.et al / Kriging-based 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 shear-deformable 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 Reissner-Mindlin plate. It is

the aim of this paper to present the development and

testing of the K-FEM 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 K-FEM

Named after Danie G. Krige, a South African mining

engineer, Kriging is a well-known 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 K-FEM 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

=∑

x N 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 K-FEM [2-11].

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 K-FEM, 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 / Kriging-based Timoshenko Beam Element for Static and Free Vibration Analyses / CED, Vol. 13, No. 1, March 2011, pp. 42–49

44

Following the previous works [3-10], 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 K-FEM 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 10-b (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 cross-section.

(11a)

(11b)

Figure 1. Coordinate system, deflection and rotation of the

beam

Using the small-strain 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 cross-section, Eq.

(14) yields

()

0

22

∫∫

where I is the moment of inertia of the cross-section.

(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 / Kriging-based 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 cross-section, Eq. (17) can be expressed as

11

22

∫∫

Finally, the work of external forces is given as.

LL

Ww 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

MEI ψ=

()

,x

QkGA w

ψ=−

Formulation of Kriging-based 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 Kanok-Nukulchai [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

&

I dx

ρ ψ=+

&

(18)

00

(19)

()()

00

00

00

,, ,,

LL

LL

xxxx

LL

EIdx w kGA wdx

wq dx

δ

m dx

δψ ψδ 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

A dx

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 selective-reduced integration

(SRI), in which the stiffness matrix is evaluated

using full integration for the bending term (the first

term in the right-hand side of Eq. (24c)) and using

reduced integration for the shearing term (the

second term in the right-hand side of Eq. (24c)). The

effectiveness of the SRI technique to eliminate the

shear locking in the Kriging-based 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:

( )( )( ) t FtDKtDM

=+

& &

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

Idx

ψ

ρ

ww

mNNNN

(24b)

dx)NN(kGA

)NNdxEINNk

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 / Kriging-based 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+=

ωtsin

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 Kriging-based

Timoshenko beam element was tested using a

number of different static and free-vibration

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, three-layer DOI (Fig.

2), and Gaussian correlation function (P3-3-G) 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 P3-3-G, the lower

and upper bound values for θ are 10-4 and 1.9,

respectively. In the subsequent analyses the

parameter θ was taken to be equal to 1, which was

nearly the mid-value between the lower and upper

bounds.

To obtain an accurate yet efficient number of the

integration sampling points, a cantilever beam

subjected to triangular-distributed 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 free-end deflection of the beam

was observed and compared to the exact solution

[12], viz.

DtD

(29)

(30)

Figure 2. Three-layer 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 length-to-thickness 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 uniformly-distributed

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