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73

74

( u w + bw) d = Ɣ t tw dɣ u = N u + N u + N 3 u 3 + N 4 u 4 + [K ] {u = {F 75

u δu L σ (L) σ dx σ + dσ x δu b δu + d(δu) ALW W = L b δu dv + Aσ (L)δu(L) δu = (= ) W = A L b δu dx + Aσ (L)δu(L) Aσ ()δu() 76

W = A L b δu dx + Aσ (L)δu(L) Aσ ()δu() = A L b δu dx + A L d(σ δu) dx dx = A L b δu dx + A L dσ dx δu dx + A L σ d(δu) dx dx = A L ( ) dσ L dx + b δu dx + A σ d(δu) dx dx = A = A L L σ d(δu) dx σ δε dx dx δu A L σ δε dx = A L b δu dx + Aσ (L)δu(L) 77

W = (b x δu + b y δv + b z δw) d + (t x δu + t y δv + t z δw) dɣ Ɣ t = (b x δu + b y δv + b z δw) d + (t x δu + t y δv + t z δw) dɣ Ɣ t +Ɣ u = (b x δu + b y δv + b z δw) d + {(σ xx n x + σ yx n y + σ zx n z )δu + (σ xy n x + σ yy n y + σ zy n z )δv Ɣ + (σ xz n x + σ yz n y + σ zz n z )δw dɣ = (b x δu + b y δv + b z δw) d + {(σ xx δu + σ xy δv + σ xz δw)n x + (σ yx δu + σ yy δv + σ yz δw)n y Ɣ + (σ zx δu + σ zy δv + σ zz δw)n z dɣ 78

W = = (b x δu + b y δv + b z δw) d { (σxx δu + σ xy δv + σ xz δw) + + (σ yxδu + σ yy δv + σ yz δw) + (σ zxδu + σ zy δv + σ zz δw) d z = (b x δu + b y δv + b z δw) d {( σxx + + σ yx + σ ) ( zx σxy δu + z + σ yy + σ ) zy δv z ( σxz + + σ yz + σ ) zz δw d z { (δu) (δv) (δw) + σ xx + σ yy + σ zz z ( (δu) + σ xy + (δv) ) ( (δw) + σ yz + (δv) ) ( (δw) + σ zx z ε x γ xy + (δu) z ) d 79

W = {( σxx = + σ yx + σ ) ( zx σxy + b x δu + z + σ yy + σ ) zy + b y δv z ( σxz + + σ yz + σ ) zz + b z δw d z + (σ x δε x + σ y δε y + σ z δε z + τ xy δγ xy + τ xy δγ xy + τ xy δγ xy ) d = (σ x δε x + σ y δε y + σ z δε z + τ xy δγ xy + τ xy δγ xy + τ xy δγ xy ) d (σ x δε x + σ y δε y + σ z δε z + τ xy δγ xy + τ xy δγ xy + τ xy δγ xy ) d = (b x δu + b y δv + b z δw) d + (t x δu + t y δv + t z δw) dɣ Ɣ t δε {σ d = δu {b d + δu {t dɣ Ɣ t 8

(σ x δε x + σ y δε y + σ z δε z + τ xy δγ xy + τ xy δγ xy + τ xy δγ xy ) d = (b x δu + b y δv + b z δw) d + (t x δu + t y δv + t z δw) dɣ Ɣ t δε {σ d = δu {b d + Ɣ t δu {t dɣ 8

. { U {σ = ε U = (σ xε x + σ y ε y + σ z ε z + τ xy γ xy + τ yz γ yz + τ zx γ zx ) = ν E ( ν)(+ν) (ε x + ε y + ε z ) + E (+ν) ( ε x + ε y + ε z + γ xy + γ yz + γ zx U εxσx U ν E = ε x ( ν)( + ν) (ε x + ε y + ε z ) + E + ν ε x = σ x VTVB { {t = V { T, {b = V B t b ) 8

{ U δε d = δu { V B ε u d + δu { V T Ɣ t u dɣ U δu d = δv B d Ɣ t δv T dɣ = δ ( U d + U d + ) V B d + V T dɣ = Ɣ t V B d + V T dɣ Ɣ t VBVT V B = u {b = b i u i, V T = u {t = t i u i Π = U d = U d u {b d u {t dɣ Ɣ t b i u i d t i u i dɣ Ɣ t 83

ΩΓu Π δπ = = (u + δu) (u) = δ + δ = δ > (u + δu) >(u) = U = ɛ {σ = ε [D] {ε ε [D] {ε d u {b d u {t dɣ Ɣ t δπ = δ = δε [D] {ε d δu {b d δu {t dɣ Ɣ t = δε {σ d δu {b d δu {t dɣ = Ɣ t 84

w j Γu (σ ij,i + b j )u j d = t j u j dɣ σ ij u j,i d = t j u j dɣ σ ij (u j,i Ɣ Ɣ + u j,i ) d = t j u j dɣ (σ ij u j,i Ɣ + σ jiu j,i ) d = t j u j dɣ (σ ij u j,i Ɣ + σ ijui, j ) d = t j u j dɣ σ ij εij d = t i ui Ɣ u +Ɣ t Ɣ dɣ σ ij εij d t 85

t i ui dɣ σ ij εij d + Ɣ t b j u j d = ti (σ ij,i + b j )u j Ɣ d (t j t j ) dɣ = t σ ij εij d = b i ui Ɣ d + t i ui dɣ t 86

= σ ij ε ij d b i u i d t i u i dɣ λ i (u i ū i ) dɣ Ɣ t Ɣ u δ = σ ij δε ij d b i δu i d t i δu i dɣ δλ i (u i ū i ) dɣ λ i δu i dɣ Ɣ t Ɣ u Ɣ u Γu δui = σ ij δε ij d t i δu i dɣ σ ij,i δu j d Ɣ u +Ɣ t δ = (σ ij,i + b j )δu j d b i δu i d + (t i t i )δu i dɣ Ɣ t (λ i t i )δu i dɣ δλ i (u i ū i ) dɣ Ɣ u Ɣ u 87

δ = δπ = σ ij,i + b j = (σ ij,i + b j )δu j d + Ɣ u (λ i t i )δu i dɣ Ɣ t (t i t i )δu i dɣ Ɣ u δλ i (u i ū i ) dɣ t i = t i λ i = t i, u i =ū i Ɣ t Ɣ u Γu δπ σ ij δε ij d = b j δu j d + t i δu i dɣ Ɣ t + (λ i t i )δu i dɣ + δλ i (u i ū i ) dɣ Ɣ u Ɣ u 88

L σ (L) x A dσ dx + b =, u() =, σ (L) = F/A L w ( ) dσ dx + b Adx = [ ] L L wσ A dw dx σ Adx + L b Adx = Hooke L dw dx E du [ ] L dx Adx = wσ A + L b Adx = 89

L dw dx E du [ ] L dx Adx = wσ A + L b Adx = 9

Δ4 e e dw dx E du dx Adx = w(l)σ (L)A w()σ ()A + e e b Adx = u u u u3 u4 u5 u u Δ Δ 3Δ 4Δ = L x 9

u 5 u(x) N a (x)u a a= N a (x) u a N a (x) x a x a x a+ x 9

e u(x) e u = N e (x)u e + N e+ (x)u e+ e N e (x)n e+ (x) N e (x) N e+ (x) N e (x) = x xe x e+ x e = xe+ x x e Δ x e+ N e+ (x) = x e u(x) x xe x e+ x e = x xe u N e (x) u e (x) u e+ (x) N e+ (x) x e Δ x e+ x 93

dw e e dx E du dx Adx = w(l)σ (L)A w()σ ()A + e a w = w(x) = N a (x)w a aa e e ee+ e b Adx = Δe w a e x e + x e dn a dx E du dx Adx = wa N a (L)σ (L)A w a N a ()σ ()A + w a e x e + x e N a b Adx = e x e + x e dn a dx E du dx dx = N a (L)σ (L) N a ()σ () + e x e + x e N a b dx = 94

e du dx = dne dx ue + dne+ u e+ = dne dx dx dn e+ dx { u e u e+ w = N e w e x e + x e dn e dx E du dx dx = x e + x e dn e dx E [ dn e dx [ x e + dn e = x e dx E dne dx dx [ k e,e k e,e ] { u e u e+ dn e+ dx ]{ u e x e + x e u e+ dn e dx dx E dne+ dx dx ]{ u e u e+ dn e dx =, dn e+ dx = k e,e = E, ke,e = E 95

w = N e+ w e+ x e + x e dw dx E du [ ] { dx dx = k e+,e k e+,e u e u e+ k e+,e = E, ke+,e = E x e + [ ] { w = N e w e N e b e b dx = x e 3 6 b e+ = S e,e w = N e+ w e+ xe+ x e N e b dx = [ 6 3 ] { b e b e+ = S e+,e 96

w = N w w = N w w = N w x = N () = ()N σ 5 (L)=(L) σ x = L Δ 4 w = N 4 w 4 w = N 5 w 5 w = N 5 w 5 x = L N 5 = (L) σ w = N w, N 3 w 3, N 4 w 4 w e (e=, 3, 4) [ ] { k e,e k e,e u e u e + [ k e,e k e,e ] { u e u e+ (S e,e + S e,e ) = [ ] k e,e k e,e + k e,e k e,e u e u e u e+ = Se,e + S e,e 97

x = w = N w [ ] { k, k, u u = S, σ () x = w = N 5 w 5 [ ] { k 5,4 k 5,4 u 4 u 5 = S 5,4 + σ (L) E E E E E E E E E E E E E u u u 3 u 4 u 5 = S, σ () S, + S, S 3, + S 3,3 S 4,3 + S 4,4 S 5,4 + σ (L) 98

ū ū () E E E E E E E E E E u u 3 u 4 u 5 = S, + S, + E ū S 3, + S 3,3 S 4,3 + S 4,4 S 5,4 + σ (L) [ E E ] [K ] {u = {F ū u u 3 u 4 u 5 = S, σ () 99

ū = u E u 3 = u 4 σ (L) u 5 u = σ (L) E, u 3 = σ (L) E, u 4 = 3 σ (L) E, u 5 = 4 σ (L) E x= ū σ () = [ E E ] u u 3 u 4 u 5 = σ (L)

L σ δε Adx = L bδu Adx + [ ] L Aσ (x)δu(x) AE x = ū Aσ () u u 3 = u 4 u 5 Aσ (L)

(i) AE u u 3 u 4 u 5 = AEū Aσ (L) (a) (ii) u Aσ () = AE u u 3 u 4 u 5 (b) (a)(b)x= σ () = σ (L)

3

(σ x δε x + σ y δε y + τ xy δγ xy ) d = (b x δu + b y δv) d + Ɣ t (t x δu + t y δv) dɣ σ x δε x + σ y δε y + τ xy δγ xy = δε x δε y δγ xy [D] [D] = ε x ε y γ xy E ( + ν)( ν) = ν ν ν ν { u v ν ε x ε y γ xy 4

e δε x δε y δγ xy [D] e = δu δv e e ε x ε y γ xy { bx b y d d + k Ɣ k t δu δv { tx t y dɣ { { u = N u v v + N { u v + N 3 { u 3 v 3 + N a a 5

3 3 3 6

3 3 3 7

ε x ε y γ xy = = 3 j= N { N u v N N + N j { N j u j N j N j N = B, N = C N = B, N = C N 3 = B 3, N 3 = C 3 v j = N N N N { u v 3 [B { j] u j j= v j + N 3 N 3 N 3 N 3 { u 3 v 3 8

e δε x δε y δγ xy [D] e = δu δv e e ε x 3 δε x δε y δγ xy = δu i δ v i [ B i] T ε y = [ B j ] { u j v γ j + xy j= ε x ε y γ xy { bx b y d d + k Ɣ k t δu δv { tx t y dɣ δε x δε y δγ xy [D] e ε x ε y γ xy d = δui δv i = δu i δv i 3 ( [ B i ][ D ][ B j] ){ u j d e v j j= 3 [ k ij ] { u j j= v j

k e j k e j k e j i i i k e j e e i

e 3 k e j i e e

j L i i i j ΔS t (δut x + δv t y ) dɣ = N i δu i Ɣ t Ɣ t N i δv i { tx t y k dɣ x = x i + + ξ (x j x i ), ξ t = t i + + ξ (t j t i ), ξ ξ (dx ) ( ) dy dɣ = + dξ = L dξ dξ dξ

N i (x, y) i j i j ξ Ɣ t (δut x + δv t y ) dɣ = δu i δv i N i (x, y) = ξ = δu i δv i ( ξ ( ξ L 3 t i x + L 6 t j x L 3 t i y + L 6 t j y ) t i x + ξ 4 t j x ) t i y + ξ 4 t j y L dξ Ɣk t (δut x + δv t y ) dɣ = δu i δv i ( ) L 3 tx i + L 6 t x j ( ) L 3 ty i + L 6 t y j = δui δv i { F i x F i y j 3

e δε x δε y δγ xy [D] e ε x ε y γ xy d = Ɣ t δu δv { tx t y dɣ δε x δε y δγ xy [D] e e Ɣk t ε x ε y γ xy (δut x + δv t y ) dɣ = δu i δv i d = δui δv i { F i x F i y n [ k ij ] { u j v j j= δu δv n [ k ij ] { u j j= v j { F i x F i y = K {u ={F

x 3 η x 4 x 4 (, ) x 3 (, ) y x ξ x x η x (, ) x (, ) u 4 u 3 ξ u u 5

ξ η = = ξ + η + ξ η ξ η = ξ η ξ η d = ξ η dξdη ξ η = J = J ξ η [ k a,b] [ j = B a ] T[ ][ D B j ] d b 6

5 ζ 8 6 7 4 η ξ 3 N = 8 ( ξ)( η)( ζ ), N = ( + ξ)( η)( ζ ) 8 N 3 = 8 ( + ξ)( + η)( ζ ), N 4 = ( ξ)( + η)( ζ ) 8 N = 8 ( ξ)( η)( + ζ ), N = ( + ξ)( η)( + ζ ) 8 N 3 = 8 ( + ξ)( + η)( + ζ ), N 4 = ( ξ)( + η)( + ζ ) 8 7

ξ η ζ = ξ η ζ z ξ η ζ z ξ z η z ζ = J z ξ η ζ = J z ( d = ξ ) dξdηdζ = det J dξdηdζ ξ η 8

f (ξ) dξ = n i= f (ξ i )w i ξ i,w i (i =,,, n) ±ξ i n w i..577 35 69 89 66..774 596 669 4 483..86 36 3 594 53.339 98 43 584 856 3.555 555 555 555 556.888 888 888 888 889 4.347 854 845 37 454.65 45 54 86 546 x = b a ξ + b + a b a f (x) dx = ( b a f ξ + b + a ) b a dξ 9

f (ξ, η) dξdη = n i= n j= f (ξ i, η j )w i w j f (ξ, η, ζ ) dξdηdζ = n i= n j= n k= f (ξ i, η j, ζ k )w i w j w k

n N j ε { x n ε y = N j u j n γ v xy j= N j N j j = [B { j] u j v j j= σ x ε x [ ] n [ ] { σ y = [D] ε y = D B j u j τ xy γ v j xy j= [D] = E ( + ν)( ν) ν ν ν ν ν

σ x σ y τ xy = [D] ε x ε y γ xy = [ D ] n [ j= B j ] { u j v j [B j ] η ξ y y y x x x

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