Analytical Solutions For Stresses Research Articles

Stresses and Displacements in Steel-Lined Pressure Tunnels and Shafts in Anisotropic Rock Under Quasi-Static Internal Water Pressure

Steel-lined pressure tunnels and shafts are constructed to convey water from reservoirs to hydroelectric power plants. They are multilayer structures made of a steel liner, a cracked backfill concrete layer, a cracked or loosened near-field rock zone and a sound far-field rock zone. Designers often assume isotropic behavior of the far-field rock, considering the most unfavorable rock mass elastic modulus measured in situ, and a quasi-static internal water pressure. Such a conventional model is thus axisymmetrical and has an analytical solution for stresses and displacements. However, rock masses often have an anisotropic behavior and such isotropic assumption is usually conservative in terms of quasi-static maximum stresses in the steel liner. In this work, the stresses and displacements in steel-lined pressure tunnels and shafts in anisotropic rock mass are studied by means of the finite element method. A quasi-static internal water pressure is considered. The materials are considered linear elastic, and tied contact is assumed between the layers. The constitutive models used for the rock mass and the cracked layers are presented and the practical ranges of variation of the parameters are discussed. An extensive systematic parametric study is performed and stresses and displacements in the steel liner and in the far-field rock mass are presented. Finally, correction factors are derived to be included in the axisymmetrical solution which allow a rapid estimate of the maximum stresses in the steel liners of pressure tunnels and shafts in anisotropic rock.

Analytical solution for stress and deformation of the mining floor based on integral transform

Following exploitation of a coal seam, the final stress field is the sum of in situ stress field and an excavation stress field. Based on this feature, we firstly established a mechanics analytical model of the mining floor strata. Then the study applied Fourier integral transform to solve a biharmonic equation, obtaining the analytical solution of the stress and displacement of the mining floor. Additionally, this investigation used the Mohr–Coulomb yield criterion to determine the plastic failure depth of the floor strata. The calculation process showed that the plastic failure depth of the floor and floor heave are related to the mining width, burial depth and physical–mechanical properties. The results from an example show that the curve of the plastic failure depth of the mining floor is characterized by a funnel shape and the maximum failure depth generates in the middle of mining floor; and that the maximum and minimum principal stresses change distinctly in the shallow layer and tend to a fixed value with an increase in depth. Based on the displacement results, the maximum floor heave appears in the middle of the stope and its value is 0.107m. This will provide a basis for floor control. Lastly, we have verified the analytical results using FLAC3D to simulate floor excavation and find that there is some deviation between the two results, but their overall tendency is consistent which illustrates that the analysis method can well solve the stress and displacement of the floor.

Dynamic mode III interface crack in aluminum matrix composites

By means of the theory of complex variable functions, dynamic extension queries on symmetrical mode III interface crack in aluminum alloys and aluminum matrix composite materials were studied. The crack extension should also appear in the format of self-similarity because interface failure is ascertained by the maximum tensile stress. The formulation involves the development of a Riemann–Hilbert problem. The queries, considered in this study, can be very readily translated into Riemann–Hilbert problem by ways of self-similar functions. The general representations of analytical solutions of stresses, displacements and dynamic stress intensity factors, for the edges of symmetrical mode III interface crack, subjected to motive enhanced loadings, Px5/t5 and Pt6/x5, respectively, were obtained. In light of the correlative material properties, the changeable law of dynamic stress intensity factor was illustrated very well. After the solutions were utilized by the use of superposition principle, the solutions of discretionally complicated problems were easily acquired.

Analytical solutions for composition and stress in spherical elastic–plastic lithium-ion electrode particles containing a propagating phase boundary

Analytical solutions are derived for the distribution of Li concentration, chemical potential, and stress in a spherical particle of an active Li insertion material containing a propagating phase boundary. Results are presented both for elastic and inelastic host materials. We focus in particular on understanding the coupling between the solution thermodynamics, transport, and mechanical forces in the particle. It is shown that solution chemistry can have a strong influence on the stress state; and conversely, the stress state may significantly alter the equilibrium conditions at the phase boundary and the driving forces for phase boundary migration.

Thermal stresses and end-bulging in the laser disc from a tetragonal crystal: the case of LiYF4

Analytical solution of the thermal stress–strain problem is presented for discs from a-cut and c-cut tetragonal crystals with a point group 4, or 4/m. Thermal stresses, strains and displacements, as well as the contribution of the end-bulging to the optical power of the thermal lens are calculated for the discs from Tm:LiYF4. The stress fracture issues are also analyzed for this crystal.

Solution of problems in dynamic elasticity via a mixed least squares method

Purpose – The purpose of this paper is to present an original mixed least squares method for solving problems in dynamic elasticity. Design/methodology/approach – The proposed approach involves two different types of unknowns: velocities and stresses. The approximate solution to the dynamic elasticity equations is obtained via a minimization of a least squares functional, consisting of two terms: a term, which includes the squared residual of a weak form of the time rate of the constitutive relationships, expressed in terms of velocities and stresses, and a term, which depends on the squared residual of the equations of motion. At each time step the functional is minimized with respect to the velocities and stresses, which for the purpose of this study, are approximated by equal order piece-wise linear polynomial functions. The time discretization is based upon the backward Euler scheme. The displacements are computed from the obtained velocities in terms of a finite difference interpolation. The proposed theoretical formulation is given the general three-dimensional case and is tested numerically on the solution of one-dimensional wave equations. Findings – To test the performance of the method, it has been implemented in an original computer code, using object-oriented logic and written from scratch. Two one-dimensional problems from the mathematical physics, with well-known exact analytical solutions, have been solved. The numerical examples include a forced vibrating spring, fixed at its both ends and a rod, vibrating under its own weight, when one of its ends is fixed and the other is traction-free. The performed convergence study suggests that the method is convergent for both: velocities and stresses. The obtained results show excellent agreement between the exact and analytical solutions for displacement modes, velocities and stresses. It is observed that this method appears to be stable for the different mesh sizes and time step values. Originality/value – The mixed least squares formulation, described in this paper, serves as a basis for interesting future developments and applications to two and three-dimensional problems in dynamic elasticity.

Fluid flow and fluid shear stress in canaliculi induced by external mechanical loading and blood pressure oscillation

The paper studies the problem of fluid flow and fluid shear stress in canaliculi when the osteon is subject to external mechanical loading and blood pressure oscillation. The single osteon is modeled as a saturated poroelastic cylinder. Solid skeleton is regarded as a poroelastic transversely isotropic material. To get near-realistic results, both the interstitial fluid and the solid matrix are regarded as compressible. Blood pressure oscillation in the Haverian canal is considered. Using the poroelasticity theory, an analytical solution of the pore fluid pressure is obtained. Assuming the fluid in canaliculi is incompressible, analytical solutions of fluid flow velocity and fluid shear stress with the Navier-Stokes equations of incompressible fluid are obtained. The effect of various parameters on the fluid flow velocity and fluid shear stress is studied.

Effect of shear deformation on adhesive stresses in plated concrete beams: Analytical solutions

In this scientific work, an improved analytical solution for adhesive stresses in a concrete beam bonded with the FRP plate is developed by including the effect of the adherend shear deformations. The analysis is based on the deformation compatibility approach where both the shear and normal stresses are assumed to be invariant across the adhesive layer thickness. The shear stress distribution is supposed to be parabolic across the depth of the adherends in computing the adhesive shear stress and Timoshenko's beam theory is employed in predicting adhesive normal stress to consider the shear deformation. Numerical results from the present analysis are presented both to demonstrate the advantages of the present solution over existing ones and to illustrate the main characteristics of adhesive stress distributions.

An analytical solution for stresses induced by a post-tensioned anchor in rocks containing two perpendicular joint sets

A new analytical approach for the analysis of stress around a post-tensioned anchor in rock with two perpendicular joint sets is presented. The solution considers nonlinear shear stresses developed along the anchor bond length as well as debonding at the tendon–grout interface. The bearing plate effects are also considered in the analyses. The following assumptions are made: (1) homogeneous and orthotropic elastic rock; (2) half-space rock mass with plane strain conditions; (3) trilinear bond-slip model for the behavior of the tendon–grout interface; (4) an elastic anchor. The employed methodology is to decompose the anchor problem into two problems of simpler loadings: stresses produced under a rigid bearing plate and stresses induced by interfacial shear stresses mobilized along the bond length. Based on the proposed solution, an illustrative example is given and the results show that a large compressive region is formed around the anchor free length. Tensile stress concentrations are also observed around the bond length and near the rock surface outside the bearing plate. A parametric study is also conducted to investigate the effects of joint properties on the induced stresses. The results show that as the number of joints intersecting the anchor axis increases, the magnitude of compressive stresses in the free zone decreases, while the size of compression zone around the anchor increases. To verify the results of the analytical approach, a comparison is made with numerical results obtained by using the finite element method. The analytical solutions compare very well with the results obtained by numerical method.

Stress field of a radially functionally graded panel with a circular elastic inclusion under static anti-plane shear loading

The elastic stress field around a circular elastic inclusion in an infinite functionally graded material (FGM) panel that is subjected to a uniform static anti-plane shear loading at infinity is considered. Assuming the rigidity modulus of the FGM panel to be exponential, the analytical solutions of stress and strain distribution around the circular elastic inclusion are obtained by using the variable separation method. Due the generality of the elastic inclusion, other geometrical discontinuities, such as hole and rigid inclusion, can be seen as special cases of circular elastic inclusion when the inclusion rigidity modulus takes a suitable value. Therefore, the present solutions ana- lytically reduce to some classical solution in some special cases. The present analytical solutions are compared with the existing exact solutions to verify them. Finally, the effects of the relative inclusion rigidity modulus ratio and the inhomogeneous rigidity modulus ratio on the stress distribution and the concentration factor are systematically investigated.

A refined laminated plate theory accounting for the third-order shear deformation and interlaminar transverse stress continuity

A variational consistent third-order shear laminated plate theory accounting for the transverse shear stress continuity at the interlaminar interfaces of laminated plates is developed in this paper. The transverse shear function used in this new laminated plate theory is based on the kinematics in the third-order shear deformation plate theory proposed by Shi (2007). The variational principle is employed to derive the variational consistent equilibrium equations in terms of displacements and the variational boundary conditions in terms of displacements and equivalent stress resultants. The continuity conditions of the in-plane displacements and transverse shear stresses at the interlaminar interfaces of laminated plates are enforced by the Heaviside step functions and continuity coefficients. The resulting new laminated composite plate theory accounting for interlaminar continuity has only five independent field variables. Furthermore, the number of the field variables in the present third-order shear laminated plate theory is the same as that used in the first-order shear deformation plate theory. The refined laminated plate theory is applied to solve the bending problems of four laminated composite plates with different lamination schemes and different aspect ratios to evaluate its reliability and accuracy. The resulting analytical solutions of both deflections and stresses agree well with the 3D elasticity solutions and the numerical results of finite element analysis. The result comparison with other laminated plate theories shows that new laminated plate theory accounting for the interlaminar continuity proposed in this paper yields more accurate displacements and stresses than other laminated plate theories with five global variables. Because only five field variables are used in this new laminated plate theory with interlaminar continuity, this refined laminated plate theory can be used as an accurate and efficient theoretical model for the finite element analysis of laminated composite plates.

Large-amplitude oscillatory shear: comparing parallel-disk with cone-plate flow

We compare the ratio of the amplitudes of the third to the first harmonic of the torque, , measured in rotational parallel-disk flow, with the ratio of the corresponding harmonics of the shear stress, |τ 3|/|τ 1|, that would be observed in sliding-plate or cone-plate flow. In other words, we seek a correction factor with which must be multiplied, to get the quantity |τ 3|/|τ 1|, where |τ 3|/|τ 1| is obtained from any simple shearing flow geometry. In this paper, we explore theoretically, the disagreement between and τ 3/τ 1 using the simplest continuum model relevant to large-amplitude oscillatory shear flow: the single relaxation time co-rotational Maxwell model. We focus on the region where the harmonic amplitudes and thus, their ratios, can be fully described with power laws. This gives the expression for , by integrating the explicit analytical solution for the shear stress. In the power law region, we find that, for low Weissenberg numbers, for the third harmonics , and for the fifth harmonics, . We verify these results experimentally. In other words, the heterogeneous flow field of the parallel-disk geometry significantly attenuates the higher harmonics, when compared with the homogeneous, sliding-plate flow. This is because only the outermost part of the sample is subject to the high shear rate amplitude. Furthermore, our expression for the torque in large-amplitude oscillatory parallel-disk flow is also useful for the simplest design of viscous torsional dampers, that is, those incorporating a viscoelastic liquid between two disks.

Stress Analysis and Parameter Optimization of an FGM Pressure Vessel Subjected to Thermo-Mechanical Loadings

The thermo-mechanical behavior and material design of functionally graded materials (FGMs) has been gaining increasing attention in the past two decades. However, little work can obtain the exact solution of FGM vessel consisting of a finite length hollow cylinder and two closed ends due to the difficult in calculating the axial strain. By constructing an exponentially function determining the material properties and based on the steady state thermo-elastic theory and Euler-Cauchy formula, the closed-form analytical solutions of the thermo-mechanical stresses for the FGM vessel subjected to an internal pressure and a thermal load were firstly established taking the effect of closed ends into consideration. And then, a numerical model of the FGM pressure vessel was developed using the finite element (FE) method for validating the analytical solution. Lastly, a double-variable optimization model was proposed for determining the optimal material distribution of the FGM pressure vessel. The results demonstrated the accuracy and efficiency of the proposed stress formulations and optimization model in this paper, which would be helpful for better understanding and scientific design of FGM vessel or related structures.

Stress Analysis of Gun Barrel Subjected to Dynamic Pressure

In the optimal design of a modern gun barrel, there are some aspects to be considered. One of the main factor is internal ballistic which consist of pressure-time, pressure-distance, velocity-time and distance-time curves. In this paper, a simple analytical solution for the plastic stress of an internally pressurized open-ended thick-walled cylinder made of hardening steel which is the closest model to gun barrel is obtained in perfectly plastic and plane stress condition by using energy method and the yield criterion of Von Mises and adding rifle grooves and choosing stress components as basic unknowns and ballistic pressure equation as known. Then results of analytical solution are compared to a numerical model and verified a very well and reliable accuracy. So the resultant can be used easily in calculation of radial expansion velocity and compressive pressure.

A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates

A new hyperbolic shear deformation theory applicable to bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates is presented. This new theory has five degrees of freedom, provides parabolic transverse shear strains across the thickness direction and hence, it does not need shear correction factor. Moreover, zero-traction boundary conditions on the top and bottom surfaces of the plate are satisfied rigorously. The energy functional of the system is obtained using Hamilton’s principle. Analytical solutions of deflection and stresses are obtained using Navier-type procedure. Free vibration frequencies are then accurately calculated using a set of boundary characteristic orthogonal polynomials associated with Ritz method. Numerical comparisons are conducted to verify and to demonstrate the accuracy and efficiency of the present theory. Excellent agreement with the known results in the literature has been obtained.

Thermoelastic bending analysis of laminated composite plates according to various shear deformation theories

AbstractThis study presents the thermoelastic analysis of laminated composite plates subjected to sinusoidal thermal load linearly varying across the thickness. Analytical solutions for thermal displacements and stresses are investigated by using a unified plate theory which includes different functions in terms of thickness coordinate to represent the effect of shear deformation. The theory presented is variationally consistent, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. Governing equations of equilibrium and associated boundary conditions of the theory are obtained using the principle of virtual work. The Navier solution for simply supported laminated composite plates has been developed. Numerical results are presented to demonstrate the thermal response of the laminated composite plates.

Steady-State Creep Deformations and Stresses in FGM Rotating Thick Cylindrical Pressure Vessels

AbstractIn the present study, a closed-form analytical solution for the steady-state creep stresses of rotating thick cylindrical pressure vessels made of functionally graded materials (FGMs) is carried out. Norton's law governs the creep response of the material. Exact solutions for stresses and strain rate are obtained under the plane strain condition. How different material parameters involved in Norton's law affect radial and circumferential stresses together with the equivalent strain rate in rotating thick-walled cylindrical vessels under internal pressure is investigated. The result obtained shows that the property of FGMs has a significant influence on the equivalent creep strain rate and stresses distributions along the radial direction.

The Regular Pattern Analysis on the Temperature Stress of Reinforced Concrete Beams Strengthened with Carbon Fiber

This paper deducing the analytical solution of temperature stress of reinforced concrete beams strengthened under the principle of compatibility of deformation. And we also analyze the influencing factors, such as elasticity modulus, thickness, coefficient of thermal expansion, the height of test beam, concrete mark and so on. The research shows that the elasticity modulus and thickness of the carbon fiber are the major influencing factors of temperature stress, the coefficient of thermal expansion and the height of test beam are the second influencing factors, and the influence of concrete mark is very tiny.

Exact elasto-plastic analysis of rotating disks made of functionally graded materials

Presented in this paper is an exact closed-form analytical solution for elasto-plastic deformations and stresses in a rotating disk made of functionally graded materials (FGMs) in which the elasto-perfectly-plastic material model is employed. As for the yield criterion, the well-known Tresca’s model is used. The material properties are assumed to vary according to the power law whereas the Poisson’s ratio is kept constant. For various values of inhomogeneity constant, the so-obtained solution is then used to study the distribution of limit angular speed, displacement and stresses versus the radial direction. Moreover, the effect of increasing the angular speed on the propagation of the plastic zone is investigated. This analysis reveals that the plasticity can occur in different regions of the disk. To the best of authors’ knowledge, so far all previous exact solutions for the elasto-plastic analysis of FGM disks address the case of constant material density. The novelty of this work is to present an exact solution by taking into account the variation of density. The results show that the variation of density has a significant influence on the distribution of deformations and stresses. Furthermore, the effect of changing Poisson’s ratio on the value of the critical material parameter is demonstrated.

Analytical elasto-plastic solution for stress and deformation of surrounding rock in cold region tunnels

In recent years, the construction of cold-region tunnels has increased in western China. However, the studies of the analytical elasto-plastic solution for the stress and deformation of surrounding rock in cold-region tunnels are limited. To solve this engineering problem and propose new theoretical insights for analysis and design, the elasto-plastic calculation model for surrounding rock in cold-region tunnels is established in this paper. In this model, the entire surrounding rock is divided into four zones, namely, the unfrozen elastic zone, the frozen elastic zone, the frozen plastic zone and the support zone. The frozen surrounding rock is assumed to conform to the ideal elasto-plastic model and the Mohr–Coulomb yield criterion. The stress and deformation fields in each zone are solved according to continuity conditions. A case is introduced to obtain the distribution of stress and strain in the surrounding rock of cold-region tunnels. The reason that the circumferential stress is less than the radial stress in the unfrozen zone is explained and the range of that is also addressed. Moreover, the location in which no displacement is generated is determined in the freezing circle. The produced high stress is mainly influenced by the linear frost strain. The results of stress field and strain field are verified by the existing elastic solution. At last, parameter analysis is performed to obtain the influence of the related parameters on plastic radius and the surrounding rock stresses at the inner and outer edges of the freezing circle. The results can guide for numerical simulations and field engineering practice.