A Dd E F F cr F d F N F Q G I I I P I i i k d x e s u N n n r n st p P q Rr r S S T t V c V v d v v v W W c W P w φ l µ ν b bs cr d e p r s u - [] []

Transcription

1 aterial echanics 00 9

2 A Dd E F F cr F d F N F Q G I I I P I i i k d x e s u N n n r n st p P q Rr r S S T t V c V v d v v v W W c W P w φ l µ ν b bs cr d e p r s u - [] []

3

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8 A B

9 - 5-6

10 (member) (strength) (failure) (rigidit) (stabilit) (external force) (internal force) (stress) (deformation) (mechanical properties) (mechanics of materials) 7

11 (model) (assumption of continuit) (assumption of homogeneit) (assumption of isotrop) (small deformation) (elasticit) (plasticit)

12 bar plate and shellsolid block (basic deformation) (axial tension or compression) -(a)(b) (torsion) -(c) (bending) -(d) (complex deformation) - (load) (reaction of constraint) (method of section) ( )

13 ( ) -a B A F F A F/ A A A0 F/A F p lim t 0 A - p (normal stress) (shear stress) -(b) L T Pa (Pa) (GPa) Pa0 6 PaGPa0 Pa0 9 Pa (strain) (linear deformation) - A AA (angular deformation) - - 5

14 - A x x - ( ) A A x x ( ) (linear strain) - x-x - ( ) x x ε x lim t 0 x ε lim t 0 x x A x (shear strain) - x0 0 γ α + β (-) ( a) ( b) A - 6

15 7

16 -??? -?? -?? -?? -5?? -6?? -7? 8

17 -? - - Aa Ab?Ba e Bb? - 9

18 s b E 0

19

20 axial force F N () control section ( ) da F -(a) -

21 (plane assumption). -(b). F da da A N A A F N (-) A A (-) (-) (SaintVenant) (aint Venant principle) - - W 8.kN AB d5mm AB ABC AB C 0 AB F 8.kN 0.6m 8. o.m sin0 N - AB. FN A FN πd N / m kn 8. 0 N π (0.05) 0.Pa m - - BC AB

22 (critical section) (stress concentration) -a -b - - α max α (-) 0 0 F/A 0 A 0-0 max - - max 0

23 ( ) -5 l α l l α α -5 l l l l F l A Fl l A E F N F FNl l - EA (Hooke) (Hooks law) (-) E ( ) (modulus of elasticit)e E E E L T Pa GPa E - EA tension or compressive rigidit F N l EA EA - F A E l l - l FN l A E N 5

24 l / l F / A ε Eε (-) E -5 α α α ε α α ε ν ε νε ε (-5) (Poisson ratio) E - E N - E EGPa 90~0 0.5~0. 7~0 0.~0.6 60~65 0.~ ~ ~0. 0.6~0..7~5 0.6~ ~ 0.9-5m 5000mm 00kN 0.5mm 0.00mm E (-) FN l 00 0 N.5m E ( l) A m (-5) m.0 0 N / m 00GPa 6

25 ν ε ε 0.00mm /00mm mm/500mm mm E0GPa A A AB BC (-) AB F N 00kN l AB 00 0 N.5m Pa m BC F N 00kN + ( 60kN) 60kN N.5m l AB Pa m 0.975mm 6 l l AB + lbc 0.75mm 0.975mm. 5mm A.5mm - -7a A E q ρ g A ρga x ((b)) m -7-7

26 F N ( x) qx ρgax d x FN ( x) ( x) ρgx ( ) A (xl ) max ρgl e (-) dx c d l df N (x) dx d( l) FN ( x) dx EA l FN ( x) dx ρgaxdx ρgal l l d( l) 0 0 EA 0 EA EA W ρgal W l WA l l l E ( ). 0.5% Q5 (standard specimen) -8 8

27 -8 AB l l d l 0 d l 5 d l A l. A l 5.65 A 0P/s F-l -9 F A l l b b elastic limit e oa a proportional limit p -0 - oa (-) (-) E - F-l ield (plastic deformation) (residual deformation) c 9

28 ield limit s (strengthening) d (strength limit) b d (necking) - F-l - - p ( e ) s b l l A A l l δ 00% l ψ A A 00% A 5% (ductile materials)5% (brittle materials) 5% -9 BA l e lp AB B BD OBD ABD p B OC AC (5%)5 Q5-0

29 0.% (offset ield stress) p ( ) 0.5%0.6% b l(.5.0)d - -5(a) p s E

30 -5-5b - -6a b %0% 55 b 5 50mm 8-7(a) -7b

31 OA - b - AC OAC

32 - ( ) s (Pa) (Pa) + b b δ (%) ψ (%) Q5 6~5 80~70 80~70 ~7 Q7 55~7 90~608 90~608 9~ 60~ n n 70~0 70~50 70~50 6~ 5~60 50~70 600~00 0.5~0.6 90~0 90~ ~ ~.5~80-0(a) -0 ( ) -0(b)(c)

33 () (fiber-reinforced composite materials) (c) E E f V f + E V m ( f ) E f Em V f 5

34 - E - (viscoelasticit) f ( ε, t) ε f (t) - - (linear viscoelasticit) (nonlinear viscoelasticit) - - (creep) (relaxation) - - (-) () 5 ( ) b s ( 0. ) u (limit stress) 6

35 (safet factor)n u [ ] (-6) n, u b u b ( 0. ) n n.5.0 n [t ] (Pa) [ c ] ~5 60~00 6~8 9~ -.5~ 0.~0.7 7~ (-) F N max max A [ ] -7 F N max (-7) A -7 5% 7

36 -7 F A N max [ ] A (-7) FN max A -5-60kN 60Pa d (-7) F N 0kN A π d FN 0 0 N 6 [ ] 60 0 Pa d5.5mm Pa c g6kn/m 0.5Pa q m (-7) mx F N A c q + ρga l A [ ] 8

37 9 m kn m m N Pa m gl A q /.8 / ] [ 6 ρ (-7) max A l ga ga l q ρ ρ / / gl ] [ gl q m m m N Pa m m m m N N A + + ρ ρ m 0.97m (redundant constraint) ( ) ( ) ( ) ( ) -7-5 AB C F EA -5-7 AB

38 F + F F a A B AB AC CB l l + l 0 b AB AC CB (-) FNACa FAa FNCBb FBb l AC, lcb (c) EA EA EA EA (c) (b) FAa FBb EA EA 0 (d) (a) (d) Fb Fa FA, Fb a + b a + b (connections) -6a ( ) ( -6-6b F m-m( ) (shearing deformation) - (method of utilit calculation) 0

39 -7a () () (bearing) () -7-7a -7b F Q (shear surface) F Q F (-8) F Q /A Q (-9) A Q d A Q d / τ F A Q Q (-0) F u (-8)(-9) u n (0.0.8) (single shearing)

40 bs -7a F bs Fbs F bs F (-) -7c Abs bs F bs / A bs (-) (-) d - A dδ F bs bs (-) Abs bs bs n ubs.7.0 bs -7a -7d F N F (-) F / A (-5) t N t bs bs A t d b A t ( b d) δ t F A N t t (-6)

41 t (-0)(-)(-6) -8-8a -8b (double shearing) F Q F/ -9a F F/ -9a -9b c -9

42 -8-0a F0kN b0mm 0mm 7mm d7mm 0Pa t 60Pa bs 00Pa F F/ -0 F / τ πd / -0-8 τ 0 0 N / N / π (0.07) m / 95.5Pa ( ) (-) P / bs bs δd 0 0 N / 6 bs N / m 5.9Pa t F N / At t bs -0b - F F F δ A t ( b d) - F N ( b )δ, (-6) A t d FN 0 0 N 6 t N / m 9. 8Pa A ( ) m 0.0 N FN 0 0 N / 6 t.0 0 N / m. 0Pa A ( ) m 0.0 N N t t

43 - -?? -?? - F - FE E dd -5 ABBC [ F] [ ]cos 0 [ ]cos 5 o A AB + ABC? -6?? -7-5 o -7-8 A A E -8 5

44 - (a) A A 50mm (b) A 850mm A 600mm A 500mm - - (a) FkNb0mmb 0 0mmmm (b) AB 80mm BC 0mm CD 0mm (c) AB 0mm BC 0mm g78kn/m mm BC 0mm AB W0kN AB BC - 5mm 00mm 58.kN 0.9mm 0.0mm E µ - -5 Q7 mm Q5 Q5 0mmmmmm 5mm Q5 s 5Pa Q7 s 7Pa Q5 6

45 -6 00mm 0000mm 00mm 0000mm F 0.mm F E 00GPaE 70GPa -7 AC g E A B B AB AD A 500mm E 00GPa CG A 500mm E 00GPaBE A 000mm E 0GPa G F60kN G F E -0 W00kN F00kN c 00Pa t 0Pa -0 - AB AB Pa AB 7

46 - - CD AB d0mm 60Pa E.00 5 Pa F - m 0.m g6kn/m g0kn/m F6kN 0.08Pa AB aa CB AB D F E A l A E BC F BC -6 F0kN d0mm 70Pa? -7 5mm 00Pa bs 80Pa 60Pa 0Pa F 8

47 F?

48 0

49

50 shaft torque x P n W T W Tα P W P Tω t T kw π rpm kw Nms rpm rads 60 P P 000 P( kw ) T 9.55 ( kn m) ω n π / 60 n( rpm) T dx

51 O O abcd abcd d ab c γ γ γ ρ dx O c c O dϕ ef ρdϕ γ ρ tanγ ρ dx dx dϕ γ ρ ρ ρθ dx dϕ θ relative angle of twist per unit length of the shaft dx γ ρ τ Gγ Hooks law in shear G shear modulus E Pa GPaG dϕ τ ρ Gγ ρ Gρ dx dϕ dx ρ ρ dϕ dx x da da O ρτ da x A ρ A

52 dϕ dϕ x Gρ da G dx dx A da A ρ ρ da I P polar moment of inertia IP ρ da A I P L mm m dϕ x dx GI P x ρ τ ρ I r τ max W I P P x r I P r X τ max WP W P section modulus of torsion WP L mm m d da πρdρ d / πd I P ρ da πρ dρ 0 W P A I P r πd d I P W P

53 d D α d / D D / πd I P ρ da πρ dρ ( α ) d / A πd πd W P ( α ) ( α ) D 6 d D d 0 r 0 D r0 + δ, d r0 δ I P πr0 δ ( ) WP πr0 δ ( ) mm A k BC D k k 5 TB 9.55.kN m 00 5 TC kN m 00 0 TD kN m 00 AC x knm AC 9 πd. (50) 0 m 6 WP.5 0 m 6 6 x max.75 0 N m 6 τ max 7. 0 N / m 7.Pa 6 WP.5 0 m d 00mm T knm 5

54 xt0knm 9 πd. (00) 0 m WP.96 0 m 6 6 x max 0 0 N m 6 τ max N / m 5.0Pa WP.96 0 m d D α d / D 0.5 π πd D ( α ) D 5mm, d 57. 7mm 9 πd. (5) 0 m W P ( α ) [ (0.5) ].8 0 m N m 6 τ max N / m 5.7Pa.8 0 m W P da A element dx, d, d, 0 ( τ dd ) dx ( τ dxd) d τ τ 6

55 theorem of coniugate shearing stress relative angle of twistϕ dx dϕ x dϕ dx GI P l l xdx ϕ dϕ 0 GI l P l x, G, I P xl ϕ GI P l GI P GI P torsional rigidit x l GI P GI P x θ GI P 7

56 T 0.8kNmT 0.5kNm 0.kNm l AB 00mm l AC 500mm d50mm G80GPa C B ABAC x 0.5kNm x 0.kNm BC A ϕ AB ϕ AC A x l ϕ AB AB GI P x l ϕ AC AC GI πd I P 500N m 0.m ϕ AB 0. 00rad 9 π 80 0 Pa (5 0 m) 0N m 0.5m ϕ AC 0. 00rad 9 π 80 0 Pa (5 0 m) A BC A T T C B ϕ T ϕ BC BC ϕ AB ϕ AB P rad 8

57 AB C T TA T B TA + TB T TA T B TA T B T C C A ϕca B ϕcb AB ϕca ϕ CB ϕca ϕ CB GI P T a TAa CA ϕ GI P GI P T b TAa ϕ CB GI P GI P b T A T B a b a TA T, TB T a + b a + b T ( τ πr δ r x T 0 ) 0 T τ πr 0 δ ϕ l r γ 0ϕ 9

58 r 0 γ ϕ l ϕ ϕ T oa τ Gγ a τ P τ P b τ s Tϕ ϕ E G E G ( + ν ) G E max max x max τ max [ τ ] W P W max x x W P W P P x max 50

59 s b u x max θ max [ θ ] GI P G Pa 0 m b AB max 6 WP x 7 0 N m mm 6 [ τ ] 0 0 Pa 6 6W mm d P mm π π 5 7 I P 7 0 N m xmac.5 0 m.5 0 mm G[ θ ] 6 π Pa 0. m 80 7 W d P mm π π 5

60 d mm D mm D0 mm Pa Pa d T F C r i T 8 i F i r i D0 ri F T 8F F T / D T πd T τ maxw P τ max 6 τ max πd F Q F D 6 D N / m π 0. m F Q 7. 0 N 0.m 6 FQ FQ τ A πd Q d / π[ τ ] F Q i 5

61 7. 0 N d,9 0 m 9. mm 6 π 60 0 N / m warping c A x τ max WT max 5

62 x θ GIT WT ab, IT βb, h b α, β α, β mh/b h m b α β m m W T b hb m I T b hb x τ max x W hb θ GI x T T Ghb x x xi xi x n + + L+ x x xn i xi 5

63 xi θ i θ θ θ LL θ n x xi θ, θ i GIT GITi h i δ i x θ θ i n G hiδ i i x τ max i δ n i hiδ i i δ i x τ max δ max n hiδ i i D0 D 0 ( a) x x τ max W πd δ P 0 55

64 h πd 0 ( b) x x τ max hδ πd0δ ( b) τ max D 0 5 τ ( a) max δ ( a) x x θ GI P GπDδ δ ( b) x x θ Ghδ GπD0δ ( b) θ D0 ( ) 75 ( a) θ δ 56

65 T ABECDF ABCD ABCD da 57

66 d TkNm G80GPa mm mm G80GPa AC AB 50mmBC 50mm 5mm BC a G80GPa G mx l Gd kw n P P 00KWP 00KW G Pa AB d BC d 58

67 AB BC d d00mm T T 80Pa900mm ϕ rad T T G Pa F0kN d0mm 50mm mm 5mm T AB C T l TkNm G80GPa 59

68 m 0.kNm G80GPa 60

69 6

70 6

71 (bending) (beam) (plane bending) (smmetric bending) F F x), ( x) Q Q ( (shearing force diagram) (bending moment diagram) df Q dx ( x) d ( x) d ( x) q( x) F Q ( x) q( x) dx dx 6

72 (pure bending) CD (bending b transverse force) - (neutral surface) (neutral axis) dx ab a d (radius of curvature) O O O O ab O O O O ρd θ ab a b a b ( ρ + ) dθ ab 6

73 a b OO ε O O ( ρ + )dθ ρdθ ρdθ ρ E E ε ρ da F d A 0 N A E ρ da 0 A d A 0 A 65

74 E ρ A d A 0 E I ρ I da A E da ρ A ρ EI EI EI (flexural rigidit) I I max max max I 66

75 I W max max W section modulus of bending L m mm ( x) ρ( x) EI ( x) I ( x) max W mm mm mm C bh W bh 6 h m 0.m 0. m 6 67

76 0 0 N 6m max 6 max N/m 9.8Pa m W d.5 0 m πd 6 5 W πd.6 0 m d 0 0 N 6m 6 max 8. 0 N/m 8.Pa m m 6 W m 0 0 N 6m 6 max. 0 N/m.Pa m mm D B 68

77 T C C I 0 I m (60 0 m (0 0 m) m) m 0 0 m 0 0 m (70 0 m (70 0 m) m) m 0 B D BD B 0 0 N m 00 0 m 6. 0 N/m m tmax.pa 0 0 N m 80 0 m 6 cmax N/m 8.6Pa m D.5 0 N m 80 0 m N/m m tmax.7pa.5 0 N m 00 0 m 6 cmax. 0 N/m.Pa m.7 Pa D 8.6Pa B D 69

78 m-m n-n dx m-m n-n +d abmncedf amdc bnfe FN F N FN F N abec df F N FN FN df amdc da da F N da da da A A A I I A amdc da S A 70

79 F N S I + d FN S I abec dx dx df τ bdx d τ dx S I b d FQ dx τ F Q I S b F I b S S S h h b h S b( ) ( + ) ( ) I bh 6 FQ h τ ( ) bh h / F Q max bh Q 7

80 m-m m-m m-m m-m τ F S I d Q d I S 7

81 S S S τ FQ h [ b( I d h h ) + d( )] / * I S * S S FQ F Q τ dx F F +d Q max Q τ τ τ A FN F N F N F N τ df 7

82 τ F F df N N F N da A S A d I I + d F N A A d I df τ δdx A A da + d S I τ τ F S τ Q I δ S u S δ u ( h δ ) S * τ u B τ τ F Q A A A τ max F Q S I max b πd FQ 8 π d 6 d π d FQ A 7

83 A F 50kN FQ τ Q max τ max I b. 0 F 6 S N/m N m m 60 0 m.pa Q τ N m 6. 0 N/m m 60 0 m.pa max W max [ ] max 75

84 F Q max τ max F Q max I S b max [ τ ] S max Pa Pa b80mm t c 0 kn/m max ql 5kN m 8 8 W max [ ] 5 0 N m Pa 5 0 m W bh (0.08m) h 6 6 h m 0.08m 0.9m 9mm h00mm 76

85 FQ max ql 0kN/m m 0kN FQ max 0 0 N 6 τ max N/m 0.9Pa < [ τ ] bh 0.08m 0.m t Pa c Pa F A-5 (A-0 I ( ( mm 0 0mm 0 mm mm + 0mm 0mm 5 mm ) + + 0mm 0mm 5 mm AB C C AB C F C c F m 6 max 5 0 Pa I m mm ) 77

86 F 9.7kN c 0.75F 95 0 m 6 c max 50 0 Pa I m F 8.6kN AB B m 6 t max 5 0 Pa I m F.0kN B m 6 c max 50 0 Pa I m F 7.87kN F [ F] 9.7kN qkn/m 0Pa Pa.Pa FQ max.5q.5m kn/m.5kn FQ maxs.5 0 N (0.m 0.05m 0.05m) τ I b 0.m 0.5 m 0.m N/m 0.Pa max ql kn/m.5 m.8kn m 78

87 max.8 0 N m 6 max N/m 9.0Pa < [ ] W 0.m 0.5 m 6 FQ max.5 0 N 6 τ max N/m 0.5Pa < [ τ ] bh 0.m 0.5m W[ ] max W W bh 6 W W/A h W A W A πd πd / W 0.5h bh / bh 0.67h A 6 h (constant strength beam) 79

88 h b(x) max ( x) [ ] W ( x) ( x) F x W ( x) b( x) h 6 F b( x) x [ ] h b(x)x b min F / τ max [ τ ] hb min F bmin h[ τ ] b h(x) Fx h ( x) b[ ] F h min b[ τ ] 80

89 max ql 8 l max ql 0 AB CD 8

90 FN da da 0 A A I I d A da 0 A A I da A I da 0 F xc 8

91 FQ FH FQ F H FQ B B FQ e F h H F H b τ 0 δdu b 0 FQδ uh du I δ F h b Q δ I h b δ e I F Q F F Q F ea+e F 8

92 FQ F xo F Q (shear center) A A -9 ( -) 8

93 τ max max s B OA BC OC (ideal elastic-plastic materials) s s bh W s s 6 (ield s bending moment) 85

94 (limit bending moment) s s (elastic plastic state) s (plastic hinge) s u FN F F F F N N N N s FN F N N F d + ( )da 0 A A s A s A A A A 86

95 T T u u da + ( s )( )da s ( da + da) s ( s s ) A s + A A A S da S A A da W s S + S W u s s bh bh W s S S W s S + S 8 bh W 6 W s W.5 87

96 Ws bh u s u s bh s bh s / / 6.5 n u / n [ ] max s 0Pa 0 h A A bδ + δ ( h δ ) δ[ a ( h δ )] δ + a b 50mm + 00mm 50mm h 75mm W ( S S ) u s s s + δ h δ S bδ ( h ) + ( h δ ) δ 50mm 50mm 50mm(75mm ) + 75mm - 50mm (75mm 50mm) 50mm mm m 88

97 89 6 m mm 50mm)] (75mm [00mm 50mm 50mm)] (75mm [00mm )] ( [ )] ( [ δ δ δ h a h a S m 77.5kN m 7788N m ) Pa( u + c A 96.mm 50mm 00mm 50mm 50mm 50mm) 00mm ( 50mm 00mm 50mm 50mm 50mm ) ( δ δ δ δ δ δ a b a a b A A h i i c S S W c I W 6 6 max 6 m mm mm] 50mm) [(00mm mm ] ) [( mm mm)] (96.mm 00mm [ 50mm 00mm mm 00 50mm ) 50mm 96.mm 50mm 00mm mm 50 50mm )] ( [ ) ( c c c c c c h a I I W h a a a h b b I δ δ δ δ δ δ δ m 59.kN m 596N Pa 0 0 m s W.7 m 59.kN m 77.5kN S u T l 6m F T F

98 6 S m, S u s ( S + S ) [ ] [ ]( S + S ) n n m [ ] Pa( ) 0 6 m 899N m 85kN m Fl F 6 m max.5fkn max [ ].5F 85kN m 85kN m [ F].kN.5 xa (deflection curve) (displacement) (deflection) C x CC w x x (angle of rotation) w f (x) (equation of the deflection curve) x w wtanθ tanθ θ θ wf(x w (equation of the angles of rotation) 90

99 ( x) ρ( x) EI w ± ρ( x ) ( + w ) / w ( x) ± / ( + w ) EI w w w w ( x) / ( + w ) EI dw w dx w ( x) w w EI (approximate differential equation of the deflection curve of beams) A Bx EI I I EIw (x) 9

100 EI w EIθ ( x) dx + C EIw [ ( x) dx] dx + Cx + D C D (boundar conditions) w A w θ CD (method of integration) F EI ( x) F( l x) EIw ( x) Fl Fx A Flx EI w EIθ Flx + C Flx Fx EIw + Cx + D x0 w0 x0 w0 C D CD A Flx Fx w θ EI EI Flx Fx W EI 6EI 9

101 B xl θ w max max Fl EI Fl EI θ max B w B max q AB EI F F ql / ql qx ( x) x A B ql x qx EI w EIθ + + C ql x qx EIw + + Cx + D x0 w0 xl w0 D D0 EIw x ql ql l + + Cl 0 ql C CD ql ql q w x + x EI EI 6EI θ ql ql q w + EI EI EI x x x xl/ w max 5ql 8EI x0 xl A B 9

102 θ A ql θ B EI ql EI xl / AB D F EI Fb Fa FA, FB l l Fb AD (0xa) ( x) x l Fb DB (axl ) ( x) x F( x a) l AD DB Fb x AD EI w EIθ + C l! Fb x EIw + Cx + D l! Fb x ( x a) DB EI w EIθ + F + C l!! Fb x ( x a) EI w + F + C x + D l!! D D D xa w w xa w w (continuit conditions) C C, D D x0 9

103 xl Fb D D 0, C C ( l b ) 6l Fb( l b ) Fb AD w θ x 6EIl EIl Fb( l b ) Fb w x 6EIl 6EIl Fb( l b ) Fb F DB w θ x + ( x a) 6EIl EIl EI w Fb( l b ) Fb F 6EIl 6EIl 6EI x x + ( x a) ab AD w 0 w l b x0 w max / Fb( l b ) 9 EIl xl / Fb w C (l b ) 8EI wmax w C F w w F max C F b0 x l l wmax w C b l b 95

104 w w max C Fbl Fbl EI EI Fbl Fbl EI EI wmax w c l F a b AB θ A Fl 6EI Fl θ B 6EI Fl w C 8EI method of superposition wc θ A EI q F w c w ( q) + w ( F) 5ql Fl 5ql + 8Fl + 8EI 8EI 8EI θ θ ( q) + θ ( F) A c A ql Fl ql + Fl + EI 6EI 8EI c A 96

105 w A BC AB B AB AB B AB F A B BC Fl F B F - - l Fl l F ( ) ( ) Fl θ B θ B( F ) + θ B( ) + EI EI 6EI l Fl l F ( ) ( ) 5Fl wb wb F w ( ) + B( ) + EI EI 96EI l θ B w B wa wb, wa θ B l F( ) AB A w A A EI l wa wa + wa + wa wb + θ B + w l F( ) 5Fl Fl l Fl EI 6EI EI 6EI A 97

106 98 ] [ max w w EI x w + EI l w EI l B B + + θ ) ( 6 l x l x EI Fl w + EI Fl w EI Fl B B + + θ l x a a x EI Fa w a x x a EI Fx w + + ), ( 6 ),0 ( 6 ) ( 6 a l EI Fa w EI Fa B B + + θ ) (6 l x l x l x EI ql w + EI ql w EI ql B B θ 5 ) ( )], ( ) [( 6EI ) ( ) ),(0 ( 6 a l x l x l a l x x l F w l x x l EIl Fax w + + ) ( ) ( 6 ) (, 6 l a EI Fa w a l EI Fl w EI a l Fa EI Fal EI Fal c D C B A + + θ θ θ

107 99 6 ) )],( ( ) ( ) ( [ EI ) ( ) ),(0 ( a l x l a l x l x l x a l x q w l x x l EIl x qa w + + ) ( 6 ) ( 6, l a EI qa w EI l qa w EI a l qa EI qla EI qla c D C B A + + θ θ θ 7 ) ( 6 l x l x EI l w B + EI l w EI l EI l B C B B B A 6,, θ θ 8 l x a a lx x EIl x l w a x b x l EIl x w + ), ( 6 ) ( 0 ), ( 6 ) ( 6 ) ( 6 ) ( 6 b a l EIl a w b a l EIl a b l EIl D B A θ θ 9 ), ( l x l x l x EI ql w + + EI ql w EI ql EI ql C B A 8 5,, + + θ θ 0 l x a a x b x b x l EIl qb w a x b x l EIl x qb w + + ] ) ( ) [( ) (0 ), ( ) ( ) ( ) ( b a l EIl a qb w b l EIl qb b l EIl qb D B A + + θ θ

108 Fl w + ( 8EI l (0 x ) Fbx w + ( l 6EIl (0 x a) a x b x l x x l Fbx w + [ ( l a) 6EIl b ( l b ) x x ] + ), b ), Fl θ A + 6EI Fl θ B 6EI Fl wc + 8EI Fab( l + b) θ A + 6EIl Fab( l + a) θ B 6EIl Fb wc + (l b ) 8EI a > b w max θ w l l w l l w l l w l l w [ θ ] 0.005rad 0.00rad F 0kN F 0kN F kn F kn 70P Pa E 5 Pa w l 00

109 F A B 8 kn, F 6kN F Q max 8kN, max 6.kN m W max [ ] 6. 0 N m Pa m 67cm W cm 6. 0 N m N/m m max max W 75Pa % I 780.cm, h 00mm, b 7 mm, d 7mm, δ mm 0

110 0 d I S F max Q max max τ ] [ 57.Pa N/m m 0 7 m m 0 ] ) (00 7 ) (00 [7 N τ +.9mm m 0.9 m Pa m N ) ( max i i i i EI b l F b w w mmm E I

111 (staticall indeterminate beam) (redundem restraint) compatibitit equation B q F B B B B B q F B w B wbq + wbf B 0 ql 8EI FBl EI 0 F B 8 ql FA 5 ql, 8 ql A 8 0

112 q A force method C A, A, B, FA FB F C C F AC CB F F C C w C AC C w CB C C 0

113 w C C w C w C l F( ) EI FCl EI l F( ) + EI w l FCl EI 5Fl FCl FCl 8EI EI EI 5 F C F 05

114 x w EI F 06

115 E.00 5 Pa mmmm 0 - dmm D600mm E00GPa c max a-a D t max 0Pa 07

116 AB 0.0mm F E00GPa m q5kn/ 0.5m 00mm b-b a-a a-a 8kN knm b-b 0mm mm kn q bh l x 08

117 da da F Q F Q P kn F t 0Pa c 60Pa I 8 m F8kNa.5m 0Pa hb d AB B d0mm BC q 09

118 AB mm EI E 5 I cm 0

119 w E00GPa AB I CD I AB w c

120

121

122

123 state of stress at a point element principal plane principle stress, 50Pa-80Pa 0 50Pa 0-80Pa unaxial stress state biaxial stress state triaxial stress state plane stress state three dimensional stress state complex stress state x τ x τ x x x x 5

124 x ef x ef ef α τ α α τ α ef da eb bf dacos dasin n t α da + ( τ xdacosα)sinα ( xdacosα)cosα + Fn 0 ( τ dasinα)cosα ( dasinα)sinα 0 τ α da + ( τ xdacosα)cosα ( xdacosα)sinα + Ft 0 ( τ dasinα)sinα ( dasinα)cosα 0 τ x τ x + x α + cos α τ x sin α x τ α sin α + τ x cos α ef x + x cos α τ sin α α 90 o + + x x τ sin α τ cos α α 90 o + x 6

125 α + o α x τ α τ α +90 o x τ x α τ α, x + x α cos α τ x sin α x + x ( α ) + τα ( ) + τ x α τ α + x x + ) (,0 ( ) + τ x stress circle ohr 895 ohrs circle Oτ OB x B B D τ x D D x D x OB B D τ D D D D C C CD CD x + OC ( OB + OB ) 7

126 x + CD ( CD ) CB + BD ( ) + τ x x α τ α CD CE E OF OC + CF OC + CE cos(α + α ) OC + CE cos α cos α CE sin α sin α OC + ( CE cos α )cos α ( CE sin α )sin α OC + Cb osα B D 0 0 sin α x + x + cos α τ x sin α α EF CE sin( α 0 + α ) CD cos α 0 sin α + CD sin α 0 cos α x sin α + τ x cos α τ α E E x x CD CD OB OB B D B D 8

127 0-0 x -Pa 60Pa x -0Pa τ τ 0 o o 0 o 0 o N / m N / m N / m N / m + cos60 6 o ( 0 0 N / m )sin 60 7.Pa N / m N / m o sin 60 6 o + ( 0 0 N / m )cos Pa N / m N / m N / m N / m + cos( 80) 6 o ( 0 0 N / m )sin( 80).Pa N / m N / m sin( 80) 6 o + ( 0 0 N / m )cos( 80) 7.Pa o o o OB x -0Pa B D τ x -0Pa D OB 60PaB D τ 0Pa D D D 9

128 C C CD CD x CD 60 E E E o 0 7Pa, τ o 0 59Pa CD E E 0 o.5pa, τ max 7Pa x OA 0 Pa A OA Pa A A A CA E E.5Pa, τ 6Pa 0 o 6 0 o 0

129 A A A A A A OA OA X + x OA OC + CA + ( ) + τ x X + x OA OC + CA ( ) + τ x X + x ± ( ) + τ x CD 0 OA A x 0 OA OA 0 tan( α ) 0 B D CB τ x ( x ) τ x tnα 0 x

130 0 CD CA x Pa x Pa OB Pa BD Pa D OD Pa D D D C C CD CD OA OA Pa Pa D CA 05 CD CA x 0 ( ) x x ± + τ x N / m ± (. Pa 7. ) N / m 6 + ( 0 N / m ) 6 6 ( 0 / ) 6 0 / tan τ x N m N m α N / m 6 0 N / m 0 x

131 x x OD τ,od τ D D D D O O OD OD OA τ OA τ OD OA x x +5 τ, 0, τ 0 x x α cos α τ α sin α α α

132 , 0, 0 α τ o max, 0 o 0 α o, τ 5 5 o τ max 0, τ 90 o 0 90 o α τ 0, 0 τ τ τ α τ cos α x, x α τ sin α α 0, τ τ τ 0 o 0 o max m-m m-m abcde a e c b d α α

133 principal stress trajectories 5

134 AE AF EF abc max, min maximum shearing stressb τ max 6

135 0Pa, -60Pa τ 0 0 N / m ( 60 0 N / m ) 6 6 max 50Pa Pa 0Pa τ N / m max 0Pa Eε E ε principal strain 7

136 ε E ε ν E ε ν E ε ε + ε + ε ν ν E E E ε [ ν ( + )] E ε [ ν ( + )] E ε [ ν ( + )] E ε x [ x ν ( + )] E ε [ ν ( x + )] E ε [ ν ( x + )] E 8

137 γ γ γ x x τ x G τ G τ x G generalied Hooks law ε ( ν ) E ε ( ν ) E ν ε ( + ) E E ( ε + νε ) ν E ( ε + νε) ν 0 Q E Pa 9

138 E. 0 N / m ( ε + νε ) ν 0..Pa E. 0 N / m ( ε + νε) ν 0. 0.Pa ν ε E N / m ( ) 0 6 ( ) ( + ) (. 0 N / m 0. 0 N / m 5-6 0mm 5-8(a) F6kN 0. 6 ) F F 6 0 N 60Pa A m x 0 5-8(b) x ε 0 (5-a) ε x ( x ν ) 0 E ν 0. ( 60 0 x 6 x N / m ) 9.8Pa, 9.8Pa, 60Pa 0 0

139 5-7 d80mm T 5-9(a) T E.00 5 Pa x x 6T τ (a) W πd P ε ( 5 o E (a) (b) 5 o ν 5 o + ν ) τ E (b) T πd E 6 + ν ε 5 o -5 E T.0kNm 5-0 dxd d V 0 dxdd V V V V 0 ( dx + ε dx)( d + ε d)( d + ε d) dxdd ( + ε )( + ε )( + ε ) dxdd dxdd V ε + ε + )dxdd ( ε (volume strain) 5-0

140 V θ ε + ε + ε V 0 (5-) (5-0) ν θ ( + + ) (5-5) E (5-5) ν θ ( x + + ) E p -p (5-5) ( ν ) θ P (5-6) E P E K (5-7) θ ( ν ) K (bulk modulus of elasticit) V W V W ε (5-8) 5-(a) F F d(l ) Fd(l ) F-l 5-(b) F l (b) OAB l W Fd( l) F 0 l

141 5- W F l (5-8) Vε Fl 5-(a) F N F l F Nl / EA F Nl Vε FN l (5-9) EA (strain-energ densit) v F l V N ε v ε (5-0) V Al (J)JNm / (J/m (5-0) 5-(a) v ε + ε + ε ε 5-

142 (5-0) v ε [ + + ν ( + + )] (5-) E (strain-energ densit of volume change) v V (strain-energ densit of distortion) vd 5-(a) 5-(b)(c) (b) m ( + + ) (5-5) (a) vv m ε m ν ε m [ m ν ( m + m )] + m E E ν ν vv m m ( + + ) (5-) E 6E (c) (5-5) V 0 vd (5-)(5-) + ν v [( ) ( ) ( ) d v vv + + ] (5-) 6E

143 5-? 5-? ( 0 0 0) ? 5

144 m 0 a-a AB AB 5-5 A 0.9Pa F

145 5-6 A ( A ) A.5Pa () A x x ()A cm cm ( ) F6kN E7.00 Pa K F E.0 5 Pa

146 5- D0mm A E.0 5 Pa0.8, F E.00 5 Pa,0. T 5- E.00 5 Pa0. 5-8

147 9

148 max max (6-) ( ) 7 theor of strength 5. (condition of failure) b b 0

149 (6-) W.J.Rankine 859. u (5-0) b u E ν + ) ( b ν + ) ( B (6-) 9. ) max s (5-9) s s τ s s (6-) (C.A.Coulomb)77 (H.Tresca) 5.

150 v d v du (5-) + ν v [( ) ( ) ( ) d + + ] 6E s s, 0 +ν vdu s E [( ) + ( ) + ( ) ] s s [( ) + ( ) + ( ) ] (6-5) (E.Beltrami) (.T.Huber)90 (R.Von ises) 900 (O.ohr)

151 5- (limit stress circle) (envelope curve) 6- (critical curve) 6-6- bt bc 6- KL PNO,O NO O PO O O N P OO O O ON OK OL ( ) bt OP O O L bc bt OO OO + OO bt + ( + ) OO OO + OO bt + bt bt bc bt 6- bt bc t c

152 [ t ] [ ] c t (6-6) 6-5 t c (6-6) (6-) (6-)(6-6) r (6-7) r (equivalent stress) r ν + ) r ( r [( ) + ( ) + ( ) ] [ t ] r (6-8) [ ] c ( ) ( )

153 max (6-9) τ max (6-0) (-0) (6-5) [( τ ) [ ] τ + ( + τ ) + ( τ τ ) ] (6-0) [ ] [ τ ] 0.577[ ] /(+) (0.50.6) (0.8.0) -0 70Pa,00Pa 6- D 0 000mm 0mm 6-a 70Pa p.6pa 5

154 6-6- (b) D p π 0 δ << D 0 D p π 0 π ( D0 + δ ) πd0 π [ ] (D0δ + δ ) δ pd 0 δ pd 0 90Pa (c) ds pdssinϕ s D p ϕ n 0 sinϕ ds p sinϕ d pd 0 0 pd 0 pd 0 δ 6

155 80Pa ((a)) t 80Pa 90Pa 0 r [( ) + ( ) + ( ) ] 55. 6Pa 6-6-5a 70Pa00Pa bc CD max 8kNm (-) W [ ] max 8 0 N m Pa m 9cm 8a W508.5cm I7.cm AC DB F Qmax 00kN (-) 8a I/S.6cm d0.85cm 7

156 F Q max 00 0 N τ max 95.6Pa < [ τ ] I m d S 8a C D f C (d) a ( ) a e a ((d)) 8 0 N m I 7. 0 m m 9.Pa * F S 6 Q 00 0 N.5 0 m τ 7.6Pa 8 I b 7. 0 m m S *.7cm S.cm.7cm (.6cm + ).5cm a + 0 ( ) ( ) + τ + τ (6-8) r r [( ) + τ + ( ) a r r (9.Pa) (9.Pa) + (7.6Pa) + (7.6Pa) + ( ) ] 09.5Pa > [ ] 96.Pa > [ ] + τ 8a a a 8

157 8 0 N m.5 0 m 0.0Pa m N m m τ 56.5Pa m m r 57.7 Pa 7. r a () () 6- Pa A -0Pa -0Pa B -0Pa Pa AB AB 6-6 A B A A B B A A B B 9

158 6- max max 6-?? 6-?? ( a) (b)??

159 6- t 60Pa r -5Pa 0Pa Pa -700Pa -900Pa 50Pa 6- A (b) x E. 0 5 Pa v0.70pa A ppa q60kn/m Dm 0mm 0Pa 6-5 ab? A-A? 5

160 .5Pa Pa A-A.Pa 6-7 (a) (b) 70Pa,00Pa a ( a a ) b / bc / ϕ? 5

161 5

162 combined deformation) oblique bending 5

163 F ϕ F F F cosϕ, F F sinϕ F F x x x F F F( l x) F F ( l x) F( l x) cosϕ cos ( l x) F( l x) sinϕ sinϕ F x A, F F A ϕ ( ) 55

164 cosϕ I I sinϕ I I F F A F F A cosϕ sinϕ + + I I cosϕ sinϕ + I I cosϕ I 0 0 sinϕ + I I tanα tanϕ I 0 I I α ϕ F α ϕ F 0 56

165 b d t max cosϕ sinϕ + I I W + max max W c max W + W D D e f [ t ] [ ] t max c max c F F C x x ω Fl EI, ω Fl EI 57

166 ω ω C ω ω + ω β ω I tan β tanϕ ω I I I β ϕ C C β ϕ F 7-6 l o m qkn/m ϕ 5 h 8cmb [ ] 0 Pa q q q cosϕ, q q sinϕ x x 58

167 q l 8 t max + + W W t max o 0 N m sin 5 m m 0 m N m 7.68Pa [ ] hb 6 q l 8 bh 6 0 N m cos 5 m 8 o 0 m m qkn/m F kn I cm W cm I cm W cm 5 E 0 Pa l q x F x A B Fl ql A + + W W W W 5 0 N m m 0 N m m m N m 07.7Pa 07. W W B 7 Pa 59

168 0 0 I I I 0 0 I 0 tanα 0 0 I I 8 0 N m m tanα N m m 80 0 m o α N m m l tanα N m m 80 0 m o α 86.0 x x ω ω ω ql 8EI N m m 6 0 N m m 0 m 60

169 6 m m m N m N EI Fl ω mm m ω ω ω F A F N x ( ) x I x A ( ) x N I A F +

170 t max min FN A max ± W max max max t max c max [ t ] [ ] c F kn AC Pa AC F F F A B Ax 5kN 60kN F Bx 0kN AC F F Ax B x AB AC AB AB AC B t max FN + A W max [ ] A W W W 6

171 W max.8 0 m 6 [ ] N m N m 8cm W cm A cm t max FN max 0 0 N 5 0 N m A W.0 0 m 09 0 m N m 70.Pa [ ] W cm A cm 0 0 N 5 0 N m N m m 5 0 m max 60.9Pa eccentric compression or tension 6

172 7- x x A, F A C e F F F C F Fe x x F F F A F F Z F x x F F, F, F N F B B FN A I I F A F I F F I B + + F F 6

173 F FF F + + A I I I, I Ai F Ai F F F + + A i i F A + + F 0 F 0 i i F 0 F i i a a i i F F D D 65

174 D D t max c max [ t ] [ ] c F kn F AB e m [ ] Pa t d ABF 7-7- c-c c-c FN F 5KN F 5KN 0.m 6KN m e AB t max FN 5 0 N 6 0 N m + + A W πd m πd m d mm F F C F N F 0KN F KN m F KN m A 66

175 t max F N + A W + W 0 0 N N m m 0.05m 0. m N m Pa N m 0.m 0.05 m 6 α α F F core of a section α α i i F, α F α 7-6 I b I h i, i A A AB 67

176 b α, α i h F 0 α i b b F α b 6 BC h F, F 0 6 CD b F 0, F 6 DA h F, F 0 6 B B B B B F B F B + + i i 0 F F B h/ b/ A d α Y, α Z

177 i i πd πd 6 d 6 (7-) d F 0, F 8 d/ 7-(c) 7-9(a) AB C F AB B F TFa 7-9(b) F AB T AB AB AB 7-9(c)(d) Fl, x Fa 7-9(e) (f) c c c 7-9g W a

178 70 p x W τ b, τ + ± 0 c r d ( ) ( ) ( ) [ ] + + r e [ ] r 7- [ ] r 7- (c) (d)(e) 7-9g [ ] τ + r 7-5 [ ] τ + r 7-6 (a) (b) (7-5) (7-6) p W W [ ] x x p x r W W W W W 7-7 [ ] x x x x p x r W W W W W 7-8 ( ) ( ) ( ) ( )

179 ( ) 7-8 d8cm Dm 5kN 7-0(a) A C 60Pa (b) AC (c) x x d e B C B B + B. ( KN m) +.5 ( KN m).58kn m C C + C.05 ( KN m) +.5 ( KN m).8kn m 7

180 B C BC B B (7-7) r W B N Bx m π Pa 6 m.58 ( KN m) +.5 ( KN m) (7-) (7-5) 7

181 7 F ( ) 7-??? ()? ( ) ( )??? p r W T W A F? 7-6

182 lm 0Pa 7- F 5 F65kNlm l 60Pa [ ω] E.00 5 Pa 500 AB F ε A ε B a Eν F 7-7- F F F 800N F 600Nl m ()b90mm h80mm (a) () d0mm (b) 7-7

183 AB lm hb0.0.m qkn/m (a) (b) b.0 kg/m H0m - d m d m W 000kN qkn/m () () hm W 000kN 0.Pa D? 7-7 A B e A B ε A ε B e ε + ε A B b 6 F 0kN F 00kN W77kN 75

184 A A F E.00 Pa Amm W Z 0 mm F AB A 000-6, B E00GPa F a? q l800mmd0mmqkn/m70pa F F T F 0.7kNF 50kNT.kNm70Pad50mml900mm 76

185 7-7- d00mm 70Pa () ABCD () h00mmb0mm 80Pa

186 78

187 8-(a) F F 8-(b) 8-(c) (stable) 8-8-(d) (unstable) (lost stabilit) F (critical force) F cr FF cr FF cr m m 8- (a) (b) (c) 8-79

188 F cr 8- F cr x (-) EIω ( x) ω x F cr ω EIω Fcrω (8-) k F cr EI (8-) 8- ω + k ω 0 8- ω Asin kx + B cos kx 8- AB k x 0 ω 0 l x ω 0 (8-) ω Asin kx (8-) A sin kl 0 A 0 sin kl 0 0 A (8-)0 ω nπ 0 sin kl k, n0,,, l nπ k (8-) l F cr n π EI l n0 n,, n 80

189 F cr π EI 8-5 l (L.Euler)77 (8-) πx ω Asin 8-6 l A l x w w 0 ω 0 ω l x A A w 0 w 0 F w F OAB w 0 F 8- OAB FF cr,w 0 w 0 8-5(a)(b) (8-5) l l F cr π EI 8-7 ( l) (c) l l l (8-5) l F cr π EI (8-8) ( 0.5l ) 8-5 (d) 0.7l 0.7l 0.7l (8-5) l 8

190 F cr π EI (8-9) ( 0.7l) π EI F cr (8-9) µ ( l) l µ (effective length) (length factor) (8-9) F cr, EI F cr () x I 8-6 I x I (8-9) I I I min 8-7 x x x x II I

191 (8-9) (8-9) F cr (8-9) F cr (critical stress) cr (8-9) I i A F cr π EI π E cr A π E ( µ l) A ( µ l) A π E µ l i cr i π E (8-0) ( µ l) ( λ) I µl λ (8-) i (slenderness) F cr cr (8-9) (8-0) p cr π E λ p (8-) (8-) 8

192 λ π E p λ p π E (8-) p λ λ p (8-) λ λ λ E λ Q5 p 00Pa E06GPa p (8-) λ 00 TC λ 0 λ 80 p p p cr p cr p p p p F cr A (8-5) cr λ cr a bλ cr (8-6) ab Q5 a0pab.patc a9.pab0.9pa (8-6) < p < (8-7) cr u cr u u s u b (8-7) λ λ > λ p > (8-8) cr u 8

193 cr u u λ (8-6) cr u λ λ u u λ u a u (8-9) b Q5 λ 60,TC λ 85 u u () p (8-0) () pu (8-6) () u (8-) λ cr λ 8-8 Q5 8- TC 8-9 p 9Pa b Pa E.00 Pa ()h0mmb90mm ()hb0mm () i I min hb b 90 imin 6. 0mm A hb min λ µl i 0 mm 6mm 5. (8-) λ p π E p π p (8-9) 85

194 F cr π π EI 0 ( µ l) ( m) 799N 79.9kN 0 N m m () i b 0mm i 0. 0mm λ µl i 0 mm 0mm 00 u p (8-6) ab 9.Pa 0.9Pa cr a bλ 9.Pa 0.9Pa Pa (8-5) 6 6 Fcr cr A 0. 0 N m 0 0 m 500N. 5kN 8- lm 0 Q5 s 5Pa E06GPa p 00Pa 8-7 x i.cm µ l 000mm λ 8. i.mm x 0.5 i.5cm µ l mm λ i 5.mm 65.8 x Q5 p 00 u s 60 u p (8-6) cr a bλ 0 Pa.Pa Pa A.cm 6 Fcr cr A 0. 0 N m. 0 m 99N 9. kn 86

195 F F cr cr n st F (8-0) cr F nst cr n st [ F ] st [ ] (8-_) st F st st n st n st (8-0)(8-) F cr F nf cr /F (8-0) n F cr nst (8-) F F cr (8-) st ϕ ϕ (8-) 87

196 F ϕ[ ] A (8-) ϕ ϕ cr [ st ] [ ] u, nst n [ st ] cr n [ ] nst u ϕ 8- u n cr u n st n ϕ 0 cr n st ϕ (GBJ788) ϕ abc ( ) 8- ϕ (GBJ588) TC7TC5 T0 λ 75 ϕ 8-5a λ λ > 75 ϕ (8-5b) λ TCTCTB7 TB5 λ 9 ϕ (8-6a) λ λ 9 ϕ (8-6b) λ (8-5)(8-6) TC7 TC5 TC TC TB0 TB7 TB5 (Pa) 8- ϕ ϕ 88

197 8- ϕ l/i Q5 6n a b a b l800mm, d0 mm E.0 5 Pa n st.0 F cr I πd πd d 0mm i 0mm A 6 µl 800mm λ 60 i 0mm 8-5 p 00 p 89

198 F cr π π EI ( µ l) N 0.7kN N m 6 ( 0.8m) π ( 0.0) m F n 0.7kN cr [ F ]. kn st 9 st 8-7m 6b 5 8- b 0mm. 70kN 70Pa() h() () h I I h 6b A5.5cm I 9.5cm I 0 8.cm, 0.75cm0mm h I I 0 + A 0 + I I 9.5cm 8.cm + 5.5cm h.75cm +.58h h h h8.cm () i i I A 9.5cm 5.5cm 6.cm 90

199 8- λ ϕ 0.08 µl i. 700cm 6.cm 9. ϕ [ ] Pa 5.Pa F 70 0 N Pa 5.7Pa A m ϕ 5% () F A 70 0 N Pa 70.5Pa 0 m ( 5.5 ) 8-5 AB CD d0mm Q5 F5kNl.5m, l 0. 55m n.5 n st.8 AB AB F NAB Fcos05kNcos0.65kN C max Fsin0 l 5kN0.5.5m5.6kNm W 0cm A.5cm W F + A N m.65 0 N + m.5 0 m max NAB max Q5 5Pa.5 [ ] 6Pa n 6Pa max 5% CD CD F N Fsin0F5kN 9

200 µ λ µ l i 0.55m 0 0.m p 0 λ 00 CD ( µ l) ( 0.55m) ( 0.0) 6 π m π 06 0 Pa π EI F 6 cr 5. 8kN F F cr [ F ] 9. kn N st nst CD I I I 8-(b) (a) 8- h I I ( 8-) 8-(b) a a I I I min 9

201 8- F cr E E E cr p 9

202 8-?? ( )? 8- d? 8-?? 9

203 d F

204 8- ((e) )? 8-8- h60mmb0mm l.0m Q5 E. 0 5 Pa (a) (b) 0.8 F cr 8-8- Q5.0 0 mm d 0.7d mm E700 Pa F

205 8-5 am d5mm a 5 70Pa F F F? 8-6 TC7 5050mm L.0m Pa Q5 AB d70mmbc a70mmab BC l.5m n st.5e.0 5 Pa 8-8 AB TC7 d00mmab Pa q ( ) b L6ma0mm 50kN Q5 70Pa 8-0 AB d0mm 5 p 00Pa E00GPa cr a-b a0pab.pa () F cr () F70kN AB n st? 97

206 AB CD 0b 665 b q9kn/m 5 70PaE.0 5 Pa 8- Q5 AB 6 BC d60mm E00GPa p 00Pa s 5Pa n n st

207 99

208 00

209 (dnamic load) (dnamic stress) (dnamic deformation) (alternating stress) (fatigue failure) 9-(a) a W A x 9-b W x F qga Nd ρga W q d a a g g a 9-(b) 0

210 9- x F W a + qx + qd x W + a + ρ gax + ax ( W + ρgax) Nd W W ρga a + + g g g g W+gAx F Nst F k 9- Nd d F Nst a k d + (9-) g (dnamic coefficient) F kd F Nd Nst d kd st A A (9-) (9-) k d max d st max d st max d max k [ ] D D 9-(a) O A D 0

211 a n ρ ga ga D qd an 9-b g g ρ ω ω D 9-9-(c) D ϕ d q d dϕ F 0 F Nd π + q 0 d D dϕ sinϕ 0 q d m-m n-n ρga ω D ρga F Nd g g υ (9-5) F Nd ρg ω D ρg d υ (9-6) A vd/ g g ρg d v [ ] (9-7) g v (9-7) [] v [ ] g (9-8) ρg 0

212 (impact) 9- () () () 9- W h 9-(a) v d F d d T V V T + V (9-9) V ε W ( ) T 0 v Wh g ( ) T 0 Wh VW d V (9-9) ε F d d 0

213 Wh + W d F d d (9-0) F d d EA Fd d C d (9-) l EA C W l W ( ) st WC st C (9-) st F d W d (9-) st F d (9-0) ± d st h 0 st d + h st st d ± st h + st st d h d ± + st kd st (9-) st k d h + + (9-) st d (9-) k d st Fd kdw (9-5) F d d 9-5 d W st k d k (9-6) d d st 9- k d W st st d d 9-(a) 9-(b) (9-) st W 05

214 k d (9-) h0 k d (sudden load) h h st k d st v v h g k d + + v g st 9-(a) W A (9-9) A v F d d 9-(b) W T g v 9- V0 Vε F d d (9-9) W g v F d d F d W d st v st d st g g st v k d st k d v 9-7 g st st W 06

215 9-(c) k d d d dmax d max k0.6kn/mm WkN h50mm C 60PaE. 0 5 Pa st W [ ] C I 0cm W cm Cst Wl 8EI N m 8. 0 N m 0 0 m 0.7mm W F RB 0.5W k 0.5 kn kN mm Bst 5 mm C st Cst + Bst 0.7mm + 6.5mm. 6mm (9-) 8 m k d h 50mm mm st C C Wl 0 N m.5 max 0 N m max.5 0 N 6 st max Pa 0. 6Pa 6 W 0 m d max kd st max kPa 59. Pa dmax < 07

216 k d k d st k d st st (impact toughness) a k (a) U 9-6b W W A W a k A ak Nm/m J/m a k a k 9-7 a W F H sinωt F H sinωt 08

217 9-7b (stress spectrum) (a) FF 9-8(b) i i ( 9-8c 0 i max i 0 i min i (stress ccle) 9-8(d)

218 b 9-8d max min (ccle performance) r min r (9-8) max (stress amplitude) max - min (9-9) max min r max r max min r - (reverse stress) max max 9-8(d)r min max 0 min 0 (pulsating stress) max 9-0(a) r min max max min 0 9-0b r- 0

219 9-0 max r N (fatigue life) r max N max N max (fatigue limit) r r - - GB78 max N max N -N (fatigue curve) N -N ( - ) -N N 0 N 0 N N 0 0 8

220 - n n r n n r (9-0) 0 60 (GBJ788) N 0 5 [ ] (9-) max min max - min max -0.7 min [ ] C N β (9-) N C 9-9- C C F min 0kN F max 00kN 0 6 () I 90mm mm 70mm 75 mm m

221 F min 0kN I N m 0 m min max min F max 00kN I 00 0 N m m max max max a max min 57.05Pa m m 6.Pa 6.9Pa () 9- C.8 0, β C (9-) C.8 0 β [ ] 0.0Pa 6 N 0

222 ?? l?? 0 9- W v 9-?? d l Dd 5 h k d + h + st h st?? F

223 () F () FF 0 +F H sint ( F 0 F H ) () 0F () F (5) F F 5

224 8 a0m/s lm A60mm 9- W 0kN b 0kN a.5m/s d0mm 60Pa ( ) n00 / AB h60mm b0mm lmr50mm 7.80 kg/m l A W E W 9-9- kn 0.65mm d0mm lm 0PaE00GPa 5kN h h? ABC C W700N h00mm E.00 Pa

225 W500kN v0.5m/s C W.00 7 mm I mm E.00 5 Pa max min r n-n max min r a 0mm0mm.5m 50kN F max 50KnF min

226 8

227 9

228 energ method V c W FN l Vε FN l EA T T ϕ OAB W Tϕ x Tϕ x l GI p 0

229 V ε W xϕ x GI l p l θ ρ l EI W θ V ε W θ l EI x dx x ( x) d x dvε EI V ε l ( x) d EI x F Q x h b 6 τ bh F Q h bdxd τ dvε Q bdxd G

230 V εq 0 L l 0 h 8F dx h Gb 6 FQ 0 Gbh Q ( x) h bd 6 h Q dx λ 0 GA ( x) dx l F ( x) 6 form coefficient for shear 5 0 A λ A 9 A f F N (x) (x) F Q (x) x (x) V ε ( x) dx l ( x) dx l FQ ( x) dx l x ( x) + + λ dx l FN EA EI GA I P I T 0 GI p A f 0- F F F i F n i n n Vε W F + F + L + Fi i + L + Fn n Fi i i B.P.E.Claperon F i generalied force generalied displacement F F C Fl 8EI l + 6EI

231 EI l EI Fl A 6 + θ F Fl l l F EI c F V A θ ε F F EI Fl EI Fl F F W CF F 96 8 A F F C F EI Fl EI l EI l F EI l F W C A θ Fl l l F EI W W V F ε F EI l EI l V EI Fl EI Fl F V A CF F θ ε ε

232 F W 0 Fd F F C 0 W df complementar work W + WC F complementar strain energv C F V C WC df 0 V C Vε virtual force complementar virtual work dw C dw dw ec ic principle of virtual force dw ec dw i principle of virtual work dw ec dv C

233 F F F i F n i n F i i F i df i dw ec df i i F F F i F n F i d F i dv V V + V + df C C C C df df L i L F F Fi df df df n df i + + V + F C n df n dv C V C dfi Fi dw ec dv C df i i V C F i i V ε F i ACastigliano F 0 0 V ε F 0 B E Pa A 90mm A 50mm F kn 5

234 FN l i i Vε i Ei Ai V F li F ε Ni Ni BV F i E A F i i F F F N N 5 FN 5 F, 6 F 6 5 FN 5 F, 6 F B BV l EA 5 5 F l EA F 6 l500mm BV. mm F 0- ABCF C EI GI P C CV Vε F BC (x) AB (x) x (x) V ε ( x) l ( x) l x ( x) + dx a dx dx EI EI 0 GI p CV Vε F a 0 EI ( x) ( x) ( x) dx + dx + F EI EI a l l ( x) ( x) ( x) ( x) ( x) F dx + l 0 EI F dx + GI l 0 x x P GI p dx F x dx F 6

235 BC ( x) Fx x F AB ( x) Fx x F CV EI F EI l ( Fx )( x ) dx + ( Fx )( x ) a 0 ( a + l ) Fa l + GI p EI x a x l 0 dx + GI F A A EI A A A 0 A Vε θ A 0 V ε ( x) l dx 0 EI Vε θ l A EI ( x) l x l V l ε θ A EI 0 0 l 6EI O ( x) ( x) ( x) 0 dx p l 0 Fa x x +, x l l l x l 0 0 x + 0 x + dx l 0 dx 7

236 F 0 ( x) ( x) V ε dx l F0 EI F0 F0 0 x F 0 ( x) ( x) + ( x) F F0 F (x) F 0 xf 0 F (x)f 0 F 0 xf 0 F ( x) 0 0 ( x) 0 ( x) F 0 F 0 (x) F (x) F 0 O ( x) ( x) dx l EI J.C.axwell 0 ( x) A EI qx x qx ( x) x l 6l A A 8

237 0 ( x) x A ql 0EI 0 ( x) ( x) A l EI l qx 0 EI 6l dx ( x) dx A A A A l qx θ A EI 0 6l 0 ( x) () ql dx EI A 0- C BD AC A 5cm A 50cm AC I60-5 m E 0 Pa C BD F N 5F F AB F N F F N N ( ) ( ) O x Fx, x x x ( ) ( ) 6 O BC x Fx F, x x 6 x C F Ni F 0 Ni EA i L i + l 0 ( x) ( x) dx EI 5F EA ( F )( ) 6 + ( F )( x ) dx + ( F F )( x )dx C x x EA EI EI 9

238 6.5F EA F + EA F + EI m 9.0mm 0

239 AB CD F C F D C F C D F D D F D C F C CD F C F D V ABC ε B F ABC B F C C C B B

240 EAEIGI P C EI C C A

241 A

242

243 A (area moment) (moment of inertia) ( ) xdv dv dv V V V xc, c, c V V V O ( A-) da da A A c, c (A-) A A (A-) S da, S A da A (static moment) S S A L m mm A- (A-) S, A c c S A (A-) S A, S A (A-) c c (A-)(A-) (centroid axis) 5

244 ( ) S n i A i ci n, S A (A-) i i ci A i ci ci (A-) (A-) A A n i A i c n A i ci i i c n n i A i n Ai ci, (A-5) A i i A- A- O A 50mm 50mm mm, c c c c 50mm 50mm + 80mm + 55mm A 80mm 50mm mm, 80mm + 50mm 0mm A 50mm 50mm.5 0 mm, 50mm 5mm 0 c c (A-5) c mm mm ( 55mm) mm ( 0mm) mm ( 5) mm mm mm mm c 0 A- J dm, J V dm V 6

245 dmda( ) J ρ da, J A ρ da A (moment of inertia) I da, I A da (A-6) A I I L m mm (radius of gration) i I I, i (A-7) A A I Ai I Ai (A-8), m mm A- A- I dabd (A-6) h bh I bd h I dahd (A-6) hb I A- A- A-5 A0 A- A-5 d da d d cosϕd (A-6) 7

246 I d sin d ϕ cos π 6 π d da ϕdϕ A π I I π 6 d πd - I p I I + I p A- (product of inertia) I da (A-9) m mm (A-9) A-6 da da (principal axis) A-6 (A-6) ( ) I A da A da + A da + L L A n n da I i A A A n A- i 8

247 9 A- (A) ( I I, ) ( i i, ) bh bh I hb I h i b i bh 6 6 bh I hb I h i b i πd 6 d I I π d i i D d D / ) ( α α π ) ( 6 α π D I I +α D i i b h bh h b bh I h b hb I 8 πd π π π π d d I d I mh J J +

248 J J m h A (parallel-axes formulas) I I I I c c + a + b A A (A-0) A-7 I I aba (A-) c + c a b I I cc A-7 A- A-8 d ( ) I I I bh I I (A-0) I I c + a d πd A 6 d + πd 5π 6 A-8 A- I bh 5πd 6 bh 5πd A-5 0a A-9a a00mm 0

249 A-9 A-5 B A-9b 0a A mm, I 8 0 mm c I mm, 0.mm c 0 I I c mm mm (A-0) a I I c A 00mm 8 0 mm mm mm mm A-0 I I I O I I I da I A cosα + sinα, cosα sin β cos α I da A A ( cosα sinα ) da sinα cosα da + sin α cos α sinα cosαi A + I + cos α cos cos α,sin α sinα cosα sin α da sin α α, I + I I I I + cos α I sin α (A-) A da

250 I + I I I I cos α + I sin α (A-) I I I sin α + I cos α (A-) (A-)(A-) (transfer formulas for rotation of axes) (A-) (A-) I + I I + I (A-5) (A-) 0 60 I (centroid principal axes of inertia) (principal moment of inertia) (centroid principal moment of inertia) 0 0 (A-) I 0 I I sin α 0 + I cos α 0 0 I tan α 0 (A-6) I I (A-) (A-) 0 60 II di dα ( I I ) α I cos α 0 sin I tan α I I (A-6) 0 (A-5) (A-6)

251 sin α 0 cos α 0 tan α 0 + tan α + tan 0 α 0 I ( I I ) + I ( I I ) + I (A-) (A-) I I 0 0 I I I max I I o min I 0 + I + I + I I I I I I + I + I (A-7) () (A-5) I I I (A-6) (A-7) () () A-6 A- mm I II III (A-5) C I I I C (A-0)(A-) I + + 0mm ( 60) mm + 00mm ( 0) ( 0) mm 00mm 0mm + 0mm ( 00) ( 0) mm 69 0 mm 00mm 0mm mm mm

252 I I 60mm + 0mm + 00mm 90 0 mm 0 + ( 0) mm + ( 69.) ( 00) mm + ( 9.) ( 0) mm + ( 50.8) + 0 ( 69.mm) ( 0mm) ( 9.mm) mm mm mm 60mm 0mm mm 0mm mm 50.8mm 00mm 0mm 0 0 mm (A-6) 0 0 mm tan α 0 mm ( 69 90) mm 0mm 0mm 00mm 00mm 0mm 0.5 tan (A-7) I I max min I + 0 ( ) ( 69 90) 50 0 mm I 0 ( ) ( 69 90) 0 0 mm 0 mm 8 0 mm 8 0 mm 8 0 mm mm 8 0 mm

253 A- (a)(b) A- A- (a)(b) S A- A- (a)(b) A- A- a A- 5

254 (a) a (b) 0 A (a)(b) A-5 A-6 A-6 A-7 A-7 A-8 A-8 A-9 6

255 A-9 7

256 8

257 9

258 50

259 5

260 5

261 5

262 5

263 55

264 - (a) 5.Pa,.7 Pa (b) 5.9 Pa,.5 Pa, 8. Pa - (a) 50 max Pa,(b) 950 max Pa (c) 0 max Pa - 8Pa,. Pa AB BC - E 7.GPa, ν φ mm -6 F 9kN -7 Fl ρgl B + EA E -8 F 0. 69mm -9 Fl l πed d -0 A 5 cm cm, A.cm, A 5 - d 6mm - F 5. kn - a 0. 57m - P 6P, A A -5 0 BC -6 τ 8.88Pa, d mm 56

265 -7 F 0kN, F 90. kn - τ.pa, τ 0, τ 7.Pa, γ rad - τ max 6Pa % - ( ) τ max 5.5Pa,() ϕ 0. 0rad -5 a 0mm 6m -6 x l ϕ Gπd -7 ) d 79mm,() d 66mm, d 79mm, d 79mm, d 50mm ( -8 ) d 9mm, d 80mm,() d 9mm ( -9 T.kN m, T 0. 5kN m 5-0 ( a) τ 09.8Pa,( b) τ. 8Pa - 8 % - l a d + ( d ) - τ.pa, θ 0.097rad / m max 80 max - τ.pa, θ 0.0rad / m max 8 max - ρ 5m, ρ m 57

266 - ρ 85.7m - max 000Pa - D 0.075Pa, t max.75pa, c max 6. 8Pa -5 ( )%,() 5.9% 8.% -6 ql ql ( a ) max ( b) max ( c) max a a -7 F 85. 8kN -8 τ a a 0 τ b b. 75Pa Pa τ 5. 9Pa -0 τ Pa - ql () F Q h - F. kn -.8Pa 6. Pa t max 8 c max -5 d 66mm -6 q 5.7kN / m -7 8 ql a -8 ( a) θ B qa qa wc 6EI EI qa qa ( b) θ D wd EI 8EI Fa Fa ( c) θ C wc EI EI ( d) w 0.8mm w 0.58mm D B -0 ( a) w D 7Fl wb EI Fl EI 58

267 Fl ( b) θ c EI 5Fl Fl ( c) wc θ B EI EI 5Fl Fl ( d) wc θ C 8EI EI - a 5 - a FB.8F b FB ql c FB F 8 - w Fl c (I I ) + E 5- a) 8.Pa τ 7. 99Pa ( 60 o 60 o ( b) ( c) ( d) 0 o 5 o 5 o 8.Pa τ 60.0Pa τ 5Pa τ 5 o 5 o 0 o.0pa 0Pa 8.66Pa 5- A.585Pa τ 0. 85Pa -70 o 0 70 o B.9Pa τ. Pa -70 o 0 70 o 5- ( a ) o 60Pa, 0Pa, α ( a) ( a) ( a) o 55Pa, 5Pa, α 55.8 o 88.Pa, 8.Pa, α 5.8 o 0Pa, 0, α A o 5.8Pa, 0.0Pa, α B 5-5 F P. 798kN o 0.08Pa,.59Pa, α

268 5-7 a ) F, F, α 0 ( 0 ( b ) F, F, α ( ) x.8pa.5pa τ x. 6Pa 0 ( ) o 7Pa 0 α , 66.5Pa F P. kn 5- F P. 8kN 5- T 5. 8kN m 5- vv 0.00Pa, vd 0. 05Pa 6- r 95Pa, r Pa 6- r 50Pa, r 9Pa 6- r 8Pa Pa r 6-5 ()( ) + τ,( ) + τ r a r b ( )( ) ( ) + τ r a r b 6-6.8Pa, τ. Pa r A A 6-7 max 68.7Pa, τ max 89.5Pa,( r ) a. 7Pa 6-8 o ϕ

269 7- max 9.799Pa 7- b 7- F ( ε + ε ) Ea /l, ( ε ε ) Ea / A B 7- ) 9.88Pa,() t 0. 5Pa ( max max 7-5 t max 5.09Pa, c max 5. 9Pa 7-6 b 5. 8m 7-7 ( ) c max 0.7Pa () D. 5m 7-9.Pa,. 7Pa 7-0 F. 9kN 7- F 9.kN, a 0. m 7- r 6Pa 7- r 6Pa 7-5 r Pa 7-6 r 80. 5Pa B A 8- F cr 58. 8kN F cr 50kN 6

270 8-5 F.9kN, F 8. 5kN 8-6 F 89. 9kN 8-7 F kn 8-8 q 60kN / m Pa 8-0 Fcr Fcr 8.5kN, F Pa, CD 0Pa 8- F kn max 9-0Pa, 7. Pa d max d Pa, 60. Pa d d 9- max 7.8Pa l 9- l ω (W + W ) EA 9-5 h.9m, h 0. 0m dmax.pa 9-7 dmax 5Pa 9-8 ( a) r 0, 00Pa,( b) r, 00Pa ( c) r, 50Pa,( b) r, 00Pa 9-9 Fa Fa 6Fa max, min, r, πd πd πd Pa 6

271 6 0- GI P a F D EI a F V c EI a V b EA a F V a 8 ),( ),( ) ( + ε ε ε 0- EI ql EI ql b EI Fl EI Fl a C C C C 8, 8 5 ),(, 6 5 ) ( θ θ 0- EA Fa b EA Fa a C C C 9 0, ),( ) ( 0- EI l EI l b EI qa EI qa a A C A C 9, 6 ),(, ) ( θ θ 0-5 0, ) ( 8 ) ( ) ( C D C A EI Fa c EI Fa b EI Fa a A A- mm mm b mm a c c c 0, 5 ),( 8.8 ) ( A- 59 ),( 500 ) ( cm s b cm s a A- 0, 6 5, 6 ) ( I d I d I a π π 0, 5580, 9. ) ( I cm I cm I b A- cm a b cm a a ),(. ) ( A-5 7 ),( 68.5 ) ( cm I I b cm I I a A-8 086, 65 ) ( cm I cm I a 6 ) (.6, 5.66 ) ( d I I c cm I cm I b π A-9 o , 7.5, 5 ) ( cm I cm I a α

272 o 0., I 7.cm, I 08 ( b) α cm A Gere J..Timoshenko S.P.echanics of materialssecond SI EditionN.Y.Van Nostrand Reinhold98 6