Elastic Solution Research Articles

Elasticity solution of fixed beams under linear temperature variation: An experimental and numerical study

This article presents the Airy stress function method for predicting the structural response of a fixed beam subjected to a linear temperature distribution, using the superposition principle. In this approach, the fixed-end beam is decomposed into three simply supported beams with unknown reactions and a linear temperature variation. For each loading condition, the suitable Airy’s stress functions were formulated to satisfy the stress equilibrium and strain compatibility equations. The results obtained using this method are subsequently contrasted with those derived from finite element modeling. To validate the analytical and experimental results, two finite element modeling approaches, namely two-dimensional and three-dimensional modeling, are implemented using the ANSYS finite element software. This numerical modeling approach utilizes a sequentially coupled steady-state thermal and static structural analysis. In addition to employing Airy’s stress function method and finite element analysis, experiments were conducted on SS304 rectangular specimens under proper insulation to investigate their thermal bowing response. Samples were tested in a simple thermal bending experimental setup under fixed support conditions, and subjected to conduction-type heating to achieve linear temperature changes along their depth. The analytical method predicts that the beam will not deflect under linear temperature variation in a fixed-beam situation. The numerical analysis results confirm this, showing a minimal deflection, that aligns well with the Airy’s stress function method. However, the deflection measured in the experimental program is significantly larger than the predictions from the analytical method. This method eliminates the complexity involved in applying boundary conditions along with thermal loading on a fixed beam.

Mechanical Behavior of Hollow Corrugated Sandwich Cylinders Under Inner Pressure Loading

Taking pressure-bearing equipment as a prototype, the mechanical behavior of hollow corrugated sandwich cylinders under inner pressure loading was investigated. Considering the hollow cylindrical shell with finite length, the elastic closed-form solutions of hollow corrugated sandwich cylinders under inner pressure loading were presented. The optimization for minimum weight was performed. The stress distribution characteristics of the structures with different topological parameters were discussed. It can be found that an inflection point of the structure loading efficiency exists during an increase in the pressure, providing that under lower inner pressure loading, the loading efficiency of hollow corrugated sandwich structures is not invariably superior to homogeneous structures, yet while subjected to higher inner pressures, the sandwich structures exhibit a significant advantage in load-bearing efficiency.

3-Dimensional numerical analysis of geosynthetic double-encased annulus stone columns

A geosynthetic double-encapsulated annulus stone column technique is proposed in this study to overcome limitations associated with conventional stone columns, such as limited lateral load-bearing capacity and potential issues related to aggregate dispersion in soft soils. Geosynthetic double-encapsulated annulus stone columns are consistent with conventional stone columns besides being confined in and around the surrounding soil. The performance of geosynthetic double-encapsulated annulus stone columns is primarily dependent upon their outer-to-inner diameter ratio. For this reason, comprehensive 3-dimensional numerical studies were carried out to evaluate the optimum outer-to-inner diameter ratio of the annulus stone column. The results suggest that the ultimate load-carrying capacity of an annulus stone column increases with an increasing ratio of outer to inner diameter until an optimum value. In addition, the double encapsulation enhances confinement, improving shear stiffness and reducing lateral bulging. However, the load-carrying capacity was substantially reduced beyond the critical outer-to-inner diameter ratio. A simple analytical model extending the theory of thin cylinders is also introduced to estimate the accumulated stresses and strains in the geosynthetic encasement. The proposed model operates within a simplified framework of elastic solutions that facilitate practising engineers to design and optimize geosynthetic encasements for enhanced structural performance.

Elastostatic analysis using the boundary element method and the Aifantis gradient elasticity theory

The increasing use of micro and nano-scale structures has sparked interest in theories incorporating the effect of scale since the classical continuum theory has limitations in capturing effects that depend on size. Therefore, three-dimensional elastostatic microstructure modeling is carried out in this work using the boundary element method (BEM).To account for microstructural effects, the simplified gradient theory proposed by Aifantis, a particularization of Mindlin's general theory, was employed. A variational argument was established to determine the governing equations and boundary conditions for the problem. This argument explains the fundamental solution of gradient elasticity, and the integral contour representation is constructed with the aid of the reciprocal identity. Curved triangular elements of Proriol spectral functions were used to approximate the geometry and the physical parameters for the discretization of the BEM. The presented formulation yielded results consistent with other analyses in the literature.

Preserving large-scale features in simulations of elastic turbulence

Simulations of elastic turbulence, the chaotic flow of highly elastic and inertialess polymer solutions, are plagued by numerical difficulties: the chaotically advected polymer conformation tensor develops extremely large gradients and can lose its positive-definiteness, which triggers numerical instabilities. While efforts to tackle these issues have produced a plethora of specialized techniques – tensor decompositions, artificial diffusion, and shock-capturing advection schemes – we still lack an unambiguous route to accurate and efficient simulations. In this work, we show that even when a simulation is numerically stable, maintaining positive-definiteness and displaying the expected chaotic fluctuations, it can still suffer from errors significant enough to distort the large-scale dynamics and flow structures. We focus on two-dimensional simulations of the Oldroyd-B and FENE-P equations, driven by a large-scale cellular body forcing. We first compare two positivity-preserving decompositions of the conformation tensor: symmetric square root (SSR) and Cholesky with a logarithmic transformation (Cholesky-log). While both simulations yield chaotic flows, only the latter preserves the pattern of the forcing, i.e. its fluctuating vortical cells remain ordered in a lattice. In contrast, the SSR simulation exhibits distorted vortical cells that shrink, expand and reorient constantly. To identify the accurate simulation, we appeal to a hitherto overlooked mathematical bound on the determinant of the conformation tensor, which unequivocally rejects the SSR simulation. Importantly, the accuracy of the Cholesky-log simulation is shown to arise from the logarithmic transformation. We also consider local artificial diffusion, a potential low-cost alternative to high-order advection schemes. Unfortunately, the artificially enhanced diffusive smearing of polymer stress in regions of intense stretching substantially modifies the global dynamics. We then show how the spurious large-scale motions, identified here, contaminate predictions of scalar mixing. Finally, we discuss the effects of spatial resolution, which controls the steepness of gradients in a non-diffusive simulation.

Analysis of elastic and rheological properties of tunnel anchorage support structure under the action of seepage flow

The large deformation and strong rheological characteristics of the underground rock body under complex geological conditions have led to the failure of the support system and even the instability of tunnels from time to time. Based on the theory of groundwater seepage, the elastic solution of the mechanical model of the tunnel anchorage support system under seepage is solved, which reveals the influence of seepage on the peripheral rock and the support parameters of the anchorage support system; the viscoelastic solution is solved according to the corresponding principle of elasticity—viscoelasticity, which analyses the influence of seepage on the rheological process of the anchorage support system; and the model is simulated with the help of finite element software, which analyses the agreement and discrepancy of the numerical solution and the theoretical solution. The results show that the analytical and numerical solutions are basically consistent. This study can provide important guidance for controlling long-term stability of water-rich tunnel rocks.

Free vibration of sandwich beams with graphene nanoplatelets reinforced piezoelectric face sheets

Owing to its exceptional physical properties, graphene is considered as an excellent reinforcement for composite materials. However, the existing higher-order models encounter significant hurdles when attempting to accurately predict the natural frequencies of sandwich beams with graphene nanoplatelets reinforced composite (GNPRC) piezoelectric face sheets. If transverse shear deformations are not accurately modeled, the dynamic behavior of piezoelectric sandwich beams will be noticeably influenced by the disparities of material properties at the interfaces of adjacent layers, as well as the inherent electromechanical coupling characteristic. To overcome these challenges, an advanced electro-mechanical coupling theory will be introduced for the vibration analysis of sandwich beams featuring GNPRC piezoelectric face sheets. Compared to previous higher-order models, the proposed beam model introduces an enhanced interlaminar shear stress field incorporating electromechanical properties. This improved stress field can be involved in the equations of motion based on the Hamilton principle, significantly enhancing the precision in analyzing the free vibration behavior of piezoelectric sandwich beams. Furthermore, a simplification of finite element implementation can be achieved by removing the second-order derivatives of in-plane displacements from the transverse shear stress field. Consequently, a C0-type three-node beam element is constructed for the dynamic analysis of sandwich beams with GNPRC piezoelectric face sheets. To validate the performance of the proposed theory, the 3D elasticity solutions and results obtained from alternative theoretical frameworks are used. Numerical results demonstrate that the present model offers superior accuracy over the existing higher-order theories. Additionally, a comprehensive parametric investigation is conducted to explore the influence of key parameters on the free vibration characteristics of piezoelectric sandwich beams.

Explicit Peck formula applied to ground displacement based on an elastic analytical solution for a shallow tunnel

Explicit Peck formula applied to ground displacement based on an elastic analytical solution for a shallow tunnel

Three-dimensional elasticity solutions for doubly-curved composite shells by extended differential quadrature method

This paper presents the three-dimensional (3D) stress analysis of doubly-curved composite shells with general boundary conditions using the strong sampling surfaces (SaS) formulation. The SaS method is based on the choice of SaS parallel to the middle surface and located at Chebyshev polynomial nodes to introduce the displacements of these surfaces as unknown functions that leads to a non-conventional shell formulation, in which strain–displacement and stress–strain relationships are represented in terms of SaS displacements. This is due to the use of Lagrange polynomials in approximating displacements, strains and stresses in the thickness direction. The outer surfaces are not included into a set of SaS that makes it possible to uniformly minimize the error used by Lagrange interpolation. The strong SaS formulation, based on direct integration of elasticity equilibrium equations in the thickness direction by the extended differential quadrature (EDQ) method, can be applied efficiently for high-precision calculations of doubly-curved composite shells with clamped and free edges. This is because in the SaS/EDQ formulation, displacements, strains and stresses of SaS are interpolated in a rectangular domain specified in a curvilinear coordinate system using a Chebyshev-Gauss-Lobatto grid and Lagrange polynomials are also used as basis functions. The proposed approach deals with equilibrium equations in terms of SaS stresses, avoiding the integration of second order differential equations in terms of SaS displacements, that greatly simplifies the implementation of the EDQ method.

Shear Capacity Prediction of Extremely-Loaded Box Culvert on Elastic Soil Using Artificial Neural Network

A box culvert, buried at shallow depths beneath roadways, may experience deflections caused by the dynamic impact of traffic loading and the vertical pressure exerted by the soil fill. A computational model commonly employed used to various engineering issues, including those in geotechnical applications, is the beam-on-elastic-foundation model. In this context, the Moment Distribution Method (MDM) must be applied to account for the elastic foundation. To achieve this, the internal forces acting on the ends of both exterior and interior walls are transferred to the beam-like bottom slab of the culvert, which rests on an elastic soil bed. Subsequently, the secondary internal forces are determined by refining the structural parameters, taking into account the characteristics of the elastic soil bed. This study presents the development and application of an Artificial Neural Network (ANN) model to predict the shear capacity of box culverts on elastic soil under traffic loading conditions. The proposed model is trained and validated using a comprehensive database of beam on elastic foundation solutions. The input parameters include the geometrical and mechanical properties of the culvert and the soil, as well as the loading conditions. The results of the ANN model show R2 values of 0.9633 and 0.9581 for the training and testing sets, respectively, indicating the model's excellent accuracy. These findings suggest that the ANN model can reliably predict the shear capacity of culverts.

Enhanced elastic stress solutions for junctions in various pipe bends under internal pressure and combined loading ([formula omitted] pipe bend, U-bend, double-bend pipe)

Piping systems in nuclear power plants normally operate under high pressure and high temperature. To efficiently manage space within these systems, pipe bends are extensively used. However, the welded joints connecting these pipes may be susceptible to flaws due to weld residual stress, transient loads, and operating environments. When flaws are present in such areas, analytical evaluation of flaws is to be carried out based on fracture mechanics parameters such as stress intensity factor. To calculate the stress intensity factor, distributions of the elastic stress are required. Therefore, this paper presents closed-form approximations of elastic stress in the junction between a pipe bend and a straight pipe under internal pressure. Review of the existing elastic stress solutions for estimating the stresses in pipe bends was carried out analyzing their limitations. Based on those limitations elastic stress solutions for thick to thin wall pipe bends were proposed and were validated against finite element (FE) analysis for thick-wall to thin-wall pipe bends under internal pressure and combined loading (i.e. internal pressure, in-plane bending, out-of-plane bending inflicted simultaneously). 90° pipe bend, U-bend and double-bend pipe configurations were considered and showed good agreement with the FE results exhibiting less than 5.8 % discrepancies. The accuracy of the elastic stress solutions for pipe bends provided in the existing code are summarized and validated in the appendix as well.

Impact noise from roof terrace - elastic ceiling in Sylomer hangers and free ceiling

Modern houses and apartments are built with flat roof and often the upper apartments contain a roof terrace. Impact noise has been measured from a roof terrace with floating floor on mineral wool, a good wooden construction with 2 layers of gypsum stiff fastened in the ceiling. This measurement shows L'n,w of 55 dB, and make it necessary with better solution for ceiling to achieve L'n,w 53 dB or better. Two measurements for improved situation in different buildings are done with ceiling in Sylomer hangers and with free ceiling. These two situations show L'n,w of 50 and 46 dB. The measurement of solution with free ceiling shows the lowest value of L'n,w for these two elastic solutions and also has the lowest levels in the low frequency range. This paper will show that for roof terrace the need for elastic or free ceiling is important. Concerning annoyance and low frequency noise, solution with free ceiling is to be preferred in future projects. There has been no complaining about impact noise from roof terrace, for the situation with free ceiling. For the situation with Sylomer hangers people living there hear some noise, but they have not complained about that.

Exact solutions for anisotropic beams with arbitrary distributed loads

The general solution of elasticity for plane anisotropic beams with arbitrary constraints at ends and arbitrary normal and tangential distributed loads on surfaces is derived, which consists of internal forces (i.e., bending moment, shearing force, axial force) and their integrals and derivatives of different orders and load-independent polynomial function sequences of longitudinal coordinates. The method for determining the function sequences is established by resolving the governing equation and boundary conditions of the stress function method. For beams with an elastic symmetry plane, a method for directly determining explicit expressions for all terms of function sequences is provided. Particular solutions of examples are solved using general solution formulas, and the results align excellently with existing exact solutions. Finally, the errors in EBT and TBT when applied to beams made of different materials are analysed.

Influence of Transverse Normal Strain on Buckling and Vibration of Sandwich Plates

This study presents the buckling and free vibration behavior of simply supported square and rectangular sandwich plates. A third-order shear deformation theory considering both the transverse shear and normal deformations is used. The condition of zero transverse shear stresses on the top and bottom surfaces of the plate is satisfied. Hence, the present theory does not require the shear correction factor generally associated with the first-order shear deformation theory. Governing equations and boundary conditions are obtained using the principle of minimum potential energy. The closed-form solutions of simply supported sandwich plates are obtained. The effect of the side-to-thickness ratio and plate aspect ratio on buckling and free vibration responses of simply supported square and rectangular sandwich plates is examined. Parametric effects of face sheet thickness ratios and load factor upon the critical buckling load for uniaxial and biaxial buckling analysis of square and rectangular sandwich plates for different staking sequences are studied. All numerical solutions are obtained using MATLAB® programs. The results obtained from the present theory are compared with those from classical plate theory, first-order shear deformation theory, higher-order shear deformation theories, and the exact three-dimensional elasticity solutions as applicable.

Evaluating the effects of nonlocality and numerical discretization in peridynamic solutions for quasi-static elasticity and fracture

The difference between the computational peridynamic results and the corresponding exact classical solutions is contributed by the error induced by numerical discretization and the nonlocality-induced difference. To evaluate and compare these contributions and investigate their dependence on the peridynamic influence functions, in this paper, we apply different peridynamic influence functions and different horizon factors (m, the ratio between the horizon size and the grid spacing) to conduct static (or quasi-static) tensile numerical tests. We calculate the difference between the peridynamic solutions and the corresponding classical solutions for static uniaxial tension of thin plates with or without hole, the J-integral of a single crack under Mode I loading condition, and the quasi-static fracture in a perforated thin plate. For the case of uniaxial tension in a thin, homogeneous plate, we separate the effects of nonlocality and numerical discretization by implementing the boundary conditions in different ways. The numerical results show that both the effects of nonlocality and numerical discretization correspond to the nonlocal constants of the influence functions (with few exceptions). For problems with the presence of the peridynamic surface effect, such as holes, influence function with weaker nonlocality is a better choice to obtain more accurate results.

Elasticity solutions for functionally graded beams with arbitrary distributed loads

This paper derives the exact general elasticity solution for functionally graded rectangular beams subjected to arbitrary normal and tangential loads and with arbitrary end constraints. The general solution consists of bending moments and their integrals and derivatives, along with load-independent function sequences of the longitudinal coordinate. The method for determining function sequences has been established based on the stress function method. General solution formulas for stresses, strains and displacements have been derived and used to solve explicit special solutions for six cases involving concentrated forces, uniformly loads, and quadratically distributed loads with different displacement constraints scenarios. The results obtained are compared with existing exact solutions and those of Euler–Bernoulli and Timoshenko beams, and the errors of the latter two are analysed.

On the time-dependent sliding contact behavior of three-phase polymer matrix smart composites

Abstract Three-phase smart composites consisting of magnetostrictive and piezoelectric reinforcements embedded with a polymer matrix can achieve specific multifunctional properties in response to external stimuli, which are well-suited for the application of sensors, actuators, and electronic devices. The materials exhibit complex behaviors characterized by electro-magneto-viscoelasticity coupling during the contact of these smart structures. This paper proposes a novel hybrid element method (HEM) for numerically analyzing the frictionless sliding contact problem stemming from the viscoelastic behavior and multiphase interactions of polymer matrix smart composites. The study aims to fully investigate the effects of material properties, sliding velocities, and action time on the contact behavior of materials via the integration of the conjugate gradient method (CGM) with the discrete convolution-fast Fourier transform (DC-FFT) algorithm. The analytical viscoelastic frequency response functions (FRFs) are derived by substituting elastic solutions with the time-dependent relaxation modulus. Numerical results show that three-phase polymer matrix smart composites exhibit lower contact pressure and higher surface electric/magnetic potential than three-phase magneto-electro-elastic composites. Sliding velocity and action time strongly influence the distribution of pressure/stress and electric/magnetic potential.

Elasticity solutions of clamped laminated arches with temperature-dependent material properties under thermo-mechanical loading

The mechanical behaviors of the clamped laminated arch with temperature-dependent material properties under the thermal and mechanical loadings are studied in this paper. An analytical method based on the theory of thermoelasticity is proposed for predicting the exact solutions of stresses and displacements in the arch with arbitrary thickness and layer numbers. The clamped support is equivalent to a simply support with a support reaction with the unit pulse function and the Dirac function. The space state method is utilized to establish the state equation by employing the displacements and stresses as the state variables. A slice model is introduced to simplify the coefficient matrix. Based on the continuities at the interfaces, the displacement and stress relationships between the internal and external surfaces of the laminated arch are derived with the transfer matrix method. Exact solutions for the displacements and stresses can be obtained according to the stress boundary conditions on the surfaces of the laminated arch. Finally, convergence and comparison studies demonstrate the excellent efficiency and accuracy of the present method. The influences of material dependency with temperature, thermal load, and thickness-to-radius ratio on the distributions of displacements and stresses are discussed in detail.

Stress-deformation analysis of the cracked elastic body

A new analytical plane Elastic Fracture Mechanics (LEFM) model of the effective elastic moduli exhibited by an isotropic elastic cracked solid in tensile and compressive stress fields is presented. The effective Young’s modulus and Poisson’s ratio E¯,v¯, respectively, relate the stresses applied to the cracked solid with its macroscopic deformation. In the case of a compressive stress field the effective elastic moduli depend solely on properly oriented cracks that are either open or closed. This model is then used for the stress and displacement estimations around a circular axisymmetric hole under internal pressure and external hydrostatic stress. The creation of the hole alters the stresses in its neighborhood, and induces a deterioration of the elastic moduli. In this manner, the effective elastic moduli depend on the radial distance from the hole’s boundary. The validation of the algorithm was done against Kirsch’s analytical solution of the axisymmetric hole problem for nearly zero density of cracks as well as for large values of the friction angle. The model predicts a lower tangential stress concentration at the wall of the hole compared to that predicted by the constant linear elasticity solution and a maximum stress concentration that is not necessarily located at the wall of the hole. This apparent enhancement of rock strength adjacent to the wall is more pronounced for larger concentrations of open cracks.

Complete Solutions in the Dilatation Theory of Elasticity with a Representation for Axisymmetry

In this paper, we present certain complete solutions in the dilatation theory of elasticity. This model can be derived as a special case of Eringen’s linear theory of microstretched elastic solids when microrotations are absent. It is also a version of the theory of materials with voids. The dilatation theory can be considered the simplest theoretical model of microstructured materials and is suitable for investigating various phenomena that occur in engineering, geomechanics, and biomechanics. We establish three complete solutions to the displacement equations of equilibrium that are the counterpart of the Green–Lamé (GL), Boussinesq–Papkovich–Neuber (BPN), and Cauchy–Kowalevski–Somigliana (CKS) solutions of classical elasticity. The links between these BPN and CKS solutions are established. Then, we present a representation of the BPN solution in the case of axisymmetry. The results presented here are useful for solving axisymmetric problems in semi-infinite and infinite domains.