Due to their simplicity, cohesive zone models (CZMs) are very attractive to describe mixed-mode failure and debonding processes of materials and interfaces. Although a large number of coupled CZMs have been proposed, and despite the extensive related literature, little attention has been devoted to ensuring the consistency of these models for mixed-mode conditions, primarily in a thermodynamical sense. A lack of consistency may affect the local or global response of a mechanical system. This contribution deals with the consistency check for some widely used exponential and bilinear mixed-mode CZMs. The coupling effect on stresses and energy dissipation is first investigated and the path-dependance of the mixed-mode debonding work of separation is analitically evaluated. Analytical predictions are also compared with results from numerical implementations, where the interface is described with zero-thickness contact elements. A node-to-segment strategy is here adopted, which incorporates decohesion and contact within a unified framework. A new thermodynamically consistent mixed-mode CZ model based on a reformulation of the Xu-Needleman model as modified by van den Bosch et al. is finally proposed and derived by applying the Coleman and Noll procedure in accordance with the second law of thermodynamics. The model holds monolithically for loading and unloading processes, as well as for decohesion and contact, and its performance is demonstrated through suitable examples.

The Generalized Differential Quadrature (GDQ) and Newmark methods are chosen to solve time integration problems such as the dynamics of composite arches and vaults with constant and variable cross-sections, under seismic impulse loading applied at the base. A 2D Equivalent Single Layer (ESL) shell theory is used to analyze the problem numerically, where the governing equations of motion are solved in a strong form without passing through any variational formulation. The total time interval is discretized in time steps, as required by a Newmark approach, and the GDQ method is applied to solve a system of linear ordinary differential equations for each time step. The accuracy of the proposed method in predicting the dynamic response of the arched or vaulted structures is demonstrated by comparing the GDQ-based results for different geometries and external loadings, with the ones obtained with a standard Finite Element Method (FEM).

Masonry arches are typical components of historic buildings throughout the world, and their damage or collapse is very often caused by earthquakes. The first-order seismic assessment of masonry structures can be represented by the equivalent static analysis method, which does not capture all of the dynamics, but provides a measure of the lateral loading that the structure can withstand before collapse. This study aims to understand the stability of unreinforced masonry arches and portals (i.e. buttressed arches) subjected to constant horizontal ground accelerations, combined with the vertical acceleration due to gravity. An analytical model based on limit analysis is developed to describe the relative stability of pointed and basket-handle arches and portals with respect to circular ones, for varying geometry parameters. The equivalent static analysis determines the value of the constant lateral acceleration needed to cause collapse of the structure, which coincides with the minimum peak ground acceleration needed to transform the vaulted system into a mechanism. Predictions of the analytical model are compared with results of numerical modelling by the Discrete Element Method (DEM). This numerical model considers masonry as an assemblage of rigid blocks with no-tension frictional joints, and is based on a time stepping integration of the equations of motion of the individual blocks. The satisfactory agreement between predictions of the two approaches validates the analytical model and verifies the potentials of the discrete element framework as a method of evaluating the quasi-static behavior of unreinforced masonry structures.

The Strong Formulation Finite Element Method (SFEM) based on the Generalized Differential Quadrature (GDQ) method is applied in the present work to estimate numerically the Stress Intensity Factor (SIF) for a single edge-notched tensile specimen (SENT) with different length-to-width ratios and composite materials, as well as for a Center Cracked Tension (CCT) and a Double Edge-Notched Tensile (DENT) specimens, under a mode-I loading condition. The basis notions of the fracture mechanics are herein analyzed numerically in order to evaluate the influence of possible cracks within materials and to relate the dimensions of cracks and the applied loading to the varying stress distribution. The numerical results, in terms of stresses and SIFs, are straightforwardly compared to those ones given by the standard Finite Element Method (FEM) and theoretical predictions available from the literature. This work presents a consistent approach for the computation of the SIF using a strong form methodology. The main aim is to demonstrate the accuracy and efficiency of the proposed methodology when treating classical plane stress problems with cracks. We show and discuss results from several numerical examples, including different composite materials and varying geometries for a mode-I SENT, CCT and DENT specimens.

This paper examines two engineering methods of evaluating the stress intensity factors for cracked beams and bars subjected to a combined loading and proposes innovative formulations, as far as the circular cross section is concerned. Based on the definition of the stress intensity factors, the compliance matrix is determined as the inverse of the stiffness matrix, modelling the cracked section of a beam through a line-spring approximation with interactive forces computed within fracture mechanics. A comparative evaluation of numerical predictions based on the proposed methods is also performed with methods available from the literature. Results for free vibration analyses of beams with transverse non-propagating open cracks are presented and compared in order to estimate the accuracy and efficiency of the proposed methods, where a good agreement is generally found. More specifically, two different coupling effects are herein analysed for circular beams subjected to a combined bending, axial and shear loading, first, and a combined bending, shear and torsion loading, subsequently.

This work aims at studying the mixed-mode delamination process in Moment-Loaded Double Cantilever Beam (MLDCB) specimens. The delamination problem is addressed both analytically and numerically, while considering the interfaces as an assemblage of two sublaminates partly bonded together by an elastic interface. Such interface is modeled as a continuous distribution of elastic-brittle springs acting along the normal and/or tangential direction depending on the interfacial mixed-mode condition. The Timoshenko's beam theory is here applied to determine the governing equations of the differential problem and the associated boundary conditions, whose solution is not straightforward. The Generalized Differential Quadrature (GDQ) method is then applied as numerical tool to solve directly the differential equations of the problem in a strong form. The capability of the proposed numerical approach is first exploited through a comparative evaluation of the results with the analytical predictions resting on a suitable change of variables for delamination test specimens. The local and global response is determined, in terms of interfacial stresses, internal forces and displacements, as well as in terms of compliance, energy release rate, mode mixity angle, and moment-rotation curves. A further check of the proposed numerical method is performed with respect to a Finite Fracture Mechanics (FFM) criterion, which is able to join both stress-and energy-based approaches. A good agreement between results confirms the good feasibility of the GDQ method when studying delamination phenomena occurring within composite materials or laminated joints, usually subjected to mixed-mode conditions.

This paper checks the consistency of some published exponential and bilinear mixed-mode cohesive zone models. The effect of coupling on traction-separation behavior and energy dissipation is investigated and the path-dependence of the debonding work of separation and failure domain is evaluated analytically and numerically. All selected models present several inconsistencies, except for the one by van den Bosch et al. (2006), which is, however, not currently formulated within a thermodynamical framework but postulated in an ad-hoc manner. We thus propose a thermodynamically consistent reformulation of this model within damage mechanics, which holds monolithically for loading, unloading, decohesion and contact.

The limitations of sophisticated analytical modelling and experimental investigations for studying complex masonry structures have increased the use of numerical modelling, such as the Discrete Element Method (DEM). Based on this approach, the discontinuous bodies can move freely in space and interact reciprocally with contact forces, leading to an automatic updating of contact detection. In this work, the potential of the DEM is first demonstrated for the assessment of the first-order seismic behaviour of generally shaped vaulted structures in 2D. The relative efficiency of different arch shapes is analysed comparatively such as circular, pointed or basket-handle shapes. The full dynamics and resistance of discrete columns subjected to simple base excitations is additionally studied, with a detailed investigation on failure domains and on modes of collapse of the structures. A parametric evaluation of the sensitivity of the response to the input parameters is numerically performed, which gives more confidence to the DEM predictions.

Elliptic geometries appear to be important components in engineering practices. The purpose of the present study is to examine the free vibration nature of laminated composite thick and moderately thick elliptic cones, cylinders and plates. A strong form approach, such as the generalized differential quadrature (GDQ) method is employed to carry out the numerical analyses. The geometric description of the structures under consideration is performed through the differential geometry which is a convenient and general mathematical tool to have parametric description of curved structures. Reference solutions are presented for laminated composite thick elliptic shells. 3D finite element models are used for proving the validity and the advantages of the present methodology. Since laminated composite structures are investigated, higher-order theories are considered to capture the nonlinear behavior of the material fibers through the shell thickness. In particular, a hierarchical expansion order of the kinematic displacements is adopted. The expansion order is a function of a free parameter. The numerical solution is found discretizing the dynamic equilibrium equations, written as functions of the displacement parameters, with GDQ method. This technique proved to have several advantages such as stability, accuracy and easy implementation as also demonstrated by the authors in the provided literature.

This paper investigates the micro- and nano-mechanical behavior of orthotropic doubly-curved shells by considering the New Modified Couple Stress Theory (NMCST). The higher order continuum assumed by the NMCST includes three material length scale parameters in order to capture the size-effect of anisotropic and orthotropic materials. The governing equations of the problem are based on the First-order Shear Deformation Theory (FSDT). According to the proposed NMCST, the expressions of the physical components for the strain and curvature tensors are obtained in an orthogonal curvilinear coordinate system. Then, the governing differential equations and boundary conditions are derived by applying the energy method and Hamilton's principle. A comparative investigation between our numerical results and the ones available in the literature proves the capability of the proposed formulation in predicting the micro- and nano-mechanical behavior of orthotropic doubly-curved shells.

By means of Non-Uniform Rational B-Splines (NURBS) curves, it is possible to describe arbitrary shapes with holes and discontinuities. These peculiar shapes can be taken into account to describe the reference domain of several nanoplates, where a nanoplate refers to a flat structure reinforced with Carbon Nanotubes (CNTs). In the present paper, a micromechanical model based on the agglomeration of these nanoparticles is considered. Indeed, when this kind of reinforcing phase is inserted into a polymeric matrix, CNTs tend to increase their density in some regions. Nevertheless, some nanoparticles can be still scattered within the matrix. The proposed model allows to control the agglomeration by means of two parameters. In this way, several parametric studies are presented to show the influence of this agglomeration on the free vibrations. The considered structures are characterized also by a gradual variation of CNTs along the plate thickness. Thus, the term Functionally Graded Carbon Nanotubes (FG-CNTs) is introduced to specify these plates. Some additional parametric studies are also performed to analyze the effect of a mesh distortion, by considering several geometric and mechanical configurations. The validity of the current methodology is proven through a comparative assessment of our results with those available from the literature or obtained with different numerical approaches, such as the Finite Element Method (FEM). The strong form of the equations governing a plate is solved by means of the Generalized Differential Quadrature (GDQ) method.

In the present work, a free vibration analysis of Carbon Nanotube-Reinforced Composite (CNTRC) conical shells is performed considering the agglomeration effect of Carbon Nanotubes (CNTs). The material properties of the nanocomposite conical shell are estimated employing the Eshelby-Mori-Tanaka approach based on an equivalent fiber assumption. The numerical results are compared with the experimental data available from the literature. The equations of motion are derived based on the First-order Shear Deformation Theory (FSDT). The Generalized Differential Quadrature (GDQ) technique is originally implemented to solve the governing equations of the problem and to obtain the natural frequencies of the structures, since it has proven to be an efficient and accurate numerical tool. A parametric study is herein developed to investigate the influence of some characteristic parameters on the vibrational behavior of the CNTRC conical shell, e.g. the CNTs volume fraction and agglomeration, or the boundary conditions and geometrical parameters like the thickness to radius ratio. Based on results, it is found that agglomeration of CNTs plays a significant role on the natural frequency of the structure.

Due to some technical issues that can appear during the manufacturing process of Functionally Graded Materials (FGMs), it can be extremely difficult to produce perfect materials. Indeed, one of the biggest problems is the presence of porosities. For this purpose, the vibrational behavior of doubly-curved shells made of FGM including porosities is investigated in this paper. With respect to previous research, the porosity has been added to the mechanical model that characterizes the through-the-thickness distribution of the graded constituents and applied to doubly-curved shell structures. Few papers have been published on this topic. In fact, it is easier to find works related to one-dimensional structures and beam models that take account the effect of porosities. The First-order Shear Deformation Theory (FSDT) is considered as the theoretical framework. In addition, themechanical properties of the constituents vary along the thickness direction. For this purpose, two power-law distributions are employed to characterize their volume fraction. Strain components are established in an orthogonal curvilinear coordinate system and the governing equations are derived according to the Hamilton's principle. Finally, Navier's solution method is used and the numerical results concerning three different types of shell structures are presented.

We use the generalized differential quadrature method (GDQ) and shell theories of different order to study free vibrations of laminated cylinders of oval and elliptic cross-sections. In the GDQ method partial derivatives of a function at a point are expressed as weighted sums of values of the function at several neighboring points. Thus, strong forms of equations of motion are analyzed. It is found that the computed frequencies rapidly converge with an increase in the number of grid points along the oval or elliptic circumference defining the cross-section of the mid-surface of the cylinder. For a clamped-free elliptic cylinder the converged frequencies match well with the corresponding experimental ones available in the literature. Furthermore, the lowest ten frequencies computed with either an equivalent single layer theory or a layer wise theory of first order and using shear correction factor are accurate.

The stress concentration in discontinuous zones is known to be a significant issue in mechanics, since the presence of a discontinuity, even in a simple structure model, makes it complicated to analyze. To this end, the application of numerical methods would require a sufficiently fine mesh for a realistic prediction of stresses around critical zones as cracks or discontinuities. Despite the large effort related to the finite element method as numerical approach to predict stress concentrations, results are still not satisfactory. In this work we propose two innovative numerical approaches to determine the stress concentration factors, with a reduced computational cost. A strong formulation finite element method, its localized version, and the isogeometric approach, are herein applied to study some classical examples, as the plane stress plates with circular holes, U-holes, or V-notches. All the numerical results obtained with both approaches in terms of stress distribution and stress concentration factors are compared to the theoretical and experimental predictions available in the literature and the numerical solutions found with finite element method. A very good agreement between the numerical and the reference results confirms the potentials and accuracy of the proposed methodologies to capture the stress concentrations in fracture mechanics, also for coarse mesh discretizations.

The lifetime of most engineering structures and components is known to depend on the presence of defects, such as holes, cracks or voids usually introduced during a manufacturing process. In many cases, the crack growth, extension and propagation within a body, still remains a challenging problem in fracture mechanics. The present paper proposes an extended analytical model based on a section method to predict the fracture direction and compute the stress intensity factors for a cracked shaft under mixed-mode loading conditions. The advantage of the present formulation is mainly related to its capability of predicting the direction of crack propagation within a shaft under coupled longitudinal, flexural and torsional loading conditions. The analytical results are straightforwardly compared with the theoretical expressions available from the handbooks and the numerical solutions found with the extended finite element method. The present approach agrees quite well with the theoretical and numerical results already proposed in the literature, thus confirming its potentials for accurate computations of the crack propagation and stress intensity factors for arbitrary configurations.

T-spline-based isogeometric analysis is applied to frictionless contact problems between deformable bodies in the context of large deformations. The continuum is discretized with cubic T-splines and cubic NURBS. A Gauss-point-to-surface formulation is combined with the penalty method to treat the contact constraints in the discretized setting. It is demon- strated that analysis-suitable T-splines, coupled with local refinement, accurately approx- imate contact pressures with far fewer degrees of freedom than NURBS. Both two- and three-dimensional examples are presented. Additionally, all T-spline analysis models are generated using commercially available T-spline modeling software without intermediate mesh generation or geometry clean-up steps.

Nowadays the isogeometric analysis (IGA) represents an innovative method that merges design and numerical computations into a unified formulation. In such a context we apply the isogeometric concept based on T-splines and Non Uniform Rational B-Splines (NURBS) discretizations to study the interfacial contact and debonding problems between deformable bodies in large deformations. More in detail, we develop and test a generalized large deformation contact algorithm which accounts for both frictional contact and mixed-mode cohesive debonding in a unified setting. Some numerical examples are provided for varying resolutions of the contact and/or cohesive zone, which show the accuracy of the solutions and demonstrate the potential of the method to solve challenging 2D contact and debonding problems. The superior accuracy of T-splines with respect to NURBS interpolations for a given number of degrees of freedom (Dofs) is always proved.

Within a setting where the isogeometric analysis (IGA) has been successful at bringing two different research fields together, i.e. Computer Aided Design (CAD) and numerical analysis, T-spline IGA is applied in this work to frictionless contact and mode-I debonding problems between deformable bodies in the context of large deformations. Based on the concept of IGA, the smooth basis functions are adopted to describe surface geometries and approximate the numerical solutions, leading to higher accuracy in the contact integral evaluation. The isogeometric discretizations are here incorporated into an existing finite element framework by using Bézier extraction, i.e. a linear operator which maps the Bernstein polynomial basis on Bézier elements to the global isogeometric basis. A recently released commercial T-spline plugin for Rhino is herein used to build the analysis models adopted in this study. In such context, the continuum is discretized with cubic T-splines, as well as with Non Uniform Rational B-Splines (NURBS) and Lagrange polynomial elements for comparison purposes, and a Gauss-point-to-surface (GPTS) formulation is combined with the penalty method to treat the contact constraints. The purely geometric enforcement of the non-penetration condition in compression is generalized to encompass both contact and mode-I debonding of interfaces which is approached by means of cohesive zone (CZ) modeling, as commonly done by the scientific community to analyse the progressive damage of materials and interfaces. Based on these models, non-linear relationships between tractions and relative displacements are assumed. These relationships dictate both the work of separation per unit fracture surface and the peak stress that has to be reached for the crack formation. In the generalized GPTS formulation an automatic switching procedure is used to choose between cohesive and contact models, depending on the contact status. Some numerical results are first presented and compared in 2D for varying resolutions of the contact and/or cohesive zone, including frictionless sliding and cohesive debonding, all featuring the competitive accuracy and performance of T-spline IGA. The superior accuracy of T-spline interpolations with respect to NURBS and Lagrange interpolations for a given number of degrees of freedom (Dofs) is always verified. The isogeometric formulation is also extended to 3D bodies, where some examples in large deformations based on T-spline discretizations show an high smoothness of the reaction history curves.

In this paper we focus on the prediction of mode-I debonding for a double cantilever beam (DCB). Among the various modeling approaches available, the Cohesive Crack Model (CCM) and Finite Fracture Mechanics (FFM) are selected for the analytical investigation, due to their ability to reconcile the stress- and energy-based approaches. The specimen is considered as an assemblage of two identical beams partly bonded together by an initially elastic interface. After the elastic stage, according to the CCM approach, it is assumed that, ahead of the physical crack tip, there exists a cohesive zone where the interface behavior is described by a stress-separation law. The interfacial stresses and length of the process zone are determined in closed form, along with the global load-displacement response. The method is first compared to the simple beam theory (SBT) and the enhanced beam theory (EBT) approaches, which are found to provide larger values of the debonding load; the difference between predictions of CCM and SBT/EBT is more pronounced for less brittle interfaces, i.e. for larger process zones. Then the analytical solution obtained by means of FFM is presented, which, despite being simply based on the elastic foundation model, closely matches the CCM results. Finally a numerical solution is achieved by a finite element analysis where generalized zero-thickness contact interface elements are adopted. An excellent agreement with these results confirms the good performance of the proposed CCM and FFM approaches.

The lifetime of most engineering structures and components is known to depend on the presence of defects, such as holes, cracks, or voids usually introduced during a manufacturing process. In many cases, the crack growth, extension and propagation within a body, still remains a challenging problem in fracture mechanics. The present paper proposes the application of the level set method combined with the numerical extended finite element method (XFEM) to predict the fracture direction of propagation within a specimen, and to compute the stress intensity factor for cracked plates under different loading conditions. This technique avoids the difficulty of remeshing when tracking the moving interface positions during the cracking process. The combined XFEM formulation is first reviewed and then applied to different examples. The numerical results provided by the XFEM are straightforwardly compared with the theoretical predictions from the handbooks and the numerical solutions found with a strong formulation finite element method (SFEM) based on the generalized differential quadrature (GDQ) approach. The good agreement between the theoretical and numerical results confirms the accuracy of the proposed formulation to treat fracture mechanics.

Much of the world’s architectural heritage consists of Unreinforced Masonry (URM) structures whose preservation is a topical subject. To prevent possible collapse of multi-block systems in hazardous con- ditions, a promising tool to investigate their structural response is represented by numerical modelling with the Discrete Element Method (DEM). Gothic buttresses of trapezoidal and stepped shapes are rst analysed comparatively under static loading, de ning the optimal con gurations. Numerical results are veri ed against the analytical predictions of overturning and sliding resistances, based on a continuum approximation of masonry. The DEM is then successfully adopted to assess the rst-order seismic be- havior of arches and buttressed arches with di erent shapes as compared to predictions based on limit analysis. A systematic investigation on dynamic behavior failure domains and on modes of collapse of URM structures is nally performed for varying input parameters, as needed to gain more con dence on the numerical results.

Earthquakes represent one of the major threats to the stability of the world architectural heritage, which is mostly constituted by unreinforced masonry (URM) structures. The dynamic behavior of these structures is complex and highly non-linear, as it involves sliding and rocking of the component blocks. As a result, numerical modeling seems to be the most appropriate predictive approach and, in particular, the Discrete Element Method (DEM) has recently emerged as a very promising tool for this purpose. Although multi- drum columns and arches on buttresses are typical components of historic URM structures, their modeling with the DEM has been the subject of relatively limited research. Moreover, a set of input parameters is required for the definition of the numerical model and, due to the uncertainty and difficulty in their experimental evaluation, these parameters are usually set in an arbitrary way. In this paper, a systematic parametric study based on the DEM is adopted to evaluate the dynamic behavior and resistance of multi-drum columns and arches on buttresses subjected to two different base motions, i.e. step and harmonic impulses. A detailed investigation on failure domains and modes of collapse is presented. The main features of the dynamic response of masonry structures and the sensitivity of the response to changes in the excitation, geometry and mechanical parameters are discussed.

This paper is aimed at studying the free vibration and thermal buckling behavior of moderately thick functionally graded material (FGM) structures including plates, cylindrical panels and shells under thermal environments. A numerical investigation is performed by applying the finite element method (FEM). A formulation based on the first-order shear deformation theory (FSDT) is proposed for the purpose, which considers the effects of the transverse shear strain and rotary inertia. A graded concept is employed to allow the material property to vary gradually inside the elements. The proposed FGM structures are characterized by two constituents (ceramic and metal) whose material properties are dependent on the temperature and vary continuously throughout the thickness according to a power law distribution proportional to the volume fraction of the constituents. Two different sets of power law distribution are used to describe the volume fraction of the constituents, based on a single, or four parameters. Based on a parametric analysis, we demonstrate the potentials of the proposed method through its comparison with results available from the literature and by means of a convergence study. Several numerical examples are further presented to investigate the effects of material compositions, geometrical parameters, specified thermal loading and boundary conditions on the free vibration and thermal buckling behavior of these structures. The effect of initial thermal stresses on the vibration behavior is also investigated for plate and shell structures.

Cohesive zone (CZ) models have long been used by the scientific community to analyze the progressive damage of materials and interfaces. In these models, non-linear relationships between tractions and relative displacements are assumed, which dictate both the work of separation per unit fracture surface and the peak stress that has to be reached for the crack formation. This contribution deals with isogeometric CZ modeling of interface debonding. The interface is discretized with generalized contact elements which account for both contact and cohesive debonding within a unified framework. The formulation is suitable for non-matching discretizations of the interacting surfaces in presence of large deformations and large relative displacements. The isogeometric discretizations are based on non uniform rational B-splines as well as analysis-suitableT-splines enabling local refinement. Conventional Lagrange polynomial discretizations are also used for comparison purposes. Some numerical examples demonstrate that the proposed formulation based on isogeometric analysis is a computationally accurate and efficient technology to solve challenging interface debonding problems in 2D and 3D.

Il presente manoscritto scaturisce dall’esperienza maturata nel corso di circa dieci anni di studio, di ricerca e di insegnamento su alcuni temi relativi alla stabilità dell’equilibrio elastico. Questi appunti e lezioni rappresentano i temi trattati in alcuni corsi di laurea in Ingegneria, quali: Scienza delle Costruzioni, Scienza delle Costruzioni II, Complementi di Scienza, Teoria delle Strutture, Dinamica delle Strutture, Piastre e Gusci, Costruzioni di Macchine e Elementi delle Macchine. Il titolo, Stabilità dell’Equilibrio Elastico, illustra il tema trattato e la prospettiva seguita nella stesura del volume. Il presente elaborato si pone come obiettivo quello di analizzare il comportamento di strutture soggette a carichi di punta o di compressione. Il libro si articola in tre capitoli, nei quali viene fornita nel dettaglio la teoria relativa ai criteri di stabilità in ambito strutturale e vengono presentati i risultati dell’applicazione di essi ai diversi problemi. Il volume nasce dall’esigenza di avere uno strumento utile ed efficace per intraprendere lo studio di uno dei temi più affascinanti e importanti della Scienza delle Costruzioni e della Meccanica Applicata in generale. L’obiettivo del presente volume è quello di agevolare gli studenti e i professionisti che intendano impegnarsi nello studio della stabilità dell’equilibrio elastico in ambito strutturale, fornendo un supporto omogeneo, diretto e comprensibile.

This study investigates the static and free vibration behavior of rotating functionally graded (FG) truncated conical shells reinforced by carbon nanotubes (CNTs) with a gradual distribution of the volume fraction through the thickness. CNTs are here selected as reinforcement, because of their noteworthy physical and chemical properties, together with their ability to enhance the mechanical properties of the whole composite structure. A two-parameter agglomeration model is considered to describe the micromechanics of such particles, which tend to agglomerate into spherical regions when scattered in a polymer matrix. From the macro-mechanical point of view, the conical structures are characterized by a gradual variation of their mechanical properties along the thickness direction, since different distributions are explored to describe the volume fraction of the reinforcing phase. The governing equations of motion for the rotating truncated composite conical shells are derived and solved numerically by means of the Generalized Differential Quadrature (GDQ) method combined with the third order shear deformation theory (TSDT) in small deformations. The GDQ approach has recently emerged as a very promising numerical tool to solve complex problems without passing through any variational formulation, but solving directly the equations of motion in a strong form. In this paper, a parametric study based on the GDQ is systematically performed to exploit the effect of some geometry parameters, i.e. the length, the radius, the thickness and the semi-vertex angle of the cone, as well as the different distribution of CNTs along the thickness, on the frequency at different circumferential wave numbers and rotating speeds. A convergence study of the numerical results is also made in terms of deflection and stress distributions of the structure, which proves the efficiency of the GDQ approach, also for coarse mesh discretizations in the meridional direction.

Despite its vital importance for the safety of masonry vaulted structures, the stability of buttresses has been the subject of a limited number of investigations. The collapse condition of masonry buttresses subjected to concentrated lateral loads has been analysed by previous researchers for the simple case of a rectangular buttress, and it has been shown that the formation of a crack prior to collapse may strongly influence the overturning resistance. Buttresses of more complex shapes, such as trapezoidal or stepped, are frequently encountered in masonry buildings or retaining walls, and yet a rigorous analysis of their failure conditions has not been developed. In this paper and a companion paper, the collapse analysis accounting for fracturing prior to overturning, as well as for possible sliding, is extended to trapezoidal and stepped geometries. In particular, this paper is devoted to the analysis of trapezoidal buttresses. Analytical formulations are developed for the determination of the fracture shape and location, and for the computation of the horizontal force at collapse. The analytical solution, obtained treating masonry as a continuum with no tension resistance, is compared with predictions obtained numerically with the discrete-element method. The numerical model considers masonry as an assemblage of rigid blocks with no-tension frictional joints, and is based on time-stepping integration of the equations of motion of the individual blocks. The satisfactory agreement between predictions of the two approaches gives confidence in both sets of results.

The stability of masonry buttresses under horizontal forces is of paramount importance for the safety of vaulted structures, and yet has been the subject of limited studies. In particular, buttresses of non-rectangular geometries such as trapezoidal and stepped buttresses, which are typical of Gothic architecture, have not been sufficiently investigated. This study follows on from a companion paper devoted to masonry with trapezoidal buttresses. Based on similar modelling approaches and assumptions, the present paper aims to predict the failure of stepped buttresses. The analytical solution, obtained by treating masonry as a continuum with no tension resistance and accounting for the formation of a fracture prior to collapse, is compared with the predictions of the discrete element method. The numerical approach considers masonry as an assemblage of rigid blocks with no-tension frictional joints and is based on time-stepping integration of the equations of motion of the individual blocks. The relative efficiency of different buttress shapes for a given total volume is also compared, and an example buttress is used as a benchmark to demonstrate the practical applicability of the proposed models.

A T-spline-based isogeometric analysis is applied to frictional contact problems between deformable bodies in the context of large deformations. The continuum is discretized with cubic T-splines and cubic NURBS (Non-Uniform Rational B- Splines) for comparison purposes. A Gauss-point-to-surface (GPTS) formulation is combined with the penalty method to treat the normal and friction contact constraints in the discretized setting. It is demonstrated that the proposed formulation combined with analysis-suitable T-spline interpolations, is a computationally accurate and efficient technology for local and global solutions of contact problems. T-spline analysis models are generated using commercially available T-spline modeling software without intermediate mesh generation or geometry clean-up steps

We study the thermal buckling behavior of cylindrical shells reinforced with Functionally Graded (FG) wavy Carbon NanoTubes (CNTs), stiffened by stringers and rings, and subjected to a thermal loading. The equilibrium equations of the problem are built according to the Third-order Shear Deformation Theory (TSDT), whereas the stiffeners are modeled as Euler Bernoulli beams. Different types of FG distributions of wavy CNTs along the radial direction of the cylinder are herein considered, and temperature-dependent material properties are estimated via a micromechanical model, under the assumption of uniform distribution within the shell and through the thickness. A parametric investigation based on the Generalized Differential Quadrature (GDQ) method aims at investigating the effects of the aspect ratio and waviness index of CNTs on the thermal buckling of FG nanocomposite stiffened cylinders, reinforced with wavy single-walled CNTs. Some numerical examples are here provided in order to verify the accuracy of the proposed formulation and to investigate the effects of several parameters-including the volume fraction, the distribution pattern of wavy CNTs, and the cylinder thickness-on the thermal buckling behavior of the stiffened cylinders made of CNT-reinforced composite (CNTRC) material.

This paper deals with the in-plane dynamic modeling of generally shaped arches with a varying cross-section in undamaged or damaged configuration, under different boundary conditions and external forces. The Generalized Differential Quadrature (GDQ) method is herein applied to solve numerically the problem without passing through any variational formulation, but solving directly the governing equations of motion in strong form. The main purpose of the work is to obtain a computationally efficient higher-order method for solving time integration problems. The total time interval is discretized in time steps and the GDQ method is applied to solve the initial value problem within each time step. At each time interval, a linear algebraic equation system has to be solved. A simple and efficient implementation scheme is presented. A wide GDQ-based numerical investigation is performed to study the linear dynamics of the arch with different geometries, boundary conditions and external forces. The numerical results based on the application of the GDQ method are compared with those ones provided by the Newmark method. A very good agreement is found between the two numerical approaches, which demonstrates the performance and feasibility of the proposed GDQ-time-stepping algorithm for transient dynamics.