Issue 59

H.A. Mobaraki et alii, Frattura ed Integrità Strutturale, 59 (2022) 198-211; DOI: 10.3221/IGF-ESIS.59.15

            6 4 2 1 1 N             7 1 2 1 2 1 N             8 4 1 2 1 N            9 2 1 2 1 N



x

y

Here,  ζ

. and  η

b are non-dimensional element coordinates and e a and e b are the element length and width,

a

e

e

respectively. Substituting Eqn.12 into Eqn.5 and Eqn.6 respectively provides:        1 2 T P e U d K d

(15)

         1 2 T P e T d M d

(16)

where   e K is the element stiffness matrix and   e M is the element mass matrix. Their expressions are provided in Appendix A. After assembling the element matrices and applying the Euler-Lagrange equations, the coupled governing equations of motion are obtained as follows:

  ¨ Δ

     

   Δ

    Δ

                                        ¨ 1 1 1 ¨ 2 2 2 3 3 q q C K q q q q q  

 

  f

M q

(17)

    ¨ q

3

where   Δ is the plate total displacement vector and   P C is the plate damping matrix, assumed as Rayleigh’s proportional damping [25]:         0 1 P P P C a M a K (a 18)

    i i

          0 i

a a

   1 1/ 2 1/

  

(b 18)

 j  

  1

j

j

 γ 1/ 2 and  β 1/ 4 .

Eqn.17 is discretized by applying the Newmark time integration method in which

N UMERICAL RESULTS

n this section, firstly, the free vibration results are compared with those available in the literature. Also, a parametric analysis is carried out to study the effects of system dynamic characteristics on the dynamic response of plate. The examples provided are based on the following material properties unless mentioned otherwise: I

203