Finite Element Procedures

1- . ( ) . . (Exact) . (Accuracy) ( Convergence). Exact solution Analytical solution Closed Form Solution Approximate Solution Numerical Solution Convergence Accuracy

1956 Clough Turner Top Martin :

Stiffness and Deflection Analysis of Complex Structures, Journal of Aeronautical Sciences, 23, 805-825 (1956).

( ) . 1-

.

. .1-

2- : :1- (Lumped parameter model) (Discrete system model)

2- (Continuum mechanics-based model) ( Continuous system) . .

:

(State Variable) ( ). .

: . 2-

:

. .

: .- . 2-

. : Ritz Galerkin . ( ).2-

3- :

4- . ( ). .

- . .- . - .4-

- (Exact) .- ( very comprehensive mathematical model) ( ).4- : . - (Effectiveness) ( Reliability) .

: . 1 : Bernoulli ( ) Timoshenko( ) ( ) ( )4-

4- 2

( ): ( ) ( )

4-

: 4-

5- Skeletal structures Continum structures Nodal points- . .

- . .5-

: . . . Clough :

. 5-

) Initial subdivision into elements( ) Dirivation of the element stiffness characteristics( - . .5-

: P=K .5-

6- ( ... ) ( Documentation) . . :

7- : . :- (Direct method)- (Variational method) :1- (System idealization) : .2- (Equilibrium of elements): .3- (Element assemblage): .4- : .)

: 3 . .: 1 4 . U1 U2 U3 . . . 7- :

:7- :Fi(j) = j Ui .

Fi(j) J=1,2,,5 i=1,2 () . U1 U2 U3 1 : 2:7- : .

K : K(i) . . 7- : :

) (Variational method)- . Ui i=1,,n (Functional) (Saddle point) . ( ) :U = W = 7- :

- (Axiom) : : (Total Potential Energy) ( ) . .(Stationary requirement) 7- :

Minimum StableSaddle point Critical stateMaximum Unstable : ( ) (Equilibrium state) .7- :

(Principle of Minimum Potential Energy) :

(Conservative) ( ) . .7- :

.: K P . . .7- :

7- :: . .

8- : . : ( ) ) ( ) . .

. . ( ) .: 8- :

: f B(x) R .8- :

: . R(t) . : ()8- :

() () () : (Boundary condition) (Initial condition)

1) (Boundary value problems) ( ) . . (Steady state) .8- : :2) (Initial value problems) . . . (Propagation) .

(stationary) . . . (Functional) . m Cm-1 .

8- : ) ( )

:

) ()Essential boundary condition) ( ( m-1 )

) () (Natural boundary conditions) ( m 2m-1 )8- :

: .8- :

) .8- :

m=2 , C1 1=m-1 : 8- :

3 5 . w : ( 4 (2m)) x =L x =L 8- : :( 3 (2m-1))

: () () .1- . ( ...) ( ...) .2- . 8- : :3- .4- .

10) Ritz Ritz . Ritz ( ). Ritz (Trial function) :ai = Ritzfi = Ritz n ai :

Ritz (fi) ( ) . .) Ritz .: w :

( ) .10) Ritz

w : ( a1 a2) : P . P .10) Ritz

Ritz (fi ) (Shape functions) (Displacement interpolation matrix) .H) (Shape function) )10) Ritz ( (u(m). Ritz (ai) U .

11) Galerkin - Galerkin . .- (Steady state) :L2m = (Differential operator) ( 2m ) = (State variable)r= (Force function)

f B(x) R : L2m :11) Galerkin

- Galerkin ( ).- : fi ( Galerkin ).(Residual) :11) Galerkin

- (Exact Solution) (R=0) R . R . ( Galerkin (Least square method) (Sub-domain) (Collocation)) ai R . Galerkin R ai n : D .11) Galerkin Galerkin ai .

1- Ritz Galerkin .2- Ritz m m Galerkin 2m 2m . 3- Ritz Galerkin . 11) Galerkin Ritz Galerkin

12) (Principle of virtual Displacements) : :

( ) .12) (Principle of virtual Displacements)

: 12) (Principle of virtual Displacements) .

: . 12) (Principle of virtual Displacements)

Ritz ( ) Galerkin ( ) . Ritz Galerkin Ritz .12) (Principle of virtual Displacements)

*************************************************************