Substructure Approach Research Articles

A multi-scale model order reduction scheme for transient modelling of periodic structures

In this paper, a Floquet Model Order Reduction (MOR) method is proposed for the modelling of finite periodic structures subjected to harmonic or transient loads. This Floquet MOR generates frequency-independent projection matrices of the full-scale structure using a Floquet expansion of the local state-vectors computed on a single unit-cell with the Forced Wave Finite Element framework. This two-step MOR method can handle large multi-scale waveguides regardless of the scale and the number of unit-cells. A comparison with a standard sub-structuring approach based on unit-cell wave-mode reduction demonstrates that Floquet MOR is able to create reduced models of finite periodic structures with an unequalled computational efficiency. The method’s ability to produce small and accurate models suitable for time-domain simulations presents a high potential for virtual sensing, real-time Structural Health Monitoring (SHM) and digital twins for large periodic structures and metamaterials.

Estimation of the Mechanical Parameters for a Reduced Coupled Flexural–Torsional Beam Model of a Tall Building by a Sub-Structure Approach

The use of equivalent beam models to estimate the dynamical characteristics of complex tall buildings has been investigated by several authors. The main reason is the structural response estimation to stochastic loads, such as wind and earthquake, using a reduced number of degrees of freedom, which reduces the computational costs and therefore gives the designer an effective tool to explore a number of possible structural solutions. In this paper, a novel approach to calibrate the mechanical and dynamical features of a complete 3D Timoshenko beam, i.e., describing bending, shear and torsional behavior, is proposed. This approach is based on explicitly considering the sub-structures of the tall building. In particular, the frames, shear walls and lattice sub-systems are modeled as equivalent beams, constrained by means of rigid diaphragms at different floors. The overall dynamic features of the tall building are obtained by equating the deformation energy of an equivalent sandwich beam with that of the selected sub-structures. Finally, the 3D Timoshenko equivalent beam parameters are calibrated by minimizing a suitable function of modal natural frequencies and static displacements. The closed form modal solution of the equivalent beam model is used to obtain the response to stochastic loads.

On the estimation of structural admittances from acoustic measurement using a dynamic substructuring approach

A reconstructed displacement field using near-field acoustic holography (NAH) serves as an alternative to conventional measurement methods when it comes to obtaining the high-resolution vibration response of a structure. The method is highly applicable as it enables direct, non-contact measurement of the 3D structural response based on a single acoustic measurement. Although useful, the method’s ill-posed nature limits its use in the field of structural dynamics. This problem can be effectively addressed by using regularization and/or field-separation techniques that can attenuate the noise and the presence of external acoustic sources. All these methods rely on the measurement of acoustic quantities; therefore, the reconstruction of structural admittances is based solely on the evaluation of the hologram(s). This article proposes an alternative approach to improving the accuracy of NAH-based structural admittances by integrating them with a few discrete response measurement on the structure itself. The formulation relies on the mixing of the high-resolution NAH measurement with accurate discrete measurements (e.g., accelerometer or laser vibrometer) using dynamic substructuring techniques. The proposed hybrid approach is a very powerful modeling methodology that can integrate high-resolution spatial measurements using NAH with the accuracy and consistency provided by precise translation discrete measurements. In order to mix two experimental response models System Equivalent Model Mixing (SEMM) method is proposed. An experimental case study on a T-shaped structure demonstrates the robustness and improved accuracy of the estimated structural admittances compared to the plain NAH formulation.

3D soil-structure interaction analysis with a combined substructure-FEM under external forces and seismic waves

This paper presents a simple methodology for studying soil-structure interaction problems. Based on the substructure approach, the unbounded soil beneath the foundation has been removed. The foundation's impedance matrix and the driving force vector due to seismic waves have been applied to the superstructure. Foundation-soil impedance functions have been reproduced by utilizing sets of springs and dashpots with elastic and damping frequency-dependent properties. The superstructures have been discretized with refined finite beam elements derived with the Carrera Unified Formulation. Vertical, horizontal, rocking and torsional vibrations of the superstructure-soil system have been investigated by considering soft, intermediate, stiff and rigid soils. Displacement and stress values of the flexible-base superstructures have been compared with those obtained for the fixed-base configuration. For validation purposes, the obtained results have been assessed with solutions available in the literature and with those provided by the ASCE-7 building code. In all cases, significant agreements have been observed. Higher-modes frequencies and responses have been obtained for the flexible-base superstructures which could be very useful for modal analyses including SSI effects.

Substructural Approach for Assessing the Stability of Higher Fullerenes.

This review describes the most significant published results devoted to the study of the nature of the higher fullerenes stability, revealing of correlations between the structural features of higher fullerene molecules and the possibility of their producing. A formalization of the substructure approach to assessing the stability of higher fullerenes is proposed, which is based on a detailed analysis of the main structural features of fullerene molecules. The developed substructure approach, together with the stability of the substructures constituting the fullerene molecule, helps to understand deeper the features of the electronic structure of fullerenes.

Ground-borne noise and vibrations in buildings induced by pile driving: An integrated approach

The use of pile driving techniques in urban environments demands accurate methods to predict and mitigate the potential discomfort of residents of neighboring buildings. In this context, a numerical model to predict ground-borne noise and vibrations in buildings induced by pile driving is proposed in the present paper. The model is based on a sub-structuring approach, in which the entire propagation medium is considered, from the vibration source to the receiver.The pile-ground domain is simulated by an equivalent linear FEM-PML approach formulated under axisymmetric conditions taking into account the pile-hammer interaction. The building simulation is performed using a 3D FEM approach, adequately adapted to include the soil-structure interaction, while the acoustic field induced by the structural vibration is analysed using a 3D MFS model. The levels of ground-borne noise and vibration inside the building resulting from pile driving are assessed through an application example, clearly showing the potential of the proposed methodology.

Towards a Non-classical Meta-theory for Substructural Approaches to Paradox

In the literature on self-referential paradoxes one of the hardest and most challenging problems is that of revenge. This problem can take many shapes, but, typically, it besets non-classical accounts of some semantic notion, such as truth, that depend on a set of classically defined meta-theoretic concepts, like validity, consistency, and so on. A particularly troubling form of revenge that has received a lot of attention lately involves the concept of validity. The difficulty lies in that the non-classical logician cannot accept her own definition of validity because it is given in a classical meta-theory. It is often suggested that this mismatch between the consequence relation of the account being espoused and the consequence relation of the meta-theory is a serious embarrassment. The main goal of the paper is to explore whether certain substructural accounts of the paradoxes can avoid this sort of embarrassment. Typically, these accounts are expressively incomplete, since they cannot assert in the object language that certain invalid arguments are in fact invalid. To overcome this difficulty I develop a novel type of hybrid proof-procedure, one that takes invalidities to be just as fundamental as validities. I prove that this proof-procedure enjoys a number of interesting properties and I analyze the prospects of applying it to languages capable of expressing self-referential statements.

On the physically consistent characterisation of a system for FRF-based substructuring

Substructuring approaches have been extensively used since long time to predict the vibroacoustic behaviour of built-up mechanical systems. In the frequency domain, these methods build on the dynamic characterisation of the subsystems via frequency response functions at the coupling interfaces. Intensively used for subsystems connected by points, dynamic substructuring still constitutes an open area of research for systems comprising continuous interfaces. Within this framework, a method which allows to characterise subsystems connected along lines is presented. The approach is based on a discretisation of the continuous interfaces into a small set of points, where the information of the subsystems is condensed. For this purpose, an inverse approach is used, which allows to characterise the dynamics of the passive subsystem using data of the assembled system. A straightforward inversion, however, does not assure the physical consistency of the characterized subsystem. In other words, the subsystem extraction procedure must not only guarantee correct dynamic behaviour of the assembled system, but the obtained subsystem must also be reciprocal and passive. The central theme of this paper is the formulation of a constrained optimisation procedure for the enforcement of reciprocity and passivity on the inversely obtained subsystem characteristics. The method is validated on a structure made of two plates connected along a common edge through a beam.

Eigensensitivitity analysis of large-scale structures by substructuring method

In sensitivity-based model updating, the eigensolutions and the associated derivatives of the analytical model need to be calculated iteratively to achieve an optimal solution, which is very time-consuming for a large-scale structure. To overcome this shortcoming, this paper proposes a new method to calculate the eigensensitivity via substructuring approach. When the design parameters in one particular substructure need to update, the sensitivity matrices to the updating parameters at this substructure are calculated while they are zeros at other substructures. The eigensensitivity at the global level is assembled by performing some constraints on the sensitivity matrices at the substructure level. The proposed method is applied in a frame and a practical bridge structure. Numerical results show that the present substructuring approach is more efficient than the traditional global method, at the satisfactory precision.

Evaluation of seismic response of inelastic structures considering soil-structure interaction

Structural deformations and soil displacements arising out of ground shaking are intertwined. This interplay is termed as seismic soil-structure interaction (SSI). Over last few decades, the conventionally perceived beneficial nature of SSI has been challenged with evidences of damaged sites from various earthquakes. This paper evaluates effects of SSI on seismic response of hardening single degree of freedom systems located in diverse geologies. The assessment is performed using substructure approach in terms of inelastic force reduction factor such that structure attains certain ductility levels. It is observed that soil-structure systems can afford a lesser reduction in design seismic forces compared to fixed-base structures. Components of total displacement arising out of structural deformation, footing translation and footing rotation are also evaluated. With increase in SSI effects, contribution of structural deformation is found to decline sharply. Footing rotation is observed to be very significant in structure-soil systems with larger SSI effects. The soil-structure system is modelled using a discrete physical model which enables inelastic response to be obtained in time domain while considering frequency dependence of impedance functions.

Effect of kinematic interaction on seismic response of offshore wind turbines on monopiles

AbstractInterest in renewable energy over the past decade has motivated a remarkable research on offshore wind energy. Due to the environmental loading conditions in the early developments of offshore wind farms, the focus of the research and engineering has been on aerodynamic and hydrodynamic subjects. However, earthquake has turned out to be a major design concern in seismic areas such as East Asia, Southern Europe, and United States. The topics include, among others, nonlinear response of foundations, soil liquefaction, and their impacts on foundation performance. An important topic in seismic soil‐structure interaction (SSI) analysis is the kinematic seismic response of foundations. This information is crucial for analyses based on the substructuring approach where both kinematic response and foundation impedances are key parameters. While considerable research has been spent on the latter topic, very little has been reported on the kinematic interaction. Large monopoles with low aspect ratios tend to rotate much more than regular piles in pile group foundations due to seismic waves. The rotation leads to additional load on the tower and the turbine. This article uses a rigorous numerical model for monopole‐soil interaction in layered soil and computes both rotational and horizontal responses at the pile head (seabed) as functions of frequency. A suite of generic homogeneous and heterogeneous soil profiles with shear moduli representative of clay (generally linear) and sand (generally parabolic) are considered in these analyses. Through representative analyses, both in frequency domain and time domain, it is illustrated how the kinematic interaction influences the earthquake loads imparted to an offshore wind turbine (OWT).

A Modified Numerical Substructure Method for Dynamic Analysis of Vehicle–Track–Bridge Systems

This paper presents a modified numerical substructure method for simulating the dynamic response of vehicle–track–bridge (VTB) systems. The method can be used to analyze large-scale VTB systems accurately and efficiently. Based on the principle of virtual work, the equations of motion are derived for two separate subsystems, i.e. a small-scale of finely modeled VTB substructure and a coarsely meshed large main bridge subsystem using different level of refinement. Different from the conventional dynamic substructuring approaches, the bridge spans close to the vehicle are modeled in both the main and substructure models, and the contradiction of repeatedly modeling is solved using a “nonlinear force corrector”. A special wheel–rail interaction (WRI) element is used to simulate the fast-moving interaction force between the vehicle and rail. In this way, the two models remain unchanged while the vehicle moves forward, and the computational accuracy is the same as the large-scale purely refined model, while the efficiency is significantly improved, particularly, for the large-scale long VTB systems. Two examples of realistic VTB systems with either smooth or un-smooth rails are used to verify the proposed method. The results demonstrate that the presented method has remarkable advantages of computational efficiency and accuracy, providing a practically useful tool for analysis of large-scale VTB systems.

Full-field FRF estimation from noisy high-speed-camera data using a dynamic substructuring approach

The use of a high-speed camera for dynamic measurements is becoming a compelling alternative to accelerometers and laser vibrometers. However, the estimated displacements from a high-speed camera generally exhibit relatively high levels of noise. This noise has proven to be problematic in the high-frequency range, where the amplitudes of the displacements are typically very small. Nevertheless, the mode shapes of the structure can be identified even in the frequency range where the noise is dominant, by using eigenvalues from a Least-Squares Complex Frequency identification on accelerometer measurements. The identified mode shapes from the Least-Squares Frequency-Domain method can then be used to estimate the full-field FRFs. However, the reconstruction of the FRFs from the identified modeshapes is not consistent in the high-frequency range. In this paper a novel methodology is proposed for an improved experimental estimation of full-field FRFs using a dynamic substructuring approach. The recently introduced System Equivalent Model Mixing is used to form a hybrid model from two different experimental models of the same system. The first model is the reconstructed full-field FRFs that contribute the full-field DoF set and the second model is the accelerometer measurements that provide accurate dynamic characteristics. Therefore, no numerical or analytical model is required for the expansion. The experimental case study demonstrates the increased accuracy of the estimated FRFs of the hybrid model, especially in the high-frequency range, when compared to existing methods.

An efficient three-dimensional method for the prediction of building vibrations from underground railway networks

This paper proposes a new efficient three-dimensional (3D) method to predict the vertical building vibration generated by trains running in tunnels in a multi-layered half-space. The tunnel-ground-building dynamic interaction involving in the proposed method is handled using the sub-structuring approach. The multi-story building is considered as rectangular floors supported by a distribution of columns. Each building floor is divided into several sub-plates. The dynamic stiffness matrix of a single sub-plate derived by the dynamic stiffness method is assembled to obtain the global one of each building floor. The free edge condition of the building floor is satisfied in this solution. The train-track-tunnel-ground dynamic interaction is modelled by a fully-coupled analytical method. The proposed method shows sufficient efficiency and accuracy, making it a suitable tool for predicting large-scale vibrations and performing parametric studies. The responses of a four-story building generated by the passage of trains in a tunnel are analyzed. The building vibrations are significantly affected by the dynamic properties of the building and the building-ground dynamic interaction. The dynamic interaction between the shallow foundations of buildings has a considerable influence on the building vibrations in the low-frequency range. The isolation effectiveness of the floating slab track and the base-isolation of buildings is highly dependent on the isolation frequency.

Vibration transmission and power flow of laminated composite plates with inerter-based suppression configurations

This paper investigates the vibration transmission and power flow behaviour of harmonically excited laminated composite plates attached with different inerter-based suppression configurations. The substructure approach based on analytical and numerical methods is employed to obtain the steady-state dynamic response. Power flow analysis is carried out to determine the time-averaged power flow input and transmission as well as the kinetic energy of the plate. The power flow density vector is used to show explicitly the vibration transmission paths within the composite plate. It is shown that the fibre orientation and stacking sequences can have significant effects on the time-averaged power flow characteristics as well as the dominant vibration transmission paths. It is also shown that passive spring, damper and inerter elements may be attached to the plate to modify its vibration response and transmission according to specific design requirements. The proposed inerter-based suppression device with two different configurations can reduce the vibration level over a wide frequency range for vibration suppression. The findings may provide insights for the enhanced dynamic designs of laminated composite plates, and the suppression of their vibrations using inerter-based devices.

Experimental updating of a segmented wind turbine blade numerical model using the substructure method

During the development of the wind turbine system, blades are considered as the most critical components. The blades’ dynamic characteristics must be investigated during the design process to improve their mechanical performance. This work presents an experimental updating of a segmented wind turbine blade numerical model. For this purpose, the blade segments were manufactured using the fused deposition 3D printing technology and assembled using a threaded spar and nut. For different values of the segments assembly load, the natural frequencies and damping ratios were identified using the eigensystem realization algorithm method. The dynamic behavior of the segmented blade was examined numerically using the three-node shell element, taking into consideration the additional stiffness generated by the applied assembly load. In this study, numerical and experimental modal analysis were used to identify the first five eigenfrequencies of the blade. To update the numerical model parameters, through the identified experimental results, an iterative method was developed based on the Craig–Bampton substructure approach. Results show that the blade natural frequencies are proportional to the segments assembly load. The application of the proposed method on the segmented blade numerical model updating shows that the present method is computationally efficient.

An intelligent nonlinear meta element for elastoplastic continua: deep learning using a new Time-distributed Residual U-Net architecture

Substructuring is a model order reduction technique that accelerates the finite element method in solid mechanics. In this improved hybrid substructuring approach, methods from computational intelligence empower a reduced-order meta element. We propose a nonlinear and inelastic intelligent meta element for history-dependent boundary value problems. Fully compatible with conventional finite elements, it can be used to assemble larger structures. Within the intelligent meta element, a new deep neural network architecture composed of convolutions and recursions, the Time-distributed Residual U-Net (TRUNet), learns to solve the history-dependent spatial regression problem. The TRUNet automatically creates and updates the internal history variables necessary for the mechanical problem. Based on a new data generation strategy, data from a wide variety of use-cases train the neural network. An interface connects the neural network and the finite element method using a new data pre- and post-processing strategy. In three numerical demonstrations of elastoplastic continua, the intelligent meta element performs well, exhibiting low errors on a separate test dataset of several thousand samples. The intelligent reduced-order models compute considerably faster and achieve excellent approximations of the displacements, stresses, and forces.

Seismic fragility analysis of URM buildings founded on piles: influence of dynamic soil–structure interaction models

The impact of adopting different dynamic soil–structure interaction (SSI) models on the assessment of seismic fragility functions for buildings with pile foundations is studied herein. Given the importance on seismic response of the coupling of structure, foundation and soil, and the challenges posed on modelling dynamic SSI, especially when soil nonlinearity plays a significant role, the linear substructure approach is initially adopted by implementing two different models, followed by the use of a nonlinear pile-head macro-element. The first model is one-dimensional and includes, between the foundation node and the ground, only a translational elastic spring and a dashpot, whose stiffness and viscous damping are retrieved from the real and imaginary parts of the dynamic impedance at the first natural frequency of the structure. The second and more refined model is a Lumped-Parameter Model accounting for frequency dependence of the coupled horizontal and rotational impedances in a two-dimensional response. A nonlinear pile-head macro-element model is introduced afterwards to explore the sensitivity of fragility functions to the linearity assumption. This approach models the soil nonlinear behaviour at near-field, as well as the far-field dynamic impedance and energy dissipation through radiation damping, by condensing the entire soil-foundation system into a single nonlinear element at the base of the superstructure. In all models, the superstructure is represented in a simplified way as a nonlinear single-degree-of-freedom system. The comparison between the adopted approaches is evaluated in terms of their effects on the characterisation of fragility functions for unreinforced masonry buildings with pile foundations.

(I can’t get no) antisatisfaction

Substructural approaches to paradoxes have attracted much attention from the philosophical community in the last decade. In this paper we focus on two substructural logics, named $${\mathsf {ST}}$$ and $${\mathsf {TS}}$$, along with two structural cousins, $${\mathsf {LP}}$$ and $${\mathsf {K3}}$$. It is well known that $${\mathsf {LP}}$$ and $${\mathsf {K3}}$$ are duals in the sense that an inference is valid in one logic just in case the contrapositive is valid in the other logic. As a consequence of this duality, theories based on either logic are tightly connected since many of the arguments for and objections against one theory reappear in the other theory in dual form. The target of the paper is making explicit in exactly what way, if any, $${\mathsf {ST}}$$ and $${\mathsf {TS}}$$ are dual to one another. The connection will allow us to gain a more fine-grained understanding of these logics and of the theories based on them. In particular, we will obtain new insights on two questions concerning $${\mathsf {ST}}$$ which are being intensively discussed in the current literature: whether $${\mathsf {ST}}$$ preserves classical logic and whether it is $${\mathsf {LP}}$$ in sheep’s clothing. Explaining in what way $${\mathsf {ST}}$$ and $${\mathsf {TS}}$$ are duals requires comparing these logics at a metainferential level. We provide to this end a uniform proof theory to decide on valid metainferences for each of the four logics. This proof procedure allows us to show in a very simple way how different properties of inferences (unsatisfiability, supersatisfiability and antivalidity) that behave in very different ways for each logic can be captured in terms of the validity of a metainference.

Dynamic soil-structure interaction models for fragility characterisation of buildings with shallow foundations

Dynamic Soil–Structure Interaction (SSI), involving the coupling of structure, foundation and soil, is a crucial and challenging problem, especially when soil nonlinearity plays an important role. This paper shows the impact of adopting different SSI models on the assessment of seismic fragility functions. The linear substructure approach is initially adopted by implementing two different models, the first of which is one-dimensional and includes, between the foundation node and the ground, a translational elastic spring and a dashpot, whose stiffness and viscous damping are retrieved from the real and imaginary parts of the dynamic impedance at the first natural frequency of the structure. The second and more refined model is a Lumped-Parameter Model (LPM) accounting for frequency dependence of the impedance. In order to explore the sensitivity of fragility functions to the linearity assumption, an additional approach, including soil nonlinearities, is employed. A nonlinear footing macro-element is adopted to model the near-field behaviour by condensing the entire soil-foundation system into a single nonlinear element at the base of the superstructure. Energy dissipation through radiation damping is also accounted for. The superstructure response is simulated in all approaches as a simple nonlinear single-degree-of-freedom (SDOF) system. The comparison between the adopted approaches is evaluated in terms of their effects on the characterisation of fragility functions for unreinforced masonry buildings (URM) on shallow foundations.