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This study aims to analyze the in-plane free vibrations of arches comprising the laterally symmetric functionally graded materials. Emphasis is placed on the circular arch whose material properties vary laterally symmetrically about the centroidal axis by a power-law function. The differential equations governing the mode shape of the arch were derived under the boundary conditions and were numerically solved to calculate the natural frequencies using the Runge–Kutta and Regula–Falsi methods. Calculation results of this study for natural frequencies compare well with those of the finite element method. The effects of various arch parameters on natural frequencies are highlighted and discussed in detail. [ABSTRACT FROM AUTHOR]

This study aims to analyze the free vibration of an ogival arch in which two symmetrical arched shapes are discontinuously met at the mid-arc. Differential equations governing the free vibration of an ogival circular arch were derived, where the rotatory inertia and shear deformation effects are included. The governing equations were numerically solved to calculate natural frequencies and mode shapes. In numerical solution methods, the symmetric and anti-symmetric boundary conditions at the mid-arc were focused rather than the boundary conditions of supported end due to the discontinuity of the mid-arc. For the first time, the free vibration problem for discontinuous arches, such as ogival arches, is solved in this study. Calculation results of this study for natural frequencies are compared well with those of the finite element method. The effects of various arch parameters on natural frequencies were highlighted and discussed in detail. [ABSTRACT FROM AUTHOR]

The paper aims to develop a unified modeling to analyze the free vibrations and buckling of laterally functionally graded material columns. The columns of rectangular cross-section, in which the mass density and Young's modulus vary along the depth as a power-law function, were considered. The differential equation governing a mode shape of a deformed column was derived with the relevant boundary conditions, including the effect of rotatory inertia. The computation results for the natural frequencies and buckling load were compared well with those available in the open scientific literature. The effects of geometrical and material properties on natural frequencies and buckling loads with the mode shapes are highlighted. [ABSTRACT FROM AUTHOR]

A novel analytical approach for predicting the buckling load with its mode shape of friction piles fully embedded in soil is proposed, in which the lateral subgrade and axial friction resistances change linearly. The proposed approach considers uniform rectangular piles that are either free or pinned or fixed at the pile and toe. The differential equation that governs the mode shape of the buckling pile along with the boundary conditions was analytically derived and numerically solved using the Runge–Kutta method and the Regula-Falsi method. The buckling loads of this study agreed well with those reported in the literature. The buckling behavior of piles was discussed as factors affecting the end condition, aspect ratio, subgrade ratio, friction ratio, subgrade parameter and friction parameter. [ABSTRACT FROM AUTHOR]

In this article, the buckling optimization of axially functionally graded (AFG) columns to maximize the buckling capacity is studied. Consideration is given to an AFG column having a tapered regular polygon cross section and variable material properties. The governing differential equation is derived based on Euler–Bernoulli beam theory with the relevant boundary conditions and is solved using the direct integration method combined with a determinant search algorithm. The computed buckling loads are compared with those presented in the literature and obtained from finite element analysis. Numerical examples for buckling load and buckled mode shape are given to highlight the effect of parameters related to the Young's modulus, cross-sectional shape, tapering and column volume. In particular, the geometry and material parameters that provide buckling optimization at constant volume of the column are analysed. [ABSTRACT FROM AUTHOR]

This paper deals with analyzing free vibrations of the symmetric arch. The boundary conditions of the stress resultants are newly derived, which can be replaced by the conventional boundary conditions of the deflections. All solutions of the natural frequency with the mode shape, using the new boundary conditions, are the same as those of the conventional deflections. The boundary conditions mixed with new and conventional conditions act correctly to calculate natural frequencies. The mode shapes of the stress resultants using the new boundary conditions are reported in two types: symmetric and anti-symmetric modes. [ABSTRACT FROM AUTHOR]

This paper presents free vibrations of the tapered horseshoe circular arch with a constant volume. The volume of the arch is constant, and the cross-sectional shape of the arch is square and circular. The taper function of the arch is a quadratic function. Differential equations with the boundary conditions that govern the free vibration of such arches are derived and numerically solved to calculate natural frequencies and mode shapes. The natural frequencies of this study agree well with those of the finite element ADINA. Parametric studies of the geometrical and cross-sectional properties of the arch on frequencies and mode shapes are performed and discussed. [ABSTRACT FROM AUTHOR]

• A unified mathematical model for free vibration and buckling of linearly tapered AFG columns is proposed. • The arbitrary functional variation of Young's modulus and mass density of AFGM are considered. • For computing the natural frequencies and buckling loads, the governing equations are numerically solved. This paper presents a unified model to analyze the free vibration and buckling of axially functionally graded Euler-Bernoulli columns subjected to an axial compressive force. The material properties vary linearly along the longitudinal direction, and column with circular and square cross sections is linearly tapered. The governing differential equations of the problem are derived and solved using the direct integral method combined with the determinant search technique. The computed results are compared with those reported in the literature and obtained from the finite element software ADINA. Numerical examples for natural frequency, buckling load and their corresponding mode shapes are given to highlight the effects of modular ratio, taper ratio and cross sectional shape as well as the end condition. [ABSTRACT FROM AUTHOR]

This paper studies the buckling of standing columns under self-weight and tip load. An emphasis is placed on linearly tapered columns with regular polygons cross-section whose volume is constant. Five end conditions for columns are considered. The differential equation governing the buckling shapes of the column is derived based on the equilibrium equations of the buckled column elements. The governing equation is numerically integrated using the direct integration method, and the eigenvalue is obtained using the determinant search method. The accuracy of the method is verified against the existing solutions for particular cases. The effects of side number, taper ratio, self-weight, and end condition on the buckling load and mode shape are investigated. The contribution of self-weight acting alone to the buckling response is also explored. For a given column volume, especially, the buckling length and its stress distribution of the columns with different geometries and end conditions are estimated. [ABSTRACT FROM AUTHOR]

This paper studies the in-plane free vibration of axially functionally graded (AFG) circular arches with non-uniform cross-section. The geometric and material properties of circular arches with regular polygon cross-section vary symmetrically about the mid-arc along the axial direction in quadratic polynomial form. The governing differential equations of the motion are derived, and the symmetric and anti-symmetric boundary conditions of the arches are developed for applying initial and boundary value problems in the solution method. The computed results agree well with the results of the finite element software ADINA. The effects of geometrical and material parameters on the natural frequency and mode shape of AFG circular arches are investigated. [ABSTRACT FROM AUTHOR]

This paper deals with the transverse free vibration of axially functionally graded (AFG) cantilever columns under the influence of axial compressive load. The columns possessing a regular polygon in their cross-section are tapered and their material properties vary along the axis of the column. An emphasis is placed on the columns with constant volume for admissible geometries and materials. The governing differential equation of the problem is derived and solved using the direct integral approach in conjunction with the determinant search technique. The obtained results are in good agreement with those in the available literature and computed by finite element analysis. Numerical examples for the natural frequency and mode shape of the columns are presented to investigate the effects of parameters related to geometrical nonuniformity and material inhomogeneity. [ABSTRACT FROM AUTHOR]