1.
27 Given the state of stress in Example 2.1·O2,-120 kPa·Gy-150 kPa, and 0%,-0 kPa, compute
the values of ,yy for all values of from 0° to 90° and plot as a function of a. Compare the values
of at which 'yy is max/min versus those found using the formula for, Repeat for ., and compare
the value of at which the shear is max/min versus that using the formula for a.
Solution
the Equiation is :
Oyy´ = ( Oxx + Oyy ) / 2 - (( Oxx - Oyy) /2) * cos ( 2* B)) - Oxy * sen ( 2*B )
With Oxx = 120 kpa , Oyy = 150 kpa Oxy = 0 kpa, so:
Oyy´ = (120+ 150 ) / 2 - (( 120 - 150) /2) * cos ( 2* B)) - ( 0 ) * sen ( 2*B )
Oyy´ = 135 + 15 * cos ( 2* B))
So :
B = 0 Oyy = 150
B = 1 Oyy = 149.94
B = 3 Oyy = 149.85
B = 4 Oyy = 149.77
B = 5 Oyy = 149.67
B = 6 Oyy = 149.55
B = 7 Oyy = 149.42
B = 8 Oyy = 149.27
B = 9 Oyy = 149.1

2.
B = 10 Oyy = 148.91
B = 11 Oyy = 148.7
B = 12 Oyy = 148.48
B = 13 Oyy = 148.24
B = 14 Oyy = 147.99
B = 15 Oyy = 147.72
B = 16 Oyy = 147.44
B = 17 Oyy = 147.14
B = 18 Oyy = 146.8
B = 19 Oyy = 146.5
B = 20 Oyy = 146.15
B = 21 Oyy = 145.79
B = 22 Oyy = 145.42
B = 23 Oyy = 145.04
B = 24 Oyy = 144.64
B = 26 Oyy = 144.23
B = 28 Oyy = 143.82
B = 29 Oyy = 143.39
B = 30 Oyy = 142.95
B = 31 Oyy = 142.5
B = 32 Oyy = 142.04
B = 33 Oyy = 141.58
B = 34 Oyy = 141.1

3.
B = 35 Oyy = 140.62
B = 36 Oyy = 140.23
B = 37 Oyy = 139.64
B = 38 Oyy = 139.23
B = 39 Oyy = 138.63
B = 40 Oyy = 138.11
B = 41 Oyy = 137.6
B = 42 Oyy = 137.1
B = 43 Oyy = 136.57
B = 44 Oyy = 136.04
B = 45 Oyy = 135
B = 46 Oyy = 134.48
B = 47 Oyy = 133.96
B = 48 Oyy = 133.43
B = 49 Oyy = 132.91
B = 50 Oyy = 132.4
B = 51 Oyy = 131.88
B = 52 Oyy = 131.37
B = 53 Oyy = 130.87
B = 54 Oyy = 130.36
B = 55 Oyy = 129.87
B = 56 Oyy = 129.38
B = 57 Oyy = 128.9

4.
B = 58 Oyy = 128.42
B = 59 Oyy = 127.96
B = 60 Oyy = 127.5
B = 61 Oyy = 127.05
B = 62 Oyy = 126.61
B = 63 Oyy = 126.18
B = 64 Oyy = 125.77
B = 65 Oyy = 125.36
B = 66 Oyy = 124.96
B = 67 Oyy = 124.58
B = 68 Oyy = 124.21
B = 69 Oyy = 123.85
B = 70 Oyy = 123.51
B = 71 Oyy = 123.18
B = 72 Oyy = 122.86
B = 73 Oyy = 122.56
B = 74 Oyy = 122.28
B = 75 Oyy = 122.01
B = 76 Oyy = 121.76
B = 77 Oyy = 121.52
B = 78 Oyy = 121.3
B = 79 Oyy = 121.01
B = 80 Oyy = 120.9

5.
B = 81 Oyy = 120.73
B = 82 Oyy = 120.58
B = 83 Oyy = 120.44
B = 84 Oyy = 120.33
B = 85 Oyy = 120.23
B = 86 Oyy = 120.15
B = 87 Oyy = 120.08
B = 88 Oyy = 120.04
B = 89 Oyy = 120.02
B = 90 Oyy = 120