Post-Buckling Analysis of FGM Beam Subjected to Non-Conservative Forces and In-Plane Thermal Loading

2012 ◽  
Vol 152-154 ◽  
pp. 474-479
Author(s):  
Feng Qun Zhao ◽  
Zhong Min Wang ◽  
Rui Ping Zhang
Keyword(s):  
Shooting Method ◽  
Thermal Loading ◽  
Governing Equations ◽  
Temperature Dependent ◽  
Tangential Load ◽  
Simply Supported ◽  
Runge Kutta Method ◽  
Buckling Behavior ◽  
Post Buckling ◽  
Fgm Beam

Based on the Kirchhoff large deformation theory, the post-buckling behavior of right movable simply supported FGM beam subjected to non-conservative forces and in-plane thermal loading was analyzed in this paper. The temperature-dependent and spatially dependent material properties of the FGM beam were assumed to vary through the thickness. The nonlinear governing equations of FGM beam subjected to a uniform distributed tangential load along the central axis and in-plane thermal loading were derived. Then, a shooting method and Runge-kutta method are employed to numerically solve the resulting equations. The post-buckling equilibrium paths of the FGM beam with different parameters were plotted, and the effects of non-conservative force, temperature, gradient index of FGM on the post-buckling behavior of right movable simply supported FGM beams were analyzed.

Post-Buckling of Piezoelectric Thin Plates

2015 ◽  
Vol 15 (07) ◽  
pp. 1540020 ◽  
Author(s):  
Michael Krommer ◽  
Hans Irschik
Keyword(s):  
Nonlinear Partial Differential Equation ◽  
Nonlinear Behavior ◽  
Dirichlet Boundary Conditions ◽  
Thin Plates ◽  
Governing Equations ◽  
Simply Supported ◽  
Buckling Behavior ◽  
Single Term ◽  
Post Buckling ◽  
Special Case

In the present paper, the geometrically nonlinear behavior of piezoelastic thin plates is studied. First, the governing equations for the electromechanically coupled problem are derived based on the von Karman–Tsien kinematic assumption. Here, the Berger approximation is extended to the coupled piezoelastic problem. The general equations are then reduced to a single nonlinear partial differential equation for the special case of simply supported polygonal edges. The nonlinear equations are approximated by using a problem-oriented Ritz Ansatz in combination with a Galerkin procedure. Based on the resulting equations the buckling and post-buckling behavior of a polygonal simply supported plate is studied in a nondimensional form, where the special geometry of the polygonal plate enters via the eigenvalues of a Helmholtz problem with Dirichlet boundary conditions. Single term as well as multi-term solutions are discussed including the effects of piezoelectric actuation and transverse force loadings upon the solution. Novel results concerning the buckling, snap through and snap buckling behavior are presented.

Comparison of Post-Buckling Behaviors of a S-S FGM Beam under Conservative and Non-Conservative Distributed Forces

2011 ◽  
Vol 250-253 ◽  
pp. 266-270
Author(s):  
Qing Lu Li ◽  
Shi Rong Li
Keyword(s):  
Deformation Theory ◽  
Shooting Method ◽  
Nonlinear Boundary ◽  
Conservative System ◽  
Functionally Graded ◽  
Governing Equations ◽  
Functionally Graded Beam ◽  
Post Buckling ◽  
Fgm Beam ◽  
Distributed Forces

Based on the large deformation theory and considering the axial extension of the beam, the governing equations of post-buckling of a simply supported elastic FGM beam subjected to conservative and non-conservative distributed forces were established. In the analysis, it was assumed that the material properties of the beam vary continuously as a power function of the thickness coordinate. By using shooting method, the nonlinear boundary-value problem was solved numerically and the equilibrium paths as well as the post- buckling configurations of the deformed beam were presented. A comparison between the results of conservative system and that of non-conservative systems were given. The results shows that the features of the equilibrium paths of the the functionally graded beam under non-conservative are evidently different from those to a conservative one.

POST-BUCKLING OF AN ELASTIC COLUMN WITH VARIOUS ROTATIONAL END RESTRAINTS

2005 ◽  
Vol 05 (01) ◽  
pp. 113-123 ◽  
Author(s):  
B. PHUNGPAINGAM ◽  
S. CHUCHEEPSAKUL
Keyword(s):  
Shooting Method ◽  
Nonlinear Differential Equations ◽  
Compressive Force ◽  
Limit Load ◽  
Technical Note ◽  
Buckling Behavior ◽  
End Conditions ◽  
Post Buckling ◽  
Secondary Bifurcation ◽  
Elastic Column

In this technical note, the post-buckling behavior of a simply supported elastic column with various rotational end conditions of the supports is investigated. The compressive force is applied at the tip of the column. The characteristic equation for solving the critical loads is obtained from the boundary value problem of linear systems. In the post-buckling state, a set of nonlinear differential equations with boundary conditions is established and numerically solved by the shooting method. The interesting features associated with this problem such as the limit load point, snap-through phenomenon and the secondary bifurcation point will be highlighted herein.

LOCAL/DISTORTIONAL/GLOBAL MODE INTERACTION IN SIMPLY SUPPORTED COLD-FORMED STEEL LIPPED CHANNEL COLUMNS

2011 ◽  
Vol 11 (05) ◽  
pp. 877-902 ◽  
Author(s):  
P. B. DINIS ◽  
D. CAMOTIM
Keyword(s):  
Shell Element ◽  
Mode Interaction ◽  
Buckling Mode ◽  
Global Mode ◽  
Simply Supported ◽  
Elastic Plastic ◽  
Buckling Behavior ◽  
Geometrical Imperfections ◽  
Post Buckling ◽  
Cold Formed Steel

This paper reports the results of a numerical investigation concerning the elastic and elastic-plastic post-buckling behavior of cold-formed steel-lipped channel columns affected by local/distortional/global (flexural-torsional) buckling mode interaction. The results presented and discussed are obtained by means of analyses performed in the code ABAQUS and adopting column discretizations into fine four-node isoparametric shell element meshes. The columns analysed (i) are simply supported (locally/globally pinned end sections with free warping), (ii) have cross-section dimensions and lengths ensuring equal local, distortional, and global (flexural-torsional) critical buckling loads, thus maximizing the mode interaction phenomenon under scrutiny, and (iii) contain critical-mode initial geometrical imperfections exhibiting different configurations, all corresponding to linear combination of the three "competing" critical buckling modes. After briefly addressing the lipped channel column "pure" global post-buckling behavior, one presents and discusses in detail numerical results concerning the post-buckling behavior of similar columns experiencing strong local/distortional/global mode interaction effects. These results consist of (i) elastic (mostly) and elastic-plastic equilibrium paths, (ii) curves and figures providing the evolution of the deformed configurations of several columns (expressed as linear combinations of their local, distortional, and global components) and, for the elastic-plastic columns, (iii) figures enabling a clear visualization of (iii1) the location and growth of the plastic strains, and (iii2) the characteristics of the failure mechanisms more often detected in this work.

Vibration of a Temperature-Dependent Thermally Pre/Postbuckled FGM Beam Over a Nonlinear Hardening Elastic Foundation

2013 ◽  
Vol 81 (1) ◽  
Author(s):  
S. E. Esfahani ◽  
Y. Kiani ◽  
M. Komijani ◽  
M. R. Eslami
Keyword(s):  
Elastic Foundation ◽  
Small Amplitude ◽  
Thermal Loading ◽  
Functionally Graded ◽  
Temperature Dependent ◽  
Graded Material ◽  
Fgm Beam ◽  
Generalized Differential ◽  
Nonlinear Hardening ◽  
Raphson Method

Small amplitude vibrations of a functionally graded material beam under in-plane thermal loading in the prebuckling and postbuckling regimes is studied in this paper. The material properties of the FGM media are considered as function of both position and temperature. A three parameters elastic foundation including the linear and nonlinear Winkler springs along with the Pasternak shear layer is in contact with beam in deformation, which acts in tension as well as in compression. The solution is sought in two regimes. The first one, a static phase with large amplitude response, and the second one, a dynamic regime near the static one with small amplitude. In both regimes, nonlinear governing equations are discretized using the generalized differential quadrature (GDQ) method and solved iteratively via the Newton–Raphson method. It is concluded that depending on the type of boundary condition and loading type, free vibration of a beam under in-plane thermal loading may reach zero at a certain temperature which indicates the existence of bifurcation type of instability.

Pre-Buckling Transverse Free Vibration of an Asymmetrically Supported Euler Beam Subjected to Distributed Follower Forces

2012 ◽  
Vol 217-219 ◽  
pp. 2640-2643
Author(s):  
Qing Lu Li ◽  
Shi Rong Li
Keyword(s):  
Mathematical Model ◽  
Free Vibration ◽  
Small Amplitude ◽  
Shooting Method ◽  
Free Vibrations ◽  
Geometrical Theory ◽  
Euler Beam ◽  
Simply Supported ◽  
Amplitude Vibration ◽  
Post Buckling

Based on the accurate geometrical theory for the extensible elastic beams, an exact mathematical model of post-buckling transverse free vibration of Euler beams subjected to a distributed tangential follower force along the central axis are established. By using shooting method, pre-buckling free vibrations of both simply supported and fixed Euler beam are solved and the responses of small amplitude vibration are obtained. The numerical results show that all the frequencies of unbuckled beams decrease continuously with the increment of the load parameters.

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