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E3S Web of Conferences 281, 04002 (2021) https://doi.org/10.1051/e3sconf/202128104002
CATPID-2021 Part 1
Hybrid finite element formulation for
geometrically nonlinear buckling analysis of
truss with initial length imperfection
*
Vu Thi Bich Quyen and Dao Ngoc Tien
Faculty of Civil Engineering, Hanoi Architectural University, Km 10, Nguyentrai, Hanoi, Vietnam
Abstract. This paper presents a novel hybrid FEM-based approach for
nonlinear buckling analysis of truss with initial length imperfection. The
two types of truss finite element
contribution deals with establishing
(perfection and imperfection element) considering large displacement
based on displacement formulation and mixed formulation. Therefore, the
hybrid global equation system is developed by assembling perfection and
imperfection truss elements. The incremental-iterative algorithm based on
the arc-length method is used to establish calculation programs for solving
geometrically nonlinear buckling analysis of truss with initial length
imperfection. Using a written calculation program, the numerical test is
presented to investigate the equilibrium path for plan truss with initial
member length imperfection.
1 Introduction
Many truss members have initial geometric imperfections as a result of manufacturing,
transporting, and handling processes. This initial member imperfection significantly
influences the buckling behaviour of the truss structure. In recent years, many research
works addressed the influence of geometrical imperfection on the behaviour of truss
structures [1-4]. For solving the buckling problem of truss structure, the finite element
method is considered the most popular and efficient method. In geometrical linear finite
element analysis, the length imperfection usually is calculated by adding equivalent loads
to the nodal external force vector. However, in geometrical nonlinear analysis, it cannot be
used. Generally, the solution of nonlinear buckling problem of truss based on displacement
finite element formulation requires the implementation of length imperfection to the mater
stiffness matrix. The operation of incorporating length imperfection considerably increases
the difficulty in constructing and solving nonlinear incremental balanced equations of the
system. For escaping difficulties of the mathematical treatment of imperfection, in [5] the
author proposed an approach to formulate the nonlinear buckling problem of truss with
imperfection based on mixed finite element formulation. The mixed model has significant
advantage over displacement-based formulation model but increases the solving system
dimension. Nowadays, the hybrid finite element approach is widely used to solve the
*
Corresponding author: bquyen1312@gmail.com
© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons
Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).
E3S Web of Conferences 281, 04002 (2021) https://doi.org/10.1051/e3sconf/202128104002
CATPID-2021 Part 1
nonlinear contact mechanic problem such as displacement-based finite elements are
difficult to solve [6-9]. In this work, the author proposes a novel hybrid finite element
approach for constructing the solving system of equation. The main idea is establishing two
types of truss finite element considering large displacement based on displacement
formulation and mixed formulation. The global equation system is developed by
assembling two types of proposed truss elements. The solving algorithm of geometrically
nonlinear buckling analysis of truss system is built by employing arc length method due to
its efficiency to predict the proper response and follow the nonlinear equilibrium path
through limit. Therefore, a new incremental-iterative algorithm for solving constructed
system of equation and calculation program is established. The numerical results are
presented to verify the efficiency of the proposed method.
2 Equilibrium equations for the truss elements considering large
displacements
For hybrid finite element formulation, the research proposed to discretize the truss system
into two types of the truss elements: first type element eI – perfection truss element; eII -
imperfection truss element with initial length imperfection (shown in Fig.1).
e
e (e )
Y II
(e ) (e )
II I
e (e ) (e )
I I
0 X
Fig. 1. Truss elements’ types
Let us consider two-node truss elements eI and eII in the global coordinate system (X0Y) as
shown in Fig.2.
Y P4 Y P4
j' j'
L P3 L P3
u4 (e ),A,E u4
(e ),A,E II
P2 I P2
u1 j x u1 Pe u5 j x
i' P1 u3 Pe i' P1 u3
y u2 y u2 (e ),A,E
(e ),A,E II
I (Y -Y ) (Y -Y )
° 2 1 ° 2 1
Le
i i
L0 e L0
( X -X ) X ( X -X )
0 2 1 0 2 1 X
Fig. 2. Truss elements e and e considering large displacements
I II
The following is designated
XY,,X,Y th th
: i and j nodal coordinates in global coordinate system before and
11 22
after deformation;
L L th th
0 và : distance between i and j node before and after deformation;
2
E3S Web of Conferences 281, 04002 (2021) https://doi.org/10.1051/e3sconf/202128104002
CATPID-2021 Part 1
nonlinear contact mechanic problem such as displacement-based finite elements are and : nodal displacements and forces in global coordinates;
uuuu,,, PPPP,,,
difficult to solve [6-9]. In this work, the author proposes a novel hybrid finite element 1234 1234
th
P : resultant external force at the i cross section after deformation;
approach for constructing the solving system of equation. The main idea is establishing two e
types of truss finite element considering large displacement based on displacement : resultant external force at the ith cross section after deformation;
uPN
formulation and mixed formulation. The global equation system is developed by A 5e E N
assembling two types of proposed truss elements. The solving algorithm of geometrically : cross sectional area of truss element; : elastic modulus of material; : axial load of
nonlinear buckling analysis of truss system is built by employing arc length method due to truss element.
its efficiency to predict the proper response and follow the nonlinear equilibrium path The length of the truss element after deformation is defined as
through limit. Therefore, a new incremental-iterative algorithm for solving constructed
22
L(XXuu)(YYuu)
system of equation and calculation program is established. The numerical results are (1)
presented to verify the efficiency of the proposed method. 2131 2142
The axial deformation of perfection truss element and imperfection truss element are
2 Equilibrium equations for the truss elements considering large obtained
displacements
()e 22
I
(e) : L LLL(XX)(YY) (2.1)
For hybrid finite element formulation, the research proposed to discretize the truss system I0 21 21 (2)
()e 22
II
into two types of the truss elements: first type element e – perfection truss element; e - (e): L LLLLL (XX)(YY)(2.2)
III II e 0e 21 21 e
imperfection truss element with initial length imperfection (shown in Fig.1).
e Work of internal axial force can be computed for each truss element as following
e(e )
YII
LL
00
e ()dx
()
(e ):V I = dV dA dx N dx
(e )(e )
III
I xx A x x dx
00
e LL
4
(e )(e )
II L
Ndx N dx NL N u ; (3.1)
i
i u
001
i
(3)
Le Le ()
e dx
()
( ): II =
e V dV dA dx N dx
0 II x x A x x dx
X 00
LL
4
LL
Fig. 1. Truss elements’ types
NL N u (3.2)
N dx N dx
ie
u
00i1
ie
Let us consider two-node truss elements eI and eII in the global coordinate system (X0Y) as
shown in Fig.2.
YP4YP4 For each truss element, the virtual external work can be defined as
j'j'
LP3LP3 ()e 4
I
(eV): δ =P uP uPuPuPu (4.1)
I 11223344 ii
u4(e ),A,Eu4 i 1 (4)
(e ),A,EII
P2I ()e 4
P2 II
(e): δV=PuPuPuPu+P P uP (4.2)
u1jxu1Pe u5jx II 11223344e e i i e e
i'P1PeP1 i 1
u3i'u3
yu2yu2(e ),A,E
(e ),A,EII
I (Y -Y ) (Y -Y )
°21°21
Le Combining equations (3) and (4), getting total work done by the applied forces and the
ii
L0eL0 inertial forces of a mechanical system
( X -X )X( X -X ) ()e 444
021021X ()e I LL
I
(eV) : + V N uPu NP u0;(5.1)
I i ii ii
uu
111
iii
ii
Fig. 2. Truss elements e and e considering large displacements
III
() 44
e
LL
() II
e
The following is designated ( ): II + P
eVVNuuP (5)
II u i e i i e e
thth
11
ii
ie
XY,,X,Y: i and j nodal coordinates in global coordinate system before and
11 22
4
LL
after deformation;
P 0 (5.2)
N uNP
e
thth u ii e
L và L : distance between i and j node before and after deformation;
i 1 ie
0
3
E3S Web of Conferences 281, 04002 (2021) https://doi.org/10.1051/e3sconf/202128104002
CATPID-2021 Part 1
Based on the principle of virtual work, in equilibrium the virtual work of the forces applied
to a system is zero, from equation (5) getting
L
( ) : P 0 ( 1,2,3,4) (6.1)
eN i
Ii
u
i
L (6)
Ni
P 0 ( 1,2,3,4)
i
u
(e ): i (6.2)
II
L
NP
e 0
e
Expressing axial force through deformation and adding deformation from the equation (2)
to equation (6), having the system (7)
()e EA L
I
q() (LL) P, (i1,2,3,4)
u
ii0
Lu
e 0 i
( I ): ()() ()
eee
q II I
( ) P (7.1)
u
or ii
i 1,2,3,4
LL
()
() EA
e 0 e
II
q LL
(, ) ( ) P
u 0
ie e i
Lu
e ei
( II ): LL (7)
()
() EA
e 0 e
II
q LL P
(, ) ( ) 0
u
50
eee
L
ee
(e )()e()e
q II (u II , ) P II (7.2)
or k ek
k 1, 2,.., 5
TT
()e(e)
I II
uuuu uuuuu P
uu,,, ; ,,,,
1234 12345 e
Input incremental loading into the equation (7) and express in matrix format
()e ()e()ee()
I III
(eI ) : k (uu)(P P) q(u) (8.1) (8)
e eee
() () () ()
II II II II
(e ): k(u,) uP( P) q(u,) (8.2)
II e e
Where the tangent stiffness matrices are written
(e ) (e ) ()e ()e
I I II II
qu( ,)
() qu() ()
ee
() ()
ee
I I II II e
(e ): ku( ) ; (e ): k(u,)
I () II e ()
ee
I II
uu
()e ()e
II II
ku( ,)
The length imperfection is considered in the tangent stiffness matrix e
e
of element eII.
4
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