5 c P 5 kn n t π (.5 P 7 MP π (.5 n t n cos π. MP 6 4 t sin π 6 cos π 6.7 MP 4 P P N i i i i N i j F j ii N i i ii F j i i N ii li i F j i ij li i i i

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1 i j ij i j ii,, i j ij ij ij (, P P P P θ N θ P P cosθ N F N P cosθ F Psinθ P P F P P θ N P cos θ cos θ cosθ F P sinθ cosθ sinθ cosθ sinθ

2 5 c P 5 kn n t π (.5 P 7 MP π (.5 n t n cos π. MP 6 4 t sin π 6 cos π 6.7 MP 4 P P N i i i i N i j F j ii N i i ii F j i i N ii li i F j i ij li i i i P F j i P j

3 BCD d d d BC BC D D BC D M ( dd d ddd M ( dd d ddd M M B C, ν (, ν(

4 d θ θ F dd dd F dd dd θ F d dcosθ d dsinθ F d dsinθ d d cosθ d cosθ d sinθ d sinθ d cosθ d d ( cosθ sinθ d d ( cosθ sinθ d d d d ( sinθ cosθ d d ( sinθ 4 d d cosθ d d d

5 d d d sinθ d cosθ ( ( ( sinθ cosθ sin θ cos θ sin θ sin θ ( cos sinθ cosθ sinθ cosθ ( sinθ cosθ sinθ cosθ (cos θ sin θ θ sinθ cosθ cos θ cosθ sin θ cosθ sin θ d F dd dd F dd dd F d dsinθ d dcosθ F d dcosθ d dsinθ d d sinθ cosθ d d cosθ sinθ d d d d d ( sinθ cosθ ( sinθ cosθ d d ( cosθ sinθ ( cosθ sinθ d 5 d d d d d θ d

6 d d d cosθ d sinθ ( sinθ cosθ cos θ ( sin θ sinθ cosθ sin θ cos θ sinθ cosθ ( cos θ sinθ cosθ ( sinθ cosθ sin θ ( sinθ cosθ (cos θ sin θ cosθ sin θ sin θ cos θ 4 MP MP MP 45 cosθ sinθ 4 MP -MP -MP 4 MP MP -MP 4 MP -MP MP MP -MP 6

7 ij, ij ij ji T cosα cos β l cosα cos β l α β β cosα cosα l l T cosβ cosβ α T p p p p p β cosα cos β α p p cosα cosβ l T l T 7 α p p ' β p

8 n l, p p, p ( ( p cosα p cosα l l p cos β p cos β p p l l p l p p l l p l l l l l l l l ( l l l l ll l ll ( l l q q q cos( π α cosα l cos β q q l T l T n l, q q, q ( ( q cosα q cosα q cos β q cos β q l q q l q p β q β q q α π α 8

9 l l l l l l l ( ll ll l l l l l l l l l l l l l l T T l l T l l l l l l l l l l l l l l l ( l l l l l l l l ( T T T T 4 MP MP MP 45 π4 π π4 π4

10 l cos π 4 cos π 4 l cos π 4 cos π 4 l l T l l MP cos θ sin θ cosθ sin θ sin θ cosθ tn θ n

11 cosθ n ± sinθ n ± principl stress plne principl stress principl is in θ sin cos θn θn inrint sin θ cos θ θ θ cos sin θ θ

12 tnθ t cosθ t ± sinθ t θ t in principl sher stress tn θ n tn θ t π π θ n ± θ t θ n ± θ t 4 π4 ( θ t θ n 4 MP MP MP

13 MP 6. MP tnθ n 4 θ n rctn rctn.49 rd MP 45 θ n 8. Mohr s circle cosθ sinθ sinθ cosθ k k R θ n O O k in k B ( P Q S

14 , (, B (, S R S k R k P Q S k R in k O k O O O O O O θ n ( k O O O O k O O cosθ n sinθ n cosθ n sinθ n 4

15 cosθ n sinθ n θ n B B (, B (, θ n θ n q q P k R O O k θ n S θ n k in B ( Q 5 MP 5 MP 5 MP θ n MP 5

16 k 5.8 MP (MP (.8, 5 (-5, 5 B 6.8 MP 6.8 MP (MP (-6.8,.8 MP -45 (6.8, 5 θ n 45 θ n.5 (5, (-.8, B 5.8cos 46.8 MP 5.8cos 46.8 MP.8sin 55.9 MP ddd i, i j ij ij, i j 6

17 p p p p p p p p l n p l n p p l n p l p p n l n l n n l p l l l p p n ( l l n n ( n n ( n l n l p l p p n l l n n (l l ( n n (n l n l l l l n n n i j i, j ij 7

18 k l i, j ki lj ij k l ki lj ij T T l T T n l n α β l cosα cos β n cos l n T T T T l ( l n T ( l n T n ( l n ( l n l ( n l ( n l n l n 8

19 ( ( ( ( ( ( { ( }( ( ( ( J J J J J ( J ( ( ( ( ( J J J ( ( ( ( ( 9 J J

20 4 4 4 ( ( ( ( } ( { } ( ( ( {( 6 J MP, MP, MP, MP, MP, MP J MP J ( (4 6 MP J MP 6 ( 6 ( 6 ( MP, MP, MP 4.5 MP,.5 MP,.5 MP

21 l l dl B dl l C D B B C B D CD CD B B {( ( }{( } ( ldl C C li ν ν ν. ν

22 ν ν ν ν ν ν ν

23 [ [ [ ν ( ν ( ν ( ] ] ] ν ( ( ν ν ν ( ν ν ν ( ν ν ( ν ( ν ( (ν (ν(ν ν ( ν ( ν ( ν λ µ ( ν ( ν ( ν ( λ µ λ λ λ (λ µ λ λ λ ( λ µ B d B C C φ d BB B d d d D

24 BB φ d tnφ d tnφ φ φ BCD d B C D B B d φ φ H B H H φ D d D d π B D φ φ B H φ φ H B H tn( φ φ tn( φ φ tnφ tnφ tn( φ φ tnφ tn tnφ tnφ tnφ tnφ φ φ C C 4

25 φ φ d C ω B B ω C ω φ φ φ d φ tnφ φ tnφ φ H D d D d ω ω ω ω ω ω ω ω ω ω G G G 5

26 G ν 45 BCD B B D D C BCD B C D d d C C C C C C C( ν B D BD BD B D BD( ν C B cos π 4 φ B (cos φ sinφ B cosφ( tnφ B D B sin π 4 φ B (cosφ sinφ B cos φ( tnφ tnφ tnφ B π4φ B φ C C C C, π4φ φ D DC 6

27 C Bcosφ B D Bcosφ C B D ν ( (ν ( ( ν ν (ν ( ν (ν ( ν G G G (ν G (ν µ G V V V V 7

28 V (V V V V V V V V V ddd d d d V V (d d(d d( d d V V V (d d(d d( d d ddd ddd ddd ddd ddd ddd ddd ddd ddd ddd V V V V V V V V V V K V K K ν V K ( V [ ν( ν ] V [ ν( ] ν V [ ν( ν ] V V 8

29 K ( ν V ( ( λ µ λ λ V V V (λ µ V K V K λ µ G [ ν( ] [ ν( ] ν ν [ ν( ] ν ν ν ν ( ν G (ν ( ν G (λ µ λ λ λ µ µ µ λ (λ µ λ µ λ µ µ λ λ (λ µ µ µ λ µ µ µ µ µ µ µ 9

30 ij k, l C ijkl kl C ijkl kl i j ij ij ij ij ij k,l S ijkl kl S ijkl kl C ijkl S ijkl l P dl dw Pdl P dl ld Pdl ld V l P dw dw dw V Pdl V d W dw d U U W d

31 O U e d d O OO U e d d d d d d U e ν( ν( ν( G ν( ( (ν ( G 6 {( ( ( ( } U (ν( 6 G (ν{( ( ( } 6 V G

32 V ( K ( ν U V K ( ν {( ( ( } 6 ( G U V V K U U U V U ( ν {( ( ( } ( 6 G U ( ν {( ( ( } 6 U ( ν ( 4 G (ν U ( ν ( G U

33 U V ( V V V ( ( ( ( ( J J J J J J J J V J J ( {( V ( V ( V ( V ( V ( V } {( ( V V } {( V } 6 {( ( ( ( } V 6 {( ( ( } J J J J ( V ( V ( V ( V ( V V J J V V

34 J J J J 6 {( ( ( } ij ji ( ( ( ( ( ( ( ( I I I I I I ( ( ( ( ( I 4

35 I I I V V V V ( ( ( ( ( I V I ( ( V ( V ( V ( V ( V ( V ( V V ( { ( } ( ( 6 {( ( ( } I 6 {( ( ( } I I I 5

36 6 (,, B(d, d, d (,, B (dd, dd, dd d d d d d d d d d d d d B ds d d d ds B α β ds d l α cos ds d β cos ds d n cos B ds n l d ds n l d ds n l d ds n l d d d B ds B B ds ds

37 7 ω ω ω ω ω ω i j j i ij i j j i ij ω,,,,

38 8 n l n l n l T l n n l n l n l det( I ( ( ( 4 ( 4 ( 4 ( 4 ( { 4 ( } 4 4

39 9 ( ( ( ( ( I I I I } ( {4( 4 ( I 4 4 I ij ij ij ij

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