Semi-analytical Solution Procedure Research Articles

Natural frequency responses of hybrid polymer/carbon fiber/FG-GNP nanocomposites paraboloidal and hyperboloidal shells based on multiscale approaches

This paper is devoted to obtain the vibrational behavior of doubly curved shells with paraboloidal and hyperboloidal geometries using an efficient semi-analytical solution method. To obtain the governing differential equations of the shells, the First-order Shear Deformation Theory (FSDT) is used. In addition, the shell structure is composed of a new hybrid three phases nanocomposite material. The material includes three parts: (1) polymer matrix, (2) carbon macroscale fiber, and (3) Graphene NanoPlatelets (GNP) nanoscale filler. Three different multiscale approaches including (1) bridging model, (2) Mori-Tanaka scheme, and (3) generalized self-consistent model are employed for homogenization of hybrid matrix and macroscale carbon fibers. The governing equation of motions are obtained using Hamilton's principles and Green-Gauss theory. Afterwards, an efficient semi-analytical solution procedure entitled Generalized Differential Quadrature Method (GDQM) is used for solving the governing differential equations of the structure. Finally, the frequency parameter of the structure is achieved for different states of boundary conditions and geometric properties. Moreover, the results are verified with the responses of other references obtained for a paraboloidal (Cap) shell structure. It is worth mentioning that the effect of homogenization approach, boundary conditions, volume fraction of carbon fibers and distribution pattern of GNP within the polymer matrix are studied numerically. It is concluded that the solution method is accurate, efficient and capable to obtain the natural frequency of doubly curved shells.

A numerical study on an infinite linear elastic Bernoulli-Euler beam on a viscoelastic foundation subjected to harmonic line loads

This paper presents a numerical study on the low-amplitude responses of an infinite Bernoulli-Euler beam resting on a viscoelastic foundation subjected to harmonic line loads. To simulate the linear response, a semi-analytical solution procedure that was theoretically proposed by Jang (2016) is utilized and several numerical experiments are conducted to investigate the influence of key model parameters characterizing stiffness and damping. The properties of the viscoelastic foundation are based on theoretical and empirical values for cohesionless sand type foundation. According to the numerical experiments, the obtained responses are compared with those from the closed-form solution and found to have a good agreement with them.

Three dimensional mechanical behaviors of in-plane functionally graded plates

A semi-analytical solution procedure to investigate the distributions of displacement and stress components in the in-plane functionally graded plates based on the scaled boundary finite element method (SBFEM) in association with the precise integration algorithm (PIA) is developed in this paper. The proposed approach is applicable to conduct the flexural analysis on functionally graded plates with various geometric configurations, boundary conditions, aspect ratios and gradient functions. The elastic material parameters of functionally graded plates discussed here are mathematically formulated as power law, exponential and trigonometric functions varied along with the in-plane directions in a continuous pattern. Only a surface of the plate parallel to the middle plane is required to be discretized with two dimensional high order spectral elements, which contributes to reducing the computational expense. By virtue of the scaled boundary coordinates, the virtual work principle and the internal nodal force vector, the basic equations of elasticity are converted into a first order ordinary differential SBFEM matrix equation. The general solution of the governing equation is analytically expressed as a matrix exponential with respect to the transverse coordinate z. According to the PIA, the stiffness matrix from the matrix exponential can be acquired. Considering that the PIA is a highly accurate method, any desired accuracy of the displacement and stress field can be obtained. The entire derivation process is built on the three dimensional elasticity equations without importing any assumptions on the plate kinematics. Comparisons with numerical solutions available from prevenient researchers are made to validate the high accuracy, efficiency and serviceability of the employed technique. Additionally, circular and perforated examples are provided to highlight the performance of the developed methodology and depict the influences of boundary conditions, thickness-to-length ratios and gradient indexes on the deformable behaviors of in-plane functionally graded plates.

Modelling of laminated composite plates with weakly bonded interfaces using scaled boundary finite element method

Scaled Boundary Finite Element Method (SBFEM), a relatively new approach, is investigated for the analysis of the weakly bonded laminated composite plates, under cylindrical bending. A plane strain formulation, which does not require any assumptions on the displacement or stress field, is developed. The interface between two plies is simulated using a linear spring layer model and incorporated in the numerical modelling with the help of zero thickness interface elements. The model is coded and postprocessed in MATLAB to capture general delamination, i.e. separation of the plies at interfaces in both tangential and normal direction. Due to the semi-analytical solution procedure of SBFEM, the model provides highly accurate results. Use of SBFEM reduces the meshing burden and complexities in the discretisation. Moreover, the method can handle hanging nodes, so one can use small interface elements without decreasing the size of adjoining domains. These qualities make the proposed numerical model computationally cheaper as well as very accurate. The presented formulation is demonstrated for a variety of examples in laminated composites including unidirectional laminates, symmetric cross ply laminates and antisymmetric cross ply laminates. The proposed model's close agreement as well as its superiority over the other methods is established for future reference.

A micromechanical approach for semi-analytical low-velocity impact analysis of a bidirectional functionally graded circular plate resting on an elastic foundation

All the rare investigations made on low velocity impact analysis of the functionally graded plates have employed the simple rule of mixture that may lead to inconsistent results in some circumstances, e.g., when the Poisson ratios of the constituent materials are quite different. On the other hand, these researches have been performed either by purely using a commercial finite element software or employing the conventional Hertz-type contact laws that are valid only if the overall plate flexibility is ignored. In the present paper, it is mainly focused on four novelties: (i) description of the apparent stiffness of the contact region of the plate based on the micromechanical Mori-Tanaka model for two-directional variations of the material properties of the functionally graded plates, (ii) employing an enhanced contact model, considering the micromechanics-based material model and plate and foundation compliances, (iii) evaluation effects of the foundation stiffness on the apparent impact stiffness and the plate responses, and (iv) using a weighted residual semi-analytical solution procedure and an iterative updating scheme to solve the nonlinear governing equations. Presence of the elastic foundation has caused very interesting, unexpected but reasonable behaviors to be observed.

Heat conduction in one-dimensional functionally graded media based on the dual-phase-lag theory

In this article, heat conduction in one-dimensional functionally graded media is investigated based on the dual-phase-lag theory to consider the microstructural interactions in the fast transient process of heat conduction. All material properties of the media are assumed to vary continuously according to a power-law formulation with arbitrary non-homogeneity indices except the phase lags which are taken constant for simplicity. The one-dimensional heat conduction equations based on the dual-phase-lag theory are derived in a unified form which can be used for Cartesian, cylindrical, and spherical coordinates. A semi-analytical solution for temperature and heat flux is presented using the Laplace transform to eliminate the time dependency of the problem. The results in the time domain are then given by employing a numerical Laplace inversion technique. The semi-analytical solution procedure leads to exact expressions for the thermal wave speed in one-dimensional functionally graded media with different geometries based on the dual-phase-lag and hyperbolic heat conduction theories. The transient temperature distributions have been found for various types of dynamic thermal loading. The numerical results are shown to reveal the effects of phase lags, non-homogeneity indices, and thermal boundary conditions on the thermal responses for different temporal disturbances. The results are verified with those reported in the literature for hyperbolic heat conduction in cylindrical and spherical coordinates.

Semi-analytical solutions for the dual phase lag heat conduction in multilayered media

A semi-analytical solution procedure for transient heat transfer in composite mediums consisting of multi-layers within the framework of the dual phase lag model is presented. The procedure is then used to derive solutions for the temperature-, temperature gradient-, and heat flux distributions in a two-layer composite planar slab, a bi-layered solid-cylinder and sphere. The solutions obtained are applicable to the classical Fourier heat diffusion, hyperbolic heat conduction, phonon–electron interaction, and phonon scattering models with perfect or imperfect contact and with layers of different materials. The interfacial contact resistance, the heat flux and temperature gradient phase lags, thermal diffusivities and conductivities, initial temperatures of the composite medium and a general time-dependent boundary heat flux enter the solutions as parameters, allowing the solutions obtained to be applicable to a wide range of arrangements including perfect and imperfect contact. Analysis of thermal wave propagation, transmission and reflection in planar, cylindrical and spherical geometries with imperfect interfaces are presented, and geometrical—as well as the temperature gradient phase lag—effects on the thermal lagging behavior in different layered media are discussed.

An experimental and analytical study of the mechanical behaviour of adhesively bonded joints for variable extension rates and temperatures

In the present paper, mechanical behaviours of adhesively bonded joints are studied with the help of double lap shear (DLS) coupon tests conducted at different extension rates and temperatures. The joint specimens are made from dual-phase steel coupons bonded with epoxy resin. Tests are also carried out to ascertain the behaviours of these component materials. It has been found that at a high temperature, the adhesive joint exhibits a greater degree of strain rate sensitivity with a perceptible fall in the joint strength. However, at a low temperature, the joint strength remains comparable to that at room temperature. A new semi-analytical solution procedure is developed considering material nonlinearity to predict mechanical behaviours of adhesively bonded DLS joints. The joint behaviours using the semi-analytical approach are predicted separately using the Von Mises (VM) and Raghava yield criteria. It has been found here that the application of the Raghava criterion yields good correlation with test load–extension behaviours for most temperatures and extension rates considered; on the other hand, the VM condition gives rise to perceptibly softer joint behaviour when compared to test data at a high temperature.

Elastodynamic Green’s function for reinforced concrete beams

A semi-analytical solution procedure for three dimensional wave propagation in reinforced concrete (RC) beams has been presented in this paper. Elastodynamic Green’s function has been derived by employing the compatibility conditions and utilizing the symmetry conditions at the loaded cross section. Numerical procedure developed for the Green’s function has been validated using results available in the literature for an infinite laminated composite plate. Three-dimensional wave propagation analysis has been performed for reinforced concrete beam sections of T and L shapes which are common forms of structural elements. Steel reinforcement has been modeled in the finite element mesh. Effect of corrosion has also been included in the finite element model. Green’s function for reinforced concrete sections affected by corrosion of steel unit normalized frequency has been evaluated for illustration. Accuracy of the solution technique has been evaluated in terms of the percentage error in energy balance between the input energy of the applied unit load and the output energy carried by the propagating wave modes. The percentage error has been found to be negligible in all the cases considered here. A simple and accurate numerical method has been presented here as a tool to evaluate Green’s function for RC beams and can be used to detect corrosion.

Nonlinear analysis of pneumatic radial tyres via piece-wise Rayleigh-Ritz technique

Based on the Sanders thin shell theory and Reddy's higher-order shear deformation shell theory, a general refined shell theory is developed in this paper for the analysis of stresses and deformations of pneumatic radial tyres of composite construction. For easy and efficient simulation of the tyre, a piece-wise Rayleigh-Ritz technique is proposed and applied to get a numerical solution to the nonlinear structural problem. Bezier polynomials are used to approximate both the geometry of the free surface of reference and displacement fields of the tyres. Stress distributions and deformations of the tyres subjected to uniform inflation pressure are computed and discussed for details. From a comparison of the present results by the piece-wise Rayleigh-Ritz approach with the numerical predictions by 3D finite element method, it has been shown that the present semi-analytical solution procedure is accurate and applicable to the much complicated time-consuming nonlinear analysis for high quality tyre.

The Instability and Vibration of Rotating Beams With Arbitrary Pretwist and an Elastically Restrained Root

The governing differential equations and the boundary conditions for the coupled bending-bending-extensional vibration of a rotating nonuniform beam with arbitrary pretwist and an elastically restrained root are derived by Hamilton’s principle. The semianalytical solution procedure for an inextensional beam without taking account of the coriolis forces is derived. The coupled governing differential equations are transformed to be a vector characteristic governing equation. The frequency equation of the system is derived and expressed in terms of the transition matrix of the vector governing equation. A simple and efficient algorithm for determining the transition matrix of the general system with arbitrary pretwist is derived. The divergence in the Frobenius method does not exist in the proposed method. The frequency relations between different systems are revealed. The mechanism of instability is discovered. The influence of the rotatory inertia, the coupling effect of the rotating speed and the mass moment of inertia, the setting angle, the rotating speed and the spring constants on the natural frequencies, and the phenomenon of divergence instability are investigated.

A semi-analytical solution for a newsboy problem with mid-period replenishment

This paper is an exploratory study of a very common variation of the classical newsboy problem: in addition to the supply at the beginning of the sales period, an additional replenishment opportunity exists sometime during the sales period. A basic model for the problem is presented and a semi-analytical solution procedure is developed. Numerical solutions obtained reveal certain interesting properties of the model. The paper shows that the replenishment extension makes the newsboy problem much more challenging managerially and also that it can be practically solved with significant profit implications.

Semi-Analytical Hydrodynamics and Motions for a TLP Structure—Part I: Enhancement of Pontoon Results Using Two of Haskind’s Theorems

Abstract A semi-analytical solution procedure for the derivation of first-order wave forces, added mass, and damping terms is presented for rectangular pontoons which may be floating, submerged, or resting on the seabed. The development is based on a solution matching technique for the scattered and radiated wave potentials and gradients across common boundaries of the subdivided solution domain. The final expressions are true for waves of arbitrary angle of incidence and singularities due to the case of normal incidence are eliminated. The approach is also generalized for the case where the pontoon is a unit of a TLP offshore production system by referring all quantities to a global coordinate system. Numerical results are extracted for a number of benchmark problems and compared with independent values where available in the literature. Excellent agreement is obtained for wave forces, added mass, and damping Values for cases involving floating, submerged, and bottom-founded pontoon configurations.

On the modeling of miscible flow in multi-component porous media

An analytical model of miscible flow in multi-component porous media is presented to demonstrate the influence of pore capacitance in extending diffusive tailing. Solute attenuation is represented naturally by accommodating diffusive and convective flux components in macropores amd micropores as elicited by the local solute concentration and velocity fields. A set of twin, coupled differential equations result from the Laplace transform and are solved simultaneously using a differential operator for one-dimensional flow geometry. The solutions in real space are achieved using numeric inversion. In addition, to represent more faithfully the dominant physical processes, this approach enables efficient and stable semi-analytical solution procedure of the coupled system that is significantly more complex than current capacitance type models. Parametric studies are completed to illustrate the ability of the model to represent sharp breakthrough and lengthy tailing, as well as investigating the form of the nested heterogeneity as a result of solute exchange between macropores and micropores. Data from a laboratory column experiment is examined using the present model and satisfactory agreement results.

A semi-analytical solution procedure for predicting damage evolution at interfaces : Zhen Chen, International Journal for Numerical & Analytical Methods in Geomechanics, 17(11), 1993, pp 807–819

A semi-analytical solution procedure for predicting damage evolution at interfaces : Zhen Chen, International Journal for Numerical & Analytical Methods in Geomechanics, 17(11), 1993, pp 807–819