PSI - Issue 13

Ramdane Boukellif et al. / Procedia Structural Integrity 13 (2018) 85–90

88

4

Boukellif et al. / Structural Integrity Procedia 00 (2018) 000–000

¯ σ xy

¯ σ xy

¯ σ xy

¯ σ yy

¯ σ yy

¯ σ yy

( x c ; y c )

¯ σ xx

¯ σ xx

( ¯ x ; ¯ y )

( + )

α

α

α

2 a

2 a

2 a

( x a ; y a )

( x a ; y a )

( x a ; y a )

( ˜ x ; ˜ y )

measuring point P m

y

y

y

x

x

x

(b)

(a)

(c)

Fig. 2: Inclined center crack in finite plate ( x c ; y c ) = (20; 20) mm under boundary loads ¯ σ xx = 30 MPa, ¯ σ yy = 90 MPa, ¯ σ xy = 20 MPa with: (a) : P m ( m = 1 , ..., 4) measuring points; (b) : P m ( m = 1 , ..., 8) measuring points; (c) : P m ( m = 1 , ..., 12) measuring points.

Table 5: Results of the crack detection and parameter identification by using eight strain gauges ( α = 60 ◦ ), see Fig. 2 (b) .

Table 4: Results of the crack detection and parameter identification by using four strain gauges ( α = 60 ◦ ), see Fig. 2 (a) .

parameters

error: given problem / obtained results (4 gauges) crack half-length a [mm] 1 3 5 7 9

parameters

error: given problem / obtained results (8 gauges) crack half-length a [mm] 1 3 5 7 9

¯ σ xx [%] ¯ σ yy [%] ¯ σ xy [%]

0 , 03 0 , 23 0 , 06 1 , 76 2 , 83 0 , 011 0 , 077 0 , 011 0 , 188 0 , 255

¯ σ xx [%] ¯ σ yy [%] ¯ σ xy [%]

0 , 03 0 , 03 0 0 , 011 0 , 011 0

0

7 , 13

0 , 011 5 , 41

0

2

0 , 15 9 , 15 14 , 45

0

0 , 05 0

0

16 , 55

a [%] α [%] x 0 [%] y 0 [%]

8 , 5

69 , 8 11

63 , 42 79 , 33

a [%] α [%] x 0 [%] y 0 [%]

0 , 1

0

0 , 2 0

9 , 22

12 , 11 53 , 81 15 , 21 8 , 98 132 , 25

0 , 05 0

0 0

0 0

34 , 85

0 , 1 0 , 1

92 , 14 0 , 1

73 , 8 81 , 7 91 , 8 45 . 9

0

0

0

12 , 4 0

0 , 1

0 , 1

0 , 1 0 , 1

0 , 1

Table 6: Results of the crack detection and parameter identification by using twelve strain gauges ( α = 60 ◦ ), see Fig. 2 (c) . parameters error: given problem / obtained results (12 gauges) crack half-length a [mm] 1 3 5 7 9

Table 7: Results of crack detection and parameter identification for di ff erent inclination angles α and a = 6 mm , see Fig. 2 (c) .

parameters

error: given problem / obtained results inclination α [ ◦ ]

0

40

80

120

160

¯ σ xx [%] ¯ σ yy [%] ¯ σ xy [%]

0 , 033 0 0 , 011 0

0 0

0 0

1 , 93 0 , 56

¯ σ xx [%] ¯ σ yy [%] ¯ σ xy [%]

0

0 0

0 , 033 0 , 033 0 , 033

0 , 02

0

0 , 011 0 , 011

0 0 0 0 0

0 , 05 0 , 05 0 , 05 4 , 55

0 , 1

0 , 05 0 , 16

0 , 05

0 , 05

0 0 0

a [%] α [%] x 0 [%] y 0 [%]

0 , 033 0

0 , 14 0 , 07

a [%] α [%] x 0 [%] y 0 [%]

0 1 0 0

0

0 0 0 0

0 , 016 0 , 016 0 , 016 0 , 5

0 , 025 0

0 , 01 0 , 1

0 0

0 , 1

0 0

0

0 , 1

0

0

0

0 , 1

0

K I ( + ) [%]

0 , 036 0 , 187 0 , 046 0 , 184 1 , 66

K I ( + ) [%] K I I ( + ) [%]

0 , 030 0 , 16

0 , 03

0 , 029 0 , 008

K I I ( + ) [%] 0

0 , 11 0 , 07 0 , 07 4 , 55

0 , 050 0 , 159 0 , 150 0 , 013 0 . 018

3.2.2. Parameter identification depending on locations of strain gauges To study the influence of the distance of strain gauges on the accuracy of the solution of the inverse problem for an inclined center crack ( a = 1 mm) in a finite plate (30 mm x 30 mm), the configurations depicted in Fig. 3 have been investigated. Five di ff erent squares indicate the positions of virtual strain gauges. The dimensions of these squares are given in the caption of Fig. 3. Along each of the squares, 12 gauges are distributed just as shown in Fig. 3. Tab. 8 shows the given problem and results of parameter identification depending on locations of strain gauges. The parameters have been determined very accurately at all strain gauge positions except for the smallest square (1).