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Tham khảo tài liệu 'theory and problems of strength of materials part 16', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | CHAP 16 COMBINED STRESSES 445 Next we observe that the ordinate of point f is given by T nf cfsin 20p - 20 c sin2ớpCos2ữ-cos20psin20 V cr - ơy Ị2 rxy 2 sin 26p cos 20 - cos 26p sin 20 Again substituting the values of Tx and ịịơy - TX from into this equation we find T Tx cos 26 ị ơx Ơ sin 26 But this is exactly the shearing stress on a plane inclined at an angle 6 to the x-axis as derived in 2 of Problem 16.13. Hence the coordinates of point f on Mohr s circle represent the normal and shearing stresses on a plane inclined at an angle 6 to the x-axis. 16.15. A plane element is subject to the stresses shown in Fig. 16-42. Determine a the principal stresses and their directions b the maximum shearing stresses and the directions of the planes on which they occur. Fig. 16-42 a In accordance with the notation of Problem 16.13 we have Ợ 12 000 lb in2 ơy 15 000 lb in2 and Txy 80001b in2. The maximum normal stress is by 5 of Problem 16.13 ỉ ơx V ị ơx - ÍT 2 tx 2 ị 12 000 15 000 V 12 000 - 15.000 2 8000 2 13 500 8150 21 650 lb in2 The minimum normal stress is given by 6 of Problem 16.13 to be ơnra ị ơ ơy - V T o x ơỹj 2 Try 2 13 500 - 8150 5350 lb in2 From 3 of Problem 16.13 the directions of the principal planes on which these stresses of 21 650 lb in2 and 53501b in2 occur are given by Txy _ 8000 an ị trx- _ j 12 000 - 15 000 - 5 33 Then 26p 79 24 259 24 and 6P 39 42 129 42 . To determine which of the above principal stresses occurs on each of these planes we return to of Problem 16.13 namely Ơ T ơy Ị ơx ơy cos 26 7 y sin 26 and substitute 6 39 42 together with the given values of TX ơr and Tx to obtain ơ ị 12 000 15 000 - ị 12 000 - 15 000 cos 79 24 8000sin79 24 21 650 Ib in2 Thus an element oriented along the principal planes and subject to the above principal stresses appears as in Fig. 16-43. The shearing stresses on these planes are zero. 446 COMBINED STRESSES CHAP. 16 Fig. 16-43 6 The maximum and minimum shearing stresses were found in 8 of Problem 16.13 to be - Ơ 2 r