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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

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

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

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

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

- 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

- 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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