Static Analysis of Thick Laminated Beams: Two-Dimensional Elasticity Solutions via Differential Quadrature

2005 ◽  
Vol 475-479 ◽  
pp. 1067-1072
Author(s):  
Chaofeng Lü ◽  
Ying Gao ◽  
W.Q. Chen
Keyword(s):  
Present Method ◽  
Matrix Theory ◽  
Laminated Composite ◽  
Static Problem ◽  
State Equation ◽  
Differential Quadrature ◽  
Two Dimensional ◽  
State Variables ◽  
State Equations ◽  
Elasticity Solutions

This paper intends to present two-dimensional elasticity solutions for static problem of thick laminated composite beams using a hybrid method of state-space-based differential quadrature. The technique of differential quadrature is employed to reduce the partial differential state equations into the ordinary differential ones at all arbitrary sampling points for each individual laminate. General solution to the assembled state equation is then obtained according to the matrix theory. Taking account of the continuity conditions at the interfaces of all the adjacent lamina, a relationship between state variables at the top and bottom surfaces of the beam is established through a global transfer matrix. After incorporating the boundary conditions at these two surfaces, an eigenvalue equation for static problem is then derived. Numerical examples are presented, through which the accuracy and convergence characteristics of the present method are investigated. It is shown that the present method is of excellent efficiency for laminated composite thick beams subjected to arbitrary end supporting conditions.

TIME DOMAIN DIAKOPTIC ANALYSIS BASED ON REDUCED-ORDER STATE EQUATIONS

2007 ◽  
Vol 17 (10) ◽  
pp. 3625-3631 ◽  
Author(s):  
MIHAI IORDACHE ◽  
LUCIA DUMITRIU
Keyword(s):  
Analog Circuits ◽  
Large Scale ◽  
State Equation ◽  
The State ◽  
State Variables ◽  
State Equations ◽  
Integration Algorithm ◽  
Reduced Order ◽  
Decomposition Procedure ◽  
Minimum Number

In this paper we present some new tearing techniques to systematically formulate the state equations in symbolic normal-form for linear and/or nonlinear time-invariant large-scale analog circuits. The excess elements of the first and of the second kind are unitarily treated in order to allow a symbolic representation of the circuit with a minimum number of state variables. A procedure to reduce the state equation number of each subcircuit is also presented. The reduced-order is based on an implicit integration algorithm and on the successive elimination of the selected state variables. Examples are given to illustrate the decomposition procedure, the assignment of the connection sources and the reduced-order technique.

A State-Space-Based Stress Analysis of a Multilayered Spherical Shell With Spherical Isotropy

2000 ◽  
Vol 68 (1) ◽  
pp. 109-114 ◽  
Author(s):  
W. Q. Chen ◽  
H. J. Ding
Keyword(s):  
Spherical Shell ◽  
Stress Analysis ◽  
Matrix Theory ◽  
Three Dimensional ◽  
The State ◽  
State Variables ◽  
State Equations ◽  
Independent State ◽  
Spherical Isotropy ◽  
Constant Coefficients

This paper presents an exact static stress analysis of a multilayered elastic spherical shell (hollow sphere) completely based on three-dimensional elasticity for spherical isotropy. Two independent state equations are derived after introducing three displacement functions and two stress functions. In particular, a variable substitution technique is used to derive the state equations with constant coefficients. Matrix theory is then employed to obtain the relationships between the state variables at the upper and lower surfaces of each lamina. By virtue of the continuity conditions between two adjacent layers, a second-order linear algebraic equation and a fourth-order one about the boundary variables at the inner and outer surfaces of a multilayered spherical shell are obtained. Numerical examples are presented to show the effectiveness of the present method.

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

2003 ◽  
Vol 125 (1) ◽  
pp. 26-35 ◽  
Author(s):  
K. Bhaskar ◽  
N. Ganapathysaran
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Shell Theory ◽  
Cylindrical Shells ◽  
Laminated Composite ◽  
Two Dimensional ◽  
Simply Supported ◽  
Elasticity Solutions ◽  
Laminated Composite Shells ◽  
Taylor’S Series

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.

scholarly journals Marine boundary layer cloud regimes and POC formation in a CRM coupled to a bulk aerosol scheme

2013 ◽  
Vol 13 (24) ◽  
pp. 12549-12572 ◽  
Author(s):  
A. H. Berner ◽  
C. S. Bretherton ◽  
R. Wood ◽  
A. Muhlbauer
Keyword(s):  
Boundary Layer ◽  
Marine Boundary Layer ◽  
Three Dimensional ◽  
Two Dimensional ◽  
State Variables ◽  
Liquid Water Path ◽  
Modeling Framework ◽  
Boundary Layer Cloud ◽  
Cloud Regimes ◽  
Layer Cloud

Abstract. A cloud-resolving model (CRM) coupled to a new intermediate-complexity bulk aerosol scheme is used to study aerosol–boundary-layer–cloud–precipitation interactions and the development of pockets of open cells (POCs) in subtropical stratocumulus cloud layers. The aerosol scheme prognoses mass and number concentration of a single lognormal accumulation mode with surface and entrainment sources, evolving subject to processing of activated aerosol and scavenging of dry aerosol by clouds and rain. The CRM with the aerosol scheme is applied to a range of steadily forced cases idealized from a well-observed POC. The long-term system evolution is explored with extended two-dimensional (2-D) simulations of up to 20 days, mostly with diurnally averaged insolation and 24 km wide domains, and one 10 day three-dimensional (3-D) simulation. Both 2-D and 3-D simulations support the Baker–Charlson hypothesis of two distinct aerosol–cloud "regimes" (deep/high-aerosol/non-drizzling and shallow/low-aerosol/drizzling) that persist for days; transitions between these regimes, driven by either precipitation scavenging or aerosol entrainment from the free-troposphere (FT), occur on a timescale of ten hours. The system is analyzed using a two-dimensional phase plane with inversion height and boundary layer average aerosol concentrations as state variables; depending on the specified subsidence rate and availability of FT aerosol, these regimes are either stable equilibria or distinct legs of a slow limit cycle. The same steadily forced modeling framework is applied to the coupled development and evolution of a POC and the surrounding overcast boundary layer in a larger 192 km wide domain. An initial 50% aerosol reduction is applied to half of the model domain. This has little effect until the stratocumulus thickens enough to drizzle, at which time the low-aerosol portion transitions into open-cell convection, forming a POC. Reduced entrainment in the POC induces a negative feedback between the areal fraction covered by the POC and boundary layer depth changes. This stabilizes the system by controlling liquid water path and precipitation sinks of aerosol number in the overcast region, while also preventing boundary layer collapse within the POC, allowing the POC and overcast to coexist indefinitely in a quasi-steady equilibrium.

The steady two-dimensional reflexion of an oblique partly dispersed shock wave from a plane wall

1973 ◽  
Vol 61 (1) ◽  
pp. 159-172 ◽  
Author(s):  
H. Buggisch
Keyword(s):  
Shock Wave ◽  
Internal State ◽  
Plane Wall ◽  
Plane Shock ◽  
Two Dimensional ◽  
State Variables ◽  
Reflected Shock ◽  
Internal State Variables ◽  
Specific Entropy ◽  
Instantaneous Values

The steady two-dimensional problem of reflexion of an oblique partly dispersed plane shock wave from a plane wall is studied analytically. Viscosity, diffusion and heat conduction are neglected. The thermodynamic state of the gas is assumed to be determined by the instantaneous values of the specific entropy s, pressure p and a finite number of internal state variables. Results for the flow field behind the reflected shock are obtained by a perturbation method which is based on the assumption that the influence of relaxation is relatively weak.

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