Hybrid-mixed stress finite element models for the
stability analysis of structures
Pedro Filipe Tiago Arruda, Luís Manuel Santos Castro1
1: Departamento de Engenharia Civil e Arquitectura
Instituto Superior Técnico
Universidade Técnica de Lisboa
Av. Rovisco Pais, 1049-001 Lisboa
e-mail: [email protected]
ABSTRACT
This work presents a hybrid mixed stress finite element model for the linear stability analysis of
beams and Reissner-Mindlin plate bending problems. This model is based on the use of
complete sets of orthonormal Legendre polynomials as approximation functions. The use of
this type of functions allows the development of analytical closed form solutions for the
computation of all integrals involved in the definition of the different structural operators. It
enables also the implementation of highly efficient p- refinement procedures.
Although the governing system may be different from the conventional finite elements, it is
possible to achieve a governing system mathematical identical, but with different physical
meanings. There for the conventional techniques of Eigen values in structures can be applied
with hybrid mixed stress finite elements.
Assuming a physically linear behaviour, the solution of the problem can be determined by
performing a non geometrical analysis in the deformed position. To validate the model and to
illustrate its potential, several numerical tests are presented and discussed. The results
obtained with the hybrid-mixed model are compared with analytical solutions and with other
numerical solutions computed with the classical displacement finite element formulation.
Keywords: Linear Stability Analysis, Hybrid-Mixed Formulation, Stress Model, Legendre
Polynomials, Beams, Reissner-Mindlin Plate Bending problems.
1
INTRODUCTION
In recent years, hybrid-mixed stress finite element models have been developed for the static
analysis of plane stretching and plate bending problems [1-4]. One of the main advantages
associated with the use of this type of formulation it is the flexibility introduced in the
selection of the approximation functions. This fact makes possible the use of functions with
special properties that cannot be implemented in the framework of conventional finite
element models, such as Legendre polynomials [1-2] and systems of wavelets and Walsh series
[3-4]. The hybrid-mixed stress models may lead also to quasi-equilibrated solutions which is
very adequate for design purposes.
This paper presents a generalization of the hybrid-mixed stress model that allows for the
geometrical non linear analysis of frame structures, and Reissner-Mindlin plate bending
1
problems. The model is called hybrid because both the stresses in the domain and the
displacements on the static boundary (which includes the boundary between elements) are
simultaneously approximated. It is termed mixed because both the stresses and the
displacements in the domain are directly approximated. All the fundamental conditions are
imposed using a weighted residual approach designed to ensure that the discrete model
embodies all the relevant properties presented by the continuum it represents, namely statickinematic duality and elastic reciprocity.
Due to the type of enforcement followed for the equilibrium and compatibility conditions in
the domain and because the connection between elements is ensured by the weighted
residual enforcement of the equilibrium conditions on the boundary, the presented
formulation is considered to lead to a stress model.
The model presented in this paper is based on the use of orthonormal Legendre polynomials
as approximation functions. The properties of these functions allow for the definition of
analytical closed form solutions for the computation of all structural operators. Numerical
integration schemes are thus completely avoided. The numerical stability associated to the use
of Legendre polynomial bases enables the use of macro-element meshes where the definition
of highly effective p-adaptive refinement procedures is simplified. A detailed presentation of
these functions can be found in [6].
The model reported in this paper makes possible the linearized stability analysis [10] for the
determination of buckling modes and geometrical non linear stresses and displacementes. This
work starts with the presentation of the fundamental relations. Next, the hybrid-mixed stress
model is discussed. The paper closes with the presentation of some numerical applications and
with the presentation of the main conclusions.
2
FUNDAMENTAL RELATIONS
Consider a set of forces acting in an elastic body. The equilibrium, compatibility and elasticity
equations governing the behaviour of that structure may be expressed as [7]:
0
(2.1)
(2.3)
(2.2)
The vectors s, e and u list the independent components of the stress, strain and displacement
fields, respectively. Assuming a geometrically linear behaviour, the differential equilibrium
operator, D, the differential compatibility operator, D*, represent linear and adjoint operators
and the vector b lists the body force components. The operator LV is a general operator that
takes in account the geometrical non linear behaviour in the domain equilibrium and N is the
normal tension stress.
The boundary of the structure may be subdivided into two complementary regions: (i) the
kinematic boundary, Γu, in which the value for the displacement fields is prescribed and (ii) the
static boundary, Γt, where the applied forces are prescribed. It can be written:
Г Г
Г
(2.4)
The vectors uΓ and t gather the components of the prescribed displacements and forces on the
list the components of the unit
kinematic and static boundaries, respectively. Matrix outward normal vector. The geometrical non linear behaviour in the boundary won’t be
considered since its contribution is small for the buckling modes with HMS [11].
2
Details of the operators of the fundamental relations for beams and plates can be found at [7].
3
HYBRID-MIXED STRESS MODEL
The hybrid-mixed stress (HMS) model presented in this work considers two independent
approximations for the stress and the displacement fields in the domain of each element,
expressed as follows:
(3.1)
(3.2)
where matrices S and UV gather the approximation functions and vectors X and qV list the
corresponding weights.
The displacement field along the static boundary (which includes the boundaries between
neighbouring elements) is also approximated:
Г Г Г
(3.3)
0
(3.4)
(3.6)
Matrix UΓ list the approximation functions and vector qΓ the corresponding weights.
The equilibrium condition is enforced in the following weighted residual form [1],
Replacing in (3.3) the approximations defined for the stress and displacement fields in the
domain (3.1), it is possible to recover the domain equilibrium condition in the discrete model,
expressed as:
(3.5)
!"
with
" (3.7)
The equilibrium conditions on the boundary, and the compatibility and elasticity relations in
the domain are also imposed in a weighted residual form [7],
Г ! Г# 0
leading to [8]
with
3
Г "Г
! 0
$% &
! 0
$% ! Г Г 'Г
(3.8)
(3.9)
Г Г Г#
"Г Г Г#
& 'Г Г Г
(3.10)
The governing system (3.12) is obtained by combining the equilibrium, compatibility and
elasticity conditions in the discrete model.
(3.11)
In chapter 4 there will be presented several possibilities for the matrix GV, in the case of beams
and plates.
&
(!Г
!Г
0
0
'Г
0 ) * + Г , + !"Г ,
!"
(3.12)
It is possible to rewrite the governing system in a condensed format similar to the one which is
typical in the classical finite element formulation. For this purpose, let’s consider [10]:
with
&' -
&
!2
'
- &
.
2
/
0
./ * 0 ' 1 0 "' 1
!"
0
. 3 4
0
'
"' 0 2 1
!"2
Condensing the linear system (3.10) it is possible to obtain:
(3.13)
' 0 1
2
5 6789 : 9":
5 678 . &';< .
" . &' ;<"' "
(3.14)
(3.15)
(3.16)
Details of the operators of the HMS for beams and plates can be found at [7].
4
4.1
GEOMETRICAL MATRIX WITH HMS
Matrix GV1 for beams and plates
It can be demonstrated [7], that if the stress resultants (M, N, V ) are written in the deformed
position relative to the bending fibber [9], and using the divergence theorem, we can obtain
the operator (4.1)disregarding the boundary effects. The new governing system takes the
respective form:
0 0
(4.1)
For beams 1 FG
H
1 1 1 0 0 0
0 0 0
For plates
4
0
N0
M
< M0
M
M0
L
0
0
0
0S
OR
R
0
OG R
OR
0
OPQ
<
< T< (4.2)
&
(!Г
!Г
0
0
'
0 ) * + Г , + !"Г , 5 678 6
!"
! <
< 89 :
"
(4.3)
With this matrix it’s possible to calculate buckling loads with shear flexibility. The procedure
for determining the buckling loads and modes is:
7 4.2
< 0
(4.4)
Matrix GV2 for beams and plates
In this matrix the stress resultants are written in the deformed shape but with the axes always
oriented in the initial position, there for it won’t be possible to achieve buckling loads with
shear flexibility.
0 0 0
0 0 0
(4.5)
For beams 2 F H
2 2 0 0
G
N0 0
M
For plates V M
0 0
M
L0 0
&
(!Г
!Г
0
0
O
O
TWX S
OG
OPR
O
O
TWX
TX R
OG
OPR
Q
0
TW
V
V '
0 ) * + Г , + !"Г , 5 678 ! 6
!"
V
V 89 :
"
(4.6)
(4.7)
The procedure for determining the buckling loads and modes is:
7 !
4.3
V 0
(4.8)
Matrix GV3 for beams
In this geometrical matrix, it will be assumed the same initial hypotheses of 4.2, but since there
is no shear flexibility it is adopted that dw/dx=-θ. There for it’s possible to demonstrate that
[7]:
0 0 0
(4.9)
For beams Y Z0 0 0[
Y Y 0 0 1
!TW
For plates Y Z!TWX
0
&
!
( Г
5
!Г
0
0
!TWX
!TX
0
0
0[
0
Y
Y '
0 ) * + Г , + !"Г , 5 678 6
!"
! Y
Y 89 :
"
(4.10)
(4.11)
The procedure for determining the buckling loads and modes is:
7 5
5.1
Y 0
(4.12)
EXAMPLES
Buckling loads and modes for a simple supported column
In this example we will compare the efficiency of the p- refinement with the h- refinement,
using HMS. These will be compared with theoretical buckling value for the successive buckling
modes
P
9m
E=210 GPa
A=0,01 m 2
I=0,001 m4
u=0,2
F=5/6
Figure 1 – Simple supported column.
Discretizations
A1 (1 element)
A12 (2 elements)
A14 (4 elements)
A18 (8 elements)
Degrees of approximation
gdl
M
N
V
56
13
2
13
64
7
2
7
80
4
2
4
96
2
2
2
Table 1 – Adopted Discretization.
For the respective degrees approximation of the displacements, these will be one time inferior
to their respective stress resultants, to avoid any dependence in the governing system.
6
100,0000%
Error (%)
10,0000%
A1
1,0000%
A12
0,1000%
A14
0,0100%
A18
0,0010%
1
2 3
4 5 6 7 8 9 10
Buckling Modes
Error (%)
Figure 2 – Error of buckling loads using the matrix GV1.
100,0000%
10,0000%
1,0000%
0,1000%
0,0100%
0,0010%
0,0001%
0,0000%
0,0000%
0,0000%
0,0000%
A1
A12
A14
A18
1
2
3
4 5 6 7 8
Buckling Modes
9 10
Error(%)
Figure 3 – Error of buckling loads using the matrix GV2.
100,0000%
10,0000%
1,0000%
0,1000%
0,0100%
0,0010%
0,0001%
0,0000%
0,0000%
0,0000%
0,0000%
0,0000%
A1
A12
A14
A18
1
2
3
4 5 6 7 8
Buckling Modes
9 10
Figure 4 – Error of buckling loads using the matrix GV3.
The results are similar in all 3 cases. The p-refinement for the first few buckling modes is
always more efficient but for higher modes the results tend to be similar. We can also
conclude that p-refinement is better with GV3 matrix
7
Figure 5 – Buckling loads and modes using the matrix GV3 with discretization A12.
It is occurred that the cinematic boundaries are violated after the 7th mode. After the 8th mode
it lost brutally the compatibility between the first and second element at mid-span.
5.2
Geometrical non-linear displacements and stress resultants
For this example we will show the efficiency of the HMS, for calculating displacements and
stress resultants in a linearized stability analysis. It is applied a p- refinement since it was most
useful in the previous example. These will be compared with conventional finite elements [8]
with h- but with the same number of degrees of freedom of the HMS. The structure will be a
cantilever column, where it will be calculated the displacements in the top, and the moment in
the bottom.
P
9m
E=210 GPa
A=0,01 m 2
I=0,001 m4
u=0,2
F=5/6
Figure 6 – Cantilever column.
8
For this example the degree of approximation related to N is constant and equal to 2. For the
respective degrees approximation of the displacements, these will be one time inferior to their
respective stress resultants, to avoid any dependence in the governing system.
Degree of M, V
gdl
HMT
EF
3-a
16
16
4-b
20
20
5-c
24
24
6-d
28
28
7-e
32
32
8-f
36
36
9-g
40
40
10-h
44
44
50-i
204
204
Table 2 – Adopted Discretization.
The theoretical values of the geometrical non linear displacements and stress resultants are
presented in [10].
Error(%)
1,00000%
GV1
0,01000%
GV2
0,00010%
GV3
0,00000%
El.Convencionais
0,00000%
a
b
c
d
e
f
Disretizations Types
g
Error(%)
Figure 7 – Error of the bottom bending moment.
1,00000%
GV1
0,01000%
GV2
0,00010%
GV3
El.Convencionais
0,00000%
a
b
c
d
e
f
Discretizations Types
g
Figure 8 – Error of the top displacement.
9
Figure 9 – Diagrams of the linear and non linear stress resultants and displacements.
The HMS obtains a better approach to the theoretical stresses and displacements than the
conventional finite elements.
5.3
Buckling loads for a rectangular bending plate
For this example we will show the efficiency of the HMS, for calculating buckling load in
bending plates with p- refinement. These will be compared with conventional finite elements
with h- but with the same number of degrees of freedom of the HMS. The structure will be a
simple supported square plate.
(0,6)
(6,6)
E=210 GPa
s h=0,005 m
F=5/6
u=0,2
s
(0,0)
(6,0)
Figure 10 – Simple supported square plate.
Discretizations Degree of mx/my/mxy Degree de vx/vy
a
3
2
b
4
3
c
5
4
d
6
5
e
7
6
Table 3 – Adopted Discretization.
10
Error(%)
100,0000%
10,0000%
1,0000%
0,1000%
0,0100%
0,0010%
0,0001%
0,0000%
GV1
GV2
GV3
EF
a
b
c
d
Discretizations Types
e
Figure 11 – Error of buckling loads.
In this case the HMS elements with GV3 matrix are clearly more efficient with a small error for
low discretizations types. The conventional finite elements are better than HMS with GV1 but
worst comparing to HMS with GV2.
6
CONCLUSION
The objective of this work consisted in the development and test of a hybrid-mixed stress finite
element stability analysis of frames, plate bending problems. The examples presented show
clearly that the model is stable, robust and leads to accurate results. Effective p- and hrefinement procedures can also be easily implemented without any numerical integration.
Other success of using the HMS is that these don’t suffer from the effect of “Shear Locking”
The most successful geometrical matrix for HMS is GV3, this one can achieve even higher order
of convergence than the conventional finite elements, either for buckling loads, or non
geometrical displacements and stress resultants, with less number of degrees of freedom.
As future work, the first step will correspond to the implementation of other types of
approximation functions in hybrid-mixed models for stability analysis, such as wavelet systems
[4]. It is expected also the generalization of the models discussed here to make possible the
static and dynamic analysis of physically non-linear structures.
REFERENCES
[1] E.M.B.R. Pereira, J.A.T. Freitas. “A mixed-hybrid finite element model based on orthogonal
functions”, Int. Journal for Numerical Methods in Engineering, vol. 39, pp. 1295-1312, 1996.
[2] E.M.B.R. Pereira, J.A.T. Freitas. “Numerical implementation of a hybrid-mixed finite
element model for Reissner-Mindlin plates”, Computers & Structures, 74, pp. 323-334, 2000.
[3] L.M.S.S. Castro, J.A.T. Freitas. “Wavelets in hybrid-mixed stress elements, Computer
Methods in Applied Mechanics and Engineering”, 190, pp. 3977-3998, 2001.
[4] L.M.S.S. Castro, A.R. Barbosa. “Implementation of an hybrid-mixed stress model based on
the use of wavelets”, Computers & Structures, 84, pp. 718-731, 2006.
[5] M.V Silva, A.R. Barbosa, E.M.B.R. Pereira, L.M.S.S. Castro. “Utilização de um modelo
Híbrido-Misto na análise dinâmica de estruturas reticuladas planas”. Congresso de Métodos
Numéricos en Ingeniería, Granada, 2005.
[6] L.A.M. Mendes, L.M.S.S. Castro. “Programa de aplicação de um modelo híbrido-misto de
tensão na análise elástica de placas”, Internal Report DTC 06/2006, ICIST, 2006.
[7] P.F. Arruda, “Análise de estabilidade de estruturas com modelos híbridos-mistos de
tensão”, MSc dissertation, November, 2009.
[8] SAP2000. Structural Analysis Program – Nonlinear version 10.1.0 CSI, 1984-2007.
[9] Timoshenko S.P ., “Theory of elastic stability”, 2ª edição, New York : McGraw-Hill, 1961.
11
[10] Reis A., Camotin D., “Estabilidade Estrutural”, Mcgraw-Hill. 2000.
[11] Arruda M.R.T.,” Análise Geometricamente não Linear de Pórticos Planos com Elementos
Finitos Híbridos-Mistos de Tensão”, Diploma de Estudos Avançados em Engenharia Mecânica,
Mecânica dos Sólidos Computacional. IST 2008.
12