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Title: Folie 1


1

University of the German Armed Forces Munich
Faculty of Civil and Environmental Engineering
Institute of Engineering Mechanics and
Structural Mechanics Laboratory of Engineering
InformaticsUniv.-Prof. Dr.-Ing. habil. N.
Gebbeken
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
2
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.
Truss structure, Railway bridge Firth of Forth
(Scotland)
Folded plate structure, Church in Las Vegas

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
3
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)
Differential quotients are substitutedthrough
difference quotients

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
4
Calculation methods - type of discretisation -
Finite Element Method (FEM)
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.
Static calculation of a concrete panel

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
5
Calculation methods - type of discretisation -
Meshfree particle solvers (e.g. Smooth Particle
Hydrodynamics (SPH)) for high velocity
impacts, large deformations and
fragmentation
Experimental und numeric presentation of a high
velocity impacta 5 mm bullet with 5.2 km/s
at a 1.5 mm Al-plate.
Aluminiumplate
Fragment cloud
PD Dr.-Ing. habil. Stefan Hiermaier

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
6
  • FEM-Advantages
  • Continua can easily be approximated with
    different elementgeometries (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 systembehaviour can be influenced with
    well-known strategies,e.g. refinement of the
    mesh, higher degrees of elementformulations,
    automated mesh adaptivity depending on stress
    gradients or local errors


University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
7
  • 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
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
8
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.
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.
?
R

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
9
Structures should not only be resistant to loads,
but also limit deformations and be stable against
local or global collapse.
Static System ActionsReaction
forces Deformation of System
Vector S of forces results from the strength of
construction. Vector U of the node displacements
depends on the system stiffness.
H
H
H
H
M
M
M
M
V
V
V
V

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
10
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
Citicorp Tower NYC
Consequences of wind-inducedvibrations on a
suspension bridge
Consequences of an earthquake
Collapse of the Tacoma Bridge at a wind velocity
of 67 km/h

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
11
FEM for the solution of structural problems
The most static and dynamic influences are
represented in thefollowing equation
static problem
dynamic problem
wind loading
- mass (M)- damping (C)- stiffness (R)
Mercedes-multistoreyin Munich

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
12
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.

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
13
Short overview of the fundamental equations for
the calculation of beam and shell structures
Beam structures
Shell structures
Differential equation for a single beam
Differential equations for a disc (expressed in
displacements)
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
1. Beam on elastic foundation
Differential equation for a plate
with n- relative stiffness of foundation, k-
coefficient of elastic foundation, b- broadness
of bearing
2. Theory of second order
with ?- shearing strain
3. Biaxial bending
with N- axial force
14
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.

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
15
Development of differential equations of beams
with or without consideration of the transverse
strain
R

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
16
Algorithm for the elaboration of a stiffness
matrixfor an ordinary beam
Basic equations
Solution
homogeneous
particular
Solution and derivatives in matrix form
D

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
17
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
Unit displacements of nodes
or
L

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
18
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
Reaction forces and internal forces
or
L1

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
19
We express the integration constants by the
displacements of the nodes
Replacing with
delivers
or in simplified form

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
20
Within means
r the relative stiffness matrix with EJ 1
the relative load column with q 1
The final stiffness matrix r and the load column
for an ordinary beam
?i
?j
wi
wj

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
21
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 foundationb broadness of
bearing
Solution

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
22
Elaboration of the stiffness matrix for a beam on
an elastic foundation
The final stiffness matrix r and the load column


University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
23
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
Basic equations
with
(shearing strain)
Solution

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
24
Theory of first order
Theory of second order
The final stiffness matrix r and the load column
for a beam element following the theory of
second order

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
25
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
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
26
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
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
27
1. Compatibility condition
Equation of equilibrium at point ?
-?x
?x
O
Element 1
Element 2
i
j
i
j
x
x
The displacement vectors and can be
expressed as Taylor rows
in the centre point O
After transformation

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
28
2. Duplication of the length of the element
x
x
Equation of equilibrium at point -?x, ?, ? x
Or in matrix form
Rearrangement of rows and columns
Application of Jordans method
with - new value of element and -
initial value of element.

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
29
2. Duplication of the length of the element

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
30
Calculation of shell structures
Wall- like girder
Loaded plate
Hall roof- like folded plate structure

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
31
Systematic approach for the development of
differential equations for a disc

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
32
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 tablewe get a
system of ordinary differential equations

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
33
Systematic approach for the development of
differential equations for a plate

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
34
Systematic approach for the development of
differential equations for a plate
Stress and internal force in plate element
Equation of equilibrium Balanced forces in
z-direction
Balanced moments for x- and y-axis
Equation of equilibrium after transformations

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
35
Partial differential equation for a plate
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

University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
36
Conclusion
  • 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


University of the German Armed Forces Munich
Faculty of Civil and Environmental
EngineeringInstitute of Engineering Mechanics
and Structural Mechanics / Laboratory of
Engineering Informatics Univ.-Prof. Dr.-Ing.
habil. N. Gebbeken
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