Some advances in open problems of isogeometric analysis

[EN]Isogeometric analysis (IGA) has arisen as an attempt to unify the fields of CAD and classical finite element methods. The main idea of IGA consists in using for analysis the same functions (splines) that are used in CAD representation of the geometry. The main advantage with respect to the traditional finite element method is a higher smoothness of the numerical solution and more accurate representation of the geometry. IGA seems to be a promising tool with wide range of applications in engineering. However, this relatively new technique have some open problems that require a solution. In this work we present our results and contributions to this issue…

Magnetohydrodynamic Effects On Mixed Convection Flows In Channels And Ducts

This work focuses on magnetohydrodynamic (MHD) mixed convection flow of electrically conducting fluids enclosed in simple 1D and 2D geometries in steady periodic regime. In particular, in Chapter one a short overview is given about the history of MHD, with reference to papers available in literature, and a listing of some of its most common technological applications, whereas Chapter two deals with the analytical formulation of the MHD problem, starting from the fluid dynamic and energy equations and adding the effects of an external imposed magnetic field using the Ohm's law and the definition of the Lorentz force. Moreover a description of the various kinds of boundary conditions is given, with particular emphasis given to their practical realization. Chapter three, four and five describe the solution procedure of mixed convective flows with MHD effects. In all cases a uniform parallel magnetic field is supposed to be present in the whole fluid domain transverse with respect to the velocity field. The steady-periodic regime will be analyzed, where the periodicity is induced by wall temperature boundary conditions, which vary in time with a sinusoidal law. Local balance equations of momentum, energy and charge will be solved analytically and numerically using as parameters either geometrical ratios or material properties. In particular, in Chapter three the solution method for the mixed convective flow in a 1D vertical parallel channel with MHD effects is illustrated. The influence of a transverse magnetic field will be studied in the steady periodic regime induced by an oscillating wall temperature. Analytical and numerical solutions will be provided in terms of velocity and temperature profiles, wall friction factors and average heat fluxes for several values of the governing parameters. In Chapter four the 2D problem of the mixed convective flow in a vertical round pipe with MHD effects is analyzed. Again, a transverse magnetic field influences the steady periodic regime induced by the oscillating wall temperature of the wall. A numerical solution is presented, obtained using a finite element approach, and as a result velocity and temperature profiles, wall friction factors and average heat fluxes are derived for several values of the Hartmann and Prandtl numbers. In Chapter five the 2D problem of the mixed convective flow in a vertical rectangular duct with MHD effects is discussed. As seen in the previous chapters, a transverse magnetic field influences the steady periodic regime induced by the oscillating wall temperature of the four walls. The numerical solution obtained using a finite element approach is presented, and a collection of results, including velocity and temperature profiles, wall friction factors and average heat fluxes, is provided for several values of, among other parameters, the duct aspect ratio. A comparison with analytical solutions is also provided, as a proof of the validity of the numerical method. Chapter six is the concluding chapter, where some reflections on the MHD effects on mixed convection flow will be made, in agreement with the experience and the results gathered in the analyses presented in the previous chapters. In the appendices special auxiliary functions and FORTRAN program listings are reported, to support the formulations used in the solution chapters.

Microfluidic device and interfacial transport: application to biomolecules and nanostructures

The aim of my dissertation is to provide new knowledge and applications of microfluidics in a variety of problems, from materials science, devices, and biomedicine, where the control on the fluid dynamics and the local concentration of the solutions containing the relevant molecules (either materials, precursors, or biomolecules) is crucial. The control of interfacial phenomena occurring in solutions at dierent length scales is compelling in nanotechnology for devising new sensors, molecular electronics devices, memories. Microfluidic devices were fabricated and integrated with organic electronics devices. The transduction involves the species in the solution which infills the transistor channel and confined by the microfluidic device. This device measures what happens on the surface, at few nanometers from the semiconductor channel. Soft-lithography was adopted to fabricate platinum electrodes, starting from platinum carbonyl precursor. I proposed a simple method to assemble these nanostructures in periodic arrays of microstripes, and form conductive electrodes with characteristic dimension of 600 nm. The conductivity of these sub-microwires is compared with the values reported in literature and bulk platinum. The process is suitable for fabricating thin conductive patterns for electronic devices or electrochemical cells, where the periodicity of the conductive pattern is comparable with the diusion length of the molecules in solution. The ordering induced among artificial nanostructures is of particular interest in science. I show that large building blocks, like carbon nanotubes or core-shell nanoparticles, can be ordered and self-organised on a surface in patterns due to capillary forces. The eective probability of inducing order with microfluidic flow is modeled with finite element calculation on the real geometry of the microcapillaries, in soft-lithographic process. The oligomerization of A40 peptide in microconfined environment represents a new investigation of the extensively studied peptide aggregation. The added value of the approach I devised is the precise control on the local concentration of peptides together with the possibility to mimick cellular crowding. Four populations of oligomers where distinguished, with diameters ranging from 15 to 200 nm. These aggregates could not be addresses separately in fluorescence. The statistical analysis on the atomic force microscopy images together with a model of growth reveal new insights on the kinetics of amyloidogenesis as well as allows me to identify the minimum stable nucleus size. This is an important result owing to its implications in the understanding and early diagnosis and therapy of the Alzheimer’s disease

Shear behavior of reinforced concrete slabs under concentrated load: an investigation through Sequentially Linear Analysis

ABSTRACT (italiano) Con crescente attenzione riguardo al problema della sicurezza di ponti e viadotti esistenti nei Paesi Bassi, lo scopo della presente tesi è quello di studiare, mediante la modellazione con Elementi Finiti ed il continuo confronto con risultati sperimentali, la risposta in esercizio di elementi che compongono infrastrutture del genere, ovvero lastre in calcestruzzo armato sollecitate da carichi concentrati. Tali elementi sono caratterizzati da un comportamento ed una crisi per taglio, la cui modellazione è, da un punto di vista computazionale, una sfida piuttosto ardua, a causa del loro comportamento fragile combinato a vari effetti tridimensionali. La tesi è incentrata sull'utilizzo della Sequentially Linear Analysis (SLA), un metodo di soluzione agli Elementi Finiti alternativo rispetto ai classici approcci incrementali e iterativi. Il vantaggio della SLA è quello di evitare i ben noti problemi di convergenza tipici delle analisi non lineari, specificando direttamente l'incremento di danno sull'elemento finito, attraverso la riduzione di rigidezze e resistenze nel particolare elemento finito, invece dell'incremento di carico o di spostamento. Il confronto tra i risultati di due prove di laboratorio su lastre in calcestruzzo armato e quelli della SLA ha dimostrato in entrambi i casi la robustezza del metodo, in termini di accuratezza dei diagrammi carico-spostamento, di distribuzione di tensioni e deformazioni e di rappresentazione del quadro fessurativo e dei meccanismi di crisi per taglio. Diverse variazioni dei più importanti parametri del modello sono state eseguite, evidenziando la forte incidenza sulle soluzioni dell'energia di frattura e del modello scelto per la riduzione del modulo elastico trasversale. Infine è stato effettuato un paragone tra la SLA ed il metodo non lineare di Newton-Raphson, il quale mostra la maggiore affidabilità della SLA nella valutazione di carichi e spostamenti ultimi insieme ad una significativa riduzione dei tempi computazionali. ABSTRACT (english) With increasing attention to the assessment of safety in existing dutch bridges and viaducts, the aim of the present thesis is to study, through the Finite Element modeling method and the continuous comparison with experimental results, the real response of elements that compose these infrastructures, i.e. reinforced concrete slabs subjected to concentrated loads. These elements are characterized by shear behavior and crisis, whose modeling is, from a computational point of view, a hard challenge, due to their brittle behavior combined with various 3D effects. The thesis is focused on the use of Sequentially Linear Analysis (SLA), an alternative solution technique to classical non linear Finite Element analyses that are based on incremental and iterative approaches. The advantage of SLA is to avoid the well-known convergence problems of non linear analyses by directly specifying a damage increment, in terms of a reduction of stiffness and strength in the particular finite element, instead of a load or displacement increment. The comparison between the results of two laboratory tests on reinforced concrete slabs and those obtained by SLA has shown in both the cases the robustness of the method, in terms of accuracy of load-displacements diagrams, of the distribution of stress and strain and of the representation of the cracking pattern and of the shear failure mechanisms. Different variations of the most important parameters have been performed, pointing out the strong incidence on the solutions of the fracture energy and of the chosen shear retention model. At last a confrontation between SLA and the non linear Newton-Raphson method has been executed, showing the better reliability of the SLA in the evaluation of the ultimate loads and displacements, together with a significant reduction of computational times.

A model for diffusive and displacive phase transitions in steels

Heat treatment of steels is a process of fundamental importance in tailoring the properties of a material to the desired application; developing a model able to describe such process would allow to predict the microstructure obtained from the treatment and the consequent mechanical properties of the material. A steel, during a heat treatment, can undergo two different kinds of phase transitions [p.t.]: diffusive (second order p.t.) and displacive (first order p.t.); in this thesis, an attempt to describe both in a thermodynamically consistent framework is made; a phase field, diffuse interface model accounting for the coupling between thermal, chemical and mechanical effects is developed, and a way to overcome the difficulties arising from the treatment of the non-local effects (gradient terms) is proposed. The governing equations are the balance of linear momentum equation, the Cahn-Hilliard equation and the balance of internal energy equation. The model is completed with a suitable description of the free energy, from which constitutive relations are drawn. The equations are then cast in a variational form and different numerical techniques are used to deal with the principal features of the model: time-dependency, non-linearity and presence of high order spatial derivatives. Simulations are performed using DOLFIN, a C++ library for the automated solution of partial differential equations by means of the finite element method; results are shown for different test-cases. The analysis is reduced to a two dimensional setting, which is simpler than a three dimensional one, but still meaningful.

Resistenza di travi metalliche a doppio T con irrigidimenti longitudinali soggette a carichi trasversali concentrati

The thesis deals with the patch loading of I-girder with two longitudinal stiffeners. The configuration with two longitudinal stiffeners is often an excellent solution for beams of higher than 3 meters but has not yet been discussed in EN 1993-1-5. It is proposed a model of resistance harmonized with the methods used in Eurocodes for the other problems of buckling. The model contains three significant parts: the yield resistance, the elastic critical load used to determine the slenderness parameter and a reduction factor that relates the resistance to the slenderness. The thesis is structured into eight chapters, in addition to Preface and Table of Contents. Chapter 3 is a list of all symbols used. Chapter 4 presents a review of earlier works. Chapter 5 details the experimental investigations conducted by Gozzi (2007) on three samples without longitudinal stiffeners. Due to the difficulty of completing a personal physical model testing during the doctorate, it was decided to carefully study the laboratory work by Gozzi and use it as a basis for the calibration of the numerical study. In Chapter 6 is presented the first part of the numerical study. At this stage, the laboratory tests conducted by Gozzi have been reproduced through a finite element model. It is observed a good agreement of numerical results with test data. In Chapter 7 summarizes the results of numerical analysis of the girder with two longitudinal stiffeners. Chapter 8 presents the procedure proposed for calculating the ultimate patch loading resistance of the girder with two longitudinal stiffeners. Chapter 9 contains a summary of work done in this thesis with suggestions for the most important issues for future development. Chapter 10 lists the references. There are also three appendices with test data by Gozzi and data obtained from literature.

Polymer properties measured by micromechanical cantilever technique

Surface stress changes induced by specific adsorption of molecules were investigated using a micromechanical cantilever sensor (MCS) device. 16 MCS are grouped within four separate wells. Each well can be addressed independently by different liquid enabling functionalization of MCS separately by flowing different solutions through each well and performing sensing and reference experiments simultaneously. In addition, each well contains a fixed reference mirror, which allows measuring the absolute bending of MCS. The effect of the flow rate on the MCS bending change was found to be dependent on the absolute bending value of MCS. In addition, the signal from the reference mirror can be used to follow refractive index changes upon mixing different solutions. Finite element simulation of solution exchange in wells was compared with experiment results. Both revealed that one solution can be exchanged by another one after a total volume of 200 µl has flown through. Using MCS, the adsorption of thiolated deoxyribonucleic acid (DNA) molecules and 6-mercapto-1-hexanol (MCH) on gold surfaces, and the DNA hybridization were performed. The nanomechanical response is in agreement with data reported by Fritz et al.1 Thus, the multiwell device is readily applicable for sensing of multiple chemical and biological recognition events in a single step. In this context controlled release and uptake of drugs are currently widely discussed. As a model system, we have used polystyrene (PS) spheres with diameters in the order of µm. The swelling behavior of individual PS spheres in toluene vapor was studied via mass loading by means of micromechanical cantilever sensors. For 4–8% cross-linked PS a mass increase of 180% in saturated toluene vapor was measured. In addition, the diameter change in saturated toluene vapor was measured and the corresponding volume increase of 200% was calculated. The mass of the swollen PS sphere decreases with increasing exposure time to ultraviolet (UV) light. The swelling response is significantly different between the first and the second exposure to toluene vapor. This is attributed to the formation of a cross-linked shell at the surface of the PS spheres. Shape persistent parts were observed for locally UV irradiated PS spheres. These PS spheres were found to be fluorescent and cracks occur after exposure in toluene liquid. The diffusion time of dye molecules in PS spheres increases with increasing chemical cross-linking density. This concept of locally dissolving non cross-linked PS from the sphere was applied to fabricate donut structures on surfaces. Arrays of PS spheres were fabricated using spin coating. The donut structure was produced simply after liquid solvent rinsing. The complete cross-linking of PS spheres was found after long exposure time to UV. We found that stabilizers play a major role in the formation of the donut nanostructures.

Micro-FEM models based on micro-CT reconstructions for the in vitro characterization of the elastic properties of trabecular bone tissue.

This master’s thesis describes the research done at the Medical Technology Laboratory (LTM) of the Rizzoli Orthopedic Institute (IOR, Bologna, Italy), which focused on the characterization of the elastic properties of the trabecular bone tissue, starting from october 2012 to present. The approach uses computed microtomography to characterize the architecture of trabecular bone specimens. With the information obtained from the scanner, specimen-specific models of trabecular bone are generated for the solution with the Finite Element Method (FEM). Along with the FEM modelling, mechanical tests are performed over the same reconstructed bone portions. From the linear-elastic stage of mechanical tests presented by experimental results, it is possible to estimate the mechanical properties of the trabecular bone tissue. After a brief introduction on the biomechanics of the trabecular bone (chapter 1) and on the characterization of the mechanics of its tissue using FEM models (chapter 2), the reliability analysis of an experimental procedure is explained (chapter 3), based on the high-scalable numerical solver ParFE. In chapter 4, the sensitivity analyses on two different parameters for micro-FEM model’s reconstruction are presented. Once the reliability of the modeling strategy has been shown, a recent layout for experimental test, developed in LTM, is presented (chapter 5). Moreover, the results of the application of the new layout are discussed, with a stress on the difficulties connected to it and observed during the tests. Finally, a prototype experimental layout for the measure of deformations in trabecular bone specimens is presented (chapter 6). This procedure is based on the Digital Image Correlation method and is currently under development in LTM.

Shear behaviour of reinforced cconcrete slab under concentrated load: an investigation through non-linear and sequentially linear analysis

English: The assessment of safety in existing bridges and viaducts led the Ministry of Public Works of the Netherlands to finance a specific campaing aimed at the study of the response of the elements of these infrastructures. Therefore, this activity is focused on the investigation of the behaviour of reinforced concrete slabs under concentrated loads, adopting finite element modeling and comparison with experimental results. These elements are characterized by shear behaviour and crisi, whose modeling is, from a computational point of view, a hard challeng, due to the brittle behavior combined with three-dimensional effects. The numerical modeling of the failure is studied through Sequentially Linear Analysis (SLA), an alternative Finite Element method, with respect to traditional incremental and iterative approaches. The comparison between the two different numerical techniques represents one of the first works and comparisons in a three-dimensional environment. It's carried out adopting one of the experimental test executed on reinforced concrete slabs as well. The advantage of the SLA is to avoid the well known problems of convergence of typical non-linear analysis, by directly specifying a damage increment, in terms of reduction of stiffness and resistance in particular finite element, instead of load or displacement increasing on the whole structure . For the first time, particular attention has been paid to specific aspects of the slabs, like an accurate constraints modeling and sensitivity of the solution with respect to the mesh density. This detailed analysis with respect to the main parameters proofed a strong influence of the tensile fracture energy, mesh density and chosen model on the solution in terms of force-displacement diagram, distribution of the crack patterns and shear failure mode. The SLA showed a great potential, but it requires a further developments for what regards two aspects of modeling: load conditions (constant and proportional loads) and softening behaviour of brittle materials (like concrete) in the three-dimensional field, in order to widen its horizons in these new contexts of study.

Numerical solutions of elliptic inverse problems via the equation error method

To estimate a parameter in an elliptic boundary value problem, the method of equation error chooses the value that minimizes the error in the PDE and boundary condition (the solution of the BVP having been replaced by a measurement). The estimated parameter converges to the exact value as the measured data converge to the exact value, provided Tikhonov regularization is used to control the instability inherent in the problem. The error in the estimated solution can be bounded in an appropriate quotient norm; estimates can be derived for both the underlying (infinite-dimensional) problem and a finite-element discretization that can be implemented in a practical algorithm. Numerical experiments demonstrate the efficacy and limitations of the method.

Analytical investigation of squeeze film dampers

Squeeze film damping effects naturally occur if structures are subjected to loading situations such that a very thin film of fluid is trapped within structural joints, interfaces, etc. An accurate estimate of squeeze film effects is important to predict the performance of dynamic structures. Starting from linear Reynolds equation which governs the fluid behavior coupled with structure domain which is modeled by Kirchhoff plate equation, the effects of nondimensional parameters on the damped natural frequencies are presented using boundary characteristic orthogonal functions. For this purpose, the nondimensional coupled partial differential equations are obtained using Rayleigh-Ritz method and the weak formulation, are solved using polynomial and sinusoidal boundary characteristic orthogonal functions for structure and fluid domain respectively. In order to implement present approach to the complex geometries, a two dimensional isoparametric coupled finite element is developed based on Reissner-Mindlin plate theory and linearized Reynolds equation. The coupling between fluid and structure is handled by considering the pressure forces and structural surface velocities on the boundaries. The effects of the driving parameters on the frequency response functions are investigated. As the next logical step, an analytical method for solution of squeeze film damping based upon Green’s function to the nonlinear Reynolds equation considering elastic plate is studied. This allows calculating modal damping and stiffness force rapidly for various boundary conditions. The nonlinear Reynolds equation is divided into multiple linear non-homogeneous Helmholtz equations, which then can be solvable using the presented approach. Approximate mode shapes of a rectangular elastic plate are used, enabling calculation of damping ratio and frequency shift as well as complex resistant pressure. Moreover, the theoretical results are correlated and compared with experimental results both in the literature and in-house experimental procedures including comparison against viscoelastic dampers.

Transformer modeling ATP : internal faults & high-frequency discretization

Power transformers are key components of the power grid and are also one of the most subjected to a variety of power system transients. The failure of a large transformer can cause severe monetary losses to a utility, thus adequate protection schemes are of great importance to avoid transformer damage and maximize the continuity of service. Computer modeling can be used as an efficient tool to improve the reliability of a transformer protective relay application. Unfortunately, transformer models presently available in commercial software lack completeness in the representation of several aspects such as internal winding faults, which is a common cause of transformer failure. It is also important to adequately represent the transformer at frequencies higher than the power frequency for a more accurate simulation of switching transients since these are a well known cause for the unwanted tripping of protective relays. This work develops new capabilities for the Hybrid Transformer Model (XFMR) implemented in ATPDraw to allow the representation of internal winding faults and slow-front transients up to 10 kHz. The new model can be developed using any of two sources of information: 1) test report data and 2) design data. When only test-report data is available, a higher-order leakage inductance matrix is created from standard measurements. If design information is available, a Finite Element Model is created to calculate the leakage parameters for the higher-order model. An analytical model is also implemented as an alternative to FEM modeling. Measurements on 15-kVA 240?/208Y V and 500-kVA 11430Y/235Y V distribution transformers were performed to validate the model. A transformer model that is valid for simulations for frequencies above the power frequency was developed after continuing the division of windings into multiple sections and including a higher-order capacitance matrix. Frequency-scan laboratory measurements were used to benchmark the simulations. Finally, a stability analysis of the higher-order model was made by analyzing the trapezoidal rule for numerical integration as used in ATP. Numerical damping was also added to suppress oscillations locally when discontinuities occurred in the solution. A maximum error magnitude of 7.84% was encountered in the simulated currents for different turn-to-ground and turn-to-turn faults. The FEM approach provided the most accurate means to determine the leakage parameters for the ATP model. The higher-order model was found to reproduce the short-circuit impedance acceptably up to about 10 kHz and the behavior at the first anti-resonant frequency was better matched with the measurements.

DEVELOPMENT OF CONFORMABLE CNG TANK FOR AUTOMOTIVE APPLICATION

The purpose of this study is to design, develop and integrate a Compressed Natural Gas (CNG) tank that will have a conformable shape for efficient storage in a light-duty pick-up truck. The CNG tank will be a simple rectangular box geometry to demonstrate capability of non-cylindrical shapes. Using CAD drawings of the truck, a conformable tank will be designed to fit under the pick-up bed. The intent of the non-cylindrical CNG tank is to demonstrate improvement in size over the current solution, which is a large cylinder in the box of a pick-up truck. The geometry of the tank’s features is critical to its size and strength. The optimized tank design will be simulated with Finite Element Analysis (FEA) to determine critical stress regions, and appropriate design changes will be made to reduce stress concentration. Following the American National Standard Institute (ANSI) guide, different aluminum alloys will be optimized to obtain the best possible result for the CNG tank.

MODEL UPDATING AND STRUCTURAL HEALTH MONITORING OF HORIZONTAL AXIS WIND TURBINES VIA ADVANCED SPINNING FINITE ELEMENTS AND STOCHASTIC SUBSPACE IDENTIFICATION METHODS

Wind energy has been one of the most growing sectors of the nation’s renewable energy portfolio for the past decade, and the same tendency is being projected for the upcoming years given the aggressive governmental policies for the reduction of fossil fuel dependency. Great technological expectation and outstanding commercial penetration has shown the so called Horizontal Axis Wind Turbines (HAWT) technologies. Given its great acceptance, size evolution of wind turbines over time has increased exponentially. However, safety and economical concerns have emerged as a result of the newly design tendencies for massive scale wind turbine structures presenting high slenderness ratios and complex shapes, typically located in remote areas (e.g. offshore wind farms). In this regard, safety operation requires not only having first-hand information regarding actual structural dynamic conditions under aerodynamic action, but also a deep understanding of the environmental factors in which these multibody rotating structures operate. Given the cyclo-stochastic patterns of the wind loading exerting pressure on a HAWT, a probabilistic framework is appropriate to characterize the risk of failure in terms of resistance and serviceability conditions, at any given time. Furthermore, sources of uncertainty such as material imperfections, buffeting and flutter, aeroelastic damping, gyroscopic effects, turbulence, among others, have pleaded for the use of a more sophisticated mathematical framework that could properly handle all these sources of indetermination. The attainable modeling complexity that arises as a result of these characterizations demands a data-driven experimental validation methodology to calibrate and corroborate the model. For this aim, System Identification (SI) techniques offer a spectrum of well-established numerical methods appropriated for stationary, deterministic, and data-driven numerical schemes, capable of predicting actual dynamic states (eigenrealizations) of traditional time-invariant dynamic systems. As a consequence, it is proposed a modified data-driven SI metric based on the so called Subspace Realization Theory, now adapted for stochastic non-stationary and timevarying systems, as is the case of HAWT’s complex aerodynamics. Simultaneously, this investigation explores the characterization of the turbine loading and response envelopes for critical failure modes of the structural components the wind turbine is made of. In the long run, both aerodynamic framework (theoretical model) and system identification (experimental model) will be merged in a numerical engine formulated as a search algorithm for model updating, also known as Adaptive Simulated Annealing (ASA) process. This iterative engine is based on a set of function minimizations computed by a metric called Modal Assurance Criterion (MAC). In summary, the Thesis is composed of four major parts: (1) development of an analytical aerodynamic framework that predicts interacted wind-structure stochastic loads on wind turbine components; (2) development of a novel tapered-swept-corved Spinning Finite Element (SFE) that includes dampedgyroscopic effects and axial-flexural-torsional coupling; (3) a novel data-driven structural health monitoring (SHM) algorithm via stochastic subspace identification methods; and (4) a numerical search (optimization) engine based on ASA and MAC capable of updating the SFE aerodynamic model.

Using Superelements in the Calculation of Lattice-Boom Cranes

In this paper a superelement formulation for geometric nonlinear finite element analysis is proposed. The element formulation is based on matrices generated by the static condensation algorithm. After defining the element characteristics, a method for the calculation of the element forces in a large displacement and rotation analysis is developed. In order to use the element in the solution of stability problems, the formulation of the geometric stiffness matrix is derived. An example shows the benefits of the element for the calculation of lattice-boom cranes.