kou05.dvi

2 C () 25

1 3 1.1........................................ 3 1.2..................................... 4 1.3..................................... 7 1.3.1................................ 7 1.3.2................................. 8 1.4................................... 1 1.5............................ 1 1.6.................................... 13 1.7.................................... 14 1.7.1 ()............ 14 1.7.2............................. 22 1.7.3 ()........... 26 1.8.................................... 32 1.8.1..................... 32 1.8.2 Castglano.............................. 38 1.9...................................... 45 1.9.1............................... 45 1.9.2 Müller Breslau................ 46 1.1.............................. 5 1.1.1..................................... 5 1.1.2................... 53 1.1.3................. 54 2 56 2.1................................... 56 2.1.1...................................... 56 2.1.2................................. 58 2.1.3 ( )......................... 61 2.2 (Bernoull-Euler )........................ 63 2.2.1............................... 63 2.2.2............................... 64 2.2.3.......................... 64 2.2.4........................... 65 1

2.2.5............................ 66 2.2.6............................... 67 2.2.7.................................... 67 2.3 ()......................... 69 2.3.1.................................. 69 2.3.2..................................... 7 2.3.3................................ 7 2.4 ()................... 72 2.5 ()........... 74 2.6................................ 74 2.7.............................. 88 2.8................................. 97 2.8.1.................................... 97 2.9............................... 18 2.9.1............................. 18 2.9.2........... 19 2.9.3............................ 113 2.9.4............. 114 3 116 3.1...................................... 116 3.2............................ 12 3.3...................................... 126 4 131 4.1 Euler................................... 131 4.1.1..................................... 131 4.1.2.................................... 132 4.1.3............................. 133 4.1.4................................ 133 4.2.................................... 139 4.2.1................................. 139 4.3.................................... 141 1

() () 1.,,, 1994. 2.,,, 23. 3., [/],, 199. 4., http://gspsun1.gee.kyoto-u.ac.jp 4. (meda) Teachng Asstant patch meda center() (nchml@meda.kyoto-u.ac.jp) : 1.1 2.5 : 2.6 4.3 : 2

1 1.1 () ( ). Hooke (2.1 ) () Hooke () = E () (1.1) E Young (1.1) N ( ) A () = (N) (A) N 1.3 δ() () = (δ) (l) (δ) = () (l) 1.2 (1.1) δ = Nl EA (1.2) 3

Dsplacement External force Reacton Pn wthout frcton Before deformaton After deformaton Reacton Fg 1.1 1.2 (dsplacement) u ( u ) u ( ) I I n I = (1.3) I Fg 1.2 Fg 1.2 1 δ 1 Fg 1.3 u 1 u 2 1 δ 1 (1.3) δ 1 = n 1 1 u 1 + n 2 1 u 2 (1.4) 4

( ) n 11 1 u 1 u 1 n 11 1 n 21 2 u 2 u 2 n 21 Fg 1.3 (1.4) n n = n 2 1 (= n 1 1 ) () = nl = l () = nl + u 2 u 1 = (nl + u 2 u 1, nl + u 2 u 1 ) = l 2 + 2(u 2 u 1 ) nl + (u 2 u 1 ) (u 2 u 1 ) u 1 u 2 l () l 1 + 2 (u 2 u 1 ) n l { l 1 + (u } 2 u 1 ) n l = l + (u 2 u 1 ) n δ 1 = (u 2 u 1 ) n δ 1 (1.4) I δ I δ I = I : n I u (1.5) 5

(1.3) n I n I = { ni I (1.6) (1.5) I () ( ) (1.5) δ I = n I u (1.7) (1.7) (nfntesmal deformaton theory) (1.7) (u, δ I ) 1Fg 1.4 u 1 1 45 45 u 3 3 1 2 x 2 2 u 2 x 1 Fg 1.4 (1.7) δ 1 = n 1 1 u 1 + n 2 1 u 2 ( ) ( ) 1/ 2 = 1/ u11 + 2 u 21 = u 11 + u 21 + u 12 u 22 2 2 2 2 ( 1/ 2 1/ 2 ) ( u12 u 22 ) δ 2 = n 2 2 u 2 + n 3 2 u 3 ( ) ( 1/ 2 1/ 2 = 1/ u 2 + 2 1/ 2 ) u 3 6

1.3 1.3.1 N ( ) () N () I f I N I j x x j I f I + f ji = x f I + x j f ji = f I f ji (x j x ) f ji = x j x f ji x j x f ji (= f I ) (f ji = )f I f ji f I f ji f I f ji (1.3) f I = n I N I (1.8) N I I N I f I f ji I ()N I Fg 1.5 1 f 2 1 = n 2 1 N 1, f 1 1 = n 1 1 N 1 7

Fg 1.5 1 1.3.2 F x 1 x 2 F 1 F 2 Fg 1.5 1 1 1 3 4 (Fg 1.6 ) Fg 1.6 1 1 f 1 1 f 1 3 f 1 4 1 f 1 1 f 1 3 f 1 4 8

F 1 1 F 1 + ( f 1 1 ) + ( f 1 3 ) + ( f 1 4 ) = F 1 = f 1I = n 1I N I I: 1 I: 1 (1.8) F = n I N I I: (1.6) F = I n I N I (1.9) (1.9) (F, N I ) 2Fg 1.7 Fg 1.7 2 (1.9) ( ) 1/ 2 F 1 = n 1 1 N 1 = 1/ N 1 2 ( ) 1/ 2 F 3 = n 3 2 N 2 = 1/ N 2 2 ( ) 1/ 2 F 2 = n 2 1 N 1 + n 2 2 N 2 = 1/ N 1 + 2 9 ( 1/ 2 1/ 2 ) N 2

1.4 F = ( ) F1 F 2 u = ( ) u1 u 2 (boundary condtonb.c.) 3Fg 1.8 Fg 1.8 1 : F 11 =, F 21 = P (u 11 =, u 21 = ) 2 : F 12 =, u 22 = (F 22 =, u 12 = ) 3 : F 13 =, F 23 = (u 13 =, u 23 = ) 4 : F 14 =, F 24 = (u 14 =, u 24 = ) 5 : u 15 =, u 25 = (F 15 =, F 25 = ) 134 5 2 34 1.5 1. 2. 3. F N uδ 1

1. (F, N) (statcally admssbles.a.) 2. (u, δ) (knematcally admssblek.a.) 4Fg 1.9 N ( -P ) Fg 1.9 2 ( ) ( ) 1/ 2 1/ 2 F 2 = 1/ N 1 + 2 1/ N 2 2 (F 12, F 22 ) = (, P ) ( ) ( ) ( ) 1/ 2 1/ 2 N = P 1/ 2 1/ 1 2 N 2 ( ) N 1 = N 2 ( 1/ 2 1/ 2 1/ 2 1/ 2 ) ( P ) = ( P/ 2 P/ 2 ) 4 4 () () 5 () () () 11

5Fg 1.1 N Fg 1.1 2 (F 12, F 22 ) = (, P ) ( ) ( ) ( ) 1/ 2 1/ 2 F 2 = 1/ N 1 + 2 1/ N 2 + N 3 = 2 1 n 2 1 n 2 2 n 2 3 ( P (F, N) 1 1 1 (F, N) N 3 N 1 N 2 ) ( 1/ 2 1/ 2 1/ 2 1/ 2 ) ( N 1 N 2 ) = ( P + N 3 ) ( N 1 N 2 ) = ( ) P N 3 / 2 ( ) P N 3 / 2 (u, δ) 6Fg 1.9 (u, δ) 12

(1.7) δ I = n I u δ 1 = n 1 1 u 1 + n 2 1 u 2 δ 2 = n 2 2 u 2 + n 3 2 u 3 u 1, u 3 = ( ) δ 1 1 = 2 1 ( ) 2 u12 δ 2 1 2 1 2 u 22 u 2 δ 1 δ 2 4 N 1 N 2 δ 1 = δ 2 = P l 2AE u 2 = ( ) u12 u 22 ( ) = P l/ae 1.6 (Prncple of vrtual work) : (F, NI ) (u, δ I ) F u = I N I δ I (1.1) (F, N ) (u, δ) [] (F, N ) (1.9) (u, δ) (1.7) = F u = ( ) n I NI u = ( ) NI n I u = NI δ I = I I I 13 []

1.7 () (F, N) 1.7.1 () (1.7.3 ) 1. () 2. 1. ((u, δ) ) 3. 1 ((F, N ) ) 4. 3. (F, N ) 2. (u, δ) 7Fg 1.11 14

1 2 3 P Fg 1.11 (Young EA) 1. (Fg 1.12 ) (4 ) Fg 1.12 2. ((1.2) δ = Nl/AE) δ 1 = δ 2 = P l 2AE (Fg 1.13 ) Fg 1.13 ( 3. 3 1) 1 F 1. Fg 1.14 Fg 1.14 15

4. 2. ( ) 3. 1 + 2 F u = I = 1 2 + u 2 3 N u 2 = P l AE N I δ I P l 2AE } {{ } δ } {{ } + 1 2 N P l 2AE } {{ } δ } {{ } 8Fg 1.15 Fg 1.15 (Young EA) 1. (Fg 1.16 ) Fg 1.16 2. δ 1 = P l δ 2 = P l (Fg 1.17 ) 2AE 2AE 16

Fg 1.17 3. x 1 (Fg 1.18 ) Fg 1.18 x 2 (Fg 1.19 ) Fg 1.19 4. 2. 3. F u = NI δ I I x 1 1 + 2 + u 1 3 = 1 2 N u 1 = P l AE P l 2AE } {{ } δ } {{ } 17 ( + 1 ) ( P l ) 2 2AE } {{ }} {{ } } N {{ δ }

x 2 1 + 2 + u 2 3 = 1 2 N P l 2AE } {{ } δ } {{ } + 1 2 N ( P l ) 2AE } {{ } δ } {{ } u 2 = 9Fg 1.2 1 2 Fg 1.2 (Young EA) 1. (Fg 1.21 ) Fg 1.21 2. (Fg 1.22 ) 18

Fg 1.22 3. x 1 (Fg 1.23 ) Fg 1.23 x 2 (Fg 1.24 ) Fg 1.24 4. 2. 3. F u = NI δ I I 19

x 1 u 1 + + + = 1 P l AE + P l AE + + ( 2) x 2 u 1 = (1 + 2 2) P l AE = u 2 = 1 P l AE ( 2P l AE ) + 1Fg 1.25 A x 2 Fg 1.25 Young EA 1. (Fg 1.26 ) Fg 1.26 2. (Fg 1.27 ) Fg 1.27 2

3. (Fg 1.28 ) Fg 1.28 4. 2. 3. F u = NI δ I I + u 2 + /2 = u 2 = 2 11Fg 1.29 t A x 1 Fg 1.29 AYoung Eα 1. (Fg 1.3 ) Fg 1.3 21

2. δ = Nl AE +δ δ αδt δ = tαl (Fg 1.31 ) Fg 1.31 3. (Fg 1.32 ) Fg 1.32 4. 2. 3. F u = I N I δ I + + u 1 = 1 2 + u 1 = 1 2 tαl ( 1 2 ) tαl 1.7.2 1. 22

2. 1. (u, δ) 3. (F, N ) 4. 3. (F, N ) 2. (u, δ) 3. (F, N ) 12Fg 1.33 Fg 1.33 AYoung E 1. n (Fg 1.34 ) Fg 1.34 2. 1. (Fg 1.35 ) δ 1 = δ 2 = (P n)l 2AE, δ 3 = nl 2AE Fg 1.35 23

3. P 1. P = n = 1 (Fg 1.36 ) Fg 1.36 4. 2. δ (F, N ) 3. (F, N ) F u = I = ( = n = N I δ I 1 ) 2 P 1 + 1 2 (P n)l 2 + 1 nl (n P )l = + nl 2AE 2AE AE 2AE 13Fg 1.37 Fg 1.37 AYoung E 1. n (Fg 1.38 ) Fg 1.38 24

2. 1. (Fg 1.39 ) δ 1 = (P n)l 2AE, (P + n)l δ 2 =, δ 3 = nl 2AE 2AE Fg 1.39 3. (Fg 1.4 ) Fg 1.4 4. 2. 3. F u = I = ( = n = N I δ I 1 2 ) ( ) (P n)l + 1 nl ( + 1 ) ( ) (P + n)l 2AE 2AE 2 2AE n = P n P ( P ) ( n) n = n n = 25

1.7.3 () 1. 2. 3. 1 () 4. 1. 2. 14Fg 1.41 x 2 Fg 1.41 AYoung E 1. (Fg 1.42 ) Fg 1.42 N 1 = N 2 = P 2 ( 1 + 2 ), N 3 = 2P 1 + 2 2. 1. (Fg 1.43 ) 26

Fg 1.43 δ 1 = δ 2 = P l 2 ( 1 + 2 ) AE, δ 3 = P l ( 1 + 2 ) AE 3.& 4. 1. P = 1 N1 = N2 1 2 = ( ), N3 = ( ) 2 1 + 2 1 + 2 (Fg 1.44 ) Fg 1.44 F u = NI δ I I = + + + u 2 P l P l 2 = ( ) 2 ( ) 2 2 + ( ) 2 1 + 2 AE 1 + 2 AE 1 + 2 u 2 = P l ( 1 + 2 ) AE () 27

1 P = 1n = 1 (Fg 1.45 ) Fg 1.45 F u = NI δ I I = u 2 = 1 P l ( ) 2 2 1 + 2 AE 2 u 2 = P l ( 1 + 2 ) AE n = 1 N = 1 N = () () 28

1 (Fg 1.46 ) Fg 1.46 () 1 Fg 1.47 ( 1 )1 Fg 1.48 2 () (3 ) (2 ) 1 Fg 1.47 1 () Fg 1.48 2 () 15Fg 1.49 A x 2 29

Fg 1.49 AYoung E 1. n (Fg 1.5 ) Fg 1.5 (Fg 1.51 ) Fg 1.51 δ 1 = δ 2 = nl AE, δ 3 = δ 4 = (n P ) l AE, δ 5 = 2 (n P ) l, δ 6 = 2nl AE AE (Fg 1.52 ) Fg 1.52 3

n F u = NI δ I I = = 1 nl AE 2 + 1 (n P ) l 2 } AE {{} 1,2 3,4 + ( 2 ) { 2 (n P ) l } + ( 2 ) { 2nl } AE AE 5 6 n = P 2 2. 1. n δ 1 = δ 2 = P l 2AE, δ 3 = δ 4 = P l 2AE, δ 5 = P l AE, δ 6 = P l AE 3. A x 2 1 () 1 ( ) (Fg 1.53 ) Fg 1.53 4. 1. 3. F u = NI δ I I u 2 = nl AE = P l 2AE 31

1.8 δ = f(n) () Castglano() 1.8.1 (Complementary Potental Energy) (F, N) Π Π (F, N) = W (N I ) u F (1.11) I: : u W (N) W (N) = δ(n)dn (1.12) δ(n) N N W (N) = N(δ)dδ (1.13) δ = Nl AE + δ (1.14) δ () (1.12) W (N) = N 2 l 2AE + δ N (1.15) Fg 1.54 W (N) = W (N) 32

Fg 1.54 W W Π (u, δ) Π(N) = I: W (δ) : F u (1.16) : () n ( = 1, 2, ) (F (n ), N(n )) Π (F (n ), N(n )) Π n = = 1, 2, (1.17) () u = Π = I W (N I ) u F = I W (N I ) W = W Π = I W (δ I ) Π 33

[] (1.11) () n Π = W (N I ) u F n n = I I W N I N I n : : u F n W = δdn W = δ I N I Π n = I δ I N I n : u F n F / n = Π = N I δ I n I n u F n ( NI n, F ) = ( ) () n (δ I, u ) Π n = [] ( ) (F (n), N(n)) F = I n I N I n ) F n = I n I N I n ( F n, N I n F n F n = ( F n, N I n 34 ) =

Π ()(δ = f(n)) 16Fg 1.55 Fg 1.55 AYoung E (Fg 1.56 ) Fg 1.56 Π = ( ) 2 l P n 2 + l/ 2 2AE 2 2AE n2 Π n = l AE l/ 2 (n P ) + AE n = P n = 1 + 1/ 2 12 35

17Fg 1.57 Fg 1.57 AYoung E Fg 1.58 Fg 1.58 Fg 1.59 Fg 1.59 Π = l ( n ) 2 ( ) 2 + n2 l 2 + 1 n 2 l n = 2AE 2 2AE 2 2 n 2AE ( ) Π 2 + 1 nl n = = 2AE 36

n = 2 AE ( ) 2 + 1 l 18Fg 1.6 t Fg 1.6 AYoung E Fg 1.61 Fg 1.61 Π = l ( n ) 2 2 + l/ 2 2AE 2AE n2 + l ( n ) 2 2 + δ ( n ) 2 2AE Π l = n 2AE 2n + l n 1 δ = 2AE 2 n = AE ( ) δ = AE α t 2 + 1 l 2 + 1 37

1.8.2 Castglano. Castglano Castglano : (N, F ) Π (N + QN, F + QF ) (1.18) Q Q= (N, F ) 1 () P P P Π (N, F ) P (1.19) [] Π (N + QN, F + QF ) Q = Q = I I W (N I + QN I ) Q W (N I + QN I ) N I N I : : u (F + QF ) u F Q = W (N I )/ N I W (1.15) δ I Π (N + QN, F + QF ) = δ Q I NI u F Q= I : (δ I, u ) (N, F ) δ I NI = I = : u F + : u F Π (N + QN, F + QF ) Q = Q= : u F 38

F P (N, F ) (N/P, F /P ) N = P N F = P F Π (N + QN, F + QF ) = Π ((P + Q) N, (P + Q) F ) Π Q = Π = Q= P Q= P Π (P N, P F ) = Π (N, F ) P [] 19Fg 1.62 (AYoung E) Fg 1.62 Fg 1.63 Fg 1.63 (x 2 ) u 2 Castglano Π = ( ) 2 l P 2 = P 2 l 2AE 2 2AE 39

u 2 = Π P = P l AE Castglano ( ) 2 Π (N + QN, F + QF l P + Q ) = 2 2AE 2 u 2 = Π = P l P Q= AE x 1 u 1 Fg 1.64 Fg 1.64 Π (N + QN, F + QF ) = l ( P 2AE 2 + Q ) 2 ( P + 2 2 Q ) 2 2 u 1 = Π Q = Q= l 2AE {(P + Q) + (Q P )} Q= = 2Fg 1.65 A (AYoung E) Fg 1.65 4

Fg 1.66 Fg 1.67 Fg 1.66 Fg 1.67 x 2 u 2 Fg 1.68 Fg 1.68 Π (N + QN, F + QF ) = = u 2 = Π Q l 2AE ( + Q ) 2 2 2 l 2AE Q2 2 Q = Q= ( Ql AE ) = 2 Q= 2 41 : u F +QF

u 2 Fg 1.69 Fg 1.69 = = Π (N + QN, F + QF ) ( l + Q ) 2 + 2AE 2 l 2AE Q2 + 2 Q u 2 = Π Q = Q= ( Q 2 ) 2 : ( Ql AE + ) = 2 Q= 2 u F +QF 21Fg 1.7 A x 2 u 2 (AYoung E, α) Fg 1.7 42

δ W = N 2 l 2AE + δ N δ = α tl Fg 1.71 Fg 1.71 Π (N + QN, F + QF ) = I = W I (N + QN ) ( l Q 2AE 2 : ) 2 + δ ( + Q 2 ) u QF + l ( + Q ) 2 2 2AE u 2 = Π Q = δ = α tl Q= 2 2 Castglano 1 1 ( 1.7.3 ) 2 P Π (N, F ) (N, F ) P n P n P n(p ) Π n n(p ) P 43

( Π ) ( Π P P ) Π P = Π (P, n(p )) + Π (P, n(p )) dn P n dp Π P P n = n(p ) 22Fg 1.72 (AYoung E) Fg 1.72 AYoung E (n = P/(1 + 1/ 2))(Fg 1.73 ) Fg 1.73 (Fg 1.74 ) Π Fg 1.74 44

x 1 u 1 Fg 1.75 Fg 1.75 u 1 = Π Q Π = = Q= = l 2AE l AE x 2 u 2 ( P n 2 + Q 2 ) 2 + l 2AE ( P n + Q ) 1 2 + 2 2 Q= l AE ( P n 2 Q 2 ) 2 + l/ 2 2AE n2 ( P n Q ) ( 2 2 Q= 1 ) 2 [2] n n P Π = ( ) 2 l P n 2 + l/ 2 2AE 2 2AE n2 l u 2 = Π P = l P (P n) = AE (1 + 2)AE 1.9 1.9.1 (Recprocty theorem) : δ = Nl/AE (u (1), F (1), δ (1), N (1) ) (u (2), F (2), δ (2), N (2) ) F (1) u (2) = 45 F (2) u (1) (1.2)

[] F (1) u (2) = = = = I I I I = N (1) I δ (2) I δ E (1) I I A I δ (1) I δ (2) 2 l I δ E (2) I I A I l I δ (1) I N (2) I u (1) F (2) [] 1.9.2 Müller Breslau Müller Breslau f ( ) x 1 AB AB A B A B l x 1 f(x 1 ) f(x 1 ) = f(a)(1 x 1 l ) + f(b)x 1 l f f Müller Breslau () : 1 [] Fg 1.76 46

Fg 1.76 (1) (2) 1 (1) (2) (1) (2) F (1) u (2) = F (2) u (1) (1) F (1) = = R 1 + 1 u (1) 2 R = u (1) 2 + [] 23Fg 1.77 A Fg 1.77 47

Fg 1.78 Fg 1.78 24Fg 1.79 A Fg 1.79 Müller Breslau () : 1 [] (1) δ (1) δ (1) (1) ln = + E (1) δ(1) A (1) N (1) = A(1) E (1)(δ(1) δ (1) ) l (1) F (1) u (2) = N (1) I δ (2) I I = A I E ( I (1) δ I δ (1) ) (2) I δ I I l I = ( (1) δ I δ (1) ) (2) I N I I = I δ (1) I N (2) I I δ (1) I N (2) I = u (1) F (2) I δ (1) I N (2) I (1.21) 48

Fg 1.8 (1) (2) Fg 1.8 (1) (2) 1 K (1.21) = u (1) 2 N (2) K (1) u (1) 2 = N (2) K Fg 1.8 u (1) 2 K Müller-Breslau [] Müller Breslau Fg 1.81 Fg 1.81 49

1. A = 1 2 2. ( ) 3. 3 Müller Breslau 1. Q 1 2. A 1.1 1.1.1 1. (F, N) 2. (δ,u) 3. (δ,u) () 1. (δ,u) 2. N 3. F u j δ I = n I u N I = E IA I l I δ I F = I n I N I 5

N I = E IA I l I n ji u j j F = j ( I n I E I A I l I n ji ) u j (1.22) n I n ji F F (1.22) () () F 1 u 1 F 2. = K u 2. F n u n F u F k F u = K ku K uu K kk K uk u u u k (1.23) k u F u (1.23) F u u ( K ku ) ( u u ) = ( F k ) ( K kk ) ( u k ) m K ku m m u u m Ax = b u 1.82 P 1 2 3 1, 2 EAl x1 2 3 x2 1 2 1 P Fg 1.82 51

n I ) n 1 1 = 1 ( 1 2 1 n 1 2 = 1 ( ) 1 2 1 n 2 1 = 1 ( ) 1 2 1 n 3 2 = 1 ( ) 1 2 1 F = j( E I n I A I I l I n ji ) u j F 11 1/2 1/2 1/2 1/2 F 21 1/2 1/2 1/2 1/2 F 12 = AE 1/2 1/2 1/2 1/2 + F 22 l 1/2 1/2 1/2 1/2 F 13 F 23 1/2 1/2 1/2 1/2 u 11 1/2 1/2 1/2 1/2 u 21 u 12 u 22 1/2 1/2 1/2 1/2 u 13 1/2 1/2 1/2 1/2 u 23 [ ] 1 1 2 2 F u F 11, F 21, u 12, u 22, u 13, u 23 F 12, F 22, F 13, F 23, u 11, u 21 ( ) = AE P l ( ) ( ) 1 u11 + AE 1 u 21 l u 12, u 22, u 13, u 23 ( ) = AE ( ) ( 1 u11 P l 1 ( 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 u 11 = u 21 = P l AE u 21 ) u 12 ) u 22 u 13 u 23 52

1.1.2 1. (δ,u) 2. ( ) 3. (δ,u ) 4. 2. 3. 1.83 P 1 2 3 1, 2 EAl x1 2 3 x2 1 2 n2 1 P n1 Fg 1.83 n 1 n 2 n 1 = 1 2 ( 1 1 ) n 2 = 1 ( ) 1 2 1 53

1. (δ,u) 2. δ 1 = n 1 u N 1 = AE n 1 u l δ 2 = n 2 u N 2 = AE n 2 u l 3. δ 1 = n 1 u δ 2 = n 2 u F u = Nδ u ( ) F = P p u P = u AE l [n 1 (n 1 u) + n 2 (n 2 u)] u P = AE l = AE l = AE l [n 1 (n 1 u) + n 2 (n 2 u)] [ ( ) 1 1 ( u 1 + u 2 ) + 1 ( ) 1 2 1 2 2 2 1 ( ) ( ) 1 u1 1 u 2 ( u 1 + u ] 2 ) 2 2 u = l AE P 1.1.3 K.A. (u, δ) Π(u, δ) Π(u, δ) = I W (δ I ) :F gven F 2 () W W (δ) = δ 54 N(δ)dδ F u

(δ,u) u Π u j = j F () u F Π u Π u = I = I W δ I F δ I u N(δ I )n I F Π = u F = N(δ I )n I I () 1. (δ,u) 2. Π 3. u Π u = 55

2 2.1 2.1.1 x 2 after deformaton x u x + u before deformaton x 1 Fg 2.1 9 ɛ j (, j = 1, 2, 3) ɛ 11 ɛ 22 ɛ 33 ɛ 11 ɛ 11 = x 1 x 1 x 1 Fg 2.2 = dx 1 = dx 1 + u 1 (x 1 + dx 1 ) u 1 (x 1 ) 56

dx 1 ɛ 11 = {dx 1 + u 1 (x 1 + dx 1 ) u 1 (x 1 )} dx 1 dx 1 u 1 x 1 x 2 x 3 ɛ 22 ɛ 33 ɛ 22 = u 2 x 2, ɛ 33 = u 3 x 3 Fg 2.2 ɛ 12 = 1 2 (θ 1 + θ 2 ) θ 1 θ 2 Fg 2.3 Fg 2.4 ɛ 12 = 1 2 ( u2 + u ) 1 x 1 x 2 Fg 2.3 57

Fg 2.4 ɛ j = 1 2 ( u + u ) j x j x 2.1.2 (Fg 2.5 ) F = x F = Fg 2.5 ( da ) n df n da ( n) 58

Fg 2.6 df t(x, n) = lm da da t (tracton) (Fg 2.7 ) t(x, n) = t(x, n) Fg 2.7 Cauchy Fg 2.8 2.9 = t(n)ds + t( 2 )ds sn θ + t( 1 )ds cos θ 59

Fg 2.8 Fg 2.9 t(n) = t( 1 )n 1 + t( 2 )n 2 t( 1 ) = (τ 11, τ 21 ) T, t( 2 ) = (τ 12, τ 22 ) T ( t1 t 2 ) = ( τ11 τ 12 τ 21 τ 22 ) ( n1 n 2 ) t = j τ j n j (2.1) Cauchy (2.1) 3 τ 11 τ 22 τ 33 (Fg 2.1 ) Fg 2.1 τ 12 = τ 21 6

τ 13 = τ 31, τ 23 = τ 32 Fg 2.1 2.1.3 ( ) Fg 2.11 Fg 2.11 (x 1 ) ɛ 11 τ 11 Fg 2.11 ɛ 11 ɛ 11 τ 11 () Hooke τ 11 (x 1 ) (Fg 2.12 )E Young ν Posson ( ν <.5) ɛ 11 = 1 E τ 11 ɛ 22 = ɛ 33 = νɛ 11 61

τ 22 τ 33 ɛ 11 = 1 E τ 11 ν E τ 22 ν E τ 33 = 1 + ν E τ 11 ν E (τ 11 + τ 22 + τ 33 ) ɛ 22 = ν E τ 11 + 1 E τ 22 ν E τ 33 = 1 + ν E τ 22 ν E (τ 11 + τ 22 + τ 33 ) ɛ 33 = ν E τ 11 ν E τ 22 + 1 E τ 33 = 1 + ν E τ 33 ν E (τ 11 + τ 22 + τ 33 ) Fg 2.12 x 1 τ 12 (= τ 21 )τ 23 (= τ 32 )τ 31 (= τ 13 ) ɛ 12 = 1 2G τ 12, ɛ 23 = 1 2G τ 23, ɛ 31 = 1 2G τ 31 E G = 2(1 + ν) ɛ j = 1 2G τ j ν E δ j 3 τ kk (2.2) k=1 Hooke δ j { 1 ( = j) δ j = ( j) 62

2.2 (Bernoull-Euler ) 2.2.1 ()x 1 (= x) x 2 u 1 (x, x 2 ) = a(x)x 2 ()x 2 u 2 (= y) x 2 u 2 (x) = y(x) ɛ 12 ɛ 12 = 1 2 ɛ 12 = ( u1 + u ) 2 = 1 ( a(x) + dy(x) ) = x 2 x 1 2 dx a(x) = dy dx u 1 (x) = dy(x) dx x 2 (2.3) u 2 (x) = y(x) (2.4) 63

Fg 2.13 2.2.2 τ 33 = τ 22 = (2.2) ɛ 11 = 1 E τ 11 ν E (τ 22 + τ 33 ) = 1 E τ 11 (2.5) Fg 2.14 2.2.3 () τ 11 M (2.3) (2.5) u 1 M = x 2 τ 11 da = E x 2 ɛ 11 da = E x 2 da A A A x 1 = E x 2 2y (x)da = Ey x 2 2dA A 64 A

(Fg 2.15 )I φ I = A x 2 2 da M φ φ = d2 y(x) dx 2 (2.6) M = EIy = EIφ (2.7) Fg 2.15 2.2.4 Q x 1 x 2 x 1 x 2 Fg 2.16 x 2 Q p(x) dx Q(x + dx) + p(x)dx Q(x) = Q (x) = p(x) (2.8) N dx N(x + dx) + N(x) = N(x) = (2.9) 65

Fg 2.16 2.2.5 Fg 2.17 A M(x + dx) + p(x)dx dx 2 M(x) Q(x)dx = dx M (x) = Q(x) (2.1) Fg 2.17 66

2.2.6 (2.7)(2.8) (2.1) (EIy ) = p (2.11) 2.2.7 (2.11) 4 4 4 4 1 N (Fg 2.18 ) 4 N ( ) Fg 2.18 y = y = M = y = Q = y = Fg 2.19 67

y = M = y = Fg 2.2 y = Q = y = Fg 2.21 y = y + M = y = M + = y + = Q = Q + y = y + Fg 2.22 68

y = y + = y = y + M = M + y = y + Fg 2.23 2.3 () 2.3.1 ((r) (l)) m l,r (M l,r!) F l,r (Fg 2.24 )M N F m l = M l, m r = M r, F l = ( N Q l ), F r = ( N Q r M Q (2.8)(2.9) (2.1) Q = p, N =, M = Q (M, Q, N, F l,r, m l,r, p) ) Fg 2.24 69

2.3.2 u 1 ()(Fg 2.25 )θ θ = y (2.12) (m ) θ l = y l, θ r = y r, u l = ( u1 y l ), u r = ( u1 θ φ (2.12) (2.6) θ = y, φ = y (y, θ, φ, u l,r, θ l,r ) B.C. y r ) Fg 2.25 2.3.3 () : (M, Q, N, F l,r, m l,r, p ) (y, θ, φ, u l,r, θ l,r ) F u + m θ + p ydx = M φdx (2.13) [] L L = M φdx = M y dx L = [M y ] L + (M ) y dx = [M y ] L + [(M ) y] L L (M ) ydx 7

L = M (L)y (L) + M ()y () + Q (L)y(L) Q ()y() (M ) ydx = M (L)y (L) + M ()y () + (Q (L)y(L) + Nu 1 ) (Nu 1 + Q ()y()) = ( ) + L p ydx m r θ r + m l θ l + F r u r + F l u l + (*) L p ydx = M (L) = m r, y (L) = θ r, M () = m l, y () = θ l, ( ) ( ) ( ) N F r =, u r u1 N =, F l =, u l = Q (L) y(l) Q () ( u1 y() ) F r u r = Q (L)y(L) + N u 1, F l u l = Q ()y() N u 1 3 [] 1 2 N = N (F u) F u = (F ) 2 (u) 2 3 (F u)(m θ) (F, u)(m, θ) Fg 2.26 Fg 2.26 71

2.4 () () I (y, θ, φ, u I, θ I ) f I I f Fg 2.27 Fg 2.28 I u I θ I u = u I, θ = θ I (2.14) u θ Fg 2.27 2 Fg 2.28 I (MI, Q I, N I, F I, m I, p I ) F m Fg 2.29 1 F 1 + ( F 1 1 ) + ( F 1 2 ) + ( F 1 3 ) = m 1 + ( m 1 1 ) + ( m 1 2 ) + ( m 1 3 ) = 72

Fg 2.29 1 F = F I, m = I: I: m I (2.15) () : (MI, Q I, N I, F I, m I, p I ) (y I, θ I, φ I, u I, θ I ) F u + m θ + p I y Idx = MI φ Idx (2.16) I I [] MI φ I dx = F I u + :I I: :I = :I : I: MI φ Idx I: = I: :I = : = : (F I u + m I θ ) + F I I: (F u + m θ ) + u + I: p I y Idx I: m I θ + p Iy I dx p I y Idx m I I: (2.15) 73 θ + I: p I y Idx

[] 2.5 () () I θ I θ m m = (2.17) (θ )(2.17) 2.6 ()( ) 1. M 2. M φ 2 φ B.C. (φ, u, θ, y) K.A. (A) 3. (F, m, p, M ) (B) 4. (A) (B) 1. M 2. M φ 2 φ B.C. (φ, u, θ, y) K.A. (A) 74

3. () (F, m, p, M ) (B) 4. (A) (B) 75

P y(l), θ(l) (EI : B.C. : y() =, θ() = ) 1. M M = P x 2. φ. M = EIφ φ = P EI x y(l) 3. M M = x 76

4. F2 y + m θ + p ydx = F u M φ dx F2 () y() + F2 (l) 1 y(l) + m () θ() = l + m (l) ( x ) ( P ) EI x M } {{ } φ θ(l) + p ydx = M φ dx dx y(l) = P 3EI l3 () y() y x= y x = θ(l) 3. M M = 1 4. F 2 y + m θ + p ydx = F u M φ dx F 2 () y() + F2 }{{ (l) } y(l) + m () θ() + m (l) θ(l) + 1 l = ( 1) ( P ) EI x M } {{ } φ p ydx = M φ dx dx θ(l) = P 2EI l2 77

p y(l), θ(l) (EI : B.C. : y() =, θ() = ) 1. M M = p 2 (x ) 2 2. φ. y(l) M = EIφ φ = p 2EI (x ) 2 3. M M = x 4. F2 y + m θ + p ydx = F u F2 () y() + F2 (l) 1 = y(l) + m () θ() l ( x ) M ( + m (l) p ) 2EI (x ) 2 φ θ(l) + M φ dx p ydx = dx y(l) = M φ dx p 8EI l4 78

θ(l) 3. M M = 1 4. F2 y + m θ + p ydx = F u M φ dx F 2 () y() + F2 }{{ (l) } = y(l) + m () θ() l ( 1) M ( + m (l) 1 p ) 2EI (x ) 2 φ θ(l) + p ydx = dx θ(l) = M φ dx p 6EI l3 79

P y(l/2), θ(l) (EI : B.C. : y() = y(l) = ) P /2 /2 1. M M = P x 2 2. φ. y(l/2) M = EIφ φ = P x 2EI P (l x) ( x l/2), (l/2 x l) 2 P (l x) ( x l/2), (l/2 x l), 2EI 3. M F* m* F*=1 m*= F* m* M* /4 M = x 2 (l x) ( x l/2), (l/2 x l) 2 4. F2 y + m θ + p ydx = F u F 2 () y() + F2 } (l/2) {{} 1 + m (l/2) = 2 y(l/2) + F2 (l) y(l) θ(l/2) + m (l) l/2 θ(l) + p + m () M φ dx θ() ydx = M φ dx ( ) ( ) x P x dx, y(l/2) = }{{ 2 }} 2EI {{} M φ P 48EI l3 8

θ(l) 3. M m*= F*=1/ m*= F*= m*=1 F*=-1/ M = x l 4. F 2 y + m θ + p ydx = F u M φ dx F2 () y() + F2 (l/2) y(l/2) + F2 (l) y(l) + m () θ() + m (l/2) θ(l/2) + m (l) θ(l) + p ydx = M φ dx 1 l ( 2 = x ) ( ) P x l ( dx + x ) ( ) P (l x) P l2 dx θ(l) = }{{ l }} 2EI l {{} l 2 2EI 16EI M φ M φ y(l) (EI : B.C. : y() =, θ() = ) 1. M M = 2. φ. φ = 81

3. M 1 m* F*=-1 x m*= F*=1 M = x 4. F2 y + m θ + p ydx = F u M φ dx F2 }{{ () y() + F2 } (l) y(l) + m () θ() + m (l) 1 1 l = θ(l) + φ p ydx = M φ dx M dx = y(l) = 82

T (EI : αb.c.:y() =, θ() = ) T T T + θ x h x θ θ + θ α Τ x φ = y = dθ dx h( θ) = α T x φ = dθ dx = α T h () 1 φ φ = α T h 83

2 3 M = x 4 y(l) = l ( ) α T ( x ) h M } {{ } φ dx α T l2 = 2h 84

1. M () 2. M φ (2 φ B.C. (φ, u, θ, y) ) 3. 4. 2 3 Rθ(l) (EI : B.C.:y() =, θ() =, y(l) = ) 1. M R M = Rx m 2. φ R. M = EIφ φ = 1 EI (Rx m) 3. = M x F * m * F * =-1 m * = 85

M = x 4. F2 y + m θ + p ydx = F u M φ dx F2 () y() +F2 (l) y(l) +m () θ() + m (l) θ(l) + p ydx = l ( ) 1 = (x ) EI (Rx m) dx M θ(l) = 1 EI φ ( R 3 l3 m 2 l2 ) R = 3m 2l 3. M M φ dx M = 1 4. F2 y + m θ + p ydx = F u M φ dx F 2 () y() + F2 }{{ (l) } = y(l) + m () θ() l ( 1) M ( 1 + m (l) 1 ) EI (Rx m) φ θ(l) + p ydx = dx θ(l) = ml 4EI M φ dx () ( ) (e.g. ) 86

R θ(l) (EI : B.C.:y() =, y(l) =, θ() = ) x R 1. M R M = Rx 2. φ R M = EIφ φ = Rx EI 3. = M x F*=1 M = x 4. F2 y + m θ + p ydx = F u M φ dx = l ( x ) ( ) Rx EI dx = Rl3 3EI R = 3EI l 3 (2.17) 87

θ(l) 3. M (R ). x F* m* m*=1 R* = - F* M = 1 + R x = 1 4. F 2 y + m θ + p ydx = F u θ(l) = l ( 1) ( ) Rx EI dx = Rl2 2EI = 3 2l M φ dx (2.18) R = R F ( R ) ( ) l Rx θ(l) R = ( 1 + R x ( ) l Rx ) dx ( ) l Rx = ( 1) dx R ( x ) dx EI EI }. {{. } θ(l)(. (2.18)) R EI }. {{. } (. (2.17)) 2.7 S.A. (ν = (M, Q, N, F, m, p)) Π (ν) = WI (M I)dx F u m θ I: : : u, θ W I 88

W (M) = φ(m)dm φ φ = M EI (+φ ) (φ : φ = ) W (M) W (M) = M 2 2EI (+M φ ) () u =, θ = Π = WI I φ = W W = M(φ)dx (W : ) Π S.A. ν ν X ν(x) X Π (ν(x)) X = R (EI : B.C.:y() =, y(l) =, θ() = ) 89

1 M = Rx m 2 Π = 1 2EI l (Rx m) 2 dx : (u ) F (θ ) : m 3 = Π R = 1 EI l (Rx m) xdx = 1 EI ( R 3 l3 m 2 l2 ) R = 3m 2l Π (ν(x)) X = I: W I (M I ) M I M I X dx : F X u : m X θ W (M) = Π (ν(x)) = φ I (M I ) M I X X dx I: φ(m)dm W (M) M : = φ(m) F X u F, m, p X : m X θ F = ( : ) X (2.19) m j = (j : ) X (2.2) p I = (I : ) X (2.21) Π (ν(x)) = φ I (M I ) M I X X dx F X u m X θ I: : : I: pi X ydx (2.22) (φ, u, θ, y) ( M, F, m, p ) X X X X (2.22) 2 Π (ν(x)) X 9 =

( M X, F X, m X ) (2.19)(2.2) ( M X, F, m X X M X = p X = ) () Π ()φ M ( ) Π R (EI : B.C.:y() =, y(l) =, θ() = ) 1 M = Rx p 2 x2 2 Π = 1 l ( Rx p ) 2 2EI 2 x2 dx 3 = Π R = 1 l EI (Rx p ) 2 x2 xdx = 1 ( R EI 3 l3 p ) 8 l4 R = 3 8 pl R (EI : B.C.:y() =, y(l) =, θ() = ) 91

1 M = Rx 2 Π = 1 2EI l (Rx) 2 dx ( R) (u F ) 3 = Π R = 1 ( ) R EI 3 l3 + R = 3 EI l 3 Castglano :,., ν = (M, Q, N, p, m, F )., () () ν = (M, Q, N, p, m, F )., () Π (ν + Xν ) X () (ν + Xν ) () X () X () X= Π (ν + Xν ) X = W I (M I + XMI ) dx I X : : (m + Xm ) (θ ) X (u ) (F + XF ) X = I W I (M I + XMI ) MI M dx I (θ ) m : : (u ) F X = Π (ν) = W I (M I ) MI X I M dx (u ) F (2.23) }{{ I } : φ I (M I ) (θ ) m (2.24) : 92

ν ν φ I MI dx = u F + I : : θ m + I y I p I dx ν p I = φ I MI dx = u F + θ m I : (2.24) Π (ν + Xν ) X = X= : u F + : : θ m () ν () ν ( ) (() ) () () () P (m ) P ( m ) () Π P, Π m X P M(x) = M (1) (x)x + M (2) (x)p M = M (2) (x) P M (2) (x) P = 1, P =, X = ν 1 P (ν ) ν = M F m p = M (1) F (1) m (1) p (1) X + M F m p P = ν + ν P 93

P j ν j Π X (ν + ν P + Xν j) = Π X= X ((P j + X)ν j + ν + ν P ) = j X= Π ((P j + X)ν j + ν + ν P ) = Π (P j ν j + ν + ν P ) = Π (ν) P j j P j X= j P j ν j, P j ν, P Π X (ν + Xν ) = Π X= P (ν) Π (ν) m A (EI :B.C.:y() =, θ(l) = ) 1 M = px2 2 2 Fg 2.3 3 M = Xx 4 Π Π = 1 2EI l ( p 2 x2 + ( Xx)) 2 dx 94

5 Castglano Π X = pl4 X= 8EI 2 Fg 2.31 3 M = m 4 Π Π = 1 l { p 2 2EI 2 x2 + ( m)} dx 5 Castglano Π m = pl3 m= 6EI 95

m θ() (EI : B.C. : y() =, y(l) =, θ(l) = ) m x 1 2 Π = M = m + Rx l (m + Rx) 2 dx 2EI 3 R = Π l R = (m + Rx) 2 dx = R 2EI 4 Castglano θ() l (m + Rx)x dx = 1 EI EI ( ) ml 2 2 + Rl3 3 R = 3m 2l R = 3m 2l Π m θ(l) = Π l m = m (m 3m 2l x)2 2EI dx = m EI l ( 1 3x 2l ) 2 dx = ml 4EI (2.25) R m Π m θ(l) = Π l m = m (m + Rx) 2 2EI dx = 1 l (m + Rx)dx = ml EI 4EI (2.26) () (2.25) dπ dm dπ dm = Π m + Π dr R dm = Π m (2.26) Π m (... Π R = ) Castglano Π, Π m P P, m 96

2.8 2.8.1 2 (1)(F (1), m (1), p (1), M (1), Q (1), N (1), u (1), θ (1), y (1), φ (1) ) (2)(F (2), m (2), p (2), M (2), Q (2), N (2), u (2), θ (2), y (2), φ (2) ) } {{ } } {{ } S.A. K.A. (M () = EIφ () [=1,2]) () = F (1) u (2) + F (2) u (1) + m (1) θ (2) + I m (2) θ (1) + I p (1) I y (2) I p (2) I y (1) I (1) (2) F (1) u (2) + m (1) θ (2) + I p (1) I y (2) I dx = I dx dx M (1) I φ (2) I dx (2.27) M (1) I φ (2) I dx = I I (EI I I φ (1) I ) (2) φ I dx = I φ (1) I ( EI I I φ (2) I ) dx = I M (2) I φ (1) I dx (2.28) (1) (2) M (2) I φ (1) I I dx = F (2) u (1) + m (2) θ (1) + I p (2) I y (1) I dx (2.29) (2.27)(2.28)(2.29) = F (1) u (2) + F (2) u (1) + m (1) θ (2) + I m (2) θ (1) + I p (1) I y (2) I p (2) I y (1) I dx dx () () 97

(Müller-Breslau ) A A B B Müller-Breslau 1 () = = = 1 ( F (1) u (2) + F (2) u (1) + R (2) (x) ) } {{ } m (1) m (2) θ (2) θ (1) + I + I + y (1) 1 R (2) (x) = y (1) (x) Müller-Breslau + p (1) I p (2) I y (2) I dx y (1) I dx 1 98

A P B B P A () A B EI EI (A) M = P (x + l) { } l P (x + l) y = ( x)dx = EI (B) 5P l3 6EI P l3 l2 P 3EI 2EI y = P l3 3EI + P l2 2EI l = 5P l3 6EI (A)(B) M = x () F (1) u (2) + = F (2) u (1) + m (1) m (2) θ (2) θ (1) + I + I p (1) I p (2) I y (2) I dx y (1) I dx 1 y (2) A = 1 y (1) y (2) A (x) = y (1) (x) Müller-Breslau 99

2 (1) (M,Q,N) (2) = F (1) u (2) + m (1) θ (2) + I F (2) u (1) + m (2) θ (1) + I p (1) I y (2) I dx p (2) I y (1) I dx N (2) A [u (1) 1 ] A Q (2) A [y (1) ] A + M (2) A [θ (1) ] A [u (1) 1 ] A (1) A [y (1) ] A (1) A [θ (1) ] A (1) A 1

() θ (1) A θ (1)+ A u (1) A u (1)+ A u (1) A = u (1)+ A = u(1) 1 y (1) 1 u(1)+ 1 y (1)+ 1 ( F (1) u (2)) + F (1) B A u (2) + ( m (1) θ (2)) + B m(1) A θ (2) A + = ( F (2) u (1)) B + F (2) A u (1) + ( m (2) θ (1)) B + m(2) A θ (1) A + ( F (1) u (2)) + F (1)+ C A u (2) + ( m (1) θ (2)) + C m(1)+ A θ (2) A + = ( F (2) u (1)) C + F (2)+ A F (1,2)+ A = u (1)+ + ( m (2) θ (1)) + C m(2)+ A θ (1)+ A + u (1,2)+ = (1,2) N A Q (1,2), A u(1,2)+ 1 y (1,2)+ p (1) y(2) dx p (2) y(1) dx p (1) y(2) dx p (2) y(1) dx F (1,2) A = F (1,2)+ A, u (1,2) = u(1,2) 1 y (1,2) m (1,2)+ A = m (1,2) A = M (1,2) A F (1) u (2) + m (1) θ (2) + p (1) y (2) dx = F (2) u (1) + m (2) θ (1) + p (2) y (1) dx N (2) ( (1)+ A u 1 u (1) ) 1 Q (2) ( A y (1)+ y (1) ) [u (1) 1 ] A [y (1) ] A +M (2) A ( (1)+ θ A θ (1) ) A [θ (1) ] A (Müller-Breslau ) 11

Müller-Breslau -1 M A x 2 ( < x < l ), 2 1 (l x) 2 ( ) l 2 < x < l () = F (1) N (2) A u (2) + F (2) u (1) + = [u(1) 1 ] A Q (2) A m (1) m (2) θ (2) [y(1) ] A θ (1) + I + I +M (2) A p (1) I p (2) I [θ(1) ] A ( 1) y (2) I dx y (1) I dx = y (1) (x) + M (2) A ( 1) y (1) (x) = M (2) A (x) : Müller-Breslau 1 12

Q A x l ( < x < l ), 2 1 (l x) l ( ) l 2 < x < l () = F (1) N (2) A u (2) + F (2) u (1) + = [u(1) 1 ] A Q (2) A m (1) m (2) θ (2) [y(1) ] A 1 θ (1) + I + I +M (2) A p (1) I p (2) I [θ(1) ] A y (2) I dx y (1) I dx = y (1) (x) Q (2) A y (1) (x) = Q (2) A (x) : () 13

Q B 1 R A, M A, M B, Q B R M A A M Q B B R A 1 M A -1 M B -1 14

R A, R B, R D, M B, M E, Q + B, Q B, Q E A /2 E B C D R A 1 R B 1 R D 1 M B -1 M E -1 + Q B 1 - Q B 1 Q E 1 15

R A, R C, M A, M C, Q B, Q + C, Q C, 16

M A Q B, M + C, M C 17

2.9 2.9.1 S.A. S.A. (M, Q) p Q = p 18

M = Q 2 S.A. K.A. (θ, φ) () y θ = y φ = θ = y K.A. K.A. 2.9.2 y θ p m() y() θ(l) θ() l y(l) m(l) x=x1 F() F(l) Fg 2.32 (, l) 3 (, l) 19

3 N 1 N 2 N 3 N 4 N 1 N 1 () = 1, dn 1() dx N 1 (l) =, dn 1(l) dx = = N 1 (x) = 1 3( x l )2 + 2( x l )3 1 N1.8.6.4.2 L Fg 2.33 N 1 N 2 N 2 () =, dn 2() dx N 2 (l) = 1, dn 2(l) dx = = N 2 (x) = 3( x l )2 2( x l )3 1 N2.8.6.4.2 L Fg 2.34 N 2 M 1 M 1 () =, dm 1() dx M 1 (l) =, dm 1(l) dx 11 = 1 =

M 1 = l[( x l ) 2(x l )2 + ( x l )3 ].16.14 M1.12.1.8.6.4.2 L Fg 2.35 M 1 M 2 M 2 () =, dm 2() dx M 2 (l) =, dm 2(l) dx = = 1 M 2 = l[ ( x l )2 + ( x l )3 ] M2 -.2 -.4 -.6 -.8 -.1 -.12 -.14 -.16 L Fg 2.36 M 2 y y()n 1 (x) + y(l)n 2 (x) + θ()m 1 (x) + θ(l)m 2 (x) (2.3) N 1 N 2 M 1 M 2 3 111

(2.3) (2.3) M (2.3) (**) l Mφ dx = F y + mθ + l py dx M (2.3) ( ) M = EI(y()N 1 + y(l)n 2 + θ()m 1 + θ(l)m 2 ) y = N 1 θ = N 1 φ = N 1 l EI(y()N 1 + y(l)n 2 + θ()m 1 + θ(l)m 2 )N 1 dx = F () + pn 1 dx y = M 1 l EI(y()N 1 + y(l)n 2 + θ()m 1 + θ(l)m 2 )M 1 dx = m() + pm 1 dx y = N 2 l EI(y()N 1 + y(l)n 2 + θ()m 1 + θ(l)m 2 )N 2 dx = F (l) + pn 2 dx y = M 2 l EI(y()N 1 EIN K e = EIM EIN + y(l)n 2 1 N 1 dx + θ()m 1 EIN 1 M 1 dx + θ(l)m 2 )M 2 dx = m(l) + EIN 1 N 2 dx pm 2 dx EIN 1 M 2 dx 1 N 1 dx EIM 1 M 1 dx EIM 1 N 2 dx EIM 1 M 2 N 1 dx EIN 2 M 1 dx EIN 2 N 2 dx EIN 2 M EIM 2 N 1 dx EIM 2 M 1 dx EIM 2 N 2 dx EIM 2 M F () y() pn1 dx m() F (l) = K θ() e y(l) pm1 dx pn2 dx m(l) θ(l) pm2 dx K e 2 dx 2 dx 2 dx (2.31) 112

2.9.3 (2.37 ) () 3 2.37 + 1 (y, y +1 )(θ, θ +1 ) (F, F +1 )(m, m +1 ) (2.31) F p F +1-1 +1 m m -1 +1 +2 +3 +1 F -1 -m -1 F -m F m -1 m -F -1 -F m Fg 2.37 1 F = F 1 m = m 1 2.38 2.38 K 1 e K e 1 K 1 e K e () (F, y )(m, θ ) b x Ax = b x + F + m 113

K e -1 y -1 θ -1 F m = - K e y θ y +1 θ +1 Fg 2.38 2.9.4 ( ) f(x) N(x) N(x+dx) f(x) Fg 2.39 N (x + dx) + f (x)dx N (x) = dn dx = f (x) N f 114

u 1 u 1 (x 1, x 2 ) = a(x 1 )x 2 + U 1 (x 1 ) U 1 (x) ε 11 ε 11 = U 1 = du 1 x 1 dx N ε 11 dx = l N du 1 dx dx = [N U 1 ] l l dn dx U 1dx = N (l)u 1 (l) N ()U 1 () + = F 1 (l)u 1(l) + F 1 ()U 1() + l l f U 1 dx f U 1 dx F 1 M φdx = F2 y + m θ + p ydx N ε 11 dx = F1 u 1 + f u 1 dx M φdx + N ε 11 dx = F u + m θ + p udx p = (f, p ) u = (U 1, y) 115

3 3.1 p /2 /2 1 + 2 p + R 1 2 5pl4, Rl3 384EI 48EI 5pl 4 384EI Rl3 48EI = R = 5pl 8 116

ml2 ml () 2EI EI P l3 P l2, () 3EI 2EI ql4 ql3, () 8EI 6EI () ml ml, () 3EI 6EI 117

P l3 48EI, () P l 2 16EI 5ql4 384EI, () ql 3 24EI 1 2 + R m 12 ml2 2EI, Rl3 3EI ml 2 2EI Rl3 3EI = R = 3m 2l 118

1 δ : () δ 1 : 1 1 1 () δ 2 : 2 1 2 () 1 = δ 1 δ 11 R 1 δ 12 R 2 2 = δ 2 δ 21 R 1 δ 22 R 2 ( δ11 δ 12 δ 21 δ 22 ) ( R1 R 2 ) ( δ1 = δ 2 ) δ 11 δ 12... δ 1n R 1 δ 21 δ 22... δ 2n R 2....... δ n1 δ n2... δ nn R n δ 1 δ 2 =. δ n δ j = δ j (... ) 119

3.2 (EI : ) p M M + + M M pl3 Ml 24EI 3EI pl3 + Ml 24EI 3EI pl3 24EI Ml 3EI = pl3 24EI + Ml 3EI M = pl2 8 12

(EI : ) ql3 24EI Ml 3EI Ml 3EI ql3 24EI Ml 3EI = Ml 3EI M = ql2 16 121

θ deformed beam θr M r snk M r θ θr θ l, θ r r φ φ φ = r l l 122

M M r M ll 3EI + M rl 6EI, M ll 6EI M rl 3EI θ l, θ r θ l = θ l + φ + M ll 3EI + M rl 6EI θ r = θ r + φ M ll 6EI M rl 3EI (3.1) M 1 M 2 M A B C beam 1 beam 2 A B C E1 I1 E2 I2 B = θ 1 r + φ 1 M Al 1 6E 1 I 1 M Bl 1 3E 1 I 1 B = θ 2 l + φ 2 + M Bl 2 3E 2 I 2 + M Cl 2 6E 2 I 2 B ( l 1 l1 M A + + l ) 2 M B + l 2 M C = θr 1 6E 1 I 1 3E 1 I 1 3E 2 I 2 6E 2 I + φ 1 θl 2 φ 2 2 123

(EI : ) p p M + M θ = = pl3 24EI + Ml 3EI + Ml 6EI M = pl2 12 124

(EI : ) M + M r θ = = l + M ll 3EI + M rl 6EI θ = = l M ll 6EI M rl 3EI M l = 6 EI, M l 2 r = 6 EI l 2 125

3.3 ) ) * * 3 () (3.1) ( θl θ r ) = ( ) ( ) l 2 1 Ml + 6EI 1 2 M r ( θl + φ θ r + φ ) θ l, θ r φ φ() before deformaton after deformaton P φ φ ( ) Ml = 2EI ( 2 1 l 1 2 M r ) ( θl θ l φ θ r θ r φ ) M l = 2EI l M r = 2EI l (2θ l + θ r 2θ l θ r 3φ) ( θ l 2θ r + θ l + 2θ r + 3φ) 126

(EI : ) 1 θ q M M = 2EI l ( = 4EIθ l θ + ql2 12 ) ql3 24EI 2ql3 24EI + M M = 2EI (2θ + ) l = 4EIθ l 4EIθ l ql2 12 = 4EIθ l θ = ql3 96EI 127

M (EI : ) 1 + 2 /2 + R 1 2 2 Rl3 6EI 2 Rl3 6EI = R = 3 EI l 3 128

θ r = + + l 6EI ( 2M B) θ l = l + + M Bl 3EI Ml 3EI = Ml 3EI + δ l M B = 3 EI 2l 2 r = M A = 2EI l (2θ l + θ) M B = 2EI l ( θ l 2θ) M B = 2EI l (2θ + θ r 3 l ) = M C = 2EI l ( θ 2θ r + 3 l ) θ = 2l, θ l = 4l, θ r = 5 4l M B = 3 EI 2l 2 129

M 13

4 4.1 Euler 4.1.1 x x x+dx y M(x) N(x) p(x) M(x+dx) Q(x) N(x+dx) Q(x+dx) x N(x + dx) N(x) = N(x) N = P (P ) y p(x)dx + Q(x + dx) Q(x) = Q (x) = p(x) O 131

M(x) y(x) y(x+dx) N(x) p(x) M(x+dx) Q(x) O N(x+dx) Q(x+dx) = M(x + dx) + p(x)dx dx 2 M(x) Q(x)dx P (y(x + dx) y(x)) M(x + dx) M(x) lm dx dx = lm dx ( Q(x) + P M (x) = Q(x) + P y y(x + dx) y(x) dx p 2 dx ) Q = M P y p = Q = M P y p = (EIy ) + P y (... ) 4.1.2,+,+,+,+ () y =, y = () M = y =, Q = EIy P y = ( ) ()y =, y = 132

4.1.3 = y + () ỹ EIỹ + P ỹ = y : EIy + P y = q q = y = ỹ : 4 4 EIy + P y = q EIy + P y EI(y y ) + P (y y ) = ỹ (y y ) ỹ = z, k = y y = ỹ y = y + ỹ P EI EIz + P z = z + k 2 z = z = A cos kx + B sn kx (A, B ) ỹ = A cos kx + B sn kx ỹ = A cos kx + B sn kx + b ỹ = A cos kx + B sn kx + a + bx A, B A, B : ỹ = A cos kx + B sn kx + a + bx (A, B, a, b ) 4 4.1.4 1 P cr (EI : ) 133

: y() = y(l) =, y () = y (l) = p = = y() = = A + a y () = = k 2 A y(l) = = A cos kl + B sn kl + a + bl y (l) = = Ak 2 cos kl Bk 2 sn kl 1 1 A k 2 B = cos kl sn kl 1 l a k 2 cos kl k 2 sn kl b K K (A, B, a, b) = (,,, ) (A, B, a, b) (,,, ) K = 1 1 k 2 K = = cos kl 1 sn kl l k 2 cos kl k 2 sn kl = k 2 k 2 l sn kl 1 1 k 2 sn kl l k 2 sn kl K = k =, sn kl = k = P =, sn kl = kl = nπ (n : ) kl = P = π2 EI l 2, 4π 2 EI l 2, y = B sn kl = B sn nπx l P EI l = nπ P = n2 π 2 EI l 2 9π 2 EI, y = l 2 n = 1 n = 2. P = π2 EI l 2 P = 4π2 EI l 2 : P cr : 2 - P cr (EI : ) 134

- - : y() = y(l) =, y () =, y (l) = q = = y() = = A + a y () = = Bk + b y(l) = = A cos kl + B sn kl + a + bl y (l) = = Ak 2 cos kl Bk 2 sn kl 1 1 A k 1 B = cos kl sn kl 1 l a k 2 cos kl k 2 sn kl b K (A, B, a, b) (,,, ) K = k 1 = K = 1 sn kl 1 l + 1 k 2 sn kl = k 2 sn kl + ( k 3 l cos kl) tan kl = kl k 1 cos kl sn kl l k 2 cos kl k 2 sn kl tan kl kl = kl = 4.5 () (kl = π) y 4. y=tan kl y= kl 2.. 2. 4.. 1. 2. 3. 4. 4.5 5. 6. kl 3 P cr (EI : ) P 135

2 P 2 P cr = π2 EI l 2l l 2 P cr = π2 EI 4l 2 4 P cr (EI : ) P * l 2 / 2 P P cr = π2 EI ( ) 2 = 4π2 EI l l 2 2 136

* l 2 - / 2 P = (4.5 )2 EI (l/2) 2 P cr 2 4π2 EI l 2 5 ( π, π) f(x) (Fourer ) f(x) = a 2 + a n cos nx + b n sn nx n=1 n=1 π π π π π π cos nx cos mxdx = sn nx sn mxdx = 2π (n = m = ) π (n = m ) (n m) { π (n = m ) (n m) cos nx sn mxdx = a n = 1 π b n = 1 π π π π π f(x) cos nxdx f(x) sn nxdx 137

x = x = l x = lx = a = ( =, 1, ) < x < l f(x) f(x) = f n sn nπ x (Fourer ) l n=1 (EI : ) q(x)y(x) q(x) = y(x) = q n sn nπ n=1 l x y n sn nπ n=1 l x y() =, y(l) = y () =, y (l) = EIy + P y = q, ( ) { nπ 2 ( ) } nπ 2 y n EI P sn nπx n=1 l l l = n=1 q n sn nπx l y n = q n ( ) { 2 nπ EI ( ) } 2 nπ l l P y = ( n=1 nπ l ) 2 { q n EI ( nπ l ) 2 P } sn nπ l x 138

n EI ( ) 2 nπ l P yn y P EIπ2 y 1 (q 1 ) y l 2 P cr = EIπ2 l 2 P cr = γei (γ :) σ = P cr l 2 A = γe l 2 A/I I (A : )r = A σ = γe (l/r) 2 l (slenderness rato) () σ r 4.2 4.2.1 P cr () (EI, A : ) P 139

-P -P/ 2 -P/ 2 P/ 2 P/ 2 P/2 P/2 P P P cr = P = π2 EI l 2 q. () R R = = 5ql4 384EI Rl3 48EI Rh EA Rh EA = 5ql4 384EI Rl3 48EI R = R = π2 EI π 2 EI h 2 = 5ql 4 384EI ( l 3 + ) h 48EI EA h 2 5ql 4 384EI ( l 3 q = 384π2 (EI) 2 14 ) + h 48EI EA ( l 3 + ) h 48EI EA 5h 2 l 4

4.3 = P l sn θ kθ θ = θ P = k l P = k θ l sn θ θ sn θ P π k/ π θ P < k θ = l P > k θ = 2 (3 ) l 141

P < k kθ P l sn θ l kθ P sn θ O θ θ = θ = θ = P > k kθ P l sn θ ( l θ = θ c,, θ c ε ) kθ θc P sn θ O θc θ θ = θ c θ = θ c εθ θ c θ = θ c + εθ θ c θ = θ c θ = θ = εθ θ c θ = εθ θ c 142

θ = θ = θ c θ = θ c εθ θ c θ = θ c + εθ θ c θ = θ c P cr = k l P < k l P > k l θ = θ = θ c, θ c θ = ( ) 143