Linear Elastic Method Research Articles

Buckling analysis of planar linear uniform deployable structures consisting of scissor-like element in space under compression

This paper comprehensively investigates the buckling load and the stability of a planar linear array deployable structure composed of scissor-like element (SLE) under compression. At present, the researches on deployable structure are mainly focused on configuration design and dynamics characteristics of the mechanisms, but less research on structural instability. In fact, when the external load exceeds the structural critical load value, the deployable structure will be permanently deformed or even collapse directly and no longer have any bearing capacity. To address this issue, a new stability model is derived using linear elastic analysis method and substructure method to evaluate the buckling characteristics of the deployable structure with n SLEs when it is carried out in space, which can accurately obtain the structural instability load and can be used quantitatively to optimize the structure for making it have the most stable configuration. In addition, the effects of the number of elements, the length, material properties and flexibility of the bar, and the deployment degree on the buckling of the scissor deployable structure are investigated, and the results of the theoretical analysis are compared with simulation and analytical results, respectively, confirming that the proposed stability model not only is able to effectively predict the structural instability load but also determine which part of the deployable structure is unstable. It can be concluded that the stability of the deployable structure gradually decreases with the increase of the number of elements or the bar flexibility. In the calculation process, the critical load of each sub-element should be considered, and the minimum value of the critical loads of all subunits can be regarded as the instability load of the whole structure.

A Mixed Discontinuous Galerkin Method for Linear Elasticity with Strongly Imposed Symmetry

In this paper, we study a mixed discontinuous Galerkin (MDG) method to solve linear elasticity problem with arbitrary order discontinuous finite element spaces in $d$-dimension ($d=2,3$). This method uses polynomials of degree $k+1$ for the stress and of degree $k$ for the displacement ($k\geq 0$). The mixed DG scheme is proved to be well-posed under proper norms. Specifically, we prove that, for any $k \geq 0$, the $H({\rm div})$-like error estimate for the stress and $L^2$ error estimate for the displacement are optimal. We further establish the optimal $L^2$ error estimate for the stress provided that the $\mathcal{P}_{k+2}-\mathcal{P}_{k+1}^{-1}$ Stokes pair is stable and $k \geq d$. We also provide numerical results of MDG showing that the orders of convergence are actually sharp.

On a relation of discontinuous Petrov–Galerkin and least-squares finite element methods

The discontinuous Petrov–Galerkin (DPG) method minimizes a residual in a non-standard norm. This paper shows that the minimization of this residual is equivalent to the minimization of a residual in a L2 norm. Since such residuals are well known from least squares finite element methods, this novel interpretation allows to extend results for least squares methods to the DPG method and vice versa. This paper exemplifies the benefits of this possibility by the verification of an asymptotic exactness result for a DPG method for the Helmholtz equation, the design of a locking-free DPG method for linear elasticity, and an investigation of the spectral condition number.

A least squares method for linear elasticity using a patch reconstructed space

We propose a discontinuous least squares finite element method for solving the linear elasticity. The approximation space is obtained by patch reconstruction with only one unknown per element. We apply the L2 norm least squares principle with the weak imposition of the continuity across the interior faces to the stress–displacement formulation. The optimal convergence order under the energy norm is attained. Numerical results of the linear elasticity are presented to verify the error estimates. Our method enjoys the advantages of discontinuous Galerkin methods, and in addition we illustrate the great simplicity in implementation, the robustness and the improved efficiency of our method.

A Staggered Cell-Centered DG Method for Linear Elasticity on Polygonal Meshes

We develop a new numerical method, namely, a locking-free staggered cell-centered discontinuous Galerkin method for linear elasticity on fairly general meshes. The method is well suited for general meshes possibly including hanging nodes; in particular, it does not deteriorate when the mesh becomes highly distorted. There are three unknowns involved in our formulation: stress, displacement, and rotation. The continuities of the three unknowns are staggered on the interelement boundaries. In addition, the symmetry of the stress tensor is imposed weakly by the introduction of Lagrange multipliers. Optimal a priori error estimates covering low regularities in $L^2$ errors of stress, displacement, and rotation are given; in addition, the locking-free error estimates are also investigated. Numerical experiments confirm the theoretical findings and verify the flexibility to rough grids and the locking-free property of the proposed method.

A Multipoint Stress Mixed Finite Element Method for Elasticity on Simplicial Grids

We develop a new multipoint stress mixed finite element method for linear elasticity with weakly enforced stress symmetry on simplicial grids. Motivated by the multipoint flux mixed finite element ...

A Mixed Formulation of Stabilized Nonconforming Finite Element Method for Linear Elasticity

A Mixed Formulation of Stabilized Nonconforming Finite Element Method for Linear Elasticity

New analytical tools for HDG in elasticity, with applications to elastodynamics

We present some new analytical tools for the error analysis of hybridizable discontinuous Galerkin (HDG) method for linear elasticity. These tools allow us to analyze more variants of HDG method using the projection-based approach, which renders the error analysis simple and concise. The key result is a tailored projection for the Lehrenfeld-Schoberl type HDG (HDG+ for simplicity) methods. By using the projection we recover the error estimates of HDG+ for steady-state and time-harmonic elasticity in a simpler analysis. We also present a semi-discrete (in space) HDG+ method for transient elastic waves and prove it is uniformly-in-time optimal convergent by using the projection-based error analysis. Numerical experiments supporting our analysis are presented at the end.

Analisis Story Drift dan Kondisi Sendi Plastis Berbasis Performa pada Gedung Bertingkat dengan Konfigurasi Struktur Persegi Panjang, U, L, H, dan T

Strength based design is commonly used in the design of resisting earthquake building. It is analyzed by linear elastic method so that it can’t describe the seismic behaviour of the structures. For that reason, the design of structure needs to consider performance based design. Irregularity of structure play important role for the results of Structure Performance. This paper uses five models of 20 storey reinforced concrete building with same height(76 m) and areas(750m2) with different plan configurations: rectangular, T, L, U, and H-shaped. Non linear time history method is used to investigate the seismic behaviour of the structures, using finite model with ETABS2016. Dimension of structures are column (600 x1000 mm), shear wall (400mm), couple-beam (400x900 mm), slab (140mm). Recorded accelorogram used in this paper are Altadena, Array, Corralit, LACC, and Yermo because it has almost same earthquake mechanism, magnitude, and epicentrum distance, so that they are used as an artificial accelorogram that is synthesized using Wavelet Function according to SNI 1726-2012.Result shows that the story drift and plastic hinges for collapse prevention (CP) are: rectangular (story drift is 0.59% and amount of CP hinges are 8), H-shaped (1.58%/23), L-shaped (2.41%/30), U-shaped (4.59%/35), T-shaped (5.8%/58).

Advanced design methodologies for steel fibre reinforced concrete and new methodology of real time quality control

Steel fibre reinforced concrete has developed over years for the use in structural applications. The material itself has undergone development into higher performing steel fibre types and thus into higher performing steel fibre concretes. Whilst in regards to standards the development went from recommendations into building codes the way of designing – linear elastic design methods or yield line theory – and the way of quality control seemed to remain unchanged from the past. This paper shall emphasize the recent development in a design tool that enables a non-linear design methodology and a new advanced way of real time quality control.

A low-order nonconforming method for linear elasticity on general meshes

In this work we construct a low-order nonconforming approximation method for linear elasticity problems supporting general meshes and valid in two and three space dimensions. The method is obtained by hacking the Hybrid High-Order method of Di Pietro and Ern (2015), that requires the use of polynomials of degree k≥1 for stability. Specifically, we show that coercivity can be recovered for k=0 by introducing a novel term that penalises the jumps of the displacement reconstruction across mesh faces. This term plays a key role in the fulfilment of a discrete Korn inequality on broken polynomial spaces, for which a novel proof valid for general polyhedral meshes is provided. Locking-free error estimates are derived for both the energy- and the L2-norms of the error, that are shown to convergence, for smooth solutions, as h and h2, respectively (here, h denotes the meshsize). A thorough numerical validation on a complete panel of two- and three-dimensional test cases is provided.

COMPARATIVE INVESTIGATION OF TWO DESIGN METHODS FOR SLAB-ON-GRADE FOUNDATIONS SUBJECTED TO CONCENTRATED LOADS

The concrete slab-on-grade is a type of shallow foundation commonly used in warehouse and other industrial facilities. The slab is constructed in formwork set into the ground with concrete poured directly onto compacted subgrade. Slab-on-grade foundations designed by the traditional linear elastic method may have overly conservative slab thicknesses. As the slab-on-grade foundation can account for up to 15% of construction costs for the projects in which they are employed, reduction in the slab thickness can result in significant cost savings. An alternate design method (Shentu et al. 1997) has been proposed but has not been widely accepted due to lack of confidence in its correlation with test results. The objectives of this study were to experimentally define a reasonable factor of safety for the Shentu method and compare with the traditional design method. Test slabs of varying thicknesses, 5 ft by 5 ft in area, were built on top of compacted local soils and aggregates inside a testing box and tested to failure under static loads to simulate storage rack post loads; companion slabs were constructed on top of compacted soils and aggregates and field-tested. Concrete mixes with and without fibers were used in the study. Factors of safety were determined. Test results indicate that the Shentu method is too liberal in its computations while the traditional design method is overly conservative.

Cut topology optimization for linear elasticity with coupling to parametric nondesign domain regions

We develop a density based topology optimization method for linear elasticity based on the cut finite element method. More precisely, the design domain is discretized using cut finite elements which allow complicated geometry to be represented on a structured fixed background mesh. The geometry of the design domain is allowed to cut through the background mesh in an arbitrary way and certain stabilization terms are added in the vicinity of the cut boundary, which guarantee stability of the method. Furthermore, in addition to standard Dirichlet and Neumann conditions we consider interface conditions enabling coupling of the design domain to parts of the structure for which the design is already given. These given parts of the structure, called the nondesign domain regions, typically represent parts of the geometry provided by the designer. The nondesign domain regions may be discretized independently from the design domains using for example parametric meshed finite elements or isogeometric analysis. The interface and Dirichlet conditions are based on Nitsche’s method and are stable for the full range of density parameters. In particular we obtain a traction-free Neumann condition in the limit when the density tends to zero.

A local mesh free method for linear elasticity and fracture mechanics

The paper presents recent developments on mesh free numerical methods, implemented at the Department of Civil Engineering and Environment of the University of Brasília.The concern is a local mesh free numerical model developed to solve two-dimensional problems in elasticity and linear elastic fracture mechanics.The model formulation is based in the work theorem of the theory of structures, kinematically formulated with a simple rigid-body displacement.The discretization considers the approximation of the elastic field with moving least squares (MLS) and implements a reduced numerical integration which improves the model accuracy. Both regular and irregular nodal distributions can be considered, which makes it a reliable model.The arbitrary discretization parameters, which determine the accuracy and efficiency of local mesh free methods, can be generated automatically, in a multi objective optimization environment, with genetic algorithms (GA), which makes it a robust model.Applications of linear elastic fracture mechanics implement the singularity subtraction technique (SST) to regularize the elastic field, before the numerical solution, introducing the stress intensity factors (SIF) as additional unknowns of the problem. These features make it an efficient model.Benchmark problems were solved for an assessment of the accuracy and efficiency of these techniques.

Back‐Calculation of Soil Modulus from PFWD Based on a Viscoelastic Model

To improve the accuracy of back‐calculation of soil modulus using the portable falling weight deflectometer (PFWD), a viscoelastic method (VEM) overcoming the limitations of the conventional linear elastic method (LEM) was proposed. A quasi‐static dynamic analysis technique of Laplace transformation and a modified Gauss–Newton optimization algorithm were adopted in the proposed method. The back‐calculation program was compiled with MATLAB. The effectiveness of the proposed method was verified with the in situ plate load test (PLT) conducted on a highway embankment. In situ test results showed that a time lag existed between the peaks of deflection and load, and load‐deflection curves were nonlinear, which indicated the viscoelastic nature of the soil. The back‐calculated modulus using the LEM and VEM was higher than that using the PLT. In the case of low stiffness soil, the average error of back‐calculation using the LEM and VEM was 53.1% and 14.8%, respectively. However, for stiffer soil, the average error of back‐calculation using the LEM and VEM method was 12.4% and 4.3%, respectively. Moduli of back‐calculation using LEM and VEM methods were used to perform flexible pavement analysis, which showed that with an 8% reduction of modulus, the pavement service life reduced by 25%. More accurate estimation of modulus can save maintenance cost in the future.

A variational framework for distance‐minimizing data‐driven method for linear elasticity

AbstractA variational framework for the distance‐minimizing data‐driven computing method is proposed based on the method of Lagrange multipliers. The data‐driven solution is defined as an optimization solution of a double‐minimization problem. A staggered scheme is employed to decompose the original problem into a sequence of single‐minimization problems. As an advantage, the variational formulation renders straightforward implementation of the high‐order polynomial interpolation. The non‐uniqueness of the data‐driven solution is the consequence of the use of the current solution strategy and non‐convexity of the data function space. A convergence study of the solution with respect to the data size is presented. The decrease of mean values and standard deviations of errors with respect to the reference solution justify the variational formulation.

Source time reversal (STR) method for linear elasticity

We study the problem of source reconstruction for a linear elasticity problem applied to seismicity induced by mining. We assume the source is written as a variable separable function f(x)g(t). We first present a simple proof a local decay result for elasticity in the case of homogeneous media. We then extend the source time reversal method, originally developed for acoustic waves, to an elastic system of waves. Additionally, we present a fast reconstruction implementation for large data sets. This is especially useful in the elastic case, in which the numerical cost is higher than in fluid acoustics. We complement this work with several 2D and 3D numerical experiments and an analysis of the results.

A lowest-order weak Galerkin method for linear elasticity

The lowest-order weak Galerkin (WG) method is considered for linear elasticity based on the displacement formulation. The new method approximates the displacement using piecewise constant vector functions both inside and on the boundary of each mesh element and its bilinear form does not require a stabilization term for the existence and uniqueness of the solution. A-priori error estimates of optimal order in the discrete H 1 - and L 2 -norms for the displacement are proved when the solution is smooth. The error estimates are independent of the Lamé constant λ , thus the performance of the new method does not deteriorate as the elastic material becomes incompressible. Further, a simple post-processing technique to obtain a numerical approximation of the stress is presented. A careful error analysis reveals that the L 2 -norm error in the stress is also optimal and independent of λ . Several numerical experiments confirm the locking-free property and optimal convergence rates of the new method.

A mixed generalized multiscale finite element method for planar linear elasticity

A mixed generalized multiscale finite element method for linear elasticity based on Hellinger Reissner principle with a strong symmetry enforcement for the stress tensor is introduced. The multiscale approximation space for the stress tensor is built on a coarse grid with carefully designed local problems so that the basis functions are also symmetric. Using eigenfunctions of local spectral problems as basis functions allows the approximation error of the multiscale finite element space to have a rapid spectral decay. Together with a properly chosen approximation space for the displacement, the method is shown to be inf–sup stable and robust as the first Lamé coefficient λ → ∞ . Numerical experiments are supplemented to demonstrate fast convergence of the method with respect to local enrichment, and robustness of the method with respect to high contrast heterogeneity of the Poisson ratio.

A superconvergent hybridisable discontinuous Galerkin method for linear elasticity

SummaryThe first superconvergent hybridisable discontinuous Galerkin method for linear elastic problems capable of using the same degree of approximation for both the primal and mixed variables is presented. The key feature of the method is the strong imposition of the symmetry of the stress tensor by means of the well known and extensively used Voigt notation, circumventing the use of complex mathematical concepts to enforce the symmetry of the stress tensor either weakly or strongly. A novel procedure to construct element by element a superconvergent postprocessed displacement is proposed. Contrary to other hybridisable discontinuous Galerkin formulations, the methodology proposed here is able to produce a superconvergent displacement field for low‐order approximations. The resulting method is robust and locking‐free in the nearly incompressible limit. An extensive set of numerical examples is utilised to provide evidence of the optimality of the method and its superconvergent properties in two and three dimensions and for different element types.