Microsoft Word - ETASR_V12_N2_pp8458-8462
Engineering, Technology & Applied Science Research Vol. 12, No. 2, 2022, 8458-8462 8458
www.etasr.com Hang et al.: Stochastic Buckling Analysis of Non-Uniform Columns Using Stochastic Finite Elements …
Stochastic Buckling Analysis of Non-Uniform
Columns Using Stochastic Finite Elements with
Discretization Random Field by the Point Method
Do Thi Hang
University of Transport and Communications
Hanoi, Vietnam
hangdo@utc.edu.vn
Nguyen Xuan Tung
University of Transport and Communications
Hanoi, Vietnam
ngxuantung@utc.edu.vn
Dao Ngoc Tien
Hanoi Architectural University
Hanoi, Vietnam
tiendn@hau.edu.vn
Received: 9 February 2022 | Revised: 9 March 2022 | Accepted: 10 March 2022
Abstract-This study examined the discretization random field of
the elastic modulus by a point method to develop a stochastic
finite element method for the stochastic buckling of a non-
uniform column. The formulation of stochastic analysis of a non-
uniform column was constructed using the perturbation method
in conjunction with the finite element method. The spectral
representation was used to generate a random field to employ the
Monte Carlo simulation for validation with a stochastic finite
element approach. The results of the stochastic buckling problem
of non-uniform columns with the random field of elastic modulus
by comparing the first-order perturbation technique were in
good agreement with those obtained from the Monte Carlo
simulation. The numerical results showed that the response of the
coefficient of variation of critical loads increased when the ratio
of the correlation distance of the random field increased.
Keywords-non-uniform column; buckling; stochastic FEM;
spectral representation; random field
I. INTRODUCTION
Science and technology revolution has produced many
high-strength and lightweight materials used in a variety of
slender structures such as steel structures [1-7] and functionally
graded beams [8]. The problem of stability calculation of these
structures is very important, e.g. in a steel truss bridge with
many slender compression members. In many cases, column or
tower structures are designed with variable cross-sections to
match the bearing characteristics of the structure and save
materials. Many studies have examined the stability of bars
with variable cross-section and many basic problems on the
stability of bars with variable cross-section were presented in
[9-12]. In addition, many researchers studied the stability of the
replacement with a changing cross-section with different forms,
calculated by different methods. In [13], a polynomial series
approximation was used to solve the problem of column
stability with variable cross-sections subjected to axial forces.
In [14], the column was calculated with a cross-sectional
variation of the ladder using the Rayleigh-Ritz method. In [15],
the stability of a column of the variable cross-section with
elastic connections was calculated by approximating the
stability differential equation. In [9], the stability of reinforced
concrete columns with elastic connections was studied. An
exact solution for some types of columns with variable cross-
sections with elastic connections under distributed axial forces
was presented in [16]. In addition to analytical methods with
exact solutions, some studies used finite element methods. In
[17], the stability of a bar with variable cross-section was
studied using the Galerkin finite element method. In [18], the
stability calculation of columns with stepped and cracked
cross-sections was studied using the finite element method. In
[19], a finite element model of circular concrete-filled steel
circular-tube columns was studied under axial compression
loading.
However, these studies were limited to deterministic
problems. In most practical engineering, structure analysis
ignores the random heterogeneity of materials. For the
advanced analysis of structures, a probabilistic model is needed
to consider random variables, random fields of geometry, and
material properties of structures. Several methods for random
field discretization were developed for stochastic finite element
methods, such as the integration point method [20, 21], the
nodal point method [22, 23], and the local averaging method
[24]. In [25], a stochastic finite element method was presented
using the weighted integration approach to analyze the static
behavior of non-uniform columns. In [26], a semi-analytical
method was applied to investigate the buckling of composite
beam-columns with random elastic stiffness and geometric
properties. In [27], the free vibration of functionally graded
beams was examined using a stochastic perturbation-based
Corresponding author: Nguyen Xuan Tung
Engineering, Technology & Applied Science Research Vol. 12, No. 2, 2022, 8458-8462 8459
www.etasr.com Hang et al.: Stochastic Buckling Analysis of Non-Uniform Columns Using Stochastic Finite Elements …
finite element method. The random finite element method was
improved for the seismic analysis of gravity dams in [28]. In
[29, 30], a probabilistic problem was developed for the seismic
analysis of cabinet facilities in nuclear power plants and soil
properties.
In this study, a stochastic finite element model was built for
the buckling of the non-uniform column with the random field
of elastic modulus.
II. THE FINITE ELEMENT MODEL OF THE NON-UNIFORM
COLUMN
Consider a non-uniform column with the coordinate system
(x, z) as shown in Figure 1:
Fig. 1. Model of non-uniform columns
The column element is assumed to have two degrees of
freedom, one rotation and one translation, at each end. The
location and positive directions of these displacements in a
typical linearly tapered column element are shown in Figure 2,
where Le is the length of the element, E and I are the area and
inertia moment of the cross-section column. The depths of the
cross-sections at the smaller and larger end of the column are
denoted as h2 and h1, respectively. The longitudinal axis of the
element lies along the x-axis. The element was assumed to
have two degrees of freedom at each end: a transverse
deflection u1, u3, and an angle of rotation of slope u2, u4.
Fig. 2. Model of a non-uniform finite element.
The displacement field is approximated by the shape
function as:
�� = ��� �� �� �
�
�������
= ��
���� (1)
where � = ��� �� �� � �, and �� is the shape function
of the i-th degree of freedom, and the Hermite polynomial
functions [31] are:
�� = 1 − 3 � ���� + 2 � ����
�� = � �1 − 2 ��� + � ���� � �� = 3 � ���� − 2 � ����
� = � �− ��� + ����� �
(2)
The deformation potential of the column is:
�� = �� � !"�# $%�&�% �� '� (���) (3)
The potential energy of load is:
*� = − +� � $%&�%� '� (���) (4)
The cross-section moment of inertia in the element is
approximated by linear interpolation:
"�#!�"�# = "�#!�� $1 − ���' + "�#!�� ��� (1)
The random field of elastic modulus is assumed as: "�# = )�1 + ,"�#
(2)
where E0, r(z) are the mean elastic modulus and a one-
dimensional Gaussian random field with a mean equal to zero
respectively. The form of the autocorrelation function of
random field r(x) is: -".# = � � ,"�#,"� + .#/�"�, � + ., .#(,"�#(,"� + .#∞1∞∞1∞ (7)
The random field elastic modulus requires discretization to
random variables for the governing equation of buckling
problem by finite element formulation.
Fig. 3. Average model for approximating random field of elastic modulus.
EI0
h
0
h1
H
P
L
z
u
1
u
2
u
3
u
4
h1
h2
w
e
(e)
EI1
EI2
Ei
E1
E2
E3
En
E
Ei
E(z)
E1
E2
E3
En
Engineering, Technology & Applied Science Research Vol. 12, No. 2, 2022, 8458-8462 8460
www.etasr.com Hang et al.: Stochastic Buckling Analysis of Non-Uniform Columns Using Stochastic Finite Elements …
By averaging random variables within the element as
illustrated in Figure 3, the random field of the elastic modulus
in the element is calculated as: ,̄ = 34 53� 5⋯5378 = ) 91 + 3453� 5⋯5378 : (8)
Taking the variation with respect to q, the resulting
equation [32] is obtained for the buckling of the column: ;�<
− =�<
>?��� = 0 (9)
[K] and [M] denote the assembled and global stiffness
respectively of the cross-section column and λi denotes the
critical load. The stiffness matrix [K]e is defined as: �<
� = ) 91 + 3453�5⋯5378 : ×
×
BC
CC
CC
DE"F4� 5F�� #��� F4� 5�F����� − E"F4� 5F�� #��� �F4� 5 F������F4� 5F���� − F4� 5�F����� F4� 5F����E"F4�5F�� #��� − �F4� 5 F�����GHI. F4� 5�F���� KL
LL
LL
M
(10)
The geometric stiffness matrix is defined as:
�<
>� = N
BC
CC
CC
D EO�� ��) − EO�� ��)��O�� − ��) − ��)��EO�� − ��)Sym ��O�� KL
LL
LL
M
(3)
III. FORMULATION OF THE STOCHASTIC FINITE ELEMENT
METHOD USING THE PERTURBATION TECHNIQUE FOR THE
BUCKLING OF THE COLUMN
The governing equation of the buckling problem in (9)
includes random variables and can be perturbed concerning the
mean of the random variables as follows:
S �<
) + ∑ U�V
U3W ,�X3�Y� −$=) + ∑ UZU3W ,�X3�Y� ' �<
> [ \�) + ∑
U]U3W ,�X3�Y� ^ = 0 (12)
Solving the stochastic equation (12), the zeroth-order and
the first-order solutions can be obtained as:
zeroth-order: ;�<
) − =)�<
> ?��)� = 0 (4)
and first-order:
;�<
) − =) �<
> ? \ U]U3W^ = − $U�V
U3W − UZU3W �<
> ' ��)� (14)
Premultiplication of (14) by \�) + ∑ U]U3W ,�X3�Y� ^_gives:
\�) + ∑ U]U3W ,�X3�Y� ^_ × × \�<
) + ∑ U�V
U3W ,�X3�Y� − $=) + ∑ UZU3W ,�X3�Y� ' �<
>^ × \�) + ∑ U]U3W ,�X3�Y� ^ = 0
(15)
Solving (15) using the orthonormal property, the first-order
partial derivatives of critical load respect random variable is
given by:
UZU3W ≈ �]a �Wb
c�d
ceW �]a��]a�Wb�f
�]a� (16)
The mean of the critical load given solved by the first-order
perturbation solutions is:
μZ = � \$=) + ∑ hZh3W ,�X3�Y� ' − =)^ i",� #(,�j1j = =) (17)
The variance of the critical load is solved by the first-order
perturbation solution as follows:
*k,Z = � �
\$=) + ∑ UZU3W ,�X3�Y� ' − =)^ ×l× �=) + ∑ UZU3m ,nX3nY� � − =)o ×× /�;,�, ,n , ,n − ,� ?(,� (,n
∞1∞∞1∞
= ∑ ∑ \ UZU3W UZU3W -".#^X3nY�X3�Y�
(18)
where R(τ) denotes the autocorrelation function of the random
field, and the relative distance vector is defined as τ=rj-ri. The
autocorrelation function was assumed in the form:
-".# = q� rsi $− t�%�' (19)
where σ, d are the Coefficient Of Variation (COV) and the
correlation distance of the random field of elastic modulus
respectively. The response variability can be represented using
the COV defined as:
uv* = wxy3z||z| (20)
IV. EXAMPLES
A. Validation of the Finite Element Approach for
Deterministic Analysis
The buckling non-uniform column in Figure 1 was
considered to validate the proposed finite element for the non-
uniform column with the moment of inertia as a formulation:
!"�# = }Fa� \√2 − �� ;√2 − 1?^� (21)
The analytical solutions of the column were given by [9]
with a critical load factor m=2.023:
N�3 = 2.023 }Fa� � (22)
where the critical load factor m is given by:
I = +�e� �}Fa (5)
Engineering, Technology & Applied Science Research Vol. 12, No. 2, 2022, 8458-8462 8461
www.etasr.com Hang et al.: Stochastic Buckling Analysis of Non-Uniform Columns Using Stochastic Finite Elements …
Fig. 4. Comparison of critical load factor between analytical and finite
element method.
Fig. 5. Error depending on mesh refinement.
As shown in Figure 4, the finite element agreed with the
analytical solution if a reasonable number of finite elements
was used in the model (approximately more than 10 finite
elements). The convergence features of the proposed finite
element are shown in Figure 5. The discrepancies between the
analytical and finite element solutions of the critical load factor
tend rapidly to zero as the mesh is refined.
B. Response Variability of Critical Load Due to the
Randomness of Elastic Modulus
A scheme of Monte Carlo Simulation (MCS) was employed
to validate the response variability of critical loads of the non-
uniform columns. MCS repeats the deterministic analysis on a
set of samples of the random field of the elastic modulus. Using
the spectral representation proposed in [33], the numerical
generation of the homogeneous univariate random field r(z),
with zero mean in one dimension, can be generated via the
summation formula of cosine functions as follows: ,"�# = √2 ∑ �� ���"�� � � �� #X1��Y)
�� � w2G33 "���#��
�� �
�]
�
, �� � ���, � � 0,1,2, . . . , � � 1
(24)
where, �], G33 are the upper cut-off frequency and the power
spectral density functio, respectively.
The stochastic buckling of linearly tapered cantilever
columns with a concrete rectangular cross-section was
considered, as shown in Figure 6. The mean modulus of
elasticity of the column was assumed to be 33.10
3
MPa. The
geometric dimensions of the example cross-section columns
were: h0 =1m, h1 =0.5 m, H=12m, and b=0.6m.
Fig. 6. Linearly tapered cantilever columns with a concrete rectangular
cross-section.
Fig. 7. Effects of the correlation distance d on the different standard
deviation σ of the critical load.
Figure 7 shows the comparison of the effect of the
correlation distance d of the random field on the variability of
the critical load of the proposed formulation with the MCS, for
the same cases of the stochastic finite element method. The
results of the MCS with 10,000 samples, presented by the
dashed-dotted line, denote the corresponding COV of the
random field of elastic modulus 0.1 and 0.2. As shown in
Figure 6, the response variability of a critical load is
converging to COV of the random field of the elastic modulus,
as the correlation distance tends to move to infinity in both
analysis schemes. As can be observed, the increasing rate of the
correlation distances is accelerated with an increase in the
coefficient of variation of the random field.
V. CONCLUSION
The paper presented a perturbation technique in conjugation
with finite element analysis that was successfully developed for
the stochastic buckling problem of non-uniform columns with a
random field of elastic modulus. MCS was performed
employing 10,000 random samples to simulate the validity of
the proposed first-order perturbation solution of the stochastic
finite element method. The efficacy of the first-order
b
h
cross-section
P
h
0
h
1
H
Engineering, Technology & Applied Science Research Vol. 12, No. 2, 2022, 8458-8462 8462
www.etasr.com Hang et al.: Stochastic Buckling Analysis of Non-Uniform Columns Using Stochastic Finite Elements …
perturbation method was verified using a homogeneous
Gaussian random field by the stochastic finite element method
and was in perfect agreement with the MCS, where the
correlation distance was as high as 5. The effect of the
correlation distance on the response COV of a critical load was
obvious, and the response COV increased when correlation
distances increased.
ACKNOWLEDGMENT
This research is funded by the University of Transport and
Communications (UTC) under grant number T2021-CT-004.
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