Airy Stress Function Method Research Articles

An innovative and simple approach for predicting tunnelling-induced ground movements semi-analytically

ABSTRACT A semi-analytical elastic solution for predicting tunnelling-induced ground movements in clayey media is presented in this study. This study modelled the non-uniform ground movements (or non-uniform ground loss) around the tunnel cavity using the Gaussian distribution function. A new approach is proposed to model the tunnel peripheral displacements other than the simplistic trigonometric functions. It is shown that by modelling the ground loss in an elliptical fashion, (i) better prediction of ground surface settlements is obtained compared to the circular ground loss in some cases, while (ii) in other cases, circular ground loss gives better predictions. This indicates that both these approaches should be used to predict the ground surface settlements. Further, the ground displacements observed in East-West Metro, Kolkata, are discussed as a case study. The semi-analytically obtained displacements give good conformity with the observed field measurements in East-West Metro. This solution is much simpler to follow and can be easily employed in a code compared to the complex variable method or airy stress function method.

Analytic formulation of elastic field around edge dislocation adjacent to slanted free surface.

Explicit and tractable formulation of the internal stress field around edge dislocations is indispensable for considering the mechanics of fine crystalline solids, because the motion of edge dislocations in a slanted direction with respect to the free surface often plays a vital role in the plastic deformation of the solids under loading. In this study, we formulated an analytical solution for the stress distribution that occurs around edge dislocations embedded in a semi-infinite elastic medium. This formulation is based on the image force method and the Airy stress function method; it describes the variation in the stress distribution with changes in the slanted angle between the traction-free flat surface of the medium and the Burgers vector of the edge dislocation. Furthermore, our analytical solution shows that the attractive force acting on the edge dislocation due to the presence of the free surface is always perpendicular to the surface, regardless of the relative angle of the Burgers vector with the surface.

Stress analysis of a bolted finite dimensions plate under bearing-bypass load interaction using the Airy stress function

Bolted joints are frequently used to connect rather thin parts in lightweight structures, e.g. air- and spacecraft. This is also motivated by inexpensive manufacturing and the ability to disassemble. However, holes need to be drilled and stress concentrations arise. Usually, multiple bolts/rows of bolts are placed. Then, the load is partly introduced into one bolt while the remaining rest stays in the plate. This problem setting is also referred to as bolted joint under combined bearing-bypass load. When there are negligible secondary bending effects a 2D model can be employed. Further, a linear approach without contact elasticity is commonly chosen, which is provided by superimposing the open- and pin-loaded hole/filled-hole problem. The latter is then idealised by sinusoidal radial tractions along half of the hole edge. The structural assessment shall be done using precise analysis means since safety-critical parts are connected by bolted joints. These means can be based on analytical methods, which are beneficial in terms of computational effort and shall be scope of the present paper. For stress field representation, use is made of the Airy stress function method. First, the solutions of the special cases open and filled hole are determined. These are then superimposed. Finite plate dimensions are taken into account using auxiliary and correction stress functions, which are based on a novel periodic arrangement technique. Effects of raised stress concentrations due to finite dimensions as well as the ratio between bearing and bypass load are extensively discussed. Validation by means of Finite Element analyses reveals excellent agreement.

On Analytic Solutions of Elastic Net Displacements around a Circular Tunnel

Displacements around a tunnel, occurring as a result of excavation, consist of the elastic and plastic parts. In this paper, we discuss the elastic part of displacements as a result of excavation, called net displacement. In general, the previous analytical solutions presented for determining the displacements around a circular tunnel in an elastic medium do not give the net displacements directly. The well-known Kirsch solution is the most widely used method for determining the induced stresses and net displacements around a circular opening in a biaxially-loaded plate of homogeneous, isotropic, continuous, linearly elastic material. However, the complete solution for obtaining the net displacements has not been presented or highlighted in the available literature. Using the linear elasticity, this paper reviews and presents three different analytical methods for determining the net displacements directly as well as induced stresses around a circular tunnel. The three solution methods are the Lame' method, airy stress function method, and complex variable method. The tunnel is assumed to be situated in an elastic, continuum, and isotropic medium in the plane strain condition. The solutions are presented for both the hydrostatic and non-hydrostatic in situ stresses in the 2D biaxial loading condition along with an internal pressure. Loading and unloading in tunneling occurring as a result of excavation and stress differences between the induced and initial ones are considered to evaluate the net displacements directly. Finally, some examples are given to demonstrate the complete solution and show the difference between the net elastic displacements as a result of excavation and total elastic displacements that are not real.

Nonlinear vibration of FGM moderately thick toroidal shell segment within the framework of Reddy’s third order-shear deformation shell theory

Nonlinear vibration and dynamic response of functionally graded moderately thick toroidal shell segments resting on Pasternak type elastic foundation are investigated in this paper. Functionally graded materials are made from ceramic and metal, and the volume fraction of constituents are assumed to vary through the thickness direction according to a power law function. Reddy’s third order shear deformation, von Karman nonlinearity, Airy stress function method and analytical solutions are used to derive the governing equations. Galerkin method is used to convert the governing equation into nonlinear differential equation, then the explicit expressions of natural frequencies and nonlinear frequency–amplitude relations are obtained. Using Runge–Kutta method, the nonlinear differential equation of motion is solved, and then nonlinear vibration and dynamic response of shells are analyzed. The effects of temperature, material and geometrical properties, and foundation parameters on nonlinear vibration and dynamic characteristics are investigated and discussed in detail.

Nonlinear buckling and post-buckling of functionally graded CNTs reinforced composite truncated conical shells subjected to axial load

This study deals with the nonlinear static analysis of functionally graded carbon nanotubes reinforced composite (FG-CNTRC) truncated conical shells subjected to axial load based on the classical shell theory. Detailed studies for both nonlinear buckling and post-buckling behavior of truncated conical shells. The truncated conical shells are reinforced by single-walled carbon nanotubes which alter according to linear functions of the shell thickness. The nonlinear equations are solved by both the Airy stress function and Galerkin method based on the classical shell theory. In numerical results, the influences of various types of distribution and volume fractions of carbon nanotubes, geometrical parameters, elastic foundations on the nonlinear buckling and post-buckling behavior of FG-CNTRC truncated conical shells are presented. The proposed results are validated by comparing with other authors.

An edge dislocation inside a semi-infinite plane containing a circular hole

This paper presents an analytical solution of the Green's function for an edge dislocation lying anywhere inside a semi-infinite plane containing a circular hole. The solution is obtained using the Airy stress function method, where the auxiliary function is expressed in series form in the bi-polar coordinate system and the convergence of the solution is ensured by an asymptotic analysis. Stress distributions are presented for different d/R ratios (where d is the distance from the free edge to the centre of the hole of radius R) and various dislocation positions to show the interactive effect of the two boundaries. The Green's function is then employed to model an edge crack growing toward the hole by a continuous distribution of dislocations.

Two-dimensional elasticity solution of elastic strips and beams made of functionally graded materials under tension and bending

This paper presents a theoretical approach to solve elastic problems of functionally graded materials (FGMs). For FGMs with exponential gradient, based on a two-dimensional theory of elasticity, a governing equation is derived by means of the Airy stress function method together with the strain compatibility equation. Simple uniaxial tension and bending are solved. For an FGM layer with transversely and/or vertically varying material properties, stress distribution and strain field under simple tension are determined according to two different assumptions. The obtained results indicate that for a thin elastic layer of thickness-wise gradient as a transition zone linking two dissimilar materials, there is a horizontal displacement difference across the transition zone due to mismatch of the material properties. In particular, when the thickness of the FGM layer reduces to zero, the horizontal displacement difference has a severe mismatch across the interface of two perfectly bonded dissimilar materials. An FGM beam subjected to a bending moment is also analyzed. The normal stress exhibits a nonlinear distribution and may arrive at its maximum tensile stress inside the beam, not at the surface. The obtained elasticity solution is useful for better understanding of the mechanical behaviors of FGMs subjected to different combined loads.

Theoretical analyses for a 2-2 cement-based piezoelectric curved composite with electrode layers

Based on the general theory of elasticity, the static behavior of 2-2 cement-based piezoelectric curved composites is investigated. The actuator consists of 2 cement layers and 1 piezoelectric layer. Considering the electrode layer between the cement layer and the piezoelectric layer as the elastic layer, the exact solutions of the mechanical and electrical fields of the curved composites are obtained by utilizing the Airy stress function method. Furthermore, the theoretical results are compared with the FEM results and good agreements (with almost no error) are obtained, thus proving the validity of this study. Furthermore, the influence of certain parameters is discussed, which can help to get the desired displacements and stresses. Finally, it is seen that the analytical model established in this paper works well, which could benefit the design of this kind of cement-based smart devices.

On a problem of the vibration of functionally graded conical shells with mixed boundary conditions

This paper presents a theoretical approach to solve vibration problems of functionally graded (FG) truncated conical shells under mixed boundary conditions. The material properties of FG shell are assumed to vary continuously through the thickness of the conical shell. The fundamental relations, motion and strain compatibility equations of FG truncated conical shells are derived by means of the Airy stress function method. Two cases of mixed boundary conditions are investigated. The basic equations are solved by using Galerkin method and fundamental cyclic frequencies of FG truncated conical shells are obtained. The results are compared and validated with the results available in the literature. The detailed parametric studies are carried out to investigate the influences of radius-to-thickness ratio, lengths-to-radius ratio, material composition and mixed boundary conditions on the fundamental cyclic frequencies of truncated conical shells.

The elliptic phenomenon effect of cross section on the torsional buckling of a nanocomposite beam reinforced by a single-walled carbon nanotube

In this article, the elliptic phenomenon effect of cross section on the torsional buckling of a nanocomposite beam reinforced by a single-walled carbon nanotube based on the continuum mechanics theory is presented. The Airy stress function method is applied to obtain the deformation of the matrix. In addition, the cross-sectional change in the single-walled carbon nanotube in torsion is considered. By using the Airy stress function, the radial and tangential displacements of the matrix and single-walled carbon nanotube, the strain energy for both the matrix and single-walled carbon nanotube is determined. Then, the stress distribution in terms of strain for different torsional curvatures, matrix thickness, single-walled carbon nanotube percentage and flatting ratio is studied. It is found from the results that with the increase in matrix thickness, the tangential and longitudinal strains in single-walled carbon nanotube decrease and vice versa for the interface stress and the dimensionless stress of the outer surface. It can be seen that in thicker matrix layers, the critical torsional buckling load of single-walled carbon nanotube decreases. Moreover, the stresses and strains of single-walled carbon nanotube and matrix will increase with an increase in the flattening ratios.

Solutions of the generalized half-plane and half-space Cerruti problems with surface effects

The generalized half-plane and half-space Cerruti problems with surface effects are analytically solved using Gurtin and Murdoch’s surface elasticity theory along with the Airy stress function method and the Papkovitch–Neuber potential function approach, respectively. The Fourier transform method is employed in the formulation. The newly derived solutions reduce to their classical elasticity-based counterparts if the surface effects are not considered. In addition, the current solution for the half-space Cerruti problem recovers that based on a simplified version of the surface elasticity theory as a special case. Further, specific solutions are obtained for the cases with loading by a concentrated tangential force and by a uniform traction acting over a circular region. The numerical results reveal that the predictions by the current solutions substantially deviate from those by their classical counterparts near the loading site, but the differences between the two sets of predictions diminish far away from the loading site. Moreover, the newly obtained solutions do not exhibit stress or displacement singularity at the loading point/boundary and predict smoother stress fields and smaller displacements than the classical solutions that do not incorporate the surface effects.

Elasticity solutions for orthotropic functionally graded curved beams

Abstract Elasticity solutions are presented for curved beams with n orthotropic functionally graded layers by means of the Airy stress function method. The beams are subjected to a uniform load on the outer surface and may have various constraints or/and loads at ends. Firstly, the stresses and displacements are expressed in terms of three unknown functions for a general model. Secondly, the unknown functions are deduced for two slightly general forms of elastic compliance parameters and represented by the generalized hypergeometric functions. Thirdly, the boundary and continuity conditions to determine integral constants are given. As the application, a curved cantilever beam, with three different variations in the elastic compliance parameters and under two types of loads, is discussed.

Exact solutions of the piezoelectric transducer under multi loads

Under the external shearing stress, the external radial stress and the electric potential simultaneously, the piezoelectric hollow cylinder transducer is studied. With the Airy stress function method, the analytical solutions of this transducer are obtained based on the theory of piezo-elasticity. The solutions are compared with the finite element results of Ansys and a good agreement is found. Inherent properties of this piezoelectric cylinder transducer are presented and discussed. It is very helpful for the design of the bearing controllers.

Exact analyses for two kinds of piezoelectric hollow cylinders with graded properties

Based on the theory of piezo-elasticity, the paper obtains the exact solutions of functionally graded piezoelectric hollow cylinders with different piezoelectric parameter <TEX>$g_{31}$</TEX>. Two kinds of piezoelectric hollow cylinders are considered herein. One is a multi-layered cylinder with different parameter <TEX>$g_{31}$</TEX> in different layers; the other is a continuously graded cylinder with arbitrarily variable <TEX>$g_{31}$</TEX>. By using the Airy stress function method with plane strain assumptions, the exact solutions of the mechanic and electrical components of both cylinders are obtained when they are subjected to external voltage (actuator) and pressure (sensor), simultaneously. Furthermore, good agreement is achieved between the theoretical and numerical results, and useful conclusions are given.

Analytical solutions of two kinds of piezoelectric actuators under shearing load

Under an external shearing load and an electric potential acting simultaneously, twopiezoelectric hollow cylinder actuators, of the expansion and contraction types, are studied.With the Airy stress function method, the analytical solutions of these two kindsof actuators are obtained based on the theory of piezo-elasticity. The solutionsare compared with the finite element results of Ansys and a good agreement isfound. Inherent properties of these two kinds of piezoelectric cylinder actuators arepresented and discussed. This is very helpful for the design of bearing controllers.

A study of the effect of impact loading on DSIFs and crack propagation in plate

In this work, a study of the effect of impact loading on dynamic crack propagation in thin and isotropic thick plates for two types of material, stainless steel, and aluminum, are investigated analytically and numerically to give a complete study of 3D crack growth of simply supported plate with internal crack at the center of plate. The dynamic stress intensity factor (DSIFs), the velocity of dynamic crack propagation, and the deep of the crack normal to the crack face for different angles from the crack tip have been calculated when the Contact of a cylindrical impactor on isotopic plates is considered, to support the analytical result for DSIFs with a numerical package result (Ansys-10). This is based on the finite element method used to investigate the stress and the values of dynamic stress intensity factors at the crack tip by full transient dynamic Analysis using 3D (element, Solid-90). The results from the Airy stress function method give good agreement with the results of Ansys for a long time duration in impact loading while the energy method is in agreement with the Ansys package for some time and then becomes limited. In addition, the DSIFs and crack propagation velocity are increasing with an angle from the crack tip and velocity impact increasing.

Nonlinear vibration analysis of isotropic cantilever plate with viscoelastic laminate

The nonlinear vibration of an isotropic cantilever plate with viscoelastic laminate is investigated in this article. Based on the Von Karman’s nonlinear geometry and using the methods of multiple scales and finite difference, the dimensionless nonlinear equations of motion are analyzed and solved. The solvability condition of nonlinear equations is obtained by eliminating secular terms and, finally, nonlinear natural frequencies and mode-shapes are obtained. Knowing that the linear vibration of this type of plate does not have exact solution, Ritz method is employed to obtain semi-analytical nonlinear mode-shapes of transverse vibration of this plate. Airy stress function and Galerkin method are employed to reduce nonlinear PDEs into an ODE of duffing type. Stability of plate and chaotic behavior are investigated by Runge–Kutta method. Poincare section diagrams are in good agreement with results of Lyapunov criteria.

Static analysis for functionally graded piezoelectric actuators or sensors under a combined electro-thermal load

Based on the theory of piezoelasticity, a functionally graded piezoelectric sandwich cantilever under an applied electric field and/or a heat load is studied. All materials may be arbitrary functional gradients in the thickness direction. The static solution for the mentioned problems is presented by the Airy stress function method. As a special case, assuming that the material composition varies continuously in the direction of the thickness according to a power law distribution, a comprehensive parametric study is conducted to show the influence of electromechanical coupling (EMC), functionally graded index, temperature change and thickness ratio on the bending behavior of actuators or sensors. The distribution of electric field or normal stress in present FGPM actuators is continuous along the thickness, which overcomes the problem in traditional layered actuators. The solution facilitates the design optimization for different piezoelectric actuators and has another potential application for material parameter identification.

Exact analysis of two kinds of piezoelectric actuator

Two kinds of piezoelectric hollow cylinder actuator are studied in this paper.One is the expansion actuator and the other is the contraction actuator. Usingthe Airy stress function method, the analytical solutions of these two kinds ofactuators are obtained based on the theory of piezo-elasticity. The solutions arecompared with numerical results and good agreement is found. Inherent properties ofthese two kinds of piezoelectric cylinder actuator are presented and discussed.Findings have applications in the field of micromechanics and microengineering.