Displacement Continuity Research Articles

Free vibration analysis of damaged laminated piezoelectric plates and composite plates with piezoelectric patch based on the Extended Layerwise Method

The dynamic analytical models for a laminated piezoelectric plate and a laminated composite plate with piezoelectric patch containing the delamination, transverse crack and debonding damages are established by the Extended Layerwise Method (XLWM). The virtual kinetic energy is introduced into Hamilton’s principle, so resulting in mass matrices and displacement-dependent second mode derivatives in the finite element (FE) governing equations. Then, the FE governing equations for the laminated piezoelectric plate, and the characteristic equations of natural frequencies for the laminated piezoelectric plate with delamination and transverse crack are derived. The coupling model of the laminated composite plate and the piezoelectric patch is established by the displacement continuity and internal force equilibrium conditions at the nodes of the contact area. Based on the degrees of freedom (DoFs) in contact, the final governing equations is deduced, followed by the analysis of free vibration responses for both plates with various damages. The accuracy of natural frequencies for each plate is verified by comparing the results with those of FE simulation by ANSYS.

Vibration analysis of piping system coupled with conical-cylindrical combined shells

Fluid-filled piping system are widely used in power cooling, fuel transportation and underwater vessel, which not only produce vibration along the system, but also transmit to the connecting structure to generate noise in ambient environment. Most investigations have concentrated on either the vibration characteristics of piping systems or of combined shells separately. In this work, an analytical impedance synthesis method (ISM) is proposed to analyze the vibration characteristics of a pipe-hull coupled system, in which the hull can be simplified to a combined conical-stiffened cylindrical-hemisphere shell. The piping system is modeled by fluid-filled beams in three-dimensional configuration. Wave function and power series solution are used to describe the displacements of cylindrical and conical shell respectively. Based on exact analytical solutions, the impedance matrix of each segment is established separately and assembled in the global coordinate by displacement continuity and force balance boundary conditions. Forced responses are compared with the experimental results to verify the correctness of the method.

Stress transfer paths and damage modes of layered coal-rock mass under static loading

In order to explore the distribution characteristics of nonlinear mechanical behavior and the tendency of damage and failure of deep in situ coal-rock mass, a layered composite thin plate coal-rock mass model is established based on thin plate theory and finite element method. The continuity of interlayer stress and displacement is maintained by transfer matrix function, and the maximum shear stress is calculated by Mohr-Coulomb strength criterion. The distribution characteristics of stress, displacement, energy and maximum shear stress of coal-rock mass under static load are obtained. The analysis results are as follows : ① The torsional stress concentration areas of τxz and τyz are formed on the boundary of coal-rock interface, and the principal stress concentration areas of σx, σy and σz are formed in the center and center of the boundary, and the shear stress distribution of τxy is formed on the diagonal line. ② The maximum shear stress of the long axis has a strengthening effect on the shear failure, and each layer forms a stress state of alternating tension-shear-compression-shear. It first destroys in the middle of the coal-rock interface and boundary, and then destroys along the long side and short side. ③ The coal-rock mass presents a ′ X ′ -type shear zone along the radial and axial. The shear failure occurs first, followed by the principal stress compression failure. The cracks are preferentially developed in the direction perpendicular to the minimum principal stress, and the stepped ′ V ′ -shaped failure characteristics are formed under the action of shear and tensile stress.

Dynamic behaviors of general composite beams using mixed finite elements

A novel mixed finite element method is developed and implemented for analyzing the vibration and buckling behavior of general composite beams which consists both transversely layered and axially jointed materials. The governing state-space equations are derived using the Hamilton's principle, where both displacements and stresses are treated as fundamental variables. This semi-analytical method uses transfer relations in the transverse direction and finite element meshing in the longitudinal direction, overcoming the difficulties for general composite beams analysis and providing computational efficiency and analyzing flexibilities. The developed mixed finite element model ensures continuity of both displacements and stresses across the material interface, thereby resolving interfacial stress singularity issues and offering more reliable simulations of boundary conditions at both ends. The proposed method is formulated and validated for the free vibration and buckling analysis of general composite beams. Additionally, it is observed that material properties such as Young's modulus and density, as well as the stiffness of the interface connecting layers, have significant effects on the free vibration and buckling responses of the composite beams. Analysis of periodically distributed and bi-directional composite beams demonstrates the versatility of this method in handling two types of combination forms. The proposed method serves as a valuable reference for obtaining accurate vibration and buckling results while ensuring stress-compatibility for composite beams in practical applications.

A unified dynamic stiffness modeling of multi-plate coupled systems with discrete spring connections

This work presents a unified dynamic stiffness modeling for the vibration analysis of multi-plate coupled systems with discrete spring connections (MPCS-DSC). First, based on the governing differential equation of the plate, the dynamic stiffness matrix (DS) of transverse and in-plane vibration for a completely free rectangular plate is separately derived by combining the generalized superposition method and the projection method. Then, according to the continuity of displacements at the connection point between the spring and the plate, the projected DS matrix of the discrete spring is developed. Then, using an element assembly concept similar to that in the finite element method (FEM), global DS matrices of various coupled systems are determined by assembling the spring's DS matrice and the transverse or in-plane DS matrices of the plate. In order to verify the accuracy and applicability of the proposed method, the free and forced vibration analysis of four types of coupling systems is carried out. The reliability and applicability of the proposed method are confirmed by comparing the present results with those from open literature and the finite element solutions. This study not only expands the application range of DS modeling theory but also provides a powerful tool to investigate the vibration characteristics of the MPCS-DSC.

A semi-analytical approach for site-city interaction under oblique incident SH waves

In this paper, a semi-analytical approach was proposed for investigating the site-city interactions (SCI). The site was modelled as a horizontal layered half-space, while the building was represented as a collection of shear wall structures fixed on the surface of half-space. The total wave field was considered as a combination of free field and scattering wave field generated by buildings, simulated by direct stiffness method and Green's function, respectively. The coupling between the site and the city is achieved through the displacement continuity conditions at the bottom of the building. Finally, the impact and some influencing factors of SCI effect on site and buildings were studied. It is found that the SCI effect has a significant impact on surface displacement and building response. The SCI effect typically reduces surface displacement near cities and the amplitude of relative displacement of buildings, while increases the seismic duration of buildings. However, this influence will be affected by the incident angle, building layout, building height, and building spacing, so these factors should be comprehensively considered when analyzing the SCI effect. Through the case studies, it is demonstrated that this method can quickly calculate SCI models with complex factors without being limited by the number and thickness of soil layers, incidence angles, and different sizes and spacing of buildings.

Nitsche's method enhanced isogeometric homogenization of unidirectional composites with cylindrically orthotropic carbon/graphite fibers

An isogeometric homogenization (IGH) technique is constructed for the homogenization and localization of unidirectional composites with radially or circumferentially orthotropic carbon/graphite fibers. The proposed theory employs multiple non-conforming Non-Uniform Rational B-Splines (NURBS) patches to depict repeating unit cells (RUCs) representative of composite microstructures. Displacements are formulated using a two-scale expansion that integrates macroscopic and microscopic contributions, with the latter addressed through the isogeometric analysis technique. Nitsche's method is utilized to apply the interfacial traction and displacement continuity and periodicity conditions. The capability and accuracy of the IGH theory were validated upon comparison with the elasticity solutions that take into account explicitly fiber morphologies, along with classical micromechanics solutions based on equivalent fiber moduli. A comparative analysis with conventional finite-element techniques showcases the developed theory's ability to accurately replicate the singular stress field at the fiber center and to capture smooth stress distributions where significant stress gradients are encountered.

Modeling of interfacial multi-cracks in dissimilar laminated structures using a nodal-based Lagrange multiplier/cohesive zone approach

Recent decades have witnessed a growing interest in simulating interfacial debonding in laminated structures using cohesive zone (CZ)-based algorithms. In the present work, we develop a nodal-based Lagrange multiplier/cohesive zone (LM/CZ) approach to achieve the end with a special focus on interfacial multi-cracks in dissimilar laminated structures. The main advantages of the developed approach are four-folded: (1) being flexible for the mesh discretization of dissimilar laminated structures and compatible with various element types; (2) being computationally more efficient for dissimilar laminated structures; (3) addressing the so-called artificial compliance issue arising in intrinsic CZ modeling; (4) providing smooth transitions from uncracked to cracking states, and from cracking to contact states. In the presented approach, nodal pairs are constructed with the aid of a searching algorithm capable of dealing with matching and non-matching meshes in a unified manner. For each nodal pair, the displacement continuity is accurately enforced using LMs that are calculated in a predictor–corrector manner with a limited number of iterations. The constraints for nodal pairs are switched from LMs to cohesive forces after a defined criterion is satisfied. The non-penetration condition between interfaces is fulfilled using penalty-based contact forces in the cracking and pure contact states, leading to a smooth transition from cracking to contact states. The accuracy and effectiveness of the presented approach are validated using three benchmark tests. Finally, the capacity of the approach to describe interfacial multi-crack behaviors of dissimilar laminated structures has been exploited and demonstrated.

Using unsupervised learning based convolutional neural networks to solve Digital Image Correlation

Digital Image Correlation (DIC) is a robust, non-contact method for full-field deformation measurement widely used in photomechanics. Based on subset image matching and an iterative optimization architecture, traditional DIC deformation field analysis requires manual selection of such calculation parameters as subset size, rendering it challenging to balance spatial resolution and measurement resolution in non-uniform field measurements. The requirement of manual parameter adjustment is lowered by a recently emerged supervised learning-based neural network DIC method, as it establishes a direct correlation between images and deformation fields using deep neural networks (DNN). However, its generalization and accuracy are contingent on the scope and variety of the dataset because it requires training with a large number of simulated speckle images of deformation fields with known standards. This paper proposes an unsupervised learning neural network based DIC measurement method that circumvents the need for standard deformation fields and achieves high precision analysis of deformed fields. It constructs a loss function based on the photometric consistency assumption of speckle images before and after deformation and allows the neural network model to directly extract displacement information from the gray data of the “reference image-deformed image” pair. Meanwhile, the model is enhanced accordingly by further introducing displacement continuity constraints to enhance accuracy and mitigate noise interference. Validation through both simulated and real experiments shows that the proposed approach surpasses traditional subset DIC in measuring non-uniform deformation fields with higher precision. It also offers superior flexibility and adaptability over supervised learning-based neural network DIC methods by eliminating the need for a large training dataset, allowing for immediate application in displacement field analysis.

Non-linear buckling analysis of delaminated hat-stringer panels using variational asymptotic method

This research proposes a computationally efficient methodology using a Constrained Variational Asymptotic Method (C-VAM) for non-linear buckling analysis on a hat-stringer panel with delamination defects. Starting with the geometrically non-linear kinematics, the VAM procedure reduces the three-dimensional (3-D) strain energy functional to an analogous 2-D plate model and evaluates the closed form warping solutions. Utilising the resulting warping solutions and recovery relations for the skin and the stringer, displacement continuity at the three-dimensional level is enforced between the stringer and the skin based on the pristine and delaminated interface regions. Consequently, the constrained matrices obtained from C-VAM is incorporated into an in-house developed non-linear finite element framework. Using the developed formulation, a stiffened panel with delamination of 40 mm between the stringer and the skin is analysed under compression. The results have been validated locally and globally, employing experimental data and 3-D finite element analysis (FEA). Experiments are carried out on the co-cured panel by applying quasi-static loading with displacement-controlled conditions, and 3-D FEA is carried out in Abaqus. Load-response plots have been obtained to validate the results at the global level, and they are in excellent agreement with experiments and 3-D FEA. Subsequently, out-of-plane displacement contour plots are obtained; the number of half waves and wave intensity in 3-D FEA and C-VAM are comparable, although there are minor differences compared to the experimental findings. The proposed framework is shown to be computationally efficient by over 55% as compared to 3-D FEA for performing non-linear buckling analysis on the stiffened composite structure considered in the current work.

Layerwise third-order shear deformation theory with transverse shear stress continuity for piezolaminated plates

Regarding laminated structures, an electromechanically coupled Finite Element (FE) model based on Layerwise Third-Order Shear Deformation (LW-TOSD) theory is proposed for static and dynamic analysis. LW-TOSD ensures the continuity of in-plane displacements and transverse shear stresses. The current LW-TOSD can be applied to arbitrary multi-layer laminated structures with only seven Degrees of Freedom (DOFs) for each element node and eliminates the use of the shear correction factors. Moreover, a shear penalty stiffness matrix is constructed to satisfy artificial constraints to optimize the structural shear strain. A dynamic finite element model is obtained based on LW-TOSD using the Hamilton’s principle. First, the accuracy of the current model is validated by comparing with literature and ABAQUS results. Then, this study carries out numerical investigations of piezolaminated structures for different width-to-thickness ratios, length-to-width ratios, penalty stiffness matrix, boundary conditions, electric fields and dynamics.

Shape sensing of the thin-walled beam members by coupling an inverse finite element method with a refined quasi-3D zigzag beam theory

In structural safety field, the inverse finite element method (iFEM) is an effective methodology to reconstruct full-field displacement on beam, plate and shell structures, independently of the loading conditions and of the material properties. However, the current iFEM in principle requires uniform shear distribution over the thickness of beam, which is practically hardly possible due to these thin-walled beam with the general cross-section shape, such as I-section beam, hat-section beam and box-section beam, and there is no effective method to realize deformation online monitoring at home and abroad. To relieve this issue, a novel iFEM strategy is proposed to establish the shape sensing model of the thin-walled beam, where the thin-walled beam is replaced with an equivalent layered composite one based on a generalized layered global-local beam (GLGB) theory, and the improved quasi-3D zigzag shear deformation theory is presented to describe deformation field of the equivalent layered composite beam. The proposed iFEM method accounts for not only thickness stretching but also interlaminar continuity of shear stresses and displacements. Besides, the proposed iFEM formulation does not need any shear correction factors. Accuracy and effectiveness of the established shape sensing model are demonstrated through several case studies. The numerical results show that the proposed iFEM can accurately reconstruct the deformation of the thin-walled structure and the reconstruction accuracy can be improved by 5 %.

A one-dimensional high-order dynamic model for twin-cell box girders with deformable cross-section

A one-dimensional high-order dynamic model for single-box twin-cell box girders is presented together with the pattern recognition algorithm. The model takes into account the deformable cross-section and can accurately predict its 3D dynamic behaviors. The cross-section deformation is captured by basis functions satisfying displacement continuity condition, which is essential to construct the initial model formulation based on the Hamilton principle. The axial variation patterns of generalized coordinates are decoupled by solving the eigenvalue problem. On this basis, the combinations of basis functions are obtained to bring out cross-section deformation. The cross-section deformation, hierarchically organized and physically meaningful, are used to update the basis functions in the reconstructed high-order model. Numerical analysis has verified the accuracy and applicability of the reconstructed one-dimensional high-order model.

A novel semi-analytical model for laterally loaded piles

A new model for laterally loaded piles is proposed, consisting of a linear part and a non-linear iterative part. The linear part is based on the Green's function method, which strictly satisfies the displacement and stress continuity conditions at the pile-soil interface and leads to a rigorous solution. The non-linear iterative part assumes that the shear modulus of the soil around the pile decreases with the average strain of the soil layer. Only two soil parameters need to be calibrated, including the initial shear modulus of the soil and the reference shear strain at half the initial modulus. Compared to the finite element model, the advantage of this model is that it converges quickly while maintaining accuracy, which greatly improves computational efficiency. The advantage of this model over the p-y curve model is that it does not have the difficulty of selecting p-y spring parameters, especially for monopiles. The accuracy of this model is verified by comparison with three-dimensional finite element models and nine existing studies, including numerical models, field tests and centrifuge tests. These cases cover various piles with diameters of D∼0.273–10 m and aspect ratios of L/D∼3–40, installed in sand or clay by driving, jacking or vibration.

Torsional vibration of tapered pile including circumferential support of surrounding soil

The tapered pile is usually divided into finite segments along the vertical direction due to its variable cross-section when investigating its torsional dynamic characteristics. However, an annular projection would be formed at the contact surface of the adjacent pile segments due to the difference in radius, through which the surrounding soil will apply the circumferential support on the pile segment. In this study, the Voigt model is adopted to model the circumferential support of the surrounding soil acting on the annular projection. On this basis, an amended impedance function transfer method is established combined with the circumferential stress equilibrium equations and circumferential displacement continuity conditions at the contact surface of the adjacent pile segments. The shear complex stiffness transfer method is employed to consider the construction disturbance during pile installation. Then, an improved analytical solution for the torsional vibration of the tapered pile considering the circumferential support of the surrounding soil is established by solving the equations for the soil and pile. Based on the solution, the torsional vibration of the tapered pile is investigated and the influence of the circumferential support of the surrounding soil is revealed under different pile parameters and construction disturbance conditions.

Boundary element method for bending-extension coupling shear deformable laminated composite plates with multiple stiffeners

In general, the laminated composite plates and stiffeners may exhibit the bending-extension coupling effect with the transverse shear deformations significantly induced. In this paper, for the first time we develop the associated boundary element method (BEM) for bending-extension coupling shear deformable laminated composite plates with multiple beams attached as stiffeners. Based upon the first order shear deformation theory, the individual boundary integral equations (BIEs) for plates and stiffeners are obtained from their corresponding matrix-form basic equations. To account for perfect bonding conditions between plate and stiffeners, we establish displacement continuity and interaction load equilibrium relations at their interfaces. To calculate the domain integrals related to unknown interaction loads, we discretize each stiffener into a series of cells using quadratic interpolation. By collecting the individual BIEs for plates and stiffeners and employing the continuity and equilibrium relations at the interfaces, we construct the entire system of linear equations to solve for all the unknown nodal quantities. Additionally, the complete solutions at the plate boundary, at internal points or along a stiffener are provided. The correctness of our present BEM formulation is verified through two representative examples by comparison with the commercial finite element software ANSYS. The numerical results demonstrate that our BEM formulation surpasses the conventional finite element method and is highly desirable for general stiffened laminated plates with bending-extension coupling and transverse shear deformations.

Research on Full-Field Dynamic Deflection Measurement of Beams Based on Dense Feature Matching and Mismatch Removal Method

To solve the problems of measurement errors led by mismatches of dense feature matching in machine vision structural deflection measurement, this paper proposes a dense feature extraction, matching, and dual-step mismatch-removal-based full-field structural dynamic deflection measurement method. First, the of dense feature detection and matching theory is introduced to extract the SIFT feature points on a structural surface in an image sequence and matched by FLANN to trace the structure movement, and the mechanisms and causes of mismatches are analyzed. Then, a dual-step mismatch removal method combining RANSAC and Structural Displacement Continuity Restriction (SDCR) is introduced to achieve full-field dynamic beam deflection measurement. The proposed method is validated through indoor cantilever beam experiments, and results show that the method can effectively eliminate a large number of SIFT feature mismatches (accounting for approximately 55% of the total matched feature points). The full-field dynamic displacement field of the beam can be measured with the correctly matched dense feature points by converting dense feature point displacements into continuous and uniform spatiotemporal deflections of the structure. A comparison with the GOM Correlate Professional DIC measurement system was conducted, and the maximum measurement error of the cantilever beam dynamic displacement of the proposed method is between 0.024 and 0.053 mm, the root mean squared error of displacement is approximately 0.01 mm, and the correlation coefficient between two deflection–time curves reaches 0.9964. The proposed algorithm is proven to be effective in full-field displacement measurement and has great potential in future structural health monitoring of bridges.

Transverse vibration of plate with multiple curved through cracks

A modeling method for transverse vibration of cracked plate containing multiple internal curved through cracks is presented and experimental studies are performed to validate its effectiveness. The cracked plate is assumed to be divided into several rectangular and curved trapezoidal domains in the global coordinates system. For each sub-domain, the transverse displacement is expanded as Chebyshev orthogonal polynomials series in local coordinates system. As for the shared edges between neighboring two sub-domains, the local collocation method is adopted and a number of Chebyshev zero points are selected to ensure the displacement continuity condition. To verify the convergence and correctness of the presented model and solutions, the natural frequencies, mode shapes and dynamic responses for several cases are compared with reference results and experimental results, with good agreement observed. In addition, the influence of several significant parameters of cracks are investigated, including the number, location and geometrical shape of the cracks. It is shown that the crack inside the plate presents less influence than that at the boundary on the mode change of the plate. The investigations in the study implies that the present model method is effective and convenient for plate with multiple curved through cracks.

Seismoelectric wave propagation through a fluid-saturated porous sandwiched interlayer

The seismoelectric waves generated in a fluid-saturated porous interlayer and the reflection/transmission characteristics at the interfaces of sandwiched structure are studied in this paper. Aiming to describe the strong attenuation and dispersion simultaneously at seismic and sonic frequencies, a new fractional viscoelastic model for the solid skeleton is introduced to the Pride's seismoelectric coupling theory. Using the interfacial continuity of stress, displacements, magnetic and electric fields, the reflection/transmission coefficients are determined. The numerical results are provided and shown graphically. The effects of seismoelectric coupling and the viscoelastic characteristics of solid skeleton on the reflection and transmission of seismoelectric waves are mainly studied. It is observed that the thickness, porosity and the viscoelastic characteristics of solid skeleton have evident effects not only on the electromagnetic waves but also on the seismic waves.

Large deformation assessment of three-tower suspension bridges: An enhanced analytical model and solution algorithm

An enhanced nonlinear approach is proposed in this study to address the limitations of traditional deflection theory in analyzing large deformations of three-tower suspension bridges. In this approach, the three-tower suspension bridge is simplified as two quasi-independent bridges for analysis, and detailed considerations are given to the vertical and longitudinal deformations of the main cable, which produce the expression for the relationship between the cable's internal force and deformations. By employing the complementary energy principle and variational calculus, the nonlinear differential equations describing bridge deformations are then established. To ensure the displacement continuity at the middle tower, this paper proposes the use of Chebyshev polynomials to approximate the solutions of the differential equations, and the weighted residual method is then used to transform these differential equations into algebraic equations. On this basis, the developed nonlinear adjustment optimization algorithm enables efficient solving of the nonlinear equation and thus determination of bridge deformation under various load conditions. The accuracy and effectiveness of the proposed scheme are validated through the analysis of two suspension bridges with equal and unequal spans. Finally, the extended applications of the proposed scheme on the other type of multi-span suspension bridges are introduced for further applications.