G. Karami

A new differential quadrature methodology for beam analysis and the associated differential quadrature element method

This paper presents a computationally efficient and an accurate new methodology in differential quadrature analysis of beam elements. The methodology would overcome the difficulties in boundary conditions implementations of fourth-order differential equations encountered in such problems. The methodology benefited from defining the second-order derivatives along the boundaries as independent degrees of freedom would enable the differential quadrature method to exactly satisfy some types of boundary conditions; where by most other conventional algorithms have to be satisfied approximately. The weighting coefficients employed are not exclusive, and any accurate and efficient method such as the generalized differential quadrature method may be used to produce the methods weighting coefficients. By solving some typical stability, deflection and frequency analysis beam problems and by comparing the results with those of exact solutions and/or those of other methodologies, accuracy, convergency and efficiency of the methodology is asserted. In order to generalize its application to large-scale beam structures and to the cases with the discontinuity in loading conditions and geometry, a new one-dimensional differential quadrature element method formulation is presented and implemented.

In-plane free vibration analysis of circular arches with varying cross-sections using differential quadrature method

Abstract A differential quadrature (DQ) methodology recently developed by the authors is used to obtain a general and a computationally efficient and accurate DQ solution for free vibration of variable cross-section circular thin arches. As an improvement to the classical theory and in order to evaluate the higher order natural frequencies more accurately, the commonly used hypothesis of “the inextensibility of the central axis” is removed. This enables one to study the effects of slenderness ratio on the natural frequencies, especially at higher order modes. Rotary inertia is included in the formulation and its influence on natural frequencies is studied. Arches with different types of boundary conditions, including those with elastic constraint against rotation at their ends, are considered. For the cases where a change in the cross-sectional or material properties of the arch occurs, a numerical domain decomposition technique in conjunction with DQ methodology is developed and incorporated. To verify the accuracy of the methodology, the results are compared with those of exact solutions and/or other approaches such as finite elements, Rayleigh–Ritz, Galerkin, cell discretization methods, and other DQ methodologies. In particular, excellent solution agreements are achieved with those of exact solutions, the generalized differential quadrature rule and the optimized Rayleigh–Ritz method solutions.

Finite Element Micromechanics for Stiffness and Strength of Wavy Fiber Composites

In this paper, a micromechanical model of a composite lamina material with fiber waviness is described. Results are presented and discussed with regard to stiffness and strength predictions for composite lamina. A micromechanical model of a unit cell from periodically distributed unidirectional waved cylindrical fibers embedded within matrix is proposed to withdraw the different material stiffness parameters. Finite element analysis of the periodic unit cell characterizing the structural stiffness of the composite material is carried out to determine the average stress and strain components. The composite stress-strain relations are then employed to determine the stiffness parameters. Numerical results for a typical composite constituted of polymer matrix and carbon fibers in the form of periodically hexagonal packing and initially sinusoidal waviness are presented for different amplitude to wavelength ratios and a range of fiber volume fractions. The results reveal the presence of local periodic-antisymmetric stresses that are usually unaccounted for in conventional structural analysis. The potential influence of these stresses on failure prediction is discussed.

AN EFFICIENT DIFFERENTIAL QUADRATURE METHODOLOGY FOR FREE VIBRATION ANALYSIS OF ARBITRARY STRAIGHT-SIDED QUADRILATERAL THIN PLATES

In this paper, a new differential quadrature (DQ) methodology is employed to study free vibration of irregular quadrilateral straight-sided thin plates. A four-nodded super element is used to map the irregular physical domain into a square domain in the computational domain. Second order transformation schemes with relative ease and less computation are employed to transform the fourth order governing equation of thin plates between the two domains. The only degree of freedom within the domain is the displacement, whereas along the boundaries, the displacement as well as the second order derivative of the displacement with respect to associated normal co-ordinate variable in computational domain are the two degrees of freedom. Implementing the method, the formulation for the DQ method for the free vibration analysis of plates of straight-sided shapes was presented together with the implementation procedure for the different boundary conditions. To demonstrate the accuracy, convergency and stability of the new methodology, detail studies are made on isotropic plates at acute angles with different geometries, boundary and loading conditions including DQ free-edge boundary condition implementations. Accurate results even with fewer degrees of freedom than for those of comparable numerical algorithms were achieved.

An efficient method to evaluate hypersingular and supersingular integrals in boundary integral equations analysis

Abstract An efficient algorithm is employed to evaluated hyper and super singular integral equations encountered in boundary integral equations analysis of engineering problems. The algorithm is based on multiple subtractions and additions to separate singular and regular integral terms in the polar transformation domain, primarily established in Refs. (Guiggiani M, Krishnasamy G, Rudolphi TJ, Rizzo FJ. A general algorithm for the numerical solution of hypersingular boundary integral equations. Trans ASME 1992;59:604–614; Guiggiani M, Casalini P. Direct computation of Cauchy principal value integral in advanced boundary element. Int J Numer Meth Engng 1987;24:1711–1720. Guiggiani M, Gigante A. A general algorithm for multidimensional Cauchy principal value integrals in the boundary element method. J Appl Mech Trans ASME 1990;57:906–915). It can be proved that the regular terms have finite analytical solutions in the range of integration, and the singular terms will be replaced by special periodic kernels in the integral equations. The subtractions involve to multiple derivatives of analytical kernels and the additions require some manipulation to separate the remaining regular terms from singular ones. The regular terms are computed numerically. Three examples on numerical evaluation of singular boundary integrals are presented to show the efficiency and accuracy of the algorithm. In this respect, strongly singular and hypersingular integrals of potential flow problems are considered, followed by a supersingular integral which is extracted from the partial differentiation of a hypersingular integral with respect to the source point.

DQM analysis of skewed and trapezoidal laminated plates

In this paper a differential quadrature method is applied for static, free vibration, and stability analysis of skewed and trapezoidal composite thin plates. To mechanize the modelling procedure, a general transformation scheme is employed to transfer the variation of the variables in the computational to the physical domain and vice versa. Examples are shown to show the accuracy and convergence of the solutions under different geometrical parameters and boundary conditions. The accuracy is demonstrated by comparing the results with those of other numerical methods.

A micromechanical hyperelastic modeling of brain white matter under large deformation

A finite element based micromechanical model has been developed for analyzing and characterizing the microstructural as well as homogenized mechanical response of brain tissue under large deformation. The model takes well-organized soft tissue as a fiber-reinforced composite with nonlinear and anisotropic behavior assumption for the fiber as well as the matrix of composite matter. The procedure provides a link between the macroscopic scale and microscopic scale as brain tissue undergoes deformation. It can be used to better understand how macroscopic stresses are transferred to the microstructure or cellular structure of the brain. A repeating unit cell (RUC) is created to stand as a representative volume element (RVE) of the hyperelastic material with known properties of the constituents. The model imposes periodicity constraints on the RUC. The RUC is loaded kinematically by imposing displacements on it to create the appropriate normal and shear stresses. The homogenized response of the composite, the average stresses carried within each of the constituents, and the maximum local stresses are all obtained. For each of the normal and shear loading scenarios, the impact of geometrical variables such as the axonal fiber volume fraction and undulation of the axons are evaluated. It was found that axon undulation has significant impact on the stiffness and on how stresses were distributed between the axon and the matrix. As axon undulation increased, the maximum stress and stress in the matrix increased while the stress in the axons decreased. The axon volume fraction was found to have an impact on the tissue stiffness as higher axon volume fractions lead to higher stresses both in the composite and in the constituents. The direction of loading clearly has a large impact on how stresses are distributed amongst the constituents. This micromechanics tool provides the detailed micromechanics stresses and deformations, as well as the average homogenized behavior of the RUC, which can be efficiently used in mechanical characterization of brain tissue.

A DQEM for vibration of shear deformable nonuniform beams with general boundary conditions

Abstract A differential quadrature element method (DQEM) for free vibration analysis of arbitrary non-uniform Timoshenko beams with attachments, i.e. concentrated mass and rotary inertia and resting on elastic supports is proposed. Using the Hamilton’s principle the governing equation and the natural compatibility conditions at the interface of adjacent elements are devised in a systematic manner. The differential quadrature (DQ) analogs to the governing equation, the compatibility conditions and the external boundary conditions implementations are explicitly formulated. The versatility, accuracy and efficiency of the presented DQEM for free vibration analysis of Timoshenko beams are tested against other solution procedures. The examples include; stepped non-uniform Timoshenko beams, non-uniform beams with attached heavy masses and supported elastically in transverse directions, and also non-uniform beams on elastic foundations. Accurate solutions are archived via few grid points for all the cases considered.

A finite element method parametric study of the dynamic response of the human brain with different cerebrospinal fluid constitutive properties

Abstract A major role for the cerebrospinal fluid (CSF) is to provide effective damping against sudden intracranial brain motions during dynamic head impact. This paper examines the roles of CSF properties on human brain responses under certain impact loadings. The brain is assumed to have a hyperviscoelastic material behaviour, while CSF is considered to be fluid-like elastic, viscoelastic, and nearly incompressible elastic with a low shear modulus and a high bulk modulus. A finite element parametric investigation on a head model under different scenarios of impact is conducted. In the study, the CSF material parameters are varied within the expected range of change, while other components of the head model are kept constant. The results indicate that the solutions from the modelling of CSF by a fluid-like medium are more realistic and support the findings of the experiment. The results also indicate that varying CSF properties did not have a major impact on the peak intracranial pressures but the impact on brain principal and shear strains are relatively significant. A sizeable impact on the relative motion of the brain, with respect to the skull, can also be observed.

A solution for the vibration and buckling of composite laminates with elastically restrained edges

A three-dimensional elasticity approach is used to develop a general free vibration and buckling analysis of composite plates with elastic restrained edges. The employed refined layerwise laminated theory guaranties an optimum and an economical solution procedure. The computation algorithm is based on finite elements analysis. The procedure permits a systematic and a straightforward modeling of restrained supports for anisotropic composite plates. By entering into the category of thick plate analysis, the accuracy and effectiveness of the method would become more evident. At each edge three translational elastic stiffnesses with no need for rotational degrees of freedom are defined and there is no limitation on the desired numbers of restraint edges.