Large displacement static analysis of a cantilever Timoshenko beam composed of functionally graded material

2011 ◽  
Vol 18 (1-2) ◽  
pp. 21-34 ◽  
Author(s):  
Turgut Kocatürk ◽  
Mesut Şimşek ◽  
Şeref Doğuşcan Akbaş
Keyword(s):  
Finite Element ◽  
Static Analysis ◽  
Functionally Graded Material ◽  
Timoshenko Beam ◽  
Functionally Graded ◽  
Element Approximation ◽  
Large Rotations ◽  
Non Linear ◽  
Graded Material ◽  
Newton Raphson

AbstractIn this study, non-linear static analysis of a cantilever Timoshenko beam composed of functionally graded material (FGM) under a non-follower transversal uniformly distributed load is studied with large displacements and large rotations. Material properties of the beam change in the thickness direction according to a power-law function. In this study, the finite element of the beam is constructed by using the total Lagrangian Timoshenko beam element approximation. The non-linear problem is solved by using the incremental displacement-based finite element method in conjunction with the Newton-Raphson iteration method. To use the solution procedures of the Newton-Raphson method, there is a need to linearize equilibrium equations, which can be achieved through the linearization of the principle of virtual work. In this study, the effects of large deflections, large rotations and various material distributions on displacements and normal stress and shear stress distributions through the thickness of the beam are investigated in detail. The convergence study is performed for various numbers of finite elements. In addition, some of the particular results of the present study which are obtained for the homogeneous material case are compared with the results of the SAP2000 packet program. Numerical results show that geometrical non-linearity and material distribution play very important roles in the static responses of FGM beams.

scholarly journals Geometrically Nonlinear Static Analysis of Edge Cracked Timoshenko Beams Composed of Functionally Graded Material

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Şeref Doğuşcan Akbaş
Keyword(s):  
Finite Element ◽  
Static Analysis ◽  
Functionally Graded Material ◽  
Functionally Graded ◽  
Nonlinear Static Analysis ◽  
Geometrically Nonlinear ◽  
Timoshenko Beams ◽  
Graded Material ◽  
Nonlinear Static ◽  
Fgm Beam

Geometrically nonlinear static analysis of edge cracked cantilever Timoshenko beams composed of functionally graded material (FGM) subjected to a nonfollower transversal point load at the free end of the beam is studied with large displacements and large rotations. Material properties of the beam change in the height direction according to exponential distributions. The cracked beam is modeled as an assembly of two subbeams connected through a massless elastic rotational spring. In the study, the finite element of the beam is constructed by using the total Lagrangian Timoshenko beam element approximation. The nonlinear problem is solved by using incremental displacement-based finite element method in conjunction with Newton-Raphson iteration method. The convergence study is performed for various numbers of finite elements. In the study, the effects of the location of crack, the depth of the crack, and various material distributions on the nonlinear static response of the FGM beam are investigated in detail. Also, the difference between the geometrically linear and nonlinear analysis of edge cracked FGM beam is investigated in detail.

Effects of Support Conditions to the Post-Buckling Behaviors of Axially Functionally Graded Material Rods

2017 ◽  
Vol 730 ◽  
pp. 502-509 ◽  
Author(s):  
Buntara Sthenly Gan ◽  
Thanh Huong Trinh ◽  
Takahiro Hara ◽  
Dinh Kien Nguyen ◽  
Thi Thom Tran
Keyword(s):  
Finite Element ◽  
Material Property ◽  
Functionally Graded Material ◽  
Functionally Graded ◽  
Support Condition ◽  
Non Linear ◽  
Graded Material ◽  
Post Buckling ◽  
Support Conditions ◽  
Axially Functionally Graded Material

The effects of support conditions to the post-buckling behaviors of rod structures made of Axially Functionally Graded Material (AFGM) are presented. The material property of the rod member is assumed to vary linearly in the axis direction of the member. The non-linear material property of the rod element is formulated in the Finite Element context. The consistent shape functions for the rod element were developed to take into account the varying material property in the finite element formulation. The geometrically non-linear behavior of the rod element is formulated in the context of the updated co-rotational formulation. The non-linear equilibrium equations are solved by using the incremental and iterative procedures in combination with the arc-length control method. The influences of the material distribution on the post-buckling behaviors of the AFGM Williams’ toggle frames under various support conditions are highlighted. As a result, the graded between two materials can increase the post-buckling behaviors of the AFGM rod element regardless of the types of support conditions. The orientation of material distributions combined with the type of support condition can increase the performance of the rod element. The fixed-fixed support condition showed the highest performance of the AFGM rod element.

A study on non-linear post-buckling behavior of tapered Timoshenko beam made of functionally graded material under in-plane thermal loadings

2016 ◽  
Vol 52 (1) ◽  
pp. 45-56 ◽  
Author(s):  
Amlan Paul ◽  
Debabrata Das
Keyword(s):  
Functionally Graded Material ◽  
Timoshenko Beam ◽  
Temperature Rise ◽  
Functionally Graded ◽  
Buckling Load ◽  
Uniform Temperature ◽  
Buckling Behavior ◽  
Non Linear ◽  
Graded Material ◽  
Post Buckling

In the present work, the non-linear post-buckling load–deflection behavior of tapered functionally graded material beam is studied for different in-plane thermal loadings. Two different thermal loadings are considered. The first one is due to the uniform temperature rise and the second one is due to the steady-state heat conduction across the beam thickness leading to non-uniform temperature rise. The governing equations are derived using the principle of minimum total potential energy employing Timoshenko beam theory. The solution is obtained by approximating the displacement fields following Ritz method. Geometric non-linearity for large post-buckling behavior is considered using von Kármán type non-linear strain-displacement relationship. Stainless steel/silicon nitride functionally graded material beam is considered with temperature-dependent material properties. The validation of the present work is successfully performed using finite element software ANSYS and using the available result in the literature. The post-buckling load–deflection behavior in non-dimensional plane is presented for different taperness parameters and also for different volume fraction indices. Normalized transverse deflection fields are presented showing the shift of the point of maximum deflection for various deflection levels. The results are new of its kind and establish benchmark for studying non-linear thermo-mechanical behavior of tapered functionally graded material beam.

scholarly journals A Lagrange Multiplier-based Technique within the Nonlinear Finite Element Method in Cracked Columns

2020 ◽  
Author(s):  
Kaveh Salmalian ◽  
Ali Alijani ◽  
Habib Ramezannejad Azarboni
Keyword(s):  
Finite Element ◽  
Lagrange Multiplier ◽  
Functionally Graded Material ◽  
Geometrical Nonlinearity ◽  
Functionally Graded ◽  
Material Inhomogeneity ◽  
Non Linear ◽  
Graded Material ◽  
Post Buckling ◽  
Conversion Matrix

In this research, two energy-based techniques, called Lagrange multiplier and conversion matrix, are applied to involve crack parameters into the non-linear finite element relations of Euler-Bernoulli beams made of functionally graded materials. The two techniques, which divide a cracked element into three parts, are implemented to enrich the secant and tangent stiffness matrices. The Lagrange multiplier technique is originally proposed according to the establishment of a modified total potential energy equation by adding continuity conditions equations of the crack point. The limitation of the conversion matrix in involving the relevant non-linear equations is the main motivation in representing the Lagrange multiplier. The presented Lagrange multiplier is a problem-solving technique in the cracked structures, where both geometrical nonlinearity and material inhomogeneity areas are considered in the analysis like the post-buckling problem of cracked functionally graded material columns. Accordingly, some case-studies regarding the post-buckling analysis of cracked functionally graded material columns under mechanical and thermal loads are used to evaluate the results.

Stability of the Size-Dependent and Functionally Graded Curvilinear Timoshenko Beams

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
J. Awrejcewicz ◽  
A. V. Krysko ◽  
S. P. Pavlov ◽  
M. V. Zhigalov ◽  
V. A. Krysko
Keyword(s):  
Differential Equations ◽  
Functionally Graded Material ◽  
Timoshenko Beam ◽  
Stability Loss ◽  
Functionally Graded ◽  
Beam Deflection ◽  
Timoshenko Beams ◽  
Size Dependent ◽  
Rigid Layer ◽  
Graded Material

The size-dependent model is studied based on the modified couple stress theory for the geometrically nonlinear curvilinear Timoshenko beam made from a functionally graded material having its properties changed along the beam thickness. The influence of the size-dependent coefficient and the material grading on the stability of the curvilinear beams is investigated with the use of the setup method. The second-order accuracy finite difference method is used to solve the problem of nonlinear partial differential equations (PDEs) by reducing it to the Cauchy problem. The obtained set of nonlinear ordinary differential equations (ODEs) is then solved by the fourth-order Runge–Kutta method. The relaxation method is employed to solve numerous static problems based on the dynamic approach. Eight different combinations of size-dependent coefficients and the functionally graded material coefficient are used to study the stress-strain responses of Timoshenko beams. Stability loss of the curvilinear Timoshenko beams is investigated using the Lyapunov criterion based on the estimation of the Lyapunov exponents. Beams with/without the size-dependent behavior, homogeneous beams, and functionally graded beams having the same stiffness are investigated. It is shown that in straight-line beams, the size-dependent effect decreases the beam deflection. The same is observed if the most rigid layer is located on the top of the beam. In the curvilinear Timoshenko beam, such a location of the most rigid layer essentially improves the beam strength against stability loss. The observed transition of the largest Lyapunov exponent from a negative to positive value corresponds to the transition from a precritical to postcritical beam state.

Sign in / Sign up

Export Citation Format

Share Document