ELASTOPLASTIC ANALYSIS OF SHELLS WITH THE
TRIANGULAR ELEMENT TRIC
J. H Argyris1, M. Papadrakakis2 and L. Karapitta2
1
Institute for Computer Applications
University of Stuttgart, D-70579 Stuttgart 80, Germany
2
Institute of Structural Analysis & Seismic Research
National Technical University of Athens, Athens 15773, Greece
SUMMARY: TRIC is a simple but sophisticated 3-node shear-deformable isotropic and
composite facet shell element suitable for large-scale linear and nonlinear engineering
computations of thin and moderately thick anisotropic plate and complex shell structures. In
the present work an elasto-plastic constitutive model based on the von Mises yield criterion
with isotropic hardening is incorporated into the element. The characteristic feature of this
formulation is that the non-linear material behaviour is taken into account entirely in the
natural system of the element. This is achieved by transforming quantities such as equivalent
plastic strain, equivalent stress, the expression of the yield surface and the components of flow
vector from the material coordinate system to the natural coordinate system. These
transformations lead to simple and elegant expressions for the respective quantities in the
natural system that eventually result to an efficient and cost effective treatment of the
nonlinear analysis of arbitrary shells including material and geometrical nonlinearities.
KEYWORDS: Natural mode method, elastoplastic shells, von Mises yield criterion
1.
INTRODUCTION
Finite element analysis of shells has been receiving continuous attention since the early days
of the development of the method. The analysis of shells presents a challenge, since their
formulation may become cumbersome and their behaviour can be unpredictable with regard to
geometry or support conditions. The pure displacement based isoparametric formulation
suffers from various kinds of locking phenomena, which are only partially dealed with
reduced or selective integration techniques. On going research efforts have been focus on
devising more elaborate element models that circumvent the deficiencies of the puredisplacement elements. Thus, a number of approaches have been proposed based on mixed or
hybrid formulations, incompatible displacement methods, stabilization methods, assumed
strain methods and free formulations.
An attempt to device a shell element with robustness, accuracy and efficiency has led to the
derivation of the TRIC shell element [1,2], a simple but sophisticated triangular, sheardeformable facet shell element suitable for the analysis of thin and moderately thick isotropic
as well as composite plate and shell structures. Its formulation is based on the natural mode
finite element method [3], a method introduced by Argyris in the 1950’s that separates the
pure deformational modes-also called natural modes-from the rigid body movements of the
element. The natural mode method in connection with the triangular shape of the element has
substantial computational advantages compared to the conventional isoparametric finite
element formulations. Appropriate treatment of the element kinematics eliminates
automatically locking phenomena while all computations are performed analytically thus
avoiding the expensive numerical computation of the stiffness matrix. Furthermore, the
inclusion of the transverse shear deformations in the formulation of the TRIC shell element
based on a first order shear-deformable beam theory is performed in a way that eliminates the
shear locking effect in a physical manner.
The derivation of TRIC’s stiffness matrix was established upon a rather physical approach
based on the observation of the element’s deformational modes and the accumulated
experience of Argyris and co-workers obtained from previous shell elements that they have
developed using physical lumping procedures. In [1], the formulation of TRIC is presented as
an evolution step emerging from three previously elements, namely TRUMP [4], TRUNC [5]
and LACOT [6]. Recently in [7], it is proved that the TRIC shell element, despite the fact that
it is based on a non-standard finite element formulation, satisfies the individual element test
and according to the non-consistent formulation convergence can be guaranteed. Furthermore,
the element’s robustness and accuracy have proved through a variety of standard benchmark
plate and shell problems where the TRIC element exhibits excellent performance.
The features of the element proposed in the past incorporate large displacements, large
rotations and small strains. The blend of the natural mode method and a path following
strategy based on arc-length methods has shown many advantages over classical formulations,
i.e. analytical and elegant expressions for all elemental matrices; a series of vector and matrix
multiplications that can be easily optimised for maximum speed; accurate location of
birfucation, limit and displacement points; computational efficiency and economy [2].
In the present study, the geometrically nonlinear formulation of the TRIC shell element is
extended to include physical nonlinearities as well. The analysis of shell structures exhibiting
material and geometrical nonlinearities has received considerable attention over the past years
[8-11]. In this work a layered elasto-plastic constitutive model based on the von Mises yield
criterion, the associated flow rule and isotropic hardening is adopted. Both formulations based
on the continuum and consistent elastoplastic constitutive matrix are presented and compared.
The main advantage of the present natural mode approach is that the elasto-plastic stiffness
matrix is formed on the natural coordinate system and can be expressed analytically for each
layer. Then, the total natural tangent stiffness matrix is computed by adding together the
tangent stiffness matrix of each layer.
A number of benchmark test examples are examined exhibiting highly nonlinear behaviour in
order to test and verify the proposed elasto-plastic large displacement formulation of TRIC
shell element. Efficiency and accuracy of the proposed element is demonstrated by a set of
numerical examples taken from the literature.
The TRIC shell element
2.1
Kinematics of the element
For the multilayered composite triangular shell element the following coordinate systems
shown in Fig. 1 are adopted. The natural coordinate system which has the three axes parallel
to the sides of the triangle. The local elemental coordinate system, placed at the triangle’s
centroid, and the global Cartesian coordinate system where global equilibrium refers to.
Finally, for each ply of the triangle, a material coordinate system 1, 2, 3 is defined with axis 1
being parallel to the direction of the fibers.
Fig. 2 depicts the three total natural axial strains that are measured parallel to the edges of the
triangle and replace the Cartesian strains in the natural mode formulation.
γ tα 
 
γ t   γ t 
γ 
 t 
(1)
Similarly, the total natural transverse shear strains γ s are defined for each one of the
triangle’s edges:
  
 
γ s   
 
 
(2)
Fig. 3 depicts the total natural transverse shear strain for side α of the triangle.
The total natural axial strains γ t are related to the three in-plane local Cartesian strains γ’
according to the expression
γ tα  c 2 αx
  
γ t  B t γ '  γ tβ   c 2 βx
 γ  c 2 γx
 tγ  
s 2 αx 
s 2 βx 
s 2 γx
2s αx c αx   γ x'x' 


2s βx c βx   γ y'y' 
2s γx c γx   2 γ x'y' 
(3)
where
c ix   cos i, x  , i  α, β, γ
s ix   sin i, x  , i  α, β, γ
(4)
and α, x  , β, x  , γ, x  are the angles that the triangle’s edges α, β and γ form with the local
x΄ axis, respectively. The total transverse shear strains γ s are related to the two out-of-plane
transverse shear strains γ s via
γ α  c αx' s αx' 
 γ x'z' 
  
γ s  Ts γ s'  γ β   c βx' s βx'  

γ  
 γ y'z' 
(5)
 γ  c γx' s γx' 
The corresponding natural stresses σ c to the total natural axial strains γ t are grouped in the
vector
σ cα 
 
σ c  σ cβ 
σ 
 cγ 
(6)
while the corresponding natural transverse shear stresses are
σ sα 
 
σ s  σ sβ 
σ 
 sγ 
(7)
The constitutive relations between the natural stresses and the total natural strains are
established by initiating the following sequence of coordinate system transformations
Material system  Local system  Natural system
With simple geometric transformations and by contemplating the invariance of the strain
energy density in the different coordinate systems, one can easily reach to an expression for
the constitutive matrix in the natural coordinate system for both axial and transverse
deformations
σ c 
κ ct   γ t 
  
  
σ s  r   χ s  r γ s  r
(8)
valid for each layer r. Matrix κ ct defines the constitutive matrices of axial straining and
symmetrical bending while matrix χ s corresponds to antisymmetrical bending and transverse
shear modes. Additional information for the derivation of the natural constitutive matrix can
be found in [1].
2.2
Natural modes and generalized forces and moments
The multilayered triangular shell element TRIC has 6 Cartesian degrees of freedom per node.
Its natural stiffness is only based on deformations and not on associated rigid body motions.
The element has 18 degrees of freedom but the actual number of straining modes is 12:
18 Cartesian d.o.f. – 6 rigid body d.o.f. = 12 straining modes
The element TRIC includes 6 rigid body and 12 straining modes which are illustrated in Figs.
4 and 5 and grouped in the vector
 ρ0 
ρe   (6x1) 
 ρN 
(18x1)
(9)
(12x1)
in which ρ 0 , ρ N represent the rigid body and the straining modes, respectively, with
corresponding entries
ρ 0  ρ 01 ρ 02 ρ 03 ρ 04 ρ 05 ρ 06 t

ρ N  γ 0tα
γ 0tβ
γ 0tγ
ψ Sα
ψ Aα
ψ Sβ
ψ Aβ
ψ Sγ
ψ Aγ
ψα
ψβ
ψγ

t
(10)
The following subvectors are contained in (10):

γ 0t  γ 0tα

ψ S  ψ Sα

ψ A  ψ Aα
γ 0tβ
γ 0tγ
ψ Sβ
ψ Aβ

t
ψ Sγ

t
ψ Aγ

t
axial straining mode
(11)
symmetric bending mode
(12)
antisymmetric bending + shearing mode
(13)

ψ z  ψα
ψβ
ψγ

t
azimuth rotational mode
(14)
The antisymmetrical mode is the sum of the antisymmetric bending mode plus the
antisymmetric shearing mode
ψ Ai  ψ bAi  ψsAi
, i=α, β, γ
(15)
while the axial straining modes
along the middle surface of the element, the total strains
γ t and the symmetric bending modes, shown in Fig. 5, are connected via
γ 0t
ψ
γ tα  γ 0tα  z  Sα
lα
ψ Sβ
γ tβ  γ 0tβ  z 
lβ
γ tγ  γ 0tγ  z 
ψ Sγ
(16)
lγ
where l i (i=α, β, γ) is the length of side i and z’ is the distance from the middle surface along
z΄ axis of the element.
The natural modes ρ N are related to the elemental Cartesian ρ via
ρN  αNρ
(17)
and the total axial strains are related to ρ N , following (16), via
γ t  α Nρ N
(18)
Matrices α N and α N are always related to the current geometry of the element only. The
local Cartesian elemental vector ρ is connected to the global Cartesian elemental vector ρ
via
ρ  T06ρ
(19)
where T06 is a matrix containing direction cosines [1]. Using (17) we may write (19) as
ρ N  α N ρ  α N T06ρ
(20)
2.3
Axial and symmetric bending stiffness terms
The natural stiffness matrix corresponding to the axial straining and symmetric bending
modes can be produced from the statement of variation of the strain energy with respect to the
natural coordinates, viz.
δU   σ ct δγ t dV
V
(21)
Following (17), (18) and (19) δγ t may be written as
δγ t  α N δρ N  α N α N δρ  α N α N T06δρ
(22)
Substitution of (22) in (21) leads to


δU    σ ct α N dV  δρ N
V

where the constitutive relation of σ c and γ t is given by (8).
(23)
The combination of (8) and (23) gives the following expression


δU  ρ tN   α tN κ ct α N dV  δρ N
V

(24)
from which the natural stiffness matrix containing contributions from the axial straining and
symmetric bending modes can be deduced
k N ( γ 0t , ψ s )   α tN κ ct α N dV
V
(25)
Transformation procedures can now be initiated to transform the natural matrix first to the
local coordinate system and then to the global coordinate system


 
 
 

  
  


 t  t  t
 


U  ρ tN T06
α
α
κ
α
dV
α
N T06  ρ N
 N  N ct N
  
   V
(26)
  
  naturalcoord.(12x12)



 
  
local
coord.
(18x18)


 




global coord.(18x18)
Details concerning the element’s full natural and Cartesian stiffness matrices are given in [1].
2.4
The geometric stiffness
The geometric stiffness, which is based on large deflections but small strains, is mainly
generated in this study by the rigid-body movements of the element. Therefore, the geometric
stiffness includes only those natural forces which produce rigid-body moments when element
undergoes rigid-body rotations.
To construct the geometric stiffness we will focus on small rigid-body rotational increments
about x΄y΄z΄ combined in the vector
dρ 02  dρ 04 dρ 05 dρ 06 t
(27)
These rigid-body rotational increments correspond to nodal Cartesian moments dM 0 . By
making use of the fact that the resultants of all forces produced by rigid-body motion must
vanish, we arrive at an expression for dM 0
dM 0  k GR dρ 02
(3x1)
(3x3) (3x1)
(28)
where k GR is the local rigid-body rotational geometric stiffness. As can be seen in [1,2], k GR
has the simple analytical form
k GR
( 3 x 3)
2
2
2
2


Pα y α2 Pβ y β Pγ y γ
Pα x α y α Pβ x β y β Pγ x γ y γ





(


)
0
l α2
l β2
l 2γ
l α2
l β2
l 2γ




2
2
2
2
2
Pα x α y α Pβ x β y β Pγ x γ y γ
Pα x α Pβ x β Pγ x γ


 (


)
 2  2
0
2
2
2
2


lα
lβ
lγ
lα
lβ
lγ


0
0
Pα  Pβ  Pγ 





(29)
where Pα, Ρβ, Ργ are the middle plane axial natural forces and
x α  lα cαx  x3  x2 , x β  lβ cβx  x1  x3 , x γ  l γ c γx  x2  x1
(30)
yα  lα cαx  y3  y2 , yβ  lβcβx  y1  y3 , y γ  l γ c γx  y2  y1
(31)
 y1 , x 2, y2 , x 3, y3 being the x΄, y΄ coordinates of the 3
are geometric expressions with x1,
vertices of the facet triangle in the local Cartesian system.
A transformation of k GR to the local coordinate system follows from
t
k G  α 0R
k GR α 0R
(18x18)
(18x3) (3x3) (3x18)
where α 0R is the transformation matrix relating the natural rigid-body rotations
Cartesian nodal displacements and rotations ρ
ρ 02
(32)
to the
ρ 02  α0R ρ
(33)
k G is the so-called simplified geometric stiffness with respect to axes x΄y΄z΄. The term
simplified refers to the fact that only the middle plane axial natural forces Pα , Pβ , Pγ are
included in k G which fully represent the prestress state within the material. Once the
simplified geometric stiffness is formed it may be transformed to the global coordinate
system.
As mentioned before, nearly all geometric stiffness arises from the rigid-body movements of
the element. However, in buckling phenomena quite often the membrane forces are relatively
large and in this case it may be worth considering an additional approximate natural geometric
stiffness arising from the coupling between the axial forces and the symmetric bending mode
(stiffening or softening effect). This natural geometric stiffness comprises the following
diagonal matrix.
1
k NG  . . . Pα . Pβ . Pγ . . . .
12
(34)
The natural geometric stiffness is then transformed first to local and ultimately to the global
coordinates. A more extensive description of the geometric stiffness matrix may be found in
[1,2].
3.
Elasto-plastic material behaviour
3.1
The continuum elasto-plastic constitutive matrix
The natural elasto-plastic stiffness of the TRIC shell element has the same structure as the
natural elastic stiffness. If we ignore the influence of the shearing stresses on the elasto-plastic
behaviour of the element, we only have to express eq.(8) in incremental form and replace the
el-pl
natural elastic material stiffness κ el
ct by the natural elasto-plastic material stiffness κ ct
[12,13]. In this case the components of antisymmetric stiffness k A remain elastic. The
primary objective in this section is to establish the explicit form of the relation between the
natural strain and stress increments for each layer r within a given load step:
pl
r
dσ cr  κ el
ct dγ t
(35)
In the following discussion the superscript r is omitted for convenience.
We start with the assumption that the total natural strain increment dγ t is the sum of dγ elt ,
representing the elastic component of the strain, and the dγ pl
t representing the irreversible
plastic component of the strain:
dγ t  dγ elt  dγ pl
t
(36)
The stress increment follows then as
dσ c  κ elct dγ t  κ elct dγ pl
t
(37)
the elastic natural strain increment is calculated as
 
dγ elt  κ el
ct
1
dσ c
κ elct
where dσ c is the incremental natural stress vector and
this formulation, it is assumed that the material is isotropic.
(38)
is the elastic constitutive matrix. In
The incremental natural plastic strain is given by the following equation:
pl
dγ pl
t  d t s N
dγ pl
t
where s N is a vector which determines the direction of the
normality flow rule as
F  F
sN 

σ c   c
F
 c
F 

 c 
(39)
and is defined by the
t
(40)
in which F is the yield criterion given in terms of natural stress σ c , and d pl
t is a positive
constant, the so-called equivalent plastic strain increment.
Using the above equations, the total strain increment dγ t can be written as
 
dγ t  κ el
ct
1
dσ c  dγ tpls N
(41)
and dσc as,
 
 
pl el
dσ c  κ el
ct dγ t  dγ t κ ct s N
(42)
In case of von Mises yield condition, the yield surface may be expressed as
F(σ c , γ tpl )  σ  σ y (γ tpl )  0
(43)
where  is the equivalent stress and  y is the yield stress. The equivalent stress is defined
explicitly by
3 
1 
σ 2  σ ct  A  E 3 σ c
2 
3 
(44)
where
1 1 1
E 3  1 1 1
1 1 1
(45)
and
 1
cos 2 α cos 2β 


A  cos 2 α
1
cos 2 γ 
cos 2β cos 2 γ
1 

where α, β, γ are the angles of triangle.
(46)
Differentiating the yield criterion, gives:
dF(σ c , γ tpl )
 d(σ
 σ y (γ tpl ))

 (σ  σ y (γ tpl ))
σ c
dσ c 
 (σ  σ y (γ tpl ))
γ tpl
dγ tpl  0
 s tN dσ c  Hd γ tpl  0
(47)
where
H
 (σ  σ y (  pl
t ))
 pl
t
H is the hardening modulus and in our case is defined as
dσ
H  plc , σ c  σ y
dγ t
From the yield condition the equivalent plastic strain increment is derived as
1
dγ tpl  s tN dσ c  0
H
(48)
(49)
(50)
Alternatively, the equivalent plastic strain can be expressed in terms of the strain increments
by substituting (39) and (41) in (50):
dγ tpl 
1
H  s tN κ elct s N
s tN κ elct dγ t  0
Finally, after substitution of (39), (51) in (37) we obtain (35) in explicit form
(51)
r
dσ cr


1
el
el
t
r
 κ el
(
κ
s
)(
κ
s
)
 dγ t
ct 
ct
N
ct
N
t
el
H  s N κ ct s N


(52)
where the expression in brackets corresponds to the standard continuum elasto-plastic material
 pl
stiffness matrix κ el
valid for every layer r:
ct
 pl r
[κ el
]
ct


1
el
el
t
 κ el
(
κ
s
)(
κ
s
)

ct 
ct
N
ct
N
H  s tN κ el
ct s N


r
(53)
The criterion for the existence of elasto-plastic natural matrix is obviously the equivalent
plastic strain increment dγ tpl which is greater than zero only if plastic yielding takes place. The
natural strain increment in the r-th layer dγ rt which is used as input to (51) is related to the
components d 0ti and d Si of the incremental natural modes vector dρ N :
dψSi
dγ rti  dγ 0ti  zr
, i  α, β, γ
li
(54)
The natural elasto-plastic stiffness of the element is obtained by summing up the natural
elasto-plastic layer stiffnesses of the element. Following the expression (26) for the natural
elastic stiffness matrix, the elasto-plastic one is given by
nl

 pl
k elNpl   α tN κ el
ct α N
r 1

(62)
where nl is the number of layers.
3.2
The tangential stiffness matrix
Following the derivation of the tangent stiffness matrix of Ref. [1,7] for geometric
nonlinearities, the corresponding element tangent stiffness for large displacement, small strain
and elasto-plastic material behaviour is given by the following expression [2]
t
t
t
k T  [Τ 06
[α Nt [k elNpl ]α N ]T06 ]  [Τ 06
k G T06 ]  [T06
[α Nt [k NG ]α N ]T06 ]
t
 Τ 06
k LT Τ 06
(63)
This expression reveals the main advantage of the present natural mode formulation: The local
elastic-plastic stiffness matrix α Nt k elNplα N along with α Nt k NG α N can be expressed
analytically. These elemental components are added together to form the local tangent
t
stiffness matrix k LT at element level. Thus, only matrix multiplications T06
k LTT06 , with T06
being a hyperdiagonal matrix, suffice for the formulation of the global tangent stiffness.
4.
Numerical Examples
Numerical examples exhibiting highly non-linear geometric and material behaviour are
presented. The performance of the TRIC element is compared to reported results in the
literature. For the solution of the non-linear finite element equations, standard Newton-
Raphson and the cylindrical arc-length iteration schemes are applied. For all examples eight
layers in the shell thickness direction are used. The computations were performed on a
Pentium III 1000 MHz processor.
4.1
Scordelis-Lo roof under gravity load
The geometry, finite element mesh and the material data are presented in Fig.6 in which the
two longitudinal edges are free and a diaphragm supports the two circular edges. The loading
is a uniform gravity load with a reference value f0=0.004 N/mm2. A 20x20 mesh with 800
TRIC elements is used to discretize one quarter of the shell. Geometrical as well as material
non-linearities corresponding to an elastic-ideal plastic material is considered. The computed
results are compared with the following models: (i) A Morley triangular element and large
strain J2 – plasticity with a plane stress condition by Owen and Peric [14]. (ii) A seven
parameters shell model and finite strain approach by Roehl and Ramm [8]. (iii) An
isoparametric finite rotation layered shell element based on Reissner Mindlin type theory and
assumed strain formulation by Montag, Kraetzig and Soric [15]. The load factor versus
vertical displacement diagrams for point A are plotted in Fig. 7, where the results obtained by
TRIC are compared with the above mentioned results taken from the literature.
4.2
Pinched Short Cylinder
A short cylinder with isotropic hardening, bounded by two rigid diaphragms at its ends and
pinched by two forces at its middle section is considered. This example was first considered
by Simo and Kennedy [16] where an elasto-plastic model formulated entirely in stress
resultants was used. The problem details (1/8 of the cylinder) are shown in Fig. 8. The
cylinder is modeled with a mesh of 32x32 resulting in 2048 TRIC elements. The geometrical
and material data are also shown in Fig. 8. Radial displacement under the pinching load for
elasto-plastic and geometrically non-linear analysis are shown in Fig. 9. The results obtained
are compared with those reported in the literature from Simo and Kennedy [16], Wriggers,
Eberlein and Reese [17] and Soric, Montag and Kraetzig [9]. A snap through mechanism is
observed when displacement under the force reaches a value of 162 mm. The response of the
shell can be decomposed into two parts: the first is characterised by bending stiffness and the
second is characterised by a membrane stiffer response of the structure after the snap-through
occurs. This can be observed in Fig. 10 which shows the decomposition of the natural strain
energy into its invariant components. It can be observed that when snap through occurs, an
interchange of the natural axial strain and symmetric bending energies take place.
5.
Conclusions
In the present analysis a layered elasto-plastic constitutive model based on the von Mises yield
criterion with isotropic hardening is incorporated into the geometrically nonlinear shell
element TRIC. The characteristic feature of this element is that the non-linear material
behaviour is taken into account entirely on the natural coordinate system and can be expressed
analytically for each layer. Then, the total natural tangent stiffness matrix is computed by
adding together the tangent stiffness matrix of each layer.
A number of benchmark test examples exhibiting highly nonlinear behaviour is selected to
test the applicability and validity of the proposed formulation. The obtained results appear to
be in good agreement with those reported in the literature. The obtained results demonstrate,
as in the case of geometrically nonlinear, that the TRIC shell element can treat geometric and
material nonlinearties of arbitrary shells in an accurately and cost-effective way.
REFERENCES
1.
J. H. Argyris, L. Tenek, and L. Olofsson, TRIC: a simple but sophisticated 3-node
triangular element based on 6 rigid body and 12 straining modes for fast computational
simulations of arbitrary isotropic and laminated composite shells, Comp. Meth. Appl. Mech. &
Engrg. 145, pp. 11-85, 1997.
2.
J. H. Argyris, L. Tenek, M. Papadrakakis and C. Apostolopoulou, Postbuckling
performance of the TRIC natural mode triangular element for isotropic and laminated
composite shells, Comp. Meth. Appl. Mech. & Engrg. 166, pp. 211-231, 1998.
3.
J. H. Argyris, H. Balmer, J. St. Doltsinis, P. C. Dunne, M. Haase, M. Muller and W. D.
Scharpf, Finite Element Method- The Natural Approach, Comp. Meth. Appl. Mech. & Engrg.
17/18, pp. 1-106, 1979.
4.
J. H. Argyris, P. C. Dunne, G. A. Malejanakis and E. Schekle, A simple triangular
facet shell element with applications to linear and nonlinear equilibrium and inelastic stability,
Comp. Meth. Appl. Mech. & Engrg. 10, pp. 371-403, 1977.
5.
J. H. Argyris, M. Haase and H.-P. Mlejnek, On an unconventional but natural
formation of the stiffness matrix, Comp. Meth. Appl. Mech. & Engrg. 22, pp. 371-403, 1977.
6.
J. H. Argyris, L.Tenek, An efficient and locking free flat anisotropic plate and shell
triangular element, Comp. Meth. Appl. Mech. & Engrg. 118, pp. 63-119, 1994.
7.
J.Argyris, M. Papadrakakis, C Apostolopoulou and S. Koutsourelakis, The TRIC shell
element: Theoretical and numerical investigation, Comp. Meth. Appl. Mech. & Engrg. 118,
pp. 63-119, 1999.
8.
D. Roehl and E. Ramm, Large elasto-plastic finite element analysis of solid and shells
with the enhanced assumed strain formulation, Int. J. Solids Struct. 33, pp. 3215-3237, 1996.
9.
J. Soric, U. Montag and W. B. Kraetzig, An efficient formulation of integration
algorithms for elastoplastic shell analysis based on layered finite element approach, Comp.
Meth. Appl. Mech. & Engrg. 148, pp. 315-328, 1997.
10.
P. Betsch and E. Stein, Numerical implementation of multiplicative elasto-plasticity
into assumed strain elements with application to shells at large strain, Comp. Meth. Appl.
Mech. & Engrg. 179, pp. 215-245, 1999.
11.
R. Eberlein and P. Wriggers, Finite element concepts for finite elastoplastic strain and
isotropic stress response in shells: theoretical and computational analysis, Comp. Meth. Appl.
Mech. & Engrg. 17 ,pp. 243-279, 1999.
12.
J. H. Argyris, H. Balmer, M. Kleiber and U. Hindenlang, Natural description of large
inelastic deformation for shells of arbritrary shape-application of TRUMP element, Comp.
Meth. Appl. Mech. & Engrg 22, pp. 361-389, 1980.
13.
J. H. Argyris, D. W. Scharpf and J. B. Spooner, Technical Report, No. 46, 1968,
University of Stuttgart, Germany.
14.
Owen DRJ. and Peric D The Morley thin shell finite element for large deformation
problems: Simplicity versus sophistication. In: Bicanic E, editor. Nonlinear engineering
computations. Swansea: Pineridge Press, 1991.
15.
U. Montag, W.B. Kraetzig and J. Soric, Increasing solution stability for finite –
element modeling of elasto – plastic shell response. Adv. in Engrg Soft. 30, pp. 607-619,
1999.
16.
J. C. Simo and J. Kennedy, On a stress resultants geometrically exact shell model. Part
V, Nonlinear plasticity: formulation and integration algorithms Comp. Meth, Appl. Mech.
Engrg. 96, pp.,133-171, 1992.
17.
P.Wriggers, E. Eberlein and S.Reese, A comparison of three dimensional continuum
and shell elements for finite plasticity, Int. J. Solids Struct., 33, pp. 3309-3326, 1996.
Figure 1: The multilayer triangular TRIC element; coordinate systems
1
1
 t
2
 t
2

1


2
 t
3
3
3
Figure 2: Total natural axial strains
1

 3
2
side α
Figure 3: Total natural transverse shear strain for side α
Figure 4: The 6 natural rigid-body modes ρ0
1 / 2  ot
1 / 2  ot
1 / 2  ot
1 / 2  ot
1 / 2  ot
1 / 2  ot
1 / 2  s
1 / 2 s
1 / 2  s
1 / 2  s
1 / 2 s
1 / 2  s
1 / 2  bA
1 / 2  bA
1 / 2  bA
1 / 2  bA
1 / 2  bA
1 / 2  bA
1 / 2 sA
1 / 2 sA
1 / 2  sA
1 / 2 sA


1 / 2 sA
1 / 2  sA

Figure 5: The 12 natural straining modes ρN
Figure 6: Geometry, finite element mesh and material data for Scoredlis-Lo roof
3.0
TRIC-Continuum Formulation
Montag et al. [15]
Roehl and Ramm [8]
Owen and Peric [14]
2.5
load factor
2.0
1.5
1.0
0.5
0.0
0.0
400.0
800.0
1200.0
1600.0
2000.0
displacement (mm)
Figure 7: Scordelis-Lo roof: Load factor –displacement curve at point A; comparison with
other reference solutions
Figure 8: Geometry, finite element mesh and material data for pinched cylinder
3000.0
TRIC
Simo and Kennedy [16]
Wriggers et al. [17]
Soric et al. [9]
force (N)
2000.0
1000.0
0.0
0.0
50.0
100.0
150.0
200.0
250.0
displacement (mm)
Figure 9: Pinched cylinder: Radial displacement under the pinching load: comparison of TRIC
with reference solutions
80.0
Axial mode
Symmetric bending mode
energy (%)
60.0
40.0
20.0
50.00
100.00
150.00
200.00
250.00
displacement (mm)
Figure 10: Pinched cylinder: Decomposition of natural strain energy