## Institut für Konstruktiven Ingenieurbau

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Institute of Structural Engineering, Institute of Structural Mechanics, as well as Institute for Computing, Mathematics and Physics in Civil Engineering at the faculty of civil engineering at the Bauhaus-Universität Weimar presented special topics of structural engineering to highlight the broad spectrum of civil engineering in the field of modeling and simulation.
The summer course sought to impart knowledge and to combine research with a practical context, through a challenging and demanding series of lectures, seminars and project work. Participating students were enabled to deal with advanced methods and its practical application.
The extraordinary format of the interdisciplinary summer school offers the opportunity to study advanced developments of numerical methods and sophisticated modelling techniques in different disciplines of civil engineering for foreign and domestic students, which go far beyond traditional graduate courses.
The proceedings at hand are the result from the Bauhaus Summer School course: Forecast Engineering held at the Bauhaus-Universität Weimar, 2018. It summarizes the results of the conducted project work, provides the abstracts/papers of the contributions by the participants, as well as impressions from the accompanying programme and organized cultural activities.

The design of engineering structures takes place today and in the past on the basis of static calculations. The consideration of uncertainties in the model quality becomes more and more important with the development of new construction methods and design requirements. In addition to the traditional forced-based approaches, experiences and observations about the deformation behavior of components and the overall structure under different exposure conditions allow the introduction of novel detection and evaluation criteria.
The proceedings at hand are the result from the Bauhaus Summer School Course: Forecast Engineering held at the Bauhaus-Universität Weimar, 2017. It summarizes the results of the conducted project work, provides the abstracts of the contributions by the participants, as well as impressions from the accompanying programme and organized cultural activities.
The special character of this course is in the combination of basic disciplines of structural engineering with applied research projects in the areas of steel and reinforced concrete structures, earthquake and wind engineering as well as informatics and linking them to mathematical methods and modern tools of visualization. Its innovative character results from the ambitious engineering tasks and advanced
modeling demands.

The proceedings at hand are the result of the International Master Course Module: "Nonlinear Analysis of Structures: Wind Induced Vibrations" held at the Faculty of Civil Engineering at Bauhaus-University Weimar, Germany in the summer semester 2019 (April - August). This material summarizes the results of the project work done throughout the semester, provides an overview of the topic, as well as impressions from the accompanying programme.
Wind Engineering is a particular field of Civil Engineering that evaluates the resistance of structures caused by wind loads. Bridges, high-rise buildings, chimneys and telecommunication towers might be susceptible to wind vibrations due to their increased flexibility, therefore a special design is carried for this aspect. Advancement in technology and scientific studies permit us doing research at small scale for more accurate analyses. Therefore scaled models of real structures are built and tested for various construction scenarios. These models are placed in wind tunnels where experiments are conducted to determine parameters such as: critical wind speeds for bridge decks, static wind coefficients and forces for buildings or bridges. The objective of the course was to offer insight to the students into the assessment of long-span cable-supported bridges and high-rise buildings under wind excitation. The participating students worked in interdisciplinary teams to increase their knowledge in the understanding and influences on the behaviour of wind-sensitive structures.

Identifying cable force with vibration-based methods has become widely used in engineering practice due to simplicity of application. The string taut theory provides a simple definition of the relationship between natural frequencies and the tension force of a cable. However, this theory assumes a perfectly flexible non-sagging cable pinned at its ends. These assumptions do not reflect all cases, especially when the cable is short, under low tension forces or the supports are partially flexible. Extradosed bridges, which are distinguished from cable-stayed bridges by their low pylon height, have shorter cables. Therefore the application of the conventional string taut theory to identify cable forces on extradosed bridge cables might be inadequate to identify cable forces.
In this work, numerical modelling of an extradosed bridge cable saddled on a circular deviator at pylon is conducted. The model is validated with the catenary analytical solution and its static and dynamic behaviours are studied. The effect of a saddle support is found to positively affect the cable stiffness by geometric means; longer saddle radius increases the cable stiffness by suppressing the deformations near the saddle. Further, accounting the effects of bending stiffness in the numerical model by using beam elements show considerable deviation from models with truss elements (i.e. zero bending stiffness). This deviation is manifested when comparing the static and dynamic properties. This motivates a more thorough study of bending stiffness effects on short cables.
Bending stiffness effects are studied using two rods connected with several springs along their length. Under bending moments, the springs resist the rods' relative axial displacement by the springs' transverse component. This concept is used to identify bending stiffness values by utilizing the parallel axis theorem to quantify ratios of the second moment of area. These ratios are calculated based on the setup of the springs (e.g. number of springs per unit length, transverse stiffness, etc...). The numerical model based on this concept agrees well with the theoretical values computed using upper and lower bounds of the parallel axis theorem.
The proposed concept of quantifying ratios of the second moment of area using springs as connection between cable rods is applied on an actual extradosed bridge geometry. The model is examined by comparison to the previously validated global numerical model. The two models showed good correlation under various changing parameters. This allowed further study of the effects of stick/slip behaviour between cable rods on an actual bridge geometry.

The design of safety-critical structures, exposed to cyclic excitations demands for non-degrading or limited-degrading behavior during extreme events. Among others, the structural behavior is mainly determined by the amount of plastic cycles, completed during the excitation. Existing simplified methods often ignore this dependency, or assume/request sufficient cyclic capacity. The paper introduces a new performance based design method that considers explicitly a predefined number of re-plastifications. Hereby approaches from the shakedown theory and signal processing methods are utilized. The paper introduces the theoretical background, explains the steps of the design procedure and demonstrates the applicability with help of an example. This project was supported by German Science Foundation (Deutsche Forschungsgemeinschaft, DFG)

The presented method for an physically non-linear analysis of stresses and deformations of composite cross-sections and members based on energy principles and their transformation to non-linear optimisation problems. From the LAGRANGE principle of minimum of total potential energy a kinematic formulation of the mechanical problem can be developed, which has the general advantage that pre-deformations excited by shrinkage, temperature, residual deformations after unloading et al., can be considered directly. Thus the non-linear analysis of composite cross-sections with layers of different mechanical properties and different preloading becomes possible and cracks in concrete, stiffness degradation and other specifics of the material behaviour can be taken into account without cardinal modification of the mathematical model. The impact of local defects on the bearing capacity of an entire element can also be analysed in this principle way. Standard computational systems for mathematical optimisation or general programs for spreadsheet analysis enable an uncomplicated implementation of the developed models and an effective non-linear analysis for composite cross-sections and elements.

A new approach to the non-linear analysis of cross-sections loaded by normal forces and bending moments is presented in the paper. The mechanical model is based on the LAGRANGE principle of minimum of total potential energy. Deformations, stresses and limit load parameters are obtained by solving a non-linear optimisation problem. The mathematical model is independent of the specifics of material. In addition to the stress strain relation and the specific strain energy W(ε) two further functions F(ε) and Φ(ε) are introduced to describe the material behaviour. Thus cracks in concrete, non-linearity of material etc. can be taken into account without basic modification of the numerical algorithm. For polygonal cross-sections the GAUSS' integral theorem is used. Numerical solutions of the non-linear optimisation problems can be found by application of standard software. Thus the analysis of reinforced concrete cross-sections or more general composite cross-sections with non-linear behaviour of material is as simple as in the case of linear elasticity. The application of the method is demonstrated for polygonal cross-sections. Pre-stresses or pre-strains can easily be included in the mathematical model.

Am Beispiel eines 3-feldrigen Durchlaufträgers wird die Versagenswahrscheinlichkeit von wechselnd belasteten Stahlbetonbalken bezüglich des Grenzzustandes der Adaption (Einspielen, shakedown) untersucht. Die Adaptionsanalyse erfolgt unter Berücksichtigung der beanspruchungschabhängigen Degradation der Biegesteifigkeit infolge Rissbildung. Die damit verbundene mechanische Problemstellung kann auf die Adaptionsanalyse linear elastisch - ideal plastischer Balkentragwerke mit unbekannter aber begrenzter Biegesteifigkeit zurückgeführt werden. Die Versagenswahrscheinlichkeit wird unter Berücksichtigung stochastischer Tragwerks- und Belastungsgrößen berechnet. Tragwerkseigenschaften und ständige Lasten gelten als zeitunabhängige Zufallsgrößen. Zeitlich veränderliche Lasten werden als nutzungsdauerbezogene Extremwerte POISSONscher Rechteck-Pulsprozesse unter Berücksichtigung zeitlicher Überlagerungseffekte modelliert, so dass die Versagenswahrscheinlichkeit ebenfalls eine nutzungsdauerbezogene Größe ist. Die mechanischen Problemstellungen werden numerisch mit der mathematischen Optimierung gelöst. Die Versagenswahrscheinlichkeit wird auf statistischem Weg mit der Monte-Carlo-Methode geschätzt.

In the paper presented, reinforced concrete shells of revolution are analyzed in both meridional and circumferential directions. Taking into account the physical non-linearity of the material, the internal forces and the deflections of the shell as well as the strain distribution at the cross-sections are calculated. The behavior of concrete under compression is described by linear and non-linear stress-strain relations. The description of the behavior of concrete under tension must account for tension stiffening effects. A tri-linear function is used to formulate the material law of reinforcement. The problem cannot be solved analytically due to the physical non-linearity. Thus a numerical solution is formulated by means of the LAGRANGE Principle of the minimum of the total potential energy. The kinematically admissible field of deformation is defined by the displacements u in the meridional and w in the radial direction. These displacements must satisfy the equations of compatibility and the kinematical boundary conditions of the shell. The strains are linearly distributed across the wall thickness. The strain energy depends on the specific of the material behavior. Using integral formulations of the material law [1], the strain energy of each part of the cross-section is defined as a function of the strains at the boundaries of the cross-sections. The shell is discretised in the meridional direction. Various methods of numerical differentiation and numerical integration are applied in order to determine the deformations and the strain energy. The unknown displacements u and w are calculated by a non-restricted extremum problem based on the minimum of the total potential energy. From mathematical point of view, the objective function is a convex function, thus the minimum can be determined without difficulty. The advantage of this formulation is that unlike non-linear methods with path-following algorithms the calculation does not have to account for changing stiffness and load increments. All iterations necessary to find the solution are integrated into the “Solver”. The model presented provides many ways of investigating the influence of various material parameters on the stresses and deformations of the entire shell structure.

An energy method based on the LAGRANGE Principle of the minimum of total potential en-ergy is presented to calculate the stresses and strains of composite cross-sections. The stress-strain relation of each partition of the cross-section can be an arbitrary piecewise continuous function. The strain energy is transformed into a line integral by GAUSS’s integral theorem. The total strain of each partition of the cross-section is split into load-dependent strain and pre-strain. Pre-strains have to be taken into account when the cross-section is pre-stressed, retrofit-ted or influenced by shrinkage, temperature etc. The unconstrained minimum problem can be solved for each load combination using standard software. The application of the method presented in the paper is demonstrated by means of examples.