Separable K-Canonical Formulation of Rectangular Element and Symplectic Integration Method for Analysis of Laminated Plates

2011 ◽  
Vol 194-196 ◽  
pp. 1496-1505
Author(s):  
Guang Hui Qing ◽  
Liang Wang ◽  
Li Zhong Shi
Keyword(s):  
Three Dimensional ◽  
Elastic Solid ◽  
Canonical System ◽  
Laminated Plates ◽  
Linear Elastic ◽  
Canonical Formulation ◽  
Single Plate ◽  
Static Responses ◽  
Rectangular Element ◽  
Symplectic Schemes

In the state space framework, a separable K-canonical formulation of rectangular element and explicit symplectic schemes for the static responses analysis of three-dimensional (3D) laminated plates are proposed in this paper. Firstly, the modified Hellinger-Reissner (H-R) variational principle for linear elastic solid is simply mentioned. Secondly, the separable J-canonical system with Hamiltonian H and the separable K-canonical formulation of rectangular element are constructed. Thirdly, on the basis of the symplectic difference schemes, the explicit symplectic schemes are employed to solve the separable K-canonical governing equation for a single plate. Then, to obtain the high accurate numerical results, a multi-scale iterative technique is also presented. Finally, based on the interlaminar compatibility condition (displacements and stresses), the excellent performance of the method presented in this paper is demonstrated by several numerical experiments of the static responses of laminated plates.

Continuous data dependence for the equations of classical elastodynamics

1969 ◽  
Vol 66 (2) ◽  
pp. 481-491 ◽  
Author(s):  
R. J. Knops ◽  
L. E. Payne
Keyword(s):  
Smooth Surface ◽  
Body Force ◽  
Three Dimensional ◽  
Elastic Solid ◽  
Time Variable ◽  
Position Vector ◽  
Governing Equations ◽  
Linear Elastic ◽  
Explicit Mention ◽  
Image Position

Consider a linear elastic solid occupying a bounded regular three-dimensional region B with smooth surface ∂B. The components of displacement ui referred to cartesian axes xi are then well known to satisfy the system of governing equationsin which t denotes the time variable, x = (x1, x2, x3) denotes the position vector, ρ(x) is the non-homogeneous density, assumed positive,. i(x, t) are the Cartesian components of body force per unit mass, and cijkl(x) are the non-homogeneous elasticities, which apart from certain smoothness conditions stated later, are assumed to possess the symmetryThroughout this paper, all suffixes range over the values 1, 2, 3 and the usual converition of summing over repeated indices is adopted. Except where it is in the interest of clarity we avoid explicit mention of the dependence of functions on their arguments.

Vibration of a Three-Dimensional, Finite Linear, Elastic Solid Containing Cracks

1995 ◽  
Vol 62 (2) ◽  
pp. 282-288 ◽  
Author(s):  
I. Y. Shen
Keyword(s):  
Integral Equation ◽  
Fredholm Integral Equation ◽  
Three Dimensional ◽  
Series Representation ◽  
Elastic Solid ◽  
Mode Shapes ◽  
Reciprocal Theorem ◽  
Shaft Length ◽  
Linear Elastic ◽  
Finite Linear

This paper is to determine vibrational eigensolutions [λm2,vm(r)]m=1∞ of a three-dimensional, finite, linear, elastic solid C containing cracks in terms of crack configuration σc and eigensolutions [ωn2,un(r)n=1∞ of a perfect elastic solid P without the cracks. Use of Betti reciprocal theorem and the Green’s function of P expands vm(r) in terms of an infinite series of un(r). Substitution of the vm(r) series representation into the Kamke quotient of C and stationarity of the quotient result in a Fredholm integral equation whose nontrivial solutions predict λm2, and vm(r) of C. Finally, natural frequencies and mode shapes of a circular shaft of finite length containing a circumferential crack under torsional vibration are predicted through a two-term Ritz approximation of the Fredholm integral equation. The results differ significantly from those predicted by the method of flexibility matrices, when the ratio of the shaft length to the shaft radius is small.

Crack-Path Effect on Material Toughness

1990 ◽  
Vol 57 (1) ◽  
pp. 97-103 ◽  
Author(s):  
Asher A. Rubinstein
Keyword(s):  
Crack Path ◽  
Numerical Procedure ◽  
Crack Tip ◽  
Elastic Solid ◽  
Singular Integral Equations ◽  
Length Change ◽  
Curvilinear Crack ◽  
Linear Elastic ◽  
System Of Cracks ◽  
Remote Stress

The material-toughening mechanism based on the crack-path deflection is studied. This investigation is based on a model which consists of a macrocrack (semi-infinite crack), with a curvilinear segment at the crack tip, situated in a brittle solid. The effect of material toughening is evaluated by comparison of the remote stress field parameters, such as the stress intensity factors (controlled by a loading on a macroscale), to effective values of these parameters acting in the vicinity of a crack tip (microscale). The effects of the curvilinear crack path are separated into three groups: crack-tip direction, crack-tip geometry pattern-shielding, and crack-path length change. These effects are analyzed by investigation of selected curvilinear crack patterns such as a macrocrack with simple crack-tip kink in the form of a circular arc and a macrocrack with a segment at the crack tip in the form of a sinusoidal wave. In conjunction with this investigation, a numerical procedure has been developed for the analysis of curvilinear cracks (or a system of cracks) in a two-dimensional linear elastic solid. The formulation is based on the solution of a system of singular integral equations. This numerical scheme was applied to the cases of finite and semi-infinite cracks.

Performance Based Evaluation of Bridge Substructure in North Mississippi Using Nonlinear FEM and Field Vibration Measurement

2000 ◽  
Author(s):  
Chris L. Mullen ◽  
Prabin R. Tuladhar
Keyword(s):  
Finite Element ◽  
Transfer Functions ◽  
Response Spectrum ◽  
Three Dimensional ◽  
Vibration Measurement ◽  
Elastic Behavior ◽  
Evaluation Procedure ◽  
Vibration Measurements ◽  
Linear Elastic ◽  
Performance Based Evaluation

Abstract Discussion of a Performance - Based Engineering evaluation procedure for an existing interstate highway bridge in north Mississippi. The bridge is in a highly trafficked location near the Memphis Metropolitan area and is reflective of modern design practices in Mississippi. Results are presented of nonlinear damage response and displacement ductility performance of the reinforced concrete bents and their foundations predicted using static finite element (FE) computations. The model considers the composite action of the concrete and the reinforcing steel materials under axial force, shear, torsion and flexure. The performance-based evaluation includes three-dimensional computational simulations of the nonlinear bridge system, including substructures and superstructure. The response spectrum dynamic analysis method will also be carried out on the linear elastic three-dimensional model to predict the linear elastic behavior. Field vibration measurements, including ambient and hammer-impact, were performed to calibrate the models. The computed transfer functions are currently being evaluated to correlate vibration measurements and the Finite element models.

On the Motion of an Elastic Body Rolling/Sliding on an Elastic Substrate

1995 ◽  
Vol 117 (2) ◽  
pp. 308-314 ◽  
Author(s):  
A. Spector ◽  
R. C. Batra
Keyword(s):  
Elastic Body ◽  
Frictional Force ◽  
Three Dimensional ◽  
The Body ◽  
Elastic Substrate ◽  
Linear Elastic ◽  
Linear Elastic Body ◽  
Body Rolling ◽  
Applied Forces ◽  
Two Phases

The three-dimensional evolutionary problem of rolling/sliding of a linear elastic body on a linear elastic substrate is studied. The inertial properties of the body regarded as rigid are accounted for. By employing an asymptotic analysis, it is shown that the process can be divided into two phases: transient and quasistationary. An expression for the frictional force as a function of the externally applied forces and moments, and inertial properties of the body is derived. For an ellipsoid rolling/sliding on a linear elastic substrate, numerical results for the frictional force distribution, slip/adhesion subareas, and the evolution of the slip velocity are given.

scholarly journals The influence of elastic deformations in high-strength structural materials on the hydrogen transport

2021 ◽  
Vol 225 ◽  
pp. 01010
Author(s):  
Polina Grigoreva ◽  
Elena Vilchevskaya ◽  
Vladimir Polyanskiy
Keyword(s):  
Chemical Potential ◽  
Potential Gradient ◽  
Elastic Solid ◽  
High Strength ◽  
Ideal Gas ◽  
Linear Elastic ◽  
Coupled Diffusion ◽  
Elastic Boundary ◽  
Linear Effects ◽  
Solid System

In this work, the diffusion equation for the gas-solid system is revised to describe the non-uniform distribution of hydrogen in steels. The first attempt to build a theoretical and general model and to describe the diffusion process as driven by a chemical potential gradient is made. A linear elastic solid body and ideal gas, diffusing into it, are considered. At this stage, we neglect any traps and non-linear effects. The coupled diffusion-elastic boundary problem is solved for the case of the cylinder under the tensile loads. The obtained results correspond to the experimental ones. Based on them, the assumptions about the correctness of the model and its further improvement are suggested.

scholarly journals Three dimensional elastic dynamic analysis of a 985 foot high freestanding communications tower

1975 ◽  
Vol 8 (3) ◽  
pp. 222-237
Author(s):  
J. Lord ◽  
M. Zayed
Keyword(s):  
San Francisco ◽  
Computer Model ◽  
Three Dimensional ◽  
Physical Structure ◽  
Design Development ◽  
Structural Configuration ◽  
Free Standing ◽  
Ground Shaking ◽  
Linear Elastic ◽  
Seismic Force

This paper reviews the design development of a 985' high free-standing communications tower recently constructed in San Francisco. Included is a description of the structural configuration of the
 tower and the criteria by which it was designed. The dynamic characteristics of a three-dimensional linear elastic mathematical computer model, devised to represent the physical structure, are presented. The dynamic response of this computer model to various levels of
ground shaking, including both horizontal and vertical excitations,
 are summarized, evaluated and compared to the seismic force levels prescribed by the 1969 edition of the San Francisco Building Code.
 Also included in the comparison are the responses derived for the tower from wind tunnel studies and static wind design criteria.

Analytical and 3D Finite Element Study of the Deflection of an Elastic Cantilever Bilayer Plate

2010 ◽  
Vol 78 (1) ◽  
Author(s):  
M. Chekchaki ◽  
V. Lazarus ◽  
J. Frelat
Keyword(s):  
Thin Films ◽  
Finite Element ◽  
Three Dimensional ◽  
Analytical Formula ◽  
Mechanical Stresses ◽  
Linear Elastic ◽  
Wide Range ◽  
Finite Element Calculations ◽  
Analytical Formulas ◽  
Three Dimensional Elasticity

The mechanical system considered is a bilayer cantilever plate. The substrate and the film are linear elastic. The film is subjected to isotropic uniform prestresses due for instance to volume variation associated with cooling, heating, or drying. This loading yields deflection of the plate. We recall Stoney’s analytical formula linking the total mechanical stresses to this deflection. We also derive a relationship between the prestresses and the deflection. We relax Stoney’s assumption of very thin films. The analytical formulas are derived by assuming that the stress and curvature states are uniform and biaxial. To quantify the validity of these assumptions, finite element calculations of the three-dimensional elasticity problem are performed for a wide range of plate geometries, Young’s and Poisson’s moduli. One purpose is to help any user of the formulas to estimate their accuracy. In particular, we show that for very thin films, both formulas written either on the total mechanical stresses or on the prestresses, are equivalent and accurate. The error associated with the misfit between our theorical study and numerical results are also presented. For thicker films, the observed deflection is satisfactorily reproduced by the expression involving the prestresses and not the total mechanical stresses.

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