PSI - Issue 6

Anatolii Bochkarev / Procedia Structural Integrity 6 (2017) 174–181

179

6

A.O. Bochkarev / Structural Integrity Procedia 00 (2017) 000–000

2.2. A case of the SSSF boundary conditions

In this more difficult case, three edges are S imply supported and the fourth is F ree (SSSF). Here we have the conditions, where the last depend on Poisson’s ratio,

w (0 , x 2 ) = 0 , w ( a, x 2 ) = 0 , w , 11 (0 , x 2 ) = 0 , w , 11 ( a, x 2 ) = 0; w ( x 1 , 0) = 0 , M ∗ 22 ( x 1 , b ) = − D ∗ ( w , 22 + ν ∗ f w , 11 ) ( x 1 , b ) = 0 , w , 22 ( x 1 , 0) = 0 , R ∗ 2 ( x 1 , b ) = − D ∗ ( w , 222 + (2 − ν ∗ f ) w , 112 ) ( x 1 , b ) = 0 .

(20)

An easier way to solve this problem employs the variational Ritz method, applied to rectangular plates by Papkovich (1941). He has shown that already a term

sin (

a )

x 2 b

πx 1

w ( x 1 , x 2 ) = A 0

(21)

sufficiently accurately describes the shape of the deflection. In the case of a coordinate function, the Ritz method reduces to the condition ∆ W = 0. From this equation we find the required critical load

h (

ha 2 (

a 2 b 2 )

a 2 b 2 )

6(1 − ν ∗ f ) π 2

π 2 D ∗

2 τ 0

3 π 2

p cr =

1 +

1 +

+

.

(22)

The first term in (22) has been used by Timoshenko as the first approximation of the critical load, and the second is its refinement under the pre-tension.

2.3. A case of the SCSF boundary conditions

This is the most complicated case of the three: two opposite loaded in-plane edges are S imply supported, one unloaded edge is C lamped supported, and the other is F ree (SCSF). We have the followoing conditions, where, as in Problem SSSF, the static part of the boundary conditions depends on Poisson’s ratio:

w (0 , x 2 ) = 0 , w ( a, x 2 ) = 0 , w , 11 (0 , x 2 ) = 0 , w , 11 ( a, x 2 ) = 0; w ( x 1 , 0) = 0 , M ∗ 22 ( x 1 , b ) = − D ∗ ( w , 22 + ν ∗ f w , 11 ) ( x 1 , b ) = 0 , w , 2 ( x 1 , 0) = 0 , R ∗ 2 ( x 1 , b ) = − D ∗ ( w , 222 + (2 − ν ∗ f ) w , 112 ) ( x 1 , b ) = 0 .

(23)

Following Papkovich (1941), as in Problem SSSF, we use the variational Ritz method. For this purpose we will expand a series where, in contrast to Problem SSSF, a sufficiently large number of terms in series has to be kept to achieve the accuracy w ( x 1 , x 2 ) = ∑ k =0 A k w k ( x 1 , x 2 ) = sin ( πx 1 a ) ( A 0 x 2 2 b 2 + x 2 b ∑ k =1 A k sin ( kπx 2 b ) ) . (24)