Research Article  Open Access
The Elastic Solution of a Radial Heterogeneous Cylinder Subjected to NonUniform Distributed Normal and Tangential Loads
Abstract
This paper presents an investigation of a cylinder with material attributes that vary arbitrarily in the radial direction under nonuniformly distributed normal and tangential pressures. A stress and displacement solution method for plane strain on a radial heterogeneous cylinder with inner and outer surfaces subjected to uneven normal and tangential pressures under two types of contact conditions (i.e., complete contact and smooth contact) is proposed based on the theory of elastic mechanics. In addition, a group of linear equations is derived, through which the analytical solution of a radial heterogeneous cylinder under a nonuniform load can be obtained. The validity of the analytical solution is verified by comparing it with the results of a numerical simulation. This research shows that the solution is more convenient and universal than traditional methods and can provide a theoretical basis for the stress state analysis, stability evaluation, and structure design of heterogeneous cylindrical structures, all of which have great significance in practical engineering.
1. Introduction
Cylinders are important structures that are widely used in civil and mining engineering in addition to other engineering fields, and they are commonly implemented in machinery and national defense applications. Therefore, research on the stress and deformation of cylinders in their applications (e.g., for shaft linings and tunnels) has important value for cylinder design and safety evaluations.
Cylinders subjected to uniform loads have been studied from many perspectives. The earliest known research on such cylinders produced the LamĆ© solution [1], which characterizes the stresses and deformation of elastic homogeneous cylinders under both internal and external uniform pressures; consequently, the LamĆ© solution has been widely utilized for cylinder design [2, 3]. In addition, numerous types of cylinders, including multilayer cylinders [4ā6] and cylinders made of composite materials [7], particularly functional gradient cylinders [8], have been developed to cope with extreme working conditions (e.g., high temperatures, high pressures, and corrosion) [9, 10]. Shi [11] obtained the exact solution for Nlayer elastic cylinders and cylinders comprising functional gradient materials by deriving a hypergeometric function. Furthermore, Sollund [12] established a group of recursive formulas and obtained the stress and deformation solution of Nlayer cylinders under the joint action of uniform pressure and temperature loads. Horgan [13] performed a mechanical analysis of a functional gradient cylinder with an elastic modulus that radially varied as a power function under internal and external pressures and discussed the differences in the stress distributions between the actions of both the internal and external pressures. In addition, Lu [14] conducted an inversion analysis of a functionally graded material cylinder under the assumption that the stress state at every point on the cylinder was either or and consequently developed an analytical solution for Youngās modulus that varies with the radius.
However, most cylinders in practical engineering applications are subjected to nonaxisymmetric loads. For instance, shaft linings and shield tunnel linings experience nonaxisymmetric loads due to variations in geological conditions and construction technologies. Uneven pressures can lead to tensile stresses in linings that are very dangerous for concrete structures [15ā21]. For example, oil well casings installed within salt rock and mudstone are more prone to instabilities and loose bearing capacities as a consequence of rock creep than to the external water pressure because rock creep applies a nonuniform load to the casing [22ā24]. In addition, the Couette flow of a viscous fluid between two cylinders applies nonuniform normal and tangential pressures on their surfaces. Therefore, it is necessary to analyze the stress and deformation of cylindrical structures under nonuniform loads. Some textbooks and related studies have provided analytical solutions for elastic homogeneous cylinders under nonuniform loads [25, 26], but little research has been performed on inhomogeneous cylinders. For example, Greenberg [27] investigated the static buckling of orthotropic composite cylindrical shells subjected to circumferentially nonuniform loads based on Fluggetype field equations. Furthermore, by assuming that the elastic modulus and the coefficient of thermal expansion are both power functions with radial coordinates, Jabbari analyzed a linear thermoelastic problem involving a functionally graded cylinder subjected to an internal nonuniform load. Rad [28] analyzed the twodimensional steadystate thermal stresses on a hollow, thick cylinder made of functionally graded materials and obtained a semianalytical solution. Alternatively, Liew [29] divided a functional gradient cylinder into a number of thin homogeneous cylinders and employed the semiinverse method to derive a solution for the thermal stresses on functional gradient cylinders under a nonuniform load that varies in circular coordinates. Batra [30] obtained the analytical solution for an incompressible functional gradient eccentric cylinder subjected to a nonuniform load and analyzed the stress and deformation variations with changes in the shear modulus according to a power function and an exponential function in the radial direction. Moreover, Li [31] determined the stress and displacement fields for a cylinder with mechanical parameters that arbitrarily changed in the radial direction under a nonuniform radial load using the theory of complex functions.
The abovementioned analytical solutions have achieved notable progress; however, those solutions are excessively complex and do not consider the comprehensive effects of different loads. Therefore, this paper proposes a mechanical model of a radial heterogeneous cylinder subjected to a new type of nonuniform load. The stress and displacement solution of the model is deduced in consideration of the contact conditions between osculant layers (complete contact and smooth contact) using elastic mechanics. In this paper, denotes the stress; and are the radial displacement and circumferential displacement, respectively; the subscripts , and represent the radial, circumferential, and shearing components, respectively; and the subscript (=1, 2, 3,āāā¦, n; as below) indicates the number of the layers in the cylinder.
2. Mechanical Model
2.1. Nondimensionalization of Parameters
To generalize the solution, we select appropriate normalized variables to transform the parameters into dimensionless forms. Regular variables with units are expressed with a symbol.
The load is used to scale the relevant stress and pressure, and the inner radius of each layer is used to scale the displacement. The load is selected according to the conditions of different problems. For example, the vertical stress at the distal end of the formation is obtained by analyzing the shaft well or the tunnel lining.
and are the dimensionless loads.
, , and represent the dimensionless forms of the stresses, and represents the dimensionless form of the shear modulus .
and are the dimensionless displacements.
() and () are nondimensional coordinates.
2.2. Mechanical Model
The pressure acting on the external cylinder can be expressed using a Fourier series dependent on the circumferential coordinates.
This study focuses on both parameter, which represents the uniform force that is selfbalanced since the resulting value along the circumference is zero, and the cosine term, which is examined because of the similarity between the sine and cosine terms in the above equation. Moreover, the cosine terms are different; through analysis, the cosine term is selfbalanced when , but the cosine term cannot balance itself when . Therefore, the distribution of the nonuniform load on the external cylinder is expressed as follows:where and reflect the nonuniform distributed normal load and tangential load, respectively, and , , and are the uniform and nonuniform distributed normal loads and the uneven tangential load, respectively. The tensile stress and outward radial displacement are positive according to the sign convention shown in Figure 1.
Figure 1 illustrates a force model of a radially inhomogeneous cylinder subjected to loads p and q. The radially inhomogeneous cylinder is regarded as an nlayered cylinder composed of several homogeneous and isotropically linear elastic layers. The inner and outer radii of the ith layer are and , respectively. The elastic modulus and Poissonās ratio are and , respectively. The loads on the inner and outer surfaces are and and and , respectively.
3. Model Solution
3.1. SingleLayer Model Solution
Mechanical model I can be decomposed into many singlelayer models (see Figure 2).
The analytic solution of the singlelayer model is given as follows [15ā17]: where , , , through are unknown coefficients, is the nondimensional radial coordinate, , and .
The stress boundary conditions of the singlelayer model include , , , and . As these stress boundary conditions are applicable to arbitrary angles, they can be solved simultaneously with (4) and (6). The following six equations can therefore be obtained.
Furthermore, coefficients through can be determined by unifying the group of equations: where , , , , , , , and .
The dimensionless stress and displacement solutions for the singlelayer model can be obtained by substituting (10) into (4)  (8).
3.2. Solution for Mechanical Model I
Each individual layer in mechanical model I is normally subjected to one of two limited contact conditions, namely, a complete contact or a smooth contact.
3.2.1. Complete Contact
In the case of complete contact, both the stresses and the displacements at the contact interface are continuous. Hence, according to the compatibility conditions of the displacements, the following equations are valid:where j=1, 2, 3,āāā¦, n1. Because the above boundary conditions are applicable to arbitrary angles , they can be simultaneously determined with (7)  (8). Thus, the following equations can be obtained.
Arranging (13)  (15), we can derive the following formulas:where , , , ,
Equations (16)(18) comprise a degree linear equation group with variables , , ,āāā¦, , , and . These equations can be directly solved using mathematical calculation software. Then, substituting the results into the stress and displacement expressions of mechanical model I, the analytical solutions of stress and displacement for the complete contact condition can be obtained.
3.2.2. Smooth Contact
In the case of smooth contact, the normal stresses and radial displacements at the contact interface are continuous, and the tangential stress is zero.
Therefore, the undetermined coefficients can be determined using (16)(18); moreover, under this condition, and . Similarly, substituting these coefficients into the stress and displacement expressions of mechanical model I, the analytical solutions of stress and displacement for the smooth contact condition can be obtained.
4. Engineering Example
The distributions of the temperature field, elastic modulus, and Poisson's ratio of a horizontal circular section of a frozen wall at a depth of 150 m are shown in Figure 3. The inner and outer radii of the frozen wall are 4.2 m and 10.2 m, respectively. The parameters of characteristic points within the frozen wall model are summarized in Table 1. The soil gravity is 0.02 MN/m^{3}. The soil inside the frozen wall is excavated one layer at one time; therefore, the inner edge of the frozen wall is completely unloaded (, ). The external side of the frozen wall is subjected to a formation pressure with a lateral pressure coefficient of . The expressions for the stresses in polar coordinates are as follows.

The proposed solution method is used to analyze the distributions of the stress and displacement fields on the frozen wall.
The frozen wall is divided in the radial direction into a cylinder with 55 layers in complete contact according to the temperature distribution. Each layer is assumed to be a homogeneous isotropic elastic body. The flat part of the temperature distribution curve (6.8 m7.4 m) in the middle of the frozen wall is considered a single layer with a thickness of 0.6 m, an elastic modulus of 601 MPa, and a Poisson's ratio of 0.22. The thickness of each remaining layer is 0.1 m, and the elastic moduli and Poisson's ratio values are obtained by linear interpolating the data shown in Figure 3 and Table 1.
Substituting the values of the physical and mechanical parameters of each layer into (16)(18), the equations are then solved to obtain the forces and coefficients for every interface. Subsequently, by substituting the results into (4)(8), the stresses and displacements of each layer can be determined. Thus, the stress and displacement fields of the entire frozen wall can be obtained.
To verify the results of the analytical solution, a finite element model consisting of onequarter of the ice wall (0) in consideration of its symmetry is established. The finite mesh elements are shown in Figure 4, where MPa, and the boundary constraint and force conditions are as follows.(1)Left boundary: sliding support restriction, the displacement in the xdirection being zero(2)Lower boundary: sliding support restriction, the displacement in the ydirection being zero(3)Inner boundary: acting loads and (4)Outer boundary: acting loads and
Figures 5, 6, and 7 indicate the distributions of the radial stress, radial displacement, and tangential stress, respectively, along both the direction of the minimum earth stress () and the direction of the maximum in situ stress () on the frozen wall. The values of the scattered points represent the results of the finite element calculations, and the curves represent the results of the theoretical calculations.
Figure 5 demonstrates that the radial stress nonlinearly increases in the radial direction and gradually transforms into the in situ stress.
As shown in Figure 6, a slight variation in the radial displacement is observed in the radial direction. The displacement in the direction of the maximum in situ stress () is much greater than that in the direction of the minimum earth stress (). This result indicates that the frozen wall experiences larger external expansion in the direction of the maximum in situ stress than in other directions.
Based on Figure 7, where , the tangential stress in the direction of the minimum earth stress () is much higher than that in the direction of the maximum in situ stress (). In the former case, the tangential stress decreases nonlinearly in the radial direction; moreover, in the latter case, the tangential stress initially increases nonlinearly and then decreases gradually in the radial direction.
Since the results shown in Figures 5, 6, and 7 are consistent with the finite element calculations, the correctness of the theoretical solution is verified.
5. Conclusions
Radial heterogeneous cylinders subjected to asymmetric loads are often encountered in practical engineering. However, it is difficult to obtain the analytical solutions of the corresponding stress and displacement. Although previous studies have developed various solutions to this problem, those solutions are relatively complex and do not consider the comprehensive effects of different loads. Consequently, this paper proposes a stress and displacement solution method for the plane strain on a radial heterogeneous cylinder with inner and outer surfaces subjected to uneven normal and tangential pressures under two types of contact conditions (i.e., complete contact and smooth contact) based on the theory of elastic mechanics. In addition, a group of linear equations ((16)(18)) is derived, through which the analytical solution for a radial heterogeneous cylinder under a nonuniform load can be obtained. Then, a finite element model of a horizontal circular frozen wall subjected to an uneven formation pressure is constructed to verify the correctness of the analytical solution, and the results indicate that the analytical solution is relatively universal and can be applied to realistic engineering applications involving cylindrical structures. Accordingly, the study provides a theoretical basis for the stress state analysis, stability evaluation, and structural design of cylindrical structures, all of which have great significance in practical engineering.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was financially supported by the National Hightech R&D Program of China (863 Program; no. 2012AA06A401) and the National Natural Science Foundation of China (nos. 41472224 and 41501075).
References
 G. LamĆ© and B. Clapeyron, āMĆ©moire sur l'Ć©quilibre intĆ©rieur des corps solides homogĆØnes,ā Journal fĆ¼r die reine und Angewandte Mathematik, vol. 7, pp. 145ā169, 1831. View at: Publisher Site  Google Scholar  MathSciNet
 ŠŖ. Š. ŠŠ°Š»ŠµŃŠŗŠøŠ½, āŠŠ°Š»ŃŃŠ¶ŠµŠ½Š½Š¾ŠµĀ Ā ŃŠ¾ŃŃŠ¾ŃŠ½ŠøŠµĀ Ā ŃŠøŠ» ŠøŠ½Š“ŃŠøŃŠµŃŠŗŠ¾Š¹Ā Ā ŃŃŃŠ±ŃĀ Ā Š²Ā Ā ŃŠæŃŃŠ³Š¾Š¹Ā Ā ŃŃŠµŠ“Šµ,ā ŃŃŃŠ“ŃĀ Ā Š»ŠøŠæŃ, Russia, 100, 1929. View at: Google Scholar
 D. C. Liu and Y. Zhao, āStability analysis of cylindrical shells under external pressures by half analytical finite element method,ā Engineering Mechanics, vol. 9, no. 1, pp. 104ā114, 1992. View at: Google Scholar
 A. P. Parker, āAutofrettage of openend tubespressures, stresses, strains, and code comparisons,ā Journal of Pressure Vessel Technology, Transactions of the ASME, vol. 123, no. 3, pp. 271ā281, 2001. View at: Publisher Site  Google Scholar
 M. Ghannad and M. Z. Nejad, āElastic analysis of heterogeneous thick cylinders subjected to internal or external pressure using shear deformation theory,ā Acta Polytechnica Hungarica, vol. 9, no. 6, pp. 117ā136, 2012. View at: Google Scholar
 H. Jahed, B. Farshi, and M. Karimi, āOptimum autofrettage and shrinkfit combination in multilayer cylinders,ā Journal of Pressure Vessel Technology, Transactions of the ASME, vol. 128, no. 2, pp. 196ā200, 2006. View at: Publisher Site  Google Scholar
 M. Schwartz, New Materials, Processes, And Methods Technology, Boca Raton: CRC Press, 2005.
 Y. L. J. Song and Z. M. Zhang, Functionally Graded Mechanics, Tsinghua University Press, China, 2003.
 L. M. Smith, Engineering with Clad Steel, Technical series no. 10064, The Nickel Institute, Brussels, Belgium, 2nd edition, 2012.
 K. Vedeld, H. Osnes, and O. Fyrileiv, āAnalytical expressions for stress distributions in lined pipes: Axial stress and contact pressure interaction,ā Marine Structures, vol. 26, no. 1, pp. 1ā26, 2012. View at: Publisher Site  Google Scholar
 Z. Shi, T. Zhang, and H. Xiang, āExact solutions of heterogeneous elastic hollow cylinders,ā Composite Structures, vol. 79, no. 1, pp. 140ā147, 2007. View at: Publisher Site  Google Scholar
 H. A. Sollund, K. Vedeld, and J. Hellesland, āEfficient analytical solutions for heated and pressurized multilayer cylinders,ā Ocean Engineering, vol. 92, pp. 285ā295, 2014. View at: Publisher Site  Google Scholar
 C. O. Horgan and A. M. Chan, āThe pressurized hollow cylinder or disk problem for functionally graded isotropic linearly elastic materials,ā Journal of Elasticity: The Physical and Mathematical Science of Solids, vol. 55, no. 1, pp. 43ā59, 1999. View at: Publisher Site  Google Scholar  MathSciNet
 A. Lu and N. Zhang, āInversion of functionally graded materials to improve the elastic ultimate bearing capacity of thickwalled hollow cylinder,ā Advanced Materials Research, vol. 378379, pp. 116ā120, 2012. View at: Publisher Site  Google Scholar
 A.Z. Lu, N. Zhang, Y.Q. Xu, and P. Cui, āStressdisplacement solution for a lined vertical borehole due to nonaxisymmetric in situ stresses,ā International Journal of Rock Mechanics and Mining Sciences, vol. 57, pp. 64ā74, 2013. View at: Publisher Site  Google Scholar
 C. CarranzaTorres, B. Rysdahl, and M. Kasim, āOn the elastic analysis of a circular lined tunnel considering the delayed installation of the support,ā International Journal of Rock Mechanics and Mining Sciences, vol. 61, pp. 57ā85, 2013. View at: Publisher Site  Google Scholar
 Y. Wang and W. Yang, āElastic stress analysis on circular frozen wall in bidirectional unequal ground stress field under unloading condition,ā Caikuang yu Anquan Gongcheng Xuebao/Journal of Mining and Safety Engineering, vol. 33, no. 3, pp. 486ā493, 2016. View at: Google Scholar
 E. Detournay and C. M. St. John, āDesign charts for a deep circular tunnel under nonuniform loading,ā Rock Mechanics and Rock Engineering, vol. 21, no. 2, pp. 119ā137, 1988. View at: Publisher Site  Google Scholar
 W. MingBin and L. ShuChai, āA complex variable solution for stress and displacement field around a lined circular tunnel at great depth,ā International Journal for Numerical and Analytical Methods in Geomechanics, vol. 33, no. 7, pp. 939ā951, 2009. View at: Publisher Site  Google Scholar
 Q.L. Wu, A.Z. LĆ¼, Y.T. Gao, S.C. Wu, and N. Zhang, āStress analytical solution for plane problem of a doublelayered thickwalled cylinder subjected to a type of nonuniform distributed pressure,ā Journal of Central South University, vol. 21, no. 5, pp. 2074ā2082, 2014. View at: Publisher Site  Google Scholar
 C.L. Hu, J. Xu, and C. Xiao, āStress analysis of thick cylinder supporting structure under eccentric load,ā Yantu Lixue/Rock and Soil Mechanics, vol. 33, no. 7, pp. 2117ā2121, 2012. View at: Google Scholar
 J. H. Nester and D. R. Jenkins, āResistances to Failure of Oilwell Casing Subjected to Nonuniform Transverse Loading,ā Drilling and Production Practice, vol. 55, pp. 374ā378, 1955. View at: Google Scholar
 F. Khalaf and U. Cairo, āIncreasing Casing Collapse Resistance Against SaltInduced Loads,ā in Proceedings of the Middle East Oil Technical Conference and Exhibition, vol. 13712, pp. 259ā264, 1985. View at: Publisher Site  Google Scholar
 A. ElSayed, F. Khalaf, and U. Cairo, āResistance of Cemented Concentric Casing Strings Under Nonuniform Loading,ā SPE Drilling Engineering, vol. 17927, no. 01, pp. 59ā64, 2013. View at: Publisher Site  Google Scholar
 S. P. Timoshenko and J. N. Goodier, Theory of elasticity, McGrawHill, New York, NY, USA, 1970.
 N. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoof Ltd., Groningen, The Netherlands, 1963.
 J. B. Greenberg and Y. Stavsky, āBuckling of composite orthotropic cylindrical shells under nonuniform axial loads,ā Composite Structures, vol. 30, no. 4, pp. 399ā406, 1995. View at: Publisher Site  Google Scholar
 A. Behravan Rad, āThermoelastic analysis of functionally graded circular plates resting on a gradient hybrid foundation,ā Applied Mathematics and Computation, vol. 256, pp. 276ā298, 2015. View at: Publisher Site  Google Scholar  MathSciNet
 K. M. Liew, S. Kitipornchai, X. Z. Zhang, and C. W. Lim, āAnalysis of the thermal stress behaviour of functionally graded hollow circular cylinders,ā International Journal of Solids and Structures, vol. 40, no. 10, pp. 2355ā2380, 2003. View at: Publisher Site  Google Scholar
 R. C. Batra and G. J. Nie, āAnalytical solutions for functionally graded incompressible eccentric and nonaxisymmetrically loaded circular cylinders,ā Composite Structures, vol. 92, no. 5, pp. 1229ā1245, 2010. View at: Publisher Site  Google Scholar
 H. Li and Y. Liu, āFunctionally graded hollow cylinders with arbitrary varying material properties under nonaxisymmetric loads,ā Mechanics Research Communications, vol. 55, pp. 1ā9, 2014. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2018 Wen Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.