Airy Stress Function Research Articles

Stress State in an Eccentric Elastic Ring Loaded Symmetrically by Concentrated Forces

The stress state from an eccentric ring made of an elastic material symmetrically loaded on the outer boundary by concentrated forces is deduced. The analytical results are obtained using the Airy stress function expressed in bipolar coordinates. The elastic potential corresponding to the same loading but for a compact disk is first written in bipolar coordinates, then expanded in Fourier series, and after that, an auxiliary potential of a convenient form is added to it in order to impose boundary conditions. Since the inner boundary is unloaded, boundary conditions may be applied directly to the total potential. A special focus is on the number of terms from Fourier expansion of the potential in bipolar coordinates corresponding to the compact disk as this number influences the sudden increase if the coefficients from the final form of the total potential. Theoretical results are validated both by using finite element software and experimentally through the photoelastic method, for which a device for sample loading was designed and constructed. Isochromatic fields were considered for the photoelastic method. Six loading cases for two different geometries of the ring were studied. For all the analysed cases, an excellent agreement between the analytical, numerical and experimental results was achieved. Finally, for all the situations considered, the stress concentration effect of the inner hole was analytically determined. It should be mentioned that as the eccentricity of the inner hole decreases, the integrals from the relations of the total elastic potential present a diminishing convergence in the vicinity of the inner boundary.

Free vibration analyses of functionally graded plates based on improved refined shear and normal deformation theory considering thickness stretching effect using Airy stress function

In this paper, an improved refined shear and normal deformation theory is used in order to investigate the vibration behavior of functionally graded rectangular plates. In this theory, displacements of various points of plate are assumed to be due to in-plane displacements of the middle plane and transverse displacement. Transverse displacement is divided into three parts: bending, shear, and thickness stretching. Using the Airy stress function, corresponding to the compatibility equation, and employing the extended Hamilton’s principle, in-plane displacements are omitted from the equations of motions. Thus, the proposed approach uses only three-unknowns in the displacement field. The results of vibration analysis using the proposed approach are in excellent agreement with three-dimensional and quasi-three-dimensional solutions containing a greater number of unknowns to consider the thickness stretching effect. Static and dynamic behavior of wide variety of thin and thick functionally graded plates can be studied using the proposed approach in which not only the number of variables is reduced, but also the contribution of bending, shear, and thickness stretching are completely clarified.

Airy-based static limit analysis of structures using stabilized radial point interpolation method

This paper presents a novel formulation for static limit analysis of structures, for which the Airy stress function is approximated using stabilized Radial Point Interpolation Mesh-free method (RPIM). The stress field is determined as second-order derivatives of the Airy function, and the equilibrium equations are automatically satisfied a priori. The so-called Stabilized Conforming Nodal Integration (SCNI) is employed to ensure a present method is truly a mesh-free approach, meaning that all constraints in problems are only enforced at nodes. With the use of the Airy function, SCNI, and Second-Order Cone Programming (SOCP), the size of the resulting problem is kept to be minimum. Several benchmark problems having arbitrary geometries and boundary conditions are investigated. The obtained numerical solutions are compared with those available in other studies to perform the computational aspect of the proposed method.

Grid-shell design and analysis via reciprocal discrete Airy stress functions

This research paper introduces a theoretical framework for the design and analysis of compression-and-tension grid-shells in static equilibrium where the states of self-stress function as design freedoms. This is based on a synthesis of reciprocal discrete Airy stress functions in the context of graphic statics and the Force Density Method (FDM). Specifically, the former is a direct method for generating 2-dimensional global static equilibrium whereas the latter allows for its 3-dimensional implementation. As a result, creative design explorations can take place directly within the equilibrium space without the need for iterative convergence algorithms to obtain equilibrium. The use of reciprocal Airy stress functions in conjunction with the lower bound theorem gives insight and explicit control with regards to the states of self-stress as design and analysis freedoms which can define the structural form and its load path.

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.

A Physics-Informed Neural Network Approach to Solution and Identification of Biharmonic Equations of Elasticity

We explore an application of the Physics-Informed Neural Networks (PINNs) in conjunction with Airy stress functions and Fourier series to find optimal solutions to a few reference biharmonic problems of elasticity and elastic plate theory. Biharmonic relations are fourth-order partial differential equations (PDEs) that are challenging to solve using classical numerical methods and have not been addressed using PINNs. Our work highlights a novel application of classical analytical methods to guide the construction of efficient neural networks with a minimal number of parameters that are very accurate and fast to evaluate. In particular, we find that enriching the feature space using Airy stress functions can significantly improve the accuracy of PINN solutions for biharmonic PDEs.

Buckling response of laminated FG-CNT reinforced composite plates: Analytical and finite element approach

A 3D finite element model using suitable user subroutines in ABAQUS finite element software and an analytical solution based on the third-order shear deformation laminated plate theory combined with von Kármán's geometric nonlinearity are proposed to simulate the functionally graded Carbon Nanotubes (CNTs) throughout a plate's thickness. The former is used for the estimation of the buckling response and the latter is used for the estimation of both buckling and post-buckling response of laminated nanocomposite plates under uniform and parabolic loading. In the analytical approach, the governing equations are solved by the Galerkin method and Airy's stress function to obtain the buckling coefficient of FG-CNT reinforced composite plates. The effect of various factors, such as boundary conditions, types of loads, distribution type and volume fraction of CNTs, geometrical parameters, etc on the buckling and post-buckling response is investigated. It has been found that a higher volume fraction of CNTs increases the buckling and post-buckling strength of laminated composite plates, whereas the laminated plate with angle-ply orientations equal to 45 degrees has the maximum buckling load factor among the CNT fiber orientations considered.

Comparison of stress separation procedures. experiments versus theoretical formulation

The article deals with stress separation using different experimental techniques and their comparison to numerical and theoretical solutions. The method is applied to a keyhole sample coupon, to which measurements of fringe order, temperature or displacement were made using photo-elasticity, TSA (Thermo-elastic Stress Analysis) and DIC (Digital Image Correlation). The results are compared to FEM simulations (Finite Element Method) using the stress concentration factor (Kt) as a benchmark. Additionally, an Airy stress function is proposed and tested against obtained measurements. The comparison of Kt shows agreement among measurements as well as numerical and theoretical results. It is concluded that the presented method can be used for isotropic materials subjected to the plane stress, where stress variation through-thickness is negligible.

Design of tension structures and shells using the Airy stress function

Discontinuities in the Airy stress function for in-plane stress analysis represent forces and moments in connected one-dimensional elements. We expand this representation to curved membrane-action structures, such as shells and cable nets, and graphically visualise the internal stresses and section forces at the boundary necessary for equilibrium. The approach enhances understanding of the interplay between form and forces and can support design decisions related to form-finding and force efficiency. As illustrative examples, the prestressing needed for three existing cable nets is determined, and its influence on the edge-beam bending moment is explored.

Vibration and nonlinear dynamic response of temperature-dependent FG-CNTRC laminated double curved shallow shell with positive and negative Poisson’s ratio

This paper reports a study on natural frequency, nonlinear dynamic response and frequency amplitude relation of temperature-dependent FG-CNTRC laminated double curved shallow shell with positive and negative Poisson’s ratio. The nonlinear motion equations and the compatibility equation are retrieved from first order shear deformation theory including the effect of von Karman nonlinearity terms, elastic foundations and initial imperfection. The system of dynamic governing equations is obtained by utilizing the Airy stress function and Bubnov–Galerkin procedure. The nonlinear dynamic response is also obtained by applying the 4th-order Runge–Kutta method. The frequency–amplitude relations of the shell is determined by using the assumption of inertia caused by a rotation. Besides, the analytical method and the finite element method are proposed to calculate the vibrational frequencies. Five types of FG-CNTRC laminate shell, three types of the structures and different thermal environments with the shell’s material properties depending on the temperature are also considered in this study. The numerical results are presented, verified with available studies in the literature. Finally, the effects of elastic foundations, initial imperfection, and temperature-dependence, typing double curved shallow shell, angle of lamination against the shell x-axis on the natural frequency and nonlinear dynamic response of FG-CNTRC laminated double curved shallow shell with positive and negative Poisson’s ratio are discussed.

Nonlinear dynamic stability analysis of clamped and simply supported organic solar cells via the third-order shear deformation plate theory

This study presents a semi-analytical framework to explore the nonlinear dynamic buckling characteristics of the imperfect organic solar cell (OSC) with clamped and simply supported restrains grounded on the third-order shear deformation plate theory (TSDT). The exerting axial displacement loading is divided into infinite and finite durations. Moreover, the spatially dependent Winker-Pasternak elastic foundation and damping effect are incorporated in the current analysis. Combining the von Kármán nonlinearity, the governing equations are derived with the aid of the Airy stress function. By applying the Galerkin method and fourth-order Runge–Kutta procedure, the dynamic buckling critical condition of the OSC subjected to these two kinds of loadings is determined by Budiansky–Hutchinson (B-H) criterion. Based on the numerical results, the influence of some critical parameters, including the geometrical dimension, boundary condition, loading configuration, initial imperfection, damping ratio, and elastic foundation coefficient are investigated. Not limited to the solar cell, this method is also suitable to the dynamic buckling behaviours of other laminated plates.

Residual stress identification in thin plates based on modal data and sensitivity analysis

Residual stresses commonly arise in manufacturing, rolling, welding and heating of mechanical structures and they have a critical influence on fatigue strength, creep resistance and other properties of the structures. Therefore, identifying the residual stress distribution is crucial for quality assessment of these structures. To this end, this paper proposes a new non-destructive residual stress identification approach for thin plates using modal data. There are two key ingredients in establishing the proposed approach. The first is the parameterization of residual stress field by resorting to the Airy stress function so that residual stress identification is recast as a parameter identification problem. The second lies in the inverse identification of stress parameters through a sensitivity-based procedure. In doing so, stress parameter identification is formulated as a nonlinear optimization problem whose goal function is just least-squares of the misfit between the measured and calculated data, and then, modal sensitivity analysis and the trust-region constraint are introduced to iteratively get the solution. Numerical examples are conducted to verify the feasibility and accuracy of the proposed approach.

A form-finding method for membrane shells with radial basis functions

The equilibrium of a membrane shell is governed by Pucher’s equation that is described in terms of the relations among the external load, the shape of the shell, and the Airy stress function. Most of the existing funicular form-finding algorithms take a discretized stress network as the input and find the shape. When the resulting shape does not meet the user’s expectation, there is no direct clue on how to revise the input. The paper utilizes the method of radial basis functions, which is typically used to smoothly approximate arbitrary scalar functions, to represent C∞ smooth shapes and stress functions of shells. Thus, the boundary value problem of solving Pucher’s equation can be converted into a least-squares regression problem, without the need of discretizing the governing equation. When the provided shape or stress function admits no solution, the algorithm recommends users how to tweak the input in order to find an approximate solution. The external load in this method can easily incorporate vertical and horizontal components. The latter part might not always be negligible, especially for the seismic hazard zones. This paper identifies that the peripheral walls are preferable to allow the membrane shells to carry horizontal loads in various directions without deviating from their original shapes. When there are no sufficient supports, the algorithm can also suggest the potential stress eccentricities, which could inform the design of reinforcing beams.

Nonlinear dynamic investigation of the perovskite solar cell with GPLR-FGP stiffeners under blast impact

The perovskite solar cell (PSC), as a potential disruptive space market entrant, has fascinated both the scientific community and aerospace industry due to the high specific power, flexibility, and low fabrication cost. With the aim of reducing structure weight while strengthening the blast-carrying capacity of the PSC, the novel graphene platelets (GPL) reinforced functionally graded porous (GPLR-FGP) stiffeners have been incorporated as enhancements against impact. This research explores the nonlinear dynamic characteristics of the PSC with GPLR-FGP stiffeners under blast load. Integrating the von-Kármán geometric nonlinearity and the first-order shear deformation theory, the governing motion equations are derived by utilizing Airy's stress function and the Galerkin method. The fourth-order Runge-Kutta approach is employed to capture the solutions of dynamic equations effectively. Diverse influences of the stiffener material, boundary condition, plate theory, porosity distribution, GPL dispersion, porosity coefficient, GPL weight fraction, GPL geometry, damping, nonlinear elastic foundation, and initial imperfection are investigated in the numerical study. Besides, the optimal parameters of the PSC with GPLR-FGP stiffeners are discovered, facilitating the following paces of space design and practical implementation in extraterrestrial circumstances.

Mixed mode equilibrium analysis of asymptotic fields around stationary sharp V-notches in polymer gels: A linear poroelastic study

In this paper, the equilibrium asymptotic stress and solvent concentration fields around stationary sharp V-notches are extracted for in-plane mixed mode loadings employing the linear poroelasticity theory. It is shown that at thermodynamic equilibrium, mechanical and chemical equilibrium equations are uncoupled and thus can be solved independently. The mechanical equilibrium equations are then solved using the Airy stress function technique. This technique replaces the coupled mechanical equilibrium relations with the compatibility biharmonic equation which can be solved using the separation of variables method. Applying traction free boundary conditions on the notch edges leads to an eigenvalue problem which gives both mode I and mode II eigenvalues. These eigenvalues can be real or complex numbers depending on the notch opening angle. Since there are infinite eigenvalues for both modes of deformation, series solutions will be obtained for the equilibrium fields. From these asymptotic solutions, it is found that the obtained stress fields are similar to their corresponding linear elasticity solution. Furthermore, from the solutions, it can be observed that the opening and shear deformations near the notch tip lead to cosine and sine variations of the solvent concentration field with respect to the angular coordinates, respectively. The accuracy of these asymptotic results is finally verified using finite element analyses of a single-edge notched (SEN) specimen subjected to far field applied small displacements. The numerical analyses are performed for two notch opening angles of 30° and 60° for both plane stress and plane strain conditions. The comparative study between the finite element results and the asymptotic solution proves that the present solution can accurately capture the near notch tip stress and solvent concentration fields by considering only the first few terms of the series solutions.

Approximate analytical solutions for piezoelectric rectangular beams by using Boley-Tolins method

The Airy stress function satisfies the bipotential equation which is a fourth order partial differential equation in two-dimensional elasticity for isotropic materials. In 1956 Bruno Boley published an iterative procedure for the solution of the Airy stress function. Nowadays this approximation method would be classified as homotopy analysis method (HAM). Boley presented analytical formulas for the deflections and the stresses of two-dimensional, isotropic, rectangular strips and beams. In this contribution Boley's method is extended by considering a piezoelectric material (PZT-5A). After defining the stress function and the electric potential and substituting the constitutive relations into the compatibility equation and Gauss’ law of electrostatics, one obtains two coupled partial differential equations. Adapting Boley's approximation method one finds an iteratively improving solution series where the higher iterations are considered as correction terms. In particular, it is interesting to note that the first iteration is identical to the one-dimensional piezoelectric beam theory within the framework of Bernoulli-Euler if the x-component of the electric displacement is neglected in the charge equation. Analytical formulas are presented for a piezoelectric rectangular beam (unimorph) and a piezoelectric bimorph in case of electrical actuation. Numerical results for the displacement and the electric potential are presented for a cantilever with various thickness-to-length ratios. The analytical outcome of this novel approach is compared to two-dimensional finite element results, showing a very good agreement even for very thick beams if the second iteration is considered.

Refined orthotropic beam models based on Castigliano’s theorem and an approximate solution of the compatibility equation

This paper presents an extension of Castigliano’s second theorem for orthotropic beams. The notion “extension” in this context means to find more accurate results not only for the vertical deflection, but also for the horizontal deflection and the rotation angle. For this purpose the Airy stress function is calculated for an orthotropic rectangular strip based on the iterative approximation method from Boley-Tolins when distributed loads or shear tractions are assumed as surface loads. Hence all stress components can be derived from the stress function and may be inserted into the complementary strain energy. Computing the partial derivative of the complementary strain energy with respect to a scalar dummy parameter, the weighted displacement field components over the cross section are obtained. Finally the analytical results from the extended Castigliano theorem (ECT) are verified by two-dimensional (2D) analytical results for an orthotropic cantilever and a clamped-hinged beam and the outcome is also compared to Bernoulli–Euler and to Timoshenko results. The accuracy of the derived results is higher than that derived by the Timoshenko theory because the normal stress in thickness direction is properly taken into account. Moreover it is demonstrated that this lower order normal stress which is caused by distributed loads yields a mean horizontal elongation or shrinkage that is not considered by the Timoshenko beam theory. Hence, highly accurate formulations for laminated composite beams, involving sandwich-structures or multi-layered beams with imperfect bonding, may be derived from the present orthotropic beam model in the future.

Nonlinear vibration, stability, and bifurcation analysis of axially moving and spinning cylindrical shells

The nonlinear vibration characteristics of the rotating axially moving circular cylindrical shells in subharmonic regions are investigated in the present paper. The motion equations are carried out based on the Hamilton principle in cylindrical coordinates utilizing Donnell’s nonlinear shell theory. By introducing the suitable airy stress function, three equilibrium equations in the cylindrical coordinates are simplified into two nonlinear coupled nonhomogeneous PDEs, including a compatibility equation and the transverse motion equation. The compatibility equation solution is obtained employing the seven degrees of freedom for the flexural mode shape of the system. By implementation of the Galerkin method, the motion equation would be projected into seven nonlinear coupled nonhomogeneous ODEs. This set of equations is solved using a direct normal form method validated by the numerical method and available data. The effect of angular velocity and axial speed is investigated employing frequency and force response curves, bifurcation diagrams, time history, and the system’s phase portraits. Always axially moving and rotation speed of the system intensifies the nonlinear behavior of the frequency responses.

A two-layer beam model with interlayer slip based on two-dimensional elasticity

The objective of this paper is the computation of the Airy stress function of a two-layer composite beam with interlayer slip. For the calculation of the upper and lower layer stress functions an iterative solution procedure from Boley-Tolins is adapted, which was originally introduced for isotropic single layer beams under plane stress assumptions. Considering the continuity equations for the interface stress, for the vertical deflection and a linear law between the interlayer slip and the shear stress, Boley-Tolins’ method may be extended to derive an ordinary differential equation for the slip. The order of the ordinary differential equation depends on the number of iterations and the desired accuracy, ranging from an elementary, second order ordinary differential equation for the slip to a more refined, fourth order ordinary differential equation for the slip. Finally the derived slip models are compared by considering a thick simply-supported two-layer beam subjected to a sinusoidal transverse load. The results from the iterative approaches are compared to two-dimensional results under plane stress assumptions for two limit cases: a perfectly bonded beam and a composite beam without bonding. It is found that the refined higher order slip model almost coincides with the elasticity solution, even for very thick composite beams, whereas the elementary slip model only yields accurate results for a thin beam.

The method of fundamental solutions for two-dimensional elasticity problems based on the Airy stress function

This paper presents a new version of the method of fundamental solutions (MFS) for two-dimensional linear elasticity problems based on the stress function (Airy stress function), which is different from the MFS utilizing the fundamental solutions of displacement. The displacement compatibilities are derived by the single-valuedness of displacements in multiply-connected region. Based on the strain and rotation of the line element on the boundary, the displacement boundary conditions are deduced in forms of the Airy stress function. Furthermore, the displacement conditions exclude pure rigid body motion, and are in the derivative forms instead of the integral ones. The interpolation equations are also reconstructed by the single-valuedness conditions of the displacements and the approximate solutions in a multiply-connected region. The numerical examples show that the proposed method has good effect and keeps high accuracy in all kinds of boundary conditions.