VIBRATION OF INITIALLY STRESSED MICRO- AND NANO-BEAMS

2007 ◽  
Vol 07 (04) ◽  
pp. 555-570 ◽  
Author(s):  
C. M. WANG ◽  
Y. Y. ZHANG ◽  
S. KITIPORNCHAI
Keyword(s):  
Boundary Conditions ◽  
Shear Deformation ◽  
Initial Stress ◽  
Transverse Shear ◽  
Virtual Work ◽  
Nonlocal Elasticity Theory ◽  
Rotary Inertia ◽  
Small Scale ◽  
Transverse Shear Deformation ◽  
Vibration Problem

This paper is concerned with the vibration problem of initially stressed micro/nano-beams. The vibration problem is formulated on the basis of Eringen's nonlocal elasticity theory and the Timoshenko beam theory. The small scale effect is taken into consideration in the former theory while the effects of initial stress, transverse shear deformation and rotary inertia are accounted for in the latter theory. The governing equations and the boundary conditions are derived using the principle of virtual work. These equations are solved analytically for the vibration frequencies of micro/nano-beams with different initial stress values and boundary conditions. The effect of the initial stress on the fundamental frequency and vibration mode shape of the beam is investigated. The solutions obtained provide a better representation of the vibration behavior of initially stressed micro/nano-beams which are stubby and short, since the effects of small scale, transverse shear deformation and rotary inertia are significant and cannot be neglected.

scholarly journals A nonlocal sinusoidal plate model for micro/nanoscale plates

2014 ◽  
Vol 228 (14) ◽  
pp. 2652-2660 ◽  
Author(s):  
Huu-Tai Thai ◽  
Thuc P Vo ◽  
Trung-Kien Nguyen ◽  
Jaehong Lee
Keyword(s):  
Boundary Conditions ◽  
Shear Deformation ◽  
Transverse Shear ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Nonlocal Elasticity Theory ◽  
Small Scale ◽  
Plate Model ◽  
Proposed Model ◽  
Small Scale Effect

A nonlocal sinusoidal plate model for micro/nanoscale plates is developed based on Eringen’s nonlocal elasticity theory and sinusoidal shear deformation plate theory. The small-scale effect is considered in the former theory while the transverse shear deformation effect is included in the latter theory. The proposed model accounts for sinusoidal variations of transverse shear strains through the thickness of the plate, and satisfies the stress-free boundary conditions on the plate surfaces, thus a shear correction factor is not required. Equations of motion and boundary conditions are derived from Hamilton’s principle. Analytical solutions for bending, buckling, and vibration of simply supported plates are presented, and the obtained results are compared with the existing solutions. The effects of small scale and shear deformation on the responses of the micro/nanoscale plates are investigated.

scholarly journals Bending analysis of laminated composite and sandwich beams according to refined trigonometric beam theory

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
A. S. Sayyad ◽  
Y. M. Ghugal ◽  
N. S. Naik
Keyword(s):  
Shear Deformation ◽  
Transverse Shear ◽  
Virtual Work ◽  
Laminated Composite ◽  
Beam Theory ◽  
Present Theory ◽  
Transverse Shear Deformation ◽  
Transverse Displacement ◽  
Sandwich Beams ◽  
Bending Analysis

AbstractA trigonometric beam theory (TBT) is developed for the bending analysis of laminated composite and sandwich beams considering the effect of transverse shear deformation. The axial displacement field uses trigonometric function in terms of thickness coordinate to include the effect of transverse shear deformation. The transverse displacement is considered as a sum of two partial displacements, the displacement due to bending and the displacement due to transverse shearing. Governing equations and boundary conditions are obtained by using the principle of virtual work. To demonstrate the validity of present theory it is applied to the bending analysis of laminated composite and sandwich beams. The numerical results of displacements and stresses obtained by using present theory are presented and compared with those of other trigonometric theories available in literature along with elasticity solution wherever possible.

A Unified Shear Deformation Theory for the Bending of Isotropic, Functionally Graded, Laminated and Sandwich Beams and Plates

2017 ◽  
Vol 09 (01) ◽  
pp. 1750007 ◽  
Author(s):  
Atteshamuddin S. Sayyad ◽  
Yuwaraj M. Ghugal
Keyword(s):  
Boundary Conditions ◽  
Shear Deformation ◽  
Transverse Shear ◽  
Deformation Theory ◽  
Virtual Work ◽  
Shear Deformation Theory ◽  
Laminated Composite ◽  
Present Theory ◽  
Functionally Graded ◽  
Shear Stresses

In this paper, a displacement-based unified shear deformation theory is developed for the analysis of shear deformable advanced composite beams and plates. The theory is developed with the inclusion of parabolic (PSDT), trigonometric (TSDT), hyperbolic (HSDT) and exponential (ESDT) shape functions in terms of thickness coordinate to account for the effect of transverse shear deformation. The in-plane displacements consider the combined effect of bending rotation and shear rotation. The use of parabolic shape function in the present theory leads to the Reddy’s theory, but trigonometric, hyperbolic and exponential functions are first time used in the present displacement field. The present theory is accounted for an accurate distribution of transverse shear stresses through the thickness of plate, therefore, it does not require problem dependent shear correction factor. Governing equations and associated boundary conditions of the theory are derived from the principle of virtual work. Navier type closed-form solutions are obtained for simply supported boundary conditions. To verify the global response of the present theory it is applied for the bending of both one-dimensional (beams) and two-dimensional (plates) functionally graded, laminated composite and sandwich structures. The present results are compared with exact elasticity solution and other higher order shear deformation theories to verify the accuracy and efficiency of the present theory.

Modal density and critical frequency of composite panels considering transverse shear deformation and rotary inertia

2020 ◽  
Vol 26 (17-18) ◽  
pp. 1503-1513
Author(s):  
K Renji
Keyword(s):  
Shear Deformation ◽  
Transverse Shear ◽  
Critical Frequency ◽  
Rotary Inertia ◽  
Transverse Shear Deformation ◽  
Bending Waves ◽  
Modal Density ◽  
The Face ◽  
Orthotropic Properties ◽  
Bending Wave

In this work, expressions for estimating the modal density, speed of the bending wave, critical frequency and coincidence frequency of panels are derived considering orthotropic properties of the face sheets, transverse shear deformation and the rotary inertia. Presence of rotary inertia results in an increase in the modal density and a reduction in the speed of the bending waves. The influence is significant at higher frequencies. The critical and coincidence frequencies increase due to rotary inertia. Results for a typical equipment panel of spacecraft are presented and they show the need for incorporating rotary inertia while determining these parameters.

A HYPERBOLIC SHEAR DEFORMATION THEORY FOR FLEXURE AND VIBRATION OF THICK ISOTROPIC BEAMS

2009 ◽  
Vol 06 (04) ◽  
pp. 585-604 ◽  
Author(s):  
YUWARAJ MAROTRAO GHUGAL ◽  
RAJNEESH SHARMA
Keyword(s):  
Boundary Conditions ◽  
Shear Deformation ◽  
Correction Factor ◽  
Displacement Field ◽  
Transverse Shear ◽  
Deformation Theory ◽  
Virtual Work ◽  
Shear Deformation Theory ◽  
Flexural Vibration ◽  
Shear Correction Factor

A Hyperbolic Shear Deformation Theory taking into account transverse shear deformation effects, is presented for the static flexure and free flexural vibration analysis of thick isotropic beams. The displacement field of the theory contains two variables and does not require shear correction factor. The hyperbolic sine function is used in the displacement field in terms of thickness coordinate to represent shear deformation. The most important feature of the theory is that the transverse shear stress can be obtained directly from the use of constitutive relation, satisfying the stress free boundary conditions at top and bottom of the beam. Hence, the theory obviates the need of shear correction factor. Governing differential equations and boundary conditions of the theory are obtained using the principle of virtual work. Results obtained for flexure and free vibration of simply supported uniform, isotropic beams are compared with those of elementary, refined, and exact beam theories to validate the accuracy of the theory.

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