Shear Correction Coefficients Research Articles

Advanced Thick FGM Plates-Cylindrical Shells in Linear Unsteady Supersonic Flow

Introduction:: Thick functionally graded (FG) plates-cylindrical shells with third-order shear deformation theory (TSDT) of displacement models under thermal vibration in linear unsteady supersonic flow are investigated by using the generalized differential quadrature (GDQ) method. Method:: The nonlinear coefficient term of TSDT models and the advanced nonlinear shear correction coefficient expression are used to study the thermal vibration of thick FG plates and cylindrical shells in linear unsteady supersonic flow. At the intersection of FG plates-cylindrical shells, the displacements, stress resultants, and moment resultants could be reasonably assumed in the continuity condition for the differential volume in linear unsteady supersonic flow, respectively. Result:: The dynamic equilibrium differential equations of thick FG plates and cylindrical shells in linear unsteady supersonic flow, respectively in terms of partial derivatives of displacements and shear rotations subjected to partial derivatives of thermal loads, mechanical loads consist of the linear supersonic aerodynamic pressure and inertia terms can be presented. Two parametric effects including environment temperature and functionally graded material (FGM) power law index on the thermal stress and center displacement of thick FG plates-cylindrical shells in linear unsteady supersonic flow are investigated. Conclusion:: The linear unsteady supersonic flow over the outer surface of thick FG platescylindrical shells provides the additional aerodynamic effect on the values of displacements and stresses.

Thermal Vibration of Thick FGM Conical Shells by Using Third-Order Shear Deformation Theory.

A time-dependent third-order shear deformation theory (TSDT) approach on the displacements of thick functionally graded material (FGM) conical shells under dynamic thermal vibration is studied. Dynamic equations of motion with TSDT for thick FGM conical shells are applied directly with the partial derivative of variable R*θ in the curve coordinates (x, θ, z) instead of y in the Cartesian coordinates (x, y, z) for thick FGM plates, where R* is the middle-surface radius at any point on conical shells. The generalized differential quadrature (GDQ) numerical method is used to solve the dynamic differential equations in equilibrium matrix forms under thermal loads. It is the novelty of the current study to identify the parametric effects of shear correction coefficient, environment temperature, TSDT model, and FGM power law index on the displacements and stresses in the thick conical shells only subjected to sinusoidal heating loads. The physical parts with values on the length-to-thickness ratio equals 5, and 10 FGMs can be used in an area of an airplane engine that usually operates near more than 1000 K of temperatures when the thermal stress is considered and affected. The important findings of the presented study are listed as follows. The values of normal stress are in decreasing tendencies with time in cases when the coefficient c1 equals 0.925925/mm2 in TSDT and length-to-thickness ratio equals 5. The shear stress values in x plane z direction on the minor middle-surface radius (r) equals the major middle-surface radius (R) over 8 and length-to-thickness ratio equals to 5 can withstand T = 1000 K of pressure.

Advanced frequency of thick FGM cylindrical shells with fully homogeneous equation

The effects of shear deformation theory and improved shear correction factors on the advanced computation of frequencies by using the fully homogeneous equation for thick functionally graded material (FGM) circular cylindrical shells are studied. It is quite reasonable to consider the extra advanced effect of third-order shear deformation theory (TSDT) of displacements on the varied shear correction coefficient. The values of advanced nonlinear shear correction coefficient are usually functions of the nonlinear coefficient term in TSDT, power-law exponent parameter, and environment temperature. The main achievements in the nonlinear case of displacements and with the varied value of shear correction coefficients, the non-dimensional fundamental frequencies are estimated, investigated, and compared with the values in the linear case. This paper aims to study the fundamental frequencies of very thick FGM cylindrical shells under the advanced effects of shear deformation theory and improved shear correction factors.

Advanced Frequency Study of Thick FGM Cylindrical Shells by Using TSDT and Nonlinear Shear

For the advanced frequency study of thick functionally graded material (FGM) circular cylindrical shells, it is interesting to consider the extra effects of nonlinear coefficient term in third-order shear deformation theory (TSDT) of displacements on the calculation of varied shear correction coefficient. The formulation for the advanced nonlinear shear correction coefficient are based on the energy equivalence principle. The values of nonlinear shear correction coefficient are usually functions of nonlinear coefficient term of TSDT, power-law exponent parameter and environment temperature. The free vibration frequencies of thick FGM circular cylindrical shells are investigated with the simply homogeneous equation by considering that simultaneous effects of the TSDT, the nonlinear shear correction coefficient of transverse shear force and the two direction of mode shapes. The novelty is more important and reasonable for the thick FGM circular cylindrical shells, especially for the ratio of length to thickness is five by considering the effects of advanced nonlinear shear correction coefficient and nonlinear coefficient term of TSDT on the advanced calculation of fundamental first natural frequencies.

Comparative study on free vibration analysis of rotating bi-directional functionally graded beams using multiple beam theories with uncertainty considerations

The present study investigates the free vibration behavior of rotating beams made of functionally graded materials (FGMs) with a tapered geometry. The material properties of the beams are characterized by an exponential distribution model. The stiffness and mass matrices of the beams are derived using the principle of virtual energy. These matrices are then evaluated using three different beam theories: Bernoulli–Euler (BE) or Classical Beam Theory (CBT), Timoshenko (T) or First-order Shear Deformation Theory (FSDT), and Reddy (R) or Third-order Shear Deformation Theory (TSDT). Additionally, the study incorporates uncertainties in the model parameters, including rotational velocity, beam material properties, and material distribution. The mean-centered second-order perturbation method is employed to account for the randomness of these properties. To ensure the robustness and accuracy of the probabilistic framework, numerical examples are presented, and the results are compared with those obtained through the Monte Carlo simulation technique. The investigation explores the impact of critical parameters, including material distribution, taper ratios, aspect ratio, hub radius, and rotational speed, on the natural frequencies of the beams is explored within the scope of this investigation. The outcomes are compared not only with previously published research findings but also with the results of 3-Dimensional Finite Element (3D-FE) simulations conducted using ANSYS to validate the model’s effectiveness. The comparisons demonstrate a strong agreement across all evaluations. Specifically, it is observed that for thick beams, the results obtained from FSDT and TSDT exhibit a greater agreement with the 3D-FE simulations compared to CBT. It is shown that the coefficient of variation (C.O.V.) of first mode eigenvalue of TSDT, FSDT and CBT are approximately identical for random rotational velocity and discernible deviations are noted in CBT compared to FSDT and TSDT in the case of random material properties. The findings suggest that TSDT outperforms FSDT by eliminating the need for a shear correction coefficient, thereby establishing its superiority in accurately predicting the natural frequencies of rotating, tapered beams composed of FGMs.

Thermal vibration analysis of cracked nanobeams submerged in elastic foundations by nonlocal continuum mechanics

Understanding the mechanical behavior of nanoscale structures is crucial in the development of advanced nanotechnologies. In this study, a novel approach to investigate the thermal lateral vibration of cracked nanobeams immersed in an elastic matrix is investigated. For this purpose, Reddy’s third-order shear deformation theory (TSDT) is considered. In contrast to Timoshenko beam theory (First-order Shear Deformation Theory, FSDT), TSDT does not depend on a shear correction coefficient. The nano-scale effect is modeled using Eringen’s nonlocal continuum mechanics theory. The nonlocal form of the governing equation is obtained through the application of Hamilton’s principle. The weak form of the finite element global mass and stiffness matrices are obtained using Lagrange linear and Hermitian cubic interpolation. To model the crack in bending vibration, two rotational springs are used for TSDT, unlike the use of a single rotational spring in traditional Bernoulli–Euler (Classical Beam Theory, CBT) and FSDT. The stiffness of the springs is adjusted based on the severity of the crack. The influences of the nonlocal parameter, beam slenderness, position of crack, crack severity, Pasternak and Winkler foundation parameters, thermal effects and boundary conditions on the natural frequencies are investigated. The model’s outcomes are compared with findings from prior publications, demonstrating a strong level of agreement. This study contributes to the growing research on nanostructures by presenting a novel approach to understanding the dynamics of cracked nanobeams using Reddy beam analysis-based solutions.

Nonlinear Thermal/Mechanical Buckling of Orthotropic Annular/Circular Nanoplate with the Nonlocal Strain Gradient Model.

This article presents the nonlinear investigation of the thermal and mechanical buckling of orthotropic annular/circular single-layer/bilayer nanoplate with the Pasternak and Winkler elastic foundations based on the nonlocal strain gradient theory. The stability equations of the graphene plate are derived using higher-order shear deformation theory (HSDT) and first-order shear deformation theory (FSDT) considering nonlinear von Karman strains. Furthermore, this paper analyses the nonlinear thermal and mechanical buckling of the orthotropic bilayer annular/circular nanoplate. HSDT provides an appropriate distribution for shear stress in the thickness direction, removes the limitation of the FSDT, and provides proper precision without using a shear correction coefficient. To solve the stability equations, the differential quadratic method (DQM) is employed. Additionally, for validation, the results are checked with available papers. The effects of strain gradient coefficient, nonlocal parameter, boundary conditions, elastic foundations, and geometric dimensions are studied on the results of the nondimensional buckling loads. Finally, an equation is proposed in which the thermal buckling results can be obtained from mechanical results (or vice versa).

An improved equivalent beam model of large periodic beam-like space truss structures

Space truss structures are essential components for space-based remote sensing loads with high spatial and temporal resolutions. To achieve high-precision vibration control, an accurate and efficient dynamics model is essential. In addition to the current equivalent beam model (EBM) based on the classical continuum theory, an improved equivalent beam model (IEBM) is proposed that considers the impact of the distinction between trusses and beams on torsional and shear deformations, as well as the impact of shear deformation on flexural rigidity. According to the displacement expressions of spatial beams, torsional, shear, and bending correction coefficients are introduced to derive expressions of strain energy and kinetic energy. The energy equivalence principle is then utilized to calculate the elasticity and inertia matrices, and dynamics equations are established using the finite element method. Subsequently, an IEBM is constructed by employing the particle swarm optimization approach to determine the correction coefficients with the truss natural frequency as the optimization target. The natural vibration characteristics of the structure are estimated for various material properties. Compared with the full-scale finite element model, the EBM reaches a maximum error of 80% for a low modulus of elasticity, while the maximum error of the IEBM is less than 2% for any given parameters, indicating its superior accuracy to the EBM.

FREE VIBRATION ANALYSIS OF COMPOSITE LAMINATED PLATES USING HOST 12

This paper presents an application of a Higher Order Shear Deformation Theory (HOST 12) to problemof free vibration of simply supported symmetric and antisymmetric angle-ply composite laminated plates.The theoretical model HOST12 presented incorporates laminate deformations which account for the effectsof transverse shear deformation, transverse normal strain/stress and a nonlinear variation of in-planedisplacements with respect to the thickness coordinate – thus modeling the warping of transverse crosssections more accurately and eliminating the need for shear correction coefficients. Solutions are obtained inclosed-form using Navier’s technique by solving the eigenvalue equation. Plates with varying number oflayers, degrees of anisotropy and slenderness ratios are considered for analysis. The results compared withthose from exact analysis and various theories from references

A new higher-order mixed four-node quadrilateral finite element for static bending analysis of functionally graded plates

In this paper, a new C0 four-node quadrilateral plate element (MiQ4) is proposed for the static bending analysis of functionally graded (FG) plates based on a C0 higher-order shear deformation theory. By introducing a quadratic shape function for higher-order terms and a constant shear result, which is eliminated at the element level, the transverse shear stresses are free on two surfaces of the plates. The present finite element model has high accurateness and a fast rate of convergence, especially for both thin and very thin plates. In addition, the proposed element is insensitive to distorted mesh, hence, it is compatible with the investigation of complex geometrical plates. Although the proposed element does not need any shear correction coefficients and selected or reduced integrations, the shear-locking phenomenon does not appear even for very thin plates. Therefore, the computation cost and time are reduced and the spurious deformation or hourglass modes are avoided. A large number of examples have been exhibited that help to validate the convergence rate, correctness, and reliability of the proposed element. Then the present element is applied for analysis of FG structures to demonstrate the potential applications of the present plate element. Significantly, the paper provides a good element that can be used in many areas of engineering for the design, testing, analysis, and optimization of complex structures to reduce the computation cost and time as well as increase the reliability of the numerical results.

Structural behaviour and bending analysis of rectangular thick plate using a new 3-D modified trigonometric displacement model

In this work, a new 3-D modified trigonometric displacement model was used to study the structural behavior and bending analysis of rectangular thick plate which was clamped in one edge and other three edges simply supported. The theoretical model whose formulation is based on static elastic principle as already reported in the literature are presented herein, obviating the shear correction coefficients while considering shear deformation effect and transverse normal strain/stress in the analysis. The equilibrium equations are obtained using 3-D kinematic and constitutive relations. An exact solution of deflection and rotation are obtained from the equilibrium equation using the general variational principle. The minimization, energy equation yields the general equation which was used to obtain the 3-D trigonometric displacement model of the plate. The percentage difference between the present work and those of 2-D Refined Plate Theory (RPT) with an assumed displacement and 2-D Refined plate theory (RPT) with derived function is 1.43% and 5.15% respectively. More so, percentage difference between the present work and those using polynomial shape function is 3.29%. The result showed that the 3-D trigonometric model for the present work predicts the vertical displacement and the stresses more accurately than RPT and polynomial displacement model. It is concluded that the 3-D trigonometric model gives an exact solution unlike polynomial and can be used with confidence in the analysis of thick plate under the particular initial condition.

Thermal Vibration of Thick FGM Circular Cylindrical Shells by Using TSDT

The generalized differential quadrature (GDQ) method is used to study the thermal vibration of thick functionally graded material (FGM) circular cylindrical shells. The nonlinear coefficient effect of third-order shear deformation theory (TSDT) model of displacements on the thermal equations of motion is considered. The parametric effects of FGM power law index and environment temperature on the time responses of stresses and displacements are considered. The linear varied effect of shear correction coefficients on the calculation of stiffness integration is also considered. The simply homogeneous equation is used to find the values of vibration frequency, also the dynamic GDQ discrete equation in matrix form are used to find the values of time response and transient response for the thick FGM circular cylindrical shells.

The 100th Anniversary of the Timoshenko–Ehrenfest Beam Model

The 100th Anniversary of the Timoshenko–Ehrenfest Beam Model

Advanced dynamic thermal vibration of thick FGM plates-cylindrical shells

The third-order shear deformation theory (TSDT) effects on the thick functionally graded material (FGM) plates-cylindrical shells under thermal vibration are investigated by using the generalized differential quadrature (GDQ) method. The nonlinear coefficient c1 term of TSDT displacement field is used to obtain the equations of motion and the advanced nonlinear shear correction coefficient expression for thermal vibration of thick FGM plates and cylindrical shells, respectively. At the intersection of FGM plates-cylindrical shells, the displacements, stress resultants and moment resultants could be considered and reasonably assumed in the continuity condition for the differential volume in the plates and cylindrical shells, respectively. The effects of advanced nonlinear shear correction, FGM power law index and environment temperature on the thermal stress and center displacement of FGM plates-cylindrical shells are investigated.

Advanced frequency analysis of thick FGM plates using third-order shear deformation theory with a nonlinear shear correction coefficient

The effects of third-order shear deformation theory (TSDT) displacements and advanced nonlinear varied shear correction coefficient on the free vibration frequency of thick functionally graded material (FGM) plates under environment temperature are studied. The nonlinear coefficient term of TSDT displacements is included to derive the advanced equation of nonlinear varied shear correction coefficient for the thick FGM plates. The determinant of the coefficient matrix in dynamic equilibrium differential equations under free vibration can be represented into fully homogeneous equation and the natural frequency can be found. The parametric effects of nonlinear coefficient term of TSDT, environment temperature and FGM power law index on the natural frequency of thick FGM plates are investigated.

Vibration analysis of cracked beam based on Reddy beam theory by finite element method

The lateral vibration of cracked isotropic thick beams is investigated. Generally, the analysis of thick beam based on line elements can be undertaken using either Timoshenko beam theory or a third order beam theory (TSDT) such as Reddy beam theory. TSDT is superior to Timoshenko beam theory, as it eliminates the need for a shear correction coefficient. However, there is no available solution for a cracked beam based on Reddy beam theory which is the main focus of this paper. The investigated beam is divided into several elements where their stiffness and mass matrices are derived using Reddy beam theory. Each element has two nodes with 3 degrees of freedom (1 lateral and 2 rotational) at each node. The crack was modeled using two rotating springs with stiffnesses that varied with crack depth. These elements are used to connect the adjacent beam elements. The impact of boundary conditions, slenderness ratio, crack location, and crack depth are investigated. In addition, experimental and finite element analyses of cracked steel beams is undertaken to validate the results of the present model. The results show that the maximum deviation between the analytical and the experimental results is less than 3% up to the third mode shape.

Static Elastic Bending Analysis of a Three-Dimensional Clamped Thick Rectangular Plate using Energy Method

Analytical formulations and solutions for the thick rectangular plate static analysis with clamped support based on a three-dimensional (3-D) elasticity theory is developed using the energy method. The theoretical model, whose formulation is based on the static elastic principle as already reported in the literature, is presented herein to obviate the shear correction coefficients while considering shear deformation effect and transverse normal strain/stress in the analysis. The equilibrium equations are obtained using 3-D kinematic and constitutive relations. The deflection and rotation functions, which are the solutions of the equilibrium equation, are obtained in closed form using a general variational technique for solving the boundary value problem. The minimization energy equation yields the general equation which was used to obtain the theoretical model for the deflection and stresses of the plate. The results are compared with the available literature and the results-computed trigonometric displacement function shows that this 3-D predicts the vertical displacement and the stresses more accurately than previous studies considered in this paper. The result showed that the percentage difference between the present work and those of 2-D Mindlin FSDT, 2-D numeric analysis, and 2-D HSDT of polynomial shape functions was about 3.02%, 0.62%, and 0.33%, respectively. It is concluded that the 3-D trigonometric model gives an exact solution, unlike other 2-D theories, and can be used for clamped-supported thick plate analysis. Doi: 10.28991/HIJ-2022-03-03-03 Full Text: PDF

Seismic analysis of a I-sectioned steel frame based on an improved time-domain spectral element method

An improved time-domain spectral element method (ITSEM) is developed for seismic analysis of I-sectioned steel structures in this paper. The developed method considers the effect of section type on spectral stiffness using shear correction coefficient. In the calculation process of the time-domain spectral stiffness and mass matrices, the invariant integral operations of the element matrices are extracted and saved in advance. Thus, the global mass and stiffness matrices of the structures can be obtained only by simple addition, subtraction, multiplication and division operations, which greatly improves computational efficiency. Next, the natural frequencies of a I-sectioned beam are computed and compared with the finite element method (FEM) and analytical results to validate the ITSEM. Then, the seismic responses of a ten-story steel frame using ITSEM are obtained and compared with those by FEM, which proves the effectiveness, accuracy and high efficiency of ITSEM in elastic seismic analysis.

Buckling analysis of thick plates using a single variable simple plate theory

Buckling analysis of thick plates has been carried out herein by using a single variable simple plate theory. Theory used herein is a third order shear deformation plate theory which uses a single displacement function for the complete formulation of plates. Plate formulation is governed by only one governing differential equation. Governing equation of the theory has close resemblance to that of Classical Plate Theory. Thus, plate problems can be solved in the similar lines as in case of classical plate theory. Plate theory used herein does not require a shear correction coefficient. To check the efficacy of the theory buckling analysis of simply supported thick rectangular plates is carried out. Critical buckling loads for simply supported plates are evaluated and the results obtained are compared to other shear deformation plate theories. Buckling load results are found to be in good agreement with other plate theory results.

An accurate and refined beam model fulfilling the shear and the normal stress traction condition

This paper presents an analytical refined beam model taking into account the axial, the shear and the transverse normal stress distributions. Based on the equilibrium equations in local form and an iterative procedure for the solution of the bipotential equation, the beam differential equations for the axial and the transverse motion are derived. From a mathematical point of view the formal structure of the obtained differential equations is identical to the Timoshenko beam equations but with increased accuracy. This is achieved by splitting the ansatzfunctions for the displacement field into three weight-averaged degrees of freedom (for the axial and the transverse deflection and the rotation) and residual deflections, so-called higher order terms which depend on the state of stress. It is shown that the shear traction condition is always fulfilled and even the normal traction condition may hold. Furthermore an analogy to the Timoshenko beam equations is given and it is shown that the shear correction coefficient depends on Poisson’s ratio. Finally analytical results of the refined beam theory are compared to several analytical beam theories and two-dimensional finite element results for statically determinate and indeterminate beam configurations.