Displacement Approximation Research Articles

Comparative Study of Skin–Stringer Connection Approaches for Stiffened Structures with Curvilinear Stiffeners

Additive manufacturing enables the deposition of material in the near-net shape at any desired location on the base plate or shell to improve the structural performance by tailoring the buckling and/or vibration mode shapes. In this study, a stiffened plate with arbitrarily shaped stiffeners is modeled using a nonconformal mesh-based finite element model. Six degrees of freedom are considered for each node of the plate and the stiffener finite element models to enable the couplings between the plate’s and stiffeners’ in-plane and out-of-plane motions. Displacement compatibility is enforced at the interface between the plate and the stiffeners where several skin–stringer connection approaches, including inverse isoparametric mapping algorithm (IIMA), radial basis function, and thin-plate spline function, are used. Various complex models available in the literature are employed for evaluating the skin–stringer connection approaches. Research studies show that the IIMA-based nonconformal mesh modeling can generate a sparse displacement approximation matrix, which makes it an efficient approach for nonconformal mesh-based modeling. For the stiffened plate with higher stiffeners, the present results agree well with those obtained from the Abaqus/CAE conventional conformal mesh and mesh tie constraint-based nonconformal mesh modeling results, but yield better results than the Abaqus/CAE wrapping mesh approach.

Peridynamic modeling of elastic instability and failure in lattice beam structures

A Simo-Reissner peridynamic (PD) beam theory is presented to model elastic instability and subsequent failure in a lattice beam structure. Governing equations of the beam are derived from the three-dimensional PD equations of motion invoking Simo’s displacement approximation. Principle of virtual work is employed to determine the deformation states. The PD force and moment resultants are related to their classical counterparts using constitutive correspondence approach. The solution method employs a geometrically exact rotation update based on exponential map. A new failure criterion is introduced using critical stretch and critical relative rotation. The Newton–Raphson and two variants of arc length method are employed to solve the governing equations under quasi-static loading conditions. Numerical simulations concern the snap-back instability of a right-angle frame and its subsequent failure in the post-buckling stage, finite deformation of an octet lattice cell and instability of a portal frame. These simulations are verified against the finite element predictions to establish the efficacy of the present approach.

Analysis of Adaptive Pushover in Confined Masonry Dwellings with Height Irregularity in the Lima Metropolitan Area

Non-engineered confined masonry dwellings are characterized by having different masonry units. One of the most common cases in the Lima Metropolitan Area is that this type of dwelling has its first floors with handmade solid units and upper floors are composed of industrial horizontal hollowed (tubular) bricks. Nonlinear static analysis by the incrementally increasing loading pattern is inadequate to estimating earthquake response of structures with height irregularity, such as non-engineered confined masonry dwellings, because the loading pattern changes during the history of reversal loads due to damaging earthquakes. This paper presents the effect of nonlinear static analysis using adaptive loading patterns in confined masonry dwellings when the type of masonry units changes drastically. Nonlinear static analyses are applied using non-adaptive and adaptive displacement-based pushover procedures. A series of nonlinear time history analyses are also conducted to measure the error in the estimation of the seismic response. The adaptive loading pattern is based on displacement approximation by the modal analysis at each step of incremental-increasing loads. The load pattern is updated considering the modal analysis of an equivalent linear model using the equivalent stiffness of the structure at the secant of the maximum previous response. The application of the adaptive loading pattern in nonlinear static analysis significantly approximates the earthquake response of confined masonry dwellings with height irregularity provided by the change of masonry bricks.

Analysis of near-interface cracks in three-dimensional anisotropic multi-materials by efficient BIEM

This paper presents the full analysis of near-interface cracks in three-dimensional linear elastic multimaterial bodies by a weakly singular boundary integral equation method (BIEM). The proposed technique was implemented in a general framework that allowed the treatment of finite cracked bodies with general configurations, material anisotropy, and mode-mixity. A system of integral equations governing unknown data on the boundary, crack surface, and material interface was established using a pair of weakly singular weak-form displacement and traction integral equations together with continuity along the material interface. A symmetric Galerkin boundary element method along with the finite element technique were implemented to solve the governing integral equations. In addition, the special near-front interpolation was employed to enhance the approximation of the relative crack-face displacement in the neighborhood of the crack front. The solved crack-face data were used directly to post-process the stress intensity factors and T-stresses along the crack front. An extensive numerical study was carried out not only to demonstrate the accuracy, convergence, and capability of the proposed technique, but also to explore the influence of material stiffness and distance to the material interface on fracture data along the crack front.

Influence of flexoelectricity on interface crack problems under a dynamic load

In the present paper, the influence of flexoelectricity on behavior of the interface crack between two dissimilar dielectric materials under a dynamic mechanical load is investigated. The induced electric field affects the distribution and evolution of mechanical fields in dielectric materials. Large strain gradients induce the electric polarization in the direct flexoelectricity. Due to the large strain gradients at the crack tip vicinity it is needed to consider the strain gradient theory model. Governing equations in this theory contain higher-order derivatives than in the conventional continuum mechanics approach. The mixed finite element method (FEM) is developed here for a general boundary value problem, where the standard C0 continuous finite elements are applied for independent approximations of displacements and strains. The constraints between them are satisfied by collocation at appropriate internal points in elements. Interface cracks are observed in layered structures frequently due to a poor adhesion of layers. The incorporation of flexoelectricity and micro-inertial effects is needed into the failure analysis of interface cracks in nano-sized structures under dynamic loading. In numerical examples, we discuss the influence of flexoelectricity coefficients as well as the ratio of elastic coefficients and the geometrical size to microstructural (micro-stiffness and micro-inertia) length scale parameters of the bilayer composite on the crack opening displacement, stresses ahead the crack tip and induced electric intensity vector.

A weak form quadrature element formulation of geometrically exact strain gradient shells

A geometrically exact strain gradient shell model and its weak form quadrature element formulation are proposed. The governing equations, boundary conditions and corresponding variational form of the shell model are presented. The generalized differential quadrature analogue is employed for derivative approximations of displacements and rotations with C1 continuity conditions brought about by the second-order strain gradient terms. A novel scheme based on auxiliary vectors is developed to represent and update the derivatives of constrained rotations on element edges for the numerical approach. The present quadrature shell element is feasible to incorporate strain gradient terms and free from shear and membrane locking problems due to its adjustable high-order approximation property in nonlinear shell analysis. To demonstrate the potential of this formulation for nonlinear analysis of shells with size effects, five benchmark numerical examples are presented. The results verify the feasibility of this formulation and illustrate the strain gradient effects on elastic shells undergoing large displacements and rotations.

Improved Operator-Splitting Method for Dynamic Real-Time Substructure Testing of Multiple-Degree-of-Freedom Structure

The operator-splitting (OS) method for real-time substructure testing (RTST) provides explicit and unconditionally stable solutions for structures with softening type stiffness and damping has been widely used for precisely disclosing the seismic performance of structures with velocity-dependent specimens. However, the OS method for RTST provides only explicit target displacement and velocity but not an explicit target acceleration, so that it is essentially an implicit method for Dynamic real-time substructure testing (DRTST) with dynamic system specimens. For precisely disclosing the seismic performance of multiple-degree-of-freedom structures with dynamic system specimens, one improved OS method is proposed for DRTST in this paper. The methodology and stability of the proposed OS method are investigated, and the effectiveness of the proposed OS method is validated by numerical simulations and experimental tests. By using the predictor displacement, velocity, and acceleration approximations, the OS method is an explicit algorithm for DRTST. The stability analysis shows that the proposed OS method is conditionally stable for DRTST, and its stability limit decreases with the increase of the mass ratio and the stiffness ratio, while it increases first and then decreases with the increase of the damping ratio. The numerical and experimental results confirm the stability and accuracy of the OS method. The OS method for DRTST provides a practical and precise investigation strategy for structures with dynamic system specimens and may have broad application prospects in civil engineering.

A comparative study on the lateral displacement of a multi-story RC building under wind and earthquake load actions using base shear method and ETABS software

The aim of this study is to propose a comparison between wind and seismic loads on a multi-story reinforced concrete (RC) building. The investigation focuses on lateral displacements and story drifts of a 6-story RC structure, which are determined manually using the base shear method and with the aid of the ETABS software tool. The study considers a linearity of the structure and uses the Chinese building design code for lateral displacement analysis. The base shear method is a simplified method that only considers one degree of freedom for each story, neglecting the non-linearity of the structures. However, the Chinese building code permits its use when considering the consistent vertical distribution of the building's mass and stiffness, which leads to shear deformation prevailing during seismic and wind reactions. The lateral displacements under wind and seismic loads are estimated, and the story drift limitation criteria of the Chinese building design code are assessed. Both the base shear method and ETABS's story drift computation (story drift 1/550) meet the design code criteria. This indicates that the base shear technique approach of RC frame structures can reasonably predict the tale drift on each story, despite relying heavily on simplifications. To compare the outcomes from the ETABS program to the internal forces under seismic and wind load activities, the story displacements and drifts are computed. The results show little discrepancy between the base shear method and ETABS software. Hence, the base shear approach can be utilized when there is no irregularity, and the building height is less than 40 m. Overall, the findings of this study indicate that the base shear method can provide a good approximation for the lateral displacements and story drifts of multi-story RC constructions under wind and seismic loads, provided that the Chinese building code criteria are met. The study highlights the importance of using appropriate design codes and tools in ensuring the safety and reliability of structures. Further research can explore the applicability of similar simplified methods in more complex and irregular structures, and the development of more accurate and efficient design tools.

An explicit total Lagrangian Fragile Points Method for finite deformation of hyperelastic materials

This research explored a novel explicit total Lagrangian Fragile Points Method (FPM) for finite deformation of hyperelastic materials. In contrast to mesh-based methods, where mesh distortion may pose numerical challenges, meshless methods are more suitable for large deformation modelling since they use enriched shape functions for the approximation of displacements. However, this comes at the expense of extra computational overhead and higher-order quadrature is required to obtain accurate results. In this work, the novel meshless method FPM was used to derive an explicit total Lagrangian algorithm for finite deformation. FPM uses simple one-point integration for exact integration of the Galerkin weak form since it employs simple discontinuous polynomials as trial and test functions, leading to accurate results even with single-point quadrature. The proposed method was evaluated by comparing it with FEM in several case studies considering both the extension and compression of a hyperelastic material. It was demonstrated that FPM maintained good accuracy even for large deformations where FEM failed to converge.

Higher-Order Accurate Explicit Time Schemes with Improved Dissipation Properties

In this paper, a novel family of fourth-order accurate explicit time integration schemes is developed by combining the new time approximations and the explicit fourth-order Runge–Kutta (RK4) method. Inaccurate predictions due to the presence of excessive numerical dissipation are often observed in practical analyses of structural dynamics when the RK4 is employed for both displacement and velocity approximations. To remedy this, novel time approximations with adjustable algorithmic parameters are employed for the displacement vectors while the velocity vectors are approximated by using the RK4. For the complete elimination of numerical dissipation, algorithmic parameters are unconventionally determined by taking the determinant of the amplification matrix as unity. A set of algorithmic parameters obtained from this process makes the new schemes completely non-dissipative while keeping the computational cost the same as the RK4. Due to this improvement, the new schemes have enhanced total energy-conserving capabilities for nonlinear systems and give noticeably more accurate predictions in practical analyses. Until now, controllable numerical dissipation and fourth-order accuracy are not attained simultaneously in a unified set of time schemes. In the new schemes, however, a systematic way to adjust the level of numerical dissipation is also presented while attaining fourth-order accuracy. The numerical results of various test problems show that the new time schemes can provide more accurate predictions for nonlinear conservative dynamic problems than the existing time schemes.

An Interface/Boundary-Unfitted EXtended HDG Method for Linear Elasticity Problems

An interface/boundary-unfitted eXtended hybridizable discontinuous Galerkin (X-HDG) method of arbitrary order is proposed for linear elasticity interface problems on unfitted meshes with respect to the interface and domain boundary. The method uses piecewise polynomials of degrees \(k\ (\ge 1)\) and \(k-1\) respectively for the displacement and stress approximations in the interior of elements inside the subdomains separated by the interface, and piecewise polynomials of degree k for the numerical traces of the displacement on the inter-element boundaries inside the subdomains and on the interface/boundary of the domain. Optimal error estimates in \(L^2\)-norm for the stress and displacement are derived. Finally, numerical experiments confirm the theoretical results and show that the method also applies to the case of crack-tip domain.

New Data on the Isomorphism in Eudialyte-Group Minerals. XII: Refined Structure of the Hydrated Mineral from the Inagli Massif

A hydrated eudialyte-group mineral from the Inagli alkaline massif (Kola Peninsula) has been investigated by electron-probe and X-ray diffraction analyses. The parameters of the trigonal unit cell are as follows: a = 14.1558(3) Å, c = 30.998(1) Å, V = 5378.6(3) Å3, sp. gr. R3; the calculated density is 2.661 g/cm3. The crystal structure is refined to the final reliability factor of R = 3.8% within the anisotropic approximation of atomic displacements using 1230 reflections with F 3σ(F). The simplified formula of the mineral (Z = 3) is (H3O)7Na7.5(K,Sr,Ba,Ce)1.5Zr3Ca6[Si24O68(OH)3(O,OH)]Si1.5Cl1(OH)2H2O1.5. The problems of symmetry of calcium-rich hydrated eudialyte-group minerals are discussed.

Solution of the plane problem of the theory of elasticity on bending of an articulated fixed multilayer panel with a circular axis

This article suggests a method for the solution of the plane problem of the theory of elasticity on the bending of an articulated fixed multilayered panel with a circular axis based on the polynomial approximation of displacements through the thickness of the panel. In contrast to the known solutions of this problem, in this case the coefficients of the approximating polynomials are calculated from the equilibrium conditions and equality of displacements and transverse stresses at the transition across the layer interface and solution of differential equations of equilibrium at several points through the thickness of the layers. Finally, the problem is reduced to the solution of a system of linear equations with respect to the coefficients of approximating polynomials. The validity of the method is confirmed by comparing the results of calculations obtained on its basis and the results obtained with the help of the reference finite element model. The problem is solved in two stages. At the first stage, for a single-layer panel, we investigate the dependence of the polynomial degree on the ratio of the average panel radius to its thickness and the ratio of the transverse shear modulus to the modulus of longitudinal elasticity, which characterize the nonlinearity of displacements. At the second stage, on the example of a three-layered panel, we consider the application of the proposed method for the calculation of multilayered panels. In such case, the results obtained at the first stage are used in selecting the initial degree of polynomials approximating displacements through the thickness of layers. The method proposed in this article makes it possible to obtain an analytical solution without introducing simplifying hypotheses about the nature of displacement of layers and their elastic characteristics in a wide range of variation in geometric dimensions and elastic characteristics of panel layers. This method can be used both for verification of numerical models and for carrying out strength calculations of multilayer panels.

Analysis of shell structures by an improved 3-node triangular flat shell element with a bubble function and cell-based strain smoothing

An enhanced 3-node triangular flat shell element with 18 degrees of freedom according to the Reissner–Mindlin theory is presented in this paper. New cell-based smoothed (CS) bending strains were derived from the displacement approximations enriched by a cubic shape function for the rotations at the bubble node located at the centroid of the triangular element. The true drilling degrees of freedom of the proposed element, which is independent from user-defined coefficients, was achieved by cell-based smoothing the Allman’s membrane strains. The membrane and bending stiffness matrices were then efficiently determined by the line integration of the smoothing domains’ boundaries. The shear-locking removal MITC3+ technique was employed to interpolate the transverse shear strains. The outstanding performance of the developed 3-node triangular flat shell element, so-called the CS-MITC18+ shell element, was demonstrated through the static and free vibration analyses of several benchmark shell structures. Numerical results shown that the CS-MITC18+ flat shell element has the spatially isotropic and zero-energy mode properties, and provides competitive accuracy, convergence, and computational cost in comparison to other conventional and strain-smoothed 3-node triangular flat shell elements.

A novel displacement amplification factor (Cd) for steel moment-resisting frames based on performance

Displacement amplification factor is one of the most important factors in seismic design of structures. Choosing the sizes of structural section in steel moment resisting frames generally depends on the story drift limitation. In current regulations, the inelastic displacement is determined by multiplying obtained displacement out of equivalent linear static analysis by a displacement amplification factor (Cd). In this paper, several moment resisting frames have been analyzed using elastic static and inelastic dynamic analyses. After post processing of the results, a relation has been suggested for displacement amplification factor which varies along the height of the frame in three different performance levels. Three sample structures illustrate that how proposed relation improves the Cd and makes more rational results than traditional Cd. Results also show that the considered displacement amplification factor in Eurocode8 makes lower approximation of displacement.

Application of the gradient theory to interface crack between two dissimilar dielectric materials

In the present paper, the interface crack between two dissimilar dielectric materials under a mechanical load is investigated with including flexoelectricity effects. Flexoelectricity is a size dependent electro-mechanical coupling phenomenon, where the electric polarization is induced by a strain gradient in dielectrics. The strain gradients may potentially break the inversion symmetry in centrosymmetric crystals and polarization is observed even in all dielectric materials. The polarization is proportional to the strain gradients in the direct flexoelectricity. Layered composite structures are frequently utilized in microelectronics. Due to a poor adhesion of protection layer and basic material, the interface crack can be created there and for the prediction of failure of these structures it becomes essential to investigate distribution of the interfacial stress and strain fields. Governing equations in the gradient theory contain higher-order derivatives than in the standard continuum mechanics. Therefore, areliable computational tool is required to solve these boundary-value problems. The mixed finite element method (FEM) is developed, where the standard C0 continuous finite elements are utilized for independent approximations of displacements and strains. The constraints between the displacement gradients and strains are satisfied by collocation at Gaussian integration points inside elements. In numerical examples, a parametric study is performed with respect to flexoelectric and elastic coefficients for both material regions. The influence of these parameters on the crack opening displacement is discussed.

A new enrichment scheme for the interfacial crack modeling using the XFEM

A new enrichment scheme is proposed for the interfacial crack modeling in the framework of extended finite element method (XFEM), in which the approximation of displacement in the element including the crack tip is reconstructed by only the Heaviside function and the material interface enrichment function. The proposed enrichment scheme can model not only the discontinuity of displacement across the crack surface but also the discontinuity of displacement gradient across the material interface in the crack tip element with the weight functions. In addition, no blending elements are introduced and it is unnecessary to define the ramp function in the new XFEM formulation. The static and dynamic stress intensity factors (SIFs) of interfacial cracks are analyzed by the new enrichment scheme and the results show that the effect of material interface in the crack tip element plays a significant role in the accuracy of fracture mechanics parameters and the proposed method can predict the deformation more accurately and obtain a more stable dynamic SIFs. The proposed new enrichment scheme can be extended to three dimensional crack modeling in future works.

The Method of Local Approximations in the Construction of High-Precision Small-dimensional MgFE for the Calculation of Composite Bodies

The method of multigrid finite elements is effectively used to analyze the stress state in composite bodies (CB). When constructing a multigrid finite element (MgFE), briefly a standard MgFE, using known procedures, a fine grid, and large ones nested in a fine one are used. The fine grid is generated by partitioning, which takes into account the heterogeneous structure of the MgFE, large grids are used to reduce its dimension. For a standard MgFE, it is characteristic that every large grid and the corresponding approximations of displacements are determined throughout its entire area. This leads to an increase in the dimension of the standard MgFE when constructing high-order approximations on large grids, which are used to increase its order of accuracy. Standard high-precision MgFE, i.e. of high order of accuracy, have a large dimension, which makes their application difficult.
 In this paper, a method of local approximations (MLA) for constructing high-precision small-dimensional MgFE (short — small-sized MgFE) is proposed. Such MgFE are used to calculate elastic CB and are designed on the basis of standard. The main idea of the MLA is that local approximations of high-order displacements are determined on large grids in the central part of the region of a small-sized MgFE, and in the vicinity of the boundary of the region — of a small order, which allows using various local approximations to vary the dimension and order of accuracy of a small-sized MgFE. Two approaches to the construction of small-sized MgFE are shown, in the case of their complex shape, forming finite elements are used. Calculations show that small-sized MgFE generate stresses in the CB, the errors of which are 15÷50 smaller than the errors of similar stresses corresponding to standard MgFE, i.e. small-sized MgFE are more effective than standard ones. The use of smallsized MgFE in calculations makes it possible to determine stresses with a small error for large CB partitions.

Кристаллохимические особенности обогащенного стронцием аквалитоподобного минерала из Кондерского массива

A hydrated Sr-rich member of the eudialyte-group mineral from Konder (Southern Yakutia) alkaline complex has been investigated by the methods of electron probe microanalysis and X-ray diffraction. The crystal structure was refined to R = 3.9 % in the anisotropic approximation of atomic displacements using 2188 independent reflections with F > 3(F). The unit-cell parameters are: a = 14.1039(2), c = 31.0743(4) Å, V = 5353.1 Å3; the space group R3. The chemical composition of the studied mineral (Z = 3): 
 Na3.87K0.06Sr1.01Ba0.16Ca4.65REE0.47Y0.3Fe0.38Mn1.27Zr2.92Hf0.3Ti0.09Nb0.12Mg0.07Si25.83O72Cl0.17. 
 The crystal chemistry features of the mineral and its place among hydrated eudialytes are discussed.

Crack Identification in Cantilever Beam under Moving Load Using Change in Curvature Shapes

Cracks in structural components may ultimately lead to failure of the structure if not identified sufficiently early. This paper presents a crack-identification method based on time-domain. Captured time-domain data are processed into central difference approximation of displacement of each node (point) in the structure. Abnormally high central difference approximation of displacement of a node relative to those of its neighborhood points indicates a crack at that point. A suite of simulation experiments and numerical calculations was conducted to find out whether the proposed identification method could accurately identify the location of a crack in a cantilever beam under moving load compared to the location found by an exact solution method, and the outcomes indicated that it was as able as the analytical method. The proposed method is an FEA analysis, an approach familiar to virtually every engineer. Therefore, the relative amount of time and effort spent on developing the proposed method for a specific application is much less than those spent on developing an analytical method. The saved time and effort should enable more engineering personnel to perform routine checks on structural elements of their interest more simply and frequently.