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(1)
(3)
(2) (3)
(𝜎𝑛−𝜎𝑥 + 𝜎𝑦
2)2 + 𝜏𝑡
2 = (𝜎𝑥 − 𝜎𝑦
2)2 + 𝜏2
(𝜎𝑛−𝑎)2 + 𝜏𝑡
2 = 𝑅2
𝑤ℎ𝑒𝑟𝑒; 𝑎 =𝜎𝑥 + 𝜎𝑦
2
𝑅 = (𝜎𝑥 − 𝜎𝑦
2)2 + 𝜏2
𝑎 𝑎𝑛𝑑 𝑟𝑎𝑑𝑖𝑢𝑠 𝑅 𝑖𝑛
(𝜎𝑥 , 𝜏) 𝜎𝑥(𝜎𝑦, 𝜏) 𝜎𝑦
=𝜎𝑥 + 𝜎𝑦
2
=𝜎𝑥 − 𝜎𝑦
2
= (𝜎𝑥 − 𝜎𝑦
2)2 + 𝜏2
𝜎𝑥 − 𝜎𝑦
2= (
𝜎𝑥 − 𝜎𝑦
2)2 + 𝜏2
(𝜎𝑥 , 𝜏)
𝐸(𝜎𝑦, 𝜏)
𝜎𝑥
𝜎𝑥 − 𝜎𝑦
2𝜏
𝜎𝑥 − 𝜎𝑦
2
𝜎𝑥 + 𝜎𝑦
2𝜏
𝜎𝑛
𝜎𝑥 − 𝜎𝑦
2𝜏
𝜏𝑡
𝜎𝑥
𝜎𝑥 + 𝜎𝑦
2
𝜏𝑡𝜎𝑛
𝜎𝑥
𝜎𝑛
𝜏𝑡𝑟
1
𝜎𝑥 𝜎𝑦 𝜏
𝜎1 𝜎2𝜏𝑚𝑎𝑥
2
𝜎𝑥 𝜎𝑦 𝜏
𝜎1 𝜎2𝜏𝑚𝑎𝑥
3
𝜎𝑥 𝜎𝑦 𝜏
𝜎1 𝜎2
𝜏𝑚𝑎𝑥
2𝜃1
2𝜃2
2𝜃1 = 72 + 180°
= 252°2𝜃2 = 72°
or 432°
4
𝜎𝑥 𝜎𝑦 𝜏
𝜎𝜎𝑛
𝜏𝑚𝑎𝑥