equilibrium and displacement elements for the design … - ea… · approximation fields (bathe et...
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EQUILIBRIUM AND DISPLACEMENT ELEMENTS FOR THE
DESIGN OF PLATES AND SHELLS
Edward A W Maunder, MA, DIC, PhD, CEng, FIStructE
College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter,UK
ORCID number: 0000-0003-3172-8566
30, Mayflower Avenue, Exeter, EX4 5DS, UK; 01392 256576; [email protected]
Bassam A Izzuddin, BEng, MSc, DIC, PhD, CEng, FIStructE Department of Civil &
Environmental Engineering, Imperial College, London, UK.
ORCID number: 0000-0001-5746-463X
Article type: paper, written 5 October 2018.
5900 words excluding Abstract and reference list
6 Tables, 20 Figures
Abstract
This paper reconsiders the finite element modelling of the linear elastic behaviour of plates
and shells as governed by the Reissner-Mindlin first order shear deformation theory.
Particular attention is given to the problems associated with locking of thin forms of structure
when modelled with isoparametric conforming elements. As a means of ameliorating or
removing these problems, three recent alternative types of element are studied. Two are
displacement elements which include different approaches to the definition of assumed
strains, and the third is based on a hybrid equilibrium formulation of a flat shell element. The
purpose of the paper is to compare and explain their performances and outputs in the context
of two benchmark problems: a trapezoidal plate and the Scordelis-Lo cylindrical shell.
Numerical examples are used to illustrate the convergence of stress-resultant contours as well
as global quantities such as strain energy. The main conclusion is that whilst all three
alternative types of element overcome locking as regards displacements, the hybrid models
are generally more efficient at providing good quality stress-resultants. This is particularly so
for those which contribute little to the total strain energy but yet may be significant in design.
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Key words: Computational mechanics, slabs & plates, shells.
List of notation
u is a vector of membrane displacements
w is the transverse deflection
θ is the rotation of a transverse fibre of a plate or shell
γ is the equivalent transverse shear strain of a plate or shell
ε is the membrane component of strain
κ is the curvature component of deformation
σ is a generalised stress composed of stress-resultants
mx, my,mxy are components of moment stress-resultants per unit length
nx, ny, nxy are components of force stress-resultants per unit length
qx, qy are components of transverse shear forces per unit length
σvM is a von Mises stress
Ub, Us are bending and transverse shear strain energies respectively.
1. Introduction
Much effort has been expended in the past to develop and enhance the performance of
elements for use in finite element models of thin plate and shell structures. Generally, such
elements were initially developed based on Reissner-Mindlin (Zienkiewicz et al, 2000, Cook
et al, 2002) theory and isoparametric conforming displacement fields, however various
locking phenomena can lead to very stiff responses and unreliable results. Later
developments have led to a plethora of elements aimed at overcoming such limitations, with
typical features being the inclusion of assumed strain fields within the element
approximation fields (Bathe et al, 2003, Izzuddin, 2007, Izzuddin et al, 2017), or the use of
selective reduced integration (Zienkiewicz et al, 2000).
This paper focuses on the isoparametric displacement element whose parent is the 9-node
quadrilateral conforming element based on biquadratic Lagrangian shape functions
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(Zienkiewicz et al, 2000, Cook et al, 2002). Two non-conforming variations on this element
are considered that include assumed strains (Izzuddin et al, 2017). On the other hand, a
hybrid equilibrium flat shell element (Maunder et al, 2013, Moitinho de Almeida et al, 2017)
has the valuable property of not suffering from any form of locking, and leads to stress-
resultants exhibiting strong forms of equilibrium which allow bounds to be determined on
some quantities when complemented by conforming solutions. In general however, care then
needs to be taken in constructing models so as to avoid the possible effects of spurious
kinematic modes, otherwise known as zero energy modes (Moitinho de Almeida et al, 2017).
A hybrid element is thus considered in this paper in the form of a flat quadrilateral formed as
a macro-element from four triangular ones. It is based on piecewise quadratic moment fields
in the interior and rotation fields along the sides, plus linear fields of transverse shear and
membrane forces in the interior and displacements along the sides.
The aim of the paper is to compare the performances of the different types of element in an
informative and objective way so as to enlighten the computational structural mechanics
community as well as practising engineers of the pros and cons of equilibrium and
displacement based elements, with clear explanations of the differences in performance.
The outline of the paper after this introduction is as follows. Section 2 explains the locking
phenomenon that can occur in isoparametric conforming elements with an example involving
a quadratic beam element. This Section then summarises how assumed strain fields are
defined within the 9-node displacement element as proposed by Izzuddin and Liang (Izzuddin
et al, 2017) in an optimised hierarchic approach, and in the MITC9 element proposed by KJ
Bathe et al (Bathe et al, 2003). Section 3 recalls the details of the hybrid equilibrium flat shell
element before comparing solutions for a trapezoidal plate as a benchmark problem in
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Section 4. Section 5 compares solutions for the Scordelis-Lo cylindrical shell benchmark
problem, whose solutions also include the hybrid equilibrium ones based on faceted models
having the form of folded plates. Conclusions and proposals for future research are presented
in Section 6.
2. Isoparametric displacement elements and locking phenomena
In the isoparametric formulation of a conforming Reissner-Mindlin plate element, it is usual
to interpolate translational and rotational components of displacement from the nodal degrees
of freedom (DOF) with the same approximation polynomials. Transverse shear strains are
thus induced directly from fields of rotation and transverse displacement, and these shear
strains can have undue influence as the plate thickness is reduced, leading to unrealistic
modes of deformation and highly overstiff behaviour. The shear locking phenomenon is
illustrated next using the quadratic Mindlin beam element. Its treatment with displacement
based assumed strain Reissner-Mindlin shell elements is then discussed along with other
locking phenomena.
2.1 Quadratic Mindlin beam element
Shear locking behaviour is simply explained and illustrated by considering a uniform
cantilever beam with shear deformation. The beam, as shown in Figure 1, is modelled with a
single 3-noded conforming element based on the same complete quadratic shape functions for
both translation w and rotation θ. The formulation of this element follows similar principles
to those of the Reissner-Mindlin plate element, and hence an investigation of its behaviour
sheds light on the performance of conforming plate elements. The single element is of length
L and has a rectangular cross section of depth t and a Poisson’s ratio of 0.3. It is loaded with
an end load P as shown in Figure 1. The model has 4 DOF consisting of the translation and
rotation at the central node and the loaded node at the right end.
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w, θ
Figure 1: Cantilever beam indicating positive senses of translation and rotation.
The relative error in strain energy of the solution is shown in Figure 2 for varying
depth/length ratios.
Figure 2: Relative error v. depth/length ratio
Figures 3 to 4 compare the theoretical solutions for translation and rotation based on
Timoshenko’s beam theory (Cook et al, 2002, Przemieniecki, 1968) with those based on the
quadratic Mindlin element for a unit load P = 1. Each Figure is in two parts, on the left for t =
L, and on the right for t → 0. In these figures the values of the abscissa have been scaled to a
non-dimensional form.
t
L
P
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Figure 3: Translations corresponding to t = L and t → 0.
Figure 4: Rotations corresponding to t = L and t → 0.
Additionally, Figures 5 and 6 compare the bending moments and shear forces for the same
thicknesses as for Figures 3 and 4.
Figure 5: bending moments corresponding to t = L and t → 0.
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Figure 6: shear forces corresponding to t = L and t → 0.
Clearly the quality of the solution degrades as the thickness is reduced, and it is noted that the
error in strain energy increases in magnitude to 25% as the thickness tends to zero.
Correspondingly the model becomes overstiff by 25% as regards the tip translation.
Shear locking would imply that fields of curvature deformation cannot exist without incurring
shear strain as the thickness tends to zero. However, in the case of the quadratic element there
is one curvature field that does not incur shear strain whatever the thickness, i.e. when fields
dw
dx= are both linear, and then curvature
d
dx
is constant. It is precisely this zero-shear
deformation mode that is excited for the cantilever as its thickness tends to zero with the
remaining shear-inducing deformation modes suppressed. Consequently, the single element
model doesn’t lock (Braess, 2007), but becomes overstiff, though locking would occur under
other boundary conditions where the zero-shear mode is either restrained or not excited by
the applied loading. Moreover, the internal stress-resultants, as illustrated in Figures 5 and 6,
are clearly in error, particularly so as regards the shear forces which are characterised by their
extreme oscillation about a correct mean value.
2.2 Assumed strain shell elements
Consideration is given first to the shear locking phenomenon with the 9-node conforming
quadrilateral shell element using quadratic Lagrange interpolation functions, which may be
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addressed with non-conforming versions of this element where the transverse shear strain
fields are replaced by other assumed strain fields. The argument for replacing the shear fields
follows from the observation that the conforming shear strains are unrealistic, being polluted
by quadratic shear strains within each element due to the rotations under a general bending
mode, becoming relatively most significant for thinner plates. These lead to excessive strain
energy due to shear, oscillating shear stresses, and overstiffness or locking, as highlighted in
the previous section for the quadratic Mindlin element.
In the assumed strain versions the strain fields may be derived as functions of the original
displacement parameters for an element, so that in effect the Bγ matrix, which relates
transverse shear strains to the DOF for the conforming element (where γ = gradw + θ), is
then transformed to a new matrix B with modified coefficients for the shear strains
corresponding to the original DOF. Consequently the assumed strain elements are in general
no longer conforming.
The first version considered was proposed by the Izzuddin (Izzudin, 2007, Izzuddin et al,
2017), where the basic idea is to correct the shear strains in each element using additional
local hierarchic transverse displacement fields towards an ‘objective’ shear strain field as
afforded by the element. To illustrate this concept on the Mindlin beam element, the objective
shear strain field afforded by the quadratic element would be linear as determined by dw
dx,
while the quadratic polluting terms arising from are filtered out with additional local
hierarchic correcting w fields of cubic order. A similar treatment is applied to the 9-noded
isoparametric shell element, where a hierarchic optimisation procedure is applied to filter out
the polluting shear strains and recover the objective assumed strain field in terms of the
element nodal displacements. This optimisation approach leads to a family of elements,
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depending on the hierarchic correction order, where the most effective variant has been
shown to be the so-called ‘H3O9’ element utilising cubic hierarchic correction transverse
displacement fields (Izzudin, 2007, Izzuddin et al, 2017). The corresponding components of
the assumed objective shear strains γxz and γyz are linear in x but quadratic in y, and linear in y
but quadratic in x, respectively, where x and y are local element coordinates.
The second version is the so-called ‘MITC9’ element, proposed by Bathe et al. (Bathe et al,
2003, 2011), which interpolates strain components in a covariant coordinate system at
selected positions, referred to as tying points, with the covariant strains subsequently mapped
to an assumed strain tensor. The locations of the tying points for the covariant strains are
depicted in Figure 7, where refers to the covariant transverse shear strains. An
implementation of the MITC9 element is employed for the comparisons undertaken in this
paper, which adopts the improvements proposed by Wisniewski et al (Wisniewski et al,
2013) and Liang et al (Liang et al, 2016) relating to the use of a constant Jacobian for the
strain mapping, allowing the MITC9 element to pass the patch test.
Figure 7: Positions of tying points for MITC9 element ( a 1/ 3= , b 3 5= , and c 1= ).
In addition to shear locking, membrane locking effects can arise in the modelling of shell
structures due to polluting membrane strains. This is readily addressed in the H3O9 element
(Izzuddin et al, 2017) using a similar optimisation procedure utilising cubic hierarchic planar
displacement fields, while it is addressed in the MITC9 element (Bathe et al, 2003, 2011,
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Wisniewski et al, 2013) with the interpolation of covariant membrane strains ( ), from
values at tying points as shown in Figure 7. The H3O9 element has the added benefit of
dealing with distortion locking (Izzuddin, 2007, Izzuddin et al, 2017), which is achieved by
expressing the objective strain fields as polynomial functions in terms of local Cartesian
coordinates.
3. Hybrid quadrilateral flat shell element.
The hybrid quadrilateral element in this paper is based on statically admissible vector fields
of stress-resultants σ defined within an element, and vector fields of displacement v defined
independently for each side (Moitinho de Almeida et al, 2017, 1991, Maunder et al, 2005).
This type of hybrid element is distinct from those proposed by Pian et al (Pian et al, 1969)
whose boundary displacement fields are continuous at the nodes.
The components of σ are the 8 stress-resultants that represent moments, transverse shear
forces, and membrane forces per unit width of a section, ordered as in Equation (1).
= = T
x y xy x y x y xym m m q q n n n 0ˆSs σ+ , (1)
where the columns of S define independent fields of stress-resultants that equilibrate with
zero loads on the interior of an element, s is a vector of stress parameters, and 0 represents
a particular field of stress-resultants that equilibrates with a load distributed over the interior
of an element.
The components of v are the 5 components of displacement (translations and rotations) which
are ordered as in Equation (2).
v = = T
x y z x yu u u ˆVv , (2)
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where the columns of V define independent displacement fields associated with displacement
parameters v . The components in Equations (1) and (2) should be taken with reference to
Cartesian axes with the z axis in the transverse direction.
The quadrilateral hybrid element is formulated as an assembly of triangular primitives as
illustrated in Figure 8(a) in order to ensure its kinematic stability (Moitinho de almeida et al,
2017). The assembly of primitive elements into a macro-element, as illustrated in Figure 8(b),
effectively condenses out the internal degrees of freedom (Maunder et al, 2013, 1996). In the
following Equations (3) to (6) we briefly summarise the formulation of a stiffness matrix of a
primitive element.
(a) primitive (b) macro
Figure 8: Hybrid equilibrium element as a macro-element.
Weak compatibility between elastic strains and side displacements of a primitive element are
expressed as follows:
( )0 0ˆ ˆ ˆ ˆ ˆ
+ = → + = e e
T T TS f Ss d S NV d v Fs e D v (3)
where 0 0ˆ , ,
e e e
T T T TF S f S d e S f d D S NV d
= = = ,
ˆSs = +0
ˆv Vv=
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and f denotes the flexibility form of the constitutive relations (as detailed in Equation (7) for
the shell element), N denotes the (8×5) matrix that transforms stress-resultants at a side to
boundary tractions Tt N= . In
Equation (3) we have omitted initial strain terms, e.g. thermal strains, for simplicity, see
(Moitinho de Almeida et al, 2017) for the complete equation.
Weak equilibrium between internal stress-resultants and boundary tractions is expressed by:
( )0 0
0 0
ˆ ˆˆ ˆ
ˆ ˆwhere and
e e
e e
T T T
T T T
V N Ss d V t d Ds t t
t V N d t V t d
+ = → + =
= =
(4)
In this Equation t denotes boundary tractions which arise from interaction with other
elements, or as prescribed tractions t , and these are represented by the vector of parameters
t .
Equilibrium at the boundary has a strong form when the approximating polynomials for σ
and prescribed tractions t are defined in terms of complete polynomials of the same degree
as for the displacements in V. For Reissner-Mindlin plate/shell elements, the degree of forces
and translational displacements is one less then the degree of moments and rotational
displacements.
It should be emphasized that the achievement of strong, i.e. complete, equilibrium comes at
the expense of full compatibility of side displacements at the vertices of elements. This is
unlike the hybrid elements of Pian (Pian et al, 1969) where the penalty of assuming
continuous boundary displacements is the lack of equilibrium of tractions between elements
The system of equations for a primitive element then has the form:
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0
0
ˆˆ
ˆ ˆˆ0
T esF D
t tvD
−=
− . (5)
The stress field parameters can be eliminated from these equations to give a stiffness matrix:
1 10 0
ˆ ˆˆ ˆ ˆTKv DF D v t t DF e
− − = − + (6)
Although any consistent set of polynomial degrees can be assumed, numerical results are
presented in this paper based on an element with quadratic moment and rotation fields, linear
fields of membrane and transverse shear forces and side translations, and a uniformly
distributed load applied in any direction. A strong form of equilibrium is thereby obtained.
As a flat shell element of thickness t, the constitutive relations are partitioned into the block
diagonal form:
( )
( ) 2 2
3 3 2
0 0 1 012 1 012
0 0 with 1 0 , ,5 120
0 0 0 0 2 1
b
s b s m b
m
ft t
f f f f f fEt Et t
f
− +
= = − = = +
. (7)
At the element level bending and transverse shear actions are uncoupled from the membrane
actions. For plate bending action the “natural” element flexibility matrix
→ = T
b b b bF F S f S d as t → 0,
where Sb contains only the 3 rows of S corresponding to the components of moment. Since fb
is positive definite and the columns of Sb remain independent, F remains positive definite and
accounts for an ever decreasing thickness in a natural way as the element behaviour becomes
dominated by flexure.
Since membrane actions are uncoupled from the flexural ones, there is no undue interaction
which could lead to membrane locking as observed with conforming elements when
modelling curved shells.
4. Trapezoidal plate bending problem
A plate in the shape of a trapezium spans between its parallel sides, and is subjected to a
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uniformly distributed load. The plate is illustrated in Figure 9, with a span of 8m, and
supported side lengths 4m and 12m. Its thickness t takes values in the range 0.25m to
0.0025m. According to Reissner-Mindlin plate theory, the obtuse corner of the plate has a
singularity for transverse shear forces, but not for moments with soft simple supports (Rössle
et al, 2011).
Four finite element models are considered whose meshes are graded in the span direction.
The initial 2×2 mesh is refined 5 times so that each refinement is contained within the
previous mesh. The models are designated according to the employed shell element as
follows, where n denotes an n×n mesh of elements :
• ETCn for the hybrid quadrilateral equilibrium element (Maunder et al, 2013);
• CONn for the 9-noded quadrilateral conforming element (Izzuddin et al, 2004);
• CASn for the H309 assumed strain element based on hierarchic optimisation (Izzuddin,
2007);
• CMIn for the MITC9 element with a constant Jacobian for the strain mapping (Bathe et
al, 2003, Wisniewski et al, 2013, Liang et al, 2016).
Figure 9: Plate modelled with the initial 2×2 mesh of elements.
4.1 Comparisons of strain energies of the plates
An initial set of analyses is carried out with the 2×2 mesh assuming soft fixed supports, i.e.
y
x
m
m
m
2
8 2
load = - kN/m
0.3
2.10 10
0.
kN m
1
/E
=
=
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there is no constraint for torsional rotations, in order to demonstrate shear locking as the plate
thickness is reduced. These support conditions ensure that the kinematic modes are
insufficient to prevent shear strain energy dominating the solution as the thickness is reduced.
Consequently, flexural deformations tend to fade in significance when compared to the shear
deformations, and the model tends to exhibit full shear locking instead of mere stiffening as
observed for the Mindlin beam example in Section 2.1.
Results with such supports are presented in Table 1 in terms of total strain energies Us and Ub
due to shear strains and flexural/bending curvatures respectively for three thicknesses. It is
clear that both assumed strain models do not suffer from shear locking.
CON CAS CMI
Thickness Us Ub Us Ub Us Ub
0.25 4.240×10-7 2.473×10-8 1.578×10-7 7.367×10-6 1.376×10-7 7.278×10-6
0.025 4.763×10-6 3.271×10-9 3.130×10-6 7.329×10-3 3.350×10-6 7.250×10-3
0.0025 4.770×10-5 3.285×10-10 3.212×10-5 7.329×100 3.664×10-5 7.250×100
Table 1: Shear and bending strain energies (kNm) for the 2×2 mesh with soft fixed supports.
In the remaining examples simple supports are assumed for the thicknesses 0.25m and
0.0025m. Although shear locking does not occur, shear stiffening is observed. Results for
relative errors in the total strain energies are presented in Table 2 and shown graphically in
Figure 10 to illustrate convergence for the thin plate. Relative error in this paper is defined as
( )−app ref refU U U where and app refU U denote approximate and reference values
respectively of strain energy. Hence a negative relative error means that the approximate
value is less than the reference value.
As expected the errors in strain energy of the equilibrium and conforming models converge
from above and below as seen in the figure. As is usually the case, the equilibrium model
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shows tighter bounds from coarser meshes when compared to the conforming model. The
16×16 mesh results from all 4 models, including the assumed strain CAS and CMI models,
agree to within about 2% (compared with 0.5% in the case of the thick plate).
In particular when the energy errors of the thin plate are considered, it is evident that shear
effects tend to stiffen the conforming displacement model for the coarser meshes, but this
stiffening is significantly reduced by the use of either of the assumed strain models. In this
case their errors, which are very similar, appear also to be lower bounds, i.e. less than zero,
though this is problem-specific.
Model
t(m): mesh ETC CON CAS CMI
0.25: 2×2 4.667 -23.222 -2.980 -4.534
0.25: 4×4 1.620 -8.971 -1.058 -1.774
0.25: 8×8 0.516 -2.244 -0.445 -0.813
0.25:16×16 0.148 -0.445 -0.179 -0.281
0.25: 32×32 0.025 -0.097 -0.056 -0.077
0.25:64×64 0.005 -0.036 -0.036 -0.036
0.0025: 2×2 6.520 -36.293 -2.051 -4.126
0.0025: 4×4 2.999 -24.243 -0.731 -1.653
0.0025: 8×8 1.407 -7.248 -0.374 -0.563
0.0025:16×16 0.631 -1.359 -0.207 -0.207
0.0025: 32×32 0.254 -0.437 -0.123 -0.123
0.0025:64×64 0.086 -0.207 -0.081 -0.102
Table 2: % relative errors in strain energies of solutions from different meshes and models.
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Reference values of strain energies are based on 128×128 mesh of equilibrium elements.
6×6×3 Gauss point integration used with all displacement elements.
Figure 10: Convergence of relative errors of different plate models of thickness 0.0025m.
While the hybrid equilibrium element (ETC) is evidently superior to the conforming element
(CON), the two assumed strain elements, H3O9 (CAS) and MITC9 (CMI), provide even
more accurate predictions of the strain energy at coarse meshes, with the H3O9 element
marginally better due to its enhanced optimisation for irregular element shapes.
4.2 Comparisons of stress fields of the plates
Comparison of the flexural stresses in the thin plate, in the form of von Mises stress in the top
of the plate, are shown in Figure 11 for the 16×16 mesh. The boundary conditions of the plate
imply that this stress field has no singularity in the neighbourhood of the obtuse corner,
though there is a singularity in the transverse shear force field.
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(a) ETC (b) CON
(c) CAS (d) CMI
Figure 11: Contour maps of von Mises stress at top of 16×16 mesh for thinner plate (t =
0.0025m) in range [0, 1.5] GPa.
The range of contours is chosen to display contours throughout the plate with the exception
of the area within the obtuse corner where a finite concentration of surface stress exists in
theory. The ETC and CMI contours are very similar apart from within the skewed boundary
layer where the mesh size is too coarse to display reliable contours. Outside the boundary
layer contours appear to be smoothly continuous in both models. However, the CON model
displays a lack of continuity at element interfaces throughout most of the plot, whereas the
CAS model generally displays better continuity at interfaces, and there is further little
improvement with the CMI case.
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The improvements in the assumed strain models are however less evident when we examine
the transverse shear stress fields, which are proportional to the corresponding strains. It
should be remembered that these stress fields are singular at the obtuse corner of the plate.
In Figure 12, the von Mises stress σvM at the mid-plane of the plate, which is directly related
to the transverse shear stresses, is compared for the 4 models based on the same 16×16 mesh.
In the absence of membrane forces, this stress component is related to the transverse shear
stress-resultants as follows in Equation (8):
( ) ( ) ( )2
2 2 2 2 2 2 2
2
3 1.5 1.53 i.e. 3
= + = + = +vM xz yz x y vM x yq q q q
t t (8)
This figure helps to discern that although oscillations remain evident with the conforming and
assumed strain models, the overall patterns of stress in the assumed strain models resemble
more closely the pattern in the equilibrium solution in Figure 12(a). This observation is
supported by the corresponding values of shear strain energy (kNm): Us = 2.253×10-4
(ETC), 7.310×10-2 (CON), 6.487×10-3 (CAS), and 7.835×10-3 (CMI). In round terms, the
shear strain energies of the assumed strain models are one order of magnitude higher, and the
shear strain energy of the conforming model is two orders of magnitude higher than the
energy of the equilibrium model. These values may be compared with a reference value of
1.625×10-3 based on a refined 128×128 ETC model. It should be noted that Us is only one
component of the total strain energy (4.771×101 kNm from the refined model), and thus the
ETC and CON values from the 16×16 mesh cannot be taken as upper and lower bounds of
the reference value. Using the reference value, the shear energy represents some 0.0034% of
the total.
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(a) ETC (b) CON
(c) CAS (d) CMI
Figure 12: Contour maps of von Mises stress at midplane of 16×16 mesh for thinner plate (t =
0.0025m) with range [0, 0.0012] GPa.
It is important to highlight that the observed oscillations in the transverse shear stress fields
with the conforming and assumed strain elements are insignificant for typical engineering
problems, given that the shear strain energy is only a very small part of the total strain
energy. This is also reflected in a relatively low maximum von Mises stress at the plate mid-
plane (Figure 12) which is generally less than 0.1% of the maximum von Mises stress at the
top surface (Figure 11), apart from the small domain in the neighbourhood of the obtuse
corner. However, for special types of problem, e.g. composite plates, it may be necessary to
have an accurate evaluation of all the stress fields, even those for which the associated strain
energy is relatively small. The hybrid equilibrium element provides clear benefits for such
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problems, with accurate stress fields obtained at relatively coarse meshes without the need for
further post-processing.
5. The Scordelis-Lo cylindrical shell problem
The second application is to a cylindrical shell based on the Scordelis-Lo benchmark problem
(Scordelis et al, 1964, Scordelis, 1971, MacNeal et al, 1985). Figure 13 illustrates the shell
which spans in the direction of axis Y between vertical end diaphragms. The diaphragms are
assumed to provide soft simple supports in the XZ plane without other constraints except to
prevent a rigid body movement in the Y direction. The longitudinal edges are assumed to be
free. The shell is loaded with a uniform conservative body force, e.g. self-weight, in the
direction –Z. The dimensions, loads, and material properties are quoted as for the benchmark
problem, and should be assumed to have a consistent set of units.
Figure 13: The Scordelis-Lo cylindrical shell as modelled by a finite element mesh. The
cylinder has a radius of 25 units and it subtends an angle of 80º. The original thickness of the
shell is 0.25 units.
The finite element models are either facetted when using the hybrid flat shell elements with
their non-structural corner nodes situated in the mid-surface of the shell, or curved when
22
using isoparametric 9-node elements with all nodes situated in the mid-surface. This
arrangement implies that the midpoints of the circumferential sides of the flat shell elements
are slightly offset from the mid-surface in the direction of the centre of the arc. The
isoparametric elements better represent the geometry of the shell. For this problem, the
various models take advantage of symmetry and only require a mesh for a quadrant. As for
the original benchmark problem (MacNeal et al, 1985), linear elastic behaviour is assumed
with a zero Poisson’s ratio.
For this paper the benchmark problem is extended to consider the effects of reducing the
thickness to a very small value. In general, three thicknesses are considered, the original 0.25
and then two thinner shells of thicknesses 0.025 and 0.0025. All meshes are loaded with a
uniform vertical pressure of 1.0 per unit area of the shell mid-surface. Five uniform n×n
meshes are used for each thickness with n = 4, 8, 16, 32 and 64. Reference solutions are
derived from ETCn models with n = 128.
Again the performances of the hybrid models are compared with those of the models
composed of conforming elements and the two types of assumed strain elements. In this type
of problem membrane locking may occur in the conforming models as the thickness is
reduced.
Two points must be emphasized with respect to the hybrid models:
• The geometries of the hybrid models are facetted, and hence they differ from the actual
shape of the shell, though this difference reduces with mesh refinement. These models
must thus have a slightly different stiffness when compared to the models based on the
isoparametric elements, which also approximate the circular shape with a curve defined
by the quadratic isoparametric mapping;
23
• The generalised displacements of each side of a hybrid element includes a drilling
freedom, but this only involves a rigid body rotation of the whole side, and independent
drilling freedoms of individual transverse fibres are not considered. Consequently the
corresponding drilling moments at an interface are released. All the other 5 components
of side traction are transmitted via sections of the interface to an adjacent element so that
each section is normally in equilibrium. This means that the resultants of normal and
transverse shear forces are codiffusive whether adjacent elements are coplanar or not.
However the same cannot be said for the pair of local torsional moments at an interface
when adjacent elements are not in the same plane because these moments then act about
different axes and the drilling moments have been released. Thus assuming such
components of traction to be codiffusive appears to violate local equilibrium. A strong
form of equilibrium could be obtained by also releasing the torsional components of the
moment tractions. However, as the mesh is refined, these releases would tend to degrade
the model as a representation of a continuous shell (Maunder et al, 2013) and will not be
considered further.
In this paper it is assumed that torsional moments are co-diffusive and rely on partial
continuity of the sections of an interface along its length to achieve an overall balance of
the complete interface, as proposed by (Maunder et al, 2013). Equilibrium tends towards
a stronger form as the mesh is refined and adjacent elements become more nearly
coplanar.
In the curved conforming elements, membrane locking may occur as the shell thickness is
reduced due to the pollution of the membrane strains derived from the transverse
displacements approximated by the element shape functions, i.e. for small displacements it is
assumed that membrane strains are defined by:
24
0 0
0 0
= +
m
zx x w
u xzy y wv
yz z
y x y x
(9)
where (x,y,z) denote local coordinates when the x,y axes lie in the local tangential plane, and
(u,v,w) denote the corresponding components of displacement. Hence the isoparametric
formulation using the same Lagrange interpolation functions for each component of
displacement can introduce significant polluting terms from w to the membrane strains which
leads to overstiffening or locking (Izzuddin et al, 2017).
As with shear locking, elements based on assumed membrane strains are derived in two
alternative formulations, denoted again by CAS and CMI with the aim of unlocking the
displacements. In the CAS (H3O9) formulation (Izzuddin 2007, Izzuddin et al 2017),
objective strain fields are derived over an element which are compatible with a bi-quadratic
(i.e. with similar polynomial forms to that of the Lagrange shape functions) membrane
displacement field in terms of local Cartesian coordinates, which are freed from the polluting
strains introduced by the transverse displacement fields of the conforming model. This
process involves membrane hierarchic corrective strains, which are derived from further
hierarchic membrane displacement fields that are one degree higher than the conforming
displacements in terms of natural coordinates.
Alternatively, the CMI (MITC9) formulation (Bathe et al, 2003, 2011) derives assumed strain
fields in terms of natural (ξ, η) coordinates by interpolating their values obtained from the
conforming fields at different patterns of tying points using Lagrange polynomials, as
illustrated previously in Figure 7.
5.1 Comparisons of deflections and strain energies of the shell
The results from the four different types of finite element model are firstly compared in Table
25
3 for the midspan vertical deflections of the thickest shell.
Mesh: model ETC CON CAS CMI
0.25: 4×4 -3.507 -2.734 -3.352 -3.335
0.25: 8×8 -3.389 -3.291 -3.344 -3.338
0.25: 16×16 -3.360 -3.337 -3.344 -3.340
0.25: 32×32 -3.351 -3.344 -3.346 -3.344
0.25: 64×64 -3.348 -3.347 -3.348 -3.347
Table 3: Convergence of midspan deflections × 103 for a unit uniformly distributed load,
assuming Young’s modulus E = 4.32×108.
It can be noted that the deflection reported by MacNeal & Harder (MacNeal et al, 1985)
based on finite element models, when scaled to correspond to the unit load, is -3.360×10-3,
and an “exact” value reported by Scordelis (Scordelis, 1971) corresponds to -3.422×10-3.
Since the results from the equilibrium and assumed strain models appear to converge to the
same value in Table 3, there are grounds for some confidence in the value of -3.348×10-3.
Percentage errors in total strain energy are presented in Table 4 for all thicknesses of shell.
26
Thickness:
model
ETC CON CAS CMI
0.25: 4×4 2.8822 -14.4921 0.0879 -0.3582
0.25: 8×8 0.7150 -1.3972 -0.0758 -0.2180
0.25: 16×16 0.2247 -0.2522 -0.0785 -0.1616
0.25: 32×32 0.0617 -0.0805 -0.0282 -0.0731
0.25: 64×64 0.0141 -0.0067 0.0074 -0.0060
0.025: 4×4 19.0888 -70.8009 5.2785 -2.4396
0.025: 8×8 1.5833 -38.4647 0.2443 -0.1284
0.025: 16×16 0.3176 -6.7847 0.0197 -0.0121
0.025: 32×32 0.1209 -0.5274 0.0017 -0.0099
0.025: 64×64 0.0528 -0.0508 -0.0016 -0.0092
0.0025: 4×4 346.8118 -94.7101 111.8258 -19.1914
0.0025: 8×8 12.7738 -87.9036 9.3305 -1.7342
0.0025: 16×16 0.8815 -67.4585 0.5082 -0.1055
0.0025: 32×32 0.1082 -27.2055 0.0050 -0.0145
0.0025: 64×64 0.0275 -3.0502 -0.0087 -0.0106
Table 4: % values of error in strain energy for the 4 different types of model.
A pattern of 6×6×3 Gaussian integration points is used for all the conforming and assumed
strain elements. Reference values for strain energies are determined by using Richardson’s
extrapolation based on the two finest meshes of the ETC models together with the uniform
mesh of 128×128 elements. Figure 14 presents plots of the errors of strain energy for the
27
different thicknesses and meshes. The errors for the thinnest shell and the coarsest mesh are
excessive for the ETC and CAS models, and are not included in the range of error for their
plots.
28
Figure 14: % error in total strain energies for the 4 models and 3 thicknesses of shell.
CON Thickness
Mesh 0.25 0.025 0.0025 0.00025
4×4 60.471 98.4 99.752 100.000
8×8 48.633 91.854 99.844 100.000
16×16 47.708 66.905 98.326 99.971
32×32 47.650 60.558 84.453 99.736
64×64 47.646 60.085 66.159 96.337
Ref:ETC128 47.631 60.057 63.164 62.984
Table 5: membrane strain energy as a % of total strain energy for the conforming models
CON.
Observations based on Tables 3 to 5, and Figures 14 and 15:
• The ETC models are strictly folded plate models of the cylinder, and thus the shape of a
model depends on the mesh, and the shape changes as the mesh is refined. These models
have no tendency to lock.
• The ETC models have a weakened form of equilibrium in that the tractions are not fully
codiffusive along interfaces parallel to the longitudinal (Y) axis of the cylinder. However,
strain energy and mid-span deflection appear to converge from above towards the shell
solution, indicating responses that are too flexible, as would be expected from fully
equilibrated models.
• It is clear from Figure 14 and Table 5 that the conforming models (CON) suffer from
membrane locking as the thickness is reduced.
• Further studies of this extended benchmark problem indicate that the proportion of
membrane strain energy tends towards a constant value of some 63% of the total strain
energy as the thickness tends to zero. This tendency is reflected in the results for all the
29
finite element models except for the conforming one, and hence this value is confidently
presented as a reference. Figure 15 illustrates the changes in deformed shape that occur as
the thickness is reduced, e.g. the curvature in the circumferential direction tends to
oscillate within an ever shrinking boundary layer. However, if the strain energy was
dominated 100% by flexure or alternatively by membrane action, the displacements
would be proportional to 1/t3 or 1/t, respectively, and no change of shape would be
observed as t is reduced. Clearly, the change in the deformed shape with thickness
reduction can be attributed to the fact that the membrane and flexural strain energies
coexist in constant asymptotic proportions as the thickness is reduced.
• Both types of assumed strain model overcome, or unlock, the excessive stiffness of the
conforming models, leading to very reliable results for energy and deflection for even the
coarsest models of the thickest shell. When the thickness is reduced, their errors increase
but even so they appear to produce the most accurate solutions of all four models in terms
of global strain energy. However, these models are no longer conforming, and so
statements concerning bounds cannot be made with confidence.
5.2 Comparison of stress fields of the shell
In this Section contour plots of selected stress-resultants are presented to supplement the
observations in Section 5.1. Contours of the membrane forces nx and ny and the
circumferential bending moment mx as solutions obtained for the “locked” CON16 model
with thickness 0.0025 are presented in Figure 16, and compared with the corresponding
“reference” quantities obtained from the ETC64 model. Values of these quantities are also
tabulated in Table 6.
30
Y
0.25 0.025
0.0025 0.00025
Figure 15: Views of deformed model CMI64 with thicknesses in the range 0.25 to 0.00025.
CON16 ETC64
Midspan deflection -7.844 -36.255
Membrane energy 154.6 305.4
Bending energy 2.614 178.1
Total energy 157.3 483.4
Table 6: Midspan deflection and strain energies of the compatible model CON16 compared
to reference values from ETC64. Thickness = 0.0025, E = 4.32×108, ν = 0.0.
X
Z
31
(i) CON16 0.0025, nx ETC64 0.0025, nx
(ii) CON16 0.0025, mx ETC64 0.0025, mx
(iii) CON16 0.0025, ny ETC64 0.0025, ny
Figure 16: Contours of stress-resultants nx, ny, mx for a “locked” compatible model compared
with a “reference” equilibrium model for thickness 0.0025.
32
It is observed that the circumferential bending moments of CON16 have a range of about
1/10th of the reference values, and the axial membrane forces have about half the reference
range. On the contrary, the circumferential membrane forces oscillate over elements with
amplitudes that are two orders of magnitude greater than the reference values. The net result
for CON16 is that, for the thickness of 0.0025, it has excessive stiffness due to the
domination of the membrane actions. The overall strain energy is only about 35% of the
reference value.
The circumferential membrane strain εx is defined in Equation (9), and it involves coupling
between the tangential displacement gradient ux
and the transverse deflection gradient
wx
which is governed mainly by the bending moments. The assumed isoparametric
quadratic displacement fields in the conforming elements imply that ux
varies linearly
with x, but the term ( ).z wx x
varies quadratically. The combination of these components
of strain leads to the locally quadratic distributions of membrane forces evident in Figure
16(i), and their corresponding strains when ν = 0. These oscillate within each element, with
the most extreme oscillation varying from approximately 260×10-5 at the edges to -127×10-5
at the centre. Clearly such strains are excessive when compared to an average of -2.2×10-5
from the reference solution. However, the axial membrane strain only involves vy
, since
the y-axis coincides with a straight generator of the shell and zy
is zero, and thus there is
no such coupling of strain terms.
Since it appears that the locking problems with the compatible models affect the quality of
the circumferential membrane stress-resultants nx the most, Figures 17 to 20 presents
convergence of its contours as obtained from the 4 different models with meshes 8×8 to
33
64×64 for a thickness of 0.025. The range of contours is -38 ≤ nx ≤ 0 for each mesh in these
figures, which corresponds to the reference solution, and the actual ranges output in each case
are also shown.
Observations on these Figures are made as follows:
• The ETC and CMI models have broadly similar patterns of stress, although the nature of
the equilibrium solution is affected by discontinuities of nx and the transverse shear
resultant qx along the fold lines of the facetted model. These discontinuities exist in order
to maintain codiffusivity of their resultants. The contours of nx as displayed in Figure 17
appear to indicate tensile tractions on the free edge when the range is -38 to 0. However,
increasing the range slightly above zero reveals that these tractions are zero as required
for strict equilibrium, but local domains of tension exist within elements along the
boundary.
• The conforming results in Figure 18 again indicate the consequences of membrane
locking with large oscillations of nx which have the right order of magnitude for their
average values. Similar patterns of stress-resultants are observed in all the meshes
considered, although the overall strain energy and the mid-span deflection have more or
less converged. It has to be noted that the reference contours of nx are one or two orders
of magnitude less than those of ny, and Poisson’s ratio has been assumed to be zero, hence
there is no linking of ny with nx in the constitutive relations.
• In Figure 19, the CAS assumed strain models produce oscillations of nx in the axial
direction, particularly in elements adjacent to the free edge, which reduce with mesh
refinement. It would appear that this is due to the process of filtering and transforming to
an objective strain distribution. It should be noted that the objective strains in the CAS
models are determined to minimise a total membrane strain energy norm over an element
34
without focusing on one particular component. Thus when x yn n , the improvement of
nx is not as apparent as that of ny as indicated in Figure 16.
• On the contrary, the CMI models in Figure 20 lead to local nx stress-resultants that are
remarkably similar to the reference solution.
• With the exception of the CON models, the stress contours from the three remaining
element types, ETC, CMI and CAS, converge for the finest 64×64 mesh.
ETC8: -72 ≤ nx ≤ 37 ETC16: -41 ≤ nx ≤ 16
ETC32: -38 ≤ nx ≤ 26 ETC64: -38 ≤ nx ≤ 2
Figure 17: Contours of circumferential membrane force nx plotted in the range -38 to 0 for 4
meshes of equilibrium elements. For each mesh the actual range of values computed in the
solution are displayed below the contour plot. Thickness = 0.025
35
CON8: -578 ≤ nx ≤ 1110 CON16: -420 ≤ nx ≤ 775
CON32: -147 ≤ nx ≤ 195 CON64: -67 ≤ nx ≤ 21
Figure 18: Contours of circumferential membrane force nx plotted in the range -38 to 0 for 4
meshes of conforming elements. For each mesh the actual range of values computed in the
solution are displayed below the contour plot. Thickness = 0.025.
36
CAS8: -107 ≤ nx ≤ 37 CAS16: -41 ≤ nx ≤ 10
CAS32: -38 ≤ nx ≤ 2.7 CAS64: -38 ≤ nx ≤ 0.5
Figure 19: Contours of circumferential membrane force nx plotted in the range -38 to 0 for 4
meshes of assumed strain elements. For each mesh the actual range of values computed in the
solution are displayed below the contour plot. Thickness = 0.025.
37
CMI8: -38 ≤ nx ≤ 2.2 CMI16: -38 ≤ nx ≤ 1.6
CMI32: -38 ≤ nx ≤ 0.8 CMI64: -37.8 ≤ nx ≤ 0.3
Figure 20: Contours of circumferential membrane force nx plotted in the range -38 to 0 for 4
meshes of assumed strain elements. For each mesh the actual range of values computed in the
solution are displayed below the contour plot. Thickness = 0.025.
6. Conclusions
In this paper a study is presented of some alternative finite element models for the linear
elastic analysis of thin plates and shells. These models are designed to avoid locking as
experienced with conventional isoparametric conforming elements governed by Reissner-
Mindlin theory. The main aim has been to compare the quality of solutions, in terms of both
38
local displacements and fields of stress-resultants, obtained from a conforming element with
those from elements with additional assumed strain fields and a hybrid equilibrium element,
which is inherently free from locking. Benchmark problems have been considered in the form
of a trapezoidal plate and the Scordelis-Lo cylindrical shell.
• As is well known, the isoparametric conforming models exhibit locking as their thickness
is reduced. A consequence that is not so often realised is that the stress-resultants
associated with transverse shear or membrane strains can oscillate significantly. The plate
model suffers from inadequate approximations of shear strains since it is based on the
same functions for interpolating both translations and rotations. As demonstrated for a
simple beam example, these approximations lead to locking and extreme oscillations of
shear forces. This may be further exacerbated by the presence of the singularity at the
obtuse corner of the trapezoidal plate. Although the oscillations are evident within
elements, shear forces also tend to be highly discontinuous at element interfaces,
throughout the interior of the plate.
• Significant overall improvements arise from using the different forms of assumed shear
strain, where the H3O9 element performs marginally better than the MITC9 element in
terms of predicting the strain energy and for irregular meshes, with both elements
performing better in this respect compared to the hybrid equilibrium element and much
more so compared to the conforming element. However, shear and membrane forces can
still remain with highly oscillatory patterns which depend on the definitions of the
assumed strains. In the examples studied in this paper, the MITC9 models generally
provide better agreement with the hybrid equilibrium models in this context.
• The main conclusion is that whilst the hybrid and assumed strain models overcome
locking as regards displacements, the hybrid models may be more efficient at providing
39
better quality stress-resultants with coarser meshes. This is particularly so for those
resultants, e.g. the previously mentioned oscillatory forces, which contribute little to the
total strain energy of a solution, but yet may be significant to the design or assessment of
a structure, It is quite common practice, in the initial design stage of a composite form of
structure, to base design on stress-resultants obtained from a linear elastic analysis
assuming homogeneous isotropic materials. In this context, the strongly equilibrating
nature of the output from hybrid equilibrating models offers distinct advantages.
• The equilibrium model suffers, as do all the models, from the presence of large stress
gradients in boundary layers where a refined mesh would be required to recover good
quality stress fields. In-spite of the presence of boundary layers, the equilibrium plate
models satisfy strongly the equilibrium conditions with uniformly distributed transverse
loading, and so they provide upper bounds to the exact strain energies. Thus these models
complement the conforming ones which provide lower bounds, and the use of both
models leads to an upper bound error estimate of either solution.
• For the hybrid equilibrium model of the shell problem, convergence of strain energy,
displacements and stress-resultants appears to be good despite two related shortcomings
for modelling shells: (i) the curved geometry is approximated by a faceted shape, and (ii)
stresses due to torsional moments are not fully defined in subdomains at the interfaces
between flat elements. Hence these models do not strictly satisfy the equilibrium
conditions and do not provide theoretical bounds on the strain energies of solutions.
However, in practice some confidence can be gained from the complementary use of
equilibrium and assumed strain models when their energies converge towards each other
with mesh refinement.
40
• Further work is required to consider the use of faceted hybrid equilibrium models for
more general shapes of shell, i.e. those where the vertices of the flat quadrilateral
elements do not lie on a surface, e.g. a warped surface with negative Gaussian curvature.
For such cases a flat triangular form of the hybrid element would be appropriate;
alternatively the stability of an assembly of triangular elements to form a warped facetted
quadrilateral macro-element should be studied.
Contribution of the paper and its practical relevance
Engineers must take responsibility for their computational models, and the selection of appropriate
finite elements from those available in commercial software can be problematic.
Selection raises many questions that need to be addressed, including most importantly whether a
plate/shell can be regarded as thick or thin. A range of formulations is commonly available based on
displacement elements with different underlying theories e.g. Reissner-Mindlin or Kirchhoff, different
numbers of nodes (typically 4 or 8 noded quadrilaterals), and different ways to compensate for
locking with “thin” plates when modelled by Reissner-Mindlin elements. It is often not a
straightforward matter to realise the details of an implementation, and the general advice should be to
examine the convergence behaviour of each case so as to determine the credibility of solutions. It still
appears to be a common fallacy that solutions from commercial software “verify equilibrium at all
points”.
This paper is intended to help civil/structural engineers to gain confidence in the use of finite element
analysis of plate and shell structures in the design of new structures or the assessment of existing
ones, particularly when the structural forms are thin. The paper explains potential problems with
conventional elements, and demonstrates the benefit of carrying out analyses of two complementary
models as a means of verification. In this context, stress based equilibrium models, which are not yet
widely available, are described and used in the numerical examples. The authors are of the opinion
that such models could have an important role to play in the design process.
41
This research did not receive any specific grant from funding agencies in the public,
commercial, or not-for-profit sectors.
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44
List of Figure captions
Figure 1: Cantilever beam indicating positive senses of translation and rotation.
Figure 2: Relative error v. depth/length ratio.
Figure 3: Translations corresponding to t = L and t → 0.
Figure 4: Rotations corresponding to t = L and t → 0.
Figure 5: bending moments corresponding to t = L and t → 0.
Figure 6: shear forces corresponding to t = L and t → 0.
Figure 7: Positions of tying points for MITC9 element ( a 1/ 3= , b 3 5= , and c 1= ).
Figure 8: Hybrid equilibrium element as a macro-element.
Figure 9: Plate modelled with the initial 2×2 mesh of elements.
Figure 10: Convergence of relative errors of different plate models of thickness 0.0025m.
Figure 11: Contour maps of von Mises stress at top of 16×16 mesh for thinner plate (t =
0.0025m) in range [0, 1.5] GPa.
Figure 12: Contour maps of von Mises stress at midplane of 16×16 mesh for thinner plate (t =
0.0025m) with range [0, 0.0012] GPa.
Figure 13: The Scordelis-Lo cylindrical shell as modelled by a finite element mesh. The
cylinder has a radius of 25 units and it subtends an angle of 80º. The original thickness of the
shell is 0.25 units.
Figure 14: % error in total strain energies for the 4 models and 3 thicknesses of shell.
Figure 15: Views of deformed model CMI64 with thicknesses in the range 0.25 to 0.00025.
Figure 16: Contours of stress-resultants nx, ny, mx for a “locked” compatible model compared
with a “reference” equilibrium model for thickness 0.0025.
Figure 17: Contours of circumferential membrane force nx plotted in the range -38 to 0 for 4
meshes of equilibrium elements. For each mesh the actual range of values computed in the
solution are displayed below the contour plot. Thickness = 0.025.
Figure 18: Contours of circumferential membrane force nx plotted in the range -38 to 0 for 4
meshes of conforming elements. For each mesh the actual range of values computed in the
solution are displayed below the contour plot. Thickness = 0.025.
Figure 19: Contours of circumferential membrane force nx plotted in the range -38 to 0 for 4
meshes of assumed strain elements. For each mesh the actual range of values computed in the
solution are displayed below the contour plot. Thickness = 0.025.
45
Figure 20: Contours of circumferential membrane force nx plotted in the range -38 to 0 for 4
meshes of assumed strain elements. For each mesh the actual range of values computed in the
solution are displayed below the contour plot. Thickness = 0.025.