@article {24165,
	title = {Simplified Method Using Homogenization Approch for Non Linear Dynamic Analysis of Dissipative Energy in Multilayered Beams},
	journal = {Arab Journal of Science and Engineering},
	volume = {39},
	number = {06},
	year = {2014},
	pages = {5073-5084},
	abstract = {Based on known theoretical developments in linear dynamics of homogeneous beams, two homogenizationapproaches of composite beams are developed further to an\&nbsp;anterior work using two equivalent properties: the physical and the mechanic{\textendash}geometrical properties. Further to the\&nbsp;assumption of Euler{\textendash}Bernoulli beams, dynamic parameters\&nbsp;are needed. Equations of a given beam structure subjected\&nbsp;to free un-damped and/or damped vibration are established.\&nbsp;The natural frequency responses of the first five modes are\&nbsp;obtained from both approaches, and then compared with\&nbsp;those obtained from a finite element model approach, taking into account different slenderness and boundary conditions. The result shows good agreement. An extension to the\&nbsp;equivalent physical parameter homogenization method using\&nbsp;the behaviour law in nonlinear state is presented here. Thus,\&nbsp;the homogenization is extended to an elastically equivalent\&nbsp;model elaboration for a system having an elastic{\textendash}plastic and\&nbsp;bilinear behaviour. The aim is to use analytical expressions\&nbsp;from an elastically equivalent model for a nonlinear system\&nbsp;of multilayer beam type to obtain the new corrected dynamicparameter. Based on the ductility factor method combined\&nbsp;with the secant method that uses a substitute structure and\&nbsp;secant stiffness to account for nonlinear behaviour, we developed a formula using global\&nbsp;

	effective\&nbsp;Es*,\&nbsp;the equvalent mass density ρ*, as well as the two types of\&nbsp;viscous damping parameter (these types include a viscous\&nbsp;resistance to transverse\&nbsp;



	displacement C* of the beam anda viscous resistance to straining of the beam material CS*)\&nbsp;which can be\&nbsp;



	incorporated into the formula without difficulty.\&nbsp;The\&nbsp;



	nonlinear analysis shows that the degradation of\&nbsp;



	rigidity decreases the frequencies response curve,\&nbsp;



	which tends to\&nbsp;increase the vibration periods giving an additional storage\&nbsp;space for the structure capacity to accumulate displacements\&nbsp;of a higher degree\&nbsp;



	compared to those of the elastic case.\&nbsp;The\&nbsp;



	importance of the nonlinearity assumption is shown,\&nbsp;



	which fits better with the real mechanical\&nbsp;



	performances of the\&nbsp;structure. \&nbsp;



	\&nbsp;

},
	url = {https://link.springer.com/article/10.1007/s13369-014-1036-4	},
	author = {Toufik Benmansour and Abdelhak Sekhri}
}
@article {24161,
	title = {Dynamic analysis of complex structure by equivalent methods: application to isotropic and orthotropic plates},
	journal = {Malaysian Journal of Civil Engineering (MJCE) },
	volume = {Vol. 25 },
	number = {01},
	year = {2013},
	pages = {82-93},
	abstract = {
	This paper intends to provide an equivalent method for the evaluation of natural frequency of isotropic and orthotropic thin rectangular plates with different restraint conditions. Starting from a simple and a general approximate formula for the frequency, which is the extension of Hearmon{\textquoteright}s expressions presented for the fundamental mode; it is shown how to calculate the fundamental mode of isotropic and orthotropic rectangular plates using the proper coefficient values already available in the scientific literature. For the higher modal frequencies, a particular form of Rayleigh{\textquoteright}s method is proposed, leading to a simple procedure to calculate the fundamental frequency. In fact the frequency calculation is reduced to the evaluation of the fundamental frequency of a special plate associated with the real one. An extensive finite element investigation was carried out to test the accuracy of this analytical short cut method. In addition a comparative study has shown good agreement between the frequencies responses obtained from both analytical (equivalent method) and finite element solution using ANSYS program. This method allows us to both avoid huge calculations, and produce a fast and simple approximate calculus of free vibration frequencies. This in turn is a necessary part of the preliminary design phase and the general and immediate verification of a construction project to its completion.

},
	url = {https://mjce.utm.my/index.php/MJCE/article/view/294},
	author = {Abdelhak Sekhri and Toufik Benmansour}
}
@article {24170,
	title = {Equivalents methods to Investigate Free Vibration of Isotropic and Orthotropic Thin Rectangular Plate with Non-Homogenous Supports},
	journal = {https://www.praiseworthyprize.org/latest_issues/IREME-latest/IREME_vol_5_n_7.html},
	volume = {5},
	number = {7},
	year = {2011},
	pages = {1196-1207},
	abstract = {Evaluating approximate frequency for isotropic and orthotropic plates is a complicated problem, thus exploiting the general formula of Hearmon, it is proposed in this investigation to calculate the fundamental mode of isotropic and orthotropic plates with two non-homogenous supports (cases: SCSC \&amp; SSCC). For the higher mode of modal frequency, a particular form of Rayleigh{\textquoteright}s method is used leading to a simple procedure for calculating the fundamental frequency. A new simple and qualitative method is proposed and has permitted a good strategy to evaluate the quality of results obtained. In order to verify the precision of the proposed qualitative method, a confrontation with finite element method using ANSYS software was done. The complementary utilization of Hearmon{\textquoteright}s principle and the qualitative method has also permitted a successful advance in evaluating higher modes. This combined procedure gives a strategy of a vibratory analysis of isotropic and orthotropic plates; it permits the satisfaction of the preliminary conception needs of the structure to be studied, and also provides a qualitative method for expertise and investigation of dynamical responses. In all cases, the dynamic investigation is based on some evaluation criteria such as: limit conditions effect, plate dimensions ratio effect, material effect and mode number effect.},
	url = {https://www.praiseworthyprize.org/latest_issues/IREME-latest/IREME_vol_5_n_7.html$\#$Equivalents_Methods_to_Investigate_Free_Vibration_of_Isotropic_and_Orthotropic_Thin_Rectangular_Plate_with_Non-Homogeneous_Supports},
	author = {Abdelhak Sekhri and Toufik Benmansour}
}
@article {24171,
	title = {Approche d\&$\#$39;homog{\'e}n{\'e}isation pour l\&$\#$39;analyse dynamique libre des poutres multicouches dissipatives d\&$\#$39;{\'e}nergie: Application aux poutres Bernoulli a sym{\'e}trie miroir},
	journal = {Revue de l{\textquoteright}universit{\'e} de Constantine Sciences \& Technologie B},
	volume = {30},
	year = {2009},
	pages = {25-32},
	author = {T. Benmansour, L. Bensmail, A. Sekhri}
}
