# Doctoral Theses

Doctoral Theses at the Institute for Structural Mechanics

### Doctoral Theses

- Roth, S. (2020).
*Algorithmen zur nichtlinearen Stabilitätsanalyse dünnwandiger Strukturen*. Doktorarbeit, Bericht Nr. 71, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-11239#### Abstract

Die wissenschaftliche Auseinandersetzung mit der Stabilität von Tragwerken reicht in der Geschichte weit zurück. Leonhard Euler befasste sich bereits im Jahr 1744 mit dem Stabilitätsversagen druckbelasteter Stäbe. Bis heute ist die Frage nach der Stabilität in vielen Bereichen, vom Bauwesen bis zur Luft- und Raumfahrt, relevant für die Bemessung und Konstruktion einzelner Bauteile sowie gesamter Tragstrukturen. Die Analyse eines Tragwerks hinsichtlich seiner Stabilität kann dabei in drei Abschnitte untergliedert werden: erstens die Untersuchung des Systems vor dem Stabilitätsversagen, zweitens die Eigenschaften des Systems an einem kritischen Punkt sowie drittens die Untersuchung des Verhaltens im Nachbeulbereich. Für eine geometrisch nichtlineare Analyse des Vor- und Nachbeulbereichs existiert eine Vielzahl inkrementell-iterativer Lösungsverfahren, die seit den 1970er Jahren entwickelt wurden. Sie ermöglichen die zuverlässige Berechnung nichtlinearer Gleichgewichtspfade einer Struktur, auch wenn diese in Last oder Verschiebung rückläufig sind. In den vergangen Jahrzehnten wurden diese Methoden stetig weiterentwickelt und verbessert, etwa durch die Erweiterung auf mehrere Parameter. So lassen sich für Systeme mit mehreren Veränderlichen die resultierenden Gleichgewichtspfade zu Gleichgewichtsflächen erweitern. Mithilfe begleitender Maßnahmen kann zudem an jedem Gleichgewichtspunkt eine Aussage über die Stabilität des Systems getroffen werden. Der Fokus der vorliegenden Arbeit liegt auf dem zweiten der oben genannten Punkte. Es werden Methoden präsentiert, die eine direkte Berechnung von Durchschlags- und Verzweigungspunkten für Systeme ermöglichen, die durch eine von außen aufgebrachte Knotenverschiebung belastet werden. Hierfür wird die für Kraftlastfälle bereits verfügbare Methode der erweiterten Systeme, die in der Literatur meistens mit dem englischen Begriff Extended Systems bezeichnet wird, modifiziert und ergänzt. Für die direkte Berechnung von Durchschlagspunkten wird die Residuumsgleichung für das Gleichgewicht um eine Stabilitätsbedingung erweitert. Diese ist ausschließlich an kritischen Punkten erfüllt und lässt sich aus dem Indifferenzkriterium nach Euler herleiten. Eine Nebenbedingung, die eine triviale Lösung vermeidet und die Anzahl der Gleichungen und die Anzahl der Unbekannten ausgleicht, ergibt sich aus der Normierung der Länge des Eigenvektors. Für die direkte Ermittlung von Verzweigungspunkten muss zusätzlich das Kriterium zur Unterscheidung von Durchschlags- und Verzweigungspunkten für den Fall inhomogener Dirichlet-Randbedingungen neu hergeleitet werden. Nach einer konsistenten Linearisierung des nichtlinearen Gleichungssystems lässt sich die Lösung mittels inkrementell-iterativer Methoden ermitteln. Mit den präsentierten Methoden wird das Anfangsnachbeulverhalten für Systeme mit inhomogenen Dirichlet-Randbedingungen numerisch analysiert, wodurch die bisherige Kategorisierung nach Koiter (1967) erweitert wird. Zur Linearisierung des nichtlinearen Gleichungssystems ist die Ermittlung einer Richtungsableitung der Steifigkeitsmatrix in Richtung des Beulvektors erforderlich. In der bisherigen Literatur wird die Ableitung mittels Differenzenquotienten angenähert. Diese Methoden benötigen eine geringe Rechenzeit, ihre Eigenschaften hängen jedoch von einem Parameter, der Größe des Differenzenschritts, ab. Wird dieser zu groß oder zu klein gewählt, ist der resultierende Fehler zu groß und die iterative Berechnung konvergiert nur langsam oder gar nicht zur gesuchten Lösung. In dieser Arbeit werden erstmals Methoden zur Ermittlung totaler und partieller Ableitungen, die auf hyperkomplexen Zahlen basieren, mit der Methode der Extended Systems kombiniert. Diese Methoden sind unabhängig von der Parameterwahl und ermöglichen eine exakte Berechnung erster und zweiter Ableitungen. In der vorliegenden Arbeit werden die beschriebenen Methoden auf die Ermittlung der Richtungsableitung übertragen. Bei der Untersuchung der Tragfähigkeit einer stabilitätsgefährdeten Struktur haben Imperfektionen einen großen Einfluss auf die Traglast. Bereits kleinste Abweichungen in der Geometrie eines Tragwerks oder Eigenspannungen im Material können, besonders bei schlanken Tragwerken, zu einer deutlichen Abminderung der Traglast führen. Um den Einfluss geometrischer Imperfektionen genauer untersuchen zu können, werden in dieser Arbeit zwei Methoden präsentiert, die sich in ihrem Umgang mit Imperfektionen grundlegend voneinander unterscheiden. Bei der ersten Methode ist die Imperfektionsform eine gegebene Größe. Durch eine effiziente, sich wiederholende Berechnung kritischer Punkte für eine Vielzahl an Imperfektionsformen oder -amplituden lassen sich damit kritische Pfade ermitteln. Dies geschieht in der vorliegenden Arbeit durch eine geeignete Wahl der Prädiktoren für die unterschiedlichen Imperfektionsformen bzw. über eine zusätzliche Erweiterung der Gleichungssysteme zur direkten Berechnung kritischer Punkte um eine Zusatzgleichung zur Pfadverfolgung. Bei der zweiten Methode werden die Imperfektionen als Unbekannte in das Gleichungssystem eingebracht. Durch eine zusätzliche Bedingung, die sich aus der Variation des Potentials nach den Imperfektionen ergibt, sowie das Einbinden der Nebenbedingung über eine Straffunktion lassen sich mit dieser Methode ungünstigste Imperfektionsformen finden. Diese Methode ist eine Weiterentwicklung der von Deml und Wunderlich (1997) vorgschlagenen Vorgehensweise. Abschließend werden die hier vorgestellten Methoden und Algorithmen an einer Auswahl an numerischen Experimenten demonstriert. Anhand der Beispiele wird u. a. gezeigt, dass die Richtungsableitung für eine Vielzahl an Elementformulierungen effizient ermittelt werden kann. Hierbei kommen vom einfachen ebenen Stabelement über isogeometrische Schalenelemente bis hin zu Volumenschalenelementen, die durch eine Erweiterung des Verzerrungsansatzes künstliche Versteifungseffekte reduzieren, zum Einsatz. Zudem wird die Anwendbarkeit der in der vorliegenden Arbeit entwickelten Methoden auf nichttriviale Systeme im Rahmen der Finite-Elemente-Methode mit vielen Freiheitsgraden nachgewiesen.#### BibTeX

- Schäuble, A.-K. (2019).
*Variationally conistent inertia templates for speed-up and customization in explicit dynamics*. Doktorarbeit, Bericht Nr. 69, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-10606#### Abstract

The present work deals with innovative numerical methods for the computer simulation of dynamic problems with explicit time integration. The proposed methods aim to increase accuracy as well as reduce the calculation effort. The basis for the development are variationally consistent inertia templates. The term ‘inertia template‘ covers both alternatives to the commonly used diagonal mass matrices and novel reciprocal mass matrices. Reciprocal mass matrices result directly from the variational formulation and are sparse, assemblable, inverse mass matrices, which allow a trivial computation of the acceleration from Newton‘s second law. In the first part of the work, the focus is on the variational consistency of reciprocal mass matrices and the therewith associated minimum requirements on the ansatz spaces. In the second part, the approach is systematically exploited not only to increase the critical time step but also to customize the inertia template to specific needs, like improved low-frequency accuracy. The third part aims at further development and investigation of reciprocal mass matrices to increase their usability for practical applications. The focus is therefore on the development of an efficient time step estimate and the treatment of contact.#### BibTeX

- Willerding, T. (2019).
*Multiscale simulation of phase transformation in metals*. Doktorarbeit, Bericht Nr. 70, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-10594#### Abstract

This thesis is about multiscale simulation of phase transformation in metals. Multiscale simulation is the simultaneous use of two or more models in order to have phenomena of different length or time scale in one simulation. Phase transformation between different lattice structures plays an important role in the formation of metals, e.g. iron or titanium. It is, therefore, of interest to simulate phase transformation in a multiscale context. In this thesis, a multiscale method for the simulation of phase transformation in metals is developed. Continuum mechanics, represented by the finite element method, is coupled with atomistics, represented by molecular dynamics. The goal is to simulate phase transformation in metals between different lattice structures such as body-centered cubic, face-centered cubic and hexagonal close-packed structure. As phase transformation requires an internal restructuring of the molecular structure, traditional multiscale methods cannot be used as these require fixed coupling at the interface between coarse scale and fine scale and very often also in the coarse scale by using the Cauchy(-Born) rule. In order to overcome these problems, a combined hierarchic-partitioned-domain method is proposed that consists of two parts. On the finite element level, a hierarchic method based on the FE2-method is used with molecular dynamics simulations as subproblems, one subproblem at each Gauss integration point. The partitioned-domain part of the method consists of dividing the domain into two parts: a molecular dynamics part and a finite element part. A part of the atoms are put into a box with the shape of a parallelepiped and with periodic boundary conditions. This box is linked to the movement of the finite element nodes.#### BibTeX

- Neukamm, F. (2018).
*Lokalisierung und Versagen von Blechstrukturen*. Doktorarbeit, Bericht Nr. 68, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-10082#### Abstract

Diese Arbeit befasst sich mit der Simulation von Blechstrukturen unter Verwendung der Finite-Elemente-Methode (FEM). Schwerpunkt ist die Vorhersage von lokalisierter Deformation und Versagen, für welche Verbesserungen und neue Ansätze vorgeschlagen werden. Die betrachteten Blechstrukturen werden üblicherweise in mehreren Prozessschritten hergestellt, beginnend mit einem Umformvorgang. Dabei können sich lokal die Eigenschaften der Blechteile verändern, was das Verhalten unter einer nachfolgenden Crashbelastung wesentlich beeinflussen kann. Zur Steigerung der Prognosegüte wird deswegen eine durchgängige Modellierung der Prozesskette mit Hilfe eines modular verwendbaren Versagensmodells eingeführt. Als Kern dieser Arbeit wird das phänomenologische Schädigungs- und Versagensmodell GISSMO (Generalized Incremental Stress-State dependent damage MOdel) entwickelt. Wesentlicher Bestandteil ist eine Evolutionsgleichung, die den Zuwachs der Schädigung in Abhängigkeit vom Deformationszustand und der aktuellen Schädigung beschreibt. Mit Hilfe eines zusätzlichen Exponenten kann die Evolution nichtlinear gestaltet werden, um die erwarteten Vorgänge im Material besser abbilden zu können. In den betrachteten Simulationen beginnt mit dem Auftreten von lokalisierter Deformation eine unphysikalische Abhängigkeit der Ergebnisse von der Netzgröße. Deswegen ist die Prognose des Lokalisierungsbeginns notwendig, um Regularisierungsmaßnahmen einzuleiten. Dazu werden zwei Methoden entwickelt: Mit Hilfe einer inkrementellen Formulierung wird das Lokalisierungsrisiko abhängig vom durchlaufenen Deformationspfad summiert. Alternativ wird der Zwei-Parameter-Ansatz (Forming Limit Surface FLS) vorgestellt, bei dem der Zustand durch die relative Blechdicke und die plastische Dehnung charakterisiert wird.#### BibTeX

- Oesterle, B. (2018).
*Intrinsisch lockingfreie Schalenformulierungen*. Doktorarbeit, Bericht Nr. 67, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-10046#### Abstract

Angesichts einer stetig steigenden Anzahl komplexer Diskretisierungsverfahren beschäftigt sich die vorliegende Arbeit mit intrinsisch lockingfreien Schalenformulierungen. Aus der Literatur bekannte Konzepte versuchen stets die durch die Diskretisierung entstehenden Locking-Effekte zu beseitigen oder abzumindern. Tritt Locking jedoch gar nicht auf, ist dessen Beseitigung obsolet. Deshalb sollen die hier vorgestellten Schalenformulierungen numerische Locking-Effekte bereits auf Theorieebene vermeiden, ungeachtet vom verwendeten Diskretisierungsschema. Die Vermeidung von Locking bereits vor der Diskretisierung verspricht ein breites Anwendungsspektrum für diverse Diskretisierungsverfahren im Bereich von Computersimulationen physikalischer Vorgänge. Der erste Teil dieser Arbeit beschäftigt sich mit der intrinsischen Vermeidung von Querschublocking in Formulierungen für Strukturtheorien. Über hierarchische Reparametrisierung der kinematischen Gleichungen kann Querschublocking im Rahmen einer primalen Methode a priori vermieden werden. Das Konzept wird gleichermaßen für schubweiche Balken-, Platten- und Schalenformulierungen demonstriert, wobei jeweils zwei hierarchische Parametrisierungen unterschieden werden. Der zweite theoretische Teil dieser Arbeit beschäftigt sich mit der intrinsischen Vermeidung aller geometrischen Locking-Effekte, vor allem aber von Membranlocking. Es wird ein neuartiges, reparametrisiertes gemischtes Prinzip vorgestellt, in dem ausschließlich Verschiebungsgrößen als Primärvariablen auftreten. Diese Reparametrisierung führt dazu, dass die für gemischte Methoden notwendige Wahl geeigneter Spannungs- oder Verzerrungsräume entfällt. Die daraus resultierende intrinsische Vermeidung geometrischer Locking-Effekte verspricht ein breites Anwendungsspektrum dieser Methode.#### BibTeX

- Wilking, C. (2017).
*Effiziente Integration und verbesserte Kontaktspannungen für duale Mortar-Formulierungen*. Doktorarbeit, Bericht Nr. 66, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-9244#### Abstract

This work deals with computer simulations of contact problems using the finite element method. Two modifications are proposed for the mortar method, which is the method applied to discretise the contact. The first modification concerns the numerical calculation of so-called contact integrals. For the corresponding integration in general cases polynomial integrands have to be integrated over polygonal areas. In order to use ordinary numerical quadratures the polygonal areas are usually subdivided into triangular integration cells. In this work an alternative subdivision into quadrilateral integration cells is suggested, which yields less integrations points. With the numerical experiments described in this work it is shown that due to this reduction the numerical effort is decreased considerably without deteriorating the accuracy of integration significantly. The second modification improves the contact stresses of the dual mortar method. This method uses dual functions to approximate the Lagrange multiplier field, yielding the advantage that the dual mortar method is more efficient than the standard mortar method. However, the contact stresses of the dual mortar method are less accurate than the contact stresses of the standard mortar method. In this work a modification of the contact stresses based on an L2 projection is presented for the dual mortar method. Numerical experiments show that with the introduced L2 projection the accuracy of the contact stress of the dual mortar method is improved and comparable to the accuracy of the standard mortar method.#### BibTeX

- Effinger, V. (2016).
*Finite nichtlinear viskoelastische Modellierung offenzelliger Polymerschäume*. Doktorarbeit, Bericht Nr. 65, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-8899#### Abstract

To be approved for European public roads passenger cars must meet the legal requirements concerning pedestrian safety. One legal impact test, assessing the compliance of the car front in the event of a collision with a pedestrian, consists in shooting the upper legform impactor onto the bonnet leading edge. In order to replicate this load case in a crash simulation with predictive capability, the material behavior of Confor ® CF-45, an open-cell polyurethane foam, and one of the constituents of the upper legform impactor is of major importance. Hereby, an adequate modeling of the energy absorption in the foam and of the contact area between impactor and bonnet is important for capturing the load transfer in the simulation. The objective of this thesis is to develop a material model of open-cell polymer foams for application in pedestrian protection, which in particular considers the nonlinear visco- elastic properties of these materials and which can be used in finite element simulations with explicit time integration. In the framework of classical continuum mechanics the material modeling follows a phenomenological approach within which the isotropic stress response is additively decomposed into an equilibrium and an over stress response. While the MFCF model (Kolling u.a. (2007a), Hallquist (2016)) is adopted as a good example of a simplified model for the equilibrium response from the finite element analysis software program LS-DYNA, the focus of this thesis is the description of the over stress in the context of finite nonlinear viscoelasticity. A possible stress contribution created by the interference of the inlet and outlet of a fluid phase due to cellullar structure of the foam can be added to the modeling. However, this stress contribution is not considered in this thesis as there were no usable measurements of the air flow in Confor CF-45 available. For the understanding and classification of the material models considered in this thesis the theory of linear viscoelasticity introduces different representations of constitutive equations of viscoelastic solids which express the stress as a functional of the complete strain or strain rate history. Rheological elements are used to illustrate the differential equations, the systems of differential equations and the integral equations as well as the relations between them. The integral representation serves as a starting point for the over stress constitutive equations in this thesis and this will require the examination of numerical convolution algorithms. Time-dependent material functions included as convolution kernels contain the material properties and their shape determines how the convolution integrals can be calculated numerically. The naïve convolution algorithm with a piecewise linear material function requires to save the complete strain history, reads it from storage during each time step and reprocesses all past values to obtain the current stress response. In contrast, the recursive convolution algorithm with a Prony series as material functions exploits the characteristics of the exponential functions in order to summarize the past strain history and to reuse it in each time step with the help of a relaxation prefactor. The local convolution algorithm is based on the work of López-Fernández u.a. (2008) and requires multiple Prony series which are valid for discrete and overlapping intervals and consequently are locally defined. Thus, the characteristics of the exponential functions can be used as well and the number of terms of the individual Prony series can be reduced in comparison to the recursive convolution algorithm. In order to reduce the simulation time this work uses the strain rate instead of the strain as input for the convolution algorithm. Furthermore, the model size of the upper legform impactor and the number of timesteps in a pedestrian impact simulation require the use of the recursive or the local convolution algorithm. In order to combine convolution integrals in a constitutive equation for open-cell polymer foams in a meaningful way it is necessary to analyze the behavior of these materials and to simplify it with appropriate assumptions. Due to their cellullar structure open-cell polymer foams put special demands on the constitutive equation. As a result of the different deformation mechanisms involved, they do not only show a distinct tension-compression asymmetry with regard to the shape and magnitude of the stress-strain curve but also with respect to the transverse strain behavior. The constitutive equations of this work are based on the assumption that longitudinal, transverse and thickness strains are uncoupled which follows from the observation of a diminishing Poisson’s ratio under uniaxial compression loading. This thesis proposes different constitutive equations for the over stress which differ besides the convolution algorithm in the material nonlinearity and the choice of stress and strain rate measures. The nomenclature of the constitutive equation is additively composed by the underlying theory, the finite linear respectively nonlinear viscoelasticity (FLVE respectively FNLVE), and the individual elements such as the stress and strain rate measures, the material nonlinearity and the convolution algorithm, each in abbreviated form. For example, FLVE-TBU-recursive represents a finite linear viscoelastic constitutive equation which uses the Biot stress and the right stretch tensor as well as the recursive convolution algorithm. Two pairs of dual stress and strain tensors are used: the second Piola-Kirchhoff stress tensor and the Green-Lagrange strain rate tensor (SE) as well as the Biot stress tensor and the rate of the right stretch tensor (TBU). Due to their definition with respect to the reference configuration all stress contributions can be summed up and converted in the same way to the Cauchy stress tensor with reference to the current configuration. The material nonlinearity of the examined models is based on two approaches. Either the results of multiple convolution integrals of different powers are summed up or the material function within one convolution integral is complemented by an additional dependency upon one or multiple functions. The sum of multiple convolution integrals results from the simplification of the Green-Rivlin model (GR) for foams which transforms the multiple integrals of fifth order into single integrals of fifth power using the product form of the material function. Materially nonlinear constitutive equations on the basis of one convolution are created with three types of rate dependent coefficients for the Prony series. The first type uses an equivalent strain rate (EQV) which corresponds to the absolute value of the excitation strain rate in an uniaxial test. The second and third type transform the constitutive equation into the principal coordinate system (HS) of the strain rate tensor and the material function is evaluated for each principal direction with the corresponding principal strain rate. The third type differs from the second type as the assessment in each principal direction is complemented by a switch which distinguishes between loading and unloading (HS+). As a result a modular material routine is established which provides a multitude of constitutive equations. In order to determine which constitutive equations are able to reproduce the test data of open-cell polymer foams the stress responses to standard excitations are examined. Hereby, the preferred strategy is to identify the parameters for a uniaxial test at one nominal strain rate with the help of a linear material model and to use the extended nonlinear material model in order to unify the parameters of the individual tests into one single parameter set. This procedure can only be applied to the constitutive equation FLVE-TBU which is able to replicate every individual tension and compression test with different parameters and which can afterwards be combined with the three different types of extended material functions into FNLVE-TBU-EQV, FNLVE-TBU-HS or FNLVE-TBU-HS+. The stress response of these models is then evaluated for tension and compression tests at different strain rates, for multiaxial tests such as the biaxial tension test, the biaxial tension compression test and shear tests as well as for cyclic uniaxial tension, tension-compression and uniaxial compression tests. Out of the three materially nonlinear constitutive equations only the model FNLVE-TBU-HS+ is able to differentiate the uniaxial tests at different strain rates, to distinguish between tensile and compressive loading and to prevent that a load reversal automatically generates a switch from tension to compression parameters. To identify the parameters of the constitutive equation a comprehensive test program was conducted for Confor CF-45. The experimental characterization reveals different influence factors on the mechanical behavior of the open-cell PU foam. Besides the influence of the air flow which induces a dependency upon the specimen size the material shows a high sensitivity in the neighborhood of the glass transition temperature upon any changes of temperature or moisture. Hence, a comparison of measurements conducted under atmospheric conditions and in vacuum cannot be used to determine the air flow. The tension and compression tests under normal conditions and at specimen size of 50x50x30mm3 show a distinct strain rate dependency in the range of strain rates between 1e-61/ms and 2e-11/ms and anomalies in comparison to other PU foams. Compression tests reveal a stress peak between the initial slope and the plateau stress and the increase of the absolute stress which is usually associated with the densification of the foam material already occurs for low compression values. In order to consider these peculiarities the parameter identification deviates form the defined stress decomposition into equilibrium stress on the basis of quasistatic tests and in the over stress as the difference between dynamic and quasistatic testing for the compression tests. With this modification it is possible to reproduce the compression tests with the model GG+FNLVE-TBU-HS(+)-recursive with high conformity and the tension tests with very high conformity. Thereby, the stress answer of the model GG+FNLVE-TBU-HS (+) -recursive is additively composed by the equilibrium response GG and the viscoelastic overstress FNLVE-TBU-HS(+)-recursive. The modeling with GG+FNLVE-TBU-HS+-recursive and the selected parameter set prove to be robust in the simulation of the material tests, of the upper legform impact test in pedestrian protection and of two component tests. Due to the decomposition of the stress response into equilibrium and over stress an instantaneous and time-delayed recovery of the foam material takes place and the elastic part of the internal energy can be recaptured. By capturing theses viscoelastic effects the new modeling exceeds by far the current state of the art. A final validation of the model with a component test is not available as Confor CF-45 is not produced anymore and hence the component test as well as the test to model the influence of the air flow could not be conducted. However, the modeling can be transferred to other open-cell polymer foams and in particular to the succession products of Confor CF-45 as part of the upper legform impactor which possess similar mechanical properties according to their specifications. Furthermore, the modular material routine represents a solid starting point to model nonlinear viscoelastic over stress for other materials at finite deformations. In particular materials with a non-zero Poisson’s ratio could be considered.#### BibTeX

- Matzen, M. E. (2015).
*Isogeometrische Modellierung und Diskretisierung von Kontaktproblemen*. Doktorarbeit, Bericht Nr. 64, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-642#### Abstract

Die wissenschaftliche Auseinandersetzung mit Kontaktvorgängen in der Mechanik begann mit Heinrich Hertz im Jahre 1882. Seine analytischen Studien sind grundlegend für unser heutiges Verständnis der Kontaktmechanik. Da analytische Lösungen nur für sehr einfache Problemstellungen existieren, werden gegenwärtig vorrangig numerische Verfahren zur Lösung strukturmechanischer Kontaktprobleme, beispielsweise bei Crash-Simulationen, eingesetzt. Im Mittelpunkt steht dabei die Finite-Elemente-Methode, die das Strukturproblem in einer schwachen integralen Weise löst. Hierzu wird die zu berechnende Struktur, bzw. das Bauteil, in einem CAD-Programm entworfen und anschließend für die Berechnung in finite Abschnitte unterteilt. In diesem Vernetzungsschritt findet eine Approximation der wahren Geometrie durch einfache, oft nur lineare, Polygonzüge statt. Die Oberfläche des Bauteils ist dadurch facettiert beschrieben, was die Formulierung von Kontaktbedingungen zwischen den Körpern erschwert. Es kann zu künstlichen Verhakungen und Verkantungen der Körper kommen, was wiederum zu unphysikalischen Ergebnissen oder numerischen Instabilitäten führen kann. Mit der isogeometrischen Analyse kann dieses Problem umgangen werden, indem der Funktionenraum der CAD-Geometrie für die Finite-Elemente-Berechnung verwendet wird. Die glatte Oberfläche bleibt bei der Vernetzung erhalten, allerdings sind nun die einfachen Polynome durch nicht-uniforme, gebrochenrationale B-Splines, kurz NURBS („non-uniform rational B-splines“), ersetzt worden. Ein neues, hoch spannendes Forschungsfeld ist eröffnet, das Raum für innovative Ideen bietet. Das Ziel dieser Arbeit ist deshalb die Entwicklung eines stabilen und leistungsfähigen isogeometrischen Kontaktalgorithmus. Da neben der Berechnung statischer Probleme dynamische Probleme betrachtet werden, wird eine modifizierte Diskretisierung in Raum und Zeit diskutiert. Um Körper am gegenseitigen Durchdringen zu hindern, müssen sogenannte Nichtdurchdringungsbedingungen formuliert werden. Der minimale Abstand zwischen zwei Körpern zeigt an, ob Regionen durchdrungen sind oder nicht. Im klassischen Ein-Schritt-Algorithmus werden dazu die zwei am Kontakt beteiligten Körper in einen „Slave“- und einen „Master“-Körper eingeteilt. Der Abstand wird vom Slave-Körper aus auf den Master-Körper, meist unter Verwendung der Master-Oberflächennormalen, gemessen. Wird der Abstand negativ, werden Kontaktspannungen aktiviert, die die durchdrungenen Regionen auseinander schieben. Weisen die Oberflächen eine gewisse Rauigkeit auf, entstehen bei einer tangentialen Relativbewegung tangentiale Spannungen, die unter Verwendung des Coulomb’schen Reibgesetzes durch die Gleitspannung begrenzt sind. Für den normalen wie tangentialen Kontakt lassen sich die Zwangsbedingungen in Form der Kuhn-Tucker-Karush-Bedingungen zusammenfassen. Die starken Zwangsbedingungen werden im nächsten Schritt in eine schwache, integrale Form überführt, um als Nebenbedingungen in den Finite-Elemente-Ansatz einzugehen. Normal zur Oberfläche wird dazu die Lagrange-Multiplikator-Methode verwendet mit der, unter Einführung neuer Unbekannten, eine exakte Erfüllung der Nichtdurchdringungsbedingung möglich ist. Für den tangentialen Kontakt wird die Penalty-Methode zur Einhaltung der Haftund Gleitbedingungen genutzt, welche die Verletzung der Zwangsbedingung mit einer Strafsteifigkeit versieht. Die hochgradig nichtlinearen Gleichungen werden konsistent linearisiert, um ein quadratisches Konvergenzverhalten zu generieren. Die so erhaltene linearisierte virtuelle Kontaktarbeit muss nun von der kontinuierlichen in eine diskrete Form überführt werden. Dies geschieht in zwei Abschnitten: Zuerst wird der Integrand durch Einsetzen der Oberflächen-Diskretisierung und des diskretisierten Lagrange-Multiplikators diskretisiert. Im zweiten Schritt wird das Integral selbst mittels einer Summe über Auswertungspunkte angenährt. Dieser unscheinbare zweite Schritt ist das Herzstück des entwickelten Kontaktalgorithmus. Je nach Anzahl und Lage der Auswertungspunkte werden die Kontaktbedingungen punktuell oder integral erfüllt. Der Integrand verbleibt in beiden Fällen unverändert, sodass die Behandlung beider Zustände in einem Kontaktelement möglich wird. Entspricht die Anzahl an Auswertungspunkten der Anzahl an Kontrollpunkten der Oberfläche, gehört das Verfahren zur Gruppe der kollokierenden, punktuellen „Node-To-Segment“-Methoden. Da die Kontrollpunkte isogeometrischer Objekte nicht zwangs- läufig auf der Geometrie selbst liegen, werden zur Auswertung der Kontaktbedingung spezielle Kollokationspunkte auf der Oberfläche des Objektes gewählt, beispielsweise Greville-, Botella- und Chebyshev-Punkte. In diesem Zusammenhang wird der Name „Point-To-Segment“-Methode, kurz PTS, für die erste Variante des entworfenen Kontaktelements vorgeschlagen. Die Nichtdurchdringungsbedingung wird an den Kollokationspunkten punktweise exakt erfüllt. Die Kontaktkräfte entstehen als Reaktionskräfte. Werden den Kollokationspunkten Kollokationsgewichte zugeordnet, ist der Übertrag von Kontaktspannungen statt -kräften in der Kontaktzone möglich. Die Kollokationsgewichte werden zu Beginn der Berechnung einmalig bestimmt und bleiben während des gesamten Simulationszeitraums konstant. Die positive Auswirkung einer Gewichtung zeigt sich vor allem bei reibbehaftetem Kontakt in der Auswertung des Reibgesetzes. Die gewichtete Kollokation, als zweite Variante des Kontaktelements, erhält den Namen PTS+. Mit der Verwendung von Gauß-Auswertungspunkten wird eine integrale Methode initiiert, welche die Kontaktbedingungen numerisch integriert. Wird zur Positionierung der Gauß-Punkte die Slave- und Master-Oberflächendiskretisierung einbezogen, wird von einer konsistenten Segmentierung gesprochen, die für gekrümmte Strukturen äußert komplex ist. Aufgrund der glatten Oberflächenbeschreibung mit NURBS sind die Integrationsfehler, die durch eine fehlende Segmentierung entstehen so gering, dass auf letztere verzichtet werden kann. Die Gauß-Punkte werden deshalb je Slave-Element auf der Slave-Oberfläche angeordnet. Die Kontaktbedingungen werden in dieser dritten Kontaktelement-Variante integral erfüllt. Da in der entwickelten Variante alle Größen auf die Kontrollpunkte der Slave-Oberfläche projiziert werden, kann von einer „Mortar“-Methode statt einer „Segment-To-Segment“-Methode (STS) gesprochen werden. Eine Gegenüberstellung der Ergebnisse der drei Kontaktelement-Varianten PTS, PTS+ und Mortar in numerischen Tests, zeichnet in puncto Rechenzeit die Kollokationsmethoden als klare Sieger aus. Da die Rechenzeit linear von der Anzahl an Auswertungspunkten abhängt, benötigen weniger Auswertungspunkte weniger Zeit. Weniger Auswertungspunkte führen aber meist zu einer geringeren Ergebnisqualität. Jedoch kann gezeigt werden, dass weder im Spannungsresultat des Hertz’schen Kontaktes noch bei den Reaktionskräften des „Ironing“ Problems markante Unterschiede zwischen der einfachen Kollokation und teureren Integration festgestellt werden können. Daraus folgend stellt sich, unter Einbezug der Resultate mit Reibung, die gewichtete Kollokation PTS+ als effizienteste Kontaktelement-Variante heraus. Werden dynamische Kontaktprobleme untersucht, wird eine stabile Zeitintegration benötigt, die neben der Diskretisierung im Raum eine Diskretisierung in der Zeit einführt. Der betrachtete Zeitraum wird dazu in Zeitintervalle unterteilt. Unter Verwendung eines Ein-Schritt-Algorithmus werden alle Feldgrößen diskret am Intervallende aus Informationen des letzten bzw. des aktuellen Zeitschritts berechnet. Werden keine Informationen des aktuellen Zeitpunktes benötigt, wird von einer expliziten Methode gesprochen. Die Bewegungsgleichung wird zum letzten bekannten Zeitpunkt ausgewertet. Die Werte zum Folgezeitpunkt können ohne Gleichungslösen daraus bestimmt werden. Das Verfahren ist allerdings nur bedingt stabil. Deshalb wird im Rahmen dieser Arbeit ein implizites Verfahren verwendet, welches Informationen des aktuellen Zeitpunktes mit einbezieht. Als Basis-Zeitintegrationsverfahren wird das Newmark-Verfahren verwendet, das durch Aufteilung der Beschleunigung in einen kontaktabhängigen und kontaktunabhängigen Anteil modifiziert wird. Die kontaktabhängigen Anteile werden dissipativ verändert, sodass nur Beschleunigungen des letzten Zeitschritts berücksichtigt werden. Die Energie des Systems wird so stabilisiert. Allerdings treten Oszillationen in den Kontaktkräften auf, deren Ursache im Unverhältnis zwischen Beschleunigung und Masse der Kontakt-Kontrollpunkte liegt. Aufgrund des Kontaktvorgangs werden die Kontrollpunkte der Oberfläche schlagartig auf null abgebremst, singuläre Beschleunigungen entstehen. Da die Kontrollpunkte massebehaftet sind, entstehen singuläre Trägheitskräfte, die zu oszillierenden Kontaktkräften führen. Wird die Masse in der Kontaktzone entfernt, kann diesem Phänomen entgegen gewirkt werden. Physikalisch betrachtet ist das Umverteilen von Oberflächenmasse legitim, da Oberflächen unendlich dünn sind und somit keine Masse besitzen. Formal wird die Oberflächenmasse durch Umverteilen der Formfunktionsanteile senkrecht zur Kontaktoberfläche auf tiefer liegende Kontrollpunktreihen entfernt. Die Kontrollpunkte der Kontaktoberfläche werden masselos und Oszillationen in den Kontaktkräften verschwinden. Zwei mögliche Verteilungsszenarien werden untersucht. Die erste Methode verteilt den Formfunktionsanteil auf tiefer liegende Kontrollpunkte des Randelements gleichmäßig, die zweite mit speziellen Faktoren, die eine Ordnungsreduktion im Randelement zur Folge haben. Beide Methoden erweisen sich als zielführend und ergeben, kombiniert mit dem dissipativen Newmark-Algorithmus, ein stabiles Zeitintegrationsverfahren für dynamische Problemstellungen.#### BibTeX

- Pohl, T. (2014).
*Adaptive Methoden zur Pfadverfolgung bei Entfestigung*. Doktorarbeit, Bericht Nr. 62, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-561#### Abstract

Until now, the determination of equilibrium paths of complex structures with strongly nonlinear behavior is a challenge for path-following schemes. Structural problems with close path sections or structures exhibiting brittle softening behavior may lead to experiment failure due to control type properties. From the early seventies, a range of methods have been developed, that are able to trace a path with snap-backs and snap-throughs in load and displacement. These arc-length methods have been modified and enhanced since then. Typical problems arising from this kind of constraint equations are loss of convergence, bisectioning while reducing step size to computer precision and unloading until reaching the undeformed state of the structure – either purely elastic or with secant stiffness due to softening behavior. In the past, arc-length methods have been applied to trace softening problems. However, they are not reliable in the post-critical range due to potential artificial unloading. For a consistently linearized cylindrical arc-length method, artificial unloading could be seen for softening behavior in several examples. Some applications of finite element programs require user intervention after computation has been started. For example, when static analysis does not converge, it has to be manually switched to a dynamic analysis. Particularly for softening behavior, a variety of methods have been developed that directly influence the softening process. Often, these methods turn out to be computationally intensive. The aim of this work is to develop a self-adapting, robust and efficient path-following technique for elastic and in particular for softening structural behavior, based on the finite element method. These control methods are derived from properties of the equilibrium path. Therefore, the main focus of this work is spilt into two parts: first, characteristic properties of equilibrium paths of elastic and softening behavior are identified. Second, control methods arising from these properties are developed by choosing appropriate control parameters. User intervention is not required. For elastic structural behavior, adaptivity criteria are presented choosing and controlling relevant degrees of freedom of the cylindrical arc-length equation. To avoid artificial unloading of softening problems, the phenomenon has first to be described: if both, elastic and dissipative energy decrease, artificial unloading can be detected. As a strategy to prevent artificial unloading, an active process zone is selected as the control region. Control parameters are evaluated in this control region. The identification of the control region is based on material parameters. A locally formulated, strain-based continuum damage model is used here for discussion. In order to be able to describe path-following methods in detail, it is necessary to distinguish between load case and control parameter. The load case is connected with the modelled structure. The control parameter is chosen independently and should be able to assess the full path. Comparing numerical and real experiments, it becomes obvious, that failure of the contol method for both types arises from the same reasons: either the control quantity decreases or the combination of control directive and equilibrium state is not unique. The form of collapse of a control method does not rely on the fact of whether the experiment is realized in reality or numerically. Decisive is the possibility of developing dynamic effects. In case of a static numerical analysis, oscillations cannot develop and the computation will not converge. Undesirable phenomena arising from failure of the control method are grouped into three kinds of mechanisms: First, dynamical snap-through or loss of convergence respectively. Second, elastic unloading on the elastic loadig path and third, unloading with secant stiffness, if softening has started. Another similarity of numerical and real experiments is the distinction between load case and control quantity. A prerequisit for this is a controlled testing machine. Continuously growing control parameters with physical meaning can trace full path until a desired deformation or damage state is reached. For complex structures with elastic or even softening behavior it is often not possible to detect suitable control parameters in advance. In adaptive schemes, constantly changing increments are therefore controlled instead of monotonically growing quantities. In this work, three adaptively modified cylindrical arc-length equations are presented. The adaptivity criterion Maximum has turned out to be the most efficient of those three. The maximum displacement increment of the previous load step enters the constraint equation quadratically. For elastic structural behavior, this method has proven to be efficient, but has shown artificial unloading in the context of softening. Alternatively, equilibrium points can be computed with an optimization algorithm or with a dynamic analysis with loading and unloading branches. The area between these branches can be understood as the energy absorbed by damping. In order to be able to reliably control the damage process of a structure, it is necessary to avoid artificial unloading. If the damage process progesses, energy plots show increasing elastic or dissipative energy. If both energies decrease, artificial unloading is detected. To avoid unloading, the control quantities are evaluated in the control region. The control region is identified in the process zone by monitoring equivalent strains of all gauss points. The Gauss point fulfilling all requirements identifying the process zone, defines the control region. Starting from this idea, three control methods for softening behavior are proposed: The first implementation restricts all possible solution points of a cylindrical arc-length method by multiplying the increment of equivalent strains of the gauss point defining the process zone. As this method is very step size sensitive and computationally intensive it will not be found in all numerical examples. The second method is based on this constrained arc-length method, just eliminating the arc-length part of the constraint equation. Only the increment of equivalent strains is prescribed in the equation. This method requires only a small number of steps compared with the methods proposed in other publications. An advantage of this adaptive strain control is the ability to trace dissipative as well as elastic parts of the equilibrium path without necessitating a switch in control types. The last proposal separates the control quantity and the parameter identifying the control region. When the control region is determined, the maximum displacement increment of the related element is chosen as the control parameter. This increment is measured as the change of a displacement in the direction of the same degree of freedom in the last converged step. Within this adaptive displacement control in process zone, the constraint equation and its linearizations have become decoupled from the choice of the equivalent strain measure. Finally, the performance of methods from other publications and the methods proposed here are predicted in relation to specific structural problems. In order to do this, structural problem types are grouped into compounds with similar properites regarding their demands on path-following schemes. For each of these categories a recommendation is given, regarding which control methods are expected to be suitable. In conclusion, the adaptive strain control as well as the adaptive displacement control in process zone have proven to be effective and robust path-following techniques in the context of softening.#### BibTeX

- Sorg, A. (2014).
*Adaptive diskret-kontinuierliche Modellierung von Materialien mit Mikrostruktur*. Doktorarbeit, Bericht Nr. 63, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-600#### Abstract

The present work focusses on the development of a method that enables simulation of fragmentation of cohesive frictional materials using the discrete element method without fully resolving the microstructure. This requires modelling of phenomena on different geometric scales. In order to link these scales in a single simulation, coupling of the finite element method (FEM) and the discrete element method (DEM) is suggested which combines the efficiency of the FEM with the accuracy of the DEM. The proposed concept transfers ideas from the quasicontinuum method in the field of atomistics to problems in structural mechanics of cohesive frictional materials. Before describing the method and analysing the results of some numerical examples, different strategies for material modelling are discussed. On different scales, materials show a different structure. Depending on the scale that the material is looked upon, different methods are applied for modelling. On the macroscopic scale, most materials can be considered continuous. When zooming in, many materials have a discrete microstructure. One example is concrete, where grains, which are also called aggregates, reside in a matrix of cement. On a very fine scale, the atomic scale, all materials are discrete. In order to accurately predict the behaviour of discrete materials until failure, mechanical models are required that are capable of representing the complex processes on the level of the discrete microstructure. Two appropriate methods are presented in this work: the discrete element method (Cundall und Strack 1979), which is often used for simulating granular or cohesive frictional materials, and the molecular dynamics (Alder und Wainwright (1957) and Alder und Wainwright (1959)) for simulations on the atomic scale. The large numerical effort limits the size of computable structures and therefore is a disadvantage of these methods. On the macroscopic level, methods based on continuum mechanics descriptions like the finite element method (FEM) are efficient and reliable as long as the solution is relatively smooth. A precondition for the application of continuum methods is scale separation. Scale separation means that the dimensions of the global structure are much larger than the dimensions of the microstructure so that homogenisation methods can be applied. Localisation phenomena of the microstructure can only be represented in an average sense and not as detailed as with discrete methods. In addition to the finite element method (Turner u.a. (1956), Argyris (1960) and Zienkiewicz und Cheung (1967)), two other continuum methods are introduced: the smoothed particle hydrodynamics (Lucy (1977) und Gingold und Monaghan (1977)) and the material point method (Sulsky u.a. 1994). To optimise accuracy and computational efficiency, an obvious idea is to adaptively combine discrete and continuous methods. Local phenomena like crack initiation and evolution require resolution with a discrete model. Therefore, a discrete method is used in areas where localisation appears. In regions where the solution is relatively smooth, a continuum method is sufficiently accurate. This allows for the simulation of large structures for which computations using a fully resolved microstructure exceed the available amount of memory. Coupling the two methods is especially efficient if the regions in which the microstructure needs to be fully resolved are relatively small and identified within an adaptive procedure. Various methods following this strategy may be found in previous publications. Three different concepts for coupling discrete and continuous methods are presented in this dissertation: The bridging domain method (Belytschko und Xiao 2003), the bridging scale method (Wagner und Liu 2003) and the quasicontinuum method (Tadmor u.a. 1996), all of which originate in the field of atomistics. Based on the quasicontinuum method, a mesh and model adaptive method is developed in order to address issues in structural mechanics. Physical particles build the material of the examined structures on a small scale. Small and large scale of the considered materials differ by only one or two orders of magnitude, but not more. The computations should be started with a coarse finite element discretisation. The transition to the fully resolved discrete model should develop adaptively in the respective areas. The emphasis is not on the development of an advanced discrete element method for certain applications but rather on the adaptive strategy. Therefore, a simple form of the discrete element method is used to model the discrete fine scale, which is basically a lattice method. The nodes of a lattice made up of equilateral triangles represent the centres of round, equally sized and rigid particles. It is assumed this is a good approximation of the microstructure of a material. Hence, the numerical solution obtained with the lattice method represents the material behaviour in a realistic way: The finite element method provides the kinematics of the system and the lattice method drives the constitutive behaviour. The finite element mesh with triangular elements is refined adaptively in areas of high strain gradients. A specific characteristic of the applied mesh refinement procedure is that new nodes are always placed at the centre of a particle in the microstructure. This ensures that at the end of mesh refinement the microstructure is fully resolved and each particle represents a FE node at the same time. Introducing three different element types, level 1 through level 3, furthermore provides an adaptive transition from the coarse scale (FEM) model to the fully resolved fine scale (DEM) model. The difference between the three kinds of elements is the method of computing the stiffness matrix, at which the microstructure is resolved with different accuracy. In regions with a coarse mesh (level 1) the material is represented by an equivalent continuum obtained from homogenisation. Where the mesh is resolved down to the small scale (level 3), the discrete particles interact according to the small scale model. Level 2 elements serve as transition between the scales. Level 2 elements do not fully resolve the microstructure. However, for computation of the element stiffness matrix, every lattice member inside the element is considered individually. In case of a homogeneous microstructure in which all members of the regular lattice have the same material properties, the displacements of particles inside an element can be obtained from interpolation of the discrete particle movements (Cauchy-Born rule) at the FE nodes. With the particle displacements at hand, the potentials of the lattice members inside the element can be computed in dependence of the nodal displacements. The sum of all these potentials yields the overall potential of the element. The second derivative of the overall potential with respect to the nodal displacements in turn yields the stiffness matrix of the level 2 element. When applying the Cauchy-Born rule in case of a heterogeneous distribution of material properties among the trusses of the microstructure, the trusses within an element are not in equilibrium. In this case, a different strategy is used. The stiffness matrix is computed from eigenvalues and eigenvectors. The eigenvectors represent a deformation state and the eigenvalues represent the stiffness of the element in the respective state. For determination of the eigenvectors, a homogeneous material law is assumed. The eigenvectors hence correspond to the eigenvectors of the stiffness matrix of a level 1 element. In order to obtain the six eigenvalues, a subproblem on the level of the microstructure is solved for each eigenmode. The eigenvectors are used as displacement boundary conditions for each state. Requiring equivalence of the energies in the fine scale and the coarse scale model provides an equation for computing the eigenvalue of the level 2 element in the applied eigenmode. Since the microstructures of all level 2 elements are independent from each other, the stiffness matrices of the level 2 elements can be computed in parallel. Applicability of the overall concept is demonstrated on four examples. All four examples feature a rectangular structure with a rhombus shaped hole in the centre which is stretched horizontally. In two of these examples, all lattice members share the same material properties (homogeneous case). In the remaining two examples, the material properties are distributed heterogeneously. In one of the two examples per case, a linear elastic material law is used for the lattice members, and in the second example, individual members can damage according to a uniaxial linear softening law. For building the heterogeneous model, an idealised section of concrete, built from round aggregates of different size, is projected on the abovementioned regular triangular lattice. The model becomes heterogeneous due to the fact that the lattice members are assigned different material properties, depending on their position (inside an aggregate, inside the matrix or building the interface). For linear elastic lattice members, stiffness is varied. In case of linear softening, strength is varied. The linear elastic examples show that the adaptive procedure performs well and that the results of the adaptive and the fine scale simulations are in good agreement. In the homogeneous case, adaptive simulation uses many degrees of freedom less than the reference solution with the fully resolved microstructure. However, during adaptive computation, several steps are necessary for refining the mesh. Nevertheless, the adaptive simulation is three times faster than the fine scale computation (wall clock time). Due to the smaller number of degrees of freedom, less memory is required. In case of a heterogeneous microstructure fine scale computation is faster than adaptive computation. Since aggregates are distributed in the whole domain, the complete microstructure is heterogeneous. The highest deformation gradients develop at the interface of aggregates and matrix. Hence, the mesh is refined in all these interfaces which leads to a huge number of degrees of freedom- even using the adaptive procedure. In combination with several refinement steps, which are required, computation time is extended. Despite the lack of efficiency, this example demonstrates that the heterogeneous microstructure is recognised and resolved by the refinement procedure and especially the level 2 elements. In this dissertation, parallelisation is not implemented for computing level 2 stiffness matrices. The materialwise nonlinear examples also show that results obtained with the adaptive simulation are in good agreement with the numerical reference solution. The adaptive procedure resolves the heterogeneous microstructure as expected and a developing crack is directed around the stronger aggregates. In the heterogeneous case computation time can be reduced by about 59% with the adaptive simulation compared to the fine scale computation. A reduction by 89% is achieved in case of a homogeneous material. The method developed is especially powerful in case localisation takes only place in a small region and continuum elements (level 1) can be used in large areas. Compared to the fine scale problem, many degrees of freedoms can be omitted. An efficient computation of large structures is possible without fully resolving the microstructure for a representation of phenomena on a microstructural level.#### BibTeX

- Echter, R. (2013).
*Isogeometric analysis of shells*. Doktorarbeit, Bericht Nr. 59, Institut für Baustatik und Baudynamik, Universität Stuttgart,. https://doi.org/10.18419/opus-510#### Abstract

The present work addresses the development of a hierarchic family of shell models and accompanying discretization schemes with NURBS (Non-Uniform Rational B-Splines) functions that are suitable for the analysis of both thick and thin shell structures. The hierarchy in the shell mechanics is based on a minimalistic 3-parameter formulation, which mechanically corresponds to the shear-rigid Kirchhoff-Love shell model. It is particularly suitable for modeling thin structures with predominantly membrane and bending action. Transverse shear effects and extensibility of the shell in thickness direction are not accounted for. The assumed linear kinematics of the thin shell can be described with three independent parameters, which correspond to the mid-surface displacement components of a material point. No rotations are defined as additional degrees of freedom such that the formulation is rotation-free. Linear-elastic and isotropic material properties are assumed. For asymptotic correctness of the model the constitutive law is modified by implementing the stress condition sigma_33 = 0 to eliminate epsilon_33 via static condensation. Additionally, Love’s first approximation, neglecting contributions with regard to curvature in thickness direction of the shell is not considered. Consequently, membrane and bending action are coupled due to nonzero off-diagonal blocks in the constitutive matrix. No pre-integration of the material law is performed. The static and kinematic variables of the shell equations are therefore stresses and strains. With increasing thickness of the structure, transverse shear effects become more pronounced, thus significantly contributing to the total strain energy of the system. For the Reissner-Mindlin shell model developed in this work, the Kirchhoff-Love assumption is relaxed by introducing additional parameters, which do not depend on the gradient of the mid-surface displacement field and thus allow for extra transverse shear effects. These parameters are introduced via a hierarchic difference vector. The inextensibility of the director in the deformed configuration reduces the number of additionally required parameters to two. In the geometrically linear case, the inextensibility constraint is con- structed by expressing the components of difference vector with respect to the in-plane base vectors of the reference shell mid-surface. In order to account for changes in thickness direction additionally, a 7-parameter shell formulation is derived, which represents an extension of the Reissner-Mindlin-type model with five parameters. The 7-parameter shell model incorporates extensibility of the director in thickness direction and enables the application of three-dimensional constitutive laws without the need of modifications. The main innovation of this thesis is based on the hierarchic parametrization of the family of 3-, 5- and 7-parameter shell formulations, which results in significant benefits both with regard to model-adaptivity and finite element technology. The common approach in FEA consists of adding a difference vector on the director of the undeformed configuration. As a result, continuity requirements on the applied function spaces are reduced. It represents the first Reissner-Mindlin-type shell formulation to be used in this work. Equal-order interpolation of both the mid-surface displacement field and the difference vector, however, results in transverse shear locking which is verified in several numerical plate bending experiments. Alternatively, a hierarchic parametrization is derived for the Reissner-Mindlin model that imposes a shear vector on the rotated director of the 3-parameter Kirchhoff-Love formulation. The procedure of adding the extra parameters is defined so that the kinematic equations of the basic Kirchhoff-Love model are gradually enhanced to obtain the shear flexible 5-parameter model, without the need of a complete new description of the shell kinematics. Although exactly representing the same shell model, with regard to finite element discretization, the hierarchic parametrization of the current director avoids transverse shear locking already in a pure displacement formulation. The ansatz to split the total deformation of the Reissner-Mindlin shell model into independent components related to bending and shear in principal follows Basar and Krätzig (1985) and was used in a similar way for shear-deformable subdivision-based shell finite elements in Long et al. (2012). In Basar and Krätzig (1985) the authors applied the decomposition of the rotation of the shell director into the contribution of the deformed shell normal (Kirchhoff-Love) and rotations related to shear. This allows to derive Kirchhoff-Love theories from shear deformation formulations by simply removing the transverse shear contribution. For FEA, this approach, moreover, offers the significant advantage of an independent parametrization of the transverse shear and consequently avoids incompatibilities of the discrete function spaces in the corresponding kinematic equations. For a decreasing shell thickness the solution asymptotically converges to the Kirchhoff-Love solution, whereas removal of the shear vector directly recovers the 3-parameter Kirchhoff-Love model. The hierarchic 7-parameter shell formulation represents an extension of the hierarchic 5-parameter Reissner-Mindlin shell model. In order to account for extensibility of the director and linear transverse normal strains, the kinematics of the Reissner-Mindlin formulation is enriched with a 6th and 7th displacement parameter, which ultimately yields linear and quadratic displacement contributions across the thickness. By switching off the linear and quadratic displacement contributions in thickness direction, the kinematics of the hierarchic 5-parameter Reissner-Mindlin model can be obtained, whereas further elimination of the difference vector yields the kinematic equations of the 3-parameter Kirchhoff-Love shell. This consequently allows for a straightforward combination of these three element types within one mesh and thus serves as an ideal basis for a model adaptive approach. Numerical experiments in this thesis demonstrate that besides transverse shear locking also curvature thickness locking is by default avoided in pure displacement-based 7-parameter shell finite elements due to the concept of a hierarchic parametrization. The non-hierarchic 7-parameter shell elements with pure displacement formulation and difference vector to be imposed on the director of the undeformed configuration are sensitive to both transverse shear and curvature thickness locking on the contrary. The continuity requirements on the displacement functions for the proposed hierarchic 5- and 7-parameter shell models are identical to those of the 3-parameter Kirchhoff-Love formulation, i.e. C1. The demand for shape functions with square integrable partial derivatives of order two, however, can be naturally satisfied with the higher-continuity NURBS discretizations used in this work. NURBS which represent the standard functions of computer-aided engineering design are applied as shape functions in an isoparametric finite element concept following the isogeometric method of Hughes and coworkers (Hughes et al. (2005), Cottrell et al. (2009)). Their higher continuity property additionally enables a pointwise exact definition of the shell director in the entire patch domain. The effect of higher-order and higher-continuity NURBS discretizations on the accuracy of the discrete solution functions is investigated and analyzed in several numerical experiments. Computational results of higher-continuity NURBS are provided to demonstrate the superior accuracy compared to C0-continuous discretizations. Additionally, analysis of the most prominent locking effects that may show up for the displacement-based isogeometric shell finite elements reveals that the in-plane part of all shell elements developed in this thesis is, in general, considerably prone to locking. Therefore, two new strategies to remove geometric locking effects from higher-order and higher-continuity NURBS discretizations were developed and applied to the membrane part of the shell elements to cure locking: First, the DSG approach of Bletzinger et al. (2000) was successfully transferred to higher-order and higher-continuity NURBS discretizations in order to remove membrane and in-plane shear locking. Second, a mixed displacement-stress formulation which is based on a two-field Hellinger-Reissner variational principle with independent displacement and stress fields is applied to the in-plane strain components of the shell elements. The modified isogeometric Kirchhoff-Love and hierarchic 5- and 7-parameter shell formulations are completely free from geometric locking. Higher-continuity NURBS shape functions to be used for the discretization of the displacement fields in general result in continuous strain and stress distributions which in the case of the NURBS-DSG method may result in a coupling of degrees of freedom that compromises computational efficiency. In several benchmark problems the performance of the newly developed hierarchic shell elements is proven. For the displacement-based element formulations the numerical results conform well with the results from literature like for example from Kiendl et al. (2009). Modification of the membrane part with the mixed displacement-stress ansatz successfully removes locking and leads to significantly faster convergence of the investigated results to the reference solutions. For multipatch analysis, the penalty-type bending strip method of Kiendl et al. (2010) is used to connect NURBS surface patches with slope continuity in a weak sense. Appropriate stiffness parameters for the bending strips are defined according to Kiendl (2011). The isogeometric analysis of highly-curved respectively thick shell structures reveals significant differences in the system response for the three different shell models (3-, 5- and 7-parameter). Simultaneously, a fast diminution of the influence of both transverse shear and higher-order mechanical effects on the investigated displacement results can be observed. For model adaptivity, analysis of the same problem setup is performed with hierarchic 5-parameter Reissner-Mindlin shell elements by systematically deacti- vating those degrees of freedom related to the shear vector. The computational results obtained, perfectly match the 3-parameter Kirchhoff-Love solution.#### BibTeX

- Irslinger, J. (2013).
*Mechanische Grundlagen und Numerik dreidimensionaler Schalenelemente*. Doktorarbeit, Bericht Nr. 61, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-511#### Abstract

The present thesis covers several significant aspects of the simulation of shell structures. First, the mechanical fundamentals of shell theories are discussed with the primary objective to establish a link between the assumptions of three-dimensional finite shell element formulations and the shell theories they are based on. Subsequently, the formulation of a robust three-dimensional solid-like shell element is presented. Finally, it is analysed, to which extent several approaches for the setup of preconditioners based on substitute problems are suitable for an efficient iterative solution of ill-conditioned equation systems arising from shell simulations. The discussion of model assumptions of classical shell theories is the first important topic of this work. Therefore, the consistent kinematic and constitutive equations of a shear-deformable shell formulation (Reissner-Mindlin kinematics) and a shear-rigid shell formulation (Kirchhoff-Love kinematics) are derived from the continuum in a first step. Kinematic assumptions and modification of the material law, which is mandatory for the asymptotic correctness of those theories, are discussed in detail. The shell theories of Love and Koiter as well as the theory of shallow shells, which can be considered historic from today’s perspective, are presented and compared with regard to the kinematic and constitutive equations. Additionally, numerical analyses of the properties of these formulations based on the Method of Manufactured Solutions are performed (see for instance Roache (2002)). It turns out, that the shell theory of Love and the theory of shallow shells are not asymptotically correct. Although this is true for the formulation of Koiter, so-called „best formulations“, as for example the one of Naghdi (1963b), often show a faster convergence and are therefore asymptotically better. In contrast to classical shell theories, three-dimensional shell theories are able to represent complete three-dimensional strain and stress states, and thus modifications of the material law can be omitted. Shell finite elements discretising these kinds of theories are nowadays usually derived on the basis of the concept of degeneration. Alternatively, standard continuum elements are optimised for shell simulations by additional element technology. Therefore, many of the assumptions made in element formulations lack the link to shell theories. Hence, a further essential objective of this work is to identify these relations in order to figure out the influence of those assumptions and thus to gain knowledge for the design of three-dimensional shell element formulations. For a 7-parameter shell kinematics it can be shown in numerical experiments, that if the accuracy of classical shell formulations shall be increased beyond the inclusion of thickness change, accounting for the quadratic strain components is crucial. Only in this case the incorporation of curvature in the material equations and in the shifter has a positive effect on the accuracy. To ensure that this additional information enters the shell formulation, it must be detected by the numerical thickness integration, which in this thesis is considered as part of the shell theory and not of an element formulation. It becomes apparent, that although a 2-point Gauss quadrature is sufficient for asymptotic correctness, this additional information is mainly lost. Only with three or more Gauss points across the thickness asymptotically better shell theories are obtained. Special focus of this work is on the development and implementation-related documentation of a robust three-dimensional solid-like shell element, whose topology is identical to a trilinear 8-node continuum element. Rank-sufficient integration substantially contributes to the robustness of the element formulation and precludes numerical instabilities (hourglassing). The major locking effects are eliminated by assumed natural strain approaches (Hughes und Tezduyar (1981), Bathe und Dvorkin (1986), Betsch und Stein (1995)) and the enhanced assumed strain method (Simo und Rifai 1990). The element is not formulated in its natural coordinate system, but in accordance with Sze und Yao (2000) in an alternative convective system. This considerably improves the performance in configurations, where the element edges in thickness direction are not orthogonal to the fictitious mid-surface. Furthermore, a method to stabilise the transverse shear first proposed by Lyly u.a. (1993) is used to optimise the performance of elements which are distorted in-plane. An adaptive approach is introduced, that determines the size of the stabilisation parameter for each element depending on its geometry. In geometrically linear and nonlinear benchmark problems from literature as well as practical problems from aircraft industry the performance of the presented element formulation is analysed and compared with existing elements of the same topology. With respect to the quality of both displacements and stresses, the new element formulation is superior in many cases and is able to provide reliable results already with relatively coarse finite element meshes. Especially the stabilisation of the transverse shear for in-plane distorted elements improves the quality of the transverse shear strains and stresses. The iterative solution of the ill-conditioned equation systems arising from the finite element simulation of shell structures represents a particular challenge. Besides elaborate multigrid methods, up to now no preconditioners are known, which enable an efficient iterative solution of such systems for arbitrary configurations. It is analysed to which extend approaches for setting up preconditioners according to Shklarski und Toledo (2008) and Avron u.a. (2009) can be adapted to finite element simulations of shell structures. Therein, substitute problems are created, which have to be fully factorised for preconditioning. This approach is in contrast to an incomplete factorisation of the initial problem, which many popular preconditioners are based on. In the first approach presented herein, the substitute problem is created by specific elimination of finite elements from the initial discretisation, taking into account that no singular systems arise. In the second approach, the stiffness matrix is first approximated by a symmetric diagonally dominant matrix, which is simplified by means of combinatorial procedures such that a similar matrix evolves, for which factorisation is cheaper. The algorithms necessary for both approaches are developed on the basis of the 8-node solid-like shell element presented in this thesis and the resulting preconditioners are analysed for problems of varying complexity. It turns out that the preconditioners based on modification of the initial discretisation have a positive impact on the convergence characteristics of the iterative solution process with respect to monotony. However, the computational effort with all presented preconditioners was in most cases significantly larger than with well-established preconditioners from literature. This approach therefore leads to no substantial increase of efficiency for the iterative solution of equation systems arising from the simulation of shell structures.#### BibTeX

- Tkachuk, A. (2013).
*Variational methods for consistent singular and scaled mass matrices*. Doktorarbeit, Bericht Nr. 60, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-517#### Abstract

Singular and selectively-scaled mass matricesare useful for finite element modeling of numerous problems of structural dynamics, for example for low velocity impact, deep drawing and drop test simulations. Singular mass matrices allow significant reduction of spurious temporal oscillations of contact pressure. The application of selective mass scaling in the context of explicit dynamics reduces the computational costs without substantial loss in accuracy. Known methods for singular and selectively-scaled mass matrices rely on special quadrature rules or algebraic manipulations applied on the standard mass matrices. This thesis is dedicated to variationally rigorous derivation and analysis of these alternative matrices. The theoretical basis of this thesis is a novel parametric HAMILTON’s principle with independent variables for displacement, velocity and momentum. The numerical basis is hybrid-mixed discretization of the novel mixed principle and skillful tuning of ansatz spaces and free parameters. The qualities of novel mass matrices are thoroughly analyzed by various tests and benchmarks. The thesis has three main parts. In the first part of the thesis, the essential fundamentals and notations are introduced. This includes the basic continuum mechanics, the local form of an initial boundary value problem for an elasto-dynamic contact problem and its treatment with finite elements. In addition, an extension of the central difference method to non-diagonal mass matrices and a theoretical estimate of speed-up with selective mass scaling is given. Besides, a motivation for implementation of alternative mass matrices is given. In the second part of the thesis, the novel variational approach for elasto-dynamic problems is presented. The corner stone of the thesis is the derivation of the novel penalized HAMILTON’s principle and an extension of the modified HAMILTON’s principle for small sliding unilateral contact. These formulations are discretized in space with the BUBNOV-GALERKIN approach. As a result, families of singular and selectively-scaled mass matrices are obtained. The correspondingshape functions are builtfor several families of finite elements. These families include truss and TIMOSHENKO beam elements for one-dimensional problems, as well as solid elements for two and three dimensions. Shape functions for singular mass matrices are derived for quadratic and cubic elements. Selectively-scaled mass matrices are given for elements up to the order three. In the third part of the thesis, the novel mass matrices are analyzed and an outlook for future work is given. Propagation of harmonic waves, free and forced vibrations and impact problems are used for evaluation of the new mass matrices. First, the propagation of harmonic waves is studied with the help of a FOURIER analysis applied to the semi-discretized equation of motion. This analysis results in a set of dispersion relations. Comparison of the analytical expressions for discrete dispersion relations with the corresponding continuous ones allows efficient error estimation. In this way, the proposed truss and beam elements are analyzed. Secondly, eigenvalue problems are solved for two-and three-dimensional problems. The error in the lowest frequencies (modes) and in the whole spectrum is computed. Thirdly, spectral response curves for forced vibrations are obtained for the new mass matrices in ranges of interest. These curves are compared with the ones obtained with consistent mass matrices via the frequency response assurance criterion. The values of the frequency response assurance criterion indicate the error for linear problems. Finally, several transient examples are solved with singular and scaled mass matrices. These examples confirm expected superiority of singular mass matrices for impact problems, i.e. spurious temporal oscillations of contact pressures are significantly reduced. Variational selective mass scaling reduces computational cost of explicit dynamic simulations. In the outlook, possible developments regarding new element types, alternative weak forms and several multi-physic applications are proposed. As by-product of this thesis, patch tests for inertia terms, an overview of parametric and non-parametric variational principles of elasto-dynamics and a derivation of the penalized HAMILTON’s principle with a semi-inverse method can be noted. Besides, the topic of finite element technology for mass matrices is posed. This can open new horizons for evolving branches of computational dynamics such as drop test and car crash simulations, phononic crystals and devices.#### BibTeX

- Cichosz, T. (2012).
*Stabile und konsistente Kontaktmodellierung in Raum und Zeit*. Doktorarbeit, Bericht Nr. 58, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-495#### Abstract

Die vorliegende Arbeit befasst sich mit verschiedenen Aspekten der Diskretisierung von Kontaktvorgängen in Raum und Zeit. Im Hinblick auf eine stabile und konsistente Modellierung werden bestehende Verfahren verglichen und Verbesserungen erarbeitet. Schwerpunkt der räumlichen Untersuchungen ist die Weiterentwicklung der in Hartmann u. a. (2007) und Hartmann und Ramm (2008) vorgestellten Kontaktdiskretisierung, die auf der dualen Mortar-Methode (Wohlmuth 2000, 2001) basiert. Durch die Verwendung der Methode der Lagrange’schen Multiplikatoren erfüllt diese Formulierung die Nichtdurchdringungsbedingung exakt. Gleichzeitig erlaubt die Diskretisierung der Multiplikatoren mit dualen Formfunktionen die einfache Kondensation der zusätzlichen Unbekannten aus dem resultierenden Gleichungssystem. Somit wird der übliche Nachteil der Methode der Lagrange’schen Multiplikatoren vermieden. Mit der herkömmlichen Definition der dualen Formfunktionen können am Rand des Kontaktbereichs inkonsistente Mortar-Matrizen entstehen. Als Folge dessen resultieren unphysikalische Werte für die Knotenklaffung und fehlerhaft übertragene Kontaktkräfte. Zur Korrektur dieses Verhaltens wird in dieser Arbeit eine modifizierte Definition der Mortar-Matrizen vorgeschlagen. Damit die Modifikation nicht die Konditionierung des resultierenden Gleichungssystems verschlechtert, wird zusätzlich eine Wichtungsprozedur für die modifizierten Mortar-Matrizen vorgestellt. Als Ergebnis ist in allen Fällen eine konsistente Übertragung der Kontaktkraft und eine konsistente Berechnung der Normalklaffung möglich, ohne dabei die Konditionierung zu beeinträchtigen. Die Betrachtungen zur zeitlichen Diskretisierung analysieren zunächst den Einfluss von Kontaktereignissen auf die Eigenschaften der dynamischen Strukturantwort. Beruhend auf den gewonnenen Erkenntnissen wird anschließend eine möglichst optimale zeitliche Kontaktdiskretisierung formuliert. Diese ist mit einer Strategie nach Kane u. a. (1999) energetisch stabil. Durch die Erweiterung einer Idee von Deuflhard u. a. (2008) auf Probleme mit großen Deformationen werden Oszillationen in der Kontaktkraft vermieden. Die Modifikation der Geschwindigkeit in einer Nachlaufrechnung stellt physikalisch sinnvolle Kontaktgeschwindigkeiten sicher. Darüber hinaus wird der Kontakt energieerhaltend modelliert, ohne die Nichtdurchdringungsbedingung zu verletzen. Hierzu kommt das Energie-Korrekturkraft-Verfahren zum Einsatz, das eine im Rahmen der vorliegenden Arbeit formulierte Weiterentwicklung des Konzepts von Armero und Petocz (1998) darstellt. Außer mit dem präsentierten Verfahren ist eine energieerhaltende Kontaktbehandlung bei gleichzeitiger exakter Erfüllung der Nichtdurchdringungsbedingung nur mit der „Velocity-Update-Method“ (Laursen und Love 2002) möglich. Im Gegensatz zu dieser gibt das Energie-Korrekturkraft-Verfahren die dissipierte Energie jedoch nicht ausschließlich in kinetischer Form zurück. Stattdessen bestimmt die Systemantwort, wie die Energie-Korrekturkraft die Gesamtenergie vergrößert. Anhand von numerischen Experimenten werden abschließend die untersuchten Verfahren bewertet. Zusätzlich wird die Leistungsfähigkeit der entwickelten Methoden demonstriert.#### BibTeX

- Etlender, R. (2012).
*Modellierung und Simulation der Wellenausbreitung in flexiblen hydraulischen Leitungen*. Doktorarbeit, Bericht Nr. 57, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-455#### Abstract

Gegenstand der vorliegenden Arbeit ist die Entwicklung ein es physikalischen Modells sowie einer passenden numerischen Berechnungsmethode zur Vorhersage der Wellenbewegungen innerhalb von flexiblen hydraulischen Leitungen. Ziel ist es ein Rechenmodell zu konstruieren, welches innerhalb von Simulationen komplexer hydraulischer Systeme eingesetzt werden kann. Aus diesem Grund liegt besonderer Fokus der Untersuchungen auf Effizienz des Modells bezüglich der Rechenzeiten. Das physikalische Modell basiert auf Erhaltungsgleichung en für das Fluid unter Hinzunahme der Bewegungsgleichungen für die Leitungswand. Für die entstehende quasi-zweidimensionale Formulierung werden mehrere numerische Schemata zur Berechnung entwickelt. Die Diskretisierung der Erhaltungsgleichungen basiert auf einer Godunov-Typ Methode zweiter Ordnung, wobei, ausgehend von unterschiedlichen Formulierungen der Erhaltungsgleichungen verschiedene numerische Schemata erarbeitet und gegen übergestellt werden. Für die Einbindung eines Berechnungsschemas für die Wandbewegung in das Strömungsberechnungsschema werden eine sequentielle, eine iterativ-gestaffelte und eine direkte Kopplungsmethode konstruiert. Zur Abbildung der Wandbewegung werden sowohl einfache Modelle betrachtet, deren Beziehungen zwischen dem Zustand in der Leitung und der momentanen Dehnung durch algebraische Abhängigkeiten ausgedrückt werden, wie auch Modelle, die die Schwingungen der Leitung in radiale und axiale Richtungen berücksichtigen. Für die entwickelten numerische Schemata wird eine Randbehandlung vorgestellt, welche es ermöglicht, das konstruierte Simulationsmodell an weitere Berechnungsmodelle hydraulischer Elemente zu koppeln, unabhängig von numerischen Methoden, die innerhalb dieser Modelle ein gesetzt werden. Hierdurch wird eine Möglichkeit für den Einsatz des Berechnungsverfahren innerhalb von Systemsimulationen geschaffen. Erhaltungsbedingungen bleiben hierbei bestehen. Eine Implementierung des Modells innerhalb einer Systemsimulationsumgebung wird zur Verifizierung- und Validierungszwecken verwendet. Bei der Verifizierung werden sowohl theoretische Aufgaben gerechnet wie auch die Simulation eines für die Vermessung der Leitungen konstruierten Prüfstands durchgeführt. Zusätzlich zum Abgleich am Prüfstand wird das Modell innerhalb von Berechnungen des Verhaltens einer vorher vermessenen Hochdruckpumpe eingesetzt.#### BibTeX

- Schneider, B. J. (2012).
*Polygonale diskrete Elemente zur Modellierung heterogener Materialien*. Doktorarbeit, Bericht Nr. 56, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-461#### Abstract

The present work addresses the modeling of the mechanical behavior of heterogeneous materials that are composed of solid particles on the microscopic or a coarser scale. To reproduce phenomena emerging on this scale, the heterogeneous material constitution out of particles is described in an explicit way. For this purpose a discrete element method is used, which models materials by separated, interacting particles. The introduced method employs rigid polygonal particles, a soft modeling of the particle interactions by contact and adhering bond as well as an explicit solution method in time. The aim of this work is, on the one hand, to enhance the models for the interaction between the particles. Since the numerical expense of the method is often high, the models shall depict the important mechanical characteristics of the interactions having acceptable numerical costs. On the other hand, the capability of the developed method is to be analyzed. For the contact of the particles an elastic normal force model is adopted to represent the repulsive force. It is supplemented by a viscous model. In tangential direction sticking as well as sliding is described by an elasto-plastic contact model. Additionally, sticking and sliding due to friction between a particle and a plane background is modeled by a new, also elasto-plastic contact model. For the adhering bond between the particles an elasto-damage beam model is presented, which is able to reproduce different types of bond failure having acceptable numerical costs. This beam model is extended by a power-law model for the rate-dependent strength so that an increase of the bond strength can be depicted for an increase of the strain rate. In order to analyze the capability of the developed method in a direct way, conceptual experiments are carried out. Those are, firstly, compression experiments on a granular model material. Using the contact models, characteristic phenomena like shear bands can be described in the simulations. Secondly, compression experiments on a model material out of glued particles are performed. It is demonstrated in the simulations that essential properties of the failure behavior can be represented using the beam model for the adhering bond. Finally, the method is used to simulate concrete. Typical properties of the concrete failure like the localization of the deformation and the failure pattern can be reproduced qualitatively. Using the model for the rate-dependent beam strength, an increase of the specimen strength can be shown for an increase of the loading velocity.#### BibTeX

- Bruss, I. (2011).
*Dreidimensionale Schädigungsmodellierung heterogener Materialien*. Doktorarbeit, Bericht Nr. 55, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-367#### Abstract

This thesis deals with three-dimensional modeling of heterogeneous cohesive materials. The investigations focus exemplarily on concrete and fiber-reinforced concrete as composite materials, which exhibit highly non-linear material behavior. Micro-mechanical failure phenomena like delamination of fibers in a matrix as well as microcracks and micropores in concrete result in narrow failure zones where strains localize and energy dissipation occurs. The elastic material stiffness is reduced due to localized damage and discrete evolving cracks. The width of the failure process zone is typically several orders of magnitude smaller than the structural dimensions, which implies displacements of multiple scales. Resolving these displacements and the high gradients in the solution functions requires either a very fine discretization or an enhancement of the approximation space. The main part of this study concentrates on a detailed numerical analysis of delamination in fiber-reinforced composites and crack evolution in concrete by means of the extended finite element method. In the second part a two-scale method is proposed that incorporates the effect of fine scale failure phenomena in the macroscopic structural behavior.#### BibTeX

- Hilchenbach, C. F. (2010).
*Optimierung von Mehrphasenstrukturen mit lokalem Schädigungsverhalten*. Doktorarbeit, Bericht Nr. 54, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-357#### Abstract

The present work addresses two-dimensional structures with stiff inclusions embedded in a soft matrix. An isotropic damage model has been chosen for the interface and matrix softening failure whereas inclusions have been assumed to be elastic for the time being. Interface and matrix failure are described by an eXtended Finite Element Method (X-FEM) applying a cohesive zone concept and levelsets for the description of the inclusion geometry as well as the enrichment functions. After reaching a crack initiation criterion the cracks propagate element by element. To enable the integration of finite elements with local discontinuities a triangulation is applied that incorporates cracks and interface contours. The aim of this work is maximizing the overall structural ductility by variation of the local geometrical layout of the inclusions for a prescribed inclusion mass. The ductility is defined as the integral of the strain energies over a specified displacement range. In the gradient based structural optimization process shape and location parameters of the inclusions are taken as design variables. If the inclusions are specified as ellipses the design variables are the semi-axes, the angle and the coordinates of the ellipses center. The variation of these parameters is restricted by a prescribed minimum distance to the boundaries of the design space and a minimum distance of the inclusions with respect to each other. Although the geometry of the inclusions in the matrix is permanently changing during optimization, a fixed structural mesh is used to avoid continuous adaptation to a material conforming mesh. The Optimality Criteria (OC) method or the Method of Moving Asymptotes (MMA) is used as optimization algorithm. Necessary gradients with respect to a variation of the optimization parameters are determined by an analytical sensitivity analysis. Using gradients of the objective function "maximization of ductility" and of the constraint "constant inclusion mass" the design of the inclusions is changed automatically in an iterative process. The calculation of sensitivities is an important issue of this work. In order to derive the sensitivities of the objective "maximization of ductility" the gradients of the state variables, i.e. the nodal displacements, have to be determined in advance. This is necessary because of the implicit dependence of the displacement field on the optimization variables. In order to determine the gradients of the nodal displacements, the derivatives of the equilibrium equations with respect to the optimization variables are computed. Here, the influence of the material interfaces, the related enrichment functions of the X-FEM, as well as the kinematic and constitutive relations have to be taken into account. The potential of the gradient based structural optimization with repect to the maximization of the overall structural ductility is demonstrated by various numerical examples.#### BibTeX

- Kato, J. (2010).
*Material Optimization for Fiber Reinforced Composites applying a Damage Formulation*. Doktorarbeit, Bericht Nr. 53, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-333#### Abstract

The present thesis proposes material optimization schemes for fiber reinforced composites, specifically for a new composite material, denoted as Fiber Reinforced Concrete (FRC) or Textile Reinforced Concrete (TRC); here a reinforcement mesh of long carbon or glass fibers is embedded in a fine grained concrete (mortar) matrix. Unlike conventional steel reinforcement, these textile fibers are corrosion free; this holds also for AR-glass due to its high alkali-proof. This favorable property allows to manufacture light-weight thinwalled composite structures. However the critical aspect of this composite is that the structural response of FRC may show brittle failure due to the material brittleness of both constituents concrete and fiber in addition to their complex interfacial behavior. This specific characteristic of FRC is an ideal target for material optimization applying the overall structural ductility as objective which ought to be maximized for a prescribed fiber volume. For this objective it is of course not sufficient to base the optimization process on a linear elastic material model, so that it is mandatory to consider material nonlinearities. In the present study a gradient enhanced isotropic damage model is applied for both matrix and fiber materials and a discrete bond model is used for their interface. The structural response of FRC depends on several parameters, e.g. fiber size, fiber length, fiber location/orientation, impregnation, surface roughness of fiber, and the kind of fiber material itself. From these the most influential parameters like fiber dimensions and locations are chosen as design variables for optimization. Conventional material optimization applying simply ‘smeared-type elements’ mostly concentrate on the fiber orientation defined at each finite element. This approach is not detailed enough when the influence of other important parameters mentioned above ought to be investigated. Considering the design requirements for the present objective, this thesis proposes three kinds of material optimization schemes, namely multiphase material optimization, material shape optimization, and multiphase layout optimization. Multiphase material optimization determines an optimal distribution of several materials over a prescribed design domain in a fixed finite element mesh. This methodology is related to topology optimization, especially to the Solid Isotropic Microstructure with Penalization (SIMP) approach. With this method optimal fiber size, fiber length, and combination of different fiber materials can be obtained. The task of material shape optimization is to improve the structural ductility of FRC with respect to ‘fiber geometry’ which is independent of the fixed finite element mesh. By applying a so-called embedded finite element formulation, the complexity of discretization for thin fibers in a conventional finite element formulation is diminished. Multiphase layout optimization provides not only optimal fiber geometry but also optimal fiber size or the kind of fiber materials simultaneously. This methodology is achieved by combining above multiphase material and material shape optimization. For the optimization problems a gradient-based optimization scheme is assumed. An optimality criteria method and a method of moving asymptotes are applied considering their numerical high efficiency and robustness. For the sensitivity analyses variational direct analytical/semi-analytical methods are utilized. The performance of the proposed methods is demonstrated by a series of numerical examples; it is verified that the ductility of FRC can be substantially improved. The proposed methods providing optimal designs are promising and methodically challenging. They are also applicable to other fiber reinforced composites, for example Fiber Reinforced Glass (FRG).#### BibTeX

- von Scheven, M. (2009).
*Effiziente Algorithmen für die Fluid-Struktur-Wechselwirkung*. Doktorarbeit, Bericht Nr. 52, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-306#### Abstract

Coupled problems, like fluid-structure interaction, arise in various areas of engineering. Often they are characterized by high complexity. Especially for three-dimensional problems this results in very long simulation times. In the present work partitioned solution schemes are used for fluid-structure interaction of thin-walled structures and incompressible fluids. The structural domain is governed by the nonlinear elastodynamic equations while the fluid dynamics is described by the incompressible Navier-Stokes equations. Both fields are discretized by finite elements in space and finite difference methods in time. The focus of this work is the development of efficient algorithms for the solution of complex, three-dimensional fluid-structure interaction problems. In the first part of this work the efficiency of the single fields is examined while the second part focuses is on the coupling. For mesh generation in large domains a two level procedure is introduced. A coarse mesh provided by the preprocessor is refined on the high performance computer taking into account the exact geometry. The calculation of element matrices of a finite element code for unstructured meshes can be accelerated on a vector computer by grouping the elements and restructuring the code making use of the advantages of vector processors. In addition, for the solution of the linear system of equations the necessary cpu time can be reduced by the correct choice of iteration scheme and preconditioner. The influence of different parameters on the efficiency of the solver is studied for a fluid simulation. The simulation of interaction between incompressible flows and thin-walled structures often requires implicit i.e. iterative staggered coupling algorithms. Especially for strongly coupled problems the additional iteration leads to very long simulation times. In the present work commonly used iterative staggered coupling schemes are presented in a consistent notation. Using the solution of the coupled problem on a coarse discretization as a predictor the coupling iteration can be accelerated. Finally, convergence rate and calculation time of these new two-level coupling algorithms are compared with other iterative staggered schemes. Numerical examples for the coupling of thin-walled shell structures and three-dimensional flows indicate possible applications of the presented efficient algorithms.#### BibTeX

- Förster, C. (2007).
*Robust methods for fluid-structure interaction with stabilised finite elements*. Doktorarbeit, Bericht Nr. 51, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-270#### Abstract

Various multifield problems and among them fluid-structure interaction applications arise in nearly all fields of engineering. The present work contributes to the development of a stable and robust approach for the numerical simulation of fluid-structure interaction problems. In particular two-dimensional and three-dimensional elastic structues interacting with incompressible flow are considered. The structural field is governed by the nonlinear elastodynamic equations while the dynamics of the fluid field are described by the incompressible Navier-Stokes equations. Both fields are discretised by finite elements in space and finite difference methods in time. An iteratively staggered partitioned coupling procedure with relaxation is applied to obtain the overall coupled solution. This work focuses on methodological aspects and contributes to a deeper understanding of the theoretical foundations of the approach. This is necessary to ensure that the formulation is stable and offers reliable results for a wide range of parameters. In particular the flow solver formulated in an arbitrary Lagrangean Eulerian approach is considered. In addition to the classical conservation laws of mass, linear momentum and energy geometric conservation has to be considered. This is a consequence of the formulation of the flow equations with respect to a moving frame of reference. The relationship of these conservation laws and the stability of the numerical scheme is investigated and stability limits in terms of maximal time step sizes for different formulations are established. It is further shown how an unconditionally stable ALE formulation has to be constructed. Another key issue is the stabilised finite element method employed on the fluid domain. The derivation of the method from a virtual bubble approach is revisited while special attention is turned to the fact that the domain is moving. A version of the stabilisation is derived which is nearly unaffected by the motion of the frame of reference. Further the sensitivity of the stabilised formulation with respect to critical parameters such as very small time steps, steep gradients and distorted meshes is assessed. At least for higher order elements where full consistency of the formulation is assured very accurate results can be obtained on highly distorted meshes. As another main issue the coupling of fluid and structure within a partitioned scheme is considered. A first concern in this context is the exchange of proper coupling data at the interface which is crucial for the consistency of the overall scheme. Subsequently the so-called artificial added mass effect is analysed. This effect is responsible for an inherent instability of sequentially staggered coupling schemes applied to the coupling of lightweight structures and incompressible flow. It is essentially the influence of the incompressibilty which excludes the successful use of simple staggered schemes. The analysis derived in the course of this work reveals why the artificial added mass instability depends upon the mass ratio but further on the specific time discretisation used on the fluid and structural field. In particular it is shown why more accurate temporal discretisation results in an earlier onset of the instability. While the theoretical considerations are accompanied by small numerical examples highlighting particular aspects some larger applications of the method are finally presented.#### BibTeX

- Hartmann, S. (2007).
*Kontaktanalyse dünnwandiger Strukturen bei großen Deformationen*. Doktorarbeit, Bericht Nr. 49, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-263#### Abstract

The present thesis is concerned with the numerical simulation of contact problems of thin-walled structures using the finite element method. A mortar-based contact formulation is presented and combined with suitable strategies for the discretization in space and time. In view of a useful coupling with the element independent contact description, a trilinear surface oriented hybrid shell element is derived on the basis of the 7-parameter shell model by Büchter and Ramm (1992). Additionally, a trilinear geometric nonlinear brick element based on the principle of Hu-Washizu is devised. Numerical tests demonstrate the performance of both element formulations. For the discretization in time, two implicit time integration algorithms are used. In addition to the existing "Generalized-α"-Method especially the "Generalized-Energy-Momentum-Method" is applied. The latter is proven to be unconditionally stable in all performed numerical analyses. The essential part of this thesis is the extension of the mortar contact formulation presented by Hüeber and Wohlmuth (2005) to the geometrically nonlinear regime. Introducing continuously approximated Lagrange Multipliers, physically representing the contact pressure, the geometric impenetrability condition is formulated in a weak, integral sense. Using dual shape functions (Wohlmuth (2000)) for the interpolation of the Lagrange Multipliers allows for a nodal decoupling of the geometric constraints. The combination with an active set strategy results in an algorithm which allows for elimination of the discrete nodal values of the Lagrange Multipliers. They can be easily recovered from the displacements in a variational consistent way. In contrast to many other formulations, the resulting contact algorithm combines two main advantages: Only the discrete nodal displacements appear as primal unknowns, thereby the size of the system of equations to be solved remains constant; there is no need for any user defined paramters like a penalty parameter. Detailed numerical analyses of dynamic contact problems illustrate the necessity of additional, algorithmic energy-conserving strategies. The "Velocity-Update"-method by Laursen and Love (2002) is characterized by the fact that it guarantees the exact conservation of energy while simultaneously satisfying the geometric impenetrability condition. This method is revised according to the presented contact formulation and generalized for combination with the "Generalized-Energy- Momentum-Method". Numerical examples are investigated to analyze and judge the effectiveness of the proposed solution strategy.#### BibTeX

- Hettich, T. M. (2007).
*Diskontinuierliche Modellierung zur Versagensanalyse von Verbundmaterialien*. Doktorarbeit, Bericht Nr. 50, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-269#### Abstract

This thesis addresses the discontinuous modeling of failure in composite materials. Composite materials are characterized by the interaction of two or more individual materials. In the present thesis such composites are analyzed which in large part consist of cohesive materials. Cohesive materials may be found in nature in form of soils or can be manufactured synthetically like concrete and ceramics. The material behavior of cohesive materials is characterized, among other things, by a failure induced anisotropic degradation of the elastic stiffness properties. At the structural level the anisotropic failure of cohesive materials often appears in the evolution of narrow zones in which deformations localize whereas the rest of the structure mostly unloads. On the basis of an enhanced continuum-mechanical description the behavior in failure of the investigated composites is modeled by the cohesive zone theory. The localization zone is approximated by a singular crack plane which can carry loads due to microscopic mechanisms as long as both crack faces are not completely separated. Since the modeling of the localization zone with a discrete crack implies a discontinuous solution, this kind to model material failure can be named discontinuous. In the present thesis numerical failure analyses of the composites are accomplished on different levels of material observation. Textile-reinforced concrete is analyzed based on a mesoscopic approach, whereas a macroscopic approach is chosen to model steel-reinforced concrete. In the mesoscopic modeling concept of textile-reinforced concrete the interface between both material constituents, the textile fiber and the concrete, has to be considered explicitly. For this reason the discontinuous failure analysis of composite structures on the mesoscopic level demands not only the consideration of discrete cracks but also of material interfaces. A main focus of this thesis is the derivation of a finite element approach to discretize these two discontinuities with different physical meanings. In addition, techniques are discussed to describe the geometry of the discontinuities. The nonlinear, softening material behavior of concrete and material interfaces is mo\-deled by appropriate traction-separation-laws in the context of the cohesive zone theory. In order to accomplish discontinuous failure analyses of steel-reinforced concrete on the macroscopic level, the material structure is homogenized so that it can be passed on the discrete modeling of the steel-reinforcement. The material behavior is described by special constitutive laws of stress-strain- and traction-separation-type. The discretization method developed in this thesis for the simulation of composite structures is included in a hierarchical two-scale concept. The resultant two-scale model affords the realization of efficient failure analyses of macroscopic structures under consideration of mesoscopic effects. Since a domain decomposition is part of the two-scale model, side conditions have to be formulated which are discussed and proven in terms of the ability to exactly model discontinuous failure with multiscale character.#### BibTeX

- Hund, A. S. (2007).
*Hierarchische Mehrskalenmodellierung des Versagens von Werkstoffen mit Mikrostruktur.*Doktorarbeit, Bericht Nr. 48, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-262#### Abstract

The present study is concerned with the analysis of the behavior of structures, made up of micro- heterogeneous materials like composites. Models, which directly resolve the material structure are too expensive, since the characteristic length of material components is much smaller than the characteristic length of the structure. On the other hand, macroscopic models capturing the local material behavior only in phenomenological sense are too inaccurate for a reliable analysis of the structural behavior within the post-failure range. An alternative are methods, which take care of the physical behavior on the scale of material heterogeneities in a detailed manner and include them on the structural scale by means of a multiscale model. The aim of this thesis is the development of an appropriate multiscale concept to solve problems of structural mechanics with multiscale characteristics for softening material behavior. In this connection an efficient and robust solution algorithm is of particular importance. The proposed multiscale method is based on a volume coupled scale transition, which is realized via a hierarchical refinement of the large scale solution. With this volume coupled scale transition, the multiscale model remains valid in the stage of softening, where the differences between scales become small due to localized material behavior. With respect to the efficiency of this method, hierarchical refinement is restricted to regions with nonlinear material behavior. Therefore a strain-based criterion for the adaptive adjustment of the multiscale region is proposed. Additionaly a local carrier is introduced for the fine scale variables, which allows a very efficient solution of the small scale equations. A simultaneous solution algorithm accounts for the strong coupling between the scales. In the context of the locality assumption for the small scale variables, different possibilities to formulate constraint conditions are presented and introduced. Using adequate test cases these possibilities are investigated and compared with respect to their pros and cons within the multiscale method. The potential of the proposed multiscale concept with respect to accuracy and efficiency is examplified by means of various applications.#### BibTeX

- Lipka, A. (2007).
*Verbesserter Materialeinsatz innovativer Werkstoffe durch die Topologieoptimierung*. Doktorarbeit, Bericht Nr. 47, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-271#### Abstract

The present study proposes a method to improve the application of materials in structural elements with the structural optimization method. Structural optimization is not used with its original objective, the determination of the topology and shape of a structure. It is rather used for the developement and application of innovative materials in light-weight structures: porous materials such as metallic and polymeric foams and reinforced composite materials, here the textile reinforced concrete. The material topology optimization is used to control the complex deformation- and failure behavior of these materials. Due to the very special properties of these materials and their resulting applications, several design criteria are investigated which describe the macroscopic behavior of structural elements. Conflicting design criteria are considered by the implication of a multicriteria optimization. Porous materials are usually utilized in topology optimization in order to relax the integer "1-0" (black and white) problem, allowing to identify zones with and without design material. In the present contribution the concept is not used as a mathematical vehicle; rather the existence of a "real" physically existing material is assumed with a varying intermediate density (grey zones); in other words, the porosity is being introduced as a design variable which is then being adjusted by the controlling optimization process. With respect to that it can be referred to "natural" material like the spongiosa in bones and tissues with a varying density. Since the mechanical behavior of porous materials is decisively influenced by the density and the shape and size of the pores, diverse micro- and macroscopic material models have been developed to display the correlation of the density and the mechanical properties. The relations between the density of the porous material and the mechanical properties are used in order to determine the optimal density distribution. The optimization is introduced for linear and non-linear material behavior based on either linear or nonlinear kinematics. Constraints concerning the producebility of optimized artificial cellular materials are discussed. The investigation of the principal behavior of cellular materials with unit cell modells opens the possibility of material design. Design criteria for the layout of the microstructure are special anisotropic macroscopic material properties but also the improvement of the ductile behavior. For the optimization of the layout of the reinforcement in thin walled conrete structures with innovative fiber materials, those fibres are determined by the classical topology optimization and a given principal fibre layout, which is necescarry for a special kind of structural behavior. During this process the nonlinear material behavior of the concrete as well as the reinforcement is taken into account. Due to the large number of design variables in material based topology optimization, mathematically orientated optimality criteria methods combined with variational adjoint methods to determine the sensitivities turn out to be efficient and robust. The results of the variety of investigated optimization problems emphasize the potential of the proposed method for the improved application of innovative materials by material topology optimization.#### BibTeX

- Huber, F. (2006).
*Nichtlineare dreidimensionale Modellierung von Beton- und Stahlbetontragwerken.*Doktorarbeit, Bericht Nr. 46, Institut für Baustatik, Universität Stuttgart. https://doi.org/10.18419/opus-238#### Abstract

This study is concerned with the modeling of plain and reinforced concrete structures including possible three-dimensional effects and nonlinear material behaviour of concrete and reinforcing steel. Based on the discretization of the concrete structure using threedimensional continuum elements the reinforcement is represented by embedded bar elements within the framework of a rebar model. Compared to discrete bar elements or smeared reinforcement layers, embedded bar elements enable a discretization of the concrete structure independent of the configuration of reinforcements. This issue is of great benefit within the discretization of the tested structures. For specific cases, a realistic limit load analysis can only be done including the flexible bond between concrete and reinforcement. In this study, the bond effect is accounted for by a discontinuity between the concrete displacements and the deformations of the reinforcing steel using additional bond degrees of freedom within the rebar model. Thereby an analogy to the Finite Element Method with enhanced kinematic description (XFEM, SDA) can be established. Alternatively, a beam element with multi-part cross section based on the work of Weimar is presented for the computation of simple reinforced concrete beams. The formulation enables to model an entire cross section composed of multiple single rectangular parts. In this context a fully three-dimensional concrete model is reduced to the beam formulation via static condensation of the stress components not considered due to the dimensional reduction process at the material point level. The reinforcement is represented by a smeared model. This formulation is very advanteagous since it allows to avoid or circumvent the need for special one-dimensional constitutive laws for concrete. In this study, the phenomenological description of the material behaviour of concrete is accomplished by two different nonlinear material models. The isotropic, multi-surface plasticity model, proposed by Menrath for two-dimensional structures, is based on plasticity and adopted in a three-dimensional form to model plain concrete. The model is formulated in the first two stress invariants and combined with a scalar damage approach in order to reproduce local unloading actions. Parallel to the multi-surface plasticity model, a damage model proposed by de Vree et al., prior used for plane structural problems, enters in a three-dimensional enhanced form the modeling of plain and reinforced concrete structures. It is based on damage theory and is formulated in the first two strain invariants. Regularization is achieved by Menrath by a mesh-adaptive softening modulus for the multi-surface plasticity model, whereas Peerlings suggests a gradient enhanced description for the damage model. Only the gradient enhancement produces fully mesh-independent results. The material description of the reinforcing steel is accomplished by a one-dimensional hardening plasticity model, the bond description between concrete and the reinforcement by a one-dimensional bond model. The application of the used concrete models in connection with the embedded reinforcement modeling is tested and compared to experimental and numerical results for several plain concrete and reinforced concrete structures. Numerical simulations are accomplished for structures obeying the plane stress condition as well as for structures under a general three-dimensional stress state.#### BibTeX

- Leukart, M. (2005).
*Kombinierte anisotrope Schädigung und Plastizität bei kohäsiven Reibungsmaterialien.*Doktorarbeit, Bericht Nr. 45, Institut für Baustatik, Universität Stuttgart. https://doi.org/10.18419/opus-205#### Abstract

The present thesis is concerned with the modeling of different failure mechanisms of cohesive frictional materials. Naturally given examples are limestone, marble, rock or clay whereas concrete or ceramics belong to the class of industrial manufactured materials. The material behavior and the material properties of cohesive frictional materials are characterized by a different tensile and compressive strength, by a failure induced anisotropy, by microcrack closure effects and by a complex interaction between an anisotropic degradation of the material properties and anisotropic irreversible strains. The presence of localized failure mechanisms induces a highly nonlinear and anisotropic material response. In the context of continuum mechanics the behavior of cohesive frictional materials is described by a combination of continuum damage mechanics and plasticity theory. The microplane concept, which became well-known for the modeling of quasi-brittle materials, marks the framework of the present work. This concept makes a compromise between a detailed micromechanical description and a structural-oriented macroscopic approach. The main characteristic of microplane modeling is the description of complex material behavior by simple constitutive laws on every microplane. The subsequent homogenization process causes a complex interaction of all microplanes. The main focus of this work is less the experimental validation of microplane models, which in the main can be considered as assured. In fact an adjustment of the microplane constitutive laws from well-known macroscopic models will be presented. Thereby, the additional information from the microplane formulations provide an advanced improvement of the classical failure mechanisms, e.g. failure induced anisotropy. Initially different simple microplane material models (pure damage as well as pure plasticity models) will be developed and implemented by means of the finite element method. In a final step a combination of microplane damage formulations with microplane plasticity models will be shown. Based on the progresses in microplane constitutive modeling the relations between the meso- and macroscale will be analyzed. Thus functional interrelations between well-known macroscopic formulations and the corresponding microplane models will be indicated. In the end this process results in mechanical sound microplane formulations based on a set of material parameters with a clear physical meaning. The microplane model as a continuum formulation requires an enhancement for the postcritical regime. Within the scope of the present work a gradient enhancement of the microplane formulation is applied as localization limiter. This regularization strategy is suitable for both friction failure and brittle material effects such as microcrack interaction. The incorporation of a so-called characteristic length scale controls the width of numerical resolvable failure zones. This length scale establishes a relationship between the size and the distance of the material inhomogeneities. Furthermore this method ensures unique and mesh independent results with finite energy dissipation. The derived anisotropic microplane models capture the complex failure characteristics of heterogeneous materials, like concrete with different material strength. In contrast to classical macroscopic formulations these models provide advanced information on the apparent failure mechanisms.#### BibTeX

- D. (2004).
*Discrete Models for Cohesive Frictional Materials.*Doktorarbeit, Bericht Nr. 42, Institut für Baustatik, Universität Stuttgart.#### Abstract

Geomaterials are widespread in nature as well as in engineering practice, for example in the form of a naturally given soil or a synthetic manufactured building material. The failure mechanisms of these materials are characterized by complex failure modes and show a highly anisotropic bias due to their inhomogeneous microstructure. Since localization phenomena like cracks or shear bands occur the material cannot be treated as continuous in the usual manner. The discontinuous nature of failure in geomaterials demands an adequate and reliable numerical simulation model like the discrete element method (DEM). The attraction of DEM simulations of continua is attributable to the fact that the appropriate complexity (localization, pattern formation, etc.) appears as an emergent feature, without the need for it to be programmed explicitly. Based on simple contact laws and a limited number of arbitrary parameters a rich behavior is obtained. Therefore, the general goal of the present thesis is to elaborate sound DEM models for the discontinuous simulation of geomaterials which are quantified by adequate homogenization techniques. The first main focus of this thesis is to advance DEM models in order to account for both the cohesive nature of materials like concrete, ceramics or rock and the cohesionless nature of materials like sand. Starting from a basic two-dimensional DEM model for non-cohesive polygonal particle assemblies, the complexity of the model is successively augmented towards the description of cohesive particle assemblies. In this context two approaches for the representation of cohesion, a beam and an interface model, are elaborated. If included into the DEM methodology by representing an attracting force between neighboring particles these approaches yield enhanced DEM models. An extensive simulation program aims at a qualitative and quantitative comparison of simulations and experiments. The scope of this confrontation is the correct representation of the crack evolution of various loading setups and the full identification of the experimentally measured softening response. The last step in the series of increasing complexity is the realization of a microstructure-based simulation environment which utilizes the foregoing enhanced DEM models. The two-phase microstructure is included, if different properties of the cohesive components (beam or interface) are assigned with respect to their position. In that, the inclusion of a microstructure regards for stiffer aggregates embedded in a less stiffer matrix. With the growing model complexity a wide variety of failure features of geomaterials can be represented and a quantification of the model is enabled. The second focal point of this thesis concerns the development and numerical implementation of adequate homogenization approaches by means of a micro to macro transition from the particle to the macro level. Homogenization procedures are developed which allow for a transfer from a simple Boltzmann continuum based particle model to a more complex continuum with microstructure according to Mindlin. The numerical realization of the transitions towards enhanced continuum theories like micropolar and gradient models is verified from a micromechanical viewpoint. The quantities of the micro or particle scale are linked to comparable continuum mechanical quantities on the macro scale and, thus, average dynamic and kinematic quantities are derived. Starting point of these homogenization approaches is the argument of scale separation between the characteristic scales of a particle assembly, namely that of a macroscopic body, a representative volume and an individual particle. Use of these arguments yields simplified equilibrium conditions for a representative volume element (RVE) on an intermediate scale.#### BibTeX

- Erhart, T. (2004).
*Strategien zur numerischen Modellierung transienter Impaktvorgänge bei nichtlinearem Materialverhalten.*Doktorarbeit, Bericht Nr. 44, Institut für Baustatik, Universität Stuttgart. https://doi.org/10.18419/opus-198#### Abstract

The present study is concerned with a special case of solid mechanics, where structures are exposed to short-time, highly concentrated loading. Such transient impact processes appear in civil and military security technology, dynamic soil compaction, vehicle crash or fastening and demolition technology. They are characterized by varying non-linearities, as e.g. large deformations and strains, highly non-linear material behavior, frictional contact between multiple bodies and stress wave propagation. The aim of this study is the development of a coherent overall strategy for the numerical simulation of such highly dynamic processes, which allows for a realistic quantitative description of associated complex physical events. Preparation and combination of different methods in the fields of adaptivity, constitutive modeling, element technology, efficient time discretization and contact are essential for the reliable computation of practical relevant engineering tasks and for predictions in industrial applications. Starting from a basis containing large deformation kinematics, a general formulation of finite plasticity, explicit time integration and an approach for contact, important aspects of different topics are examined and new methods are developed. Accuracy, robustness and efficiency are the authoritative requirements for the solution of those complex problems. For the spatial discretization of plane strain and axisymmetric situations, Finite Element formulations are provided, which are especially qualified for the problems at hand. Particularly, a reduced integrated quadrilateral with stabilization is extended to geometrical and material non-linear cases. Since large deformations occur and a Lagrangean description is used, repeated remeshing of individual domains is essential. To achieve quality controlled solutions and an optimal distribution of used computational resources at the same time, an adaptive strategy is applied. The core of this strategy is the assessment of discretization errors by adequate error indicators. For this purpose, different possibilities are presented and new methods are developed, which are appropriate for the simulation of transient impact processes. The used explicit time integration procedure is enhanced by introducing adaptive time step control and spatially varying time step ratios - so-called subcycling. Based on the theory of finite plasticity, constitutive models for thermoviscoplastic metals and cohesive as well as non-cohesive frictional materials are presented and developed. Here, the main focus will be on a new formulation for the transition of compact brittle materials into loose, granular media under high pressure loadings as it appears in impact situations. Therefore, a Drucker-Prager-Cap model is modified and enhanced concerning the evolution of material parameters and a modified hardening behavior in case of powderization. The properties and effects of the developing powder will be examined in detail. The proposed methods are verified individually and in combination for model problems and their performance in practical relevant applications is evaluated.#### BibTeX

- Gee, M. (2004).
*Effiziente Lösungsstrategien in der nichtlinearen Schalenmechanik.*Doktorarbeit, Bericht Nr. 43, Institut für Baustatik, Universität Stuttgart. https://doi.org/10.18419/opus-193#### Abstract

Numerical and algorithmic aspects of simulating the behaviour of thin-walled shell structures using finite element methods are discussed. Attention is focussed upon choice and layout of methods allowing for an efficient treatment of equations resulting from the 7-parameter shell model by Buechter and Ramm (1992) applying parallel computers. Parallel iterative solution methods are considered in a software framework proposed here. However, their efficiency is significantly influenced by the underlying mechanics of the shell structure. Shells discretised by finite elements result in ill-conditioned systems of equations and therefore pose a great challenge to the preconditioner of the iterative solver. With the shell model mentioned above the situation further deteriorates, as the thickness change of the shell is taken into account and the parametrisation of rotations and thickness change is done by relative displacements between the upper- and the mid-surface of the shell. A preconditioner is presented combining two approaches in a parallel, domain decomposition based framework. The first approach is a mechanically motivated improvement of the conditioning of the resulting element matrices by scaling the director field of the shell. By this the extra ill-conditioning from the above mentioned parametrisation can be removed. The second approach is a semi-algebraic multigrid preconditioner based on Smoothed Aggregation following Vanek et al. (2002). Disjoint so-called aggregates of nodal blocks of the system matrix are formed which then can be represented on a coarser level by a single nodal block. The method abandons the need for explicit coarse discretisations and constructs interlevel transfer operators upon the rigid body modes of the unconstrained structure. Consideration of thickness change in the shell formulation allows for contact constraints on the shell surfaces. Based on the continuum contact framework in Laursen (2002), a node-to-segment discretisation of (self-)contact for the 7-parameter shell model including finite deformations and friction is presented. The required regularisation of contact constraints is done by augmented Lagrangean multipliers. A multi-phase concept for the contact search algorithms is proposed that makes use of the slenderness assumption of the structure to improve efficiency but also considers the challenges resulting from this property. A series of numerical examples demonstrates the applicability and effectiveness of the considered approaches.#### BibTeX

- Kemmler, R. (2004).
*Stabilität und große Verschiebungen in der Topologie- und Formoptimierung*. Doktorarbeit, Bericht Nr. 41, Institut für Baustatik, Universität Stuttgart. https://doi.org/10.18419/opus-182#### Abstract

Structures generated by structural optimization are optimal with regard to the assumptions made for the optimization process. Therefore, all effects that could have a significant influence on the result have to be included in the optimization process. Taking non-linear kinematics into consideration, two phenomena can be described. Firstly, the non-linear relation between effect and structural response. Secondly, critical points of a structure can be determined. A main focus of this thesis is the assumption of large displacements in topology optimization. Several stiffness criteria can be developed on the basis of the non-linear kinematics. For material topology optimization a proposal is made that allows local design criteria to be taken into consideration without explicit dependency on the design variables. In order to generate flexible structures two types of the problem will be investigated. For transport mechanisms, the object is to transfer the actuator power in the best way as possible to a work piece. It is shown that the enhancement towards non-linear kinematics yields a better relationship between the actuator power and the strain energy stored in the work piece. Based on the same assumption path tracing mechanisms are designed, which can follow a curved path. To achieve the clearest possible material distribution in the design space, various stiffness criteria are discussed that describe approximately the contact of actuator and work piece. To create structures that do not show unstable behavior for a prescribed safety level, critical loads are determined using two different methods. The critical load level is estimated by a linear eigenvalue analysis and is incorporated into the optimization problem as a constraint. In the next step, critical points are calculated directly with an extended system in order to obtain a good prediction even for non-linear pre-buckling behavior. This also allows taking into consideration the response of imperfect structures, which generally show a strong non-linear pre-buckling behavior. The method presented is also utilized to generate the decisive imperfection shape. Non-linear optimization problems are solved using gradient-based methods. For all problems the sensitivities are determined on the basis of the adjoint approach. Further more, the use of potential character in finding efficient solutions to the design problems are commented. Finally, the design cases presented and the algorithmic implementation are verified by selected examples.#### BibTeX

- Gravemeier, V. (2003).
*The Variational Multiscale Method for Laminar and Turbulent Incompressible Flow.*Doktorarbeit, Bericht Nr. 40, Institut für Baustatik, Universität Stuttgart. https://doi.org/10.18419/opus-181#### Abstract

The present thesis addresses a new approach for solving numerically instationary, incompressible flow governed by the appropriate set of Navier-Stokes equations. This approach named 'variational multiscale method' has recently been introduced as a powerful means for problems of computational mechanics having to deal with large ranges of scales. Such notably widened scale ranges emerge in various flow situations. At large, this study aims firstly at the development of a general framework for the numerical solution of the Navier-Stokes equations based on the variational multiscale method. Secondly, a specific implementation within a Galerkin finite method is realized based on this general framework. Finally, an extensive test program for the developed method in form of sample laminar and turbulent flow situations is conducted. The variational multiscale method enables the separation of the complete scale range into various scale groups. In this work, the distinction of three different scale groups is mainly proposed in view of the underlying fluid flow problems. These are especially large resolved scales, small resolved scales and unresolved scales. The scale separation is initially applied to the linear model problem of a scalar convection-diffusion-reaction equation and some solution strategies are suggested in this context. The transition to the more complicated problem of the nonlinear set of instationary, incompressible Navier-Stokes equations is performed and reasonable solution strategies for this problem are picked up again. Additional considerations have to be taken into account as soon as the challenging phenomenom of turbulent flow regimes is encountered. As a starting point for these particular considerations, two classical procedures for the numerical simulation of turbulent flows, Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES), are adapted to the numerical method of choice in this work, the Galerkin finite element method, in a straightforward manner. It is, in fact, possible to identify these classical approach as special cases of the variational multiscale method employing a separation of two scales. Nevertheless, the variational multiscale method gives rise to further approaches going beyond this, e.g. based on the three-scale separation proposed in this work. As a practical method based on the three-scale separation, a three-level finite element method is developed. A major objective of this practical method has to be on computational efficiency inevitably due to the usual imbalance of necessary and available computer resources in the context of turbulent flow situations. Therefore, a local approximation in form of residual-free bubbles is exploited as a first approach for the three-level method. The name of the method especially accounts for three different types of discretizations linked with the respective levels. Starting with a basic discretization (level 1), submeshes (level 2) are introduced on each element of this basic mesh. These two levels constitute a two-level finite element method for the time being, which is tested in numerical examples. With the help of these submeshes, approximate solutions for residual-free bubbles on these elements are pursued. Particular attention is also paid to the adequate consideration of the continuity equation on level 2. The proposed three-level method is not achieved until a third level is introduced. In particular, a dynamic way of modeling the still unresolved scales of the problem is proposed as the manifestation of third level. According to this, a third type of discretization, namely elementwise sub-submeshes, i.e. slightly refined submeshes with regard to the submeshes on the second level, is employed into the method. The performance of the three-level finite element method is demonstrated for several laminar and turbulent flow examples.#### BibTeX

- Hörmann, M. (2002).
*Nichtlineare Versagensanalyse von Faserverbundstrukturen.*Doktorarbeit, Bericht Nr. 39, Institut für Baustatik, Universität Stuttgart. https://doi.org/10.18419/opus-168#### Abstract

The present thesis is concerned with the numerical simulation of nonlinear behavior in composite structures by means of the Finite Element method. Thereby special attention is payed to nonlinear material behavior. For this phenomenological material models are developed and presented, which are capable of describing the overall structural response including local effects like delaminations. After the presentation of an anisotropic plasticity model combined with anisotropic hardening the frequently observed and very dangerous type of failure delamination is considered in particular. On the basis of a multilayered, three-dimensional oriented shell theory including higher-order displacement kinematics across the thickness first-ply-failure indicators available from literature are presented at first. They are compared among themselves in particular with respect to their use as yield criterion or damage criterion in a materially nonlinear analysis. In what follows the anisotropic Hoffman failure criterion is used in the context of a plasticity formulation. Thereby an anisotropic hardening law is included allowing to simulate a nonlinear anisotropic hardening response of the laminate. In order to study the impact of the anisotropic hardening plasticity model structural examples are compared and assessed. For their simulation the progressing delaminations are described in this thesis in a smeared manner and not as a geometrical discontinuity. Due to this continuum mechanics based description it is possible to represent delamination within a thin process layer. Hence four different material models are developed and presented. Some are formulated in the context of a softening plasticity theory while others are formulated in the context of continuum damage mechanics. All models in common is, that they use the critical fracture energy within the description of softening material response. Using those models in combination with a standard continuum mechanics approach leads to the well known problems of mesh sensitivity and loss of ellipticity of the underlying partial differential equations. In order to overcome these problems a mesh adjusted softening modulus is used and moreover a visco-plastic respectively visco-damage regularization is applied where required. In case of the delamination plasticity model the viscosity parameter is adapted automatically with the objective of minimal effect on numerical results. In contrary the delamination damage models utilize a constant viscosity parameter. Both plasticity and some damage models employ the Brewer-Lagace criterion in its original or modified form as yield or damage criterion, respectively. Additionally a further delamination damage model is developed, which has some similarities to the model proposed by Ladeveze and co-workers. It uses the definition of an equivalent energy release rate as damage criterion. The equivalent energy release rate is thereby composed of different energy release rates which are related to the fracture modes I, II and III, respectively. Those are the work conjugated variables to the damage parameters. By means of model examples and appropriate structural examples the different models are compared and assessed.#### BibTeX

- Krausz, K. (2002).
*Tragverhalten gemauerter Tonnengewölben mit Stichkappen.*Doktorarbeit, Bericht Nr. 38, Institut für Baustatik, Universität Stuttgart. https://doi.org/10.18419/opus-165#### Abstract

Zur Einwölbung langer rechteckiger Räume wurden im Barock vorzugsweise gemauerte zylindrische Tonnengewölbe verwendet. Im Bereich der im Regelfall hochgezogenen Fenster wurden Stichkappen angeordnet. Die Standsicherheit solcher Gewölbe wird mit der Methode der Stützlinie untersucht. Dabei werden die Stichkappen vernachlässigt. Im Rahmen vorliegender Arbeit wurden zahlreiche numerische Untersuchungen an Modellen von Bogentragwerken und von zylindrischen Tonnengewölben mit Stichkappen durchgeführt. In den verwendeten Werkstoffmodellen wurde auch Rissbildung berücksichtigt. Die Ergebnisse der Untersuchungen zeigen, wie Stichkappen das Tragverhalten der Tonne beeinflussen. Aufbauend auf die Ergebnisse wird ein zur Untersuchung gemauerter Tonnengewölbe mit Stichkappen entwickeltes Konzept vorgestellt.#### BibTeX

- Steeb, H. (2002).
*Fehlerschätzer für FE-Berechnungen bei entfestigenden Materialien.*Doktorarbeit, Bericht Nr. 37, Institut für Baustatik, Universität Stuttgart.#### Abstract

In the following contribution a posteriori error estimation techniques for adaptive Galerkin finite elements are developed and analyzed. After a respective comparison of the well-known classical explicit and implicit residual based error estimators and gradient-based error estimators, the presented methods are extended to goal-oriented error estimation techniques. Goal-oriented error estimators are based on duality techniques. These techniques are formulated within this work and numerically evaluated using Galerkin finite element methods. Comparing goal-oriented and classical error estimators, the advantage of the first technique is the higher flexibility with respect to the analyzed error norm. This means that the error functional can be adjusted to the boundary value problem in a goal-oriented way. Thus, from a view point of a practical engineer, this highly flexible technique is very attractive for various numerical applications in the industry. In the second part of this contribution the introduced goal-oriented techniques are extended to inelastic material response. The extension to softening materials, here described within the framework of the Perzyna-viscoplasticity, is especially considered. In a first step the previously presented goal-oriented techniques are developed in the tangential space of the Perzyna-viscoplasticity model, based on the canonical equations of the initial value problem in strong form. In a second step the methods are extended to space-time error estimations techniques. In the last two sections of this contribution the applicability and reliability of the presented error estimation techniques are shown in various numerical examples. Thus the error estimators are extended to refinement indicators and are implemented in an adaptive finite element code.#### BibTeX

- Haufe, A. (2001).
*Dreidimensionale Simulation bewehrter Flächentragwerke aus Beton mit der Plastizitätstheorie.*Doktorarbeit, Bericht Nr. 35, Institut für Baustatik, Universität Stuttgart. https://doi.org/10.18419/opus-139#### Abstract

Based on a layered 3D-oriented shell formulation structural as well as material models for reinforced concrete structures are developed. Finite element analyses of experiments taken from the literature are used for validation. First the special properties of the layered 3D shell formulation which can be employed with C0-or C1-continuous displacement fields across the thickness direction will be pointed out. An overview of various possibilities to describe the full three dimensional constitutive relationship of cohesive frictional materials is given and discussed in view of the application in the structural model. It will be shown that the phenomenological description of the material behaviour by advanced plasticity models in combination with a simple approach from damage theory is a suitable means for the numerical simulation of concrete structures. On the one hand an associated multi-surface plasticity-model for plain concrete which has been proposed by Menrath for 2D-oriented structures obeying the plane stress condition and which is based on two invariants of the stress tensor, is expanded for the 3D shell formulation. On the other hand the non-associated single-surface plasticity-model of Kang which incorporates all three stress invariants is developed further. Aspects of the time discretization of the single-surface model and its implementation are extensively discussed since the required derivatives are extremely involved. Both models exhibit softening evolution laws based on the fracture energy approach. The local unloading effects are taken into account through a simple scalar damage formulation. The main problem of softening materials in numerical investigations is the loss of ellipticity of the underlaying differential equation which comes along with undesired effects like dependencies on the mesh orientation and the mesh density. Using a mesh adjusted internal length parameter is a common remedy of the problem, yet the results are slightly dependent on the mesh orientation. The hierarchy of failure indicators is discussed using established model problems. Numerical in vestigations of both concrete models show the expected loss of uniqueness already in the hardening regime of the non-associated model. The approach of viscoplastic regularisation is investigated to maintain the well-posedness of the boundary value problem. Here regularising effects can be shown for viscoplastic materials yet for concrete materials under static loading this approach has been proven to be unsuitable. The reinforcement is accounted for through an orthotropic constitutive model which is also based on classical plasticity theory with nonlinear or multilinear evolution laws. Different evolution laws are defined for steel and textile reinforcement and are discussed in view of the tension-stiff ening-effect. The three dimensional load carrying behaviour of textile reinforcement structures (fabrics etc.) is accounted for within a shear deformable approach. All necessary model parameters are identified by means of model problems. Numerical simulations of appropriate experiments of plates, shells and folded plates are used to verify the developed constitutive models and to show their significance for practical applications.#### BibTeX

- Maute, A. (2001).
*Fehlerkontrolle bei Finite-Element-Methoden in der linearen Strukturdynamik.*(p. 190). Doktorarbeit, Bericht Nr. 33, Institut für Baustatik, Universität Stuttgart.#### Abstract

Errors in numerical analysis of structural dynamics are due to the spatial and temporal discretization. A few methods from literature will be presented for estimating the average spatial discretization error in the energy norm at a specific point in time. The average temporal error of a time interval can be measured accordingly. These methods for controlling the error are based on the semidiscrete formulation and the Newmark-time integration. The efficiency of the adaptive error control can be improved by specifying the variable and the domain for error control instead of using the average error in the energy norm. The concept of dual problem is introduced for controlling the error in user specified spatial and temporal domains. The problem must be formulated by a finite-element method in space and time, such as the Discontinuous Time Galerkin method. The dual problem, which is derived for linear structural dynamics, represents the influence of the local error control variable in space and time. By introducing the dual concept it is possible to control the error in different variables, as well as the evolution of the error in time. Examples verify that the error can be measured in different quantities in user defined spatial domains by introducing dual problems. Also, the evolution of the error in time can be controlled. By comparing the dual approach with the traditional methods for error control, the potential of error control based on the dual concept is illustrated.#### BibTeX

- Mok, D. P. (2001).
*Partitionierte Lösungsansätze in der Strukturdynamik und der Fluid-Struktur-Interaktion.*Doktorarbeit, Bericht Nr. 36, Institut für Baustatik, Universität Stuttgart. https://doi.org/10.18419/opus-147#### Abstract

The area of partitioned analysis techniques for the numerical simulation of coupled systems has experienced increasingly intense research efforts in many areas of application over the last few years. This is due to a series of advantages over simultaneous, monolithic solution methods. Partitioned solution approaches allow the independent use of suitable modelling, discretization and solution methods for the individual subsystems, a modular and parallel software architecture, as well as the reduction of the numerical solution and memory efforts by decomposing the fully coupled system into subsystems which are easier to handle separately. The present report studies methods for the partitioned solution of coupled dynamical systems, particularly with regard to geometrically nonlinear structural dynamics and the transient interaction of instationary, incompressible flows with flexible structures featuring large deformations. Formulations developed in different scientific disciplines are compiled within a unified framework and classifying survey. They are differentiated according to the type of spacial partitioning, the time integration methods used in the subsystems, and the coupling strategy, as these criteria are of fundamental relevance for construction and numerical properties of the specific partitioned schemes. Within the class of non-overlapping Dirichlet-Neumann methods, which are especially suitable for the simulation of the problems in mind, simple staggered and iterative staggered coupling strategies, i.e. coupling strategies leading to algorithmically loose and strong coupling, respectively, are further investigated and compared theoretically and numerically. It is shown that simple staggered schemes are numerically cheap and - if formulated appropriately - sufficiently accurate, yet they are in principle weakly unstable. This weak instability is caused by their inherent violation of the basic balance equations for mass and energy due to a lack of kinematic continuity at the interface between adjacent subsystems, and amplified by the ärtificial added mass"-effect in the case of fluid structure interaction with incompressible flows. Iterations over the subsystems, i.e. iterative staggered schemes, have proven to be the most advisable stabilization technique. These techniques are traced back to classical domain decomposition methods (iterative substructuring methods). By this means it has become possible to derive convergence statements, and to develop efficient, robust and user-friendly convergence acceleration schemes. In particular, the application of the gradient method and Aitken's method are proposed. Finally, a series of numerical examples demonstrates the specific applicability and effectiveness of the analyzed partitioned solution approaches.#### BibTeX

- Schwarz, S. (2001).
*Sensitivitätsanalyse und Optimierung bei nichtlinearem Strukturverhalten.*Doktorarbeit, Bericht Nr. 34, Institut für Baustatik, Universität Stuttgart. https://doi.org/10.18419/opus-140#### Abstract

To generate meaningful and reliable structures by using the methods of structural optimization it is essential to gather the real, in general nonlinear structural behavior already for the optimization process. The design problems of interest often lead to nonlinear optimization problems which can be solved efficiently by gradient based methods. Thus, it is necessary to compute the sensitivities, that are the gradients of the objective and the constraints with respect to the optimization variables. Because the optimization criteria depend in general on the structural response, the theoretical and computational complexity of the sensitivity analysis is dominated by the underlying, mechanical and numerical model. The present work focuses on developing and implementing an analytical approach for the sensitivity analysis. The structural response is characterized by an elastoplastic material behavior with strain hardening and softening as well as geometrical nonlinearities, and is simulated by a Finite Element method. A Prandtl-Reuss model is applied to describe the elastoplastic material which leads to path-dependent response. In turn, this requires a special procedure to treat the also path-dependent sensitivity analysis. A variational and direct formulation for the analytical sensitivity analysis is presented. The advantages of this formulation in the context of elastoplasticity are discussed. The proposed procedure is consistent with the one for computing the structural response. This refers to the integration method of the constitutive equations as well as the path following strategy. Due to its generality, the proposed sensitivity analysis can be applied to diverse optimization problems including shape and topology optimization. The influence of the mechanical and numerical model on the optimization procedure and the optimization results is demonstrated with selected examples. For maximizing the ductility and minimizing the weight the influence of the material model on the optimum topology and shape is studied. The importance of subsequent shape optimization following a topology optimization step is shown. Additional constraints on the structural stability are imposed in order to avoid structural failure. The influence of these constraints on the optimum shape is studied.#### BibTeX

- Kuhl, E. (2000).
*Numerische Modelle für kohäsive Reibungsmaterialien*. Doktorarbeit, Bericht Nr. 32, Institut für Baustatik, Universität Stuttgart.#### Abstract

Not only the classical geomaterials like sand, clay, stone and rock but also a number of engineering materials, for example concrete, can be classified as cohesive frictional materials. Their mechanical behavior is governed by their pronounced pressure-sensitivity manifesting itself in entirely different failure mechanisms under tensile and compressive loadings. Consequently, the tensile and compressive strength of these materials can vary by several orders of magnitude. During their failure process, the formation of highly localized zones of concentrated straining, such as microcracks, slip planes, macroscopic crack planes or shear bands, can be observed. In general, this phenomenon of strain localization induces a highly anisotropic material response. The development of finite element-based numerical models which take into account the above mentioned failure characteristics is the basic concern of the present work. To set the stage, the classical, local isotropic models of elasto-plasticity and elasto-damage are briefly reviewed. In a second step, the basic features of these models will be transferred to an anisotropic material characterization embedded in the so-called microplane concept. Within the framework of microplane theory, the response of a material point can be understood as the volume integral of its behavior on all material planes in space integrated over the solid angle. A general, thermodynamically consistent concept of formulating microplane-based constitutive laws will be presented. Its basic features are illustrated by means of the constitutive equations of microplane elasticity, microplane elasto-plasticity and microplane elasto-damage. Especially in the post-critical regime, the solution of classical local continuum approaches shows the tendency to form highly localized failure zones. In a numerical simulation, the width of these failure zones corresponds to the width of a single finite element and thus tends to zero with an infinite mesh refinement. This disability of classical continuum approaches to model correctly the material behavior in the softening regime is caused by the fact, that local continuum models disregard the effects of changes in the microstructure. From a mathematical point of view, the resulting mesh dependency of the numerical solution is caused by a change of type of the governing equations. Insufficient boundary conditions lead to an ill-posed problem, the result of which is primarily determined by the underlying discretization. In the literature, several strategies have been proposed to remedy this deficiency through the introduction of an internal length scale. By doing so, microstructural changes can be taken into account. Within the present work, the microstructural length scale will be introduced in terms of a gradient enhanced continuum approach. Through the incorporation of higher order gradients in the constitutive equations, the problem remains well-posed and a finite number of solutions can be guaranteed. The existing gradient enhanced damage approaches in the literature are generalized to capture not only the classical isotropic material models but also the new anisotropic microplane-based material formulation. The performance of the proposed models is demonstrated and discussed by means of several selected examples.#### BibTeX

- Bischoff, M. (1999).
*Theorie und Numerik einer dreidimensionalen Schalenformulierung*. Doktorarbeit, Bericht Nr. 30, Institut für Baustatik, Universität Stuttgart. https://doi.org/10.18419/opus-126#### Abstract

The 7-parameter shell model as proposed by Büchter and Ramm (1992) is an extension of conventional shear deformation theories with five degrees of freedom. Application of this model is especially sensible if complete three-dimensional constitutive laws ought to be applied allowing also to solve problems involving large deformations and large strains. Based on a mixed formulation the seventh degree of freedom can be eliminated on the element level. Thus, the numerical effort is only slightly larger compared to `usual' shell elements. The model can consequently also be utilized for analyses of such shell structures, where a conventional shell formulation would be sufficient. Büchter and Ramm (1992) describe the 7-parameter shell model along with a finite element formulation. Here, the seventh degree of freedom is introduced on the element level by means of a hybrid-mixed formulation. In the present work the 7-parameter model is derived independent from a finite element formulation. In this context it can be interpreted as semi-discretization of the shell continuum in thickness direction. The decisive difference to most of the conventional shell theories is that this discretization is based on a multifield variational formulation. By this procedure a system of partial differential equations can be obtained, describing the 7-parameter model as a two-dimensional, continuous theory with seven kinematical degrees of freedom per material point of the reference surface. It is also intended to give a physical interpretation of the kinematic and static variables appearing in the model. Here, the main emphasis is put on those quantities, which do not show up in a conventional 5-parameter formulation. The investigations provide some insight into the mechanical behavior of the model. For the linear part of the transverse shear strains a new shear correction factor is proposed, which can reduce the error with respect to the three-dimensional solution in membrane dominated situations. It is shown that the 7-parameter model is `optimal' concerning the number of kinematic and static variables involved. This means that exactly those components are involved which are necessary for a complete three-dimensional material description. Finally, a concept for the formulation of efficient finite elements for the 7-parameter model is presented. Here, established methods from the literature are combined with own developments. A special feature of the proposed concept is that a unified formulation for triangular and rectangular elements of arbitrary polynomial order is obtained. In addition, an improvement in the treatment of shells with kinks is proposed. In the course of the present study the concept is realized for linear and quadratic triangular and rectangular elements. The features of the proposed elements are investigated in numerical experiments, including both linear and geometrically and materially non-linear problems.#### BibTeX

- Menrath, H. (1999).
*Numerische Simulation des nichtlinearen Tragverhaltens von Stahlverbundträgern.*Doktorarbeit, Bericht Nr. 29, Institut für Baustatik, Universität Stuttgart.#### Abstract

Basis of the present thesis is the challenge to simulate the real load carrying performance of composite steel-concrete constructions. It describes the numerical simulation of "classical" steel-concrete beams, which are composed of an upper concrete slab and a lower steel beam, connected at the interface by studs transmitting shear and normal forces. Rate-independent elastoplasticity for small strains is applied individually to each of the three material components (steel, concrete and interface layer). The local integration procedure is based on a Return-Backward-Euler algorithm for multisurface yield functions. For the steel-beam the classical von-Mises-elastoplasticity model with combined linear work-hardening (kinematic and isotropic hardening) is applied. The algorithmic constitutive law for cracked reinforced-concrete combines several features commonly accepted for modelling plain concrete, reinforcement and tension-stiffening (the so called interaction stress contribution). The concrete is modelled by softening plasticity with fracture-energy, Gf for tensile behaviour and Gc for compressive behaviour. Due to the underlying approach mesh-independent results can be obtained with respect to the postcritical regime. Having two Drucker-Prager regions and a spherical cap the geometry of the 3D-multisurface yield criterion is defined by four additional parameters to describe the combined yield function. A third Drucker-Prager yield function is used to handle the singularity problem at the first one. For reinforcement perfect bond is assumed on the one hand, on the other hand tension-stiffening is considered as an additional stress in rebar direction. The constitutive law for reinforcement is based on a one-dimensional elastoplastic model with hardening. Interface layers are introduced to model the load carrying behaviour of the studs and the geometrical discontinuities between beam and slab. Here the studs are modelled in a smeared approach instead of being modelled by discrete connectors. This thesis describes the materially non-linear interface behaviour using the Coulomb friction law. The proposed numerical simulation is verified by two-dimensional examples. The 3D-constitutive law for concrete and the related stress tensor are condensed to 2D on the Gaussian point level.#### BibTeX

- Wall, W. A. (1999).
*Fluid-Struktur-Interaktion mit stabilisierten Finiten Elementen*. Doktorarbeit, Bericht Nr. 31, Institut für Baustatik, Universität Stuttgart. https://doi.org/10.18419/opus-127#### Abstract

Strictly speaking, most technical-scientific problems and especially real engineering tasks belong to so called multifield or multiphysics problems. One important subgroup of such coupled problems is characterized through the interaction of fluids and flexible structures. The present study aims at the development of numerical methods that allow a realistic description of the behavior of such coupled fluid structure interaction problems and in a level of complexity that is necessary for many applications. Because of this complexity of problem classes in mind, common simplifications and limitations of many existing approaches have to be waived and the approach has to be based on models with appropriate complexity. Thereby this study on fluid structure interaction (FSI) should concentrate mainly on methodical aspects. The different physical fields are herein characterized by the instationary, incompressible Navier-Stokes equations and through the field equations of geometrically nonlinear elastodynamics, respectively. Main concern of this study on fluid structure interaction, that is undertaken within an Institute of Structural Mechanics, is the prediction of the instationary, nonlinear response of mainly thin-walled structures. However, even in such situations the flow part is dominating the whole coupled problems simulation. Hence, one main focus of this study is the development of a method for computational fluid dynamics. The presented approach for the coupled fluid structure interaction problem should be purely based on a finite element concept for all fields under consideration. For this reason a new stabilized finite element method for the simulation of instationary, incompressible viscous flows is developed. Essential aspects concern the inserted, partly newly-developed and improved, stabilization approaches and new combinations of algorithmic approaches. The developed method is able to deal with low- and higher-order elements with equal order interpolation for velocities and pressure. For instationary flow simulations it also allows simple adaptive time-stepping procedures. The extension of the flow solver to domains with moving boundaries, based on an Arbitrary Lagrangean Eulerian (ALE) description, establishes the second main focus. Thereby a consistent derivation for the stabilization approaches and a respective algorithmic implementation are the main concerns. Hence, a new semi-discrete, consistent, fully stabilized ALE-FEM is introduced. In order to be able to tackle a broad range of applications a powerful tool for the moving grid description within the CFD-ALE-formulation is developed. For the structural part of this multifield problem, existing methods for geometrically nonlinear elastodynamics are adopted for the coupled problem requirements. Together with the CFD solver and the mesh solver these are embedded in a newly developed environment for the simulation of multifield problems. The coupled problems are then treated through simple partitioned analysis procedures. The combination of these individual building blocks finally results in the newly-introduced three-field FSI solver. The performance of the coupled solver as well as of the individual methodical components is demonstrated through a number of numerical examples.#### BibTeX

- Cirak, F. (1998).
*Adaptive Finite-Element-Methoden bei der nichtlinearen Analyse von Flächentragwerken.*Doktorarbeit, Bericht Nr. 27, Institut für Baustatik, Universität Stuttgart.#### Abstract

This thesis presents a-posteriori error estimators and adaptive mesh refinement techniques for the linear and nonlinear analysis of plate, membrane and shell structures. The developed framework enables the error control and mesh refinement with respect to different locally and globally defined variables. This thesis presents a-posteriori error estimators and adaptive mesh refinement techniques for the linear and nonlinear analysis of plate, membrane and shell structures. The developed framework enables the error control and mesh refinement with respect to different locally and globally defined variables. Typical for the presented error estimators is the introduction of duality techniques as used in the classical influence surface concept. For the linear case the error estimator is evaluated solving the discretized boundary value problem for an additional right hand side and applying the classical energy norm error estimators two times. In the nonlinear regime the linearized problem at the equilibrium point of the discretized prob lem is utilized for error estimation. As numerical and analytical studies indicate the error esti mators based on the linearized problem are asymptotically exact. In a first step the presented methods are also applied to the instationary J2 - plasticity problem in an incremental sense. The performance of the proposed a-posteriori error estimators and the related mesh refinement strategies is demonstrated by various examples.#### BibTeX

- Maurer, B. (1998).
*Karl Culmann und die graphische Statik.*Doktorarbeit, Bericht Nr. 26, Institut für Baustatik, Universität Stuttgart.#### Abstract

Karl Culmann (1821 - 1881) is one of the most important engineers of the 19th century. He contributed a great deal towards putting engineering on a scientific footing. His work was based on the assumption that "drawing" is "the language of the engineer"; to him, the quintessence of science was mathematics and, consequently, he developed graphical statics on the basis of Poncelt's projective geometry. It was to be the fundamental discipline on which engineering research was to be based. Graphical statics is almost unknown today, but up to the beginning of our century, lectures were being held on graphical statistics at every university college of technology - in some universities, there were even professorships for this discipline. This thesis contains a Culmann biography, a history of graphical statics and a comprehensive appendix of Culmann texts. For the biography, numerous hitherto uncovered sources have been tapped, concerning, in particular, the period in which he was involved in the construction of the Bavarian railways and his time in England and America. The chapters on the history of graphical statics examine the roots of this discipline and describe its development from the first approaches by Lamé and Clapeyron to Culmann's main work, Graphical Statics. They trace the rapid spread of graphical statics at university colleges of technology all over the world up to the beginning of the 20th century when its methods were absorbed by engineering mechanics and by structural analysis and graphical statics ceased to exist as an independent discipline. The appendix contains over a hundred pages of so far unpublished Culmann texts, including reports he wrote during his time as an engineering apprentice with the Bavarian railways, an expertise on the system of education in the Zürich canton and, above all, the technical reports on his visits to England and America. The appendix contains, in addition, a compilation of all the lectures held by Culmann at the ETH (Swiss College of Technology) in Zürich, short biographies of graphical statics authors, a detailed examination of the position of graphical statics in German-speaking colleges of technology and, finally, a collection of about a thousand textbooks and original works on graphical statics.#### BibTeX

- Maute, K. (1998).
*Topologie- und Formoptimierung von dünnwandigen Flächentragwerken.*Doktorarbeit, Bericht Nr. 25, Institut für Baustatik, Universität Stuttgart.#### Abstract

The present thesis deals with topology and shape optimization of continuous, thin-walled structures. The task of topology optimization is to determine the basic layout of a structure in a given design space. The shape of contours and surfaces of structures, e.g. generated by topology optimization, is optimized in detail by shape optimization. The topology optimization problem is formulated as a "0-1" material distribution problem, in order to generate arbitrarily complex geometries. This leads to a nonconvex optimization problem, which is regularized by introducing porous materials with variable density. Materials with optimal properties regularize correctly the material distribution problem, however they do not lead to definite "0-1" results. Therefore, suboptimal materials are preferred for real world design problems. These materials lead to almost pure "0-1" material distributions, but the solutions have to be additionally stabilized. Today, topology optimization procedures solve robustly the standard optimization problem "maximum stiffness for given mass in the design space". There are still numerical instabilities for maximum stiffness problems with stress constraints and linear eigenvalue problems arising in dynamics and stability analyses. Studies on maximizing the critical load for geometrical nonlinear response and on maximizing the ductility for an elastoplastic material behavior show the necessity to include the nonlinear structural response within topology optimization. The material distribution generated by topology optimization describes only roughly the geometry of the optimum structure. The detailed shape can be efficiently determined by CAGD-orientated shape optimization procedures. Since topology optimization often generates geometrically complex structures, a flexible method is presented describing the structural layout on given plane and spatially curved surfaces for shape optimization. The quality and the numerical efficiency of the optimization results can be improved by adaptive procedures. In structural optimization two kinds of adaptivity have to be distinguished: adaptive discretization of the mechanical model and adaptive discretization of the geometrical model. Using adaptive finite element methods in structural optimization, additionally to the discretization error of the structural response the discretization error of the sensitivities of the structural response ought to be considered. An adaptive discretization of the geometrical model is most often of secondary importance in shape optimization. However in topology optimization the presented adaptive procedure considerably improves the numerical efficiency and leads to geometrically more refined results. The adaptive combination of topology and shape optimization methods is a promising and methodically challenging approach.#### BibTeX

- Schmidts, H. (1998).
*Zur effizienten Modellierung und Analyse von Hochhaustragwerken.*Doktorarbeit, Bericht Nr. 23, Institut für Baustatik, Universität Stuttgart.#### Abstract

This paper deals with the modeling and the calculation of sectionally regular high-rise structure. The aim is to develop efficient macro-elements. Especially in the structural design these will allow for a numerical calculation which, by using the finite-element-method, will meet the requirements and permits a practice-oriented evaluation of the results. Due to the structure's regularity substitute systems may be used for the calculation and therefore the high-rise structure consists of continuous wall panels. Reasonable, i.e. realistic approaches with respect to the course of deformation in the structure's cross-section "generalized cross-section values" are derived, reducing the three dimensional problem to a one-dimensional reflection. The discretization of the deformation along the "bar center line" finally leads to the stiffness-matrix of the respective generalized beam element (macro-element) This permits a more efficient modeling and calculation compared to the classical beam- and wall-elements, because the special characteristics for high-rise structures were taken into consideration from the beginning Especially the simple interpretation of the results allows a review of the structures quality and sensible variation of the system parameters. This also considers the influence of second-order effects according to DIN 1045 approximately with horizontal forces. Statements for the vibration behavior respectively the vibration sensitivity according to DIN 1055 are possible. The described solution method is realized in the computer program HIGH-RISE.#### BibTeX

- Trautz, M. (1998).
*Zur Entwicklung von Form und Struktur historischer Gewölbe aus der Sicht der Statik.*Doktorarbeit, Bericht Nr. 28, Institut für Baustatik, Universität Stuttgart.#### Abstract

In this study researches of the influence of geometry and structural details on the structural behaviour of historical domes and vaulted structures are presented. Statical calculations and simulations are carried out in order to investigate the principal statical behaviour of masonry vaults and also in order to get information about the state of prevalent empirical and theoretical knowledge, design methods and construction methods. Historical research subjects are: the oriental barrel-vaults, the construction process and the structural behaviour of Roman domes in composite design, a comparative study of the structural behaviour of Byzantine and Ottoman pendentif domes, structural behaviour and structural design of barrel vaults from Renaissance, structural behaviour of ovaloid domes in the late Baroque period, the early application of special forms with statical implication to domes. Traditional thrustline-methods as well as nonlinear finite-element-methods are applied to the statical investigations. The load case of main interest is 'deadload'. Thrustline-methods help to investigate problems of formfinding and to evaluate different forms of vaults with special statical characteristics. The features of the catenary in general termns, i.e. for other load configurations than uniformal distributed loads, are discussed. The finite-element-method allows the simulation of cracking of brittle material and the investigation of the change of the stress distribution in masonry structures due to the materially nonlinear behaviour. The mechanical parameters of historical masonry and their application to nonlinear finite-element-analysis are discussed particularly. The simulation results are verified by means of basic mechanical models. Finally some of the typical features of the cracking behaviour of simple masonry structures are pointed out.#### BibTeX

- Wang, H. (1998).
*Interaktion des lokalen und globalen Stabilitätsverhaltens dünnwandiger Stäbe.*Doktorarbeit, Bericht Nr. 24, Institut für Baustatik, Universität Stuttgart.#### Abstract

This thesis deals with the stability analysis of thin-walled beams considering cross sectional distortion applying both beam and shell elements. The objective is the development of a finite element model for thin-walled beams coupling beam and shell elements. For global buckling of thin-walled beams the beam element is used, and the shell element is employed in areas of local stability failure. For an efficient use of the shell model a mesh generator DBGEN is written specifically for thin-walled beams. With respect to the local character of the cross sectional distortion the coupled beam and shell elements are applied. The multipoint constraints are derived satisfying the geometric compatibility within the cross-section. The system equation with multipoint constraints is solved using a straightforward elimination. As expected for the analysis of the interaction of local and global buckling the coupling method turns out to be much more efficient. The automatical output of the multipoint constraints by DBGEN allows to conveniently analyse the coupled model for complex frames with deformable joint areas.#### BibTeX

- Hausser, C. (1996).
*Effiziente Dreieckselemente für Flächentragwerke.*Doktorarbeit, Bericht Nr. 21, Institut für Baustatik, Universität Stuttgart.#### Abstract

This thesis deals with the efficient formulation of triangular plate, membrane and shell finite elements in the linear elastic range. The aim of this study is the development of low order elements that achieve high convergence rates and a stable solution without any locking defects. For an efficient formulation of shear-deformable plate elements with Reissner-Mindlin kinematics, the problem of shear locking needs to be completely resolved. In order to avoid locking, different methods, like the assumed natural strain method (ANS), mixed functionals, the Kirchhoff-mode concept or enhanced rotations are adapted to the special needs of triangular geometry. To improve the convergence of triangular membrane elements, drilling rotations are introduced using Allmann-interpolation or the Free Formulation by Bergan. These elements can be further improved with the enhanced assumed strain method (EAS). By combining the most powerful plate and membrane elements, efficient facet elements for the analysis of shell structures can be constructed. Besides the mathematical conditions for stable and convergent finite element formulations, an understanding of the mechanical aspects of the applied methods is of primary interest. The performance of the different element formulations is tested by numerous numerical examples.#### BibTeX

- Kuhl, D. (1996).
*Stabile Zeitintegrationsalgorithmen in der nichtlinearen Elastodynamik dünnwandiger Tragwerke.*Doktorarbeit, Bericht Nr. 22, Institut für Baustatik, Universität Stuttgart.#### Abstract

The objective of this thesis is the development of numerical methods which guarantee the stable integration of the initial boundary value problem given by the geometrically nonlinear elastodynamics especially the nonlinear dynamics of thin walled structures. The semi-discretization technique with Finite Element discretization in space and implicit time integration schemes for discretization and integration in the time domain is used. Spatial discretization of thin walled structures is done by a finite 8-node Serendipity shell element with Reissner-Mindlin kinematics which includes the thickness stretch of the shell. The element is realized by an extensible shell director and its update from the reference to the current configuration by the difference vector. This element is able to handle finite displacements and rotations and complete three dimensional constitutive equations. A further feature of the shell element is the numerical efficient calculation of the inertia term of the 'Principle of Virtual Work' with acceleration vector and mass matrix being independent of the deformation. In order to avoid artificial stiffening caused by the three dimensional displacement field, termed 'Thickness Locking', the 'Enhanced Assumed Strain' concept is used. The spatial discretization results in the nonlinear semidiscrete initial value problem. An essential aspect of the development of algorithms for the numerical solution of the nonlinear initial value problem is the numerical stability, which is equivalent to the conservation of total energy within a typical time step. In principle there are three numerical properties which guarantee the satisfaction of the energy condition for numerical stability: * the algorithmic dissipation (e.g. 'Generalized-a Method'), * the enforced conservation of energy 'Constraint Energy Method') * and the algorithmic conservation of energy ('Energy-Momentum Method'). These principles will be discussed. Further the 'Constraint Energy Momentum Algorithm' will be designed. The new algorithm combines controllable algorithmic dissipation and enforced conservation of total energy, linear and angular momentum. This new implicit, second order accurate algorithm is developed by the extension of the 'Generalized-a Method' with constraints via Lagrange multipliers. The constraints result from the global conservation theorems by spatial discretization an A set of examples is examined to illustrate the application of time integration methods to the analysis of snap-through and buckling of structures. Furthermore the features of different time integration schemes will be shown.#### BibTeX

- Rehle, N. (1996).
*Adaptive Finite Element Verfahren bei der Analyse von Flächentragwerken.*Doktorarbeit, Bericht Nr. 20, Institut für Baustatik, Universität Stuttgart.#### Abstract

In the present thesis adaptive methods in finite element analyses are developed and investigated. The objective is to obtain finite element meshes with a relatively low number of degrees of freedom and high accuracy for the results of plate and shell structures. For that strategies are used calculating the quality of the finite element discretisation and opimizing the mesh, along with the standard finite element algorithms. Out of the variety of published different adaptive strategies some were selected and are introduced. On this basis a new procedure is developed handling the adaptive generation of almost ideal finite element meshes. A mesh generator of the advancing front method type is the core of this strategy generating adaptive refined FE-meshes on result of the error estimation. Chapter 2 - 4 presents the adaptive concept including the new development. Chapter 5 displays expressive examples. For the design of real world structures several loadcases have to be examined. This means that the acceptance of an adaptive concept is only guarantied if all loadcases are considered. Therefore some strategies are discussed in order to integrate a variety of different loadcases.#### BibTeX

- Braun, M. (1995).
*Nichtlineare Analysen von geschichteten, elastischen Flächentragwerken.*Doktorarbeit, Bericht Nr. 19, Institut für Baustatik, Universität Stuttgart.#### Abstract

The reason to develop nonlinear, three-dimensional shell theories based on an adequate kinematics for anisotropic laminates is caused by the increasing demand for fiber reinforced materials for shell structures. Considering several shell models with different kinematic approximations, a suitable analysis for every individual problem can be carried out. This procedure represents the basis for physically reasonable defined failure and damage criteria. Fiber reinforced materials are mostly used for high loaded structures. As large strain effects are becoming dominant, the thickness change of the laminate should not be neglected. The required extension of a conventional 5-parameter shell theory to a 6-parameter formulation leads to a significant error for bending dominated cases which depends on the anisotropic material properties. This thickness locking is due to the resulting linear distribution of the normal stress in thickness direction, not being balanced by the constant strain. The locking phenomenon is avoided if the shell formulation is extended by a linear thickness strain term which can be introduced by an extra independent variable (7-parameter theory) via the enhanced assumed strain concept based on a hybrid mixed approach. The above mentioned single layered theories (singledirector theories) are based on the assumption of a straight director which cannot be maintained for definite composites and laminates with respect to the physics. With a higher order kinematics, a nonlinear, layer-wise theory (multidirector theory) which includes the singledirector theories as a special case with one kinematic layer has been developed. This layer-wise model is able to simulate the structural behaviour of anisotropic laminates with extreme differences in the thicknesses and material properties of each layer. Furthermore there are two more advantages by analyzing the shell as a three-dimensional continuum: Firstly it is possible to use arbitrary complete three-dimensional constitutive laws without reduction or manipulation in nonlinear plate and shell analysis. Secondly the location within the shell body where the loading is applied can exactly be considered. For all presented formulations the typical characteristics of shell structures (reference surface, explicit or pre integration, higher efficiency) are preserved.#### BibTeX

- Suanno, R. (1995).
*Ein dreidimensionales Simulationsmodell für Stahlbeton mit Plastizität und Schädigung.*Doktorarbeit, Bericht Nr. 18, Institut für Baustatik, Universität Stuttgart.#### Abstract

The analysis of reinforced concrete structures is still a challange in computational mechanics. Despite a long and successful tradition of this öld" material in the construction practice many obscure points in its mechanical behaviour still exist. Several simulation models are described in an enormous amount of literature, but a compromise between simplicity an realistic modelling seems to be difficult to achieve. The theory of continuum damage mechanics (CDM) has opened a new horizon in the numerical representation of brittle materials. Since the pioneering work of Kachanov, several improved models have been proposed. It is well known that CDM alone is somewhat restrictive if applied to quasi-brittle materials. Therefore the combination of the theory of plasticity with CDM is more appealing. It offers a broader spectrum to describe different phenomena of concrete structures. In the present work an elasto-pastic CAP-model recently developed by Hofstetter et al. 38 in the framework of consistent algorithmic material tangent and closestpoint return algorithms is combined with a scalar damage model. The local integration procedure is based on a generalized mid point rule satisfying the consistency condition at the mid point. As shown by Simo and Govindjee 107, this scheme is of second order accuracy for the true mid point, B-stable and retains a symmetric elastotropic consistent tangent for the associative plasticity. In this work the associative version of the CAP-Model as well as its nonassociative counterpart (nonassociative flow and hardening rules) are employed. The required material parameters for the analysis are obtained applying optimization procedures. For the reinforcement the embedded approach is used. Two versions are implemented. In the first version the reinforcement is modelled by discrete bars: it is utilized for members with low reinforcement ratio. In the second version the reinforcement is smeared by embedded layers; this model is sufficiently accurate for highly reinforced members. The proposed material model is applied to several examples of three-dimensional concrete stuctures.#### BibTeX

- Reitinger, R. (1994).
*Stabilität und Optimierung imperfektionsempfindlicher Tragwerke.*Doktorarbeit, Bericht Nr. 17, Institut für Baustatik, Universität Stuttgart.#### Abstract

Structural optimization is always based on various disciplines. This characteristic becomes obvious when a computer is used for the solution of structural optimization problems. In a general and direct appraoch, several disciplines contribute to the solution of the entire problem, which can be distinguished in: The subject of this thesis is the optimization of buckling and imperfection sensitive structures. This becomes important if structures are designed to obtain a maximum load carrying capacity for a minimum of material. Such structures are thin and slender and the loads are carried mainly by membrane stresses rather than by bending action. In order to be able to consider the buckling failure phenomena and the imperfection sensitivity geometrically nonlinear structural analyses have been carried out. The presentation of the required methods and techniques starts with path-following schemes. Then, extended systems of equations are discussed to directly compute bifurcation and limit points. Branch switching techniques or the asymptotic approach by Koiter are used for die post-buckling range. The influence of imperfections is treated at the end of the first part of the thesis. The second part starts with the introduction into structural optimization. Firstly, stability phenomena are approximated by eigenvalue analyses. Then, an optimization method is presented which contains fully geometrically nonlinear behaviour and also considers the imperfection sesitivity. The method is based on the techniques for geometrically nonlinear analyses, discussed in the first part of the thesis. The sensitivity analysis of great importance and its relation to the imperfection sensitivity is pointed out. Finally, examples are demonstrating the sizing and shape optimization of stuctures to mazimize the buckling load for a given volume of material. The examples show the capabilities of the proposed opimization method.#### BibTeX

- Roehl, D. (1994).
*Zur Berechnung von großen elastoplastischen Deformationen bei Flächentragwerken und Kontinua.*Doktorarbeit, Bericht Nr. 16, Institut für Baustatik, Universität Stuttgart.#### Abstract

The present work deals with the analysis of three dimensional and shell structures undergoing large elastoplastic deformations with aid of the finite element method. Herefore is the material model based on the multiplicative split of the deformation gradient. The elastic behaviour follows a hyperelastic description and the evolution laws for the internal variables are derived from the principle of maximum plasic dissipation. The solution of the nonlinear mechanical equations is achieved through a Newton iteration proceedure for which die material tangent consistent with the time integration algorithm employed - here the implicit Euler method - is determined. As an altenative to the displacement finite element model, which proves to be inadequate for the analysis of large elastoplastic deformations, the Enhanced Assumed Strain Method was chosen. Some numerical examples validate the theory and the developed numerical procedure whereby special atention is given to problems with large local deformations.#### BibTeX

- Büchter, N. (1992).
*Zusammenführung von Degenerationskonzept und Schalentheorie bei endlichen Rotationen.*Doktorarbeit, Bericht Nr. 14, Institut für Baustatik, Universität Stuttgart.#### Abstract

This thesis deals with the efficient formulation of triangular plate, membrane and shell finite elements in the linear elastic range. The aim of this study is the development of low order elements that achieve high convergence rates and a stable solution without any locking defects. For an efficient formulation of shear-deformable plate elements with Reissner-Mindlin kinematics, the problem of shear locking needs to be completely resolved. In order to avoid locking, different methods, like the assumed natural strain method (ANS), mixed functionals, the Kirchhoff-mode concept or enhanced rotations are adapted to the special needs of triangular geometry. To improve the convergence of triangular membrane elements, drilling rotations are introduced using Allmann-interpolation or the Free Formulation by Bergan. These elements can be further improved with the enhanced assumed strain method (EAS). By combining the most powerful plate and membrane elements, efficient facet elements for the analysis of shell structures can be constructed. Besides the mathematical conditions for stable and convergent finite element formulations, an understanding of the mechanical aspects of the applied methods is of primary interest. The performance of the different element formulations is tested by numerous numerical examples.#### BibTeX

- Hofmann, T. J. (1992).
*Beitrag zur verfeinerten Balkentheorie.*Doktorarbeit, Bericht Nr. 15, Institut für Baustatik, Universität Stuttgart.#### Abstract

An extended and more consistent beam theory is introduced and compared to the elementary theory. The contradiction between shear stresses and shear strains, existing in the beam theory of Timoshenko, is removed by the present extension. The inconsistency is eliminated by a more exact representation of the shear strains, leading to a more accurate calculation of the longitudinal stresses. Essentially warping functions are introduced into the bending theory in analogy to the well known warping torsion. The derivation of the extended beam theory for plane and spatial beams is given. Its accuracy is estimated, and the method is investigated with regard to its range of application. It turns out that the new theory is almost as simple as the traditional beamtheiries. The result is a geometrically linear theory for linear elastic homogeneous isotropic material with an extension to the linearized second order theory and to dynamics. It is shown that the developed beam theory is suitable to describe the "shear lag effect" and the problem of "effective flange width" of beams having usual dimensions and arbitary thin-walled cross sections.#### BibTeX

- Andelfinger, U. (1991).
*Untersuchungen zur Zuverlässigkeit hybrid-gemischter finiter Elemente für Flächentragwerke.*Doktorarbeit, Bericht Nr. 13, Institut für Baustatik, Universität Stuttgart.#### Abstract

The thesis is dealing with four-noded hybrid-mixed membrane, plate and shellfinite elements. The term hybrid-mixed encloses formulations which besides a displacement approximation have one or more additional field assumptions (for strains, stresses or incompatible displacements). The additional fields are assumed discontinuously fromelement to element, so that the corresponding unknows can be eliminated on the element level. Therefore all the hybrid-mixed methods are leading to an element stiffness matrix. Typical formulations are the so-called assumed natural strain concept (in this method the strain fields do not possess independent degress of freedom but are linked directly to the displacement dependent strains at certain sampling points), formulations based on a Hellinger-Reissner principle with stress (or strain) and displacement asuumptions, three-filed approximations based on the principle of Hu-Washizu and the enhanced assumed strain method. For all the hybrid-mixed methods main points of discussion are stability and locking. Guidelines for safe stress and strain assumptions are given through a table-like investigation of the polynomial expansion of the displacement derivatives. The membrane, plat and shell finite elements (the shell formulation is based on the three-dimensional degenerated approach and a local coordinate formulation) with stress and/or strain assumptions are compared to numerous other elements known from the literature.#### BibTeX

- Bletzinger, K.-U. (1990).
*Formoptimierung von Flächentragwerken.*Doktorarbeit, Bericht Nr. 11, Institut für Baustatik, Universität Stuttgart.#### Abstract

The subject of this thesis is shape optimization of plates and shells. Such kind of structural shapes are to be developed which are determined by certain objectives (e.g. minimal weight, minimal strain energy) where the stress and displacement behaviour of linear elastic material is considered. For that purpose methods of * finite elements (to analyze the structural behaviour) * mathematical programming (to optimize) * Computer Aided Geometric Design, CADG (to describe the geometry) are used together. The interactions of these different methods and of the sub-models which can be described by them decides about the flexibility and practicability of the overall model of structural optimization. Detailed knowledge about the underlying fundamental concepts is necessary to utilize the manifold capabilities of the purely mathematically formulated methods of mathematical programming and CADG. Therefore, chapters two to four are dealing extensively with those methods with special regard to applications and increased efficiency for problems in the field of structural optimization. The sub-models are brought together in chapter five. They are structured to improve applicability in a way that even complex problems can be formulated in a flexible manner by a few characteristic optimization variables. This will be demonstrated by different examples in chapter six: applications to fundamental questions of form finding of minimal surfaces and shells well as the practical design of a Vierendeel girder.#### BibTeX

- Kimmich, S. (1990).
*Strukturoptimierung und Sensibilitätsanalyse mit finiten Elementen.*Doktorarbeit, Bericht Nr. 12, Institut für Baustatik, Universität Stuttgart.#### Abstract

Structural optimization is always based on various disciplines. This characteristic becomes obvious when a computer is used for the solution of structural optimization problems. In a general and direct appraoch, several disciplines contribute to the solution of the entire problem, which can be distinguished in: * programming techniques * problem formulation * geometric modelling * structural and sensitivity analysis * mathematical programming A satisfactory solution of structural optimization problems presumes the knowledge of teh principles of the individual disciplines. Bayond this the interdependence of these disciplines in the optimization process has to be considered. Tehrefore an impression of the overall procedure as well as detailed knowledge of the individual solution strategies is required, when real optimization tasks are faced. Due to this, the chapters one to three are dedicated to some basic considerations of programm concepts and programming techniques. For each individual optimization task the according optimization problem has to be formulated. This subject is considered in chapter four, where the principles of light weight constructions are suitably used. These principles finally lead to the characteristic problems of optimal stiffness, minimal weight and optimal strenght. Here the introduction of variables, objective and constraint functions become important for the statment of the problem. The direct solution of an optimization task with the strategies of mathematical programming requires the construction of an appropriate optimization model. On this level there exists a relation between geometric modelling (design model) and structural or sensitivity analysis respectivly (structural model). All three disciplines are tightly joined together by the optimization model. Chapter five especially deals with the independence of structural and sensitivity analysis. The derivation of an efficient problem formulation and the discrete sensitivity analysis is done in shapter six for three dimensional truss and shell elements. Several numerical examples show the advantages of the under- lying integrated programm concept. The program option can be usefull for the verification of the assumptions introduced in the different modelling levels.#### BibTeX

- Tao, D. (1989).
*Die Technik der reduzierten Basis bei nichtlinearen finiten Element-Berechnungen.*Doktorarbeit, Bericht Nr. 9, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Weimar, K. (1989).
*Ein nichtlineares Balkenelement mit Anwendung als Längsstreifen axialbelasteter Kreiszylinder.*Doktorarbeit, Bericht Nr. 10, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Matzenmiller, A. (1988).
*Ein rationales Lösungskonzept für geometrisch und physikalisch nichtlineare Strukturberechnungen.*Doktorarbeit, Bericht Nr. 8, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Burmeister, A. (1987).
*Dynamische Stabilität nach der Methode der finiten Elemente mit Anwendungen auf Kugelschalen.*(p. 233). Doktorarbeit, Bericht Nr. 6, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Kammler, G. (1987).
*Ein finites Elementmodell zur Berechnung von Trägern und Stützen mit offenem, dünnwandigem Querschnitt unter Berücksichtigung der Interaktion zwischen globalem und lokalem Versagen*. Doktorarbeit, Bericht Nr. 7, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Forde, B. (1986).
*Iteration Procedure for Sudden Local Alteration of Structural Stiffness*. Doktorarbeit, Mitteilung Nr. 6, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Stegmüller, H. (1985).
*Grenzlastberechnungen flüssigkeitsgefüllter Schalen mit “degenerierten” Schalenelementen.*(p. 143). Doktorarbeit, Bericht Nr. 5, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Diack, A. (1983).
*Beitrag zur Stabilität diskret längsversteifter Kreiszylinderschalen unter Axialdruck.*(p. 191). Doktorarbeit, Bericht Nr. 3, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Kompfner, T. A. (1983).
*Ein finites Elementmodell für die geometrisch und physikalisch nichtlineare Berechnung von Stahlbetonschalen.*Doktorarbeit, Bericht Nr. 2, Institut für Baustatik und Baudynamik, Universität Stuttgart.#### BibTeX

- Osterrieder, P. (1983).
*Traglastberechnung von räumlichen Stabwerken bei großen Verformungen mit finiten Elementen.*(p. 186). Doktorarbeit, Bericht Nr. 1, Institut für Baustatik und Baudynamik, Universität Stuttgart.#### BibTeX

- Andrä, H.-P. (1982).
*Zum Tragverhalten des Auflagerbereichs von Flachdecken.*Doktorarbeit, Bericht Nr. 82-3, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Häfner, L. (1982).
*Einfluß einer Rundschweißnaht auf die Stabilität und Traglast des axialbelasteten Kreiszylinders.*Doktorarbeit, Bericht Nr. 82-1, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Schweizerhof, K. (1982).
*Nichtlineare Berechnung von Tragwerken unter verformungsabhängiger Belastung mit finiten Elementen.*Doktorarbeit, Bericht Nr. 82-2, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Gaizo, R. I. D. (1980).
*Liegende zylindrische Behälter und Rohre auf Sattellagern endlicher Breite.*Doktorarbeit, Bericht Nr. 80-1, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Rembold, R. W. (1980).
*Beitrag zum Tragverhalten ausgewählter Plattentragwerke unter Berücksichtigung der Reissnerschen Theorie und der Methode der gemischten finiten Elemente.*Doktorarbeit, Bericht Nr. 80-2, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Sättele, J. M. (1980).
*Ein finites Elementkonzept zur Berechnung von Platten und Schalen bei stofflicher und geometrischer Nichtlinearität.*Doktorarbeit, Bericht Nr. 80-3, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Berg, H. G. (1979).
*Tragverhalten und Formfindung versteifter Kuppelschalen über quadratischem Grundriß auf Einzelstützen.*Doktorarbeit, Bericht Nr. 79-2, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Brendel, B. (1979).
*Geometrisch nichtlineare Elastostabilität.*Doktorarbeit, Bericht Nr. 79-1, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Fujii, F. (1978).
*Anwendung der Methode der finiten Elemente auf die Berechnung von Stahlbetonplatten.*Doktorarbeit, Bericht Nr. 78-1, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Sulke, B.-M. (1977).
*Berechnung dünnwandiger prismatischer Faltwerke mit verformbarem mehrzelligen Querschnitt.*Doktorarbeit, Bericht Nr. 77-1, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Müller, G. (1975).
*Numerische Behandlung der Kirchhoffschen und Reissnerschen Plattentheorie nach einer diskretisierten und erweiterten Trefftz-Methode.*Doktorarbeit, Bericht Nr. 75-2, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- O., E. A. C. (1975).
*Beitrag zur Berechnung langer dünnwandiger dreizelliger Träger unter Berücksichtigung der Profilverformung.*Doktorarbeit, Bericht Nr. 75-3, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- KlausTompert. (1974).
*Berechnung kreiszylindrischer Silos auf elastischer Unterlage.*Doktorarbeit, Bericht Nr. 74-3, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Riehle, W. (1974).
*Studie über verallgemeinerte Variationsfunktionale und ihre Anwendung bei der Methode der finiten Plattenelemente.*Doktorarbeit, Bericht Nr. 74-4, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Werner, G. (1974).
*Experimentelle und theoretische Untersuchungen zur Ermittlung des Tragverhaltens biege- und verdrehbeanspruchter Stäbe mit I-Querschnitt.*Doktorarbeit, Bericht Nr. 74-2, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- El-Masri, M. (1973).
*Theorie und Berechnung nichtlinearer Schwingungen vorgespannter Membrane nach der Methode der Finiten Elemente*. Doktorarbeit, Bericht 1973, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Seidel, J. (1973).
*Beitrag zur geometrisch nichtlinearen Theorie dünner Schalen unter Annahme kleiner Verzerrungen und großer Rotationen*(pp. 1–114). Doktorarbeit, Bericht 1973, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Becker, M. (1972).
*Beitrag zur Schalenberechnung nach einem verallgemeinerten Variationsprinzip und nach der überbestimmten Kollokation*(pp. 1–119). Doktorarbeit, Bericht 1972, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Ramm, E. (1972).
*Beitrag zur praktischen Berechnung dünner Kugelschalen bei nicht rotationssymmetrischer Berandung nach der linearen Biegetheorie.*(pp. 1–157). Doktorarbeit, Bericht 1972, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Tamhankar, M. G. (1969).
*Die Berechnung von Parallelogrammplatten unter Verwendung modifizierter Steifigkeitsmatrizen*(pp. 1–127). Doktorarbeit, Bericht 1969, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Flegel, R. (1968).
*Beiträge zu den HRENNIKOFFschen Stabmodellen für Scheiben- und Plattenprobleme*. Doktorarbeit, Bericht 1968, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Ludwig, W. (1968).
*Beitrag zur Berechnung polygonal begrenzter Platten nach dem Differenzenverfahren*. Doktorarbeit, Bericht 1968, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Ringelstein, K. H. (1968).
*Die Berechnung von Kreiskegelschalen mit beliebigem Wanddickenverlauf und beliebiger Belastung mit Hilfe eines elektronischen Analogrechners*(pp. 1–123). Doktorarbeit, Bericht 1968, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Feder, D. (1967).
*Spannungsoptische und theoretische Untersuchungen zur Kraftverteilung in Kehlnaht - Laschenverbindungen*(pp. 1–155). Doktorarbeit, Bericht 1967, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Steinle, A. (1967).
*Torsion und Profilverformung - Berechnung eines in Feldmitte durch ein Torsionsmoment belasteten Trägers mit einem verformbaren biegesteifen Rechteckkastenprofil mit auskragenden Gurten*(pp. 1–131). Doktorarbeit, Bericht 1967, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Krüger, U. (1966).
*Beitrag zur Biegetheorie der Kegelschale mit linear veränderlicher Wanddicke*(pp. 1–94). Doktorarbeit, Bericht 1966, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Hees, G. (1965).
*Eine Verallgemeinerung des Kraftgrößen- und des Formänderungsgrößenverfahrens aufgrund einer mathematischen Analyse dieser Verfahren*(pp. 1–72). Doktorarbeit, Bericht 1965, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Stein, E. (1965).
*Beiträge zu den direkten Variationsverfahren in der Elastostatik der Balken und Flächentragwerke*. Doktorarbeit, Bericht 1965, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Anheuser, L. (1964).
*Beitrag zur Berechnung des Kreisträgers mit offenem dünnwandigem Profil*. Doktorarbeit, Bericht 1964, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Mast, P. E. (1964).
*Ein Beitrag zur praktischen Berechnung prismatischer Faltwerke*. Doktorarbeit, Bericht 1964, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Müller, R. K. (1964).
*Ein Beitrag zur Dehnungsmessung an Kunstharzmodellen*. Doktorarbeit, Bericht 1964, Institut für Baustatik, Universität Stuttgart.#### BibTeX

- Andrä, W. (1963).
*Beitrag zur Berechnung zweistegiger Plattenbalken mit veränderlichem Querschnitt bei Torisionsbelastung*. Doktorarbeit, Bericht 1963, Institut für Baustatik, Universität Stuttgart.#### BibTeX