Effect Of Transverse Shear Deformation Research Articles

Geometrically exact aeroelastic stability analysis of helicopter composite rotor blades in forward flight

In this paper, a sophisticated method for analyzing the geometrically exact aeroelastic stability of composite rotor blades in forward flight has been proposed. The unique advantage of the proposed method is to use the geometrically exact beam model with the updated Variational Asymptotic Beam Sectional analysis (VABS) for structural modeling to not only calculate the blade cross-sectional properties and blade spanwise deformations more accurately, but also take into account the effect of transverse shear deformation and blade initial twist and curvatures sophisticatedly. The Peters finite state airloads theory and the Peters-He finite state dynamic inflow model are coupled to calculate the three-dimensional unsteady airloads applied on the blade. The auto-pilot trim scheme is adopted to determine the blade pitch controls. The finite element spatial discretization method, the Newmark numerical integration method, the Newton-Raphson method and the moving-block analysis method are combined to solve the established geometrically exact aeroelastic equations and obtain the geometrically exact aeroelastic stability of composite rotor blades. The proposed method is validated by correlation with existing analytical results. The investigations show that the effect of transverse shear deformation on the aeroelastic stability of composite hingeless rotors in forward flight is non-negligible. Blade initial curvatures have significant effect on the aeroelastic stability of composite hingeless rotors in forward flight, which is not only related to the blade elastic couplings, but also to the advance ratio.

Stress analysis of laminated and sandwich beams subjected to concentrated load by using quasi-two-dimensional theory

This study presents a quasi-two-dimensional higher-order shear deformation theory for stress and displacement analysis of isotropic, laminated, and sandwich beams subjected to concentrated load. The assumed displacement field includes the effect of transverse shear and normal deformations. The condition of zero transverse shear stresses on the upper and lower surface of beams is satisfied, hence the present formulation does not require the shear correction factor generally associated with first-order models. By applying the principle of virtual work, the governing equations and boundary conditions for laminated and sandwich beams are derived. Transverse shear and normal stresses are determined from the stress-equilibrium equations of elasticity theory. The transverse stresses obtained using this approach satisfy the continuity condition at layer interfaces and the stress boundary conditions at the external surfaces. The outcomes of the present theory are compared with those of third-order, and first-order shear deformation models, and the classical beam model. The results highlight the significant deviation in displacements and stresses of laminated and sandwich beams when normal strain is included in the model, as compared to the prediction made by lower-order models.

Parameter Identification of Frame Structures by considering Shear Deformation

This paper presents a method to identify the damages in frame structures with slender beams. This method adjusts the parameters of the structure to match the analytical and the measured displacements. The effect of transverse shear deformation on the nodal analytical displacement is analyzed, and the parameter identification of frame structures with slender beams is performed. The results demonstrate that parameter-identification accuracy can be considerably improved by considering the transverse shear deformation in the frame structure with slender beams. The proposed method can accurately identify the damages in frame structures with slender beams using displacement measurements.

Propagation characteristics of guided waves in CNTRCs plates resting on elastic foundations in a thermal environment

This paper investigates the wave propagation behavior of carbon nanotube-reinforced composites (CNTRCs) plates resting on elastic foundations in a thermal environment. The first-order shear deformation plate theory is applied to establish the mathematical model for the CNTRC plates. The effective material properties are calculated using the rule of mixture. Considering the effects of transverse shear deformation and rotary inertia, the governing equations of the CNTRC plates are derived using Hamilton’s principle. Then, the equations of motion are discretized by the Galerkin method and solved analytically by the eigenvalue method. Finally, the effects of various parameters including the temperature variations, the volume fraction of carbon nanotubes (CNTs), elastic foundations and the CNTs’ distribution patterns on the dispersion relations are investigated, it can be found that these parameters have significant effects on the wave propagation of CNTRCs plates.

Evaluation of static and dynamic responses considering thickness stretching effect for layered composite and sandwich arches using exponential shear and normal deformation theory

Present theory investigates the dimensionless vibration frequencies, deformations, and stresses for multilayered and sandwich arches using exponential shear and normal deformation theory (ESNDT). Present studies consider the effect of [1+(z/R)]radius of curvature while selecting the displacement field of arches under the action of uniform load. It includes the effects of transverse shear (γxz≠0) and transverse normal deformation(εz≠0)i.e., effect of thickness stretching. The governing relations are obtained using the Navier's method, and Hamilton's virtual work principle. The proposed layered arches are completely free from shear correction, and its top and bottom surfaces is satisfies zero traction free end conditions. Present study obtained very accurate natural frequencies for layered sandwich arches than other theories because, available literature not considered the thickness stretching effect i.e.,(εz=0). Hence, the present scientific investigation accounts the effect of thickness stretching i.e.,(εz≠0). However, the non-availability of exact elasticity results of bending and dynamic responses for layered sandwich arches. Present theory is obtained vibration frequencies, displacements, and stresses of layered composite and sandwich arches; the results are compared and validates through prior published literature.

Modeling the Thermoelastic Bending of Ferric Oxide (Fe2O3) Nanoparticles-Enhanced RC Slabs.

Nanoparticles, by virtue of their amorphous nature and high specific surface area, exhibit ideal pozzolanic activity which leads to the formation of additional C-S-H gel by reacting with calcium hydroxide, resulting in a denser matrix. The proportions of ferric oxide (Fe2O3), silicon dioxide (SiO2), and aluminum oxide (Al2O3) in the clay, which interact chemically with the calcium oxide (CaO) during the clinkering reactions, influence the final properties of the cement and, therefore, of the concrete. Through the phases of this article, a refined trigonometric shear deformation theory (RTSDT), taking into account transverse shear deformation effects, is presented for the thermoelastic bending analysis of concrete slabs reinforced with ferric oxide (Fe2O3) nanoparticles. Thermoelastic properties are generated using Eshelby's model in order to determine the equivalent Young's modulus and thermal expansion of the nano-reinforced concrete slab. For an extended use of this study, the concrete plate is subjected to various mechanical and thermal loads. The governing equations of equilibrium are obtained using the principle of virtual work and solved using Navier's technique for simply supported plates. Numerical results are presented considering the effect of different variations such as volume percent of Fe2O3 nanoparticles, mechanical loads, thermal loads, and geometrical parameters on the thermoelastic bending of the plate. According to the results, the transverse displacement of concrete slabs subjected to mechanical loading and containing 30% nano-Fe2O3 was almost 45% lower than that of a slab without reinforcement, while the transverse displacement under thermal loadings increased by 10%.

Bending analysis of FGM plates using sinusoidal shear and normal deformation theory

This paper presents the bending analysis of functionally graded material (FGM) plates using sinusoidal shear and normal deformation theory. The in-plane displacements include sinusoidal functions in the thickness coordinate to consider the effect of transverse shear deformation, and transverse displacement includes the effect of transverse normal strain using the cosine function in thickness coordinate. The displacement field of the theory enforces to satisfy shear stress-free boundary conditions on the top and bottom surfaces of the plate with realistic variations across the thickness. Plate material properties vary across thickness directions according to a power law. The boundary value problem of the theory is derived using the principle of virtual work. Simply supported plate bending problems are solved using the Navier solution technique. Response of the plate is obtained with respect to the type of load, type of plate, aspect ratio, and power law index. The results of present theory are compared with those of quasi-3D discrete layer theory and semi-analytical solutions based on the theory of elasticity to ensure the accuracy of theory. The current theory showed excellent agreement with more exact theories in bending response.

Displacement in Deep Propped Cantilever Beam Subjected to Semi-elliptical Load

A fifth order shear deformation theory considering transverse shear deformation effect is presented for displacement in propped cantilever beam. The considered displacement field accounts for non-linear variation corresponding to depth of beam. Governing equations and boundary conditions of the theory are obtained using the standard of virtual work. Numerical results for static deflection analysis for aspect ratio (length to depth) 2 to 50 for isotropic beam is obtained. The results of present theory are in close agreement with those of higher order shear deformation theories.

On the Static Flexural Response of Shear Deformable Isotropic Rectangular Nanobeams under the Parabolic Loading

It is well known that size-dependent effects, which are insignificant for macroscopic structures, become significant for small-scale structures like nanobeams. In addition, effects of the beam transverse shear deformation, which are insignificant for slender beams, become significant for shear deformable beams. Hence the need arises for a beam theory which is simple-to-use as well as which accounts for just-mentioned features of shear deformable nanobeams. Recently, the authors have developed a single variable new first-order shear deformation nonlocal beam theory (NFSDNBT, doi: 10.1007/s40430-019-2128-6). The NFSDNBT is applicable for the flexure of linear isotropic nanobeams undergoing small deformations. Displacement functions of the NFSDNBT give rise to the constant transverse shear strain through the beam thickness. Hence similar to the nonlocal Timoshenko beam theory (NTBT), the NFSDNBT also requires a shear correction factor. In the NFSDNBT, nonlocal differential stress-strain constitutive relations of Eringen are utilized, through which size-dependent effects have been taken into account. These relations relate not only the beam axial stress with the beam axial strain but also the beam transverse shear stress with the beam transverse shear strain. The governing equation of the NFSDNBT is obtained by utilizing beam gross equilibrium equations and it has a strong resemblance with the governing equation of the nonlocal Bernoulli-Euler beam theory (NBEBT). In this paper, the NFSDNBT is utilized for finding the static flexural response of shear deformable isotropic rectangular nanobeams under the action of parabolically distributed transverse loading. For the simply-supported, clamped-clamped and cantilever nanobeams, effects of variations in values of the nonlocal parameter of Eringen and beam thickness-to-length ratio on the maximum non-dimensional beam transverse displacement are presented. Obtained results are compared with corresponding results obtained by utilizing the NTBT and NBEBT so as to demonstrate the efficacy of the NFSDNBT. Along-the-length profiles of the non-dimensional beam transverse displacement for the just-mentioned cases of nanobeams, for various values of the nonlocal parameter and beam thickness-to-length ratio are also presented.

Steady-state and transient vibration analysis of a thermally loaded power law-based functionally graded rotor system with induced porosities

The dynamic analysis of a power law-based functionally graded (FG) rotor-bearing system with induced porosities for non-uniform porosity distributions has been performed using the finite element method. A novel porous rotor element considering the effects of transverse shear deformation, gyroscopic moments, translational and rotational inertial has been developed for non-uniform porosity distributions based on the Timoshenko beam theory. Material properties are graded based on the power law along the radial direction of the FG shaft. The nonlinear temperature distribution law has been considered in conjunction with the power law to vary the temperature across the cross-section of the FG shaft. Material is graded in such a way that the stainless steel is comprised at the core of the FG shaft and the outer periphery of the FG shaft is made of Zirconia. The steady-state and transient time and frequency responses of a thermally loaded porous FG rotor-bearing system have been computed for both types of uneven porosity distribution using the Houbolt time marching technique and Fast Fourier Transform, respectively. The free vibration analysis has also been performed on the porous FG rotor-bearing system for various parameters such as power law index, the volume fraction of porosity and porosity distribution under different thermal gradients. The reduction of critical speed with increased amplitude is observed from the steady-state and transient frequency responses with an increase in volume fraction of porosity and temperature gradients

ДОСЛІДЖЕННЯ ДЕФОРМУВАННЯ БАГАТОШАРОВОЇ ТРАНСВЕРСАЛЬНО-ІЗОТРОПНОЇ ПЛИТИ НА ЖОРСТКІЙ ОСНОВІ ЗА БЕЗЗГИНОВОЮ УТОЧНЕНОЮ КОНТИНУАЛЬНОЮ МОДЕЛЛЮ

The high-precision estimation of the stress-strain state (SSS) of multilayered plates on a rigid foundation under the action of stationary transverse loading is an urgent task. As its includes the calculations of strength and deformability of various homogeneous and multilayer coatings. This is the calculation of road surface on relatively rigid bridge structures, or on a non-deformable underlying layer or calculation protective multilayer coatings of flat structural elements of greater rigidity than coatings, etc. The combining of materials with isotropic and transversal-isotropic physical characteristics into a multilayer package allows creating of the multifunctional designs. The SSS of such structures due to their structural heterogeneity and the relatively low transverse stiffness of the individual layers is significantly associated with the effect of transverse shear deformations and transverse compression deformations. Therefore, the problem of refined modelling of SSS of plates, which takes into account these types of deformations, is an urgent one. Based on the decomposition the SSS of plate into the flexural and unflexural components, it is proposed to optimize the design diagram of deformation a rectangular multilayer plate on a rigid foundation. The essence of optimization is to consider such a design diagram of the plate, in which the SSS of plate would be fully described by only one component, namely the unflexural component of SSS. To do this, instead of the actual design of the multilayer plate, which is deformed without separation from the foundation, it is suggested to consider the design diagram of the plate, which is formed by supplementing it with a symmetric one about the contact surface of the foundation. In this case, the plate will be symmetrically loaded with respect to the middle surface of the plate, and the thickness of the plate will double. The SSS of plate will be unflexural, which greatly simplifies its modeling. For unflexural SSS, a twodimensional and high-degree iterative approximation but three-dimensional by the nature reflection of SSS, model of deformation of multilayer rectangular plates on a rigid foundation with isotropic and transverse-isotropic layers is constructed in an elastic formulation. This model takes full account deformations of transverse shear and of transverse compression at transverse loading of a plate. Calculations of homogeneous and two-layer transverse-isotropic plates on a rigid foundation under the action evenly distributed and localized transverse loads on the surface of a plate are performed by the finite difference method.

Free vibration analysis of multi-layer rectangular plate containing magnetorheological fluid and flexible core layers rested on Winkler-Pasternak foundation

Magneto-rheological (MR) fluids viscosity can be varied by changing the magnetic field intensity. Therefore, they can improve structural rigidity and damping property. The current study presents a free vibration analysis of a multilayer rectangular plate with two layers of MR fluid  and a flexible core layer, rested on a Winkler-Pasternak foundation based on exponential shear deformation theory (ESDT). In this theory, the exponential functions are applied in terms of thickness coordinates to include the effect of transverse and inertial rotational shear deformation. The flexible core displacement is modeled via the second-order Frostig model. The Hamilton’s principle is used to derive the equations of motion. These equations are solved using the Navier method to obtain the natural frequencies of the plate. The accuracy of the derived equations is validated, and the obtained results are compared with a specific case. Finally, the results show that by applying and increasing the magnetic field intensity, the natural frequencies and loss factor increase. With the increase in the mode number for each specific magnetic field intensity, the natural frequency increases and the loss factor decreases. The natural frequencies and loss factor decrease with the increase of the MR layer thickness to overall thickness ratio and the flexible core layer thickness to overall thickness ratio. The natural frequencies increase when the parameters of the foundation increase.

On the static deformation and frequency analysis of functionally graded porous circular beams

Due to the availability of limited literature on functionally graded porous circular beams, the static deformation and free vibration analysis of simply supported functionally graded porous circular beams considering the effects of even and uneven porosities are presented in this paper. The Navier type closed-form solutions are obtained for both static and vibration problems within the framework of the higher-order hyperbolic circular beam theory. The theory considers the effects of transverse shear deformation and rotary inertia. The theory satisfies the transverse shear-stress free boundary conditions at the upper and lower surfaces of the beam. The circular beam is made up of the functionally graded material in which material characteristics vary across the thickness direction using the power-law technique which has been adjusted to approximate the porosity effects with even and uneven distributions. A dynamic version of the virtual work principle is employed to derive the governing differential equations of motion, which are then solved analytically using the Navier solution technique. The non-dimensional values of displacements, stresses, and fundamental frequencies are presented for various values of the power-law index (pi), the radius of curvature (R), porosity volume fraction (α), and type of porosity distribution (even and uneven). The present results are compared in few problems with existing literature and found in good agreement with those. This study also highlights some new results for the reference of future researchers, especially on circular beams.

Bending Analysis of Sandwich Plate Under Concentrated Force Using Quasi-Three-Dimensional Theory

This study presents the quasi-three-dimensional, higher-order shear-deformation theory for the stress and displacement analysis of sandwich plate subjected to concentrated force. Using expansion of thickness coordinate in the displacement field, the effects of both transverse shear and normal deformations are included. The theory satisfies the zero shear stress conditions on the top and bottom surfaces of sandwich plate. Hence, the shear correction factor is not required, generally associated with the First order shear deformation theory (FSDT). The boundary-value problem of the theory is derived using the principle of virtual work. Transverse shear and normal stresses are recovered from the stress-equilibrium equations of the theory of elasticity, and the continuity condition is satisfied at the interface between layers and stress boundary condition at the external surfaces. In addition, this study provides the effects of transverse normal strain on flexural response of sandwich plate. Closed-form solutions for simply supported sandwich plate subjected to concentrated force are obtained for various side-to-thickness ratios. The results of the present theory are compared with third-order shear and normal deformation theory, first-order shear-deformation theory, and the classical plate theory. The inclusion of the effect of transverse normal strain in the theory showed significant deviations in displacements and stresses.

Elastic-plastic buckling analysis of stiffened panel subjected to global bending in forming process

A new method has been proposed in this study for the elastic-plastic buckling analysis of stiffened panels under global bending. In this method, a simplified model of the stiffened panels has been built with the application of two theories of plasticity, the incremental theory (IT) and the deformation theory (DT). The effect of transverse shear deformation through the stiffener thickness has been considered using the Mindlin-Reissner plate theory. The governing differential equations have been solved by the differential quadrature (DQ) method and an iteration process has been adopted due to the non-linearity of material properties in the elastic-plastic buckling analysis. Non-linear finite element (FE) modelling of elastic-plastic buckling analysis has been carried out, and the FE results are between those based on DT and IT in general. When the reciprocal of strain hardening exponent mt increases to 20, the FE results are in a good agreement with DT results. Based on the proposed method and FE simulations, the effect of geometric parameters of stiffened panels (stiffener thickness to height ratio, stiffened panel length to height ratio, width to height ratio, and skin thickness to stiffener thickness ratio) on buckling behaviour in the elastic-plastic region has been investigated and discussed. The proposed method provides an efficient way for parameter optimisation in the structure design of stiffened panels for the aerospace applications.

Modeling of peridynamic beams and shells with transverse shear effect via interpolation method

Peridynamics is a nonlocal theory which can easily handle discontinuities such as cracks, while it is computationally more expensive compared with finite element method (FEM). Beams and shells, as the main units of structural idealization, can tremendously improve the computational efficiency for complex structures. This study presents new peridynamic (PD) beam and shell models with the effect of transverse shear deformation based on the micro-beam bond and the Timoshenko beam theory. Interpolation method is used for describing axial, torsional, bending and transverse shear deformations as well as the corresponding strain energy densities of the micro-beam bond. The micro moduli for PD beam and shell models can be solved spontaneously. The deflections of the PD models for thin/thick beams and shells are captured accurately due to the high-order relationships between the transverse forces/bending moments and the deflections/rotational angles of the micro-beam bond. Also, no restrictions are needed on the material parameters in the PD models. The simulation results prove the high precision of the presented PD beam and shell models.

A couple stress model in non-polynomial framework to examine structural responses of laminated composite micro-plates: An analytical solution

In the present article, the couple stress models in the non-polynomial framework are developed and employed to examine the structural response characteristics of composite micro-plates. The length scale effects in the micro-laminates are considered using the couple stress theory. The trigonometric, inverse-trigonometric, hyperbolic, inverse-hyperbolic and exponential shear shape functions are considered to model transverse shear deformation effects in micro-plates. The Hamilton principle is employed to formulate the governing equations assuming linear structural kinematics. The governing equations are solved for simply supported micro-plates using the Navier solution for the prediction of analytical results. The structural responses of the composite micro-plates are examined for bending, free vibration, and buckling responses. The nondimensionalized deflection, stresses, frequencies, and axial buckling loads are evaluated using the present methodology. The higher buckling modes and free vibration modes are also obtained. The response characteristics due to five plate theories are compared and the applicability of the considered transverse shear deformation theories to model and analyze composite micro-plates is assessed. The effects of various parameters on the structural responses of micro-laminates are examined.

Effect of initial stresses on vibration behavior of functionally graded materials

Abstract Functionally Graded Materials (FGM) is the result of continuous gradation of two or more constituent materials over a certain volume. This type of material overcomes many problems, particularly delamination, cracking and damages which are very frequently found in laminated composite materials. The FGMs are commonly found in space structures, and these structural elements are subjected to various kinds of loadings during its service period, in which in-plane loading is the one which significantly affects the vibration characteristics of the structural elements. In this investigation, the effect of tensile and compressive stresses on vibration characteristics of FG panels has been studied by using finite element technique (FE). For mathematical modelling Poisson's ratio is assumed constant and Young’s modulus of elasticity is assumed to vary according to the power-law distribution of volume fraction of the constituents. The plate is modelled by using 8-noded isoparametric element by considering the effect of transverse shear deformation and rotary inertia. The effect of different factors such as volume fraction index, the thickness of the panel, boundary condition and tensile as well as compressive edge loads are considered to study the vibration behaviour of the FGM plate under tensile and compressive uniaxial edge loads.

Free Vibration Analysis of Multi-Layer Rectangular Plate with Two Magneto-Rheological Fluid Layers and a Flexible Core

In the present article, the free vibration analysis of a multi-layer rectangular plate with two magneto-rheological (MR) fluid layers and a flexible core is investigated based on exponential shear deformation theory for the first time. In exponential shear deformation theory, exponential functions are used in terms of thickness coordinate to include the effect of transverse shear deformation and rotary inertia. The displacement of the flexible core is modeled using Frostig’s second order model which contains a polynomial with unknown coefficients. MR fluids viscosity can be varied by changing the magnetic field intensity. Therefore, they have the capability to change the stiffness and damping of a structure. The governing equations of motion have been derived using Hamilton`s principle. The Navier technique is employed to solve derived equations.  To validate the accuracy of the derived equations, the results in a specific case are compared with available results in the literature, and a good agreement will be observed.  Then, the effect of variation of some parameters such as magnetic field intensity, core thickness to panel thickness ratio and MR layer thickness to panel thickness ratio  on natural frequency of the sandwich panel is investigated.

Stress analysis of smart composite plate structures

In the present study, the electro-mechanical responses of smart laminated composite plates with piezoelectric materials are derived using a two-dimensional (2 D) displacement-based non-polynomial higher-order shear deformation theory. The kinematics of the mathematical model incorporates the deformation of laminates which account for the effects of transverse shear deformation and a non-linear variation of the in-plane displacements using inverse sine hyperbolic function of the thickness coordinate. The equilibrium equations are obtained using the minimization of energy principle known as the principle of minimum potential energy (PMPE) which is also based on a variational approach and the solutions are obtained using Navier’s solution technique for diaphragm supported smart laminated composite plates. The responses obtained in the form of deflection and stresses are compared with three dimensional (3 D) solutions and also with different polynomial and non-polynomial based higher-order theories in the literature. The transverse shear stresses are obtained using 3 D equilibrium equations of elasticity to enhance the accuracy of the present results. Various examples are numerically solved to establish the efficiency of the present model.