AmitRamji10241445FEA Report

Amit Ramji – A4 – University of Hertfordshire
1
COURSEWORK ASSIGNMENT
Module Title: Mechanics and Properties of Materials Module Code: 6ACM0003
Assignment Title: Finite Element Analysis of a Wind
Tunnel Model
Individual
Tutor: Dr Y Xu/Dr A Chrysanthou Internal Moderator: Dr. Yong Chen
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2
ASSIGNMENT BRIEF
Students, you should delete this section before submitting your work.
This Assignment assesses the following module Learning Outcomes:
4. Examine existing designs and actual components in engineering situations, using methods such as
finite element analysis, photoelasticity, non-destructive testing and fractography.
5. Limit the occurrence of failure in materials by appropriate modelling, design and materials selection.
6. Apply analytical methods to structural components subjected to complex stress/strain fields.
Assignment Brief:
In this assignment, you will create a finite element model of a wind tunnel model of a wing to be tested at
supersonic speeds in order to assess its structural integrity. You should carry out the modelling and write a
short report on your findings. Further details are provided on the attached sheets.
Submission Requirements:
Submission shall be through StudyNet
This assignment is worth 50 % of the overall in- course assessment for this module.
Marks Awarded for:
Validity of the FE results - 40%; Extent to which reporting requirements are met - 20%; concluding
remarks section – 20%; presentation of report - 20%.
A note to the Students:
1. For undergraduate modules, a score above 40% represent a pass performance at honours level.
2. For postgraduate modules, a score of 50% or above represents a pass mark.
3. Modules may have several components of assessment and may require a pass in all elements.
For further details, please consult the relevant Module Guide or ask the Module Leader.
Typical (hours) required by the student(s) to complete the assignment: 15 hours
Date Work handed out:
7
th
November 2013
Date Work to be handed in:
29
th
November 2013
Target Date for the return of
the marked assignment:
7
th
January 2014
Type of Feedback to be given for this assignment:
General feedback on StudyNet and individual feedback for each report.

3
Finite Element Analysis
ANSYS Report
Group E
Amit Ramji
10241445
University of Hertfordshire - Aerospace Engineering
Year 4 – Mechanics and Properties of Materials - 6ACM0003
26th
November 2013

4
Contents

Introduction .......................................................................................................................................................................... 5

Preliminary Analysis............................................................................................................................................................ 5

Finite Element Analysis (FEA) Procedure........................................................................................................................... 5

Analysis:............................................................................................................................................................................... 8

Alternative methods of analysis ........................................................................................................................................... 9

Conclusions:....................................................................................................................................................................... 10

Discussion .......................................................................................................................................................................... 11

References .......................................................................................................................................................................... 12

Table
of
Figures

Figure 1 – Cantilever Plate geometry....................................................................................................................................... 5

Figure 2 – Material Properties Input ........................................................................................................................................ 6

Figure 3 - Uniform stiffness properties.................................................................................................................................... 6

Figure 4 - Uniform thickness properties .................................................................................................................................. 6

Figure 5 - Selection of fine mesh ............................................................................................................................................. 6

Figure 6 - Post meshing operation............................................................................................................................................ 6

Figure 7 - Boundary conditions (Fix all DOF between Node ID’s E to F).............................................................................. 7

Figure 8 - Applying Pressure Loading onto wind surface ....................................................................................................... 7

Figure 9 – Nodal Displacements stating Max Deflection at Wing Tip (CD) is 1.596mm....................................................... 7

Figure 10 - Stress Intensity Showing Highest Stress at Node F, lowest at Node A as expected. ............................................ 8

Figure 11 - Von-Misses Stress peak at Node F as expected. ................................................................................................... 8

Data
Tables

Table 1 - Provided Parameters and calculated pressures ......................................................................................................... 5

Table 2 - Cantilever Plate Coordinates .................................................................................................................................... 5

Table 3 - Summary Material Properties ................................................................................................................................... 6

Table 4 - Raw material properties from sample tests............................................................................................................... 6

Table 5 - Post Processor Output Summary Table of results .................................................................................................... 8

Table 6 - Roark's Stress and Strain simplification into Rectangular Plate............................................................................... 9

5
Introduction

This
study
considers
a
single
edge
cantilever
plate
bending
for
interpretation
and
simplification
of
a

model
aircraft
wing
at
supersonic
speeds
for
wind
tunnel
testing.
The
objectives
are
to
evaluate
the

pressure
loads
and
analyse
the
structural
integrity
of
the
wing
model
to
be
tested.
The
relation
to
a
real

wing
allows
for
identification
of
stress
concentration
centre
and
maximum
stress
regions
which

attention
can
be
paid
in
more
detail
by
further
analysis,
reinforcement
or
alternative
modelling

techniques.
The
Finite
Element
methods
used
in
this
report
provide
a
basis
for
understanding
the

properties
of
thin
plates
under
uniform
pressure
loads,
however
in
reality
the
loading
conditions
and

boundary
conditions
are
very
complex.
Simple
analytical
methods
do
not
exist
for
complex
shapes
such

as
a
wing
plan-‐form,
hence
a
comparative
study
is
shown
using
rectangular
cantilever
plates
to
display

that
the
theory
agrees
but
the
values
obtained
are
non
comparable.
Thus
providing
further
significance

to
Finite
Element
Modelling
(FEM)
methods
in
order
to
analyse
the
part
geometry
chosen
for
design
in

haste,
compared
to
non-‐conservative
approximations
such
as
rectangular
plate
methods
described
later

in
this
report.

Table 1 - Provided Parameters and calculated pressures

Preliminary
Analysis

Sample
Calculation
of
Pressure
Loads

∆𝑝 = 𝑞!
4
𝑀!
! − 1
𝛼
Calculation
of
Max
Pressure
Loading
from
Mach
No:

𝑀!" = 1.5 ∴ 𝑞!" = 260 𝑀𝑃𝑎 ∴ Δ𝑝!" = 𝑞!"
4
𝑀!"
!
− 1
8!
360
2𝜋
= 129.881 𝑀𝑃𝑎
𝑀!" = 2.5 ∴ 𝑞!" = 275 𝑀𝑃𝑎 ∴ Δ𝑝!" = 𝑞!"
4
𝑀!"
!
− 1
8!
360
2𝜋
= 67.032 𝑀𝑃𝑎
∴ Δ𝑝!"# = 𝑆. 𝐹 × 𝑀𝐴𝑋 Δ𝑝!" Δ𝑝!" = 𝟏𝟗𝟒. 𝟖𝟐𝟐 𝐌𝐏𝐚 (𝑊ℎ𝑒𝑟𝑒 𝑆. 𝐹 = 1.5 𝑔𝑙𝑜𝑏𝑎𝑙)
Finite
Element
Analysis
(FEA)
Procedure

Initially
set
up
the
geometry
of
a
flat
plate
with
the
dimensions
as
shown
in
Figure 1
and
tabulated

coordinates
in
Table 2.
Later
set
up
the
plate
geometry
by
selecting
the
points
and
creating
the
lines
as
in

Figure 1.

Table 2 - Cantilever Plate Coordinates
Figure 1 – Cantilever Plate geometry
Given
Parameters

Angle
of
attack, 𝛼
(deg)
8.0
7.0
6.0
5.0
4.0
3.0
2.0
1.0

Angle
of
attack, 𝛼
(rad)
0.13963
0.12217
0.10472
0.08727
0.06981
0.05236
0.03491
0.01745

M01
1.5

M02
2.5

S.F
(Safety
Factor)
1.5

Δ𝑝!"For
M01
129881.0
113645.9
97410.8
81175.6
64940.5
48705.4
32470.3
16235.1

Δ𝑝!"
For
M02
67031.7
58652.8
50273.8
41894.8
33515.9
25136.9
16757.9
8379.0

Node
ID
X
(m)
Y
(m)

A
0
0.180

B
0
0

C
0.080
0.080

D
0.030
0.080

E
0
0.150

F
0
0.025

6
Secondly
set
up
the
linear
elastic
isotropic
material
properties
as
shown
below
in
Figure 2
and
Table 3,

where
the
steel
plate
is
assumed
to
perfectly
manufactured
with
uniform
(isotropic)
properties
in
all

directions,
perfectly
elastic,
and
without
consideration
of
thermal
expansion
as
the
plate
dimensions
are

relatively
small.

Table 3 - Summary Material Properties
Table 4 - Raw material properties from sample tests
Figure 2 – Material Properties Input
Next
set
up
the
thickness
properties
of
the
plate
to
account
for
the
stiffness
properties,
again
this

assumes
the
material
stiffness
is
uniform
in
all
directions,
which
isn’t
the
case
for
all
material
scenarios.

Figure 3 - Uniform stiffness properties
Figure 4 - Uniform thickness properties
Subsequently
set
up
a
Fine
4
Node
quadrilateral
mesh
using
the
surface
selection
tool.
This
type
of
mesh

can
be
used
for
analysis
of
plane
stress
or
strain,
thin
plate
bending
and
for
shear
analysis
of
plates.

Other
mesh
types
are
explained
in
Table
5.1
of
literature
[1],
also
in
[2]
and
new
approaches
found
in
[3-‐
6].
Package
specific
FEA
guides
also
explain
the
use
and
types
of
mesh’s
and
their
applications
whereas

tools
such
as
Abaqus,
Hypermesh
and
MSC
PATRAN
can
allow
specific
mesh
optimisation
in
key
areas,

however
is
out
of
the
scope
in
this
application
of
simple
cantilever
plate
bending.

Figure 5 - Selection of fine mesh
Figure 6 - Post meshing operation

Steel
Plate

[Rolled:
tmin<3mm<tmax]

Units

Young’s
modulus,
E
200
Gpa

Poisson’s
Ratio,
v
0.30
-‐

Tensile
strength
(Yield),
𝝈 𝑻𝒀𝑺
600
Mpa

Tensile
strength
(Ultimate)
𝝈 𝑼𝑻𝑺
800
Mpa

7
Following
meshing,
apply
boundary
conditions
and
load
cases
into
the
pre-‐processor
menu,
where
the

wing
root
is
treated
as
fixed
in
all
Degrees
of
Freedom
(DOF)
and
apply
the
uniform
maximum
pressure

load
(Δp!"#)
as
calculated
in
the
Preliminary Analysis
section
of
this
report.
The
direction
of
the
pressure

load
is
to
be
applied
normal
to
the
wing
surface,
however
as
the
analysis
is
linear,
geometry
above
and

below
the
mid-‐plane
is
symmetric,
weight
direction
is
not
considered,
therefore
the
loading
direction

does
not
affect
the
results.

Figure 7 - Boundary conditions (Fix all DOF between Node ID’s E to F)
Figure 8 - Applying Pressure Loading onto wind surface
Finally
proceed
with
simulation
of
the
load
case
and
view
the
solution
as
nodal
for
Displacements,

Stress
intensity
and
Von-‐Misses
stress.

Figure 9 – Nodal Displacements stating Max Deflection at Wing Tip (CD) is 1.596mm

8
Figure 10 - Stress Intensity Showing Highest Stress at Node F, lowest at Node A as expected.
Figure 11 - Von-Misses Stress peak at Node F as expected.
Post
Processor
Outputs
Output
(unit)
Output
(Required
unit)

Displacement
(Normal
to
Wing
surface)
0.001596
(m)
1.596
(mm)

Stress
Intensity
(Node
F)
729,000,000
(Pa)
729
(MPa)

Von-‐Misses
(Node
F)
694,000,000
(Pa)
694
(MPa)

Table 5 - Post Processor Output Summary Table of results
Analysis:

From
the
Finite
Element
method
above,
the
results
state
that
the
maximum
stress
is
seen
at
Node
F,

which
agrees
with
classical
static
mechanics.
Node
F
is
the
point
that
is
surrounded
by
most
of
the

perpendicular
area
from
the
root
chord
therefore
will
have
the
highest
stress
concentration.
Additionally

the
method
of
constraint
for
the
FE
modelling
means
that
this
area
is
showing
to
fail
and
go
beyond
the

elastic
region
of
the
material,
however
the
truth
may
be
that
the
part
will
not
fail
in
this
region
and
is
a

property
of
the
constraint
method
used
in
modelling.

Furthermore
the
stress
distribution
will
become
smaller
as
the
leading
edge
(AC)
is
approached
until
the

stress
is
the
same
as
the
wing
loading
pressure.
In
the
lateral
direction
moving
from
the
root
to
the
tip,

the
stress
levels
will
decrease
as
the
constraint
is
moved
further
away
from
the
area
of
interest.
Thus
the

stress
distribution
in
the
lateral
direction
will
decrease
to
the
pressure
loading
value
(wing
tip)
as
this

case
is
considering
a
cantilever
solution.
The
maximum
deflection
nodes
are
also
as
expected
as
the
wing

tip
is
furthest
from
the
support
of
the
root
structure,
hence
will
be
prone
to
relatively
high
levels
of

deflections.
The
slope
of
deflection
at
the
root
will
be
greatest
as
once
again
the
constraint
is
present
in

this
region
from
EF.

9
Overall
the
FE
model
does
correctly
describes
what
is
true
regarding
cantilever
plate
bending
and

justifies
a
reason
to
conduct
further
analysis
in
the
constraint
region
at
Node
F.
The
possibility
of
gradual

increased
cross
section
can
be
justified
or
a
reinforcement
wing
spar
added.
However
for
this
analysis,

the
Von
Misses
stress
is
shown
to
be
694
MPa
(Table 5),
the
material
Tensile
Yield
Strength

(σ!"#)=600MPa
(Table 3),
therefore
suggests
this
cantilever
plate
has
exceeded
its
elastic
limits
and
could

potentially
fail
rapidly
under
plastic
fast
fracture.
This
is
the
most
extreme
case,
judging
from
the
stress

difference
between
beginning
of
plastic
region
and
maximum
Von-‐Misses
stress,
the
difference
is
94

MPa,
therefore
requires
further
investigation
if
reinforcement
is
not
to
be
carried
out.

Alternative
methods
of
analysis

There
can
be
many
methods,
which
attempt
to
calculate
the
stress
for
a
non-‐rectangular
cantilever
plate

to
calculate
the
stress
at
different
locations.
Using
integration
methods
to
work
out
the
areas
in
different

locations
over
the
wing
and
discretizing
the
areas
into
dy
and
dx
of
which
the
product
is
a
small
element

area.
The
slope
of
the
leading
edge
and
trailing
edge
will
lead
one
to
encounter
a
function
of
geometry
for

the
integration
limits.
Later
the
pressure
loading
is
multiplied
to
acquire
a
loading
function
per
dydx

area.
Subsequently
a
function
for
moment
arm
is
required
which
cantilever
beam
bending
theory

provides
for
uniform
load
distribution.
However
this
is
the
exact
same
as
discretising
the
problem
into
4

Node
Quad
Elements,
which
FEA
has
provided
a
solution
for
in
a
shorter
time.

Other
methods
found
in
chapter
7
of
reference
[7]
by
Megson,
considers
a
pure
analytical
approach
is

methods
using
Kirchhoff-‐Love
plate
representation,
Navier
Solutions,
Mildlins
methods
for
thicker

plates,
or
the
most
appropriate
for
the
current
case
would
be
to
utilise
Reissner-‐Stein
Cantilever
plates.

[1,
7-‐13]

Rectangular
plate
approximation

An
attempt
to
show
the
trend
has
been
made
below
which
utilises
methods
based
on
stiffness
constants

of
flat
plates
through
experimental
means
and
problem
simplification
into
a
rectangular
plate.
The
same

solution
is
not
reached,
as
the
real
solution
would
require
an
iterative
and
element
wise
approach
as
FE

provides
and
is
explained
above.
This
does
however
provide
a
justification
of
classic
mechanics
of

perpendicular
moment
arms
being
kept
constant
and
the
wing
root
(parameter
“a”
below)
being

increased.

Table 6 - Roark's Stress and Strain simplification into Rectangular Plate.
From
the
above,
increasing
root
length
has
very
shallow
stress
climb
rate
compared
with
perpendicular

distance
(Span)
increase,
thus
confirming
classic
beam
bending
theory.

Roarks
[Table
11.4]
(Simplification
of
wing
into
Flat
Rectangular
Cantilever
Plate
to
observe
trend)
[1]

t
(mm)
3

a
(mm)
50
80
120
160
240

b
(mm)
80

a/b
0.625
1
1.5
2
3

Δ𝑝!"#
(MPa)
-‐0.194821529

𝛽!
0.38
0.665
1.282
1.804
2.45

𝛽!
0.386
0.565
0.73
0.688
0.434

𝛾!
0.541
0.701
0.919
1.018
1.055

𝛾!
0.526
0.832
1.491
1.979
2.401

𝜎 =
−𝛽!Δ𝑝!"# 𝑏!
𝑡!

(at
centre
of
fixed
edge)
(MPa)
52.6451
92.1289
177.6080
249.9257
339.4224

𝑅 = 𝛾!Δ𝑝!"# 𝑏

(at
centre
of
fixed
edge)
(mm)
8.4319
10.9256
14.3233
15.8663
16.4429

𝜎 =
−𝛽!Δ𝑝!"# 𝑏!
𝑡!

(at
centre
of
free
edge)
(MPa)
-‐53.4763
-‐78.2750
-‐101.1340
-‐95.3154
-‐60.1263

𝑅 = 𝛾!Δ𝑝!"# 𝑏

(at
end
of
free
edge)
(mm)
8.1981
12.9673
23.2383
30.8441
37.4213

10
Conclusions:

By
observing
the
Maximum
Von
Misses
stress
of
694
MPa
from
Table 5,
one
can
observe
that
some
areas

have
indeed
surpassed
the
elastic
limit
of
steel
with
the
current
geometry
of
which
the
yield
limit
is

shown
in
Table 3
of
600
MPa.
This
may
be
due
to
modelling
constraints
as
discusses
earlier,
where
the

maximum
stressed
area
is
amplified
by
the
presence
of
constraint
features.
This
has
the
same
effect
as

stress
concentration
factors
where
stresses
are
amplified
based
on
geometry,
further
reading
can
be

found
in
Petersons
et
al
[14]
for
stress
concentrations
factors
(Kt)
based
on
geometry.

The
structural
integrity
of
the
plate
wing
model
in
the
wind
tunnel
will
be
effected
as
the
stressed

material
would
yield
in
some
places
as
indicated
by
FEA,
therefore
may
exhibit
fast
fracture.
To
reduce

the
chances
of
failure
and
damage
to
the
wind
tunnel,
this
case
should
be
investigated
further
by;
Non-‐
Linear
FEA,
a
better
representation
of
constraints,
possible
structural
improvements
such
as
a
thicker

plate,
thicker
wing
root,
adding
a
stiffener
spar
or
spreading
the
loads
through
the
cantilever
over
a

larger
root
chord
distance
if
possible.
In
actuality
the
structural
integrity
of
the
wind
tunnel
will
be
fine

as
the
safety
factor
will
not
cause
the
part
to
fail
during
testing,
however
the
wing
plate
will
need
to
be

checked
regularly
if
left
as
is,
there
will
be
some
permanent
deformation
which
could
progressively

worsen.

Non-‐strength
related
issues
identified
with
the
FEA
is
with
the
use
of
constrains.
Use
of
nodal
elements

to
constrain
the
root
of
the
wing
has
identified
and
amplified
a
potential
highly
stressed
region
as
stated

previously.
Material
is
said
to
be
failing
by
surpassing
the
steel’s
elastic
yield
limit.
This
indicates
a
high

stress
concentration
when
in
actual
fact
more
information
is
required
on
the
part.
A
static
deflection

experiment
can
be
conducted
to
prove
or
disprove
the
FEA
and
is
usually
what
is
done
on
actual
aircraft

skins.
The
use
of
strain
rosettes
and
bonded
strain
gauges
on
aircraft
test
skins
can
identify
experimental

strains
which
can
be
later
input
back
into
the
FEA
model
for
comparison.

Checking
by
analytical
methods
is
a
lengthy
task
and
the
solution
will
not
always
be
accurate
as
the

number
of
iterations
can
never
be
matched
to
that
carried
out
by
FEA.
Some
analytical
methods
are

shown
above
and
their
limitations
on
the
real
geometry
of
the
plate.

The
objective
of
this
FE
investigation
was
to
determine
the
structural
integrity
of
a
test
wing
to
be
tested

at
supersonic
speeds
in
a
wind
tunnel.
The
analysis
shows
that
further
investigation
is
required
as
the

material
would
be
prone
to
yielding
in
some
areas.
Therefore
simply
based
on
that
evidence,
justification

for
FEA
is
complete
as
it
avoids
potential
damage
to
the
wind
tunnel,
saves
on
costs
for
development
as

components
can
be
sized
to
withstand
the
loads
imparted
on
them
without
testing.
For
the
case
of

Cantilever
thin
plates,
analytical
methods
exist
for
rectangular
shapes
however
the
boundary
conditions

for
simply
supported
plates
cannot
be
applied
in
this
application
directly.
Cantilever
beam
bending

theory
can
be
used
to
determine
the
maximum
Bending
Moment
(BM)
at
the
root;
however
often
in
most

cases
the
faster
and
accurate
solution
will
be
by
FEM
methods.

The
maximum
deflection
of
this
plate
during
peak
loading
conditions
was
approximately
1.6mm
as

shown
in
Table 5.
Which
means
it
has
a
deflection
ratio
of
approximately
2%
over
the
span
of
80mm.
This

is
a
reasonable
deflection
and
can
be
calculated
with
cantilever
beam
bending
functions.
(A
clamped
15

cm
steel
rule
will
also
help
to
understand
the
static
mechanics
of
this
wing
loading
problem).
To
improve

deflection
the
real
wing
is
given
a
thickness,
with
reinforcement
stringers,
spars
and
ribs,
thus
provides
a

larger
Second
Moment
of
Area
(I)
and
enables
the
skins
to
be
in
pure
tension
or
compression.

Composites
can
therefore
be
introduced
in
order
to
use
this
geometry
and
loading
condition
to
one’s

advantage.

11
Discussion

The
direction
of
pressure
loads
as
described
in
the
Finite Element Analysis (FEA) Procedure
section
of
this

report
explains
how
direction
is
important.
However
for
this
simplification
of
a
thin
cantilever
plate,
the

mid
plane
symmetry
and
combination
of
not
considering
weight
means
this
analysis
is
valid
if
a
+ve
or
–
ve
pressure
is
applied.
Other
investigations
including
sinusoidal
loading
due
to
shock
waves,
ground

interference,
weather
gusts
or
microbursts
may
be
considered
for
wing
loading
along
with
frequency

analysis
of
sustained
engine
imbalance
and
its
effects
on
fatigue.

Plate
modelling
can
be
used
for
fuselage
and
wing
skin
loading
analysis
alongside
other
applications

using
composite
structures
and
sandwich
panels
[15-‐18].
Consideration
of
cantilever
plate
methods
have

also
been
made
in
gear
tooth
analysis
as
reported
by
Wellauer
et
al
[19].
Composite
modelling
with

isotropic
properties
can
also
be
made
simple
with
thin
plate
analogies
where
minimum
strength
values

are
input
into
the
model
to
identify
stress
hot-‐spots,
delamination
and
surface
effects
for
further

consideration
and
fibre
orientation
and
design
sizing.
[16,
18,
20,
21]

From
Table 1,
one
can
see
how
the
pressure
loading
is
increasing
as
the
angle
of
attack
is
increased
and
at

lower
Mach
No’s.
The
benefit
of
FEM
methods
is
that
one
can
input
all
these
complex
combinations
into

the
model
as
separate
load
cases
and
run
the
analysis
in
a
very
short
amount
of
time.
Therefore
the

sizing
and
analysis
can
be
carried
out
on
the
components
with
consideration
to
a
wide
range
of
input

variables/cases.
The
same
applies
for
frequency
and
vibration
analysis,
a
range
band
can
be
set
for
each

load
case
and
studied
further
in
the
post-‐processor
or
numerically
through
direct
output
files.
It
may
be

interesting
to
investigate
increases
in
aircraft
pitch
angle,
thus
Mach
No
decreases,
meaning
from
the

range
considered
in
Table 1,
the
combination
of
these
flight
characteristics
could
worsen
the
loading
on

the
wing.

12
References

[1] Young, W., Budynas R. Roark’s formulas for stress and strain. 7. 2002, McGraw-Hill.
[2] Jaehwan, K., et al., Finite-element modeling of a smart cantilever plate and comparison with experiments. Smart
Materials and Structures, 1996. 5(2): p. 165.
[3] Clough, R.W. and C.A. Felippa, A refined quadrilateral element for analysis of plate bending. 1968, DTIC
Document.
[4] Batoz, J.L., An explicit formulation for an efficient triangular plate-bending element. International Journal for
Numerical Methods in Engineering, 1982. 18(7): p. 1077-1089.
[5] Batoz, J.L., K.J. Bathe, and L.W. Ho, A study of three-node triangular plate bending elements. International
Journal for Numerical Methods in Engineering, 1980. 15(12): p. 1771-1812.
[6] Hughes, T.J.R., R.L. Taylor, and W. Kanoknukulchai, A simple and efficient finite element for plate bending.
International Journal for Numerical Methods in Engineering, 1977. 11(10): p. 1529-1543.
[7] Megson, T.H.G., Chapter 7 - Bending of thin plates, in Aircraft Structures for Engineering Students (Fifth
Edition), T.H.G. Megson, Editor. 2013, Butterworth-Heinemann: Boston. p. 233-266.
[8] Wang, W. and M.X. Shi, Thick plate theory based on general solutions of elasticity. Acta mechanica, 1997.
123(1): p. 27-36.
[9] Arnold, D.N. and R.S. Falk. Edge effects in the Reissner-Mindlin plate theory. in Presented at the winter annual
meeting of the American Society of Mechanical Engineers. 1989.
[10] Karam, V.J. and J.C.F. Telles, On boundary elements for Reissner's plate theory. Engineering Analysis, 1988.
5(1): p. 21-27.
[11] Fo-van, C., Bending of uniformly cantilever rectangular plates. Applied Mathematics and Mechanics, 1980. 1(3):
p. 371-383.
[12] Reissner, E. and M. Stein, Torsion and transverse bending of cantilever plates. 1951: National Advisory
Committee for Aeronautics.
[13] Reissner, E., On bending of elastic plates. Quart. Appl. Math, 1947. 5(1): p. 55-68.
[14] Pilkey, W.D. and D.F. Pilkey, Peterson's stress concentration factors. 2008: John Wiley & Sons.
[15] Thomsen, O.T. and W. Rits, Analysis and design of sandwich plates with inserts—a high-order sandwich plate
theory approach. Composites Part B: Engineering, 1998. 29(6): p. 795-807.
[16] Lu, P., et al., Thin plate theory including surface effects. International Journal of Solids and Structures, 2006.
43(16): p. 4631-4647.
[17] Barbero, E.J., J.N. Reddy, and J. Teply, An accurate determination of stresses in thick laminates using a
generalized plate theory. International journal for numerical methods in engineering, 1990. 29(1): p. 1-14.
[18] Barbero, E.J. and J.N. Reddy, Modeling of delamination in composite laminates using a layer-wise plate theory.
International Journal of Solids and Structures, 1991. 28(3): p. 373-388.
[19] Wellauer, E.J. and A. Seireg, Bending strength of gear teeth by cantilever-plate theory. Journal of Engineering
for Industry, 1960. 82: p. 213.
[20] Newman Jr, J.C. and I.S. Raju, Analyses of Surface Cracks in Finite Plates Under Tension or Bending Loads.
1979, DTIC Document.
[21] Reddy, J.N., E.J. Barbero, and J.L. Teply, A plate bending element based on a generalized laminate plate theory.
International Journal for Numerical Methods in Engineering, 1989. 28(10): p. 2275-2292.

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