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Module Linear Structural Computational Mechanics for Wind Energy Systems i
Lecture Notes
Linear Computational Structural Mechanics for Wind Energy Systems
Prof. Dr.-Ing. habil. Detlef Kuhl
Unive
Online M.Sc. Wind Energy Systems University of Kassel and Fraunhofer IWES
www.uni-kassel.de/wes
ii D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel
Lecture Notes
Linear Computational Structural Mechanics for Wind Energy Systems
Prof. Dr.-Ing. habil. Detlef Kuhl
1st Edition, December 2013
Online M.Sc. Wind Energy Systems (wes.online)
University of Kassel
Department of Civil and Environmental Engineering
Institute of Mechanics and Dynamics
Prof. Dr.-Ing. habil. Detlef Kuhl
Mönchebergstraße 7
34109 Kassel, Germany
www.uni-kassel.de/fb14/mechanics
© Prof. Dr.-Ing. habil. Detlef Kuhl, 3. November 2015
All rights reserved. In particular, the right to translate the text of this document into an-
other language is reserved. No part of the material protected by this copyright notice
may be reproduced or utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any other information storage and retrieval sys-
tem, without written permission of the author.
Module Linear Structural Computational Mechanics for Wind Energy Systems iii
Lecture Linear Computational Structural Mechanics for Wind Energy Systems
Abbreviation
LCSM
Abstract The present set of lecture notes is designed to assist the students of the online master’s study
Wind Energy Systems with their learning in linear finite element methods and linear structural
dynamics of wind energy systems. For this reason it includes sections on the theory develop-
ment, application of methods in selected examples and program flowcharts, as well as coding
instructions supporting the homework and the final case study of the course. After an introduc-
tion to numerical methods for the static and dynamics simulation of structures, a brief review of
the history and a first course classification of the applied models and methods the finite element
method will be newly invented for the simple case of one dimensional continua. This all ows for
an artless but also completed representation of the main ideas of the finite element method as
well as the comparison of numerical and analytical solutions. Afterwards advanced topics of the
one-dimensional finite element method will be extended in order to enable calculation of space
frameworks, to obtain higher order accurate p finite element methods and also residual based
error estimates, to include inhomogeneous DIRICHLET boundary, to have a first idea about static
and dynamic solution procedures and, finally, to prepare the development of the finite element
method for the simulation of general three dimensional structures. The development of the
general n dimensional p finite element method starts with a brief repetition of linear continuum
mechanics, then the finite element representations of virtual work term are realized and afte r-
wards specialized for a family of three and two dimensional finite elements. The following chap-
ter 'eigenvalue analysis' will provide methods the analyze the dynamic characteristics of struc-
tures and to provide a analytical solution of simple structural dynamics with con later on com-
pared with numerical results to validate the program development of time integrations schemes
of the central difference and NEWMARK-type discussed in the following two chapters.
iv D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel
Online M.Sc. Wind Energy Systems (wes.online)
Since the foundation of the University of Kassel in 1971, an awareness of the environment has
always been an important part of science and education. More than 60 professors and their
scientific employees work on environmental questions within departments, interdisciplinary
research centers and institutes. The number of environmental research projects and environ-
mental study programs has been increasing continuously over the years. Bicultural and interna-
tional on-campus study programs with innovative teaching concepts are part of this portfolio.
The Online M.Sc. Wind Energy Systems is another milestone in this story of a green university.
The specific expertise of the University in the fields of computational science and engineering
with regards to renewable energy systems is incorporated into the learning content of this wind
energy study program.
These competencies are extended by the Fraunhofer Institute of Wind Energy and Energy Sys-
tem Technology (IWES). The Fraunhofer IWES is one the largest institutes for wind energy and
energy system technology in Europe. Lecturers from the Fraunhofer IWES introduce further as-
pects into the study program, such as the economic integration of a large amount of wind ener-
gy into the energy supplier system. Students also gain knowledge about how to design and de-
velop innovative concepts for individual components of the wind energy converter systems, like
the nacelle systems, rotor blades or support structures.
Beside lecturers from Fraunhofer IWES and University Kassel leading experts from industry and
cooperating universities enrich the team of Online M.Sc. Wind Energy Systems.
The teaching methods of the Online M.Sc. Wind Energy Systems are new and innovative. The
program is a part-time, extra-occupational Master's program. It is explicitly developed for stu-
dents who would like to study alongside their job or family responsibilities. Our aim is also to
provide a worldwide global student body with the knowledge of the se two institutions in the
area of renewable and wind energy. We would like to extend the knowledge of male and female
engineers from regions where this technology is not easily accessed. For this reason we teach
the 28 modules of our program 100% online. We welcome you in our program.
Partners offering the Online M.Sc. Wind Energy Systems (wes.online)
Module Linear Structural Computational Mechanics for Wind Energy Systems v
Online M.Sc. wind Energy Systems
Accredited by
Reputational partners from research and industry
Project founding and project alliance for the development of premium online master ’s courses
vi D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel
Curriculum Vitae
Prof. Dr.-Ing. habil. Detlef Kuhl
University of Kassel
Faculty of Civil and Environmental Engineering
Institute of Mechanics and Dynamics
Moenchebergstrasse 7
34109 Kassel
Email: kuhl@uni-kassel.de
Webpage: www.uni-kassel.de/fb14/mechanics
Scientific Vitae
Prof. Dr.-Ing. habil. Detlef Kuhl has studied aerospace engineering at the University of Stuttgart
with the main focus on regenerative energy systems and wind energy. 1992 he has finished his
master’s thesis at M.A.N Technology in Munich about the mechanical analysis and experimental
verification of the wind turbine WKA 60 on the island Helgoland. His professional career has
started as designing engineer at the wind turbine manufacturer Enercon in Aurich. Knowing the
demands of wind engineering he has decided to devote his life to the investigation, application
and teaching of methods of computational mechanics, in particular of structures, wind turbines
and general multifield problems.
The scientific qualification of Prof. Dr.-Ing. habil. Detlef Kuhl has started with the PhD study at
the University of Stuttgart about dynamics of shell structures, finished in 1996. The years 1996
to 1998 he has spend his post doc as head of the research group Thermomechanical Modeling
Group at the German Aerospace Center in Lampoldshausen, as postdoctoral fellow at the De-
partment of Aeronautics, Imperial College of Science, Technology and Medicine in London
(1997) and as senior scientist and lecturer at the Institute of Structural Mechanics, Ruhr Univer-
sity Bochum. He finished his Habilitation about the simulation of time dependent multi eld prob-
lems in 2004. Beside the thesis work he has researched about the thermo mechanical modeling
and simulation of rocket combustion chambers and the computational analysis of tensegrity
structures.
Since 2007 Detlef Kuhl is Professor for Mechanics and Dynamics at the Faculty of Civil and Env i-
ronmental Engineering, University of Kassel. He is teaching bachelor courses on soli d mechanics,
master’s courses on computational solid mechanics and is member of the teaching team of the
lecture series Simulation of Wind Energy Systems. His research is related to the computational
analysis of dynamics of structures, the modeling and simulation of thermo mechanical, electro
magneto thermo mechanical and the fluid structure interaction, the simulation of tensegrity
structures and wind turbines and didactical concepts of online university teaching. Furthermore,
Prof. Dr.-Ing. habil. Detlef Kuhl is the Academic Director of Online M.Sc. Wind Energy Systems
(wes.online), Dean of Students of Faculty of Civil and Environmental Engineering, University of
Kassel, Head of Chair of Mechanics and Dynamics, University of Kassel and
Guest Professor at Vietnamese German University (VGU), Binh Duong New City, Vietnam. In
Module Linear Structural Computational Mechanics for Wind Energy Systems vii
2015 he was as visiting Professor at the Department of Mathematics, University of Auckland.
Lectures held in Online M.Sc. Wind Energy Systems
Solid Mechanics of Wind Energy Systems
Linear Computational Structural Mechanics
Nonlinear Computational Structural Mechanics
Research Interests
Computational structural dynamics
Non-linear multi-field finite element methods
Adaptive time stepping schemes
Computational tensegrity mechanics
Simulation of wind turbines
Projects
Modeling and simulation of electro magneto dynamics coupled with heat conduction
problems
Modeling and simulation of electro magneto mechanical interactions
Simulation of thermomechanical fluid structure interaction
Form finding of tensegrity structures
Higher order accurate integration of multifield elastoplasticity
Recent Publications
D. Kuhl, G. Meschke: Numerical Analysis of Dissolution Processes in Cementitious Materials Us-
ing Discontinuous and Continuous Galerkin Time Integration Schemes. International
Journal for Numerical Methods in Engineering, Vol. 69, No. 9, 1775-1803, 2007
S. Carstens, D. Kuhl: Higher Order Accurate Implicit Time Integration Schemes for Transport
Problems. Archive of Applied Mechanics, Vol. 82, 1007{1039, 2012
T. Gleim, D. Kuhl: Higher Order Accurate Discontinuous and Continuous p-Galerkin Methods for
Linear Elastodynamics. Zeitschrift f•ur Angewandte Mathematik und Mechanik, Vol. 93,
177-194, 2013
P. Birken, T. Gleim, D. Kuhl, A. Meister: Fast Solvers for Unsteady Thermal Fluid Structure
Interaction. International Journal for Numerical Methods in Fluids, DOI: 10.1002/d.4040,
2015
B. Schröder, D. Kuhl: Small Strain Plasticity: Classical Versus Multi field Formulation. Archive of
Applied Mechanics, DOI 10.1007/s00419-015-0984-9, 2015
T. Gleim, B. Schröder, D. Kuhl: Nonlinear Thermo-Electromagnetic Analysis of Inductive Heating
Processes. Archive of Applied Mechanics, DOI 10.1007/s00419-014-0968-1, 2015
Die Scientific Vitae kann in Stichworten dargestellt werden (siehe nächste Seite).
viii D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel
Curriculum Vitae
Prof. Dr.-Ing. habil. Detlef Kuhl
University of Kassel
Faculty of Civil and Environmental Engineering
Institute of Mechanics and Dynamics
Mönchebergstraße 7
34109 Kassel
Email: kuhl@uni-kassel.de
Webpage: www.uni-kassel.de/fb14/mechanics
Current Positions
Academic Director of International Online Master’s Course Wind Energy Systems
Dean of Students of Faculty of Civil and Environmental Engineering
Head of Chair of Mechanics and Dynamics
Guest Professor at Vietnamese German University (VGU), Binh Duong New City, Vietnam
Scientific Vitae
1985-1992 Study of Aerospace Engineering, University Stuttgart
1992 Mechanical Engineer, Enercon, Aurich
1992-1996 PhD student, PhD degree 1996, Department of Civil Engineering, Institute of
Structural Mechanics, University of Stuttgart
1996-1998 Head of Thermomechanical Modelling Group, Institute of Space Propulsion,
German Aerospace Center, Lampoldshausen
1997 Postdoctoral fellow at Department of Aeronautics, Imperial College of Science,
Technology and Medicine, London
1998-2007 Senior scientist and lecturer, Habilitation 2004, Institute of Structural Me-
chanics, Ruhr University Bochum
since 2007 Professor at Chair of Mechanics and Dynamics, University of Kassel
Lectures held in Online M.Sc. Wind Energy Systems
Solid Mechanics of Wind Energy Systems
Linear Computational Structural Mechanics
Nonlinear Computational Structural Mechanics
Research Interests
Computational structural dynamics
Non-linear multifield finite element methods
Adaptive time stepping schemes
Computational tensegrity mechanics
Simulation of wind turbines
Module Linear Structural Computational Mechanics for Wind Energy Systems ix
Projects
Modeling and simulation of electro magneto dynamics coupled with heat conduction
problems
Modeling and simulation of electro magneto mechanical interactions
Simulation of thermomechanical fluid structure interaction
Form finding of tensegrity structures
Higher order accurate integration of multifield elastoplasticity
Recent Publications
D. Kuhl, G. Meschke: Numerical Analysis of Dissolution Processes in Cementitious Materials Us-
ing Discontinuous and Continuous Galerkin Time Integration Schemes. International
Journal for Numerical Methods in Engineering, Vol. 69, No. 9, 1775-1803, 2007
S. Carstens, D. Kuhl: Higher Order Accurate Implicit Time Integration Schemes for Transport
Problems. Archive of Applied Mechanics, Vol. 82, 1007{1039, 2012
T. Gleim, D. Kuhl: Higher Order Accurate Discontinuous and Continuous p-Galerkin Methods for
Linear Elastodynamics. Zeitschrift f•ur Angewandte Mathematik und Mechanik, Vol. 93,
177-194, 2013
P. Birken, T. Gleim, D. Kuhl, A. Meister: Fast Solvers for Unsteady Thermal Fluid Structure
Interaction. International Journal for Numerical Methods in Fluids, DOI: 10.1002/d.4040,
2015
B. Schröder, D. Kuhl: Small Strain Plasticity: Classical Versus Multi field Formulation. Archive of
Applied Mechanics, DOI 10.1007/s00419-015-0984-9, 2015
T. Gleim, B. Schröder, D. Kuhl: Nonlinear Thermo-Electromagnetic Analysis of Inductive Heating
Processes. Archive of Applied Mechanics, DOI 10.1007/s00419-014-0968-1, 2015
x D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel
Contents
Introduction to Linear Structural Computational Mechanics for Wind Energy Systems 1
1 Finite Element Method for One Dimensional Continua and Truss Elements ............ 7
1.1 Learning Goals ......................................................................................................... 8
1.2 Required Prior Knowledge (Empfehlung) ................................................................... 8
1.3 Section 2 ................................................................................................................. 8
1.3.1 Section 3 .......................................................................................................... 8
1.3.2 Formula ........................................................................................................... 9
1.3.3 Essenz (Empfehlung) ......................................................................................... 9
References ....................................................................................................................... 10
2 Finite Element Method for One Dimensional Continua and Truss Elements .......... 13
2.1 Introduction .......................................................................................................... 14
2.1.1 Learning goals ................................................................................................ 14
2.1.2 Section 3 ........................................................................................................ 14
2.1.3 Sections 3....................................................................................................... 14
2.1.4 Formula ......................................................................................................... 15
2.2 Essenz ................................................................................................................... 15
References ....................................................................................................................... 15
Bibliography (Möglichkeit)................................................................................................ 17
Appendix ............................................................................................................................ 18
Glossary (Möglichkeit) ...................................................................................................... 18
Module Linear Structural Computational Mechanics for Wind Energy Systems xi
Index (Möglichkeit) ........................................................................................................... 19
Nomenclature (Möglichkeit) ............................................................................................. 20
xii D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel
List of Figures Figure 1.1: Tension of a truss: Geometry and loading cases ..................................................... 11
Figure 1.3: Wind power Plant ................................................................................................ 10
Figure 2.1: Wind power Plant ................................................................................................ 15
Module Linear Structural Computational Mechanics for Wind Energy Systems xiii
List of Tables
Table 0.1: Embedding of the Module in Online M.Sc. Wind Energy Systems ............................... 4
Table 1.1: Nomenclature for one dimensional linear continuum mechanics and linear truss
mechanics ............................................................................................................................ 10
Module Linear Structural Computational Mechanics for Wind Energy Systems 1
Introduction to
Linear Structural Computational Me-
chanics for Wind Energy Systems Jedes Kapitel beginnt mit einem Deckblatt, auf welchem die Kapitelüberschrift, eine kurze
Zusammenfassung des Kapitels sowie die Key Words zu finden sind. Das erste Kapitel des Skripts bein-
haltet eine Zusammenfassung des Moduls: Learning Goals of the Module, Motivation, Prior Knowledge
Required for the Module, Embedding in Online M.Sc. Wind Energy Systems, Learning Schedule
Abstract In the present chapter continuum mechanical for the simulation of wind turbine components are
briefly reviewed. In particular, linear, physically nonlinear and geometrically nonlinear models
are characterized. Furthermore, the significance of dynamical effects on the deformation of
wind turbines is demonstrated and, consequently, continuum mechanical models for stationary
and transient analysis of wind turbines are distinguished. Since wind power plants are using
components made of different kind of materials also the modeling of isotropic and transversal
isotropic as well as elastic and inelastic material models are brief ly discussed.
Above reviewed continuum mechanical models of wind turbine components constitute time
dependent or time independent partial differential equations. In general these model can nor be
solved analytically. Therefore, sequences of mathematical reformulations and numerical meth-
ods for the solution of linear dynamics, linear statics and non-linear dynamics are sketched. Lin-
ear dynamics is numerically solved by the spatial weak formulation, the finite element method
and time integration schemes. These principal solution steps can also used for non-linear dy-
namics. Only the linearization and an iterative solution procedure must be used additionally. For
the particular reason to become familiar with continuum mechanics and later also with the line-
ar finite element method also the differential equation of one dimensional continuum mechan-
ics is presented, the solution procedures for dynamics and statics are shown and, finally, the
analytical solution of static one dimensional continua is deviated.
Beside the technical aspects of computational mechanics for wind turbines also the history of
mechanics, the finite element method and also the time integration method is briefly reviewed.
Key Words linear and non-linear elasticity, finite element method, computational mechanics, time integra-
tion, history of mechanics and computational mechanics
2 D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel
Motivation (Empfehlung) Modern engineering structures as vehicles, air planes and wind turbines are subjected to me-
chanical as well as non-mechanical external actions. Due to external loading deformations and
internal stress states, which may lead to the failure of the structure, are observed. In order to
design wind turbines with a high level of safety and a long life time, the deformations and
stresses in the structure should be known in advance for standard operation and extremely co n-
ditions. The prognosis of the mechanical behavior, including collapse, low and high cycle fatigue,
of wind turbines and their components is based on their adequate mechanical models, consider-
ing for the applied materials, dynamic and static actions, wind and death loads and also temper-
ature changes. Only very simple models, far away from a realistic description, but, nevertheless,
valid as basis for a first design of estimation of mechanical behavior of wind turbines or interac-
tions of components, can be solved analytically. More realistic mechanical models taking into
account realistic geometries, materials and mechanical effects require numerical solution proce-
dures for an approximated solution of these highly sophisticated models. During the last six de c-
ades the finite element method has been developed to a powerful tool for the mechanical anal-
ysis of structures of civil and environmental, mechanical, aerospace, electrical and wind energy
engineering. A broad range of strong commercial tools have been developed for the linear me-
chanical analysis of structures using a more or less automated procedure for the meshing, the
calculation of deformations and stresses and the post-processing of engineering relevant results.
Beside these basic calculation competences several commercial finite element programs have
strong capabilities on selected advanced simulation methods. For example for advanced materi-
al modeling, dynamic analysis, contact problems, soil and structural analysis, stability and mult i-
field analyses. However, a general tool for the adequate mechanical analysis of wind turbines is
not available. It is worth to mention that not only the highly sophisticated mechanical models of
wind turbines needs real experts for the application of commercial programs, but also the pa-
rameter identification, the decision of adequate algorithms and finite elements and the interpre-
tation of results and errors. Already linear models requires a deep knowledge of the underlying
numerical methods for not only the reason of watching colorful pictures but also providing a
serious and tough prognosis of expected deformations and stresses. It is self evident that more
advanced mechanical models and computational methods require a strong knowledge for the
educated decision for problem specific software packages and of course to overcome the limita-
tions of commercial finite element programs for special applications in wind turbine mechanics.
Beside the classical engineering prognosis of the mechanical behavior of wind turbines the nu-
merical solution procedure consisting of spatial and temporal discretization methods are powe r-
ful tools as basis for the classical engineering design and optimization method. Within these
process simulations of wind turbines or components are used to study the influence of design
modifications. Obviously, the applied numerical methods can also be used together with a sens i-
tivity analysis and gradient based of evolutionary optimization algorithms for the systematic and
computer oriented improvement of the design. Furthermore, computational wind turbine me-
chanics can be used as ingredient for the operation control and the damage detection of wind
turbines using simplified or reduced models and inverse analysis, respectively.
Above sketched applications, requirements and limitations of computational wind turbine me-
chanics motivate to reach a strong knowledge of numerical methods for the simulation, optimi-
zation and control of high tech wind power plants. In order to be able to achieve this goal, two
Module Linear Structural Computational Mechanics for Wind Energy Systems 3
lectures about the computational solid mechanics of wind turbines are included in the schedule
of the master’s course 'Wind Energy Systems'. The first one 'Linear Computational Structural
Mechanics for Wind Energy Systems' is carefully limited to linear computational analysis of static
and dynamic deformation of wind turbines. The main focus of this part is to understand the
methods of spatial and temporal discretization, to know disadvantages and advantages of se-
lected numerical methods and to be able to select algorithms and finite elements and to capable
interpret results of commercial software packages. Simultaneously the competence will be
reached to overcome limitations of commercial codes and to develop more advanced, special-
ized and realistic computational models of wind turbines. In the second part 'Non-Linear Compu-
tational Structural Mechanics for Wind Energy Systems' non-linear continuum mechanical mod-
els, non-linear finite element methods and algorithms for non-linear statics and dynamics will be
studied.
Learning Goals for the Module
Reviewing linear continuum mechanics
Knowing different, also non-linear, models of continuum mechanics
Having a idea of numerical methods applied for the solution of continuum mechanical
models
Having fun with the histories of the finite element method and the time integration
schemes
Prior Knowledge Required for the Module Requirements according
to examination:
Module Mathematics, Module Solid Mechanics of Wind Energy
Systems
Recommended prior
learning:
Module Application of Software Tools
Modules:
Solid Mechanics for Wind Energy Systems
Mathematics for Wind Energy Systems
Practice of Software Tools for Wind Energy Systems
Design of Mechanical and Electrical Components of Wind Energy Systems
Competencies:
Vector and tensor analysis
Basic knowledge on differential equations
Integration and Differentiation in one to three spatial dimensions
Linear systems of equations
Mechanical forces and stress resultants
Linear continuum mechanics in one to three spatial dimensions
Beam and truss models of mechanics
Programming in MATLAB
4 D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel
How this fits into the Online M.Sc. Wind Energy Systems The present module 'Linear Computational Structural Mechanics for Wind Energy Systems' is
one six-credit module of the specialist study 'Simulation and Structural Technology for Wind
Energy Systems'. This specialist study will enable the students to understand the structural com-
ponents of wind energy systems, to permit prognoses of their life time for working and extreme
conditions, and to design future wind turbines with an optimized use of foundations, materials,
structural components and design concepts. As basis for this, a deep knowledge of the mechan-
ics and technology of structural components is provided. Advanced fluid and solid mechanics
and related novel simulation methods are thought as basis for studying the aerodynamic and
mechanical behavior of wind turbines and their components. Together with the technological
knowledge about on- and offshore foundations, towers, rotor blades and safer materials the
generation of efficient and reliable wind turbines can be designed.
Present course Strong basis Strong interaction
Master’s Thesis (in academia or industry)
Specialization:
Simulation and Structural Technology
(each 6 ECTS-Credits)
Specialization:
Energy System Technology
(each 6 ECTS-Credits)
Additional Key Competencies:
Energy and Law
(each 3 ECTS-Credits)
Rotor Aerody-
namics
Strength Dura-
bility and
Reliability
Rotor
Blades
Wind Energy
Meteorology Energy Storage
Contract
Law
Occupational
Safety On and
Offshore
Energy Law
Computational
Fluid Dynam-
ics
Nonlinear
Computational
Structural
Mechanics
Theoreti-
cal Fluid
Mechanics
Construction
and Design of
Nacelle
Systems
Control and
Operational
Management of
Wind Turbines
and Wind Farms
Project
Manage-
ment
Planning and
Constructions
of Wind
Farms
Business
Administration
and Manage-
ment of Wind
Turbines and
Wind Farms
On and Off-
shore Founda-
tions
Linear Compu-
tational Struc-
tural Mechan-
ics
Towers
Reliability,
Availability
Maintenance
Strategies
Technical and
Economic As-
pects of Grid
Integrations
Personal
Management
Fundamentals of Mathematics and Engineering for Wind Energy Systems (each 6 ECTS-Credits)
Design of Mechanical and
Electrical Components Electrical Engineering Mathematics Solid Mechanics
Application of
Software Tools Fluid Mechanics
Table 0.1: How this Module fits into the Online M.Sc. Wind Energy Systems
Table 0.1 shows the present module Linear Computational Structural Mechanics for Wind Energy
Systems embedded in the specialist studies Simulation and Structural Technology for Wind Ene r-
gy Systems and the master’s course Wind Energy Systems. The present lecture is based on the
knowledge of the modules of Fundamental Studies of Mathematics and Engineering. In particu-
lar, very good knowledge of Mathematics for Wind Energy Systems, Design of Mechanical and
Electrical Components of Wind Energy Systems and Practice of Software Tools for Wind Ene rgy
Systems is essential for the successful graduation of the present module. Since in the present
module almost all continuum and structural mechanical problems, previously presented in mod-
ule Solid Mechanics for Wind Energy Systems, are solved numerically, it is quite important to
understand the topics of this fundamental module. The present module is extended to the nu-
Module Linear Structural Computational Mechanics for Wind Energy Systems 5
merical analysis of non-linear static and dynamic problems in module Non-Linear Computational
Structural Mechanics for Wind Energy Systems (NCSM) and to the valuation of strength, failure,
low and high cycle fatigue in lecture Strength Durability and Reliability for Wind Energy Systems.
The present module can be combined with the fluid mechanics modules of specialist studies
Simulation and Structural Technology for Wind Energy Systems in order to obtain the knowledge
to overcome traditional borders between solid and fluid mechanics with study of both and finally
with the analysis of fluid structure interaction. Furthermore, it can be combined with the tech-
nology modules of the specialist study in order to use numerical analysis of towers, foundations
and rotor blades to improve or optimize these components of wind turbines.
Learning Schedule (Beispiel) The simulation of wind turbines under real operating conditions enforces the consideration of
time dependent loads and inertial forces. These simulations are performed by applying time
integration schemes. Since these schemes are requiring a large numerical effort and significantly
influencing the quality of the prognosis of the dynamic behavior of structures, it is worth to care-
fully develop these methods in Chapters 6 to 8 and to enrich the basic time integrations schemes
by error measures and adaptive time stepping procedures. Methodologically oriented we will
review continuum mechanics and we will discuss the dynamic characteristic and analytical sol u-
tion of structural dynamics.
Basics Static
analysis
Spatial
discreti-
zation
Dynamic
analysis
Tem-
poral
discreti-
zation
Chapter 1, page 1: Introduction to Linear Com-
putational Structural Mechanics
Chapter 2, page 25: Finite Element Method for
One Dimensional Continua
Chapter 3, page 35: Advanced Topics and Spa-
tial Truss Structures
Chapter 4, page 7: Generalized Finite Element
Method for n-Dimensional Continua
Chapter 5, page 13: Dynamic Characteristics
and Analytical Solution of Dynamics
Chapter 6, page 17: Central Difference Method
Chapter 7, page 21: Newmark Time Integration
Schemes
Chapter 8, page 25: Galerkin Time Integration
Schemes
Figure 1: Learning Schedule of Linear Computational Structural Mechanics for Wind Energy Systems
Afterwards, as main tasks of the present lecture, methods for the numerical solution of statics
and dynamics are presented. In particular, the spatial and temporal discretization methods are
6 D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel
thought and intensively studied by means of analytical analyses and representative and illustra-
tive examples. The simulation of wind turbines under real operation condition enforces the con-
sideration of time dependent loads and inertial forces. These simulations are performed by ap-
plying time integration schemes. Since these schemes are requiring a large numerical effort and
significantly influencing the quality of the prognosis of the dynamic behavior of structures, it is
worth to carefully develop these methods in Chapters 6 to 8 and to enrich the basic time inte-
grations schemes by error measures and adaptive time stepping procedures. Methodologically
oriented we will review continuum mechanics and we will discuss the dynamic characteristic and
analytical solution of structural dynamics. Afterwards, as main tasks of the present lecture,
methods for the numerical solution of statics and dynamics are presented. In particular, the spa-
tial and temporal discretization methods are thought and intensively studied by means of analyt-
ical analyses and representative and illustrative examples.
Module Linear Structural Computational Mechanics for Wind Energy Systems 7
1 Finite Element Method for One
Dimensional Continua and Truss
Elements
Abstract In the present chapter ... Kurze Zusammenfassung des Kapitels.
Jedes Kapitel beginnt mit einem Deckblatt, auf welchem die Kapitelüberschrift, eine kurze
Zusammenfassung des Kapitels sowie die Key Words zu finden sind.
Key Words linear elasticity, finite element method, history of mechanics
8 D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel
1.1 Learning Goals
reviewing linear continuum mechanics
knowing different, also non-linear, models of continuum mechanics
having a idea of numerical methods applied for the solution of continuum mechanical
models
having fun with the histories of the finite element method and the time integration
schemes
1.2 Required Prior Knowledge (Empfehlung) Welche Voraussetzungen müssen erfüllt sein, um dieses Kapitel zu verstehen.
1.3 Section 2 Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invi d-
unt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo
duo dolores et ea rebum. Stet clita kasd gubergren, no sea taki mata sanctus est Lorem ipsum
dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod
tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et
accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est
Lorem ipsum dolor sit amet.
1.3.1 Section 3 Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invi d-
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duo dolores et ea rebum. Stet clita kasd gubergren, no sea taki mata sanctus est Lorem ipsum
dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod
tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et
accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est
Lorem ipsum dolor sit amet.
Example for citation:
"Fluid ows at and below the earth's surface are the cause and the cure for problems of water
and soil pollution" (Wendland & Efendiev, 2003, S. 37).
Section 4 The section 4 will not be consecutively numbered.
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unt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo
duo dolores et ea rebum. Stet clita kasd gubergren, no sea taki mata sanctus est Lorem ipsum
dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod
tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua1. At vero eos et
1 This is a footnote.
Module Linear Structural Computational Mechanics for Wind Energy Systems 9
accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est
Lorem ipsum dolor sit amet.
Example for citation:
"Fluid ows at and below the earth's surface are the cause and the cure for problems of water
and soil pollution" (Wendland & Efendiev, 2003, S. 37).
1.3.2 Formula
(𝒙 + 𝒂)𝒏 = ∑ (𝒏𝒌
)𝒙𝒌𝒂𝒏−𝒌𝒏
𝒌=𝟎 (1.1)
(𝟏 + 𝒙)𝒏 = 𝟏 +𝒏𝒙
𝟏!+
𝒏(𝒏−𝟏)𝒙𝟐
𝟐!+ ⋯ (1.2)
𝒙 =−𝒃±√𝒃𝟐−𝟒𝒂𝒄
𝟐𝒂 (1.3)
1.3.3 Essenz (Empfehlung)
Chapter Checks 1. (Question/Task 1 of the paragraph 1.1)
2. (Question/Task 1 of the paragraph 1.1)
3. (Question/Task 1 of the paragraph 1.1)
Special texts like examples, excursions or tips are framed in a box: At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet.
Enumeration 1. level - Enumeration 2. level
* Enumeration 3. level
Memotechnic verse: Field shaded in gray to give short (!) memos or advices.
10 D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel
Figure 1.1: Wind power Plant
Nomenclature
Symbol Equivalent Uni Explanation
T s Time
𝚯 Κ temperature
𝚾𝟏 m position
𝝊𝟏 m displacement
Table 1.1: Nomenclature for one dimensional linear continuum mechanics and linear truss me-
chanics
References Ehlers, W. & Bluhm, J. (2002). Porous Media. Theory, Experiments and Numerical Applications.
Berlin: Springer.
Wendland, W. L. & Efendiev, M. (2003). Analysis and Simulation of Multield Problems. Berlin:
Springer.
Verschiedene Darstellungsweisen möglich!
[1] M. Ameen. Computational Elasticity. Theory of Elasticity and Finite and Boundary Element
Methods. Alpha Science International, Harrow, 2005.
[2] T. L. Anderson. Fracture Mechnics. Fundamentals and Applications. Taylor & Francis Group,
Broken, 3. edition, 2005.
Module Linear Structural Computational Mechanics for Wind Energy Systems 11
[3] Archimedes. De planorum aequilibriis. 285-212 v.Chr.
[4] J. Argyris. Dynamics of Structure. Elsevier, Amsterdam, 1991.
[5] V. I. Arnold. Lectures on Partial Differential Equations. Springer & Phasis, Berlin & Moscow,
2004.
[6] G. Galilei. Discorsi e dimostrazioni matematiche intorno a due nuove scienze. Leiden, 1638.
Homework (Möglichkeit)
Hausaufgaben können auch in Moodle oder in anderer Form den
Studierenden zur Verfügung gestellt werden.
Figure 1.2: Tension of a truss: Geometry and loading cases
In the present homework your own finite element program for the static analysis of one dime n-
sional continua should be extended in order to allow for the application of the p finite element
method. Therefore, higher order (𝜌 = 1; 2; 3; 4; 5; 6), one dimensional continuum elements
should be applied together with the Gauss-Legendre integration. The correct implementation of
the finite element and finite element procedure on the structural level should be verifie d by
means of above sketched model problems. These examples are described by a truss loaded by
load cases i, ii and iii. They should be analyzed using ΝΕ = 1; 2; 4; 8; 16; 32 p finite elements for
the discretization of the truss. For these reasons the following working stages are proposed:
Develop a finite element routine for calculation of the element stiffness 'tensors' 𝑘𝑒𝑖𝑗
and the consistent load 'tensors' 𝑟𝑒𝑖 for all load cases using the Gauss-Legendre integra-
tion with GAUSS point coordinates and weights as given in the file gauss.f provided in the
Moodle course.
Chose the number of GAUSS points 𝑁𝐺 such that the stiffness tensors and the load ten-
sors for load cases i and ii are exactly integrated. The load tensors according to load case
iii cannot integrated exactly. For these integrations please use a integration rule
with 𝑁𝐺 = 𝑝 + | 5.
Develop finite element procedure for analyses with 𝑁𝐸 = 1; 2; 4; 8; 16; 32 finite el-
ements of polynomial degrees 𝑝 = 1; 2; 3; 4; 5; 6 and check your solutions for all
load cases.
Perform all forthcoming tasks only for load case iii, but for all implemented polynomial
degrees p.
12 D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel
Extend your p finite element program by a post-processing procedure, calculating the
approximations of the displacement 𝑢1, stress 𝜎11 and residuum 𝜎11,1 + 𝑝𝑏1
Calculate the local (at position X1) and global (of the hole system) displacement errors
with respect to the analytical solution.
Plot diagrams of the displacements, stresses, the residuum and the local displacement
error.
Your homework submission should include
a brief report documenting your results in form of diagrams
your program code
Module Linear Structural Computational Mechanics for Wind Energy Systems 13
2 Finite Element Method for One Di-
mensional Continua and Truss Ele-
ments
Abstract In the present chapter ... Kurze Zusammenfassung des Kapitels
Key Words linear elasticity, finite element method, history of mechanics
14 D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel
2.1 Introduction
2.1.1 Learning goals
reviewing linear continuum mechanics
knowing different, also non-linear, models of continuum mechanics
having a idea of numerical methods applied for the solution of continuum mechanical
models
having fun with the histories of the finite element method and the time integration
schemes
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invi d-
unt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo
duo dolores et ea rebum. Stet clita kasd gubergren, no sea taki mata sanctus est Lorem ipsum
dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod
tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et
accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est
Lorem ipsum dolor sit amet.
2.1.2 Section 3 Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invi d-
unt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo
duo dolores et ea rebum. Stet clita kasd gubergren, no sea taki mata sanctus est Lorem ipsum
dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod
tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et
accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est
Lorem ipsum dolor sit amet.
Example for citation:
"Fluid ows at and below the earth's surface are the cause and the cure for problems of water
and soil pollution" (Wendland & Efendiev, 2003, S. 37).
2.1.3 Sections 3
Memotechnic verse: Field shaded in gray to give short (!) memos or advices.
Special texts like examples, excursions or tips could be framed in a box: At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet.
Enumeration 1. level - Enumeration 2. level
* Enumeration 3. level
Module Linear Structural Computational Mechanics for Wind Energy Systems 15
2.1.4 Formula
(𝒙 + 𝒂)𝒏 = ∑ (𝒏𝒌
)𝒙𝒌𝒂𝒏−𝒌𝒏
𝒌=𝟎 (2.1)
(𝟏 + 𝒙)𝒏 = 𝟏 +𝒏𝒙
𝟏!+
𝒏(𝒏−𝟏)𝒙𝟐
𝟐!+ ⋯ (2.2)
𝒙 =−𝒃±√𝒃𝟐−𝟒𝒂𝒄
𝟐𝒂 (2.3)
Chapter Checks 1. (Question/Task 1 of the paragraph 1.1)
2. (Question/Task 1 of the paragraph 1.1)
3. (Question/Task 1 of the paragraph 1.1)
Figure 2.1: Wind power Plant
2.2 Essenz Example for citation:
"Fluid ows at and below the earth's surface are the cause and the cure for problems of water
and soil pollution" (Wendland & Efendiev, 2003, S. 37).
References Ehlers, W. & Bluhm, J. (2002). Porous Media. Theory, Experiments and Numerical Appli-
cations. Berlin: Springer.
Wendland, W. L. & Efendiev, M. (2003). Analysis and Simulation of Multield Problems.
Berlin: Springer.
16 D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel
Homework (Möglichkeit)
Hausaufgaben können auch in Moodle oder in anderer Form den
Studierenden zur Verfügung gestellt werden.
Module Linear Structural Computational Mechanics for Wind Energy Systems 17
Bibliography (Möglichkeit)
Beinhaltet die gesamte Literatur im Text. Die Literatur sollte jedoch in
jedem Kapitel aufgelistet sein. Die Gesamtdarstellung stellt ein Zusatz dar.
Capital 1
Ehlers, W. & Bluhm, J. (2002). Porous Media. Theory, Experiments and Numerical Applications .
Berlin: Springer.
Wendland, W. L. & Efendiev, M. (2003). Analysis and Simulation of Multifield Problems. Berlin:
Springer.
Verschiedene Darstellungsmöglichkeiten. Diese müssen einheitlich im Dokument sein!
Capital 1
[1] M. Ameen. Computational Elasticity. Theory of Elasticity and Finite and Boundary Element
Methods. Alpha Science International, Harrow, 2005.
[2] T. L. Anderson. Fracture Mechnics. Fundamentals and Applications. Taylor & Francis Group,
Broken, 3. edition, 2005.
[3] Archimedes. De planorum aequilibriis. 285-212 v.Chr.
Module Linear Structural Computational Mechanics for Wind Energy Systems 18
Appendix
Glossary (Möglichkeit)
Der Glossary stellt ein Zusatz dar.
Actuator
Actuator is a device to convert an electrical control signal to a physical action. Actuators may be
used for flow-control valves, pumps, positioning drives, motors, switches, relays and meters.
Floating-Point Operations Per Second (FLOPS)
Floating-Point Operations Per Second (FLOPS) is a measurement of performance of capability
assigned to a floating-point processor. It is usually noted as MFLOPS or Million FLOPS.
Local Area Network
A Local Area Network is a group of interconnected devices that share common processing and
file management resources, usually within a specific physical area. An example would be an of-
fice computer network.
Resolution
Resolution is a measure of accuracy or dynamic range of an A/D or D/A converter.
D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel 19
Index
(Möglichkeit)
Der Index stellt ein Zusatz dar.
Actuator 8
Formula 5
Internet adress 7
Questions/Tasks 2, 5
D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel 20
Nomenclature (Möglichkeit)
Beinhaltet die gesamte Nomenklatur aller Kapitel. Diese sollte jedoch in
jedem Kapitel aufgelistet sein. Die Gesamtdarstellung stellt ein Zusatz dar.
𝝂 Poisson ratio
𝝂 Poisson ratio
𝝈𝟏𝟏 normal stress / normal stress component in direction 𝑒1
𝝈𝟏𝟏 normal stress / normal stress component in direction 𝑒1
𝝈𝟏𝟏 normal stress / normal stress component in direction 𝑒1
𝜺𝟏𝟏 normal stress / normal strain component in direction 𝑒1
𝜺𝟏𝟏 normal stress / normal strain component in direction 𝑒1
Module Linear Structural Computational Mechanics for Wind Energy Systems 21
Online M.Sc. Wind Energy Systems www.uni-kassel.de/wes University of Kassel and Fraunhofer IWES
Lecture Notes
Linear Computational Structural Mechanics for Wind Energy Systems Prof. Dr.-Ing. habil. Detlef Kuhl These lecture notes are designed to assist students of the online master’s study wind energy systems with their learning process in linear finite ele-ment methods and linear structural dynamics of wind energy systems. For this reason it includes various elements: the theoretical development, ap-plication of methods in selected examples and program flowcharts, as well as coding instructions supporting the homework and the final case study of the course.
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