A method for the development and control of
stiffness matrices for the calculation of beam and
shell structures using the symbolic programming
language MAPLE
N. Gebbeken, E. Pfeiffer, I. Videkhina
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics
Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
Relevance
of
the
topic
In
structural
engineering
the
design
and
calculation
of
beam
and
shell
structures
is
a
daily
practice
.
Beam
and
shell
elements
can
also
be
combined
in
spatial
structures
like
bridges,
multi

story
buildings,
tunnels,
impressive
architectural
buildings
etc
.
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
Truss
structure,
Railway bridge Firth of Forth (Scotland)
F
olded plate structure,
Church in Las Vegas
Calculation
methods
In
the
field
of
engineering
mechanics,
structural
mechanics
and
structural
informatics
the
calculation
methods
are
based
in
many
cases
on
the
discretisation
of
continua,
i
.
e
.
the
reduction
of
the
manifold
of
state
variables
to
a
finite
number
at
discrete
points
.
Type
of
discretisation
e
.
g
.:

Finite Difference Method (FDM)
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
X
Y
x
i,j
i,j+1
i+1,j+1
i

1,j+1
i

1,j
i

1,j

1
i,j

1
i+1,j

1
i+1,j
x
y
y
Inside points
of grid
Outside points
of grid
Boundary of continuum
Center
point
Δ
Δ
Δ
Δ
i 1,j i,j
i,j
i , j 1 i,j
i,j
O( x)
x x
O( y)
y y
f f
f
f f
f
Differential quotients are substituted
through difference quotients
Continuum
u
1
v
1
u
3
v
3
u
2
v
2
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
Calculation
methods

type
of
discretisation

Finite
Element
Method
(FEM)
Static calculation of a concrete panel
First calculation step: Degrees of freedom in nodes.
Second calculation step: From the primary unknowns the state variables at
the edges of the elements and inside are derived.
Calculation
methods

type
of
discretisation

Meshfree
particle
solvers
(e
.
g
.
Smooth
Particle
Hydrodynamics
(SPH))
for
high
velocity
impacts,
large
deformations
and
fragmentation
Aluminiumplate
Fragment cloud
Experimental und numeric presentation of a high velocity impact:
a 5 [mm] bullet with 5.2 [km/s] at a 1.5 [mm] Al

plate.
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
PD Dr.

Ing. habil. Stefan Hiermaier
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
FEM

Advantages
:
Continua
can
easily
be
approximated
with
different
element
geometries
(e
.
g
.
triangles,
rectangles,
tetrahedrons,
cuboids)
The
strict
formalisation
of
the
method
enables
a
simple
implementation
of
new
elements
in
an
existing
calculus
The
convergence
of
the
discretised
model
to
the
real
system
behaviour
can
be
influenced
with
well

known
strategies,
e
.
g
.
refinement
of
the
mesh,
higher
degrees
of
element
formulations,
automated
mesh
adaptivity
depending
on
stress
gradients
or
local
errors
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
Aspects
about
FEM
Extensive
fundamentals
in
mathematics
(infinitesimal
calculus,
calculus
of
variations,
numerical
integration,
error
estimation,
error
propagation
etc
.
)
and
mechanics
(e
.
g
.
nonlinearities
of
material
and
the
geometry)
are
needed
.
Unexperienced
users
tend
to
use
FEM

programmes
as
a
„black
box“
.
Teaching
the
FEM

theory
is
much
more
time
consuming
as
other
numerical
methods,
e
.
g
.
FDM
At
this
point
it
is
helpful
to
use
the
symbolic
programming
language
MAPLE
as
an
eLearning
tool
:
the
mathematical
background
is
imparted
without
undue
effort
and
effects
of
modified
calculation
steps
or
extensions
of
the
FEM

theory
can
be
studied
easier!
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
The Finite Element Method (FEM) is mostly used for the analysis of
structures.
Basic
concept
of
FEM
is
a
stiffness
matrix
R
which
implicates
the
vector
U
of
node
displacements
with
vector
F
of
forces
.
R U F
Of
interest
are
state
variables
like
moments
(
M
),
shear
(
Q
)
and
normal
forces
(
N
),
from
which
stresses
(
,
)
and
resistance
capacities
(
R
)
are
derived
.
It
is
necessary
to
assess
the
strength
of
structures
depending
on
stresses
.
[ ] [ ]
cal allowable
F
A
R
F
l
l
l
l
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
Static System
Actions
Reaction forces
Deformation of System
M
V
M
H
V
H
F
M
V
M
H
V
H
1
T
A S F
A U
B S
R U F
Vector
S
of forces results from the
strength of construction.
Vector
U
of the node displacements
depends on the system stiffness
.
Structures should not only be
resistant to loads
, but also limit
deformations
and
be
stable against local or global collapse
.
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
In
the design process of structures we have to take into account not
only
static actions
, but different types of
dynamic influences
.
Typical
threat
potentials for
structures
:

The stability against earthquakes

The
aerodynamic
stability
of
filigran
structures

Weak
spot
analysis,
risk
minimisation
Collapse of the Tacoma Bridge
at a wind
velocity of 67 [km/h]
Consequences of an earthquake
Consequences of wind

induced
vibrations on a suspension bridge
Citicorp Tower NYC
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
( )
M U C U R U F t
dynamic problem
static problem

mass (
M
)

damping (
C
)

stiffness (
R
)
The most static and dynamic influences are represented in the
following equation
:
Mercedes

multistorey
in Munich
wind loading
FEM
for
the
solution
of
structural
problems
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
Research
goals
:
1
.
The
basic
purpose
of
this
work
is
the
creation
of
an
universal
method
for
the
development
of
stiffness
matrices
which
are
necessary
for
the
calculation
of
engineering
constructions
using
the
symbolic
programming
language
MAPLE
.
2. Assessment of correctness of the obtained stiffness matrices.
Short overview of the fundamental equations for the
calculation of beam and shell structures
u
i
u
j
j
w
j
i
w
i
Beam structures
Shell structures
Differential equation for a single beam
4
4
d w q
dx EJ
with
w

deflection,
EJ

bending stiffness (
E

modul of elasticity,
J

moment of inertia),
x

longitudinal axis,
q

line load
Beams with arbitrary loads
and complex boundary conditions
2. Theory of second order
3 2
3 2
,
d q dw d
dx EJ dx dx
4
4
4
4
4,with
4
d w q kb
n w n
dx EJ EJ
1. Beam on elastic foundation
with
n

relative stiffness of foundation,
k

coefficient of elastic
foundation,
b

broadness of bearing
with

shearing strain
3. Biaxial bending
4 2
4 2
d w N d w q
dx EJ dx EJ
with
N

axial force
Differential equations for a disc
(expressed in displacements)
2 2 2
2 2
2 2 2
2 2
1 1
0
2 2
1 1
0
2 2
u u v
x y x y
v v u
y x x y
Differential equation for a plate
4 4 4
4 2 2 4
2
w w w p
x x y y D
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
Calculation
of
beam
structures
For
the
elaboration
of
the
stiffness
matrix
for
beams
the
following
approach
will
be
suggested
:
1. Based on the differential equation for a beam the stiffness matrix
is developed in a local coordinate system.
2. Consideration of the stiff or hinge connection in the nodes at the
end of the beam.
3. Extension of element matrix formulations for beams with different
characteristics, e.g. tension/ compression.
4. Transforming the expressions from the local coordinate system
into the global coordinate system.
5. The element matrices are assembled in the global stiffness matrix.
FEM
equations
Equations from the strength of materials
Type of the
d
evelopment
Beam
structure
Tension

compression
Bending without
consideration of the
transverse strain
Bending with
consideration
of the transverse strain
Equation of
equlibrium
A S F
A
N dA
A
M ydA
A
M ydA
1
Geometrical
relations
T
A U
du
dx
2
2
du d w
y
dx dx
mit
du d dw
y
dx dx dx
2
Material law
1
B S
E
E
E
3
(2)
(3)
1
T
B A U S
du
E
dx
2
2
d w
E y
dx
mit
d dw
E y
dx dx
4
1
T
A B A U F
du
N EA
dx
2
2
d w
M EJ
dx
mit
d dw
M EJ
dx dx
5
(4)
(1)
1 1
( )
T
A B A F U
du N
dx EA
2
2
d w M
dx EJ
mit
M d dw
EJ dx dx
6
(6)
(4)
1 1 1
( )
T T
B A A B A F S
N
A
M
y
J
M
y
J
7
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
R
Development of differential equations of beams with or
without consideration of the transverse strain
4
2 3
3
1
'
2
2
''
2
3
'''
4
24
1
0 1 2 3
6
0 0 2 6
0 0 0 6
2
0 0 0 0
1
IV
x
w w
x x x
C
x
w
x x
C
q
w M
x
C
EJ
x
w Q
C
w q
x
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
Basic equations:
4
4
d w q
dx EJ
2
2
d w
M EJ
dx
3
3
d w
Q EJ
dx
Solution:
4
2 3
1 2 3 4
( )
24
qx
w x C C C x C x
EJ
Solution and derivatives in matrix form:
Algorithm for the elaboration of a stiffness matrix
for an ordinary beam
D
homogeneous
particular
i
j
u
q
u L C L
u
EJ
4
2 3
2
3
1
2
3
4
0
1 0 0 0
0
0 1 0 0
1
24
0 1 2 3
6
i
i
j
j
w C
C
q
w C
EJ
C
l
l l l
l l
l
L
u
L
C
4
2 3
3
1
'
2
2
''
2
3
'''
4
24
1
0 1 2 3
6
0 0 2 6
0 0 0 6
2
0 0 0 0
1
IV
x
w w
x x x
C
x
w
x x
C
q
w M
x
C
EJ
x
w Q
C
w q
x
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
Unit displacements of nodes
Substituting
in
the
first
two
rows
of
the
matrix
D
the
coordinates
for
the
nodes
with
x
=
0
and
x
=
l
we
get
expressions
corresponding
to
unit
displacements
of
the
nodes
:
D
or
L1
4
2 3
3
1
'
2
2
''
2
3
'''
4
24
1
0 1 2 3
6
0 0 2 6
0 0 0 6
2
0 0 0 0
1
IV
x
w w
x x x
C
x
w
x x
C
q
w M
x
C
EJ
x
w Q
C
w q
x
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
Substituting
in
the
second
two
rows
of
the
matrix
D
the
coordinates
for
the
nodes
with
x
=
0
and
x
=
l
follow
the
shear
forces
and
moments
at
the
ends
of
a
beam
corresponding
with
the
reactions
:
f
i
l
f
wi
f
j
f
wj
M
i
l
Q
i
M
j
Q
j
Reaction forces and internal forces
2
2
6
1
2
3
4
0
0 0 0 0
0
0 0 2 0
0 0 0
0 0 2 6
2
wi i
i i
w j j
j j
f Q C
f M C
EJ q
f Q C
f M C
l
l
l
f
C
1 1
i
j
f
f EJ L C q L
f
1
L
or
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
1 1
q
C L u L L
EJ
We
express
the
integration
constants
by
the
displacements
of
the
nodes
:
i
j
u
q
u L C L
u
EJ
Replacing
with
delivers
C
1 1
f EJ L C q L
1 1 1 1
1 1 1 1 1
q
f EJ L L u L L q L EJ L L u q L L L L
EJ
or
in
simplified
form
:
q
f EJ r u q f
r
q
f
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
Within
means
q
f EJ r u q f
r
the
relative
stiffness
matrix
with
EJ
=
1
the
relative
load
column
with
q
=
1
q
r
The final stiffness matrix
r
and the load column for an ordinary beam:
q
f
w
i
i
w
j
j
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
4
4
4
4
4,with
4
d w q kb
n w n
dx EJ EJ
Solution:
1 2 3 4
4
( ) cos( ) sin( ) cos( ) sin( )
4
nx nx nx nx
q
w x Ce nx C e nx C e nx C e nx
n EJ
Elaboration of the stiffness matrix for a beam on an
elastic foundation
In analogous steps the development of the stiffness matrix for a beam
on an elastic foundation leads to more difficult differential equations:
Basic equations:
n
relative stiffness of foundation
k
coefficient of elastic foundation
b
broadness of bearing
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
Elaboration of the stiffness matrix for a beam on an
elastic foundation
The final stiffness matrix
r
and the load column :
q
f
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
Basic equations:
3
3
2
2
d q
dx EJ
dw d
dx dx
d
M EJ
dx
2
2
d
Q EJ
dx
Solution:
3
2
1 2 3
4 2
0 1 2 3
( )
6
2 3
( ) ( 2 )
2 3 24 2
qx
x C C x C x
EJ
x x EJ qx qx
w x C C x C C x
GF EJ GF
Algorithm for the elaboration of a stiffness matrix for a
beam element following the theory of second order
Considering transverse strain the algorithm changes substantially.
Instead of only one equation two equations are obtained with the two
unknowns bending and nodal distortion:
with
EJ
GF
(shearing strain)
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
The final stiffness matrix
r
and the load column for a beam element
following the theory of second order:
q
r
w
i
i
w
j
j
resultmatr_r
:=
12
E
J
l
(
)
12
l
2
6
E
J
12
l
2
12
E
J
l
(
)
12
l
2
6
E
J
12
l
2
q
l
2
6
E
J
12
l
2
4
E
J
(
)
3
l
2
l
(
)
12
l
2
6
E
J
12
l
2
2
E
J
(
)
6
l
2
l
(
)
12
l
2
q
l
2
12
12
E
J
l
(
)
12
l
2
6
E
J
12
l
2
12
E
J
l
(
)
12
l
2
6
E
J
12
l
2
q
l
2
6
E
J
12
l
2
2
E
J
(
)
6
l
2
l
(
)
12
l
2
6
E
J
12
l
2
4
E
J
(
)
3
l
2
l
(
)
12
l
2
q
l
2
12
x
z
m
n
o
m
1
o
1
n
1
x
xz
x
z
m
n
o
m
1
o
1
n
1
Axis of beam
(bended)
x
x
Axis of beam
(unformed)
Theory of first order
Theory of second order
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
Single beam
Beam on elastic
foundation
Harmonic oscillation
Biaxial bending
Theory of second order
4
4
d w q
dx EJ
4
4
1
4
4
1
4,
mit
4
d w q
n w
dx EJ
kb
n
EJ
g
4
4
2
4
2
4
2
,
mit
d W q
n W
dx EJ
F
n
EJ
4 2
2
3
4 2
3
mit
d w d w q
n
dx dx EJ
N
n
EJ
3
3
2
2
d q
dx EJ
dw d
dx dx
The formulas of the moment (M) and the shear force (Q)
2
2
d w
M EJ
dx
3
3
d w
Q EJ
dx
4
4
d w
q EJ
dx
2
2
d w
M EJ
dx
3
3
d w
Q EJ
dx
4
4
1
4
4
d w
q EJ n w
dx
2
2
d W
M EJ
dx
3
3
d W
Q EJ
dx
4
4
2
4
d W
q EJ n W
dx
2
2
d w
M EJ
dx
3 2
2
3
3 2
d w d w
Q EJ n
dx dx
4 2
2
3
4 2
d w d w
q EJ n
dx dx
mit
d dw
M EJ
dx dx
2
2
d
Q EJ
dx
3
3
d
q EJ
dx
Fundamental equations for the calculation of beam
structures used in the development of the stiffness matrix
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
Assessment
of
correctness
of
the
stiffness
matrices
Derivations
of
stiffness
matrices
are
sometimes
extensive
and
sophisticated
in
mathematics
.
Therefore,
the
test
of
the
correctness
of
the
mathematical
calculus
for
this
object
is
an
important
step
in
the
development
process
of
numerical
methods
.
There
are
two
types
of
assessment
:
1
.
Compatibility
condition
2. Duplication of the length of the element
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
1
.
Compatibility
condition
i
j
x
Element 1
i
j
Element 2
O

x
x
x
0
r z r r z r z F
o
x x
ji jj ii ij
Equation of equilibrium at point
О
:
The displacement vectors and can be expressed as Taylor rows:
z
x
z
x
'
w
z
o
w
in the centre point
O
''''''
2 3 4
5
'
''2!'''3!4!
''''''
2 3 4
5
'
''2!'''3!4!
IV
w
w w w w
x x x
z x o x
x
IV v
w
w w w w
IV
w
w w w w
x x x
z x o x
x
IV v
w
w w w w
''''''
2 3 4
5
0
'
''2'''6 24
IV
w
w w w w
x x x
r r r r x r r r r r r r r o x F
ji ii jj ij ji ij ji ij ji ij ji ij
IV v
w
w w w w
After transformation:
r
ij
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
2
.
Duplication
of
the
length
of
the
element
x
x
i
j
Element 1
i
j
Element 2
O

x
x
Equation of equilibrium at
point

x,
О
,
x
:
0
( )
( ) 0
( ) ( )
0
( )
r z r z r
o
x
ii ij iq x
r z r r z r z r r
o
x x
ji jj ii ij jq o iq o
r z r z r
o
x
ji jj jq x
Or in matrix form:
r r r
ii ij
iq x
r r r r r r
ji jj ii ij
jq o iq o
r r r
ji jj
jq x
Rearrangement of rows and columns
Application of Jordan’s method with

new value of element
and

initial value of element.
*
r r r r
ij ij i j
*
r
ij
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
Calculation
of
shell
structures
Wall

like girder
Loaded plate
Hall roof

like folded plate structure
Panel
Plate
F
olded plate structure
x
y
p
–
Boundary load in plane
A
B
A and B
–
Reaction force in plane
y
x
P
Reaction force
Plane
Load
Plane
x
z
y
Boundary of panel
+
=
+
=
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
Systematic approach for the development of differential
equations
for a disc
Type of the development
Equ
ation of equ
i
librium
0 0
yx xy y
x
xy xy
x y x y
1
Geometrical relations
x y
u v u v
x y x y
2
Material law
2
2
1
1
1
1
x x y x x y
y y x y y x
E
E
E
E
3
(2)
(3)
2 2
1 1
2 1
x y
E u v E u v
x y x y
E u v
y x
4
(4)
(
1
)
2 2 2
2 2
2 2 2
2 2
1 1
0
2 2
1 1
0
2 2
u u v
x y x y
v v u
y x x y
5
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
The
system
of
partial
differential
equations
for
discs
changes
to
a
system
of
ordinary
differential
equations
if
the
displacements
are
approximated
by
trigonometric
rows
:
Inserting the results of this table into
equation (5) from the previous table
we get a system of ordinary differential equations:
2
1 1
2
0
2
2 2
2
1 1
2
0
2
2 2
d V dU
V
dy
dy
d U dV
U
dy
dy
( ) cos with
n
u U y x
L
( ) sin with
n
v V y x
L
( ) sin
u
U y x
x
( ) cos
v
V y x
x
( )
cos
u dU y
x
y dy
( )
sin
v dV y
x
y dy
2
2
( ) cos
u
U y x
x
2
2
2
( ) sin
v
V y x
x
2 2
2 2
( )
cos
u d U y
x
y dy
2 2
2 2
( )
sin
v d V y
x
y dy
2
( )
sin
u dU y
x
x y dy
2
( )
cos
v dV y
x
x y dy
Type of the development
Bending of plate
Equation of equilibrium
2
2
h
x x
h
m zdz
2
2
h
y y
h
m zdz
2
2
h
xy xy
h
m zdz
1
Geometrical relations
w w
u z v z
x y
2 2 2
2 2
2
x y xy
w w w
z z z
x y x y
2
Material law
2
1
x x y
E
2
1
y y x
E
2 1
xy xy
E
3
(2)
(3)
2 2
2 2 2
1
x
E w w
z
x y
2 2
2 2 2
1
y
E w w
z
y x
2
1
xy
E w
z
x y
4
2 2
2 2
x
w w
m D
x y
2 2
2 2
y
w w
m D
y x
2
1
xy
w
m D
x y
5
(4)
(1)
2 2
2 2
x
m
w w
x y D
2 2
2 2
y
m
w w
y x D
2
1
xy
m
w
x y D
6
(6)
(4)
3
12
x
x
m
z
h
3
12
x
y
m
z
h
3
12
xy
xy
m
z
h
7
3
2
with
12 1
E h
D
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
Systematic approach for the development of differential
equations for a plate
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
Systematic approach for the development of differential
equations for a plate
Stress and internal force in plate element
x
dx
dy
x
y
dx
dy
m
x
m
y
dx
dy
yz
xz
dx
dy
q
x
q
y
dx
dy
xy
yx
dx
dy
m
xy
m
yx
Shearing stress
Torsion with shear
Shear force
Torsional moment
dx
dy
x
z,
w(x,y)
p(x,y)
h/
2
h/
2
Equation of equilibrium
Balanced forces in z

direction:
y
x
q
q
p
x y
Balanced moments for x

and y

axis:
xy yx
x x
y x
m m
m m
q q
y x x y
Equation of equilibrium after transformations:
2 2
2
2 2
2 (1)
xy y
x
m m
m
p
x x y y
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken
Partial
differential
equation
for
a
plate
:
4 4 4
4 2 2 4
2
w w w p
x x y y D
This changes to an ordinary
differential equation if the
displacements are
approximated by
trigonometric rows.
Inserting the results of the
table in the above equation
we get the ordinary
differential equation:
4 4 4
2 4
4 2 4
2
d W d W d W p
dx dy dy D
( ) sin with
n
w W y x
L
( ) cos
w
W y x
x
( )
sin
w dW y
x
y dy
2
2
2
( ) sin
w
W y x
x
2 2
2 2
( )
sin
w d W y
x
y dy
2
( )
cos
w dW y
x
x y dy
2
( )
cos
w dW y
x
x y dy
3
3
3
( ) cos
w
W y x
x
3 3
3 3
( )
sin
w d W y
x
x dy
3
2
2
( )
sin
w dW y
x
x y dy
3 2
2
( )
cos
w d W y
x
y x dy
4
4
4
( ) sin
w
W y x
x
4 4
4
4 4
( )
sin
w d W y
x
y dy
4 2
2
2 2 2
( )
sin
w d W y
x
x y dy
4 2
2
2 2 2
( )
sin
w d W y
x
y x dy
University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and Structural Mechanics / Laboratory of Engineering Informatics
Univ.

Prof. Dr.

Ing. habil. N. Gebbeken

MAPLE permits a fast calculation of stiffness matrices for different element
types in symbolic form

Elaboration of stiffness matrices can be automated

Export of the results in other computer languages (C, C++, VB, Fortran)
can help to implement stiffness matrices in different environments

For students
‘
education an understanding of algorithms is essential to test
different FE

formulations

Students can develop their own programmes for the FEM
Conclusion:
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