Caùc öùng suaát chính 1 max i j k 3 min i j k 33 Voøng troøn Mohr öùng suaát

Caùc öùng suaát chính 1 max i j k 3 min i j k 33

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Caùc öùng suaát chính: 1 = max[ i , j , k ]; 3 = min[ i , j , k ]. 3.3. Voøng troøn Mohr öùng suaát cho traïng thaùi öùng suaát phaúng: 3.3.1. Voøng troøn Mohr: a G A xy M 0 P 2 D H y 2 / ) ( y x x ) a c N B 1 B A M 0 P 1 max D N 1 ) b 2 1 n E max n c 3 3 n 2 min n min F J Hình 2.10
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Giaùo Trình CÔ CHÖÔNG 2: ÖÙNG SUAÁT VAØ BIEÁN DAÏNG 82 TS. Vuõ Coâng Hoøa Töø 2 coâng thöùc (5.45) vaø (5.46) ta coù: 2 xy 2 y x 2 2 y x 2 2 (2.52) const I 2 I 2 I 2 2 1 2 2 1 (2.53) Töø (2.52) vaø (2.53) ta thaáy raèng: Neáu choïn heä truïc toaï ñoä phaúng vuoâng goùc ) , ( thì ta coù theå bieåu dieãn traïng thaùi öùng suaát phaúng cuûa 1 ñieåm baèng 1 ñöôøng troøn, goïi laø voøng troøn Mohr öùng suaát. Trong coâng thöùc (2.45 ) vaø (2.46 ), cho bieán thieân töø 0 tôùi 2 , ta khaûo saùt söï bieán thieân cuûa a , a , tìm ñöôïc caùc maët caét nôi a hoaëc a ñaït cöïc trò. Tuy nhieân phöông phaùp giaûn tieän khaûo saùt a , a theo ñöôïc aùp duïng nhôø vieäc döïng moät voøng troøn (goïi laø voøng troøn Mohr öùng suaát) theo trình töï sau, (hình 2.10.a). - Laäp heä truïc toaï ñoä saùt ( , - ) coù truïc song song vôùi truïc x treân hình; - Töø goác toaï ñoä ñaët caùc ñoaïn 0M = y vaø 0N = x ; - Chia ñoâi ñoaïn MN, ta ñöôïc ñieåm C; - Xaùc ñònh vò trí ñieåm P coù toaï ñoä P ( y , - xy ). Ñieåm P ñöôïc goïi laø ñieåm cöïc cuûa voøng troøn. - Laáy ñieåm C laøm taâm, veõ voøng troøn baùn kính R baèng ñoaïn CP. Ñeå tìm öùng suaát treân maët caét xaùc ñònh bôûi goùc . Thoûa maõn ñieàu kieän (2.45) vaø (2.46) ta laøm nhö sau: Töø P keû ñöôøng song song vôùi truïc hoaønh, caét voøng troøn taïi ñieåm D. Deã daøng nhaän thaáy DN song song vôùi PM. Töø P keû tia P a hôïp vôùi truïc goùc . Giao cuûa P a vôùi voøng troøn ta kí hieäu laø G. Toaï ñoä G treân truïc hoaønh vaø truïc tung laàn löôït cho ta caùc giaù trò , . Sau ñaây ta seõ chöùng minh ñieàu nhaän ñònh treân. Töø G keû GH vuoâng goùc vôùi truïc hoaønh, caét truïc hoaønh taïi ñieåm H. Ta caàn chöùng minh OH = a vaø GH = a . Kyù hieäu goùc <DCB = . Töø caùch döïng voøng troøn ta coù 1 soá heä thöùc sau: ; 2 GCD (2.54)
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Giaùo Trình CÔ CHÖÔNG 2: ÖÙNG SUAÁT VAØ BIEÁN DAÏNG 83 TS. Vuõ Coâng Hoøa 2 xy 2 y x 2 2 y x y x y y x y x 2 / ) ( MP CM R ; 2 / ) ( 2 / ) ( CM OM OC ; 2 / ) ( 2 / MN CN MC ; MN (2.55) 2 y x 2 xy ) ( 4 2 1
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