Analysis of the dynamic behavior on structures is one vital aspect of designing structures such as buildings and bridges. Determination of the correct damping factor is of critical importance as it is the governing factor of dynamic design. Damping on structures exhibits a very complex behavior. Different models are suggested in literature to explain damping behavior. The usefulness of a valid damping model depends on how easily it can be adopted to analyze the dynamic behavior. Ease of mathematically representing the model and ease of analyzing the dynamic behavior by using the mathematical representation are the two determining aspects of the utility of the selected model. This thesis presents a parametric representation of non-linear damping models of the form presented by [Jea86] and the mathematical techniques to use the parametrically represented damping model in dynamic behavior analysis. In the damping model used in this thesis, the damping factor is proportional to the amplitude of vibration of the structure. However, determination of the amplitude again depends on the damping of the structure for a given excitation. Also, the equations which explain the behavior of motion are differential equations in a matrix form that is generally linearly inseparable. This thesis addresses these challenges and presents a numerical method to solve the motion equations by using Runge-Kutta techniques. This enables one to use a given non-linear model of the form proposed by [Jea86] to analyze the actual response of the structure to a given excitation from wind, seismic or any other source. Several experiments were conducted for reinforced concrete and steel framed buildings to evaluate the proposed framework. The non-linear damping model proposed by [Sat03], which conforms to [Jea86] is used to demonstrate the use of the proposed techniques. Finally, a new damping model is proposed based on the actual behavior and the serviceability criteria, which better explains the damping behavior of structures.

Submitted by Willian Minnemann Kuhnert null (willian.kuhnert@feb.unesp.br) on 2016-08-29T20:53:53Z No. of bitstreams: 1 VERSAO_FINAL_WILLIANMK.pdf: 7510529 bytes, checksum: d0f36cb6fb5aeb748d7d26333a6c853c (MD5) / Approved for entry into archive by Ana Paula Grisoto (grisotoana@reitoria.unesp.br) on 2016-08-31T12:32:11Z (GMT) No. of bitstreams: 1 kuhnert_wm_me_bauru.pdf: 7510529 bytes, checksum: d0f36cb6fb5aeb748d7d26333a6c853c (MD5) / Made available in DSpace on 2016-08-31T12:32:11Z (GMT). No. of bitstreams: 1 kuhnert_wm_me_bauru.pdf: 7510529 bytes, checksum: d0f36cb6fb5aeb748d7d26333a6c853c (MD5) Previous issue date: 2016-07-19 / Fundação para o Desenvolvimento da UNESP (FUNDUNESP) / O isolamento de vibração é a técnica mais utilizada atualmente para a proteção de mecanismos e estruturas que sofrem excitação, seja ela por choque/impacto, seja ela harmônica. Este trabalho adiciona ao isolador de vibração comum, composto por molas e amortecedores, um elemento conhecido como inerter, que recentemente tem chamado bastante a atenção da comunidade científica, e também, separadamente, adiciona amortecedores não lineares, com o intuito de avaliar a influência destes elementos no isolamento. As curvas de transmissibilidade obtidas, que indicam a performance do isolamento à excitação harmônica, para os isoladores com inerter são comparadas à de um isolador comum composto somente por uma mola, e entre elas, enquanto que as curvas obtidas para os isoladores com amortecedores não lineares são comparadas entre si e à de um isolador comum com amortecimento linear. Os resultados obtidos mostram que a adição do inerter aos isoladores de vibração pode ser muito benéfica para o isolamento em determinadas faixas de frequência, mas em outras não, e tais faixas dependem de como o isolador é construído. Além disso, os isoladores com inerter são benéficos principalmente para sistemas subamortecidos. Os isoladores subamortecidos com inerter apresentaram características de isolamento diferentes uns dos outros, o que os leva a serem aplicados em diferentes situações. Os resultados obtidos para os isoladores com amortecedores não lineares mostraram que tais sistemas também podem melhorar ou piorar o isolamento em determinadas faixas de frequência quando comparados à um isolador com amortecimento linear. / The vibration isolation is currently the most used technique for protecting mechanisms and structures which are under shock/impact or harmonic excitation. This work presents to the common vibration isolator, consisted by springs and dampers, an element known as inerter, which recently has took great attention in the scientific community, and also presents the use of non-linear dampers to analyze the influence of these elements on isolation. The transmissibility curves obtained, which indicate the performance of the isolation for systems under harmonic excitation, for the isolators with the inerter element are compared with the spring-damper isolator frequency response as well the isolators with non-linear damping. The results obtained show that the addition of the inerter element can be beneficial for the isolation performance in a frequency range, but degrades the high frequency isolation, and they depend on how the isolator is built. Besides, the isolators with inerter are beneficial mainly for underdamped systems. The different underdamped systems with inerter presented unique isolation characteristics. The results obtained for the isolators with non-linear dampers presented that such systems can also improve the isolation in certain frequency ranges when compared to an isolator with linear damping. / FUNDUNESP: 110/2013-CCp/PIB

The work is focused in investigating the effectiveness of discretized damping tapes applied to a cantilever beam subjected to free and forced vibrations. The work is divided into three main sections. First, we performed material characterization of the viscoelastic (VE) pressure sensitive adhesive layer of the damping tapes. To do so, we designed a novel quad shear specimen to measure shear storage and loss moduli, and tan delta from dynamic mechanical analyzer measurements. Second, the optimal discretization length for different damping tapes was experimentally determined and analytically verified using linear viscoelasticity and basic strength of materials and vibrations principles. These results showed a mean to improve the damping of a structure without increasing the weight of the added damping layer. Third, a nonlinear analysis was performed for cantilever beams with damping layers subjected to parametric excitation. Comparison of the response amplitude of the parametrically excited beam was performed for different discretization lengths, and system identification of the nonlinear parameters was carried out. The effects of large deflections of a beam under parametric excitation were analyzed; large deflections were found to induce localized buckling of the stiff constraining layer of the damping tape that would invalidate some of the assumptions and analytical solutions that do not take such phenomena into account. / Master of Science

The Vlasov technique is used to model the electron cooling force. Limitations of the applicability of the method is obtained by considering the perturbations of the electron plasma. Analytical expressions of the electron cooling force, valid beyond the Coulomb logarithm approximation, are derived and compared to numerical calculations using adaptive Monte Carlo integration. The calculated longitudinal cooling force is verified with measurements in CELSIUS. Transverse damping rates of betatron oscillations for a nonlinear cooling force is explored. Experimental data of the transverse monochromatic instability is used to determine the rms angular spread due to solenoid field imperfections in CELSIUS. The result, θrms= 0.16 ± 0.02 mrad, is in agreement with the longitudinal cooling force measurements. This verifies the internal consistency of the model and shows that the transverse and longitudinal cooling force components have different velocity dependences. Simulations of electron cooling with applications to HESR show that the momentum reso- lution ∆p/p smaller than 10−5 is feasible, as needed for the charmonium spectroscopy in the experimental program of PANDA. By deflecting the electron beam angle to make use of the monochromatic instability, a reasonable overlap between the circulating antiproton beam and the internal target can be maintained. The simulations also indicate that the cooling time is considerably shorter than expected.

<p>Within the past decade, advances in small-scale electronics have reduced power consumption requirements such that mechanisms for harnessing ambient kinetic energy for self-sustenance are a viable technology. Such devices, known as energy harvesters, may enable self-sustaining wireless sensor networks for applications ranging from Tsunami warning detection to environmental monitoring to cost-effective structural health diagnostics in bridges and buildings. In particular, flexible electroelastic materials such as lead-zirconate-titanate (PZT) are sought after in designing such devices due to their superior efficiency in transforming mechanical energy into the electrical domain in comparison to induction methods. To date, however, material and dynamic nonlinearities within the most popular type of energy harvester, an electroelastically laminated cantilever beam, has received minimal attention in the literature despite being readily observed in laboratory experiments. </p><p>In the first part of this dissertation, an experimentally validated first-principles based modeling framework for quantitatively characterizing the intrinsic nonlinearities and moderately large amplitude response of a cantilevered electroelastic generator is developed. Nonlinear parameter identification is facilitated by an analytic solution for the generator's dynamic response alongside experimental data. The model is shown to accurately describe amplitude dependent frequency responses in both the mechanical and electrical domains and implications concerning the conventional approach to resonant generator design are discussed. Higher order elasticity and nonlinear damping are found to be critical for correctly modeling the harvester response while inclusion of a proof mass is shown to invigorate nonlinearities a much lower driving amplitudes in comparison to electroelastic harvesters without a tuning mass.</p><p>The second part of the dissertation concerns dynamical systems design to purposefully engage nonlinear phenomena in the mechanical domain. In particular, two devices, one exploiting hysteretic nonlinearities and the second featuring homoclinic bifurcation are investigated. Both devices exploit nonlinear magnet interactions with piezoelectric cantilever beams and a first principles modeling approach is applied throughout. The first device is designed such that both softening and hardening nonlinear resonance curves produces a broader response in comparison to the linear equivalent oscillator. The second device makes use of a supercritical pitchfork bifurcation wrought by nonlinear magnetic repelling forces to achieve a bistable electroelastic dynamical system. This system is also analytically modeled, numerically simulated, and experimentally realized to demonstrate enhanced capabilities and new challenges. In addition, a bifurcation parameter within the design is examined as a either a fixed or adaptable tuning mechanism for enhanced sensitivity to ambient excitation. Analytical methodologies to include the method of Harmonic Balance and Melnikov Theory are shown to provide superior insight into the complex dynamics of the bistable system in response to deterministic and stochastic excitation.</p> / Dissertation

<p dir="ltr">Shape memory alloys (SMAs) belong to the category of smart materials due to their unique shape memory properties induced by a thermomechanically-triggered phase transformation. This phase changing process is also associated with a pronounced energy dissipation capacity. In recent years, the shape-recovery and energy-dissipating capabilities of SMAs have been object of extensive studies with particular focus on the opportunities they offer for the design of smart composites. The restoring stress of constrained SMAs as well as the modulus change, following thermal loading, can be leveraged to improve the static and dynamic performance, such as the pre/post-bulking behavior, the aerodynamic stability, and the impact resistance of composite materials embedded with SMA wires or fibers. The nonlinear damping resulting from the nonlinear material behavior associated with the ferro-elastic and pseudo-elastic phases was explored in a few studies focusing on vibration suppression in composites. Nonetheless, existing research mainly focused on either SMA wire or fiber reinforced composites, while the understanding of the dynamics of hybrid composites integrating SMA layers still presents several unexplored areas. In part, this technological gap might be explained by the fact that the most common SMA alloy, the so-called Nitinol, is expensive and hence not amenable to be deployed in large scale applications. With the most recent advancements in low-cost SMAs (e.g. Fe-based and Cu-based alloys), new applications that make more extensive use of SMAs are becoming viable. It follows that the understanding of the dynamic response of composites integrating SMA laminae becomes an important topic in order to support the development of innovative hybrid composite structures.</p><p dir="ltr">This dissertation explores the design and the nonlinear dynamic response of hybrid composites integrating SMA laminae, with a particular emphasis on the damping performance under different operating conditions. The dynamic properties of SMA monolithic beams and hybrid composite beams integrated with SMA laminae are investigated via one-dimensional constitutive models. Monolithic SMA beams are investigated to understand the fundamental aspects of the damping capacity of the material as well as possible bifurcation phenomena occurring under different types of harmonic excitations and different levels of pre-strain. The study then focuses on hybrid composite beams, highlighting the effects of design parameters, such the thickness, position, and pre-strain level of SMA layers on the transient and forced dynamic characteristics.</p><p dir="ltr">To further explore the potential of embedding SMA laminae to tailor the damping capacity of the hybrid composite and optimize the distribution of SMA materials, hybrid composite plates (HCPs) assembled by stacking fiber composites and SMA layers (either monolithic or patterned) are explored. The damping capacity of the HCP is assessed under different operating conditions, with emphasis on the effect of pre-strain levels in the SMA layers. The optimization study focuses on understanding the distribution of SMA materials and the synergistic role of patterning and pre-straining individual SMA layers within the HCP. The damping capacity of the HCP is also estimated as a function of the SMA total transformed volume fraction in order to identify the types of patterns and the pre-strain profiles capable of improving the overall damping capacity of the HCP.</p><p dir="ltr">The investigation on the dynamics of SMA hybrid composites continues with the optimal design of sandwich composite beams with elastic face sheets and SMA cellular cores. A deep learning-based surrogate model is proposed to efficiently predict the nonlinear mechanical response of the SMA sandwich beams subject to transverse loading, hence enabling the optimization of the SMA cellular core. The multi-objective optimization of the energy-dissipating capacity and of the overall stiffness is then performed by taking advantage of evolutionary algorithms. Once the optimal geometric parameters of the SMA cellular cores are obtained, finite element simulations are conducted to numerically validate the optimal configurations of the sandwich beams.</p><p dir="ltr">Finally, the numerical models are validated via experimental measurements conducted on monolithic SMA beams. Tests include both tensile and vibration measurements in both the ferro-elastic and pseudo-elastic regimes. The stress-strain relations obtained from tensile tests are used to calibrate the constitutive model of SMAs. Subsequently, experimental vibration tests are performed on clamped-clamped SMA beams to assess the effect of pre-strain levels on the damping capacity of SMA beams via a dedicated experimental setup to apply and maintain the pre-strain levels. The theoretical, numerical, and experimental results provided in this dissertation can serve as important guidelines to design lightweight SMA smart composites with customizable dynamic behavior.</p>

No description available.

Energy harvesting of ambient and aeroelastic vibrations is important for reducing the dependence of wireless sensing and networks on batteries. We develop a configuration for a piezoelectric energy harvester with the capability to wirelessly communicate vibration measurements while using those vibrations to power the sensing and communication devices. Particularly, we perform experiments that aim at identifying challenges to overcome in the development of such a configuration. Towards that objective, we successfully tested a self-powered real-time point-to-point wireless communication system between a vibration sensor and transmission and receiving modules. The sensing device and transmission module are powered by the vibrating object using a piezoelectric energy harvester. The communication is established by using two XBee modules. In the second part of this dissertation, we address the optimization of the output power of piezoelectric energy harvesters of aeroelastic vibrations. Given the complexity of high-fidelity simulations of the coupling between the fluid flow, structural response and piezoelectric transduction, we develop and experimentally validate a phenomelogical reduced-order model for energy harvesting from wake galloping. We also develop a high-fidelity simulation for the same phenomena. The modeling and high-fidelity simulations can be a part of a multi-disciplinary optimization framework to be used in the design and operation of galloping-based energy harvesters. / Doctor of Philosophy / Energy harvesting of ambient or flow-induced vibrations is important for reducing the dependence on batteries in wireless sensing and networks to monitor deterioration conditions, environmental pollution or wildlife conservation. Balancing the benefits and shortcomings of a specific approach, namely piezoelctric transduction, for energy harvesting from vibrations, we address a specific challenge related to the development of a configuration that allows for communicating measured vibrations using their power. Furthermore, given the low levels of output power from piezoelectric transduction, we address the need to optimize power output levels through the development of predictive models that depend on geometry and speed of the fluid flow.

This Thesis comprises of two parts containing similar studies of Nonlinear Localization induced Base Isolation of structural systems. The present method of base isolation,like other nonlinear vibration isolation methods, enjoys certain merits like capability of absorbing broad band vibrations, attenuating heavy shocks etc. The research in this thesis is an extension of this base isolation strategy first proposed by Vakakis and co-author. The strategy involves augmenting an appendage referred to as the secondary system with the main structural unit or the primary system, which we want to isolate from disturbances at the base. The primary system is coupled to the secondary system through a stiffness element. Both the primary and secondary systems have nonlinear dynamic behavior. It is seen that for certain choice of values of the coupling element, steady state vibration of very small magnitude is induced in the primary system. This result was established by considering a general discrete nonlinear system with viscous damping. Now it is a well known fact that viscous damping, though being widely used in literature as well as in practice doesn't turn out to be accurate enough to capture structural damping behaviors. Moreover, the actual damping mechanism if governed by some nonlinear function of the system variables, may influence the physics governing the nonlinear localization phenomenon in a manner rendering the present method not suitable for structural systems at the very outset. So in the present study we focus our attention in establishing the robustness and hence utility of the method by considering technically more defensible models of structural damping. These models efficiently capture certain complex phenomena which structures are known to exhibit. The occurrence of localization induced vibration isolation in structural systems in the presence of these damping models is taken as a proof of the efficacy of the method and its applicability to a wide range of situations. The present study establishes existence of localization through relevant analytical and numerical exercises. In the first part of the thesis we take up the study of nonlinear localization induced base isolation of a three degrees of freedom system having cubic nonlinearities under sinusoidal base excitation. The damping forces in the system are hysteretic in nature. In the present setting this is captured by Bouc-Wen model of hysteresis. Bouc-Wen model is one of the most widely used phenomenological model of hysteresis to have a ready-to-use mathematical description of hysteretic patterns appearing in structural engineering systems. The nature of responses of the different degrees of freedom as excitation frequency varies is a better way of analyzing the performance of the vibration isolation system. We adopt this line of approach for the present study. Normally Harmonic Balance Method (HBM) serves this purpose very well but in the present case as the hysteretic variable is not explicitly related to the system variables, HBM cannot be straightway implemented. Moreover, the hysteretic variable is related to other state variables through a relation which contains non-smooth terms. As a result, Incremental Harmonic Balance (IHB) method is used to obtain amplitude frequency relationship of the system response. The stability analysis of the solution branches is done by using Floquet Theory. Direct numerical simulation is then made use of to support our results that are obtained from this approximate numeric-analytic estimate of the amplitudefrequency relationships of the system, which helps us to analyze the efficacy of this method of base isolation for a broad class of systems. In the next part we consider a similar system where the damping forces in the system are described by functions of fractional derivative of the instantaneous displacements. Fractional Derivative based damping model has been found to be very effective in describing structural damping. We adopt half-order fractional derivative for our study, which can capture damping behavior of polymeric material very well. Typically linear and quadratic damping is considered separately as these are the two most relevant representations of structural damping. Under the assumption of smallness of certain system parameters and nonlinear terms an approximate estimate of the response at each degree of freedom of the system is obtained using Method of Multiple Scales. We then consider a situation where the nonlinear terms and certain other system parameters are no longer small. For the case where asymptotic methods are no longer valid, the assessment of performance of the vibration isolation system is made from amplitude-frequency relations. As a result, we take recourse to the Harmonic Balance Method in conjunction with arc length based continuation technique for obtaining the frequency amplitude plot for linear damping and Incremental Harmonic Balance method for quadratic damping, each of which is validated against results obtained from direct numerical simulation of the system. It needs to be appreciated that base isolation obtained this way has no counterpart in the linear theory.

The aim of the current dissertation is the development of finite element models capable of predicting the damping and the damped structural dynamic response of laminated composite blades and beams. The present thesis is divided into two main parts, of which the first one studies the material coupling effect on the static and modal characteristics of composite structures. New damping coupling terms are formulated and incorporated into a linear beam finite element to better capture the composite material and structural coupling effects. The second part describes the theoretical framework for predicting the nonlinear damping and damped vibration of laminated composite structures due to large in-plane tensile and compressive forces. A nonlinear beam finite element for composite strips is developed capable of capturing the effects of geometric nonlinearity on the damping of composite laminates. The damping mechanics consider a strain based Kelvin viscoelastic model and Green-Lagrange nonlinear strain expressions, which introduce geometric nonlinearity into the formulation. Incorporation of first-order shear deformation theory into the equations of motion provides the linear and new nonlinear cross-section stiffness and damping terms. Within each element, the stain field is approximated by linear interpolation shape functions. An incremental-iterative methodology is formulated into the finite element solver, based on the Newton-Raphson technique in order to obtain the system solution at each iteration, till the final convergence is achieved. For the sake of completeness, a series of experimental measurements were carried out for the composite strip, subject to tensile and buckling loads. Correlations with theoretical predictions gave credence to the ability of the nonlinear finite element to predict damping of composite structures undergoing large displacements and rotations in the nonlinear regime. The finite element was further extended to include the nonlinear analysis of large-scale hollow composite structures. New first- and second-order stiffness and damping terms were developed and incorporated into an updated nonlinear beam finite element, capable of capturing the effect of rotational stresses on the static and modal characteristics of composite beams and blades. / Σκοπός της παρούσας διδακτορικής διατριβής με τίτλο: "Ανάπτυξη Μη-Γραμμικού Προτύπου Πεπερασμένου Στοιχείου με Απόσβεση για τη Στατική και Δυναμική Ανάλυση Πτερυγίων Ανεμογεννητριών" είναι η ανάπτυξη προτύπων πεπερασμένων στοιχείων με απόσβεση ικανών να προβλέπουν τη στατική και δυναμική απόκριση δοκών και πτερυγίων από σύνθετα υλικά. Η εργασία επικεντρώνεται σε δύο κύριες κατευθύνσεις, που αφορούν τόσο την εισαγωγή νέων όρων στο μητρώο απόσβεσης ενός πεπερασμένου στοιχείου δοκού, όσο και την ανάπτυξη ενός μη-γραμμικού κώδικα πεπερασμένου στοιχείου για τη μελέτη της μη-γραμμικής συμπεριφοράς δοκών και πτερυγίων από σύνθετα υλικά που υπόκεινται σε μεγάλες μετατοπίσεις και περιστροφές. Στο πρώτο μέρος της διατριβής μελετάται η επίδραση της σύζευξης, λόγω της ανισοτροπίας του σύνθετου υλικού, τόσο στη στατική απόκριση όσο και στα μορφικά χαρακτηριστικά κατασκευών από σύνθετα υλικά, διαφόρων διατομών και γεωμετριών. Διατυπώνονται νέοι όροι απόσβεσης που εκφράζουν την εν λόγω σύζευξη και οι οποίοι καθιστούν το γραμμικό πεπερασμένο στοιχείο δοκού πιο πλήρες στην επίλυση προβλημάτων όπου η σύζευξη υλικού επηρεάζει τη συμπεριφορά της κατασκευής. Στο δεύτερο και πλέον σημαντικό μέρος της παρούσας διατριβής αρχικά περιγράφεται το θεωρητικό υπόβαθρο για την πρόβλεψη της μη-γραμμικής δυναμικής απόσβεσης λεπτών δοκών κατασκευασμένα από σύνθετα υλικά οι οποίες υπόκεινται σε μεγάλα συν-επίπεδα εφελκυστικά φορτία ή φορτία λυγισμού. Αναπτύσσεται νέο πεπερασμένο στοιχείο ικανό να περιγράψει την επίδραση της γεωμετρικής μη-γραμμικότητας στην απόσβεση και τη δυσκαμψία της δοκού. Εφαλτήριο για την ανάπτυξη αυτής της μεθοδολογίας ήταν η ανάγκη της πρόβλεψης της δυναμικής απόσβεσης σε κατασκευές από σύνθετα υλικά με πιο πολύπλοκη και εύκαμπτη γεωμετρία, όπως αυτή των πτερυγίων ανεμογεννητριών. Η ανάπτυξη του μη-γραμμικού πεπερασμένου στοιχείου ξεκινά από το επίπεδο της στρώσης του υλικού, όπου διατυπώνονται οι καταστατικές εξισώσεις θεωρώντας το ιξωδοελαστικό πρότυπο του Kelvin για το υλικό της κατασκευής. Στη συνέχεια εισάγονται οι Green-Lagrange εξισώσεις συμβιβαστού οι οποίες εκφράζουν τη γεωμετρική μη-γραμμικότητα καθώς και οι εξισώσεις κίνησης. Σε επίπεδο διατομής, οι κινηματικές υποθέσεις βασίζονται στις παραδοχές της διατμητικής θεωρίας δοκού πρώτης τάξης. Η πρόβλεψη της μη-γραμμικής απόκρισης μιας κατασκευής από σύνθετα υλικά επιτυγχάνεται μέσω της προσομοίωσης της με έναν επαρκή αριθμό πεπερασμένων στοιχείων. Στο εσωτερικό κάθε στοιχείου οι παραμορφώσεις προσεγγίζονται από γραμμικές συναρτήσεις μορφής, οι οποίες οδηγούν στη μητρωική μορφή των μη-γραμμικών εξισώσεων του συστήματος. Λόγω του γεγονότος ότι οι εξισώσεις αυτές εξαρτώνται από τη λύση, δεν μπορούν να λυθούν απευθείας κάτι που καθιστά αναγκαία τη χρήση μιας σταδιακής-επαναληπτικής τεχνικής. Στην παρούσα διατριβή εισάγεται στο λύτη του μη-γραμμικού κώδικα η Newton-Raphson τεχνική. Το επόμενο βήμα αφορά τη σύνθεση των ολικών δομικών μητρών του συστήματος και την εφαρμογή των συνοριακών συνθηκών. Σε κάθε επανάληψη λαμβάνει χώρα η επίλυση των γραμμικοποιημένων εξισώσεων και ο υπολογισμός των πραγματικών και εφαπτομενικών μη-γραμμικών μητρώων δυσκαμψίας και απόσβεσης της κατασκευής, τα οποία τελικώς επιλύονται με τη μέθοδο της αριθμητικής ολοκλήρωσης κατά Gauss. Το πεπερασμένο στοιχείο δοκού εξελίχθηκε περαιτέρω ώστε να συμπεριλάβει τη μη-γραμμική ανάλυση μεγάλων λεπτότοιχων κατασκευών από σύνθετα υλικά, όπως αυτά των πτερυγίων ελικοπτέρων και ανεμογεννητριών. Η εισαγωγή της πλήρους έκφρασης της αξονικής μη-γραμμικής Green-Lagrange παραμόρφωσης στη διατύπωση των κινηματικών υποθέσεων οδηγεί στην πλήρη έκφραση των πραγματικών και εφαπτομενικών δομικών μητρών της κατασκευής. Οι νέοι μη-γραμμικοί όροι δυσκαμψίας και απόσβεσης πρώτης και δεύτερης τάξης μπορούν να περιγράψουν την επίδραση των εσωτερικών εφελκυστικών τάσεων στα μορφικά χαρακτηριστικά δοκών και πτερυγίων. Το μη-γραμμικό πεπερασμένο στοιχείο είναι ικανό να χαρακτηρίσει τη στατική συμπεριφορά και την αποσβενυμένη ταλάντωση δοκών από σύνθετα υλικά. Η επαλήθευση του μη-γραμμικού κώδικα πραγματοποιήθηκε μέσω μιας σειράς πειραματικών μετρήσεων που αφορούσαν τη μέτρηση της φυσικής συχνότητας και της μορφικής απόσβεσης σε λεπτές δοκούς από σύνθετα υλικά τόσο σε εφελκυσμό όσο και σε συνθήκες λυγισμού. Τα πειραματικά αποτελέσματα έρχονται σε πολύ καλή συμφωνία με τις θεωρητικές προβλέψεις του κώδικα κάτι που εξασφαλίζει την αξιοπιστία του μη-γραμμικού πεπερασμένου στοιχείου.