Plate/shell structure topology optimization of orthotropic material for buckling problem based on independent continuous topological variables

The purpose of the present work is to study the buckling problem with plate/shell topology optimization of orthotropic material. A model of buckling topology optimization is established based on the independent, continuous, and mapping method, which considers structural mass as objective and buckling critical loads as constraints. Firstly, composite exponential function (CEF) and power function (PF) as filter functions are introduced to recognize the element mass, the element stiffness matrix, and the element geometric stiffness matrix. The filter functions of the orthotropic material stiffness are deduced. Then these filter functions are put into buckling topology optimization of a differential equation to analyze the design sensitivity. Furthermore, the buckling constraints are approximately expressed as explicit functions with respect to the design variables based on the first-order Taylor expansion. The objective function is standardized based on the second-order Taylor expansion. Therefore, the optimization model is translated into a quadratic program. Finally, the dual sequence quadratic programming (DSQP) algorithm and the global convergence method of moving asymptotes algorithm with two different filter functions (CEF and PF) are applied to solve the optimal model. Three numerical results show that DSQP&CEF has the best performance in the view of structural mass and discretion.

Key Techniques and Applications of Adaptive Growth Method for Stiffener Layout Design of Plates and Shells

The application of the adaptive growth method is limited because several key techniques during the design process need manual intervention of designers. Key techniques of the method including the ground structure construction and seed selection are studied, so as to make it possible to improve the effectiveness and applicability of the adaptive growth method in stiffener layout design optimization of plates and shells. Three schemes of ground structures, which are comprised by different shell elements and beam elements, are proposed. It is found that the main stiffener layouts resulted from different ground structures are almost the same, but the ground structure comprised by 8-nodes shell elements and both 3-nodes and 2-nodes beam elements can result in clearest stiffener layout, and has good adaptability and low computational cost. An automatic seed selection approach is proposed, which is based on such selection rules that the seeds should be positioned on where the structural strain energy is great for the minimum compliance problem, and satisfy the dispersancy requirement. The adaptive growth method with the suggested key techniques is integrated into an ANSYS-based program, which provides a design tool for the stiffener layout design optimization of plates and shells. Typical design examples, including plate and shell structures to achieve minimum compliance and maximum bulking stability are illustrated. In addition, as a practical mechanical structural design example, the stiffener layout of an inlet structure for a large-scale electrostatic precipitator is also demonstrated. The design results show that the adaptive growth method integrated with the suggested key techniques can effectively and flexibly deal with stiffener layout design problem for plates and shells with complex geometrical shape and loading conditions to achieve various design objectives, thus it provides a new solution method for engineering structural topology design optimization.

A NEW APPROACH OF PREDICTING BIFURCATION POINTS OF ELASTIC-PLASTIC BUCKLING OF PLATES AND SHELLS

The paper proposes a new approach of predicting the bifurcation points of elastic-plastic buckling of plates and shells,which is obtained from the natural combination of the Lyaponov's dy- namic criterion on stability and the modified adaptive Dynamic Relaxation(maDR)method developed recently by the authors.This new method can overcome the difficulties in the applications of the dy- namic criterion.Numerical results show that the theoretically predicted bifurcation points are in very good agreement with the corresponding experimental ones.The paper also provides a new means for further research on the plastic buckling paradox of plates and shells.

h-ADAPTIVE ANALYSIS BASED ON MESHLESS LOCAL PETROV-G ALERKIN METHOD WITH B SPLINE WAVELET FOR PLATES AND SHELLS

Using the two-scale decomposition technique, the h-adaptive meshless local Petrov- Galerkin method for solving Mindlin plate and shell problems is presented. The scaling functions of B spline wavelet are employed as the basis of the moving least square method to construct the meshless interpolation function. Multi-resolution analysis is used to decompose the field variables into high and low scales and the high scale component can commonly represent the gradient of the solution according to inherent characteristics of wavelets. The high scale component in the present method can directly detect high gradient regions of the field variables. The developed adaptive refinement scheme has been applied to simulate actual examples, and the effectiveness of the present adaptive refinement scheme has been verified.

Three Dimensional Coupling between Elastic and Thermal Fields in the Static Analysis of Multilayered Composite Shells

This new work aims to develop a full coupled thermomechanical method including both the temperature profile and displacements as primary unknowns of the model.This generic full coupled 3D exact shell model permits the thermal stress investigation of laminated isotropic,composite and sandwich structures.Cylindrical and spherical panels,cylinders and plates are analyzed in orthogonal mixed curved reference coordinates.The 3D equilibrium relations and the 3D Fourier heat conduction equation for spherical shells are coupled and they trivially can be simplified in those for plates and cylindrical panels.The exponential matrix methodology is used to find the solutions of a full coupled model based on coupled differential relations with respect to the thickness coordinate.The analytical solution is based on theories of simply supported edges and harmonic relations for displacement components and sovra-temperature.The sovra-temperature magnitudes are directly applied at the outer faces through static state hypotheses.As a consequence,the sovra-temperature description is assumed to be an unknown variable of themodel and it is calculated in the sameway as the three displacements.The final systemis based on a set of coupled homogeneous differential relations of second order in the thickness coordinate.This system is reduced in a first order differential relation system by redoubling the number of unknowns.Therefore,the exponential matrix methodology is applied to calculate the solution.The temperature field effects are evaluated in the static investigation of shells and plates in terms of displacement and stress components.After an appropriate preliminary validation,new benchmarks are discussed for several thickness ratios,geometrical data,lamination sequences,materials and sovra-temperature values imposed at the outer faces.Results make evident the accordance between the uncoupled thermo-mechanical model and this new full coupled thermo-mechanical model without the need to separately solve the Fourier heat conduction relation.Both effects connected with the thickness layer and the related embedded materials are included in the conducted thermal stress analysis.

Thermodynamic Consistency of Plate and Shell Mathematical Models in the Context of Classical and Non-Classical Continuum Mechanics and a Thermodynamically Consistent New Thermoelastic Formulation

Inclusion of dissipation and memory mechanisms, non-classical elasticity and thermal effects in the currently used plate/shell mathematical models require that we establish if these mathematical models can be derived using the conservation and balance laws of continuum mechanics in conjunction with the corresponding kinematic assumptions. This is referred to as thermodynamic consistency of the mathematical models. Thermodynamic consistency ensures thermodynamic equilibrium during the evolution of the deformation. When the mathematical models are thermodynamically consistent, the second law of thermodynamics facilitates consistent derivations of constitutive theories in the presence of dissipation and memory mechanisms. This is the main motivation for the work presented in this paper. In the currently used mathematical models for plates/shells based on the assumed kinematic relations, energy functional is constructed over the volume consisting of kinetic energy, strain energy and the potential energy of the loads. The Euler’s equations derived from the first variation of the energy functional for arbitrary length when set to zero yield the mathematical model(s) for the deforming plates/shells. Alternatively, principle of virtual work can also be used to derive the same mathematical model(s). For linear elastic reversible deformation physics with small deformation and small strain, these two approaches, based on energy functional and the principle of virtual work, yield the same mathematical models. These mathematical models hold for reversible mechanical deformation. In this paper, we examine whether the currently used plate/shell mathematical models with the corresponding kinematic assumptions can be derived using the conservation and balance laws of classical or non-classical continuum mechanics. The mathematical models based on Kirchhoff hypothesis (classical plate theory, CPT) and first order shear deformation theory (FSDT) that are representative of most mathematical models for plates/shells are investigated in this paper for their thermodynamic consistency. This is followed by the details of a general and higher order thermodynamically consistent plate/shell thermoelastic mathematical model that is free of a priori consideration of kinematic assumptions and remains valid for very thin as well as thick plates/shells with comprehensive nonlinear constitutive theories based on integrity. Model problem studies are presented for small deformation behavior of linear elastic plates in the absence of thermal effects and the results are compared with CPT and FSDT mathematical models.

ITERATIVE METHOD FOR LARGE DEFLECTION NONLINEAR PROBLEM OF LAMINATED COMPOSITE SHALLOW SHELLS AND PLATES

Based on the results by Wang,in this paper, the iterative method is presented for the study of large deflection nonlinear problem of laminated composite shallow shells and plates. The rectangular laminated composite shallow shells have been analyzed. The results have been compared with the small deflection linear analytical solution and finite element nonlinear solution. The results proved that the solution coincide with small deflection linear analytical solution in the condition of the low loads and finite element nonlinear solution in the condition of the high loads.

The deformation and failure mechanism of cylindrical shell and square plate with pre-formed holes under blast loading

The deformation and failure mechanism of cylindrical shells and square plate with pre-formed holes under blast loading were investigated numerically by employing the Ansys 17.0 and Ls-Dyna 971.To calibrate the numerical model,the experiments of square plates with pre-formed circle holes were modeled and the numerical results have a good agreement with the experiment data.The calibrated numerical model was used to study the deformation and failure mechanism of cylindrical shells with pre-formed circle holes subjected to blast loading.The structure response and stress field changing process has been divided into four specific stages and the deformation mechanism has been discussed systematically.The local and global deformation curves,degree of damage,change of stress status and failure modes of cylindrical shell and square plate with pre-formed circular holes are obtained,compared and analyzed,it can be concluded as:(1)The transition of tensile stress fields is due to the geometrical characteristic of pre-formed holes and cylindrical shell with arch configuration;(2)The existence of preformed holes not only lead to the increasing of stress concentration around the holes,but also release the stress concentration during whole response process;(3)There are three and two kinds of failure modes for square plate and cylindrical shell with pre-formed holes,respectively.and the standoff distance has a key influence on the forming location of the crack initiating point and the locus of crack propagation;(4)The square plate with pre-formed holes has a better performance than cylindrical shell on blast-resistant capability at a smaller standoff distance,while the influence of pre-formed holes on the reduction of blast-resistant capability of square plate is bigger than that of cylindrical shell.

ON THE STABILITY ANALYSIS OF PLATES AND SHELLS USING A QUADRILATERAL,16-DEGREES OF FREEDOM PLAT SHELL ELEMENT DKQ16

The linear buckling problems of plates and shells were analysed using a recently developped quadrilateral,16-degrees of freedom flat shell element (called DKQ16).The geometrical stiffness matrix was established.Comparison of the numerical results for several typical problems shows that the DKQ16 element has a very good precision for the linear buckling problems of plates and shells.

ANALYTICAL FORMULAS OF SOLUTIONS OF GEOMETRICALLY NONLINEAR EQUATIONS OF AXISYMMETRIC PLATES AND SHALLOW SHELLS

Starting from the step-by-step iterative method, the analytical formulas of solutions of the geometrically nonlinear equations of the axisymmetric plates and shallow shells, have been obtained. The uniform convergence of the iterative method, is used to prove the convergence of the analytical formulas of the exact solutions of the equa- tions.

The Hamilton System and Hamilton Type Generalized Variational Principle for the Laminated Composite Plates and Shells

By introducing the Hamilton theory and algorithms into the problems of laminated composite plates andshells, the Hamiltion type generalized variational principle is established, and the Hamilton canonical equations andthe boundary conditions for the static and elastoplastic analysis of composite plates are presented. With thetransformation of phase variables, the Hamilton canonical equations and their boundary conditions for thecylindrical shells and doubly curved shells in the curvilinear coordinate are given.

UNIQUENESS FOR THE SOLUTIONS OF ELASTIC THIN PLATES AND SHALLOW SHELLS AS WELL AS THEIR CONTACT PROBLEM WITH HALF SPACE

The uniqueness for the solutions mentioned in the subject is proved by using the uniqueness of the solution for the internal boundary problem of Laplace and bi-Laplace equations of the first kind as well as of the second.

A POTENTIAL-HYBRID/MIXED FINITE ELEMENT SCHEME FOR ANALYSIS OF PLATES AND CYLINDRICAL SHELLS

Based on the potential-hybrid/mixed finite element scheme, 4-node quadrilateral plate-bending elements MP4, MP4a and cylindrical shell element MCS4 are derived with, the inclusion of splitting rotations. All these elements demonstrate favorable convergence behavior over the existing counterparts, free from spurious kinematic mode's and do not exhibit locking phenomenon in thin platef shell limit. Inter-connections between the existing modified variational functionals for the use of formulating C0-and C1-continuous elements are also indicated. Important particularizations of the present scheme include Prathop's consistent field formulation, the RIT/SRIT-compatible displacement model and so on.

Stochastic Finite Element Analysis of Plate and Shell Construction

The response of random plate and shell construction is analyzed with the stochastic finite element method (SFEM). Random material properties and geometric dimensions of construction are involved in this paper. A simplified isoparametric local average model is used to describe the random field. Numerical results of the examples indicate that the approach presented herein is an economical and efficient solution for such an analysis compared with Monte Carlo simulation (MCS).

A new method determining safe thickness of karst cave roof under pile tip

Cave roofs are used to support pile foundation in many engineering projects. Accurate stability analysis method of cave roof under pile tip is important in order to ensure the safety of the pile foundation structure. Firstly the mechanical model to analysis the stability of cave roof under pile tip is founded aiming to solve the problems that the simplified mechanical model has. Secondly, the boundary of cave roof is simply supposed to be supported according to the integrity of the rock mass in the boundary of cave roof. Thirdly, based on the theory of plates and shells, the simplified model is calculated and the theoretical calculation formula to determine the safe thickness of cave roof under pile tip can be obtained when the edges of the cave roof are simply supported. In the end, the analysis of the practical engineering project proves the feasibility and the rationality of the method which can be a new method to calculate the safe thickness of cave roof under pile tip.

A FINITE ELEMENT ANALYSIS OF THE FLANGE EARRINGS OF STRONG ANISOTROPIC SHEET METALS IN DEEP-DRAWING PROCESSES

Flange earrings of strong anisotropic sheet metals in deep-drawing process are numerically analyzed by the elastic-plastic large deformation finite element formulation based on a discrete Kirchhoff triangle plate shell element model. A Barlat-Lian anisotropic yield function and a quasi-flow corner theory are used in the present formulation. The numerical results are compared with the experimental ones of cylindrical cup drawing process. The focus of the present researches is on the numerical analysis and the constraining scheme of the flange earring of circular sheets with strong anisotropy in square cup drawing process.

Analytic solutions of a class of nonlinear partial differential equations

An approach is presented for computing the adjoint operator vector of a class of nonlinear （that is, partial-nonlinear） operator matrices by using the properties of conjugate operators to generalize a previous method proposed by the author. A unified theory is then given to solve a class of nonlinear （partial-nonlinear and including all linear） and non-homogeneous differential equations with a mathematical mechanization method. In other words, a transformation is constructed by homogenization and triangulation, which reduces the original system to a simpler diagonal system. Applications are given to solve some elasticity equations.

An interlaminar damage shell model for typical composite structures

Using the plate/shell elements in commercial software,accurate analysis of interlaminar initial damage in typical composite structures is still a challenging issue.To propose an accurate and efficient model for analysis of interlaminar initial damage,the following work is carried out:(A)A higher-order theory is firstly proposed by introducing the local Legendre polynomials,and then a novel shell element containing initial damage prediction is developed,which can directly predict transverse shear stresses without any postprocessing methods.Unknown variables at each node are independent of number of layers,so the proposed model is more efficient than the 3D-FEM.(B)Compression experiment is carried out to verify the capability of the proposed model.The results obtained from the proposed model are in good agreement with experimental data.(C)Several examples have been analyzed to further assess the capability of the proposed model by comparing to the 3D-FEM results.Moreover,accuracy and efficiency have been evaluated in different damage criterion by comparing with the selected models.The numerical results show that the proposed model can well predict the initial interlaminar damage as well as other damage.Finally,the model is implemented with UEL subroutine,so that the present approach can be readily utilized to analyze the initial damage in typical composite structures.

Technological Perspectives for Propulsion on Nuclear Attack Submarines

This work aimed at proposing a new combination of technologies to improve military performances and reduce costs of nuclear attack submarines, without overlooking safety constraints. The last generation of nuclear attack submarines increased size to meet safety and operational requirements, imposing huge burden on costs side, reducing fleet size. The limitations of current Technologies employed were qualitatively discussed, explaining their limitations. There are new technologies (plate and shell heat exchangers) and architectural choices, like passive safety, and segregation of safety and normal systems, which may lead to reduction of costs and size of submarines. A qualitative analysis was provided on this combination of technologies, stressing their commercial nature and maturity, which reduced risks. The qualitative analysis showed the strong and weak points of this proposal, which adopted the concept of strength in numbers. Concluding, new Technologies enabled the existence of 3800 t nuclear attack submarines with powerful propulsion systems and good acoustic discretion.

Microstructure and Element Composition of Shell Plates of Liolophura japonica

The microstructure characteristics of shell plates of chiton Liolophura japonica Lischke were analyzed using environmental scanning electron microscopy （ESEM）. The results show that the internal structure of the shell plate of chiton Liolophurajaponica is composed of seven calcium layers which were crossed lamellar crystallites, homoge- neous structure, granular crystallites, and trabecular type crystallites. The element compositions of shell plates were analyzed using X-ray photoelectron spectroscopy （XPS）. The results determined by XPS show that the surface chemical states of various layers are different. There are 13 elements （Na, O, N, C, S, P, Ca, C1, Si, A1, K, Fe, Mg） on the shell plate of chiton L. japonica. Among the 13 elements, 11 are on the outside pigment layer except K and S. Among seven layers of shell plate, there is the largest quantity of elements on the outside pigment layer, and the least quantity of elements is on layer C （Articulamentum auctorum）. There are 7 elements on the layer C （Articulamentum auctorum）. Be- sides calcium carbonate, there are some inorganic compounds on the shell plate, such as NaC1, MgO, A1203, silicate, sulfate and phosphate. There are some organic chemical components, such as carbohydrate, organic sulfide, and organic nitride on the shell plates.