scholarly journals A Shell Description of Beams Incorporating Transverse Thickness Strain Energies for Receding Contact

2020 ◽  
Vol 87 (5) ◽  
Author(s):  
Adam R. Brink ◽  
Allen T. Mathis ◽  
D. Dane Quinn
Keyword(s):  
Shell Theory ◽  
Weak Form ◽  
Momentum Balance ◽  
Rigid Surface ◽  
Thickness Strain ◽  
Cambridge University ◽  
Receding Contact ◽  
Order Representation ◽  
Shell Equations ◽  
Geometrically Exact Shell

Abstract The geometrically exact nonlinear deflection of a beamshell is considered here as an extension of the formulation derived by Libai and Simmonds (1998, The Nonlinear Theory of Elastic Shells, Cambridge University Press, Cambridge, UK) to include deformation through the thickness of the beam, as might arise from transverse squeezing loads. In particular, this effect can lead to receding contact for a uniform beamshell resting on a smooth, flat, rigid surface; traditional shell theory cannot adequately such behavior. The formulation is developed from the weak form of the local equations for linear momentum balance, weighted by an appropriate tensor. Different choices for this tensor lead to both the traditional shell equations corresponding to linear and angular momentum balance, as well as the additional higher-order representation for the squeezing deformation. In addition, conjugate strains for the shell forces are derived from the deformation power, as presented by Libai and Simmonds. Finally, the predictions from this approach are compared against predictions from the finite element code abaqus for a uniform beam subject to transverse applied loads. The current geometrically exact shell model correctly predicts the transverse shell force through the thickness of the beamshell and is able to describe problems that admit receding contact.

A Review of Tubular Conical Transition Strength Design Equations

2020 ◽  
Author(s):  
Albert Ku ◽  
Jieyan Chen ◽  
Bernard Cyprian
Keyword(s):  
Thin Shell ◽  
Shell Theory ◽  
Yield Criterion ◽  
Plasticity Theory ◽  
Transition Strength ◽  
Ultimate Capacity ◽  
Design Standards ◽  
Fem Simulations ◽  
Column Type ◽  
Shell Equations

Abstract This paper consists of two parts. Part one presents a thin-shell analytical solution for calculating the conical transition junction loads. Design equations as contained in the current offshore standards are based on Boardman’s 1940s papers with beam-column type of solutions. Recently, Lotsberg presented a solution based on shell theory, in which both the tubular and the cone were treated with cylindrical shell equations. The new solution as presented in this paper is based on both cylindrical and conical shell theories. Accuracies of these various derivations will be compared and checked against FEM simulations. Part 2 of this paper is concerned with the ultimate capacity equations of conical transitions. This is motivated by the authors’ desire to unify the apparent differences among the API 2A, ISO 19902 and NORSOK design standards. It will be shown that the NORSOK provisions are equivalent to the Tresca yield criterion as derived from shell plasticity theory. API 2A provisions are demonstrated to piecewise-linearly approximate this Tresca yield surface with reasonable consistency. The 2007 edition of ISO 19902 will be shown to be too conservative when compared to these other two design standards.

On the theoretical foundations of thin solid and liquid shells

2016 ◽  
Vol 22 (3) ◽  
pp. 343-371 ◽  
Author(s):  
Roger A Sauer ◽  
Thang X Duong
Keyword(s):  
Weak Form ◽  
Material Behavior ◽  
Momentum Balance ◽  
Power Balance ◽  
Equilibrium Equations ◽  
Material Stability ◽  
Out Of Plane ◽  
Mechanical Dissipation ◽  
Theoretical Foundations ◽  
Unified Description

This paper gives a concise summary of the general theoretical framework suitable to describe shells with solid-like and liquid-like behavior. Thin-shell kinematics are considered and used to derive the equilibrium equations from linear- and angular-momentum balance. Based on the mechanical power balance and the mechanical dissipation inequality, the constitutive equations for the hyperelastic material behavior of constrained shells are derived and their material stability is examined. Various constitutive examples are considered and assessed for their stability. The governing weak form of the formulation is derived and decomposed into in-plane and out-of-plane components. The presented work provides a very general framework for a unified description of solid and liquid shells and illustrates what leads to their loss of material stability. This framework serves as a basis for developing computational shell formulations based on rotation-free shell discretizations. Therefore the full linearization of the formulation is also presented here.

A Weak Form Implementation of Nonlinear Axisymmetric Shell Equations With Examples

2019 ◽  
Vol 86 (12) ◽  
Author(s):  
Matteo Pezzulla ◽  
Pedro M. Reis
Keyword(s):  
Weak Form ◽  
Surface Shape ◽  
Energy Potential ◽  
Spherical Shells ◽  
Axisymmetric Shell ◽  
Axisymmetric Shells ◽  
Shell Equations ◽  
Thickness Variations ◽  
Nonlinear Deformations ◽  
Dynamic Community

Abstract We present a weak form implementation of the nonlinear axisymmetric shell equations. This implementation is suitable to study the nonlinear deformations of axisymmetric shells, with the capability of considering a general mid-surface shape, non-homogeneous (axisymmetric) mechanical properties and thickness variations. Moreover, given that the weak balance equations are arrived to naturally, any external load that can be expressed in terms of an energy potential can, therefore, be easily included and modeled. We validate our approach with existing results from the literature, in a variety of settings, including buckling of imperfect spherical shells, indentation of spherical and ellipsoidal shells, and geometry-induced rigidity (GIR) of pressurized ellipsoidal shells. Whereas the fundamental basis of our approach is classic and well established, from a methodological view point, we hope that this brief note will be of both technical and pedagogical value to the growing and dynamic community that is revisiting these canonical but still challenging class of problems in shell mechanics.

Recommended Stress Formulae for Tubular Conical Transition Designs

2020 ◽  
Vol 143 (2) ◽  
Author(s):  
Albert Ku ◽  
Jieyan Chen
Keyword(s):  
Stress Concentration ◽  
Stress Concentration Factor ◽  
Shell Theory ◽  
Plate Thickness ◽  
Design Standards ◽  
Shell Equations ◽  
Offshore Industry ◽  
Hoop Stresses ◽  
Code Provisions

Abstract For the design of tubular conical transitions, the axial, bending, and hoop stresses at the junctions are required. Among the offshore design standards, API RP-2A, ISO 19902, and NORSOK N-004, various equations exist for the same stress quantity which may cause confusions. The quality of these existing stress formulae will be examined in this paper. The tubular conical stress equations used in the offshore industry started from Boardman’s studies in the 1940s. Recently, Lotsberg re-formulated this problem and applied the results to stress concentration factor (SCF) applications. This paper solves the same set of shell equations but the formulations are cast in a different form. This new format allows for an in-depth examination of existing code equations. In addition, the formulation as presented can be used for modifications to gain higher accuracy. Several recommended new stress formulae are provided. It is observed that the existing code provisions’ accuracy quickly deteriorates for cases where plate thickness in tubular and cone differ. The recommended approach is based on theoretical framework of shell mechanics, which better facilitate tubular/cone force balances when compared with existing equations. The sectional relationships among moment, shear, and hoop loads are also treated consistently using shell theory. The resulted improvements make the recommended formulae more accurate than the existing provisions.

On Boundary Layers in a Pressurized Mooney Cylinder

1987 ◽  
Vol 54 (2) ◽  
pp. 280-286 ◽  
Author(s):  
L. A. Taber
Keyword(s):  
Boundary Layer ◽  
Shell Theory ◽  
Circular Cylindrical Shell ◽  
Small Strain ◽  
Axisymmetric Deformation ◽  
Transverse Shear Deformation ◽  
Edge Zone ◽  
Secondary Layer ◽  
Shell Equations ◽  
Pressurized Cylinder

Asymptotic expansions are developed for the equations governing large axisymmetric deformation of a circular cylindrical shell composed of a Mooney material. The shell equations allow large normal strains and thickness changes but ignore transverse shear deformation. For a pressurized cylinder with rigid end plugs, results are presented to illustrate the development of a primary and a secondary boundary layer as generalizations of those that occur in small-strain shell theory. The form of the WKB-type expansion divides the secondary layer into bending and stretching components, which lie within the wider primary boundary layer. While the bending component of the secondary layer can become significant when strains are still small, the stretching component emerges as a consequence of large geometry changes in the edge zone, becoming significant as strains grow large and material nonlinearity becomes important.

Impulsive Deformation of Mirsky-Herrmann’s Thick Cylindrical Shells by a Numerical Method

1973 ◽  
Vol 40 (4) ◽  
pp. 1009-1016 ◽  
Author(s):  
M. Ziv ◽  
M. Perl
Keyword(s):  
Shell Theory ◽  
Pulse Velocity ◽  
Normal Strain ◽  
Finite Difference Technique ◽  
Characteristics Method ◽  
Transverse Normal Stress ◽  
Infinite Cylindrical Shell ◽  
Shell Equations ◽  
Thickness Parameter ◽  
Transverse Normal Strain

The transient response of a thick elastic semi-infinite cylindrical shell subjected to impulsive step loads is obtained. This work presents solutions to two boundary-value problems. First, the shell is exposed to an axially step pulse velocity load while second, the pulse is applied radially to the shell. The influence of the transverse normal stress and the transverse normal strain on the deformation is being studied in great detail. The shell theory employed is based on the thick-shell equations derived by Mirsky and Herrmann, which comprise these transversal effects. These equations are solved by the characteristics method, while integration is carried out by the finite-difference technique. The results also present the influence of the thickness parameter (h/R) on the solution. Comparisons are made to solutions of other shell theories which neglect the transversal effects. Major conclusions show the existence of an important influence of the transverse normal strain on the deformation.

On a consistent dynamic finite-strain shell theory and its linearization

2018 ◽  
Vol 24 (8) ◽  
pp. 2335-2360 ◽  
Author(s):  
Zilong Song ◽  
Jiong Wang ◽  
Hui-Hui Dai
Keyword(s):  
Shell Theory ◽  
Finite Strain ◽  
Circular Cylindrical Shell ◽  
Shell Element ◽  
Momentum Balance ◽  
Orthotropic Materials ◽  
Bottom Surface ◽  
Position Vector ◽  
Field Equations ◽  
Order Error

In this paper, a dynamic finite-strain shell theory is derived, which is consistent with the three-dimensional (3-D) Hamilton’s principle with a fourth-order error under general loadings. A series expansion of the position vector about the bottom surface is adopted. By using the bottom traction condition and the 3-D field equations, the recursive relations for the expansion coefficients are successfully obtained. As a result, the top traction condition leads to a vector shell equation for the first coefficient vector, which represents the local momentum-balance of a shell element. Associated weak formulations, in connection with various boundary conditions, are also established. Furthermore, the derived equations are linearized to obtain a novel shell theory for orthotropic materials. The special case of isotropic materials is considered and comparison with the Donnell–Mushtari (D-M) shell theory is made. It can be shown that, to the leading order, the present shell theory agrees with the D-M theory for statics. Thus, the present shell theory actually provides a consistent derivation for the former one without any ad hoc assumptions. To test the validity of the present dynamic shell theory, the free vibration of a circular cylindrical shell is studied. The results for frequencies are compared with those of the 3-D theory and excellent agreements are found. In addition, it turns out that the present shell theory gives better results than the Flügge shell theory (which is known to provide the best frequency results among the first-approximation shell theories).

An Upper Bound on the Error in Linear Quasi-Shallow Shell Theory

2008 ◽  
Vol 75 (1) ◽  
Author(s):  
J. G. Simmonds
Keyword(s):  
Upper Bound ◽  
Shallow Shell ◽  
Shell Theory ◽  
Elastic Shell ◽  
Energy Norm ◽  
Shell Equations

For a completely clamped elastic shell, an explicit upper bound is derived for the error in the energy norm of a solution of the linear, quasi-shallow shell equations as compared to the corresponding solution of the Sanders-Koiter equations.

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