aterial echanics 00 9
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 - [] []
A B
- 5-6
(member) (strength) (failure) (rigidit) (stabilit) (external force) (internal force) (stress) (deformation) (mechanical properties) (mechanics of materials) 7
(model) (assumption of continuit) (assumption of homogeneit) (assumption of isotrop) (small deformation) (elasticit) (plasticit)
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) ( )
( ) -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
- 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
7
-??? -?? -?? -?? -5?? -6?? -7? 8
-? - - Aa Ab?Ba e Bb? - 9
s b E 0
axial force F N () control section ( ) da F -(a) -
(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 0. 0 6 N / m kn 8. 0 N π (0.05) 0.Pa m - - BC AB
(critical section) (stress concentration) -a -b - - α max α (-) 0 0 F/A 0 A 0-0 max - - max 0
( ) -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
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.~0.7 7 0.6~0. 8 0.6~0. 0.6~0..7~5 0.6~0.8 0.0078 0.7 9~ 0.9-5m 5000mm 00kN 0.5mm 0.00mm E (-) FN l 00 0 N.5m E ( l) A 0.5 0 m 50 00 0 6 (-5) m.0 0 N / m 00GPa 6
ν ε ε 0.00mm /00mm 0. 0.5mm/500mm - -6 00mm E0GPa A A AB BC (-) AB F N 00kN l AB 00 0 N.5m 9 0 0 Pa 00 00 0 6 m BC F N 00kN + ( 60kN) 60kN -6-60 0 N.5m l AB 9 0 0 Pa 00 00 0 0.000975m 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
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
-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 - -0-9 -0 b b elastic limit e oa a proportional limit p -0 - oa (-) (-) E - F-l ield (plastic deformation) (residual deformation) 50 5 5 5 c 9
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 (5%)5 Q5-0
0.% (offset ield stress) p0. 0. - - - - - - - - - ( ) 0.5%0.6% b l(.5.0)d - -5(a) p s E
-5-5b - -6a b - -6 5%0% 55 b 5 50mm 8-7(a) -7b
-7-8 - -8 OA - b - AC OAC - - -9 - -9 -
- ( ) s (Pa) (Pa) + b b δ (%) ψ (%) Q5 6~5 80~70 80~70 ~7 Q7 55~7 90~608 90~608 9~ 60~70 5 0 0 5 50 50 5 50 6 0 5n 00 50 50 50 6n 70~0 70~50 70~50 6~ 5~60 50~70 600~00 0.5~0.6 90~0 90~600 568.5~0 755 0 98 0.~.5~80-0(a) -0 ( ) -0(b)(c)
() (fiber-reinforced composite materials) 60 0 - - - -(c) E E f V f + E V m ( f ) E f Em V f 5
- E - (viscoelasticit) f ( ε, t) ε f (t) - - (linear viscoelasticit) (nonlinear viscoelasticit) - - (creep) (relaxation) - - (-) () 5 ( ) b s ( 0. ) u (limit stress) 6
(safet factor)n u [ ] (-6) n, u b u b ( 0. ) n n.5.0 n.0.5 - - [t ] (Pa) 70 70 0 0 [ 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
-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 - -5 - -6-6 -.Pa c g6kn/m 0.5Pa q m (-7) mx F N A c q + ρga l A [ ] 8
9 m kn m m N Pa m gl A q /.8 / 0 6 0. 0.8 ] [ 6 ρ (-7) max A l ga ga l q ρ ρ + + 6 97 0. / 0 6 0 0.5 0.8 / 0 6 0.8 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
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
-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)
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)
t (-0)(-)(-6) -8-8a -8b (double shearing) F Q F/ -9a F F/ -9a -9b c -9
-8-0a F0kN b0mm 0mm 7mm d7mm 0Pa t 60Pa bs 00Pa F F/ -0 F / τ πd / -0-8 τ 0 0 N / 6 95.5 0 N / π (0.07) m / 95.5Pa ( ) (-) P / bs bs δd 0 0 N / 6 bs 5.9 0 N / m 5.9Pa 0.0 0.07 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 9.8 0 N / m 9. 8Pa A (0. 0.07) m 0.0 N FN 0 0 N / 6 t.0 0 N / m. 0Pa A (0. 0.07) m 0.0 N N t t
- -?? -?? - 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
- (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 - - 0000mm 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
-6 00mm 0000mm 00mm 0000mm F 0.mm F E 00GPaE 70GPa -7 AC g E A B B -6-7 -8 AB AD A 500mm E 00GPa CG A 500mm E 00GPaBE A 000mm E 0GPa G F60kN G -8-9 -9 F E -0 W00kN F00kN c 00Pa t 0Pa -0 - AB AB Pa AB 7
- - 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 - -5-5 l A E BC F BC -6 F0kN d0mm 70Pa? -7 5mm 00Pa bs 80Pa 60Pa 0Pa F 8
F? -6-7 9
0
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
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
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
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 9.55 0.80kN m 00 0 TD 9.55 0.96kN 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
xt0knm 9 πd. (00) 0 m WP.96 0 m 6 6 x max 0 0 N m 6 τ max 5.0 0 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 6 6 0 0 N m 6 τ max 5.7 0 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
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
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 0.000 rad 8
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
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
s b u x max θ max [ θ ] GI P 0 0 0 G Pa 0 m b AB max 6 WP x 7 0 N m 0.75 0 mm 6 [ τ ] 0 0 Pa 6 6W mm d P 6 0.75 0 96 mm π π 5 7 I P 7 0 N m xmac.5 0 m.5 0 mm G[ θ ] 6 π 8 0 0 Pa 0. m 80 7 W d P.5 0 6 mm π π 5
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 0 0 6 70 0 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
7. 0 N d,9 0 m 9. mm 6 π 60 0 N / m warping c A x τ max WT max 5
x θ GIT WT ab, IT βb, h b α, β α, β mh/b.0..5.0.5.0.0 6.0 8.0 0.0 0.08 0.6 0.6 0.9 0.65 0.80.50.789.56. 0.0 0.90 0.9 0.57 0.6 0.790..789.56..00 0.90 0.858 0.796 0.766 0.75 0.75 0.7 0.7 0.7 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
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
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
T ABECDF ABCD ABCD da 57
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
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
m 0.kNm G80GPa 60
6
6
(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
(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
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
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
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 5.67 0 m 0.m 0. m 6 67
0 0 N 6m max 6 max 9.8 0 N/m 9.8Pa 5.67 0 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 5.6 0 m m 6 W m 0 0 N 6m 6 max. 0 N/m.Pa 6 080 0 m mm D B 68
T C C I 0 I 60 0 - m (60 0 m (0 0 m) + 60 0 m) + 60 0 m 0 0 m 0 0 m (70 0 m (70 0 m) m) 86.6 0 6 + m 0 B D BD B 0 0 N m 00 0 m 6. 0 N/m 6 86.6 0 m tmax.pa 0 0 N m 80 0 m 6 cmax 8.6 0 N/m 8.6Pa 6 86.6 0 m D.5 0 N m 80 0 m 6.7 0 N/m 6 86.6 0 m tmax.7pa.5 0 N m 00 0 m 6 cmax. 0 N/m.Pa 6 86.6 0 m.7 Pa D 8.6Pa B D 69
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
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
m-m m-m m-m m-m τ F S I d Q d I S 7
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
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
A F 50kN FQ τ Q max τ max I b. 0 F 6 S N/m 9 50 0 N 80 60 90 0 m 6 86.6 0 m 60 0 m.pa Q τ 9 50 0 N 0 60 70 0 m 6. 0 N/m 6 86.6 0 m 60 0 m.pa max W max [ ] max 75
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 6 0 0 Pa 5 0 m W bh (0.08m) h 6 6 h 6 5 0 m 0.08m 0.9m 9mm h00mm 76
FQ max ql 0kN/m m 0kN FQ max 0 0 N 6 τ max 0.9 0 N/m 0.9Pa < [ τ ] bh 0.08m 0.m t Pa c Pa F A-5 (A-0 I ( ( 88 0 0mm 0 0mm 0 mm mm + 0mm 0mm 5 mm ) + + 0mm 0mm 5 mm AB C C AB C F C 0.75 5 0 c F m 6 max 5 0 Pa I 88 0 0 m mm ) 77
F 9.7kN c 0.75F 95 0 m 6 c max 50 0 Pa I 88 0 0 m F 8.6kN AB B 0.5 95 0 m 6 t max 5 0 Pa I 88 0 0 m F.0kN B 0.5 5 0 m 6 c max 50 0 Pa I 88 0 0 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 6 0. 0 N/m 0.Pa max ql kn/m.5 m.8kn m 78
max.8 0 N m 6 max 9.0 0 N/m 9.0Pa < [ ] W 0.m 0.5 m 6 FQ max.5 0 N 6 τ max 0.5 0 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
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
max ql 8 l max ql 0 AB CD 8
FN da da 0 A A I I d A da 0 A A I da A I da 0 F xc 8
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
FQ F xo F Q (shear center) A A -9 ( -) 8
τ max max s B OA BC OC (ideal elastic-plastic materials) s s bh W s s 6 (ield s bending moment) 85
(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
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
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 6 9065mm 90.6 0 m 88
89 6 m 0 765.6 76565mm 50mm)] (75mm [00mm 50mm 50mm)] (75mm [00mm )] ( [ )] ( [ δ δ δ h a h a S m 77.5kN m 7788N m 0 765.6) Pa(90.6 0 0 6 6 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 0 66. mm 0 66. 96.mm] 50mm) [(00mm mm 0 0.9 ] ) [( mm 0 0.9 50mm)] (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 0 66. 6 6 s W.7 m 59.kN m 77.5kN S u T l 6m F T F
6 S 90.6 0 m, S u s ( S + S ) [ ] [ ]( S + S ) n n 6 765.6 0 m [ ] 60 0 6 Pa(90.6 + 765.6) 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
( 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
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
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
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
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 x0 0. 577l l wmax w C b l b 95
w w max C Fbl Fbl 0.06 9 EI EI Fbl Fbl 0.065 6EI 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
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 + + 96EI 6EI EI 6EI A 97
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 8 6 + + 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
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,, 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
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 500 600 l l w 00 00 l l w 500 750 l l w 5000 0000 [ θ ] 0.005rad 0.00rad F 0kN F 0kN F kn F kn 70P Pa E 5 Pa w l 00
F A B 8 kn, F 6kN F Q max 8kN, max 6.kN m W max [ ] 6. 0 N m 6 70 0 Pa 67 0 6 m 67cm W cm 6. 0 N m 6 75 0 N/m 6 56 0 m max max W 75Pa % I 780.cm, h 00mm, b 7 mm, d 7mm, δ mm 0
0 d I S F max Q max max τ ] [ 57.Pa N/m 0 57. m 0 7 m 0 780 m 0 ] ) (00 7 ) (00 [7 N 0 8 6 8 9 τ +.9mm m 0.9 m 0 780 Pa 0 0. 8 m N 0.77 8 ) ( 8 6 5 6 max i i i i EI b l F b w w mmm E I
(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
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
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
x w EI F 06
E.00 5 Pa mmmm 0 - dmm D600mm E00GPa c max a-a D t max 0Pa 07
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
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
AB mm EI E 5 I cm 0
w E00GPa AB I CD I AB w c
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
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
α + o α x + +90 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
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 0 0 0 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
0-0 x -Pa 60Pa x -0Pa τ τ 0 o o 0 o 0 o 0 6 6 0 0 N / m + 60 0 N / m 6 6 0 0 N / m + 60 0 N / m + cos60 6 o ( 0 0 N / m )sin 60 7.Pa 6 6 0 0 N / m + 60 0 N / m o sin 60 6 o + ( 0 0 N / m )cos60 58.97Pa 6 6 0 0 N / m + 60 0 N / m 6 6 0 0 N / m + 60 0 N / m + cos( 80) 6 o ( 0 0 N / m )sin( 80).Pa 6 6 0 0 N / m + 60 0 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
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
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
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 6 6 0 N / m ± (. Pa 7. ) 6 6 0 N / m 6 + ( 0 N / m ) 6 6 ( 0 / ) 6 0 / tan τ x N m N m α 6 6 6 0 N / m 6 0 N / m 0 x
0 0 5 067.5 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
, 0, 0 α τ 0 0 0 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
principal stress trajectories 5
AE AF EF abc max, min maximum shearing stressb τ max 6
0Pa, -60Pa τ 0 0 N / m ( 60 0 N / m ) 6 6 max 50Pa Pa 0Pa τ 6 60 0 N / m max 0Pa Eε E ε principal strain 7
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
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
E. 0 N / m ( ε + νε ) ν 0..Pa E. 0 N / m ( ε + νε) ν 0. 0.Pa ν ε E 6. 0 0.. 0 N / m (0 0. 60) 0 6 ( 60 + 0. 60) 0 6 6 ( + ) (. 0 N / m 0. 0 N / m 5-6 0mm 5-8(a) F6kN 0. 6 ) 5-8 5-6 F F 6 0 N 60Pa A 0.0 0.0m 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
5-7 d80mm T 5-9(a) -5 60 T E.00 5 Pa 0. 5-9 5-7 x x 6T τ (a) W πd P -5 5 5- -5 5-5-0 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
V θ ε + ε + ε V 0 (5-) (5-0) ν θ ( + + ) (5-5) E (5-5) 0 - + + 0 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
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-
(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
5-? 5-? 5-5- 5-5 ( 0 0 0) 0 0 + 5-6? 5
5-5- 5-0.0.5m 0 a-a AB 5-5- 5-5- AB 5-5 A 0.9Pa F 5-5-5 6
5-6 A ( A ) 5-7 5-6 5-7 5-8 A.5Pa () A x x ()A 5-8 5-9 cm 0.950.95cm ( ) F6kN E7.00 Pa0. 5-0 K 5 -.60-5 F E.0 5 Pa0.8 5-0 5-7
5- D0mm A 70.0 - E.0 5 Pa0.8, F 5-5.00 - E.00 5 Pa,0. T 5- E.00 5 Pa0. 5-8
9
max max (6-) ( ) 7 theor of strength 5. (condition of failure) b b 0
(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.
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)
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
[ 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 ( ) ( )
max (6-9) τ max (6-0) (-0) 6-- 6-6- 0 - (6-5) [( τ ) [ ] τ + ( + τ ) + ( τ τ ) ] (6-0) [ ] [ τ ] 0.577[ ] 0.5 6-6- /(+) (0.50.6) (0.8.0) -0 70Pa,00Pa 6- D 0 000mm 0mm 6-a 70Pa p.6pa 5
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
80Pa ((a)) t 80Pa 90Pa 0 r [( ) + ( ) + ( ) ] 55. 6Pa 6-6-5a 70Pa00Pa 6-5 6- bc CD max 8kNm (-) W [ ] max 8 0 N m 6 70 0 Pa 9 0 6 m 9cm 8a W508.5cm I7.cm AC DB F Qmax 00kN (-) 8a I/S.6cm d0.85cm 7
F Q max 00 0 N τ max 95.6Pa < [ τ ] I.6 0.85 0 m d S 8a C D f C (d) a ( ) a e a ((d)) 8 0 N m.6 0 8 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 0.85 0 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
8 0 N m.5 0 m 0.0Pa 8 075.5 0 m 6 00 0 N m 67. 0 m τ 56.5Pa 8 075.5 0 m 0.95 0 m r 57.7 Pa 7. r a () () 6- Pa A -0Pa -0Pa 0 - -6 B -0Pa 0 - -9-00Pa -7 - -9 AB AB 6-6 A B A A B B A A 6-6 6- B B 9
6- max max 6-?? 6-?? 6-6-5 ( a) (b)?? 6-5 50
6- t 60Pa r -5Pa 0Pa 6-6- 6- -650Pa -700Pa -900Pa 50Pa 6- A (b) x.880-7.70 - E. 0 5 Pa v0.70pa A 6-6- 6- ppa q60kn/m Dm 0mm 0Pa 6-5 ab? 6-5 6-6 6-6 A-A? 5
.5Pa Pa A-A.Pa 6-7 (a) (b) 70Pa,00Pa a ( a a ) 6-7 6-8 6-8 b / bc / ϕ? 5
5
combined deformation) oblique bending 5
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
cosϕ I I sinϕ I I F F A F F A cosϕ sinϕ + + I I 0 0 0 0 0 cosϕ sinϕ + I I cosϕ I 0 0 sinϕ + I 0 0 0 0 I tanα tanϕ I 0 I I α ϕ F α ϕ F 0 56
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
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
q l 8 t max + + W W t max o 0 N m sin 5 m 8 8 0 m 0 m 6 6 7.68 0 N m 7.68Pa [ ] hb 6 q l 8 bh 6 0 N + 8 0 6 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 + 6 6 8.8 0 m 0.9 0 m 6 07.7 0 N m 07.7Pa 07. W W B 7 Pa 59
0 0 I I I 0 0 I 0 tanα 0 0 I I 8 0 N m 50.5 0 m tanα 7. 8 8 5 0 N m m 80 0 m o α 8. 8 0 N m 50.5 0 m l tanα. 5 8 5 0 N m m 80 0 m o α 86.0 x x ω ω ω ql 8EI 8 0 5 5 0 N m m 6 0 N m 50.5 0 0.995 8 m 0 m 60
6 m m m N m N EI Fl 8 6 5 0 9.5 0 80 0 0 0 ω mm m 57 9. 0 9.57 + ω ω ω F A F N x ( ) x I x A ( ) x N I A F +
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
7-0 7- W max.8 0 m 6 [ ] 5 0 60 0 N m N m 8cm W cm A cm t max FN max 0 0 N 5 0 N m + + 6 A W.0 0 m 09 0 m 70. 0 6 N m 70.Pa [ ] W cm A cm 0 0 N 5 0 N m 6 + 60.9 0 N m 6 6. 0 m 5 0 m max 60.9Pa eccentric compression or tension 6
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
F FF F + + A I I I, I Ai F Ai F F F + + A i i 0 0 0 0 F A + + F 0 F 0 i i F 0 F 0 + + i i 0 0 0 0 0 0 a a 0 0 0 0 0 0 i i F F D D 65
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 0.05 0.5KN m F 0.05 0.5KN m A 66
t max F N + A W + W 0 0 N 0.5 0 N m + 0. 0.05m 0.05m 0. m 6 6 0 N m Pa + 0.5 0 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
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/ 7-7 7-8 A d α Y, α Z 7-8 7-7 68
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 7-9 69
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 [ ] + + + 0.75 x x x x p x r W W W W W 7-8 ( ) ( ) ( ) ( )
( ) 7-8 d8cm Dm 5kN 7-0(a) A C 60Pa 7-0 7-8 (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
B C BC B B (7-7) r W B 59. 0 6 + N Bx m π 8 0 59.Pa 6 m.58 ( KN m) +.5 ( KN m) (7-) (7-5) 7
7 F ( ) 7-??? ()? ( ) ( )??? 7-5 + + p r W T W A F? 7-6
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
AB lm hb0.0.m qkn/m (a) (b) b.0 kg/m 7-5 7-6 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
7-8 7-9 A A 5000-6 F E.00 Pa Amm W Z 0 mm F AB A 000-6, B -6000-6 E00GPa F a? 7-0 7-7- q l800mmd0mmqkn/m70pa F F T F 0.7kNF 50kNT.kNm70Pad50mml900mm 76
7-7- d00mm 70Pa () ABCD () h00mmb0mm 80Pa 7-5 7-6 77
78
8-(a) F F 8-(b) 8-(c) (stable) 8-8-(d) (unstable) (lost stabilit) F (critical force) F cr FF cr FF cr 907 58m 909 60 m 8- (a) (b) (c) 8-79
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
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 0 8-8- 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
F cr π EI (8-9) ( 0.7l) π EI F cr (8-9) µ ( l) l µ (effective length) (length factor) 0.5 0.7 (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 8-6 8-7 8
(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
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
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 0 9 0.7 p (8-9) 85
F cr π π EI 0 ( µ l) ( m) 799N 79.9kN 0 N m 0 90 0 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 00 0. 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 0.5 000mm λ i 5.mm 65.8 x Q5 p 00 u s 60 u p (8-6) cr a bλ 0 Pa.Pa 65.8 0. Pa A.cm 6 Fcr cr A 0. 0 N m. 0 m 99N 9. kn 86
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 5.05.5.8. (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
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 + 80 000 λ > 75 ϕ (8-5b) λ TCTCTB7 TB5 λ 9 ϕ (8-6a) λ + 65 800 λ 9 ϕ (8-6b) λ (8-5)(8-6) TC7 TC5 TC TC TB0 TB7 TB5 (Pa) 8- ϕ ϕ 88
8- ϕ l/i Q5 6n a b a b 0.000.000.000.000.000 0 0.995 0.99 0.99 0.989 0.97 0 0.98 0.970 0.97 0.956 0.9 0 0.96 0.96 0.950 0.9 0.8 0 0.9 0.899 0.90 0.86 0.69 50 0.96 0.856 0.88 0.80 0.57 60 0.88 0.807 0.85 0.7 0. 70 0.89 0.75 0.75 0.656 0. 80 0.78 0.688 0.66 0.575 0.6 90 0.7 0.6 0.570 0.99 0.0 00 0.68 0.555 0.87 0. 0.6 0 0.56 0.9 0.6 0.7 0 0.9 0.7 0.58 0. 0 0. 0.87 0.0 0.8 0 0.8 0.5 0.7 0.9 50 0.9 0.0 0.9 0. 60 0.0 0.76 0. 0.97 70 0.70 0.9 0.89 0.76 80 0. 0.5 0.69 0.59 90 0.0 0.0 0.5 0. 00 0.99 0.86 0.8 0. 8-5 8-0 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
F cr π π EI ( µ l). 0 079N 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 + 85.5h 566.8 0 h h8.cm () i i I A 9.5cm 5.5cm 6.cm 90
8- λ ϕ 0.08 µl i. 700cm 6.cm 9. ϕ [ ] 0.08 70Pa 5.Pa F 70 0 N 7 5.7 0 Pa 5.7Pa A 5.5 0 m ϕ 5% () F A 70 0 N 7 7.05 0 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 8-8-5 5.6 0 N m.65 0 N + m.5 0 m max NAB max 6 0 0 Q5 5Pa.5 [ ] 6Pa n 6Pa max 5% CD CD F N Fsin0F5kN 9
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
8- F cr E E E cr p 9
8-?? ( )? 8- d? 8-?? 9
8-6 8-7 d F 8-7 95
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 8-5050mm E700 Pa F 8-8- 96
8-5 am d5mm a 5 70Pa F F F? 8-6 TC7 5050mm L.0m Pa 8-5 8-7 8-7 Q5 AB d70mmbc a70mmab BC l.5m n st.5e.0 5 Pa 8-8 AB TC7 d00mmab Pa q 8-8 8-9 75756( ) 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
8-9 8-0 8- 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 8-8- 98
99
00
(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
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
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
(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
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
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
9-(c) k d d d dmax d max 9-9-5 6 k0.6kn/mm WkN h50mm C 60PaE. 0 5 Pa st W 9-5 9- [ ] C I 0cm W cm Cst Wl 8EI 0.7 0 0 N m 8. 0 N m 0 0 m 0.7mm W F RB 0.5W k 0.5 kn 6. 0.6kN mm Bst 5 mm C st Cst + Bst 0.7mm + 6.5mm. 6mm (9-) 8 m k d h 50mm + + + +.98.6mm st C C Wl 0 N m.5 max 0 N m max.5 0 N 6 st max 0.6 0 Pa 0. 6Pa 6 W 0 m d max kd st max.98 0.6kPa 59. Pa dmax < 07
k d k d st k d st st (impact toughness) a k 9-6 9-6(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
9-7b (stress spectrum) 9-7 9-8(a) FF 9-8(b) i i ( 9-8c 0 i max i 0 i min i (stress ccle) 9-8(d) 9-8 09
9-9 9-9 9-7b 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
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) 9-9- -N -N ( - ) -N N 0 N 0 N 0 0 7 N 0 0 8
- 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 900 860 0 0 0 0 0 0 9-9- F min 0kN F max 00kN 0 6 () I 90mm mm 70mm 75 mm 7.8 0 6 m
F min 0kN I 9-9- 0 0 N 750 0 m 0 m min max min 6 7.8 0 F max 00kN I 00 0 N 750 0 m 0 6 7.8 0 m max max max a 6.9 6. 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
?? l?? 0 9- W v 9-?? d l Dd 5 h k d + h + st h st?? F
9-5 9-7 () F () FF 0 +F H sint ( F 0 F H ) () 0F () F (5) F F 5
8 a0m/s lm A60mm 9- W 0kN b 0kN a.5m/s d0mm 60Pa ( ) 9-9- 9- 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 9-5 9-6 6
W500kN v0.5m/s C W.00 7 mm I5.00 9 mm E.00 5 Pa max min r n-n max min r 9-7 9-8 9-9 a 0mm0mm.5m 50kN F max 50KnF min 0 0 6 7
8
9
energ method V c W FN l Vε FN l EA T T ϕ OAB W Tϕ x Tϕ x l GI p 0
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
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 + 0 0 0 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
EI l EI Fl A 6 + θ F + + + 6 6 96 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 6 6 6 + + + θ + + + 6 6 96 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 6 96 8 θ ε ε
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
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
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 6 0-8 0- B BV l EA 5 5 F + 6 6 l EA 5 6 5 F 6 l500mm BV. mm F 0- ABCF C EI GI P C CV Vε F 0-9 0- BC (x) AB (x) x (x) V ε ( x) l ( x) l x ( x) + dx a dx dx EI + 0 0 EI 0 GI p CV Vε F a 0 EI ( x) ( x) ( x) dx + dx + F EI EI a l l 0 0 0 ( 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
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 0-0-0 A A EI A A A 0 A Vε θ A 0 V ε ( x) l dx 0 EI Vε θ l A EI 0 0 0 ( 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 +, + 0 0 x l l l x l 0 0 x + 0 x + dx l 0 dx 7
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
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 0 5 0 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 5.5 + EA ( F )( ) 6 + ( F )( x ) dx + ( F F )( x )dx C x x 6 6 0 EA EI EI 9
6.5F EA 9.0 0 F + EA F + EI m 9.0mm 0
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
EAEIGI P C EI C C A
A
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
( ) 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 7.5 0 mm, c c c c 50mm 50mm + 80mm + 55mm A 80mm 50mm 9.0 0 mm, 80mm + 50mm 0mm A 50mm 50mm.5 0 mm, 50mm 5mm 0 c c (A-5) c 7.5 0 0mm mm ( 55mm) + 9.0 0 mm ( 0mm) +.5 0 mm ( 5) 7.5 0 mm + 9.0 0 mm +.5 0 mm mm c 0 A- J dm, J V dm V 6
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
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
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 8 8 8 π π π π d d I d I mh J J +
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
A-9 A-5 B A-9b 0a A 8.8 0 mm, I 8 0 mm c I 780. 0 mm, 0.mm c 0 I I c 780.0 mm 560.80 mm (A-0) a I I c + + 0 A 00mm 8 0 mm + + 0.mm 8.8 0 mm 089. 0 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
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)
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
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 + 0 00mm 0mm + 0 + 0mm 50.8mm 00mm 0mm 0 0 mm (A-6) 0 0 mm tan α 0 mm ( 69 90) 80 9 0 60mm 0mm 0mm 00mm 00mm 0mm 0.5 tan 0 0 0 7.5-80-5.7 0-76. 76. 0 0 0 (A-7) I I max min I + 0 ( 69 + 90) ( 69 90) 50 0 mm I 0 ( 69 + 90) ( 69 90) 0 0 mm 0 mm 8 0 mm 8 0 mm 8 0 mm 8 + 0 + 0 8 0 mm 8 0 mm 0 0 0 0 0 0 8 8
A- (a)(b) A- A- (a)(b) S A- A- (a)(b) A- A- a A- 5
(a) a (b) 0 A-5 70708 (a)(b) A-5 A-6 A-6 A-7 A-7 A-8 A-8 A-9 6
A-9 7
8
9
50
5
5
5
5
55
- (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 0. 6-5 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
-7 F 0kN, F 90. kn - τ.pa, τ 0, τ 7.Pa, γ 0.59 0 rad - τ max 6Pa - 6.67% - ( ) 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
- 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 -9 55.8Pa 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
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 0. 56 ( a) ( a) ( a) o 55Pa, 5Pa, α 55.8 o 88.Pa, 8.Pa, α 5.8 o 0Pa, 0, α 5 0 0 0 5- A o 5.8Pa, 0.0Pa, α 0. 86 B 5-5 F P. 798kN o 0.08Pa,.59Pa, α 0 8. 66 59
5-7 a ) F, F, α 0 ( 0 ( b ) F, F, α 0 5-8 ( ) x.8pa.5pa τ x. 6Pa 0 ( ) o 7Pa 0 α 0 6. 90 5-9, 66.5Pa 0 5-0 F P. kn 5- F P. 8kN 5- T 5. 8kN m 5- vv 0.00Pa, vd 0. 05Pa 6- r 95Pa, r 86. 75Pa 6- r 50Pa, r 9Pa 6- r 8Pa 6-56. 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 6.6 60
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 8-.9.05.6 8- F cr 50kN 6
8-5 F.9kN, F 8. 5kN 8-6 F 89. 9kN 8-7 F kn 8-8 q 60kN / m 8-9 70Pa 8-0 Fcr Fcr 8.5kN, F. 8 8-75Pa, CD 0Pa 8- F kn max 9-0Pa, 7. Pa d max d 7 9-5.Pa, 60. Pa d d 9- max 7.8Pa l 9- l ω (W + W ) EA 9-5 h.9m, h 0. 0m 0 9-6 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 9-0 7.7Pa 6
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 90 0. ),(. ) ( 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 0 5 86., 7.5, 5 ) ( cm I cm I a α
o 0., I 7.cm, I 08 ( b) α cm A 99 000 00 997 5 999 6 990 7 998 8Gere J..Timoshenko S.P.echanics of materialssecond SI EditionN.Y.Van Nostrand Reinhold98 6