Analytical 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

An FFT based approach to account for elastic interactions in OkMC: Application to dislocation loops in iron

Object kinetic Monte Carlo (OkMC) is a fundamental tool for modeling defect evolution in volumes and times far beyond atomistic models. The elastic interaction between defects is classically considered using a dipolar approximation but this approach is limited to simple cases and can be inaccurate for large and close interacting defects. In this work a novel framework is proposed to include exact elastic interactions between defects in OkMC valid for any type of defect and anisotropic media. In this method, the elastic interaction energy of a defect is computed by volume integration of its elastic strain multiplied by the stress created by all the other defects, being both fields obtained numerically using a FFT solver. The resulting interaction energies reproduce analytical elastic solutions and show the limited accuracy of dipole approaches for close and large defects.The OkMC framework proposed is used to simulate the evolution in space and time of self-interstitial atoms and dislocation loops in iron. It is found that including the anisotropy has a quantitative effect in the evolution of all the type of defects studied. Regarding dislocation loops, it is observed that using the exact interaction energy results in higher interactions than using the dipole approximation for close loops.

Elastic analytical method with machine learning for predicting the stratum displacement field induced by shallow tunneling

Support vector regression (SVR) with sparrow search algorithm (SSA) is developed as the machine learning (ML) model to predict maximum surface settlement smax caused by tunneling. A novel method for calibrating boundary conditions of analytical solution is proposed, where the maximum surface settlement derived by the analytical method is equal to smax predicted by SSA-SVR method. The elastic analytical solution for stratum displacement of a shallow tunnel is presented by the complex variable method, when the calibrated nonuniform displacement function is applied as the tunnel displacement boundary condition. The proposed analytical solution-machine learning (AM) method can predict the stratum displacement field prior to the tunnel excavation. Seventy-three tunnel engineering cases are employed to verify the rationality of the proposed SSA-SVR method in predicting smax. The value of R2 in the training and test process is 0.894 and 0.877, respectively. Taking Heathrow Express Trial Tunnel as an example, the potential of the proposed AM method in predicting stratum displacement is presented where the influence of cohesion strength, internal friction angle, Young's elastic modulus of stratum, tunnel radius and depth are considered. The proposed AM method can well predict the stratum surface settlement trough curve, vertical and horizontal displacement at different positions.

Analytical solutions for bolted tunnel structures with voids behind the lining

Understanding the mechanical behaviour of tunnels reinforced by end-anchored bolts and lining with voids behind the lining is crucial in determining the appropriate supporting measures. In this study, we constructed a geometric model for a tunnel with a single or double void. We used the substructure method and point matching method to obtain the elastic analytical solutions for the model. We examined the stress, internal force, and deformation characteristics of the tunnel and support structure under the influence of single and double voids. To validate the analytical model, we established a numerical model using the finite difference method. Additionally, we compared and analyzed the mechanical behaviour of tunnels reinforced by bolts and lining versus lining-only, and determined the contribution factor of the bolts.

Analytical solution of circular tunnel-lining interaction with elastic contact

Closed-form solutions for tunnel lining are useful to estimate the member forces in the first stages of lining design. A key element in these solutions is the interaction between the lining and the ground. Existing solutions have limited this interaction to two limited cases: perfect bonding, in which there is no separation between the lining and the ground, and perfect slip in which there is no friction between them. In this study, a new analytical solution is established for an elastic contact between the lining and the ground in which the interaction forces depend on the relative displacement of the two elements. This model can cover more realistic cases of the lining-ground interactions in tunnels, for instance, the existence of tail void partially filled with grouting materials. It has been shown that the new solution extends well the previous ones, which can be considered as its limit cases, and also corrects certain errors inherent in these solutions. The analytical solution and compared between them and also to numerical results obtained by the Finite Element Analysis code Disroc. The differences between the solutions are estimated in some typical cases and explained based on their different simplification assumptions. Finally, a simple linear transformation of stresses around the tunnel is shown to provide an efficient method to assess the frictional slip possibility for the results obtained from an analytical elastic solution.

Three-dimensional analytical elasticity solutions in isosceles-right triangular composite laminates

In this study, analytical solutions are derived for three-dimensional, transversely isotropic and multilayered isosceles-right triangular plates under simply supported edge conditions. For any homogeneous layer, we construct the general solution in terms of a simple formalism that resembles the Stroh formalism and the dual-variable and position method. For multilayered isosceles-right triangular plates, we derive the analytical solution by the new propagating method in terms of the symmetry theory. Numerical results clearly show the influence of load distribution, stacking sequence, and mechanical and geometric parameters on the elastic fields. The proposed solutions can also serve as benchmarks for other numerical methods such as the finite element method.

Fractional viscoelastic solution of stratum displacement of a shallow tunnel under the surface slope condition

The unified displacement function (UDF) is presented to describe the deformation behaviours of the tunnel profile along with time under the surface slope condition. Based on the discrete Fourier method, the third-order UDF in the physical plane is expanded to the Laurent series in the complex variable plane. The complex variable method is employed to derive the elastic analytical solution of stratum displacement, when the third-order UDF is taken as the displacement boundary condition of tunnel cross-section (DBCTC). The proposed elastic solution agrees well with the results of the finite element method for the consistent model, which verifies the correctness of the proposed analytical solution. Combining the corresponding principle and fractional Generalized Kelvin viscoelastic constitutive model, the fractional viscoelastic solution under the surface slope condition is determined. The time effect of stratum displacement is presented in two aspects: time-dependent DBCTC and time-dependent material parameters. The parameter analysis is performed to investigate influences of deformation modes of the third-order UDF, slope angle, tunnel radius and fractional order on the time effect of stratum vertical and horizontal displacement.

Three-dimensional analytical elasticity solution for the mechanical analysis of arbitrarily-supported, cross and angle-ply composite plates under patch loads

Patch-type discontinuous loads lead to highly localized stresses which are three-dimensional (3D) in nature and can initiate failures. Their accurate characterization is of primal importance for composite and sandwich structures. The present contribution elaborates a 3D analytical elasticity solution for the mechanical analysis of cross and angle-ply composite plates under transverse patch loads and arbitrary support conditions. For the solution, the Reissner-type, mixed variational principle is used to derive the governing equations. Analytical solutions are obtained using the multiterm, extended Kantorovich method, by dividing the plate into loaded and unloaded parts. The present analytical solution exactly satisfies inter-segment, inter-layer continuity and edge support conditions, treating stresses and displacements as primal variables. Its accuracy is established by direct comparisons with existing analytical – where available – and 3D finite element results, both for cross-ply and angle-ply designs, for different boundary conditions. Extensive analysis is performed to quantify the effect of thickness, angle-ply and patch load position on the composite mechanical performance. Through thickness stress asymmetries are identified and quantified, along with angle-ply designs that can smoothen stress concentrations in the vicinity of patch loads. The methodology can be used as a reference solution in the design of multilayer anisotropic plates under discontinuous loads.

A Simple and Efficient eight node Finite Element for Multilayer Sandwich Composite Plates Bending Behavior Analysis

In this paper, a C0 simple and efficient isoparametric eight-node displacement-model based on higher order shear deformation theory is proposed for the bending behavior study of multilayer composites sandwich plates. Difficult C1-continuity requirement is overcome by extracting the seven degrees of freedom from strain relations for each element node: two displacements for in-plane behavior and five bending unknowns namely: a transverse displacement, two rotations and two shear angles, which results in a kinematic approximation formulation having only first order derivative requirement. The governing equations of the element (constitutive, virtual work and equilibrium equations) are implemented for the prediction of approximate solutions of deflections and stresses of sandwich plates linear elastic problems. Thereby, the formulation element is able to present a cubic in-plane displacement along both core and faces sandwich thickness, as well as, the shear stresses are found to vary as quadratic field without requiring shear correction factors and independent from any transverse shear locking problems. The accuracy and validity of the proposed formulation is verified through the numerical evaluation of displacements and stresses and their comparison with the available analytical 3D elasticity solutions and other published finite element results.

Analytical elasticity solution for accurate prediction of localized stresses in laminated composites under patch loading

Discontinuous patch-type loads cause highly localized stresses which can initiate failures like crack and delamination, and thus must be predicted accurately for efficient design of structures. This paper presents an iterative analytical elasticity solution for accurate stress analysis of laminated composite beams with arbitrary support conditions under a transverse patch loading. The beam is divided into segments with and without the patch loading. The governing equations are derived in the mixed form by employing the Reissner-type mixed variational principle and solved analytically for each segment using the multiterm extended Kantorovich approach. As both displacements and stresses are primary variables, the solution ensures the point-wise satisfaction of inter-layer and inter-segment continuity and boundary conditions. The accuracy of the solution is established by comparing the present numerical results with exact elasticity solutions available in literature for simply-supported edge conditions and compared with finite element solutions of Abaqus for other boundary conditions. An extensive numerical study is performed, and benchmark results are provided for various composite and sandwich laminates, thickness ratios, and boundary conditions under different cases of distributed patch load. The behavior of laminated metal-composite beams is also studied for patch loading, which reveals that thin aluminum layers on the composite beam are highly efficient and effective in reducing the stress concentration in the vicinity of patch load. The presented benchmark results will be beneficial to assess the accuracy and efficacy of different beam theories and elasticity-based approximate models for the analysis of multilayered anisotropic beams subjected to discontinuous loads.

A Generic Analytical Elastic Solution for Excavation Responses of an Arbitrarily Shaped Deep Opening under Biaxial In Situ Stresses

This paper presents a generic analytical solution for stresses and displacements developed around an arbitrarily shaped underground opening excavated in an elastic soil/rock mass with biaxial in situ stresses. A series of separable functions satisfying the governing biharmonic equation is employed as the basic stress function for the problem considered. With the basic stress function and the stress rotation equation, the normal stress and the shear stress on the opening boundary surface are expressed in terms of separable function series with unknown coefficients to make use of the boundary condition. Based on the variational theory, the unknown coefficients are determined by minimizing the difference between the real stress boundary condition and the assumed stress function on the opening surface. The excavation responses around different shapes of openings, including elliptical-shaped, ovaloid-shaped, horseshoe-shaped, and D-shaped openings, are investigated and compared with those from the finite-element simulations. The results show that both the stress field and the displacement field agree well with those from the corresponding numerical model, which verify the proposed solving technique. The proposed solution provides an easy, generic, and feasible approach to assess the stresses and displacements developed around an arbitrarily shaped opening, which is simpler and more straightforward than the complex theory–based solutions.

An Analytical Elastic Solution for Right-Angle Trapezoidal Opening in Steeply Inclined Coal Seam

In the process of underground mining, steeply inclined rocks or coal seams are often encountered, forming the openings of right-angle trapezoid. According to the geological conditions of a mining project in China, an analytical elastic solution of stress and displacement around right-angle trapezoidal opening in a homogeneous, isotropic, and linear elastic geomaterial is presented, which is based on the evaluation of the conformal mapping representation by an appropriate numerical calculation and the complex potential functions. The different results from other shaped openings are shown as follows. In a right-angle trapezoidal opening, the maximum displacements of roof falling occur on the low side, while the most horizontal displacements on the low side are around the roof and the most horizontal displacements on the high side are around the middle of the high side in this opening. These results are also compared with the numerical calculations in FLAC software, illustrating that the solution may be easily applied to rock mechanics or rock engineering for understanding the deformation of floor heave and roof falling down. The solution is also suitable for optimum design of bolt supporting in a right-angle trapezoidal opening, which is different from the traditional concept of symmetrical bolt supporting. Finally, a methodology is proposed for the estimation of conformal mapping coefficients for a given cross-sectional shape of an opening without symmetrical axis.

Magma Chamber Formation by Dike Accretion and Crustal Melting: 2D Thermo‐Compositional Model With Emphasis on Eruptions and Implication for Zircon Records

AbstractWe present a 2D thermo‐elastic model of a magma body formation in a granitic crust by injection of rhyolitic or basaltic dikes and sills. Elastic analytical solutions enable the computation of rock displacement in response to magma intrusion and evacuation during volcanic eruptions. Phase diagrams for magma and rocks govern melting/crystallization behavior and temperature evolution in 16 million computational cells. Calculated temperature histories are used to predict zircon crystallization/dissolution, their ages, and isotopic ratios within individual batches of magma and rocks. Incremental dike injection naturally generates magma batches of melt that form kilometer‐wide clusters appearing in different parts of the domain with diverse shapes, horizontal and vertical interconnectivity. The volume of eruptive melt strongly depends on magma influx rates Q, the width of the injection region W, and eruptions. For example, rhyolitic dike injection with Q = 0.125 m3/s with W = 5 km during 100 ka generates ∼50 km3 of eruptive melt while no significant melt forms if W = 10 km. Injection of basaltic dikes into the granitic crust generates comparable amounts of rhyolitic melt from dike‐residual and country rocks. High injection intensities produce magma reservoirs capable of large eruptions, while repetitive eruptions lead to shrinkage of magma bodies. High heat input causes host rock zircons to lose a significant portion of their old cores. Magmatic zircons in the periphery form quickly due to rapid cooling of intruded dikes while in the central part crystals can grow during several hundreds of ka. The model predicts highly heterogeneous O and Hf isotopic ratios recorded in zircons.

Shrink-fit shear and seal integrity of tubular connections

Make-up of threaded connections generates shrink-fit interference along the engaged thread length of pin and box. Pressure on the pin-box interface, together with Poisson's ratio, shortens the box and stretches the pin, creating shear and affecting sealability. This paper presents an analytical elastic solution of make-up with linear shear to help understand the physics of make-up and the unexpected consequences on leak. The model is driven and supported by results from FEA (finite element analysis) on an example API LTC connection. The main conclusion from the analysis is that make-up shear reduces leak resistance.

Numerical simulation of 2-D fluid-structure interaction with a tightly coupled solver and establishment of the mooring model

In this study, a newly enhanced Fluid-Structure Interaction (FSI) model which incorporates mooring lines was used to simulate a floating structure. The model has two parts: a Computational Fluid Dynamics (CFD) model and a mooring model. The open-source CFD OpenFOAM® v1712 toolbox was used in the present study, and the convergence criteria and relaxation method were added to the computational procedure used for the OpenFOAM multiphase flow solver, interDyMFoam. A newly enhanced, tightly coupled solver, CoupledinterDyMFoam, was used to decrease the artificial added mass effect, and the results were validated through a series of benchmark cases. The mooring model, based on the finite element method, was established in MATLAB® and was validated against a benchmark analytical elastic catenary solution and numerical results. Finally, a model which simulates a floating structure with mooring lines was successfully constructed by connecting the mooring model to CoupledinterDyMFoam.

Ambient multiaxial creep: shakedown analysis of structures using stereo digital image correlation

A new framework to utilize stereo-vision digital image correlation (DIC) as a means of identifying the multiaxial cyclic shakedown behavior of structures is demonstrated on high strength steel bars. AISI 1144 carbon steel cylindrical bars are subjected to cyclic tension with nonzero mean stress and constant torque under ambient conditions. An elastic analytical solution is used in a post-processing procedure to extract inelastic strains and estimate the accumulated inelastic strain during cycling. DIC results are used to understand the cyclic strain evolution and determine if the accumulation of plastic strain stabilizes (shakedown). The results are used to construct a load interaction diagram (Bree Diagram) that displays the loading combinations resulting in purely elastic, safe shakedown, and undesirable cyclic inelastic behavior. The manifestation of ambient creep, during the multiaxial cyclic loading, is demonstrated through dedicated experiments. The shakedown criteria has been adapted to account for the ambient creep behavior. Depending on target structural lifetimes and allowable total strains, it is found that the design space can be enhanced by allowing shakedown to occur. An illustration is given for which the design space is enlarged 1.25 times the typical yield-limited approach.

Analytical solution of stress and displacement for a circular underwater shallow tunnel based on a unified stress function

A unified stress function is represented in a Fourier series to capture the complex stress state of an underwater shallow tunnel perimeter. The first-order unified stress function is employed as stress boundary condition of the tunnel perimeter (SBCTP) to describe the non-symmetrical stress with respect to the horizontal and vertical centerlines of the tunnel. The mechanical analysis model of an underwater shallow tunnel is established. Combining the complex variable theory, the elastic analytical solution of the stress and displacement are obtained. The rationality of the elastic solution is verified by comparing the stress fields between the proposed method and the finite element method (FEM). Based on the obtained elastic solution, the fractional viscoelastic solution is determined by the correspondence principle. The time-dependent ground displacement are investigated considering the effect of stress release. A series of parameter analyses are carried out to discuss the ground vertical and horizontal stress distributions under the influences of SBCTPs.

Analytical investigation of elastic and plastic behavior of rotating double-walled FGM-homogenous hollow shafts

Cylindrical rotating components have been of special interest in different industries and due to their wide applications such as grinding wheel, drive shaft; the analyses of elastic and plastic stress and strains have been an interesting topic for investigation. Therefore, in this study an analytical elastic and elasto-plastic solution to evaluate the stress field in axisymmetric thick-double-walled cylindrical hollow shafts made of functionally graded materials and homogeneous layers subjected to pressure, temperature gradient, and angular speed are presented. In the first step, by considering the combined different loading condition, a closed-form analytical thermo-elastic solution for radial and circumferential stresses as well as the normalized effective stresses are presented. Then, the starting radius of the plastic deformation by using a completely elastic solution and a failure criterion is determined. In the second stage, the relations for determining the plastic zone radius as well as the radial, circumferential, and effective stresses in both elastic and plastic zones are obtained for three types of functionally graded layers under different combined loading condition. Finally, it will be shown that by using the functionally graded layer, the stress distribution and consequently the yield pattern in the thick-double-walled cylindrical hollow shaft can be improved.

Verification of the iterative procedure for solving the elastoplastic Kirsch problem on the Lame problem

An iterative procedure for solving the elastoplastic Lame problem was proposed to further develop an iterative solution of the elastoplastic Kirsch problem. A ring under an external pressure is considered at which a certain plastic zone arises in the ring. The proposed iterative procedure for solving the elastoplastic Lame problem consists in an explicit analytical representation for stresses in the plastic region and the iterative technique for an elastic analytical solution. Furthermore, the plastic zone radius is beforehand unknown. The convergence of the iterative procedure for the elastoplastic Lame problem is shown in comparison with well-known analytical solution to this problem in elastoplastic formulation. A numerical-analytical iterative solution of the Lame problem in elastoplastic formulation is given for various external pressure and internal radius of the ring. Shows the convergence of numerical-analytical iterative solutions the problem in comparison with numerical solutions in elastoplastic formulation.

Theoretical Analysis of the Mechanical Effects of Water Jet

The mechanical characteristics and stress state caused by water jet loading are influenced by many factors that have restricted studies on the jet mechanism. Using the basic theory of elasticity and fluid dynamics, the mechanical effects of water jet loading are analyzed. Based on the simplified jet distribution pressure, the elastic analytical solution of the stress distribution in a semi-infinite plane under the jet distribution pressure is derived, and the stress distribution characteristics are discussed. In addition, the Mohr–Coulomb yield criterion served as a fracture criterion, enabling establishment of the relationship between jet pressure and rock strength. The results indicate that the compressive stress is symmetrically distributed around the jet axis with the maximum at the center. Conversely, the shear stress is Anti-symmetrically distributed around the jet axis with the maximum at the point (0.5r, ± 0.65r). The initial fracture of rock under the impact of water jet is mainly shear compression failure. The Mohr–Coulomb yield criterion is useful for estimating the relationships between the rock failure zone diameter with the water jet impact pressure.