• Coupled systems mechanics
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• v.7 no.4
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• pp.441-453
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• 2018

Longitudinal fracture in a functionally graded beam configuration was studied analytically with taking into account the non-linear behavior of the material. A cantilever beam with two longitudinal cracks located symmetrically with respect to the centroid was analyzed. The material was functionally graded along the beam width as well as along the beam length. The fracture was studied in terms of the strain energy release rate. The influence of material gradient, crack location along the beam width, crack length and material non-linearity on the fracture behavior was investigated. It was shown that the analytical solution derived is very useful for parametric analyses of the non-linear longitudinal fracture behavior. It was found that by using appropriate material gradients in width and length directions of the beam, the strain energy release rate can be reduced significantly. Thus, the results obtained in the present paper may be applied for optimization of functionally graded beam structure with respect to the longitudinal fracture performance.

• Steel and Composite Structures
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• v.23 no.3
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• pp.263-271
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• 2017

A theoretical study was carried-out of mode II delamination fracture behavior of the End Loaded Split (ELS) functionally graded beam configuration with considering the material non-linearity. The mechanical response of ELS was modeled analytically by using a power-law stress-strain relation. It was assumed that the material is functionally graded transversally to the beam. The non-linear fracture was investigated by using the J-integral approach. Equations were derived for the crack arm curvature and zero axes coordinate that are needed for the J-integral solution. The analysis developed is valid for a delamination crack located arbitrary along the beam height. The J-integral solution was verified by analyzing the strain energy release rate with considering material non-linearity. The effects of material gradient, non-linear material behavior and crack location on the fracture were evaluated. The solution derived is suitable for parametric analyses of non-linear fracture. The results obtained can be used for optimization of functionally graded beams with respect to their mode II fracture performance. Also, such simplified analytical models contribute for the understanding of delamination fracture in functionally graded beams exhibiting material non-linearity.

• Earthquakes and Structures
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• v.13 no.3
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• pp.255-265
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• 2017

In this paper, an efficient shear deformation theory is developed for wave propagation analysis in a functionally graded beam. More particularly, porosities that may occur in Functionally Graded Materials (FGMs) during their manufacture are considered. The proposed shear deformation theory is efficient method because it permits us to show the effect of both bending and shear components and this is carried out by dividing the transverse displacement into the bending and shear parts. Material properties are assumed graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents; but the rule of mixture is modified to describe and approximate material properties of the functionally graded beams with porosity phases. The governing equations of the wave propagation in the functionally graded beam are derived by employing the Hamilton's principle. The analytical dispersion relation of the functionally graded beam is obtained by solving an eigenvalue problem. The effects of the volume fraction distributions, the depth of beam, the number of wave and the porosity on wave propagation in functionally graded beam are discussed in details. It can be concluded that the present theory is not only accurate but also simple in predicting the wave propagation characteristics in the functionally graded beam.

• Structural Engineering and Mechanics
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• v.64 no.4
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• pp.487-494
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• 2017

The present paper deals with a theoretical study of delamination fracture in the Crack Lap Shear (CLS) functionally graded beam configuration. The basic purpose is to analyze the fracture with taking into account the material non-linearity. The mechanical behavior of CLS was described by using a non-linear stress-strain relation. It was assumed that the material is functionally graded along the beam height. The fracture was analyzed by applying the J-integral approach. The curvature and neutral axis coordinate of CLS beam were derived in order to solve analytically the J-integral. The non-linear solution of J-integral obtained was verified by analyzing the strain energy release rate with considering material non-linearity. The effects of material gradient, crack location along the beam height and material non-linearity on fracture behavior were evaluated. The J-integral non-linear solution derived is very suitable for parametric studies of longitudinal fracture in the CLS beam. The results obtained can be used to optimize the functionally graded beam structure with respect to the fracture performance. The analytical approach developed in the present paper contributes for the understanding of delamination fracture in functionally graded beams exhibiting material non-linearity.

• Wind and Structures
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• v.27 no.1
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• pp.59-70
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• 2018

In this paper, geometrically non-linear analysis of a functionally graded simple supported beam is investigated with porosity effect. The material properties of the beam are assumed to vary though height direction according to a prescribed power-law distributions with different porosity models. In the nonlinear kinematic model of the beam, the total Lagrangian approach is used within Timoshenko beam theory. In the solution of the nonlinear problem, the finite element method is used in conjunction with the Newton-Raphson method. In the study, the effects of material distribution such as power-law exponents, porosity coefficients, nonlinear effects on the static behavior of functionally graded beams are examined and discussed with porosity effects. The difference between the geometrically linear and nonlinear analysis of functionally graded porous beam is investigated in detail. Also, the effects of the different porosity models on the functionally graded beams are investigated both linear and nonlinear cases.

• Wind and Structures
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• v.23 no.1
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• pp.59-73
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• 2016

In this paper, a hyperbolic shear deformation beam theory is developed for free vibration analysis of functionally graded (FG) sandwich beams. The theory account for higher-order variation of transverse shear strain through the depth of the beam and satisfies the zero traction boundary conditions on the surfaces of the beam without using shear correction factors. The material properties of the functionally graded sandwich beam are assumed to vary according to power law distribution of the volume fraction of the constituents. The core layer is still homogeneous and made of an isotropic material. Based on the present refined beam theory, the equations of motion are derived from Hamilton's principle. Navier type solution method was used to obtain frequencies. Illustrative examples are given to show the effects of varying gradients and thickness to length ratios on free vibration of functionally graded sandwich beams.

• Coupled systems mechanics
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• v.6 no.1
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• pp.97-111
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• 2017

The present paper reports an analytical study of delamination fracture in the Mixed Mode Flexure (MMF) functionally graded beam with considering the material non-linearity. The mechanical behavior of MMF beam is modeled by using a non-linear stress-strain relation. It is assumed that the material is functionally graded along the beam height. Fracture behavior is analyzed by the J-integral approach. Non-linear analytical solution is derived of the J-integral for a delamination located arbitrary along the beam height. The J-integral solution derived is verified by analyzing the strain energy release rate with considering the non-linear material behavior. The effects of material gradient, crack location along the beam height and material non-linearity on the fracture are evaluated. It is found that the J-integral value decreases with increasing the upper crack arm thickness. Concerning the influence of material gradient on the non-linear fracture, the analysis reveals that the J-integral value decreases with increasing the ratio of modulus of elasticity in the lower and upper edge of the beam. It is found also that non-linear material behavior leads to increase of the J-integral value. The present study contributes for the understanding of fracture in functionally graded beams that exhibit material non-linearity.

• Structural Engineering and Mechanics
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• v.60 no.4
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• pp.693-705
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• 2016

Functionally graded material (FGM) plates can be bonded to the soffit of a beam as a means of retrofitting the RC beam. In such plated beams, tensile forces develop in the bonded plate and these have to be transferred to the original beam via interfacial shear and normal stresses. In this paper, an interfacial stress analysis is presented for simply supported concrete beam bonded with a functionally graded material FGM plate. This new solution is intended for application to beams made of all kinds of materials bonded with a thin plate, while all existing solutions have been developed focusing on the strengthening of reinforced concrete beams, which allowed the omission of certain terms. It is shown that both the normal and shear stresses at the interface are influenced by the material and geometry parameters of the composite beam. This research is helpful for the understanding on mechanical behavior of the interface and design of the FGM-RC hybrid structures.

• Geomechanics and Engineering
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• v.35 no.2
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• pp.181-194
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• 2023

The primary objective of this study is to examine the influence of geometry on the stability characteristics of cylindrical microstructures. This investigation entails a stability analysis of a bi-directional functionally graded (BD-FG) cylindrical imperfect concrete beam, focusing on the impact of geometry. Both the first-order shear deformation beam theory and the modified coupled stress theory are employed to explore the buckling and dynamic behaviors of the structure. The cylinder-shaped imperfect beam is constructed using a porosity-dependent functionally graded (FG) concrete material, wherein diverse porosity voids and material distributions are incorporated along the radial axis of the beam. The radius functions are considered in both uniform and nonuniform variations, reflecting their alterations along the length of the beam. The combination of these characteristics leads to the creation of BD-FG configurations. In order to enable the assessment of stability using energy principles, a numerical technique is utilized to formulate the equations for partial derivatives (PDEs).

• Structural Engineering and Mechanics
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• v.77 no.5
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• pp.587-599
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• 2021

Based on Euler-Bernoulli beam theory and continuous element method, the free vibration of bi-dimensional functionally graded beams is investigated. It is assumed that the material properties vary exponentially along the beam thickness and length. The characteristic frequency equations of beams with different boundary conditions are obtained by transfer matrix method. The validity of the proposed method is assessed through comparison with available results. Parametric studies are carried out to analyze the influences of the gradient indexes and the beam slenderness ratio on the natural frequencies of bi-dimensional functionally graded beams.