• Structural Engineering and Mechanics
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• v.25 no.2
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• pp.201-217
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• 2007

A harmonic type differential quadrature approach for nonlinear dynamic analysis of multi-degree-of-freedom systems has been developed. A series of numerical examples is conducted to assess the performance of the HDQ method in linear and nonlinear dynamic analysis problems. Results are compared with the existing solutions available from other analytical and numerical methods. In all cases, the results obtained are quite accurate.

• Structural Engineering and Mechanics
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• v.17 no.1
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• pp.1-14
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• 2004

Numerical solution to linear bending analysis of circular plates is obtained by the method of harmonic differential quadrature (HDQ). In the method of differential quadrature (DQ), partial space derivatives of a function appearing in a differential equation are approximated by means of a polynomial expressed as the weighted linear sum of the function values at a preselected grid of discrete points. The method of HDQ that was used in the paper proposes a very simple algebraic formula to determine the weighting coefficients required by differential quadrature approximation without restricting the choice of mesh grids. Applying this concept to the governing differential equation of circular plate gives a set of linear simultaneous equations. Bending moments, stresses values in radial and tangential directions and vertical deflections are found for two different types of load. In the present study, the axisymmetric bending behavior is considered. Both the clamped and the simply supported edges are considered as boundary conditions. The obtained results are compared with existing solutions available from analytical and other numerical results such as finite elements and finite differences methods. A comparison between the HDQ results and the finite difference solutions for one example plate problem is also made. The method presented gives accurate results and is computationally efficient.

• Structural Engineering and Mechanics
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• v.28 no.2
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• pp.221-238
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• 2008

Numerical solution to buckling analysis of beams and columns are obtained by the method of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for various support conditions considering the variation of flexural rigidity. The solution technique is applied to find the buckling load of fully or partially embedded columns such as piles. A simple semi- inverse method of DQ or HDQ is proposed for determining the flexural rigidities at various sections of non-prismatic column ( pile) partially and fully embedded given the buckling load, buckled shape and sub-grade reaction of the soil. The obtained results are compared with the existing solutions available from other numerical methods and analytical results. In addition, this paper also uses a recently developed technique, known as the differential transformation (DT) to determine the critical buckling load of fully or partially supported heavy prismatic piles as well as fully supported non-prismatic piles. In solving the problem, governing differential equation is converted to algebraic equations using differential transformation methods (DT) which must be solved together with applied boundary conditions. The symbolic programming package, Mathematica is ideally suitable to solve such recursive equations by considering fairly large number of terms.

• Steel and Composite Structures
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• v.27 no.1
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• pp.35-48
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• 2018

In this paper, the response of a sandwich cylindrical shell over any sort of boundary conditions and under a general distributed static loading is investigated. The faces and the core are made of some isotropic materials. The faces are modeled as thin cylindrical shells obeying the Kirchhoff-Love assumptions. For the core material it is assumed to be thick and the in-plane stresses are negligible. The governing equations are derived using the principle of the stationary potential energy. Using harmonic differential quadrature method (HDQM) the equations are solved for deformation components. The obtained results primarily are compared against finite element results. Then, the effects of changing different parameters on the stress and displacement components of sandwich cylindrical shells are investigated.

• Structural Engineering and Mechanics
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• v.17 no.6
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• pp.789-817
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• 2004

This paper deals with the modal analysis of rotational shell structures by means of the numerical solution technique known as the Generalized Differential Quadrature (G. D. Q.) method. The treatment is conducted within the Reissner first order shear deformation theory (F. S. D. T.) for linearly elastic isotropic shells. Starting from a non-linear formulation, the compatibility equations via Principle of Virtual Works are obtained, for the general shell structure, given the internal equilibrium equations in terms of stress resultants and couples. These equations are subsequently linearized and specialized for the rotational geometry, expanding all problem variables in a partial Fourier series, with respect to the longitudinal coordinate. The procedure leads to the fundamental system of dynamic equilibrium equations in terms of the reference surface kinematic harmonic components. Finally, a one-dimensional problem, by means of a set of five ordinary differential equations, in which the only spatial coordinate appearing is the one along meridians, is obtained. This can be conveniently solved using an appropriate G. D. Q. method in meridional direction, yielding accurate results with an extremely low computational cost and not using the so-called "delta-point" technique.

• Structural Engineering and Mechanics
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• v.19 no.4
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• pp.459-475
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• 2005

This paper deals with a novel application of the Generalized Differential Quadrature (G.D.Q.) method to the linear elastic static analysis of isotropic rotational shells. The governing equations of equilibrium, in terms of stress resultants and couples, are those from Reissner-Mindlin shear deformation shell theory. These equations, written in terms of internal-resultants circular harmonic amplitudes, are first put into generalized displacements form, by use of the strain-displacements relationships and the constitutive equations. The resulting systems are solved by means of the G.D.Q. technique with favourable precision, leading to accurate stress patterns.

• Steel and Composite Structures
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• v.35 no.3
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• pp.449-462
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• 2020

This work aims to study effects of the crack and the surface energy on the free longitudinal vibration of axially functionally graded nanorods. The surface energy parameters considered are the surface stress, the surface density, and the surface Lamé constants. The cracked nanorod is modelled by dividing it into two parts connected by a linear spring in which its stiffness is related to the crack severity. The surface and bulk material properties are considered to vary in the length direction according to the power law distribution. Hamilton's principle is implemented to derive the governing equation of motion and boundary conditions. Considering the surface stress causes that the derived governing equation of motion becomes non-homogeneous while this was not the case in works that only the surface density and the surface Lamé constants were considered. To extract the frequencies of nanorod, firstly the non-homogeneous governing equation is converted to a homogeneous one using an appropriate change of variable, and then for clamped-clamped and clamped-free boundary conditions the governing equation is solved using the harmonic differential quadrature method. Since the present work considers effects of all the surface energy parameters, it can be claimed that this is a comprehensive work in this regard.

• Steel and Composite Structures
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• v.42 no.5
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• pp.699-710
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• 2022

This work presents a comprehensive study on the surface energy effect on the axial frequency analyses of AFGM nanorods in cylindrical coordinates. The AFGM nanorods are considered to be thin, relatively thick, and thick. In thin nanorods, effects of the inertia of lateral motions and the shear stiffness are ignored; in relatively thick nanorods, only the first one is considered; and in thick nanorods, both of them are considered in the kinetic energy and the strain energy of the nanorod, respectively. The surface elasticity theory which includes three surface parameters called surface density, surface stress, and surface Lame constants, is implemented to consider the size effect. The power-law form is considered for variation of the material properties through the axial direction. Hamilton's principle is used to derive the governing equations and boundary conditions. Due to considering the surface stress, the governing equation and boundary condition become inhomogeneous. After homogenization of them using an appropriate change of variable, axial natural frequencies are calculated implementing harmonic differential quadrature (HDQ) method. Comprehensive results including effects of geometric parameters and various material properties are presented for a wide range of boundary condition types. It is believed that this study is a comprehensive one that can help posterities for design and manufacturing of nano-electro-mechanical systems.

• Steel and Composite Structures
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• v.49 no.3
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• pp.349-359
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• 2023

Dynamic response and economic of a laminated porous concrete beam reinforced by nanoparticles subjected to harmonic transverse dynamic load is investigated considering structural damping. The effective nanocomposite properties are evaluated on the basis of Mori-Tanaka model. The concrete beam is modeled by the sinusoidal shear deformation theory (SSDT). Utilizing nonlinear strains-deflection, energy relations and Hamilton's principal, the governing final equations of the concrete laminated beam are calculated. Utilizing differential quadrature method (DQM) as well as Newmark method, the dynamic displacement of the concrete laminated beam is discussed. The influences of porosity parameter, nanoparticles volume percent, agglomeration of nanoparticles, boundary condition, geometrical parameters of the concrete beam and harmonic transverse dynamic load are studied on the dynamic displacement of the laminated structure. Results indicated that enhancing the nanoparticles volume percent leads to decrease in the dynamic displacement about 63%. In addition, with considering porosity of the concrete, the dynamic displacement enhances about 2.8 time.

• Steel and Composite Structures
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• v.43 no.1
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• pp.129-137
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• 2022

Dynamic response of a laminated porous concrete beam reinforced by nanoparticles subjected to harmonic transverse dynamic load is investigated considering structural damping. The effective nanocomposite properties are evaluated on the basis of Mori-Tanaka model. The concrete beam is modeled by the sinusoidal shear deformation theory (SSDT). Utilizing nonlinear strains-deflection, energy relations and Hamilton's principal, the governing final equations of the concrete laminated beam are calculated. Utilizing differential quadrature method (DQM) as well as Newmark method, the dynamic displacement of the concrete laminated beam is discussed. The influences of porosity parameter, nanoparticles volume percent, agglomeration of nanoparticles, boundary condition, geometrical parameters of the concrete beam and harmonic transverse dynamic load are studied on the dynamic displacement of the laminated structure. Results indicated that enhancing the nanoparticles volume percent leads to decrease in the dynamic displacement about 63%. In addition, with considering porosity of the concrete, the dynamic displacement enhances about 2.8 time.