1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45 W = 1.4.1) P cos 6 P 1 cos 45 = 1.4.) P = P 1 cos 45 / cos 6 ) = P 1 1/ )/1/) = P 1 1.4.1) P1 / + P 1 1/ W = P 1 6 + )/ = W P 1 = /{ + 1})W P = P 1 = / + 1)W 1.5 R A + R B 4 = 1.5.1) B R A 1 7 4 4 = R A = N 1.5.1) R B = N
1.6 T 1.6.1)/ 1.6.) F T sin θ = F = T sin θ 1.6.1) T cos θ W = W = T cos θ 1.6.) tan θ = F/W θ = tan 1 F/W a A a/) sin α W a cos α F = tan α = F/W α = tan 1 F/W. 1.7 A 1. + 8.4 R BX.4 = R BX = 6.8 kn. R BX R AX = R AX = R BX = 6.8 kn. R AY 8 kn kn = R AY = 8 kn. 1.8 A + 6 + + 8 4 + 4 R BY 6 = R BY = 11. kn. R AX + + = R AX = 5 kn. R AY 6 4 8 + R BY = R AY = 18 R BY = 6.7 kn.
.1 1) P = kgf = 164 N. P 4 = P 1 + P P = + 18 194 N) = 14 N) = 1.8 kgf) ) 1 P 1 N P 1 + P 41 N 4 P 1 + P P 14 N.. σ = P A, ε = σ E, λ = P l = εl EA 4 1 N) σ 1 = π.) m ) = 18 16 Pa) = 18 MPa) 1 18 MPa) ε 1 = = 1.54 1 6 GPa) λ 1 = 1.54 1 1 mm) = 1.54 mm) 4 1 N) σ = π.) m ) = 141 16 Pa) = 141 MPa) 141 MPa) ε = =.1 1 7 GPa) λ =.1 1 1 mm) =.1 mm) 4 1 N) σ = π.5) m ) = 5.9 16 Pa) = 5.9 MPa) 5.9 MPa) ε = = 5.9 1 1 GPa) λ = 5.9 1 1 mm) = 5.9 mm) ) 1 N mm = 1 N 1 ) m = 1 16 N m = 1 MPa N mm MPa. P N) = M kg) g m/s ) = Mg N)..1).17) λ S = Mg N) l m) E S 1 9 Pa) A S 1 6 m ) = λ A = Mgl 1 m) = Mgl mm)..) E S A S E S A S Mgl E A A A mm)..)
4..)..) A A = E S E A A S..4)..4) E S A S = E A A A..5) ).17).4 ε =.5 mm) 5 mm) = 1 1. ε = νε =. 1. λ = ε b =. 1 1 mm) =. mm)..5.17) 1 λ 1 = P l 1 E 1 A, λ = P l E A λ = λ 1 + λ = l1 + l ) P E 1 E A.6 5 1 6 Pa).5 m).1m) = 5 kn) 7 5/7 kn) S m ) 5 1 N) 7 = 4 16 N/m ) S m ) S m ) = 5 1 N) 7 4 1 6 N/m ) = 68 1 6 m ) 4 d = 68 mm) = 18.5 mm) π
5 in ) 4 in 1 in = 5.4 mm ) in 4 in 5 mm 1 mm.7 σ a = σ B S = 84 MPa) W = π 4 D p 1 F F = π 4 d σ a n n W F = D p d σ a = 5.6 1 84 = 1.4 1 1) ) M M mm M1 1 M1 1.8 mm 1 mm ) 1 1 15 16 )
6.1 l m) W = ρgal N).1.1) σ = W A = ρgl N/m ).1.).1.) l = σ ρg specific strength). =.11).7) σ ) σ = mg A A) x=l = A, ) ρgl A) x=l = C exp = A.11) σ C = A exp ρgl ) σ A = A exp ) ρg x l) σ. P 1 P 1 P 1 λ P 1 + P = P..1) λ = P 1 E 1 A 1 l = P E A l..)
7..1)..) E 1 A 1 + E A ) λ l = P λ = P l E 1 A 1 + E A.7 parallel model).9 series model).4 α 1 > α 1 1 1.) P =.).5 λ = l P + α 1 T l = E 1 A 1 1 + 1 E 1 A 1 E A α 1 α ) T 1 + 1 = E 1 A 1 E A l E A P + α T l ) P = α 1 α ) T E 1 A 1 E A α 1 α ) T E 1 A 1 + E A l P + α 1 T l = E A α 1 α ) T l E 1 A 1 E 1 A 1 + E A = E 1A 1 α 1 + E A α E 1 A 1 + E A C A 1a) ) T l + α 1 T l P AC = 1 P, P BC = P ) a) b) A 1
8 A-1b) ) AC λ AC = P AC l EA BC λ BC = P BC l = EA P l = EA P l EA C δ V ) δ H δ V = λ AC cos θ + λ BC cos θ 1 = + 4 δ H = λ AC sin θ λ BC sin θ 1 = 4 P l EA P l EA.6 A AB, AC A-a) A A P P A AB, AC P A AB, AC P AB, P AC AB A AC A a) A b) B c) C A A b) B B P AB BC, BD BC BD A c) C AC BC P AC, P BC ) CD, CE A A P.7.45) K = 1 + = 7. 5 MPa) 7 = 71 MPa).
9 A.8 n 1 + n = 1 N 1 = 1 1 6, N = 5 1 6, n 1 = 1 5 n =.7. N 1 N N n =.5 1 6.9 A 4 A 4
1 4 4.1 P = τ S A = 4 1 6 π.5 1 6 = 1884 N) 4. 4. I p = πd4 = π 5 1 ) 4 Z p = πd 16 = π 5 1 ) 16 = 6.1 1 7 m 4 ) =.45 1 5 m ) I p = πd4 d 4 1) Z p = πd4 d 4 1) 16d = π 54 4 ) 1 8 = 5.4 1 7 m 4 ) = π 54 4 ) 1 8 16 5 1 =.14 1 5 m ) 4.4 1) L 1 M t. πgd 4 = 1 π 8 1 9 6 1 ) 4 = 8.846 1 rad) ) L M t. πgd 4 = π 8 1 9 1 ) 4 = 1.414 1 1 rad) ) 1 : 16 4) 8.846 1 + 1.414 1 1 = 1.5 1 1 rad) 4.5 τ max = 16M t πd τ max τ u 16Mt d πτ u ) 1 / = ) 1 / 16 8 π 6 1 6 =.679 1 ) 1 / =.878 m)
11 5 5.1 1 MPa 5.6) 1 1 6 = 16M t π.5 [1 ) ] 4 18 5 M t =.917 1 5 N m.5 =.46 rad 5.4).46 = M t 8 1 9 π.5 4 [1 ) ] 4 18 5 M t = 1. 1 5 N m 5. AB τ a = 5 MPa 5.6) 5 1 6 = [ 1 16 9 ) ] 4 d1 π.14 [1.14 ) ] 4 d1 =.477.14 d 1 d 1 =.119 m AB ψ AB ψ AB = 9.5 8 1 9 π.14 4.464 =.15 rad CD τ a = 5 MPa 5 1 6 = 16 9 π d d =.19 m
1 CD ψ CD ψ CD = BC ψ BC ψ BC = ψ 9. 8 1 9 =.8 rad π.194 9.15 8 1 9 =.44 rad π.144 ψ = ψ AB + ψ BC + ψ CD =.44 rad =.5 5. B ψ AB M t1 BC M t M t1 = Gπd4 1ψ l 1 M t = Gπd4 ψ l M t1 + M t = M t 8 1 9 [ πψ. 4 1 +.4 ] =.5 ψ =.56 1 rad =. 5.4 1.5 =.6 rad.6 = 1.5 [ ) ] 4 1 8 1 9 π d 4 1 d =.6 m 8 MPa 8 1 6 = 16 [ ) ] 4 1 π d 1 d =.5 m d d =.6 m
1 5.5 5.1 b) π 18 = 7.15 17 H 5 8 1 1 4 H = 144 5.1 a) τ max =.57 1 7 144 = 4.9 MPa) 1 1 5.6 5.1 a) 75 =.57 1 7 1 d d = 9.6 mm 5.7 5.1 a) 75 =.57 1 7 8 [ 1 d d = 44 mm 1 ) ] 4 1 d 1 = 17 mm 5.8 5.1).4 = 64 1.5 8 1 9.5 4 P P =.565 N 5.8) τ max = 16.565.5 π.5 1 + ) 5 4 5 = 5.5 MPa)
14 5.9 δ = 6 mm 5.1) [ 64P na Ra G d 4 + n br ] b a d 4 = δ b 64P 8 1 9 [ 1.5.1 4 + P = 1.7 N 15.1.15 4 ] =.6 a τ a 5.8) 16 1.7.5 1 +.1 ) 4.5 τ a = π.1 =. MPa) b τ b 5.8) 16 1.7.1 1 +.15 ) 4.1 τ b = π.15 =.5 MPa)
15 6 6.1 R A + R B = P 1. P 1 l + d) = R A l R A = l + d)p 1 l, R B = l 1 d)p 1 l 6. x l/ l/ x l Q = P 1, M = P 1 x Q = P 1 + f x l/), M = P 1 x + f x l/) / SFD BMD Q P 1 +f l/ M P 1 l+ f l /8 P 1 P 1 l/ l/ l l/ l SFD BMD 6. M max M max = f l = 5 1 = 1 kn m) a σ max = M max / a = 1 mm σ max = 6M max a = 6 14 1 9 a = 6 1 6 Pa) 6.4 M max = P l/4 σ max = M max /z z z = πd4 d 4 1) d
16 σ a P MPa) = 8d lp πd 4 d4 1 ) P = 16 πd 4 d 4 1) 8d l 87 kg = 859 N) = 87 kgf) 6.5 M 1, M M M 1 = τ max 6.6 1 Q dx = Q. 8.4 6.6 =.Q Q = 6 kn) τ max = Q bh = 6 1 = MPa) 15 1 6 P = 15. = 18 kgf) = 1764 N) f = 1764/ = 88 N/m) 6.7 v = 5f l 4 84 = 5 88 4 1 84 1 1 9..5 = 5.88 1 m) P ρ P = C D v wa = 1.11 1. π.6 4 = 11.4 kgf) = 1.1 kn) v max I = πd4 d 4 1) = 8.7 1 8 m 4 ) 64 v max = P l = 1.1 1 1 1 9 =.111 m) = 1.1 mm) 8.7 1 8 6.8 x a M = P x dv 1 dx = P x + C 1 ), v 1 = P x 6 + C 1x + C )
17 a x l a M = P a dv dx = P ax + C ), v = P ax + C x + C 4 ) x = a x = l/ v 1 = v =, dv /dx =, dv 1 dx = dv dx, v 1 = P 6 {x al a)x + a l 4a)}, dv 1 dx = P {x al a)}, v = P a {x lx + al a)} A x = ) v 1 dv 1 /dx C x = l/) v v 1 ) A = P ) a l 4a) dv1 P al a) P al a), =, v ) C = 6 dx 8 A 6.9 B R B R B = f l BD Mx) = f x lx) dv dx = f v = f x 4 x x = l 6.9.) lx + l 1 l x + l x v) C = 5f l 4 4 B x = 6.9.1) A dv dx = f l ) ) 6.9.1) 6.9.) v) A = f l l) = f l 4
18 6.1 6.45 I h 1 = 54, h = 6, h = 6, b 1 = 4, b = 8, b = 5, e 1 = 9., e =.7, c = 14.7) I = 1 b e b 1 c + b e 1) = 65 mm 4 ) =.65 1 7 m 4 ) C ) dv = P l dx 6 =.81 14 rad), C A v) A = v) C +.5 C v) C = P l 8 = 1.4 1 4 m) ) dv = 1.4 1 4 +.5.81 1 4 =.4 1 4 m) dx C
19 7 7.1 x = 1 l R A = R B = P, M A = M B < x < l/ x M M = M A 1 P x d v dx = 1 M A 1 ) P x dv dx = 1 14 ) P x + M A x + C 1 v = 1 11 P x + 1 ) M Ax + C 1 x + C x dv dx = v = C 1 = C = x = 1 dv l dx = 1 16 P l + 1 M Al =, M A = M B = 1 8 P l dv dx = v max v max = P l 19 1/4 7. R A + R B = P R A l + M B P l = a), b) a) A b) A v a = R Al v b = P l l )l 6 a) b) A v a + v b =, R A = P l l l ) l R B = P P l l l ) l = P l ll + l) l
M B = P l R A l = P l l ll + l) l = P l 1l l1) l 7. A, C M A = M C = M B l 1 + l ) = f l 1 4 8M B = 6 1 4 + P b l l b ) + 5 1.5.5 ) M B =.9 kn m AB B.9 1 + R A 6 1 1 = R A = 5 kn BC B y R C +.9 1 1.5 5 1 = R C = 1 75.9) 1 = 7.1 kn) R A + R B + R C = 6 1 + 5 1 = 17 1 R B = 17 5 7.1) 1 = 9.9 kn) M B =.9 kn m, R A = 5 kn, R B = 9.9 kn, R C = 7.1 kn 7.4 M M = f x b h σ σ = 6M bh = f x bh b x /h = h h x /b = b x
1 7.5 σ x = l σ l σ = σ l M = P x I = b h 1 σ = 6P x b h σ l = 6P l b h l h = h l x l h l I l I = b h 1 = b h l 1 ) x = I lx x l l l x = l dv/dx = v = d v dx = P xl l l x x = P l l x dv dx = P l ) x 1 + C1 l v = P l ) 4 l x + C1 x + C x = v max C 1 = l, C = l v = P l ) 4 l x lx + l v max = 1P l EB h l l 8P l = Eb h l b h l 4P l /Eb h l
7.6 h 1 E 1 y 1 A 1 + E y A = 7 GPa)1 mm )h 1 5 mm) + 1 GPa)1 mm )h 1 6 mm) = h 1 = 7.4 mm, h = 7.6 mm y 1 =.4 mm, y =.6 mm I 1 = bt 1 1 + A 1y 1 = 1 1 m)1 1 m) 1 = 1.58 1 6 m 4 + 1 1 1 1 m ).4 1 m) I = 1.44 1 6 m 4, I Z = I 1 + I = 14.5 1 6 m 4 E i I i = E 1 I 1 + E I = 7 1 9 1.58 1 6 + 1 1 9 1.44 1 6 n=1 = 8.5 kn m ) σ i = E iy ρ = E i M E 1 I 1 + E I y σ 1 = 4 kn m)7 16 kn/m )7.4 1 m) 8.5 kn m = 5. 1 kn/m = 5. MPa σ = 4 kn m)1 16 kn/m ) 7.6 1 m) 8.5 kn m = 1.9 MPa σ 1 = 5. MPa, σ = 1.9 MPa 7.7 a) AB G P AB A, B θ 1 θ 1 = P l 16 AB b) M AB A, B θ θ = Ml
AB A θ = θ 1 θ AC, BD M AC, BD A, B θ θ = Ml 16 c) θ = θ P l 16 Ml = Ml M = P l 16 7.8 C y, z I y, I z I y = hb 1 = 6 4 1 = cm 4 ), I z = hb 1 = 4 6 1 = 7 cm 4 ) y, z I yz = Y, Z I Y, I Z, I Y Z I Y = 1 I y + I z ) + 1 I y I z ) cos θ = 1 + 7) + 1 7) cos ) = 4. cm 4 ) I Z = 1 I y + I z ) 1 I y I z ) cos θ = 1 + 7) 1 7) cos ) = 6 cm 4 ) I Y Z = 1 I y I z ) sin θ = 1 7) sin ) = 17. cm 4 ) 7.9 A A = l 1 t + l t)t = 8 1 + 1 1) 1 = 17 cm ) Y, Z S Y, S Z S Y = t l 1 S Z = l 1 t) t C y, z + l t)t + t l = 1 8 + = 8 1) 1 1 1) 1 + 1 1 y = S Z A = 5.5 17 =.15 cm) z = S Y A = 6.5 17 =.15 cm) = 6.5 cm ) = 5.5 cm )
4 C y, z I y, I z I yz I y = I Y z A = tl 1 + l t)t = 1 8 1 1) 1 + I z = I Z y A = l 1 t)t z A 1.5 17 = 17.1 cm 4 ) + tl y A 8 1) 1 = + 1 1.15 17 = 8.1 cm 4 ) I yz = I Y Z y za = l 1t + l t t4 4 4 4 y za = t 4 l 1 + l t ) y za = 1 4 8 + 1 1 ).15.15 17 = 74. cm 4 ) z θ θ = 1 tan 1 I yz = 1 74. I z I y tan 1 8.1 17.1 =.9 θ 1 =.9, θ = 67.1 z θ 1 =.9 Z I 1 θ = 67.1 Y I 7.1 y, z I y, I z I y = hb 1, I z = bh 1 AB 7.18 M = P l M y = P l sin θ, M z = P l cos θ A C σ max 7.7) y = h, z = b σ max = M y z I y = 6P l bh M z y I z = sin θ b + cos θ h P l sin θ) b ) ) hb 1 ) h P l cos θ) bh 1
5 7.76) tan β = I z I y tan θ = bh 1 hb 1 tan θ = h b tan θ
6 8 8.1 ) 15 / P = 1 π cos 6 = 54 N) σ = 1 cos 6 = 5 MPa) τ = 1 sin 6 = 87 MPa) 8. x = OC = OE + EC y = OD = AF AE OE = AO cos θ = x cos θ EC = AG = AP sin θ = BO sin θ = y sin θ AF = PG = AP cos θ = BO cos θ = y cos θ AE = AO sin θ = x sin θ x = x cos θ + y sin θ y = x sin θ + y cos θ 8. 8.1) σ 1 = 1 σ x + ) + 1 σ x ) + 4τxy = 1 σ x + 1 σx + 4τxy 8.) τ max = 1 σ x ) + 4τxy = 1 σx + 4τxy 8.4 8.) ε x = 1 ε x + ε y ) + 1 ε x ε y ) cos θ + 1 γ xy sin θ ε y = 1 ε x + ε y ) 1 ε x ε y ) cos θ 1 γ xy sin θ γ xy = ε y ε x ) sin θ + γ xy cos θ ε x = 1, ε y = 1, γ xy = 1 4
7 ε x = 1 cos θ 1 1 4 sin θ ε y = 1 cos θ + 1 1 4 sin θ γ xy = 1 sin θ + 1 4 cos θ 8.5 1) σ x, σ y, τ xy ) = 5, 5, 5) σ 1, σ = 1,, θ = 45 ) σ x, σ y, τ xy ) = 5, 5, 5) σ 1, σ = 1,, θ = 45 ) σ x, σ y, τ xy ) = 5, 5, 5) σ 1, σ = 5, 5, θ =.5 4) σ x, σ y, τ xy ) = 5, 5, ) σ 1, σ =,, θ = 8.6 G = E 1 + ν) ν = E G 1 ν = 11.55 1 4.4 1 ) 1 =. 8.7 8.59) ε x = σ x νσ y E, ε y = νσ x + σ y E 1 + ε x 1 + ε y = E + σ x νσ y E E νσ x + σ y E E + σ x νσ y E νσ x + σ y = E + σ x νσ y E νσ x + σ y = = γ + γ 1 γ 1 + γ γ)e νσ x + σ y ) = + γ)e + σ x νσ y ) E νσ x + σ y ) γe νσ x + σ y ) = E + σ x νσ y ) + γe + σ x νσ y ) γe νσ x + σ y + E + σ x νσ y ) = σ x + νσ x σ y νσ y ) γ{e νσ x + σ y ) + σ x + σ y } = {1 + ν)σ x 1 + ν)σ y } γ{e νσ x + σ y ) + σ x + σ y )} = {1 + ν)σ x σ y )} σ x + σ y =, σ x σ y = σ
8 Eγ = 1 + ν) σ) γ = 1 + ν)σ E = 1 + ν)τ E
9 9 9.1 9.) S = σ t σ a 5 = σ t 9. σ t = 45 MPa. 9. 1) 1) ) 1) ) 45 45 )) ) ) 1) σ ult = P A = P max πd 4
T τ τ = T ) d = T ) d I p πd 4 σ 1 σ 1 = τ = 16T πd = 16T πd = σ ult = 4P max πd T = d 4 P max. ) P σ x T τ 8 σ 1 = σ x + 1 σx + 4τ, θ 1 = 1 ) τ tan 1 σ x σ 1 = σ ult = 4P max πd 9. 1) σ θ = pr t θ 1 = p 1 = p, σ z = pr t = p 1 = 15p σ r = σ θ = σ 1, σ z = σ, σ r = σ 9.1) σ 1 σ ) + σ σ ) + σ σ 1 ) = σ Y σ θ σ z ) + σ z ) + σ θ ) = σ Y 15p) + 15p) + p) = ), p = = 11.5 MPa) ) 9.1) σ 1 σ = σ Y σ θ σ r = σ Y p = MPa), p = 1 MPa)
1 9.4 1) 1) ) 45 1) ) ) ) ) σ max = τ, σ min = τ 4 τ = T Z p = 16DT πd 4 d 4 ) τ = RT πr 4 r 4 ) 9.4.1) 8 Hooke ε x = 1 E {σ x νσ y + σ z )} ε x = ε, σ z =, σ x = σ max, σ y = σ min ε = 1 E {σ max νσ min } = τ {1 + ν}. E
9.4.1) G = ε = 1 E {σ max νσ min } = 1 + ν) E RT πr 4 r 4 ) = RT GπR 4 r 4 ) E 1 + ν) 9.5 M t 9.5) C τ = 16M t πd M 9.7) C N σ N = σ M = M πd N πd 4 = 4N πd M t M N τ = 16M t πd σ = σ M + σ N = M πd + 4N πd = M πd + 4Nd πd 4 = πd M + Nd ) 8 9.7) M M + Nd 8 9.8) M M + Nd 8 9.6 9.4) ε a = 1 ε 1 + ε ) + 1 ε 1 ε ) cos θ ε b = 1 ε 1 + ε ) + 1 ε 1 ε ) cos θ + α) ε c = 1 ε 1 + ε ) + 1 ε 1 ε ) cos θ + α) α = 6 ε 1 + ε = A, ε 1 ε = B ε a = A + B cos θ ε b = A + B cos θ + 6 ) ε c = A + B cos θ + 1 )
ε a = A + B cos θ ε b = A + B cos θ cos 1 B sin θ sin 1 ε c = A + B cos θ cos 4 B sin θ sin 4 9.6.1) 9.6.) 9.6.) 9.6.) 9.6.) 9.6.) 9.6.) ε a = A + B cos θ 9.6.1) cos θ sin θ ε b = A B B 9.6.) cos θ sin θ ε c = A B + B 9.6.) ε a + ε b + ε c = A ε b + ε c = A B cos θ ε b + ε c A = B cos θ 9.6.4) ε a A = B cos θ ε c ε b = B sin θ ε c ε b = B sin θ 9.6.5) 9.6.4) 9.6.5) ) ε a A) εc ε b + = B B A = ε a + ε b + ε c B = tan θ = ε a A) + sin θ cos θ = εb ε c ) = ε b ε c ε a A εa ε b ) + ε b ε c ) + ε c ε a )
4 ε 1 + ε = 1 ε a + ε b + ε c ) ε 1 ε = εa ε b ) + ε b ε c ) + ε c ε a ) tan θ = ε c ε b ε a ε b ε c 9.7 1) 8 ε x = ε x cos θ + ε y sin θ + γ xy sin θ cos θ = 1 ε x + ε y ) + 1 ε x ε y ) cos θ + 1 γ xy sin θ ε a = ε x cos θ 1 + ε y sin θ 1 + γ xy sin θ 1 cos θ 1 ε b = ε x cos θ + ε y sin θ + γ xy sin θ cos θ 9.7.1) ε c = ε x cos θ + ε y sin θ + γ xy sin θ cos θ ) 9.7.1) θ 1 =, θ = 6, θ = 1 ε a = ε x cos + ε y sin + γ xy sin cos ε b = ε x cos 6 + ε y sin 6 + γ xy sin 6 cos 6 ε c = ε x cos 1 + ε y sin 1 + γ xy sin 1 cos 1 ε a = ε x ε b = ε x 4 + ε y 4 + ε c = ε x 4 + ε y 4 γxy 4 γxy 4 ε x = ε a ε y = 1 ε b + ε c ε a ) γ xy = ε b ε c )
5 1 1.1 ) q N/mm ) x σ = qax A = qx N/mm ) x dx 1.6) du = σ E Adx = q x E Adx q = mg Al U = du = q A E U = mg/al) Al 6E x dx = q Al 6E = m g l 6AE 1. a)b)c) U a, U b, U c σ e U a = σ E Al = σ e E π 4 d) l = σ eπd l E b) d P b / π 4 d ) σ e d P b / π 4 d) ) σ e /4 U b = σ E Al = σ e E = σ eπd l 8E U c = σ E Al = σ e E U a : U b : U c = σ eπd l E σe ) π 4 d) l + 4 π E 4 d) l + σ eπd l E = 5σ eπd l E π 4 d) l = σ eπd l 4E : 5σ eπd l E : σ eπd l 4E = : 5 : 8 1. x d A σ A = π 4 d, σ = P π 4 d
6 dx du ) 4P πd du = σ E Adx = E π 4 d dx = P dx πed d = d 1 + d d 1 )x/l U 1.4 U = = P πe = P πe du = P πe dx d = P dx πe d 1 + d d 1 )x l l l 1 { d d 1 d 1 + d d 1 ) x } l [ l 1 1 ] = P l d 1 d d d 1 πe d 1 d A x M x = R A x + M A + f x U R A =, U = U R A M x M x = 1 R A U M A =, U = = 1 = 1 = 1 [ RA M x dx M x M x dx R A R A x + M A + f x x M A x f RA l M A l f 8 l4 8 x4 ] l ) ) = ) x dx 8R A l 1M A l f l 4 = 1M A + 8R A l f l = 1.4.1) U = U M A M x M x = 1 M A = 1 = 1 M x M x dx M A R A x + M A + f x [ RA x + M A x + f 6 x ) dx ] l = 1 6 R Al + 6M A + f l ) =
7 1.4.1) 1.4.) 6M A R A l + f l = 1.4.) R A = f l = R B, M A = f l 1 = M B 1.5 x C M M M = M = x l ) x l/) l/ x l) M = x l/) M = 1 l/ x l) δ = = i = = / / / / 1.6 U U = A δ A MM P x MM P x dx + P x) dx + l/ dx + dx + δ A = U P = dx + l/ l/ MM dx l/ P x MM x l ) P x 1 dx dx = dx = P l 8 5P l 48 ) l P l) dx = P + l = 4 P l U = P P l P l ) = 4 1.7 AB M AB = P x T AB = P b BC M CB = P x 1 U U = a M AB dx + a = P a 6 + P ab + P b GI p 6 TAB b dx + GI p P l M CB dx 1
8 C δ C 1.8 δ C = U P = P a + b ) + P ab GI p AB T 1 BC T T 1 cos θ + T =, T 1 sin θ = P T 1, T T 1 = P sin θ, T = P tan θ T 1, T δ B = T 1 l 1 P AE + T ) l = T 1L 1 AE AE T 1 P + T l AE T P l 1 = l cos θ, l = l, T 1 P = 1 sin θ, T P = 1 tan θ 1.9 δ B = 1 AE = P l AE P sin θ 1 + cos θ sin θ cos θ l cos θ 1 sin θ + 1 AE P tan θ ) l 1 tan θ AC CB M 1, M, U 1, U U = U 1 + U M 1 = R A x + 1 f x, U 1 = 1 / A R A δ A δ A = M = R A x + f l M 1 dx, U = 1 l/ x l ) 6 M dx U = U 1 + U = U 1 M 1 + U M R A R A R A M 1 R A M R A = 1 { / M 1 M 1 R A dx + l/ M M R A dx M 1, M M 1 = M = x A x = δ A = R A R A / R A x 1 ) { f x x dx + R A x f l x l )} x dx = l/ 6 R A = 115 648 f l } )
9 1.1 1) M = P x M t = P s ) U B = 1 U T = 1 GI p M dx = P l 6 M t dx = P s l GI p A M = 1 x A M t = 1 δ = 1 ψ = 1 GI p MM dx = 1 M t M t dx = 1 GI p P x dx = P l P s dx = P s GI p l
4 11 11.1 I = 1 8 1 6 4 1 = 1184 cm 4 ) = 1184 1 8 m 4 ) P cr = π 4 l = π 4 1 9 1184 1 8 1 = 59π = 19476 N) = 19.5 kn) 11. I = π 14 64 cm 4 = π 64 1 4 m 4 P cr = 4π l = 4π σ cr = 4π Al = 4π 8 19 5 π 8 1 9 π.5 5 64 1 4 = π 1 4 = 6. 1 5 N) = 6 kn) π 64 1 4 = 8π 1 6 = 78.96 1 6 Pa) = 79 MPa) 11. v = C 1 sin α x l d4 v dx 4 + P d v dx = + C cos α x l + C x l + C 4 C + C + C 4 = α l C 1 + 1 l C = α l C 1 cos α α l C 1 sin α + C = l α ) α ) C1 cos α + C sin α = l l C + C 4 = C = C 4 C 1 =, C = C sin α = n = 1 α = nπ P cr = π l
41 11.4 vx) x M = d v dx vx) v max M P Mx) = P v max v) d v dx + P v = P v max ) v h = v max ) v = C 1 sin α x l + C cos α x l + v max x = v =, k dv dx = P v max α k l vl) = v max C + v max = ) C 1 = P v max C 1 sin α + C cos α = α tan α = kl k α = π/ k α = kl/ P cr = k/l 11.1 11.5 A = 1 8 11 6 = cm ) = 1 4 m ) I = 1 8 1 σ cr = π l A 11 6 1 = 14 cm 4 ) = 14 1 8 m 4 ) = π E 14 1 8.8 1 4 = 1.654 1 π E
4 σ cr σ cr = 1 1 + σ cr /G 1 = 1 + 1.654 1 π E/E/1 +.5))) 1 = 1 +.5 1.654 1 π =.961 4% 11.6 σ cr = 1974 MPa > σ Y / σ cr = 4 [ r = 1 I πd 4 A = 64 4 4π 8 4 πd = d 4, l e = l ) ] [ 1 = 4 1.5 4 16 4π 8 = 4.797 = 19 MPa) ]
4 1 1.1 1.4) { σ θ ) max = ρr ω + ν) 4 1 + 1 ν + ν r1 r ) } = 7.8 1.5 {4 4 π 6 )} +.) { 1 + 1. +..5.5 ) } = 8.9 1 6 Pa) = 8.9 MPa) 1. 1.9) 8σ θ ) max v = ωr = + ν)ρ = 8 1 6 =.4 m/s) +.) 8.8 1 1. σ a = σ θ ) max 1.47) 1.4 p 1 = r r 1)σ θ ) max r + r 1 =.1.1 ).1 +.1 5 1 6 = 1.8 1 6 Pa) = 1.8 MPa) σ θ ) max σ θ ) max = p 1r 1 r r 1 = p 1 n 1 σ θ ) max 1.47) ) + 1} σ θ ) max = r 1 + r )p 1 r r 1 = { r r 1 r ) 1 p 1 = n + 1)p 1 n 1 r 1 σ θ ) max σ θ ) max = n + 1 n + 1 1.5 1.6) δ = 4p m E r r r 1) r r )r r 1 ) = 4 16 6 1 9.6.7.5 ).7.6 ).6.5 ) =.141 1 m) =.141 mm)
44 1.6 1) 1.61) σ θ = σ θ + σ θ = r + r r p m + r =.1 +.11.1.11 16 + = 479 1 6 Pa) = 479 MPa) ) 1.6) r r 1 r r 1 r.1.1.1 + r 1 r ) p 1.1.11 +.1.1 ) 5 1 6 σ θ1 = σ θ1 + σ θ1 = r r p m + r + r1 r 1 r p 1 r 1 =.11.11.1 16 +.1 +.1.1 5 16.1 = 46.8 1 6 Pa) = 46.8 MPa) 1.7 1.47) 1.78) σ θ ) max = r 1 + r )p 1 r r 1 = 1 + n 1 n p 1 σ max = r + r 1 r r 1 ) p 1 = 1 + n 1 n ) p 1 σ θ ) max σ max 1.8 1.78) 1.9 p 1 = r r 1) r + r 1 = 1 + n 1 n 1 n ) 1 + n = 1 +.7 1.7 1.7 ) 1 +.7 =.8 σ max 1 =.14.8 ).14 +.8 16 1 = 118 16 Pa) = 118 MPa) 1.84) + ν) t = p a 8 σ max +.) = 8 6 1 6 16.1 =. m) = mm) 1.1 1.8) w w = 1 D p 64 r4 + C 1 r + C + C r log r + C 4 log r)
45 p = w = 1 D C1 r + C + C r log r + C 4 log r ) C 4 = 1.79) 1.8) d w Q = D dr + 1 r w d w dr 1 ) dw r = D d dr dr Q = 4C r { 1 r r Q P P + πrq = d dr r dw )} dr C = P 8π w w = 1 D C 1 r + C + P ) 8π r log r r = a w = M θ = 1.8) C 1 = P 16 log a + + ν ), C = P a 1 + ν 16π 1.8) M θ M r M θ = P 4π 1 + ν) log a r, M r = P 4π + ν 1 + ν {1 + ν) log a r + 1 ν) } σ θ σ r 1.15) 1.16) σ θ = E 1 ν ε θ + νε r ) = Ez d w 1 ν dr + ν ) dw = 1M θ r dr t z σ r = E 1 ν ε r + νε θ ) = Ez ) 1 dw 1 ν r dr + ν d w dr = 1M r t z σ θ σ r σ θ = 1M θ t σ r = 1M r t z = 1z t P 4π 1 + ν) log a r = 1P z 1 + ν) 4πt z = 1z P {1 t + ν) log a } 4π r + 1 ν) = 1P z { 4πt 1 + ν) log a } r + 1 ν)