66 σ σ (8.1) σ = 0 0 σd = 0 (8.2) (8.2) (8.1) E ρ d = 0... d = 0 (8.3) d 1 NN K K 8.1 d σd σd M = σd = E 2 d (8.4) ρ 2 d = I M = EI ρ 1 ρ = M EI ρ EI

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65 8. K 8 8 7 8 K 6 7 8 K 6 M Q σ (6.4) M O ρ dθ D N d N 1 P Q B C (1 + ε)d M N N h 2 h 1 ( ) B (+) M 8.1: σ = E ρ (E, 1/ρ ) (8.1)

66 σ σ (8.1) σ = 0 0 σd = 0 (8.2) (8.2) (8.1) E ρ d = 0... d = 0 (8.3) d 1 NN K K 8.1 d σd σd M = σd = E 2 d (8.4) ρ 2 d = I M = EI ρ 1 ρ = M EI ρ EI EI I 2 1 2 1 2 K d 1 1 2 d 2 2 K (8.1) (8.5) σ() = M I (8.5) (8.6)

67 σ I 2 8.1 M > 0 = h 1 σ 1 = h 2 σ 2 σ 1 = Mh 1 I = M Z 1, σ 2 = Mh 2 I = M Z 2 (8.7) Z 1, Z 2 Z 1 Z 2 2 σ ma = M, Z = I/h h (8.8) Z σ ma 2 K 1 2 K 1 2. 1 1 S, S S, S S = d, S = d (8.9) O 0 G 0 1 d X 2 8 2 6 10 5 1 G 1 5 G 2 2 = 6 2 (2.5, 3.5) 1 = 10 2 Y 8.2: 1 ( 0, 0 ) d 0 = = 1 d, 0 = d d d = 1 d

68 1 0 = S, 0 = S (8.10) 1 S = 1 0 1 + 2 0 2 + = d, S = 1 1 0 + 2 2 0 + = d (8.11) 8.2 = 1 + 2 = 32cm 2 1 S = 1 0 1 + 2 0 2 = 5 (10 2) + 1 (6 2) = 112 cm 2 S = 1 1 0 + 2 2 0 = 1 (10 2) + 5 (6 2) = 80 cm 2 0 = S = 112 32 = 2.5 cm, 0 = 80 = 3.5 cm 32 mm cm mm cm=10mm, cm 4 = 10 4 mm B. 2, 2 I = 2 d, I = 2 d (8.12) G( 0, 0 ) X 2 I X I = I X + 2 0 (8.13) 2 (0, 0) X d 1 I = 2 d = ( 0 + 1 ) 2 d = 1d 2 + 2 0 1 d + 0 2 d 1 1d 0 I = 2 d = 2 1d + 2 0 d = I X + 2 0

69 h h/2 h/2 G X Y I X = 1 12 h3 d dy 8.3: 2 K OK 2 8.3 2 I, I X I = 0 2 d = 1 3 h3, I X = 2 h/2 0 Y 2 dy = 1 12 h3 (8.14) 0 2 = (h/2)2 h = h 3 /4 I = I X + 0 2 (8.13) 8.2 2 I 2 2 I 1 I2 I 1 I = I 1 + I 2, I 1 = I 1 X + 2 0, I 2 = I 2 X + 2 0 I 1 X = 1 12 h3 = 2 103 12 = 166.7, 2 0 = 5 2 10 2 = 500 I 1 = I 1 X + 2 0 = 666.7 cm 4 I 2 IX 2 = 1 12 h3 = 6 23 12 = 3, 2 0 = 1 2 6 2 = 12, I 2 = 15 I = I 1 + I 2 = 515 cm 4 K B V, V B V + V B = 20 8, 4 V = 4 V B... V = V B = 80 kn

70 20kN/m 20 C 8m B 300mm 600mm /2 M M = V 2 20 = 102 + 80 = 10( 4) 2 + 160 M = 4 C M =4 = 160 knm Z Z = I h/2 = h3 6 = 300 600 6 = 18 10 6 mm 3 (I = 1 12 h3 ) σ ma σ ma = M =4 Z 160 106 = = 8.89 N/mm2 18 106 B M, M B M M = M B M Q M B M B M B V l V B M + 8.4: Y = (V + V B ) = 0, M = M M B l V B = 0 Q M Y = Q V = 0, M = M M + V = 0... V = M B M, V B = M M B l l... Q = M B M, M = M + l ( MB M l ) M = M B Q = 0, M = M (0 < = < = l) M K

71 300mm 2mm 30mm Z 1 =20mm 3 Z 2 = 300mm 3 2mm 30mm Z 1 = h 2 /6 = 30 2 2 /6 = 20mm 3 Z 2 = 2 30 2 /6 = 300mm 3 15 M = σz M 15 σ Z M K 8.5 I Z I Z h I = 1 12 h3 Z = 1 6 h2 I = π 64 (d4 2 d 4 1) Z 1 = π 32 (d4 2 d 4 1) d 2, d 1 h I = 1 36 h3 Z 1 = 1 24 h2 Z 2 = 1 12 h2 2 I = π 4 a3 Z = π 4 a2 2a I = π 64 d4 Z 1 = π 32 d3 d 8.5: 2 I Z h G X X 2 I X 2 I I h I = 2 2 (h ) d = d = h3 h 12 0

72 (8.13) I X = I 0 2 = 2 d = h3 12 ( ) h 2 h 3 2 = h3 36 h 0 d d a G : a = h : (h ) (h ) (h ) a = d = d h h X 8.6: 2 Z 1 Z 2 (8.8) Z 1 = I X 2h/3 = h2 24 Z 2 = I X h/3 = h2 12 K K P σ σ = P K 8.7 P a d c d a d 1 = d a d 2 = d ad d c 8.7:

73 a ( d 2 ) ad ( d 1 ) = 0, d 1 = d a, d 2 = d ad = (8.15) conjugate shearing stress K 1 1.21 (1.5) M Q Q = dm d (8.16) M Q Q M Q = Q S I (8.17) I 2 S 1 (8.17) K 8.8 d 2 B CD B CD Q, Q + dq M, M + dm C h 2 h 1 Q M 1 σ E F Q + dq M + dm σ + dσ h 1 1 G H 0 B d D ϕ 8.8: 1 EF EBDF EB Q EF

74 EB M σ FD M + dm σ + dσ X = h1 1 {(σ + dσ) σ} d d = 0 (8.6) σ = M I, dσ = dm I h1 1 = 1 dm I d h1 1 d = Q h1 d (8.18) I 1 d = 1 = h 1 1 S (8.17) 1 S (8.10) 8.8 G 0 K H (8.17) K 8.9 8.9: TOP 0 s (8.17) = s cos ϕ, s = cos ϕ = QS I cos ϕ (8.19)

75 8.8 K h Q (8.18) = Q I S = h/2 1 h/2 1 d = 2 (8.14) ϕ = 0 = s d = QS I h/2 1 d = 8 (h2 4 2 1) I = h3 12 = s = 3 Q 2 h 3 (h2 41) 2 (8.20) 2 = 0 h 1 d h/2 h/2 Q M 1 σ E F Q + dq M + dm ma = s d d d = Q = 0 8.10: Q h = Q (8.20) h = s = 3 ( ) 2 1 42 1 h 2 (8.21) 1 = 0 ma = 3 Q 2 h = 1.5 1 = ±h/2 = 0 Q/

76 Q d d = d d = 3 Q h/2 ) (1 2 h 3 42 h/2 h 2 d = 3 Q 2 h 3 2h3 = Q (8.22) 3 Q K r d 1 r θ 1 θ d 1 /2 π 2 ϕ r /2 = s = 0 ma s sin(π/2 ϕ) = cos ϕ = 2r ϕ = s = 0 8.11: 8.11 2 = r sin θ, d = r cos θdθ, d = 2r cos θd = 2r 2 cos 2 θdθ 2 ( ) π/2 π/2 I = 2 d = 2 2r 4 sin 2 θ cos 2 θdθ = 4r 4 sin 2 θ(1 sin 2 θ)dθ = πr4 4 = 2r cos θ 1 (8.18) 0 0 r = Q Q d = I 2r cos θ 1 πr4 4 cos θ = t sin θdθ = dt, π/2 π/2 2r 3 sin θ cos 2 4Q θdθ = θ 1 πr 2 sin θ cos 2 θdθ cos θ 1 θ 1 cos 2 θ sin θdθ = t 2 dt = 1 3 t3 = 1 3 cos3 θ = 4Q 3πr 2 cos2 θ 1 = 4 Q 3 πr 2 ( ) 1 2 1 r 2 (8.23) = Q/πr 2 = 4 ( ) 3 1 2 1 r 2 (8.24) 2 1 = 0 ma = 4 3 (8.25)

77 K (8.19) s = cos φ = 2 ( ) 4 3 1 2 1 r 2 = 43 ( ) 1/2 1 2 1 r 2 (8.26) K (8.26) 2 2 s 2 2 1 B 2 = 1, 1/ = 3/4, 1/B = 1/r, 2 s 2 s 1 = 0 ma = s ma = 4 3 (8.27) Q = 10 4 N 2 I I BCD 2 D 10cm 10cm 15cm 30cm = 8.9 0. 0.0143 2 ma = 26.1 0.0143 2 B C h/2 10cm 40cm 8.12: 2 (8.18) I = 40 503 12 2 = Q I 15 303 12 h/2 d = 3.492 10 5 1... = Q I h/2 h/2 d = d = 15 10d + 25 15 40d = 9125 5 2 10 4 10 3.492 10 5 (9125 52 ) = 26.1 0.0143 2

78... = Q I h/2 h/2 d = d = 25 40d = 12500 20 2 10 4 40 3.492 10 5 (12500 202 ) = 8.9 0.0143 2 ma = 0 ma = 26.1N/cm 2 ************************* 8 9 9 2017.10.28

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