Hencky Bar-Chain Model for Buckling and Vibration of Beams with Elastic End Restraints

2015 ◽  
Vol 15 (07) ◽  
pp. 1540007 ◽  
Author(s):  
C. M. Wang ◽  
H. Zhang ◽  
R. P. Gao ◽  
W. H. Duan ◽  
N. Challamel
Keyword(s):  
Flexural Rigidity ◽  
Beam Theory ◽  
Length Scale ◽  
Chain Model ◽  
Beam Model ◽  
Small Length ◽  
First Order ◽  
Small Length Scale ◽  
Nonlocal Beam ◽  
Central Finite Difference

This paper presents the Hencky bar-chain model (HBM) for buckling and vibration analyses of Euler–Bernoulli beams with elastic end restraints. The Hencky bar-chain comprises rigid beam segments (of length a = L/n where L is the total length of beam and n the number of beam segments) connected by frictionless hinges with elastic rotational springs of stiffness EI/a where EI is the flexural rigidity of the beam. The elasticity and the mass of the beam are concentrated at the hinges with rotational springs. The key contribution of this paper lies in the modeling of the elastic end restraints of the Hencky bar-chain that will simulate the same buckling and vibration results as that furnished by the first-order central finite difference beam model (FDM) which was earlier shown to be analogous to the HBM. The establishment of such a physical discrete beam model allows one to obtain solutions for beam-like structure with repetitive cells (or elements) as well as to calibrate the Eringen's coefficient e0 in the nonlocal beam theory that captures the small length scale effect.

POSTBUCKLING OF NANO RODS/TUBES BASED ON NONLOCAL BEAM THEORY

2009 ◽  
Vol 01 (02) ◽  
pp. 259-266 ◽  
Author(s):  
C. M. WANG ◽  
Y. XIANG ◽  
S. KITIPORNCHAI
Keyword(s):  
Differential Equation ◽  
Scale Effect ◽  
Shooting Method ◽  
Beam Theory ◽  
Length Scale ◽  
Small Length ◽  
Concentrated Load ◽  
Small Length Scale ◽  
Nano Rods ◽  
Nonlocal Beam

This paper is concerned with the postbuckling problem of cantilevered nano rods/tubes under an end concentrated load. Eringen's nonlocal beam theory is used to account for the small length scale effect. The governing equation is derived from statical and geometrical considerations and Eringen's nonlocal constitutive relation. The nonlinear differential equation is solved using the shooting method for the postbuckling load and the buckled shape. By comparing with the classical postbuckling solutions, the sensitivity of the small length scale effect on the buckling load and buckled shape may be observed. It is found that the small length scale effect decreases the postbuckling load and increases the deflection of the rod.

Visualization of buoyancy opposing flow structures in a small length scale fluidic junction

2005 ◽  
Vol 8 (4) ◽  
pp. 288-288 ◽  
Author(s):  
P. A. Walsh ◽  
M. R. D. Davies ◽  
T. Dalton
Keyword(s):  
Flow Structures ◽  
Length Scale ◽  
Small Length ◽  
Small Length Scale

Nonlocal Approaches for the Vibration of Lattice Plates Including Both Shear and Bending Interactions

2018 ◽  
Vol 18 (07) ◽  
pp. 1850094 ◽  
Author(s):  
F. Hache ◽  
N. Challamel ◽  
I. Elishakoff
Keyword(s):  
Natural Frequencies ◽  
Scale Effects ◽  
Dynamical Behavior ◽  
Mindlin Plate ◽  
Length Scale ◽  
Small Length ◽  
Simply Supported ◽  
Small Length Scale ◽  
Plate Models ◽  
Scale Coefficient

The present study investigates the dynamical behavior of lattice plates, including both bending and shear interactions. The exact natural frequencies of this lattice plate are calculated for simply supported boundary conditions. These exact solutions are compared with some continuous nonlocal plate solutions that account for some scale effects due to the lattice spacing. Two continualized and one phenomenological nonlocal UflyandMindlin plate models that take into account both the rotary inertia and the shear effects are developed for capturing the small length scale effect of microstructured (or lattice) thick plates by associating the small length scale coefficient introduced in the nonlocal approach to some length scale coefficients given in a Taylor or a rational series expansion. The nonlocal phenomenological model constitutes the stress gradient Eringen’s model applied at the plate scale. The continualization process constructs continuous equation from the one of the discrete lattice models. The governing partial differential equations are solved in displacement for each nonlocal plate model. An exact analytical vibration solution is obtained for the natural frequencies of the simply supported rectangular nonlocal plate. As expected, it is found that the continualized models lead to a constant small length scale coefficient, whereas for the phenomenological nonlocal approaches, the coefficient, calibrated with respect to the element size of the microstructured plate, is structure-dependent. Moreover, comparing the natural frequencies of the continuous models with the exact discrete one, it is concluded that the continualized models provide much more accurate results than the nonlocal Uflyand–Mindlin plate models.

Small Length-Scale Analysis of Transport Phenomena in Oil-Wells Formation with Drilling Fluid Losses

2009 ◽  
Vol 27 (3) ◽  
pp. 345-356 ◽  
Author(s):  
G. Espinosa-Paredes ◽  
R. Vázquez-Rodriguez ◽  
R. Ramos-Alcantara ◽  
R. Varela-Ham ◽  
H. Romero-Paredes ◽  
...  
Keyword(s):  
Drilling Fluid ◽  
Transport Phenomena ◽  
Oil Wells ◽  
Length Scale ◽  
Scale Analysis ◽  
Small Length ◽  
Small Length Scale ◽  
Fluid Losses
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