(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