scholarly journals STRAIN–DISPLACEMENT EXPRESSIONS AND THEIR EFFECT IN DEFLECTION OF RECTANGULAR PLATE

2020 ◽  
pp. 2-2
Author(s):  
O. M. IBEARUGBULEM ◽  
F. C. ONYEKA ◽  
V.A. BALOGUN ◽  
JOHN. WASIU ◽  
S. E. NNOCHIRI
Keyword(s):  
Mild Steel ◽  
Yield Stress ◽  
Rectangular Plate ◽  
Thick Plate ◽  
Shear Deformation Theory ◽  
Rectangular Plates ◽  
Linear Strain ◽  
Simply Supported ◽  
Nonlinear Stress ◽  
Bending Behaviour

Due to enormous application of thick plate and its relevance in engineering, various theories for plate analysis have been developed using linear strain–displacement expressions. It is proven from previous studies that results obtained using linear strain–displacement expressions may be unreliable for nonlinear stress and bending analyses. In the present paper, nonlinear strain– displacement expressions are employed for rectangular plates subjected to uniform distributed loads to suggest a more reliable refined plate theory that satisfies the continuity of all of the transverse stress components. This theory, which is based on traditional third-order shear deformation theory of plate is presented and applied in a bending analysis of rectangular thick plate. Governing equations and associated boundary conditions of the theory are obtained using the principle of variational calculus. From the formulated expression, the formula for calculation of the actual critical lateral imposed load, q𝑖𝑤, on the plate before deflection reaches the specified maximum specified limit and critical lateral imposed load, q𝑖𝑝, before plate reaches an elastic yield stress were obtained. By solving using the formulated expression, the effect of deflection and crack in a mild steel rectangular plates with opposite edge clamped and the other edge simply supported (CSCS) and simply supported at first and fourth edge and clamped at second edge and free of support at third edge (SCFS) were analysed and compared. This approach overcomes the challenges of the conventional practice in the structural analysis/design, which involves checking of deflection and shear; the process which is proved unreliable. In the result of CSCS plate, the positive value of the critical lateral imposed load, q𝑖𝑤(between 31.08735 N/mm to 155.4414 N/ mm) before deflection reaches the maximum specified limit and the critical lateral imposed load, q𝑖𝑝 (between 193.8246 N/mm to 193.8246 N/mm) before mild steel plate reaches the elastic yield stress, reveals that the plate neither failed in q𝑖𝑤 nor in q𝑖𝑝 for plate span (a) of 1000mm at allowable deflection, wa of 1mm to 5mm. Also, the positive value of critical lateral imposed load q𝑖𝑤 (between 16.23992 N/mm to 81.20424 N/mm) 𝑎𝑛𝑑 q𝑖𝑝(between 115.3523 N/ mm to 115.3523 N/mm) reveals that the plate neither fail in q𝑖𝑤 nor in q𝑖𝑝 for plate span (a) of 1000mm at allowable deflection, wa of 1mm to 5mm for SCFS. This means that the plate structure is safe. It is observed that the value of q𝑖𝑝 is constant at any value of wa for SCFS plate. This means that change in specified deflection limit does not affect the overall performance of SCFS rectangular plate unlike CSCS plate. Hence, it also reveals that the values of critical lateral imposed loads q𝑖𝑤 𝑎𝑛𝑑 q𝑖𝑝 decrease as the length-width ratio increases. This continues until failure occurs. This means that increase in plate width increases the chance of failure in a plate structure.It is concluded that the values of critical lateral load obtained by this theory gives realistic variation of transverse shear stress through the thickness of plate and satisfied the transverse flexibility of the rectangular plate’s condition while predicting the bending behaviour of isotropic thick rectangular plate. Therefore, using this theory it is possible to predict actual load that cause the bending behaviour of isotropic rectangular plate.

Solving the Deflection Equations of Thick Plate under Concentrated Load

2012 ◽  
Vol 594-597 ◽  
pp. 2659-2663
Author(s):  
Dan Zhang
Keyword(s):  
Rectangular Plate ◽  
Thick Plate ◽  
Numerical Results ◽  
Reciprocal Theorem ◽  
Rectangular Plates ◽  
Concentrated Load ◽  
Simply Supported ◽  
Thick Rectangular Plate ◽  
Reciprocal Theorem Method

According to reciprocal-theorem method (RTM), the deflection equations of thick rectangular plate with two edges simply supported and two edges free under concentrated load are obtained in this paper. Simultaneously through the programming computation, the numerical results with actual value are obtained, which further showed the accuracy and superiority of RTM to solve the bending of thick rectangular plates.

Buckling of Rectangular Plates With General Variation in Thickness

1973 ◽  
Vol 40 (3) ◽  
pp. 745-751 ◽  
Author(s):  
D. S. Chehil ◽  
S. S. Dua
Keyword(s):  
Rectangular Plate ◽  
Variable Thickness ◽  
Perturbation Technique ◽  
Thickness Variation ◽  
Rectangular Plates ◽  
Bending Vibration ◽  
Simply Supported ◽  
Buckling Stress ◽  
General Variation ◽  
Plate Of Variable Thickness

A perturbation technique is employed to determine the critical buckling stress of a simply supported rectangular plate of variable thickness. The differential equation is derived for a general thickness variation. The problem of bending, vibration, buckling, and that of dynamic stability of a variable thickness plate can be deduced from this equation. The problem of buckling of a rectangular plate with simply supported edges and having general variation in thickness in one direction is considered in detail. The solution is presented in a form such as can be easily adopted for computing critical buckling stress, once the thickness variation is known. The numerical values obtained from the present analysis are in excellent agreement with the published results.

Fundamental Frequency of Simply Supported Rectangular Plates With Linearly Varying Thickness

1965 ◽  
Vol 32 (1) ◽  
pp. 163-168 ◽  
Author(s):  
F. C. Appl ◽  
N. R. Byers
Keyword(s):  
Rectangular Plate ◽  
Fundamental Frequency ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Varying Thickness

Upper and lower bounds for the fundamental eigenvalue (frequency) of a simply supported rectangular plate with linearly varying thickness are given for several taper ratios and plan geometries. These bounds were determined using a previously published method which yields convergent bounds. In the present study, all results have been obtained to within 0.5 percent maximum possible error.

Stability of Rectangular Plates Under Shear and Bending Forces

1936 ◽  
Vol 3 (4) ◽  
pp. A131-A135 ◽  
Author(s):  
Stewart Way
Keyword(s):  
Critical Load ◽  
Rectangular Plate ◽  
Bending Moment ◽  
The Other ◽  
Rectangular Plates ◽  
Bending Stress ◽  
Simply Supported ◽  
Tension And Compression ◽  
The Way

Abstract The author first discusses the problem of a plane, simply supported rectangular plate loaded by shearing forces in the plane of the plate on all four edges. There are two stiffeners attached one third and two thirds of the way along the plate. The critical load is calculated for various stiffener rigidities. Also, the rigidity necessary to keep the stiffeners straight when the plate buckles is found. This stiffener rigidity is found to be slightly larger than that necessary for a plate with one stiffener and the same panel dimensions as the plate with two stiffeners. The second problem discussed by the author is that of a plane, simply supported rectangular plate loaded by uniformly distributed edge shearing forces in the plane of the plate and linearly distributed tension and compression in the plane of the plate at the ends. The end forces vary from tension hσo, at one corner to—hσo, at the other corner, so that their resultant is a bending moment. The presence of the edge shearing forces is found to diminish the critical bending stress in this case. Calculations are made for various magnitudes of bending and shearing forces for plates of various proportions.

scholarly journals CRITICAL LATERAL LOAD ANALYSIS OF RECTANGULAR PLATE CONSIDERING SHEAR DEFORMATION EFFECT

2020 ◽  
pp. 16-27
Author(s):  
F. C. Onyeka
Keyword(s):  
Potential Energy ◽  
Shear Deformation ◽  
Rectangular Plate ◽  
Thick Plate ◽  
Shear Deformation Theory ◽  
Lateral Load ◽  
Width Ratio ◽  
Total Potential ◽  
Transverse Loads ◽  
Total Failure

This work present flexural analysis of rectangular plate subjected to uniform distributed transverse loads using displacement and third-order shear deformation theory. The aim of this study is to establish the formula’s for calculation of the critical lateral imposed load of the plate before deflection reaches the specified maximum specified limit q𝑖𝑤 and critical lateral imposed load before plate reaches an elastic yield point q𝑖𝑝. The essence is to ensure that deflection does not exceed specified maximum limit and the plate shear not exceeding the elastic yielding point. Furthermore, this approach overcomes the challenges of the conventional practice in the structural analysis/design which involves checking of deflection and shear; the process which is proved unreliable. Total potential energy equation of a thick plate was formulated from the static elastic theory of the plate. The formulated potential energy was in the same way used by the method of direct variation to obtain the coefficient of deflection and shear deformation. This expression was applied to solve bending problem of two different types of rectangular thick plates. The plates has one edge clamped and other three edges simply supported (CSSS). From the result obtained in this work among the two types of plate, it is observed that the value of q𝑖𝑝 if greater than that of q𝑖𝑤. It can be said that the failure of plate in q𝑖𝑤 is like a warning requesting maintenance whereas failure in q𝑖𝑝 means total failure and cannot be maintained. Hence, failure in deflection (q𝑖𝑤) is seen in the plate into consideration. The numerical analysis obtained, it is found that if the value of critical lateral imposed load (q𝑖𝑤 and q𝑖𝑝) increase as the specified thickness (t) of plate increases and decrease as the length to width ratio increases. This implies that as we increase the thickness and allowable deflection improve the safety in the plate, whereas an increase in the span (length) of the plate increases the failure tendency of the plate structure. Furthermore, effects of aspect ratio of the critical lateral load of isotropic plates are investigated and discussed. It is concluded that the values of critical lateral load obtained by this theory achieve accepted transverse shear stress to the thickness of plate variation and satisfied the transverse flexibility of the condition of the plate while predicting the be characteristics for the CSSS isotropic rectangular thin or thick plate.

Free Vibration of Rectangular Plates Stiffened With Viscoelastic Beams

1985 ◽  
Vol 52 (2) ◽  
pp. 397-401 ◽  
Author(s):  
K. Ohtomi
Keyword(s):  
Free Vibration ◽  
Rectangular Plate ◽  
Convenient Method ◽  
Laplace Transformation ◽  
Equation Of Motion ◽  
Frequency Equation ◽  
Rectangular Plates ◽  
Dirac Delta ◽  
Simply Supported ◽  
Delta Functions

This investigation treats the free vibration of a simply supported rectangular plate, stiffened with viscoelastic beams. Using a convenient method in which the effects of beams are expressed with Dirac delta functions, the equation of motion can be expressed by only one equation. The frequency equation is obtained by applying the Laplace transformation to the equation of motion. The effects of the volume and the number of beams on the frequency and the logarithmic decrement are clarified.

Vibration of a multiphase magneto-electro-elastic simply supported rectangular plate subjected to harmonic forces

2016 ◽  
Vol 28 (4) ◽  
pp. 451-467 ◽  
Author(s):  
Alireza Shooshtari ◽  
Soheil Razavi
Keyword(s):  
Rectangular Plate ◽  
Shear Deformation Theory ◽  
Time History ◽  
Phase Portraits ◽  
Multiple Time ◽  
Closed Form Solutions ◽  
Simply Supported ◽  
Frequency Responses ◽  
Plate Size ◽  
Nonlinear Forced Vibration

Nonlinear forced vibration of a magneto-electro-elastic rectangular plate is studied based on the first-order shear deformation theory. The excitation force is harmonic, the boundary condition is considered to be immovable simply supported, and the plate rests on a viscoelastic foundation. The electric and magnetic fields are assumed to be applied along the thickness direction, and different magneto-electric boundary conditions are considered. Magneto-electric behavior of the plate is modeled using Gauss’ laws for electrostatics and magnetostatics. The system is discretized using Galerkin method and then multiple time scale method is used to solve the obtained equation analytically. As a result, closed-form solutions are obtained for the frequency responses of the plate in the primary and subharmonic resonances. Time history and phase portraits of the plate are also obtained numerically. Some examples are carried out to validate the proposed model and to investigate the effects of electric and magnetic potentials, material properties, and plate size on the frequency responses of these smart multiphase plates.

Transverse Vibration Characteristics of Simply Supported Moving FGM Rectangular Plate

2011 ◽  
Vol 243-249 ◽  
pp. 5374-5380 ◽  
Author(s):  
Ruan Miao ◽  
Zhong Min Wang ◽  
Jing Hu Feng
Keyword(s):  
Rectangular Plate ◽  
Transverse Vibration ◽  
Differential Quadrature Method ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Mapping Technique ◽  
Power Law Distribution ◽  
Simply Supported ◽  
Moving Speed

The present paper investigates the dynamic characteristics and stability of moving functionally graded material rectangular thin plate. Based on Voigt model, the material properties are assumed to vary continuously through their thickness according to a power-law distribution of the volume fractions of the plate constituents. By the first order shear deformation theory, the differential equations of motion of the moving FGM rectangular plate are derived. The vibration frequencies are obtained from the solution of a generalized eigenvalue problem. Entire computational work is carried out in a normalized square domain obtained through an appropriate domain mapping technique. Results of the reduced problem revealed excellent agreement with other studies. The dimensionless complex frequencies of the moving FGM rectangular plate with four edges simply supported are calculated by the differential quadrature method. The effects of gradient index, aspect ratio, and dimensionless moving speed on the transverse vibration and stability of the moving FGM rectangular plate are analyzed. Results are furnished in dimensionless amplitude–frequency vs. dimensionless moving speed in the form of curves and pictorial representations of some vibration mode shapes are made.

scholarly journals DYNAMICS OF RECTANGULAR ORTHOTROPIC PLATES USING CHARACTERISTIC ORTHOGONAL POLYNOMIAL – GALERKIN’S METHODS

2016 ◽  
Vol 36 (1) ◽  
pp. 50-56
Author(s):  
NN Osadebe ◽  
CM Attama ◽  
OA Oguaghamba
Keyword(s):  
Orthogonal Polynomial ◽  
Rectangular Plate ◽  
Natural Frequencies ◽  
Rectangular Plates ◽  
Trial And Error ◽  
Shape Functions ◽  
Orthotropic Plates ◽  
Approximate Methods ◽  
Simply Supported ◽  
Polynomial Theory

The assumed deflection shapes used in the approximate methods such as in the Galerkin’s method were normally formulated by inspection and sometimes by trial and error, until recently, when a systematic method of constructing such a function in the form of Characteristic Orthogonal Polynomial (COPs) was developed by Bhat in 1985. In the vibrational analyses of orthotropic rectangular plates with different boundary conditions, the study used the characteristic orthogonal polynomial theory to obtain satisfactory approximate shape functions for these plates. These functions were applied to Galerkin indirect varational method to obtain new set of fundamental natural frequencies for these plates. The results were reasonable when compared with those in the previous work. All round simply supported thin rectangular plate (SSSS), rectangular clamped plated (CCCC) and rectangular plate with one edge clamped and all others edges simply supported (CSSS) gave 5.172, 9.429 and 6.202 natural frequencies in rad /sec respectively at 0.05%, 0.0% and 22.93% difference with the previous[3] results5.170rad/sec, 9.429rad/sec and 8.048rad/sec  for SSSS, CCCC and CSSS. For others like: rectangular plate with one edge simply supported and all other edges clamped (CCSC), rectangular plate simply supported at two opposite sides and clamped at the others (CSCS) and rectangular plate clamped at two adjacent sides and simply supported at the others (CCSS) with no available results, their natural frequencies obtained are 8.041rad/sec, 6.272rad/sec and 7.106rad/sec respectively. http://dx.doi.org/10.4314/njt.v36i1.8

Bending of Thick Rectangular Plates with Different Boundaries under Concentrated Load

2010 ◽  
Vol 163-167 ◽  
pp. 1440-1444
Author(s):  
Ying Jie Chen ◽  
Gang Li ◽  
Zhen Xian Zhang ◽  
Bao Lian Fu
Keyword(s):  
Analytical Solutions ◽  
Thick Plate ◽  
Engineering Application ◽  
Reciprocal Theorem ◽  
Rectangular Plates ◽  
Concentrated Load ◽  
Simply Supported ◽  
The Third ◽  
Thick Rectangular Plate ◽  
Reissner’S Theory

Reciprocal theorem method (RTM) is generalized to solve the problem of bending of thick rectangular plate under concentrated load with four edges fixed and with two opposite edges fixed, the third edge simply supported, and the fourth edge free based on Reissner’s theory. The analytical solutions of the thick plate are given, and the relevant date and diagram are given to guidance engineering application.

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