Analysis of Heterogeneous Anisotropic Plates

1969 ◽  
Vol 36 (2) ◽  
pp. 261-266 ◽  
Author(s):  
J. M. Whitney ◽  
A. W. Leissa
Keyword(s):  
Thermal Stresses ◽  
Anisotropic Plate ◽  
Plate Theory ◽  
Flexural Vibration ◽  
Governing Equations ◽  
Anisotropic Plates ◽  
Closed Form Solutions ◽  
Basic Assumptions ◽  
Thin Plate Theory ◽  
Arbitrary Thickness

Using the basic assumptions of thin-plate theory, including nonlinear terms in the von Karman sense, the governing equations of a laminated anisotropic plate are formulated. In particular, the type of plate under discussion consists of n layers of orthotropic sheets bonded together. Each layer has arbitrary thickness, elastic properties, and orientation of orthotropic axes with respect to the plate axes. The governing equations are obtained by integrating the equations of nonlinear elasticity. Inertia terms and thermal stresses are included. Closed-form solutions to the linearized equations are obtained for bending, flexural vibration, and buckling of special, but important, classes of laminates for which coupling between bending and stretching is unavoidable.

Nonlinear Vibration Analysis of Prestressed Double Layered Nanoscale Viscoelastic Plates

2019 ◽  
Vol 24 (3) ◽  
pp. 394-407
Author(s):  
Farzad Ebrahimi ◽  
S. Hamed S. Hamed S. Hossei
Keyword(s):  
Plate Theory ◽  
Differential Quadrature Method ◽  
Geometrical Nonlinearity ◽  
Flexural Vibration ◽  
Nonlocal Elasticity Theory ◽  
Small Scale ◽  
Nonlinear Frequency ◽  
Governing Equations ◽  
Viscoelastic Coefficient ◽  
Vdw Interaction

In the present study, the nonlinear flexural vibration behavior of a double layered prestressed viscoelastic nanoplate under shear in-plane load is investigated based on nonlocal elasticity theory. Using nonlinear strain-displacement relations, the geometrical nonlinearity is modeled. Both nonlocal plate theory and Hamilton’s principle are utilized for deriving the governing equations. The differential quadrature method (DQM) is employed for the computation of nonlinear frequency of the nanoplate. The detailed parametric study is conducted, focusing on the influences of small scale, aspect ratio of the plate, Winkler and Pasternak effects, van der Walls (vdW) interaction, temperature, the effect of pre-stress under shear in-plane load, and the viscidity of the plate. The influence of the viscoelastic coefficient is also discussed. The plots for the ratio of nonlinear to linear frequencies versus maximum transverse amplitude for double layered viscoelastic nanoplate are presented.

Shear Deformation in Heterogeneous Anisotropic Plates

1970 ◽  
Vol 37 (4) ◽  
pp. 1031-1036 ◽  
Author(s):  
J. M. Whitney ◽  
N. J. Pagano
Keyword(s):  
Shear Deformation ◽  
Plate Theory ◽  
Flexural Vibration ◽  
Laminated Plates ◽  
Transverse Load ◽  
Laminated Plate ◽  
Anisotropic Plates ◽  
Reinforced Composite ◽  
Classical Elasticity Theory ◽  
Good Agreement

A bending theory for anisotropic laminated plates developed by Yang, Norris, and Stavsky is investigated. The theory includes shear deformation and rotary inertia in the same manner as Mindlin’s theory for isotropic homogeneous plates. The governing equations reveal that unsymmetrically laminated plates display the same bending-extensional coupling phenomenon found in classical laminated plate theory based on the Kirchhoff assumptions. Solutions are presented for bending under transverse load and for flexural vibration frequencies of symmetric and nonsymmetric lamninates. Good agreement is observed in numerical results for plate bending as compared to exact solutions obtained from classical elasticity theory. For certain fiber-reinforced composite materials, radical departure from classical laminated plate theory is indicated.

scholarly journals Semi-analytical static analysis of nonlocal strain gradient laminated composite nanoplates in hygrothermal environment

2021 ◽  
Vol 43 (5) ◽  
Author(s):  
Giovanni Tocci Monaco ◽  
Nicholas Fantuzzi ◽  
Francesco Fabbrocino ◽  
Raimondo Luciano
Keyword(s):  
Thin Plate ◽  
Strain Gradient ◽  
Plate Theory ◽  
Laminated Composite ◽  
Equilibrium Equations ◽  
Thin Plate Theory ◽  
Bending Behavior ◽  
Hygrothermal Environment ◽  
Load Types ◽  
Novel Applications

AbstractIn this work, the bending behavior of nanoplates subjected to both sinusoidal and uniform loads in hygrothermal environment is investigated. The present plate theory is based on the classical laminated thin plate theory with strain gradient effect to take into account the nonlocality present in the nanostructures. The equilibrium equations have been carried out by using the principle of virtual works and a system of partial differential equations of the sixth order has been carried out, in contrast to the classical thin plate theory system of the fourth order. The solution has been obtained using a trigonometric expansion (e.g., Navier method) which is applicable to simply supported boundary conditions and limited lamination schemes. The solution is exact for sinusoidal loads; nevertheless, convergence has to be proved for other load types such as the uniform one. Both the effect of the hygrothermal loads and lamination schemes (cross-ply and angle-ply nanoplates) on the bending behavior of thin nanoplates are studied. Results are reported in dimensionless form and validity of the present methodology has been proven, when possible, by comparing the results to the ones from the literature (available only for cross-ply laminates). Novel applications are shown both for cross- and angle-ply laminated which can be considered for further developments in the same topic.

A Three-Dimensional Finite Difference Solution for the Thermal Stresses in Railcar Wheels

1969 ◽  
Vol 91 (3) ◽  
pp. 891-896 ◽  
Author(s):  
G. E. Novak ◽  
B. J. Eck
Keyword(s):  
Computer Program ◽  
Thermal Stresses ◽  
Three Dimensional ◽  
Transient Temperature ◽  
Shear Stresses ◽  
Radial Temperature ◽  
Closed Form Solutions ◽  
Brake Application ◽  
History Of ◽  
Thin Disks

A numerical solution is presented for both the transient temperature and three-dimensional stress distribution in a railcar wheel resulting from a simulated emergency brake application. A computer program has been written for generating thermoelastic solutions applicable to wheels of arbitrary contour with temperature variations in both axial and radial directions. The results include the effect of shear stresses caused by the axial-radial temperature gradients and the high degree of boundary irregularity associated with this type of problem. The program has been validated by computing thermoelastic solutions for thin disks and long cylinders; the computed values being in good agreement with the closed form solutions. Currently, the computer program is being extended to general stress solutions corresponding to the transient temperature distributions obtained by simulated drag brake applications. When this work is completed, it will be possible to synthesize the thermal history of a railcar wheel and investigate the effects of wheel geometry in relation to thermal fatigue.

Study on Gradual Deformation and Prediction Model of the Mined Overburden

2012 ◽  
Vol 524-527 ◽  
pp. 699-704
Author(s):  
Xiao Gang Xia ◽  
Yun Feng Yang
Keyword(s):  
Boundary Condition ◽  
Plate Theory ◽  
Surface Expression ◽  
Rock Deformation ◽  
Double Trigonometric Series ◽  
Mechanics Model ◽  
Thin Plate Theory ◽  
Gradual Deformation ◽  
Exchange Condition ◽  
Gradual Transformation

Based on the overburden three caving feature, the deformation of mining rock process was devided and the criterion of gradual transformation of each stages deformation were given. Then , combined the thin-plate theory, the differential models were derived for rock deformation in level and similar to level bured condition. The boundary condition of each models and exchange condition between different models were put forward and the gradual mechanics model was set up.The subsidence model before roof collapse was solved by Navier double trigonometric series and the deflection surface expression of rock deformation was put forword. At last, the reliability and practicality of the models was verified by engineering examples.

Normal Loading on a Wedge-Shaped Plate

1955 ◽  
Vol 6 (3) ◽  
pp. 196-204 ◽  
Author(s):  
D. E. R. Godfrey
Keyword(s):  
Thin Plate ◽  
Mellin Transform ◽  
Plate Theory ◽  
Normal Loading ◽  
Thin Plate Theory

SummaryThe equations of thin plate theory are expressed in polar co-ordinates and transformed using the Mellin transform. Problems involving discontinuous and isolated normal loadings may then be solved in the case of the built-in or freely supported wedge-shaped boundary.

Size-Dependent Geometrically Nonlinear Forced Vibration Analysis of Functionally Graded First-Order Shear Deformable Microplates

2016 ◽  
Vol 32 (5) ◽  
pp. 539-554 ◽  
Author(s):  
R. Ansari ◽  
R. Gholami ◽  
A. Shahabodini
Keyword(s):  
Size Effect ◽  
Nonlinear Equations ◽  
Forced Vibration ◽  
Plate Theory ◽  
Couple Stress Theory ◽  
Continuation Method ◽  
Functionally Graded ◽  
Geometrical Parameters ◽  
Governing Equations ◽  
First Order

AbstractIn this paper, a non-classical plate model capturing the size effect is developed to study the forced vibration of functionally graded (FG) microplates subjected to a harmonic excitation transverse force. To this, the modified couple stress theory (MCST) is incorporated into the first-order shear deformation plate theory (FSDPT) to account for the size effect through one length scale parameter, only. Strong form of nonlinear governing equations and associated boundary conditions are obtained using Hamilton's principle. The solution process is implemented on two domains. The generalized differential quadrature (GDQ) method is first employed to discretize the governing equations on the space domain. A Galerkin-based scheme is then applied to extract a reduced set of the nonlinear equations of Duffing-type. On the second domain, through a time differentiation matrix operator, the set of ordinary differential equations are transformed into the discrete form on time domain. Eventually, a system of the parameterized nonlinear equations is acquired and solved via the pseudo-arc length continuation method. The frequency response curve of the microplate is sketched and the effects of various material and geometrical parameters on it are evaluated.

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