# stiffness matrices for the calculation of beam and

Urban and Civil

Nov 15, 2013 (4 years and 6 months ago)

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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
-
:

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

depending

on

stress

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

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

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
, 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

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
-

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
-

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

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

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

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

Hall roof
-

like folded plate structure

Panel

Plate

F
olded plate structure

x

y

p

A

B

A and B

Reaction force in plane

y

x

P

Reaction force

Plane

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: