Elastic Solution Research Articles

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

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.

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.

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.

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.

Stress solution of static linear elasticity with mixed boundary conditions via adjoint linear operators

We revisit stress problems in linear elasticity to provide a perspective from the geometrical and functional-analytic points of view. For the static stress problem of linear elasticity with mixed boundary conditions we present the associated pair of unbounded adjoint operators. Such a pair is explicitly written for the first time, despite the abundance of the literature on the topic. We use it to find the stress solution as an intersection of the (affinely translated) fundamental subspaces of the adjoint operators. In particular, we treat the equilibrium equation in the operator form, which involves the spaces of traces on a part of the boundary, known as the Lions-Magenes spaces. Our analysis of the pair of adjoint operators for the problem with mixed boundary conditions relies on the properties of the analogous pair of operators for the problem with the displacement boundary conditions, which we also include in the paper.

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.

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 ca 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.

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.

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.

A refined model for functionally graded graphene nanoplatelets reinforced composite beams under thermal loads

Graphene is highly regarded as a promising material for various engineering purposes due to its remarkable physical characteristics. However, existing higher-order shear deformation theories may struggle to accurately capture the interlaminar mechanical response of functionally graded graphene nanoplatelets (FG-GNP) reinforced composite beams subjected to thermal loads. This arises from the pronounced discrepancies in material properties and thermal expansion coefficients among the various layers, making the interlaminar mechanical behavior of composite structures exceedingly complex under thermal loading. As a result, a meticulous analysis of interlaminar stresses becomes necessary for the structural assessment of GNP-reinforced composite beams under temperature variations. To overcome this limitation, we propose an advanced model incorporating transverse normal thermal deformation, tailored specifically for FG-GNP reinforced composite beams subjected to thermal loads. The developed model primarily boasts two advantages. Firstly, although transverse normal deformation is introduced into the displacement field, no additional displacement variables are increased, thereby maintaining model simplicity and efficiency. Secondly, the model achieves a high-precision interlaminar shear stress field taking into account the interlaminar continuity conditions and temperature effects. Furthermore, the elimination of second-order derivatives of in-plane displacement parameters from the transverse shear stress components leads to a simplification in the finite element implementation. Based on the proposed model, a simple three-node beam element is constructed. The 3D elasticity solutions and the results from alternative modeling frameworks are used to assess the performance of the developed model. Additionally, a comprehensive investigation is conducted to explore the effects of various parameters on the deformations and stresses of GNP-reinforced composite beams.

Vertical Crustal Deformation Due To Viscoelastic Earthquake Cycles at Subduction Zones: Implications for Nankai and Cascadia

AbstractDespite significant progress in studying subduction earthquake cycles, the vertical deformation is still not well understood. Here, we use a generic viscoelastic earthquake‐cycle model that has recently been validated by horizontal observations to explore the dynamics of vertical earthquake‐cycle deformation. Conditioned on two dimensionless parameters (i.e., the ratio of earthquake recurrence interval T to mantle Maxwell relaxation time tM (T/tM) and the ratio of downdip seismogenic depth D to elastic upper‐plate thickness Hc (D/Hc)), the modeled viscoelastic deformation exhibits significant spatiotemporal deviations from the simple time‐independent elastic solution. Caution thus should be exercised in interpreting fault kinematics with vertical observations if ignoring Earth's viscoelasticity. By systematically exploring these two parameters, we further investigate three metrics that characterize the predicted vertical deformation: the coastal pivot line (CPL), the uplift zone (UZ) landward above the downdip seismogenic extent, and the secondary subsidence zone (SSZ) in the back‐arc region. We find that these metrics can all be time‐dependent, subject to D/Hc and T/tM. The CPL location and the UZ width are mainly controlled by D/Hc and T/tM, respectively. The presence of the SSZ is prevalent during the interseismic phase due to viscous mantle flow driven by ongoing plate convergence. Contemporary vertical deformation in Nankai and Cascadia is largely consistent with the model predictions and features differences mainly related to contrasting D/Hc values in the two margins. These findings suggest that vertical crustal deformation bears fruitful information about subduction‐zone dynamics and is potentially useful for inversions of key subduction‐zone parameters, deserving properly designed monitoring.

Time-Dependent Reliability Analysis of Anhydrite Rock Tunnels under Swelling Conditions: A Study on Stress, Deformation, and Engineering Solutions

This study presents an analytical approach for evaluating the reliability of anhydrite rock tunnels, focusing on their characteristic swelling behavior. Anhydrite rocks, prone to significant expansion upon moisture exposure, pose a challenge in tunnel construction, potentially leading to structural issues such as floor heave and lining damage. To address this, this research develops an elastic swelling analytical solution based on humidity stress field theory, enabling the assessment of time-dependent stress and deformation changes in anhydrite tunnels. The solution’s applicability is demonstrated through its application to the Lirang tunnel. The investigation into the effects of support pressure, swelling time, and reserved deformation on tunnel reliability reveals that circumferential stress at the tunnel wall increases by 13.94% and 21.86% for swelling periods of 30 and 365 days, respectively. Similarly, radial displacement escalates by 22.97% and 35.93% over these periods, highlighting the significant impact of swelling behavior. Using a spreadsheet-based First Order Reliability Method (FORM) for analysis, this study finds that the original design of the Lirang tunnel did not meet the desired reliability standards under swelling conditions. However, strategic adjustments in construction variables, such as increasing support pressure to 1.2 MPa or enhancing reserved deformation to 59 mm, elevated the tunnel’s reliability to meet safety requirements. This research provides a vital framework for assessing and enhancing the reliability of anhydrite rock tunnels, considering the long-term effects of swelling. It underscores the importance of incorporating swelling behavior in the design and construction of tunnels in anhydrite rock formations, offering valuable insights for optimizing tunnel stability in such challenging geological conditions.

Prediction of internal stress in alumina laminates induced by shrinkage mismatch based on elastic model modification

AbstractCracks induced by internal stresses compromise the structural integrity of ceramic laminates. Such stresses can be classified into thermal stress derived from thermal expansion mismatch and shrinkage stress derived from shrinkage mismatch. Analytical models have been proposed for the former, whereas only numerical predictions have been explored for the latter. In this study, an equation is constructed to predict the shrinkage stress by modifying an elastic solution proposed in previous studies. We fabricated laminates with three‐layer sandwiched structures using only alumina specimens with high and low densities to control the shrinkage and avoid thermal mismatch. Surface cracks were initiated when the shrinkage stress exceeded the flexural strength of the surface layer. Although the original strength of the surface layer was 308.8 MPa, the laminates’ flexural strength was lower than that because of shrinkage constraint. Our results demonstrate that the shrinkage stress was successfully estimated analytically using a few material properties measured in an ordinary environment. This study will ensure the structural integrity of ceramic laminates in laminate designs.