Nonlinear Analysis of RC Shell Structures Using Laminated Element. I

The variation-difference method is a convenient numerical method for shells of complex forms. It is enough when only cinematic boundary conditions are satisfied because the method is based on the principle of Lagrange. Another advantage of the variation-difference method is the better opportunity to create computer programs based on it. For shell analysis in orthogonal coordinate system as well as for shell analysis in principal curvatures the system of equations describing stress-strain state can be simplified. In this paper the difference between analysis in orthogonal coordinate system and analysis in principal curvatures of the surface is considered. The main distinction of the analysis of shells in orthogonal curvilinear coordinate system is the necessity of determination of components which include curvature of torsion of coordinate lines. The addition of these components in the equations of the theory of shells for the coordinate system in principal curvatures gives possibility to analyze shells in common orthogonal coordinate system. In this article shell analysis in orthogonal coordinate system is applied to shells based on normal cyclic surfaces.

Compared with 8-node and 4-node isoparametric element, the 9-node isoparametric element is more flexible and higher rate of convergence. Based on state-space techniques, the 9-node isoparametric element of piezothermoelastic material under orthogonal curvilinear coordinate system is established. Finally, the example shows the practicability and accuracy of the 9-node isoparametric element of piezothermoelastic material under orthogonal curvilinear coordinate system.

The paper considers the regularity problem on three-dimensional incompressible Navier-Stokes equations in general orthogonal curvilinear coordinate systems. We establish one regularity criteria of the weak solutions involving only in a vorticity component ω 3 and one a priori estimate on the solution that H 3 u 3 L ∞ 0 , T ; L p ℝ 3 is bounded for 1 ≤ p ≤ ∞ to three-dimensional incompressible Navier-Stokes equations in orthogonal curvilinear coordinate systems. These extent greatly the corresponding results on axisymmetric cylindrical flow.

In Part I, an analytical examination of electromagnetic boundary conditions at the interface between two different media is presented in an orthogonal curvilinear coordinate system which conforms locally and instantaneously to the interface. Subsequently, an original natural coordinate system is introduced to resolve the uncertainty in the choice of an orthogonal curvilinear coordinate system. This natural coordinate system is also appeared to reduce the number of the nonvanishing discontinuity relations, and to determine local and instantaneous natural aspects of the e.m. boundary conditions problem.

A pedagogically pleasing and mnemonically useful method for expressing the velocity and acceleration vectors in orthogonal systems of coordinates is shown. The method relies on expressing the time derivatives of the unit vectors in matrix form. It is also noted that the spatial derivatives of the unit vectors obey a similar rule.

Preface. Introduction. Vectors and Tensors. Kinematics of Fluid Motion. Laws of Fluid Dynamics. Dynamics of Discontinuity Surfaces. Velocity Representations and Associated Theorems. Vorticity Transport Theorems. Pressure Theorems. Two-Dimensional Potential Flows. Forces on Bodies in Two-Dimensional Flows. Two-Dimensional Flows with Vorticity. Three-Dimensional Potential Flows. Axisymmetric Vortex Flows. Vortex Tubes. Interfacial Wave Motion. Stability of Fluid Flows. Appendix A: Common Expressions in Orthogonal Curvilinear Coordinate Systems. Index.

This paper investigates the globally dynamic stabilizing effects of the geometry of the domain at which the flow locates and of the geometrical structure of the finite-energy solutions to three-dimensional (3D) incompressible Navier-Stokes and Euler systems. We have established the global existence and uniqueness of the smooth solution to the Cauchy problem for 3D incompressible Navier-Stokes and Euler equations for a class of smooth large initial data in orthogonal curvilinear coordinate systems, which has no smallness assumption on initial data for the Cartesian coordinate system. Moreover, we have also established the existence, uniqueness, and exponential decay rate of the global strong solution to the initial-boundary-value problem for 3D Navier-Stokes equations for a class of smooth large initial data and a class of special bounded domain in orthogonal curvilinear coordinate systems. Regarding application, some new classes of large-amplitude global smooth solutions with very complex geometric structures, and that are either non-axisymmetric or non-helical in $ \mathbb{R}^3 $, have been established for 3D incompressible Navier-Stokes and Euler equations. This is the first result on the global existence and uniqueness of the large smooth solution to 3D incompressible Navier-Stokes and Euler equations in general orthogonal curvilinear coordinate systems.

UDC 539.3 Investigations of stability of deformable bodies carried out from the three-dimensiona l elasticity theory point of view [3, 4] have shown that the application of classical theories of plates and shells to thin-walled construction elements consisting of modern anisotropic materials does not always produce reliable results. Yet in many cases [3] a refinement of the theory which takes into account the effects of cross shear determines satisfactorily the critical loads. Though sufficient attention has been focused on the stability of multilayered isotropic plates and shells with a poor shear rigidity [1, 5, 6, 9], constructions with anisotropic properties characteristic for composite materials have not been sufficiently investigated. It is also noted that under specific relations between the elastic characteristics of the fillers and of the carrier layers some effects may also appear which are related to cross compression. 1. Equations for neutral equilibrium of multilayer plates and shells can effectively be obtained by linearizing the nonlinear equations for large deflections (of the Ks type). To this end a multilayer shell is considered of the total thickness h = hi + h2 in an orthogonal curvilinear coordinate system (xl, x2, z) subjected to normal loads q§ and q-(xi) applied to the upper z = +h 2 and the lower z = -h 1 surfaces, respectively. The coordinate surface (xl, x2) is identical with the reduction surface z = 0, its coordinate lines being the lines of principal curvature. Similarly as in [7, 8] the following rules are adopted for changes along the thickness of the layered shell in the cross tangential stresses crj~ or in the normal one ors3 and in the cross strain e33:

The state equations for orthotropic, doubly curved shells are established in an orthogonal curvilinear coordinate system. Simplifying hypotheses about displacement models or stress distribution that were assumed in early work are not introduced in this paper. The continuity conditions of stresses and displacements at the interfaces between the contacting layers are satisfied, and analytical solutions for statics and dynamics of the simply supported, doubly curved shells with ortho‐tropic layers are presented by means of the Cayley‐Hamilton theorem. For any number of layers considered, the problem reduces to solving a set of linear algebraic equations with three unknowns. A unified analytical solution is given for thin, moderately thick, and thick laminated shells. The solution satisfied all the equations of elasticity, and each of nine elastic constants is considered. Arbitrary precision of the desired order can be obtained. Numerical results are given to compare with solutions available in the literature.

Post buckling analysis of smart laminated doubly curved shells

This article develops an efficient second‐order reduced multiscale (SORM) method to study the nonlinear shell structure with orthogonal periodic configurations. The heterogenous shell structure is periodically distributed in orthogonal curvilinear coordinate systems. At first, the nonlinear problems for the shell structure are introduced, and the detailed higher‐order nonlinear multiscale formulas based on the asymptotic homogenization approach are given including microscale unit cell functions, effective material parameter and the homogenized equation. Also, since it requires a large number of computation cost to solve the nonlinear multiscale problems by the traditional high‐order homogenization methods, the novel reduced order multiscale model is constructed. Further, according to the reduced‐order multiscale models and higher‐order nonlinear formulas, an effective SORM algorithm is provided for studying the nonlinear shell structures. The main characteristics of the proposed algorithm are that the novel reduced forms established to investigate the nonlinear shell structures and an efficient higher‐order homogenized solution evaluated by postprocessing that does not need higher‐order continuities of the homogenization solutions. Finally, according to some typical nonlinear examples including block structures, cylindrical shell and double‐curved shallow shell, the availabilities of the SORM algorithm are confirmed.

This article investigates the pressure-driven flow of a magnetized hybrid nanofluid confined between two curved corrugated walls in the presence of slip constraints. The base fluid, water, is augmented with single-walled and multiwalled carbon nanotubes (SWCNTs and MWCNTs) to enhance thermal conductivity, evaluated using the Xue and Yamada–Ota models. The channel geometry is mathematically modeled in an orthogonal curvilinear coordinate system, with sinusoidal functions describing the corrugation grooves and phase differences between the walls. An analytical solution for velocity and temperature profiles is derived using the boundary perturbation method, with the corrugation amplitude as the perturbation parameter. The effects of various physical parameters on the flow fields are thoroughly analyzed and presented using graphs and tables. Additionally, key quantities such as volumetric flow rate, skin friction coefficient, and local Nusselt numbers are numerically computed for different parameter values and comprehensively tabulated. The results reveal a significant reduction in the temperature profile with increasing curvature parameter. Comparisons indicate that the Yamada–Ota model outperforms the Xue model in terms of predictive capability and analytical robustness. Finally, the trapping process is examined in detail through the construction of contour streamlines.

The paper considers three-dimensional isentropic flows of a polytropic gas under gravity. A system of gas dynamics equations for a polytropic gas is used as a mathematical model. With the polytropic exponent $ \gamma = 7.02 $, this system describes the motion of water. An orthogonal curvilinear coordinate system is introduced into the system. To formulate the problem of the decay of a special discontinuity in the system, a degenerate change of variables is made, namely: the dependent and independent variables swap roles. In the new variables, an initial-boundary value problem with data on the sound characteristic and an additional condition is formulated for the system. The solution to the initial-boundary value problem is constructed in the form of power series. The convergence of the constructed series in the region from the surface of the weak discontinuity to the gas-vacuum boundary inclusive is proved. To determine the law of motion of the gas-vacuum boundary, a quasi-linear system of partial differential equations is written out, which is reduced to a system of ordinary differential equations using a characteristic parameter.

By modifying the Rodi assumption to take account of the influence of flow curvature, a new curvature modified algebraic stress model (CMASM) is developed from the second moment closure in the generalized curvilinear coordinate system. And the explicit form of the ASM, a new curvature modified nonlineark-e model (CMNKE), is derived in the orthogonal curvilinear coordinate system. This new nonlineark-e model is further validated by a numerical simulation of a two-dimensional U-type turnaround duct flow. The results show that the CMNKE can effectively capture the main characteristic of this curvature flow and simulate the damping effect of the shear stress by a convex curvature and the enhancing effect by a concave curvature. So, this model is a rational and effective simplification to the second moment closure.

The deformation problem of elasticity theory with regard to nonlinear deformations is examined. The expressions of deformations through displacements in the orthogonal curvilinear coordinate system are recorded. The relations for finite deformations in cylindrical and polar coordinate systems are derived. Physical relations for finite deformations and corresponding generalized stresses are recorded.