@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}
}
