Buckling of initially curved and imperfectly supported beam

Nosilca, ki je idealno vpet, idealno obremenjen in idealne geometrije v praktičnih aplikacijah ne moremo pričakovati. Imperfekcije, pa čeprav majhne, lahko povzročijo precejšnje odstopanje od teorije, ki opisuje idealen nosilec. V tej diplomski nalogi predstavimo napravo za preizkušanje vpliva različnih imperfekcij vpetja in obremenitve na odziv osno obremenjenega nosilca. Problem si najprej ogledamo z analitičnega vidika. Začnemo z eksaktno diferencialno enačbo upogibnice ukrivljenega nosilca, ki je izpeljana po teoriji 3. reda za velike premike ter iz nje izpeljemo enačbo upogibnice po teoriji 2. reda za majhne premike. Nato uporabimo eksperimentalno napravo, s katero preizkušamo vpliv začetne ukrivljenosti nosilca ter ekscentričnost vpetja pri uklonu. Rezultati meritev na preizkuševališču so primerjani z rezultati teoretičnih izračunov.Beam with ideal support, ideal geometry and ideally loaded cannot be expected in real engineering applications. Imperfections, though small, may cause substantial deviation from the theory, which describes ideal beams. In this diploma thesis, we present experimental apparatus for testing effects of different support and load imperfections on the response of axially loaded beam. The problem is first approached analytically with the derivation of the deflection equation of initially curved beam according to the exact and linearized theories. We use our experimental apparatus and test the effect of initial curvature and eccentric application of the load. The results of the measurements at the test site are compared with the results of theoretical calculations

Verjetnostno modeliranje in posodabljanje modelov visokih lesenih stavb

Technological advances in engineered timber structural elements have led to an increase in the popularity of mid-rise and high-rise timber buildings. The governing design criterion of such buildings is very often serviceability under wind-induced vibration. Accurate finite element models for predicting their modal properties (i.e. natural frequencies and mode shapes) are crucial for designing buildings that satisfy the current serviceability criteria. It is a challenge for structural engineers to decide what to include in the structural modelling. This is because elements that are typically considered non-structural (partition walls, plasterboards, screed, façade, etc.) have been shown to act structurally and can significantly influence the modal properties of timber buildings. This thesis presents three case studies of timber buildings, where finite element models were developed. Using experimental modal properties, the models were updated to learn about the as-built stiffness characteristics of the analysed buildings. A probabilistic framework based on Bayesian inference was applied to account for the uncertainties. A polynomial chaos expansion was used to develop a surrogate model allowing for a more timeefficient conducting of stochastic analyses such as uncertainty quantification, sensitivity analysis, and Markov chain Monte Carlo posterior sampling. Deterministic and probabilistic approaches were compared and assessed for their effectiveness.Tehnološki napredek pri lesenih konstrukcijskih elementih je privedel do večje priljubljenosti srednje visokih in visokih lesenih stavb. Vodilni kriterij za projektiranje takih stavb je pogosto uporabna ustreznost stavbe pri vibracijah, ki jih povzroča veter. Natančni modeli končnih elementov za napovedovanje njihovih modalnih lastnosti (t.j. lastne frekvence in oblike) so ključni za projektiranje stavb, ki izpolnjujejo sedanja merila uporabnosti. Gradbeni inženirji morajo sprejeti določene predpostavke o tem, kaj vključiti v konstrukcijsko modeliranje. Izkazalo se je namreč, da elementi, ki običajno veljajo za nekonstrukcijske (predelne stene, mavčne plošče, estrih, fasada idr.), delujejo konstrukcijsko in lahko bistveno vplivajo na modalne lastnosti lesenih stavb. V tej disertaciji so predstavljene tri študije primerov lesenih stavb, za katere so bili razviti modeli končnih elementov. Z uporabo eksperimentalnih modalnih lastnosti so bili modeli posodobljeni, da bi spoznali učinke različnih gradbenih elementov. Uporabljen je bil verjetnostni okvir, zasnovan na Bayesovem sklepanju, z namenom upoštevanja negotovosti v formulaciji. Razvit je bil nadomestni model na podlagi polinomske razširitve kaosa. Ta omogoča časovno učinkovitejše izvajanje stohastičnih analiz, kot so kvantifikacija negotovosti, analiza občutljivosti in vzorčenje posteriorne verjetnosti z Markovsko verigo Monte Carlo. Deterministični in verjetnostni pristop sta bila primerjana in ocenjena glede na njuno uspešnost

Probabilistic Modelling and Model Updating of Tall Timber Buildings

Tehnološki napredek pri lesenih konstrukcijskih elementih je privedel do večje priljublje-nosti srednje visokih in visokih lesenih stavb. Vodilni kriterij za projektiranje takih stavb je pogosto uporabna ustreznost stavbe pri vibracijah, ki jih povzroča veter. Natančni modeli končnih elementov za napovedovanje njihovih modalnih lastnosti (t.j. lastne frekvence in oblike) so ključni za projektiranje stavb, ki izpolnjujejo sedanja merila uporabnosti. Gradbeni inženirji morajo sprejeti določene predpostavke o tem, kaj vključiti v konstrukcijsko modeliranje. Izkazalo se je namreč, da elementi, ki običajno veljajo za nekonstrukcijske (predelne stene, mavčne plošče, estrih, fasada idr.), delujejo konstrukcijsko in lahko bistveno vplivajo na modalne lastnosti lesenih stavb. V tej disertaciji so predstavljene tri študije primerov lesenih stavb, za katere so bili razviti modeli končnih elementov. Z uporabo eksperimentalnih modalnih lastnosti so bili modeli posodobljeni, da bi spoznali učinke različnih gradbenih elementov. Uporabljen je bil verjetnostni okvir, zasnovan na Bayesovem sklepanju, z namenom upoštevanja negotovosti v formulaciji. Razvit je bil nadomestni model na podlagi polinomske razširitve kaosa. Ta omogoča časovno učinkovitejše izvajanje stohastičnih analiz, kot so kvantifikacija negotovosti, analiza občutljivosti in vzorčenje posteriorne verjetnosti z Markovsko verigo Monte Carlo. Deterministični in verjetnostni pristop sta bila primerjana in ocenjena glede na njuno uspešnost.Technological advances in engineered timber structural elements have led to an increase in the popularity of mid-rise and high-rise timber buildings. The governing design criterion of such buildings is very often serviceability under wind-induced vibration. Accurate finite element models for predicting their modal properties (i.e. natural frequencies and mode shapes) are crucial for designing buildings that satisfy the current serviceability criteria. It is a challenge for structural engineers to decide what to include in the structural modelling. This is because elements that are typically considered non-structural (partition walls, plasterboards, screed, façade, etc.) have been shown to act structurally and can significantly influence the modal properties of timber buildings. This thesis presents three case studies of timber buildings, where finite element models were developed. Using experimental modal properties, the models were updated to learn about the as-built stiffness characteristics of the analysed buildings. A probabilistic framework based on Bayesian inference was applied to account for the uncertainties. A polynomial chaos expansion was used to develop a surrogate model allowing for a more time-efficient conducting of stochastic analyses such as uncertainty quantification, sensitivity analysis, and Markov chain Monte Carlo posterior sampling. Deterministic and probabilistic approaches were compared and assessed for their effectiveness

Finite element model calibration procedures for strucutral systems

V magistrski nalogi raziskujemo postopke za kalibriranje numeričnih modelov konstrukcijskih sistemov glede na eksperimentalne podatke odziva sistema. Izmed možnih pristopov podrobneje obravnavamo dva - metodo odzivne površine in Bayesov pristop. Z metodo odzivne površine aproksimiramo odziv v odvisnosti od parametrov numeričnega modela ter iščemo take vrednosti parametrov, pri katerih se aproksimacija odziva najbolj ujema z eksperimentalnim. Bayesov pristop pa temelji na predpostavki, da so vrednosti parametrov naključne spremenljivke z določeno porazdelitvijo verjetnosti in išče rešitve, ki so najbolj verjetne. V nalogi obravnavamo dva primera: enostaven nosilec in gradbeno konstrukcijo, ki jo sestavljata jeklen okvir in sovprežna plošča iz profilirane pločevine in armiranega betonaza slednjo imamo na voljo eksperimentalne rezultate. Numerične analize izvedemo s komercialnim programom Ansys, kalibriranje numeričnega modela po obeh metodah pa z lastnim programom, napisanim v programskem jeziku Python. Izkaže se, da je zadovoljiva kalibracija z odzivno površino mogoča le, če je začetni numerični model dovolj dober približek realnega sistema. Zadovoljive rezultate z Bayesovim pristopom h kalibriranju pa lahko dosežemo tudi ob zmerni modelski napaki. Ne glede na izbrano metodo pa je kalibriranje numeričnega modela smiselno le takrat, ko je že osnovni model dober približek realnega stanja (t.j. ima majhno diskretizacijsko in modelsko napako).In this thesis, we study the procedures for calibrating numerical models of structural systems to match the experimental data. Two possible approaches are discussed in more detail - the response surface method and the Bayesian approach. Using the response surface method, we approximate the response with a function of chosen parameters and search for values that minimize the error between the approximation and the experiments. The Bayesian approach, however, is based on the assumption that parameter values are a random variable with a certain probability distribution and searches for the most probable solutions. Two examples are considered: a simple beam and a structure consisting of a steel frame and a steel-reinforced concrete slab supported on a corrugated steel plateexperimental results are available for the latter. Numerical analyses are performed with the commercial software Ansys. Calibration of the numerical model by both methods is performed using our own computer code written in the Python programming language. It turns out that successful calibration with a response surface is only possible if the initial numerical model is a good approximation of the real system. However, satisfactory results when calibrating a numerical model with moderate model error can also be achieved with the Bayesian approach. Regardless of the chosen method, calibration of a numerical model should be used only after discretization and model error is minimized

Surrogating and calibrating finite element models of tall timber buildings

As the number of tall wooden buildings increases, a good understanding of their dynamic behaviour becomes important. This calls for the collection of empirical data, namely in-situ measured dynamic responses, to enable the calibration of finite element models, the use of surrogates, Bayesian structural identification and uncertainty quantification.As the number of tall wooden buildings increases, a good understanding of their dynamic behaviour becomes important. This calls for the collection of empirical data, namely in-situ measured dynamic responses, to enable the calibration of finite element models, the use of surrogates, Bayesian structural identification and uncertainty quantification

Model updating of seven-storey cross-laminated timber building designed on frequency-responsefunctions-based modal testing

Based on the experimental estimation of the key dynamic properties of a seven-storey building made entirely of cross-laminated timber (CLT) panels, the finite element (FE) model updating was performed. The dynamic properties were obtained from an input-output full-scale modal testing of the building in operation. The chosen parameters for the FE model updating allowed the consideration of the timber connections in a smeared sense. This approach led to an excellent match between the first six experimental and numerical modes of vibrations, despite spatial aliasing. Moreover, it allowed, together with the sensitivity analysis, to estimate the stiffness (affected by the connections) of the building structural elements. Thus, the study provides an insight into the as-built stiffness and modal properties of tall CLT building. This is valuable because of the currently limited knowledge about the dynamics of tall timber buildings under service loadings, especially because their design is predominantly governed by the wind-generated vibrations

Bayesian updating of tall timber building model using modal data

A framework for the probabilistic finite element model updating based on measured modal data is presented. The described framework is applied to a seven-storey building made of cross-laminated timber panels. The experimental estimates based on the forced vibration test are used in the process of model updating. First, a generalized Polynomial Chaos surrogate model is derived representing the map from the model parameters to the eigenfrequencies and the eigenvectors. To overcome the difficulties caused by mode switching, we propose a novel approach to mode tracking based on partitioning an extended and low-rank representation of the k mode shapes resulting from different setups of the finite element model into k clusters by the k-means clustering algorithm. Second, the surrogate model derived with the help of mode pairing is used to efficiently perform sensitivity analysis and uncertainty quantification of the first five frequencies and the correspon ding mode shapes. Finally, the surrogate-based Bayesian update of the model parameters is efficiently performed, providing engineers not only with a finite element model that gives a good fit to the experimental modal data, but also a stochastic model that represents the uncertainties originating from the initial model and the uncertainties of measuring modal properties