scholarly journals A hybrid variationally consistent homogenization approach based on Ritz's method

2013 ◽  
Vol 94 (7) ◽  
pp. 625-647 ◽  
Author(s):  
N. Bleier ◽  
J. Mosler
Keyword(s):  
Homogenization Approach ◽  
Ritz’S Method

Vibration of Rectangular Plates by the Ritz Method

1950 ◽  
Vol 17 (4) ◽  
pp. 448-453 ◽  
Author(s):  
Dana Young
Keyword(s):  
Ritz Method ◽  
Normal Modes ◽  
Approximate Solutions ◽  
The Other ◽  
Rectangular Plates ◽  
Elastic Plates ◽  
Modes Of Vibration ◽  
Ritz’S Method ◽  
Square Plate ◽  
Set Up

Abstract Ritz’s method is one of several possible procedures for obtaining approximate solutions for the frequencies and modes of vibration of thin elastic plates. The accuracy of the results and the practicability of the computations depend to a great extent upon the set of functions that is chosen to represent the plate deflection. In this investigation, use is made of the functions which define the normal modes of vibration of a uniform beam. Tables of values of these functions have been computed as well as values of different integrals of the functions and their derivatives. With the aid of these data, the necessary equations can be set up and solved with reasonable effort. Solutions are obtained for three specific plate problems, namely, (a) square plate clamped at all four edges, (b) square plate clamped along two adjacent edges and free along the other two edges, and (c) square plate clamped along one edge and free along the other three edges.

Convergence–confinement analysis of a bolt-supported tunnel using the homogenization method

2006 ◽  
Vol 43 (5) ◽  
pp. 462-483 ◽  
Author(s):  
Henry Wong ◽  
Didier Subrin ◽  
Daniel Dias
Keyword(s):  
Input Parameter ◽  
Uniaxial Stress ◽  
Homogenization Method ◽  
External Input ◽  
Proposed Model ◽  
Homogenization Approach ◽  
Parametric Studies ◽  
Perfect Bonding ◽  
The Moment ◽  
Grouted Bolts

The behaviour of tunnels reinforced with radially disposed fully grouted bolts is investigated in this paper. Perfect bonding and ideal diffusion of bolt tension are assumed, so that the bolt tension can be assimilated to an equivalent uniaxial stress tensor. An analytical model of the convergence–confinement type is proposed that accounts for the delayed action of bolts due to ground decompression prior to bolt installation. This factor leads to nonsimultaneous yielding, and more generally, a different stress history for each constituent, requiring special treatments in the incremental elastoplasticity calculations. Nonetheless, the resulting model remains sufficiently simple, and an analytical solution is still accessible. Charts are provided to allow for parametric studies and quick preliminary designs. Comparisons with 3D numerical calculations show that the model gives precise results if the correct convergence at the moment of bolt installation is used as an "external" input parameter, validating the homogenization approach. An approximate methodology based on previous works is proposed to determine this parameter to render the proposed model "self-sufficient." Its predictions are again compared to 3D numerical computations, and the results are found to be sufficiently accurate for practical applications.Key words: reinforcement, anisotropy, analytical, lining, yield, elastoplasticity.

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