Two-dimensional Shell Theories Research Articles

Legendre-based node-dependent kinematics shell models for the global–local analysis of homogeneous and layered structures

The present work demonstrates the use of the node-dependent kinematics method to derive and compare several two-dimensional shell theories. The three dimensional displacement field is expressed in terms of generalized coordinates, which are subsequently expanded along the shell thickness using arbitrary functions. The in-plane unknowns, are then discretized through classical finite element approximation. Based on the Carrera Unified Formulation, the proposed method combines in a unique manner the theory of structures and the finite element method; thickness interpolation functions are defined node-wise. As a consequence, the resulting finite element model represents diverse approximation theories at each single node. In this work Taylor-based kinematics (including the Murakami Zig-Zag function) and Legendre-type nodal kinematics are incorporated at the element level without adopting mathematical artifices leading to the global–local strategy, where refined theories are selectively employed in specific areas, while maintaining acceptable computational costs. Numerical cases from the existing literature are employed to establish the effectiveness of node-dependent models in bridging a locally refined theories to global kinematics when local effects need to be considered. The analyses focus on localized loads for both homogeneous and multi-layered structures.

Effects of Transverse Normal Stress on Hygrothermomechanical Analysis of Laminated Shallow Shells

Hygrothermomechanical analysis of doubly curved laminated composite shallow shells is presented in this paper using a two-dimensional higher-order shell theory considering the effects of transverse normal stress. The present theory accounts for the fifth-order expansion of the thickness coordinates in the in-plane displacements and the fourth-order expansion in the transverse displacement. The main focus of this work is to study the effects of both transverse normal stress and the fifth-order expansion of thickness coordinates on the hygrothermomechanical responses of doubly curved laminated shallow shells. The governing equations of the theory are derived based on the principle of virtual work and solved by using Navier’s method. The theory ensures traction-free boundary conditions using constitutive relations and does not require a shear correction factor. Interlaminar transverse shear stresses are recovered using the three-dimensional equilibrium equations of the theory of elasticity to ascertain the stress continuity at the layer interface. Three-dimensional elasticity solutions are taken as a basis for comparing the present results in thermal problems, whereas this study contributes some useful results to the hygrothermomechanical analysis of doubly curved laminated composite shallow shells.

Stress intensity factor for a circumferential crack in a coke drum under two-dimensional transient temperature distribution

Coke drums are subjected to extreme temperature cycling in service, leading to circumferential cracking in the drum wall. For a deeper understanding in the cracking mechanism, stress intensity factor (SIF) for a circumferential crack in a coke drum was investigated in this paper. Based on two-dimensional heat conduction theory and classical thin shell theory, analytical solutions for the transient temperature distribution, displacement, and thermal stress in the drum wall during hot feed filling and water quench steps were determined. The weight function method (WFM) was applied to investigate the mode I SIFs based on the analytical thermal stress solutions. Numerical simulations using finite element method (FEM) were also performed to validate the analytical solutions for both the thermos-elastic field and the SIFs. The effects of shell geometry, operating parameters and material properties on the SIF during the water quench step were discussed. It is shown that the SIF increases significantly as crack depth increases. A greater shell thickness or lower diameter to thickness ratio results in lower SIFs. Lower rising rate and higher temperature of quench water can reduce the SIF. The SIF increases as the thermal conductivity and thermal expansion coefficient of the drum wall materials increase. Finally, the SIFs for Cr–Mo steel and carbon steel coke drums are also discussed.

Analyzing FG shells with large deformations and finite rotations

Purpose The purpose of this study is dedicated to use an efficient mixed strain finite element approach to develop a three-node triangular shell element. Moreover, large deformation analysis of the functionally graded material shells is the main contribution of this research. These target structures include thin or moderately thick panels. Design/methodology/approach Due to reach these goals, Green–Lagrange strain formulation with respect to small strains and large deformations with finite rotations is used. First, an efficient three-node triangular degenerated shell element is formulated using tensorial components of two-dimensional shell theory. Then, the variation of Young’s modulus through the thickness of shell is formulated by using power function. Note that the change of Poisson’s ratio is ignored. Finally, the governing linearized incremental relation was iteratively solved using a high potential nonlinear solution method entitled generalized displacement control. Findings Some well-known problems are solved to validate the proposed formulations. The suggested triangular shell element can obtain the exact responses of functionally graded (FG) shell structures, without any shear locking, instabilities and ill-conditioning, even by using fewer numbers of the elements. The obtained outcomes are compared with the other reference solutions. All findings demonstrate the accuracy and capability of authors’ element for analyzing FG shell structures. Research limitations/implications A mixed strain finite element approach is used for nonlinear analysis of FG shells. These structures are curved thin and moderately thick shells. Small strains and large deformations with finite rotations are assumed. Practical implications FG shells are mostly made curved thin or moderately thick, and these structures have a lot of applications in the civil and mechanical engineering. Social implications The social implication of this study is concerned with how technology impacts the world. In short, the presented scheme can improve structural analysis ways. Originality/value Developing an efficient three-node triangular element, for geometrically nonlinear analysis of FG doubly-curved thin and moderately thick shells, is the main contribution of the current research. Finite rotations are considered by using the Taylor’s expansion of the rotation matrix. Mixed interpolation of strain fields is used to alleviate the locking phenomena. Using fewer numbers of shell elements with fewer numbers of degrees of freedom can reduce the computational costs and errors significantly.

Comparisons of Paraboloidal Shells and Sinusoidal-Shaped Shells in Natural Frequencies

Natural frequencies and mode shapes are obtained for a sinusoidal-shaped shell of revolution by using the Ritz method from a three-dimensional (3-D) analysis instead of a mathematically two-dimensional (2-D) thin shell theory or high order thick shell theory. The present analysis uses circular cylindrical coordinates instead of 3-D shell coordinates, which have been used in traditional shell analyses. Convergence studies can analyze the first five frequencies to four-digit exactitude. Results are given for a variety of shallow and deep sinusoidal-shaped shells with different boundary conditions. The sinusoidal-shaped shells are very similar to paraboloidal shells in shape. The frequencies of the sinusoidal-shaped shells from the present 3-D method are compared with those from 2-D thin shell theories for paraboloidal shells. The present 3-D method is applicable to very thick as well as thin shells.

3D Vibration Analysis of Combined Shells of Revolution

A three-dimensional (3D) method of analysis is presented for determining the natural frequencies and the mode shapes of combined hemispherical–cylindrical shells of revolution with and without a top opening by the Ritz method. Instead of mathematically two-dimensional (2D) conventional thin shell theories or higher-order thick shell theories, the present method is based upon the 3D dynamic equations of elasticity. Mathematically, minimal or orthonormal Legendre polynomials are used as admissible functions in place of ordinary simple algebraic polynomials which are usually applied in the Ritz method. The analysis is based upon the circular cylindrical coordinates instead of the shell coordinates which are normal and tangent to the shell mid-surface. Strain and kinetic energies of the combined shell of revolution with and without a top opening are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the Legendre polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. Numerical results are presented for the combined shells of revolution with or without a top opening, which are completely free and fixed at the bottom of the combined shells. The frequencies from the present 3D Ritz method are compared with those from 2D thin shell theories by previous researchers. The present analysis is applicable to very thick shells as well as very thin shells.

Three-dimensional vibration analysis of arbitrary angle-ply laminated cylindrical shells using differential quadrature method

On the basis of three-dimensional elasticity theory, semi-analytical solutions that combine the state space technique (SST) and differential quadrature method (DQM) were developed for the free vibration of arbitrary angle-ply laminated cylindrical shells with simply-simply, simply-clamped and clamped-clamped support boundary conditions. The SST was applied in the radial direction while the DQM was employed to discretize the longitudinal direction of the cylindrical shells. The derived equation of motion was solved for the free vibration of general cylindrical shells, including the isotropic, orthotropic, or monoclinic material behavior. The theoretical solutions were verified by comparing the reported results of two-dimensional shell theory in the literature and the finite element results from commercial software. The effects of layer angle on natural frequencies were also investigated.

Semi-analytical research on acoustic-structure coupling calculation of partially submerged cylindrical shell

Vibroacoustic analysis of a partially submerged cylindrical shell-liquid coupling system is a typical acoustic-structure interaction problem in an acoustic half-space. Generally, the vibration and acoustic solutions of this problem almost depend on numerical method in previous studies. However, whether from the aspect of verifying the numerical methods or from the aspect of revealing the vibroacoustic mechanism of the acoustic-structure interaction system, the development of an analytical or semi-analytical method is indispensable. In this study, a semi-analytical method is proposed to address the vibroacoustic response of a horizontal cylindrical shell partially immersed in water The acoustic coordinate system is established on the free surface, and the sine series function is introduced to express the sound pressure to meet the pressure release boundary condition on the free surface automatically. Then based on the two-dimensional Flgge shell theory, the motion equation of the shell-liquid coupling system is established with using the center of the shell as the origin of the coordinate system. By using the Galerkin method, the velocity continuous condition on the fluid-structure interface is established. Then the relation matrix between the acoustic pressure amplitude and the shell displacement amplitude is derived after converting the sound pressure from different coordinate systems into that in the same coordinate system by using the geometry relationship of two coordinate systems. Then the vibration and sound radiation of this coupling system are predicted by solving the coupled matrix equations, and the coupled free vibration can also be addressed by solving the characteristic equation after setting the determinant of the coefficient matrix to be zero. To verify the accuracy and reliability of the present method, the coupled natural frequencies, the vibration response and the distribution of the sound pressure obtained from the present method are compared with those obtained from the finite element method, showing that they are in good agreement with each other. Then the shapes of the first four order modes for the coupled system are presented and compared with those of the system submerged in unbounded fluid field, and the couple between different modes, and the couple between the symmetric and antisymmetric modes are observed due to the effect of the free surface. The characteristics of the root mean square velocity of the shell with different immersed depths are discussed, which reveals that the peaks of response curves shift to the lower frequencies with increasing immersed depth. The characteristics of far field sound pressure directivity are presented and explained by the image method in detail. This study provides a novel method to analytically predict the vibroacoustic response of an elastic structure partially coupling with the fluid field when bounded in a sound half space.

Vibrations of a composite shell of hemiellisoidal-cylindrical shell having variable thickness with and without a top opening

Natural frequencies and mode shapes of a composite shell of revolution consisting of a circular cylindrical shell and a deep or shallow hemi-ellipsoidal shell with variable thickness having a circular cylindrical hole or not are determined by the Ritz method using a mathematically three-dimensional analysis instead of two-dimensional thin shell theories or higher order thick shell theories. The present analysis is based upon the circular cylindrical coordinates while in the traditional shell analyses three-dimensional shell coordinates have been commonly used. Using the Ritz method, the Legendre polynomials, which are mathematically orthonormal, are used as admissible functions instead of ordinary simple algebraic polynomials. Natural frequencies are presented for different boundary conditions. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the composite shells. The frequencies from the present three-dimensional method are compared with those from three types of two-dimensional thin shell theories (finite element method, finite difference method, and numerical integration) by previous researchers. The present analysis is applicable to very thick shells as well as thin shells; and to shallow shells as well as deep shells.

Free vibration analysis of shallow and deep ellipsoidal shells having variable thickness with and without a top opening

A three-dimensional (3-D) method of analysis is presented for determining the natural frequencies and the mode shapes of hemi-ellipsoidal domes having non-uniform thickness with and without a top opening by the Ritz method. Instead of mathematically two-dimensional (2-D) conventional thin shell theories or higher-order shell theories, the present method is based upon the 3-D dynamic equations of elasticity by the Ritz method. Mathematically minimal or orthonormal Legendre polynomials are used as admissible functions in place of ordinary simple algebraic polynomials which are usually applied in the Ritz method. The analysis is based upon the circular cylindrical coordinates instead of the shell coordinates which are normal and tangential to the shell mid-surface. Potential (strain) and kinetic energies of the hemi-ellipsoidal dome having variable thickness with and without a top opening are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the Legendre polynomials is increased, the frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. Numerical results are presented for a variety of shallow and deep hemi-ellipsoidal domes having variable thickness of five values of aspect ratios with and without a top opening, which are completely free and fixed at the bottom. The frequencies from the present 3-D analysis are compared with those from other 3-D analysis and a 2-D thin shell theory.

Three-Dimensional Vibration Analysis of a Hermetic Capsule with Variable Thickness

The natural frequencies of a hermetic capsule of a circular cylinder closed with hemispheroidal caps at both ends and having variable thickness are determined by the Ritz method using a three-dimensional analysis. However, in the traditional shell analysis, mathematically two-dimensional thin-shell theories or higher-order thick-shell theories, which make very limiting assumptions about the displacement variation through the shell thickness, have usually applied. Although most researchers have used three-dimensional shell coordinates that are normal and tangent to the shell midsurface, the present analysis is based upon the circular cylindrical coordinates. Using the Ritz method, the Legendre polynomials, which are mathematically orthonormal, are used as admissible functions instead of ordinary simple algebraic polynomials. The potential and kinetic energies of the hermetic capsule are formulated, and upper bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the Legendre polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. The frequencies from the present three-dimensional method are compared with those from other three-dimensional approaches and two-dimensional shell theories by previous researchers. The present three-dimensional analysis is applicable to very thick shells as well as thin shells.

Vibration analysis of a circular cylinder closed with a hemi-spheroidal cap having a hole

The natural frequencies and mode shapes of a circular cylindrical shell closed with a hemi-spheroidal dome are determined by the Ritz method using a three-dimensional (3-D) analysis instead of two-dimensional (2-D) thin shell theories or higher-order thick shell theories. The present analysis is based upon the circular cylindrical coordinates, while in the traditional shell analyses 3-D shell coordinates have been usually used. Using the Ritz method, the Legendre polynomials, which are mathematically orthonormal, are used as admissible functions instead of ordinary simple algebraic polynomials. Natural frequencies are presented for different boundary conditions. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the combined shell. The frequencies from the present 3-D method are compared with those from 2-D thin shell theories. The present method is applicable to very thick shells as well as thin shells.

Vibrations of a cylindrical shell closed with a hemi-spheroidal dome from a three-dimensional analysis

The natural frequencies of a circular cylindrical shell closed with a hemi-spheroidal dome are determined by the Ritz method using a three-dimensional (3-D) analysis instead of two-dimensional (2-D) thin shell theories or higher-order thick shell theories. The present analysis is based upon the circular cylindrical coordinates,while in the traditional shell analyses 3-D shell coordinates have been usually used. Using the Ritz method, the Legendre polynomials, which are mathematically minimal or orthonormal, are used as admissible functions instead of ordinary simple algebraic polynomials. Natural frequencies are presented for different boundary conditions. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the combined shell. The frequencies from the present 3-D method are compared with those from 2-D thin shell theories. The present method is applicable to very thick shells as well as thin shells.

Free Vibration of Sandwich Plates and Shells by Using Zig-Zag Function

This paper analyses the free vibration response of sandwich curved and flat panels by introducing the zig-zag function (—1)kζk(ZZF) in the displacement models of classical and higher order two-dimensional shell theories. The main advantage of ZZF is the introduction of a discontinuity in the first derivative, zig-zag effect, of the displacements distribution with correspondence to the core/faces interfaces. Results including and discarding ZZF are compared. Several values of face-to-core stiffness ratio (FCSR) and geometrical plate/shell parameters have been analyzed. Both fundamental vibration modes and those corresponding to high wave numbers are considered in the analysis. It is concluded that: (1) ZZF is highly recommended in the free vibration analysis of sandwich plates and shells; (2) the use of ZZF makes the error almost independent by FCSR parameter; (3) ZZF is easy to implement and its use should be preferred with respect to other `more cumbersome' refined theories.

Free vibration of sandwich plates and shells by using Zig-Zag function

This paper analyses the free vibration response of sandwich curved and flat panels by introducing the zig-zag function (—1)k ζ k (ZZF) in the displacement models of classical and higher order two-dimensional shell theories. The main advantage of ZZF is the introduction of a discontinuity in the first derivative, zig-zag effect, of the displacements distribution with correspondence to the core/faces interfaces. Results including and discarding ZZF are compared. Several values of face-to-core stiffness ratio (FCSR) and geometrical plate/shell parameters have been analyzed. Both fundamental vibration modes and those corresponding to high wave numbers are considered in the analysis. It is concluded that: (1) ZZF is highly recommended in the free vibration analysis of sandwich plates and shells; (2) the use of ZZF makes the error almost independent by FCSR parameter; (3) ZZF is easy to implement and its use should be preferred with respect to other `more cumbersome' refined theories.

Closed-form solutions for the free-vibration problem of multilayered piezoelectric shells

This paper addresses the free-vibration problem of multilayered shells with embedded piezoelectric materials. A series of hierarchic, two-dimensional axiomatic shell theories are presented within the “Unified Formulation” introduced by the last author. Shells of constant curvature are considered, and no simplifying assumptions on the curvature terms are made in the geometric relations. Closed-form solutions are given for the free-vibration problem of simply supported, orthotropic piezoelectric laminates. The formulations are applied to study the influence of the electro-mechanical coupling on the resonant frequencies. It is demonstrated that the slenderness of plates with through-thickness polarized piezoelectric layers increases the electro-mechanical coupling. For comparison purposes, the fundamental axisymmetric mode of hollow cylinders has been exemplarily considered: with respect to flat plates, the thickness and the curvature of the shells have a less important effect on the piezoelectric coupling.

On the Interaction of Closely Spaced Circular Openings in a Transversely Isotropic Shallow Cylindrical Shell

A transversally isotropic shallow cylindrical shell with two different circular openings under axial tension is examined. The centers of the openings are aligned along a generatrix. Stresses are evaluated by finding an approximate analytical solution to differential equations of a two-dimensional shell theory that takes transverse shears into account. The results are compared with those obtained by the finite-element method

Elasticity Solutions for Laminated Orthotropic Cylindrical Shells Subjected to Localized Longitudinal and Circumferential Moments

The purpose of this work is to present baseline elasticity solutions for laminated composite shells subjected to localized moments. For simply supported cross-ply cylindrical shells, the problem reduces to one of coupled ordinary differential equations which are solved in terms of Taylor’s series. Results, in the form of tables and graphs, are presented for the cases of longitudinal and circumferential moments. These results would be very useful for judging the accuracy of approximate two-dimensional shell theories. They are used herein to study the errors of a shell theory based on the classical Love-Kirchhoff hypothesis.

Assessment of shell theories for hybrid piezoelectric cylindrical shell under electromechanical load

The assessment of classical lamination shell theory and first-order shear deformation theory is presented for simply supported finite circular cylindrical hybrid shell with cross-ply composite laminate as elastic substrate under electromechanical static load. Navier-type solutions are obtained and used in threedimensional equilibrium equations and transverse strain—displacement relation to obtain transverse stress components and improved value of deflection. These solutions are assessed by comparison with the threedimensional solution. The error in the two-dimensional shell theories increases as the shell becomes thicker and it is more for the patch loads in comparison to the uniformly distributed and sinusoidal loads.

FREE VIBRATION OF THICK CIRCULAR CYLINDRICAL SHELLS SUBJECTED TO AXIAL STRESSES

A two-dimensional higher-order shell theory is applied to the free vibration problems of a simply supported cylindrical shell subjected to axial stresses. The effects of higher-order deformations such as shear deformations with thickness changes and rotatory inertia on natural frequencies of thick elastic circular cylindrical shells are studied. Based on the power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher-order shell theory is derived through Hamilton's principle. Several sets of truncated approximate theories which can take into account the complete effects of higher-order deformations are applied to solve the vibration problems of a simply supported thick circular cylindrical shell. In order to assure the accuracy of the present theory, the convergence of the natural frequencies is examined and the results are compared with those obtained in existing theories.