γ b =2 γ e In case there is no infiltration under the dam, the angle α is given by In case with infiltration under the dam, the angle a is given by

Transcription

1 O γ e γ b α H γ b γ e Sol A B In case thee is no infiltation unde the dam, the angle α is given b So that Fom whee In case with infiltation unde the dam, the angle a is given b So that Fom whee A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit utoial (continued) : Semi infinite plane unde point load he Ai stess function associated with this loading is : A(, ) cp sin Show that A(,) is bihamonic. Detemine the components of the stess tenso. Detemine the constant c as a function of angle α. Calculate the stesses when απ/. A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit

2 A(, ) cp sin A A A ( A) he stess function A(,) is well bihamonic A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit 3 A(, ) cp sin he stesses ae defined b the following epessions he balance of foces is witten to detemine the constant c : If απ/, the constant c is -/π and the stess is the given b : A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit 4

3 utoial (continued) : Stess field in a plate loaded in tension and pieced with a tin hole he Ai stess function associated with this loading is : f A b c d e (, ) ln cos M Show that A(, ) is bihamonic Detemine the stess,, τ ( a is the adius of the hole) A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit 5 infinite plate loaded in tension and pieced b a cicula hole of adius a A b c d e f (, ) ln cos A A A A A c f b e cos 3 A c 6 f b e 4 cos A f d e 4 cos A 8d 3 cos A 4d 4 cos A 6d cos M 4d A 4b cos ( A) 0 A(, ) is well bi-hamonic A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit 6 3

4 A A A τ A,,,,, A b c d e f (, ) ln cos A c f b e cos 3 M A c 6 f b e 4 cos A f d e sin A f d e 4 cos c d f b e 4 6 cos 4 τ d 6 f e sin 4 c 6 f b e cos 4 A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit 7 Bounda conditions at, ie fa fom the hole (, ) cos sin 0 (, ) P (, ) t P P sin cos sin ( cos ) cos ( cos ) at >> a τ sin cos sin M b e A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit 8 4

5 Bounda conditions at a ( a, ) τ ( a, ) 0 c d f 4 6 cos 4 a a a M τ d 6 f sin 4 a a d f 4 6 d a 4 a a c a et d 6 f 3 6 f a 4 a a A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit 9 4 c d f b e 4 6 cos 4 c 6 f b e cos 4 d 6 f τ e sin 4 b e c a d a 3 6 f a 4 4 a a a 4 3 cos 4 4 a a 3 cos 4 τ 4 a a 3 sin 4 ( a,0) 3 3 A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit 0 5

6 Ai stess function fo few loadings () A A A,,, A (, ) a Beam in taction A(, ) a Beam in shea 3 Beam subjected to A(, ) a bending moment A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit Ai stess function fo few loadings () A(, ) a b 3 A(, ) a b c d e A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit 6

7 Ai stess function fo few loadings (3) Aismmetic loading A, A, τ 0 A(, ) A( ) C A(, ) A( ) a ln c A(, ) A( ) a ln b ln c A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit 3 Ai stess function fo few loadings (4) A A A τ A,,,,, A(, ) c sin A(, ) c cos A(, ) a bsin A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit 4 7

8 Comple fomulation of the Ai stess function - Holomophic functions (o analtical functions) M(,) S z i z i S z z z z i (, ) Plan g g(, ) (, ) ( z, z) g g( z, z) g g ig he deivation ules ae g g ig S P P(, ) Q Q(, ) ( ), z,, ( ), z,, g P iq g S g g g,, z, z g i( g g ),, z, z g is holomophic if 0 z E g g g'( z) i A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit 5 * Popeties o analtic functions dg g g g P iq avec i dz P Q P Q P Q i i P Q Cauch conditions he eal o imagina pats of an analtic function, ae hamonic E P Q 0 Convesel, if P(, ) and Q(, ) veif the Cauch conditions g P iq is an analtic function - If g is an analtical function, its deivative and its integal ae also A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit 6 8

9 Eamples of analtic functions e inz, z n and ln z ae analtic functions. hei eal and imagina pats that ae hamonic, can be detemined. i f z e e e n i n inz in( i) n ( ) (cos sin ) df inz f in( i) df f in( i) df ine ine ne i dz dz dz n n he associated hamonic fonctions ae e cos n and e sin n Echanging n b n, it is seen that e n cosn and e n sinn ae also hamonics. It follows that sinhn sinn, coshn sinn, sinhn cosn and coshn cosn, obtained b linea combination of the peceding hamonic functions, ae also hamonic. he hpebolic sine and hpebolic cosine functions ae defined b e e e e sinh n cosh n n n n n A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit 7 n n ei n n i f ( z) z ( i) ( ) (cos n i sin n ) df n f n df f n df nz n( i) in( i) i dz dz dz n n he associated hamonic fonctions ae cos n and sin n i i f ( z) ln z ln( i) ln( e ) ln i df f df f i df i dz z i dz i dz he associated hamonic fonctions ae ln and A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit 8 9

10 Α Epessions of the Ai stess function b g 0 If P A then P 0 P is hamonic P Q f ( z) P iq is analtic with P Q Calculating the hamonic function Q(, ) Q Q dq d d P P Q dq d d p q ϕ ( z) f ( z) dz p iq is also analtic function P If p Α p q then p 0 χ( z) p iq is analtic function A p q p S bg bg bg bg bg bg Α e zϕ z χ z Α zϕ z χ z zϕ z χ z A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit 9 Epession of the stess function Stesses epessions fom the potential comple ϕ( z) and ( ) Α zϕ ( z) χ ( z) zϕ ( z ) χ ( z ) Α g, z g, ig, χ z S g, z g, ig, d i S d i c h c h i Α iα i Α iα i Α Α Α,,,,, z,, zz, zz ( ) ( ) ( ) ( ) i ϕ ' z ϕ ' z zϕ '' z χ '' z d i,,, A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit 0 Α Α g g g,, z, z g i( g g ),, z, z i Α, iα, dα, iα, i cα z Α zz Α,, h c,,, zzh ( ) ( ) ( ) ( ) i ϕ ' z ϕ ' z zϕ '' z χ '' z e bg dij c bgh ϕ ' z ϕ ' z 4 e ϕ ' z ( ( ) ( ) ) i zϕ '' z χ '' z 0

11 S Displacements epessions b g d i b g d i b g d i b g d i S λ µ ε λ µ λ µ λ µ λ µ ε λ µ λ µ λ µ p q A P 4 4 ( λ µ ) µ u p Α, α λ µ ( λ µ ) µ u q Α, β λ µ λ µ µ ε Α Α, λ µ b g λ µ µ ε Α Α, λ µ ( ) ( ) avec b g bg A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit S α c d βbg c d λ µ g, z g, ig, µ ( u iu ) ( p iq) ( Α, iα, ) λ µ S g, z g, ig, A dα, iα, i d i z λ µ Α λ µ µ ( u iu ) ϕ ( z) ϕ ( z) ϕ ( z) zϕ '( z) χ '( z) λ µ z λ µ ( U iu ) ( z) z ( z) ( z) µ κ ϕ ϕ ' χ ' d i λ 3µ κ 3 4 v fo plane stain λ µ with 3 ν κ fo stess plane ν developed in this cuvilinea sstem. e e i u iu e u iu S e cos. sin. e sin. cos. F HG u u I KJ F H G F HG I e e I F KJ H G I K J F P avec P : HG F u cos u sini HG u sin u coskj u u KJ P cos sin i ( ) µ ( u iu ) e κ ϕ ( z) zϕ '( z) χ '( z) h Ph t e, e, P sin cos ( ) A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit I KJ ( ) i i e i ( ( ) ( ) ) i e zϕ z χ z * Sstem coodinates change Because we will use of pola coodinates in the solution of man poblems in elasticit, the pevious govening equations will now be i '' ''

12 Let ( O;,, ) denote the Catesian coodinates sstem and ( M ; α, β, ) 3 3 a coodinate sstem associated with cuvilinea coodinates α, β. he comple numbe z i is associated with the, coodinates and the comple ζ α iβ is associated with the cuvilinea coodinates α, β. As ( α, β ) and ( α, β ), then we have : dz z f ( ζ ) and f '( ζ ) dζ A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit 3 dz z f ( ζ ) and f '( ζ ) dζ i We easil show that f '( ζ ) f '( ζ ) e, in othe wods that the agument of the comple numbe is equal to, the angle between the two coodinate sstems espectivel associated with, and α, β. dz ag '( ) ag ag ag f ζ dz dζ u, u, α dζ So we have '( ) '( ) i i f f e ζ ζ and f '( ζ ) f '( ζ ) e so that f '( ζ ) e f '( ζ ) i A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit 4

13 Summa of ke findings he esolution of a plane elasticit poblem comes down to the seach fo a stess function, called the Ai function A, which is bihamonic, that is to sa ( A)0. he epession of this stess function, fom the comple potentials ϕ and χ which ae analtical functions of the comple vaiable z, is given b : Α ezϕ ( z) χ ( z) zϕ ( z) χ ( z) zϕ ( z) χ ( z) he seach fo the Ai stess function is theefoe to find these comple potentials. he components of the stess tenso and the displacement vecto ae then detemined b the following elationships : A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit 5 In a Catesian coodinates sstem (,) In a cuvilinea coodinates sstem associated to vaaibles (α,β) κ 3 4 v fo plane stain with 3 ν κ fo stess plane ν A. Zeghloul Factue mechanics, damage and fatigue Plane elasticit 6 3