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

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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


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


meshes of assumed strain elements. For each mesh the actual range of values computed in the


37

CMI8: -38 ≤ nx ≤ 2.2 CMI16: -38 ≤ nx ≤ 1.6

CMI32: -38 ≤ nx ≤ 0.8 CMI64: -37.8 ≤ nx ≤ 0.3




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.


meshes of equilibrium elements. For each mesh the actual range of values computed in the



meshes of conforming elements. For each mesh the actual range of values computed in the





45




equilibrium and displacement elements for the design … - ea… · approximation fields (bathe et...

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