Kul Models for beam, plate and shell structures, MT

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1 Kul Models fo eam, plate and shell stuctues, MT- 4. Mapping (, φ, z) = cosφi + sinφ j + zk (in detail) the geneic fomula defines the cylindical φ z coodinate system. Use e e eφ= ( [ F])[ F] eφ α α ez ez, whee α {, φ, z} to deive the deivatives of the asis vectos.. Deive the expessions of linea stain components ε, ε φ, εφ and ε φφ of the pola coodinate system. Use the displacement epesentation u= ue + ue φ φ, asis vecto deivative expessions e / φ = e φ, eφ / φ = e and definition ε = + [ u ( u) ] c, whee = e + eφ. φ 3. Conside the simply suppoted xy plane eam of length shown. Mateial popeties E and G, cosssection popeties A, S =, I, and loading ae constants. Wite down the equiliium equations, constitutive equations, and ounday conditions accoding to the Benoulli eam model. Afte that, solve the equations fo the tansvese displacement. y x 4. Vitual wok expession of cetain eam model is given y δw = ( δw EIw + δwkw + δwf ) dx Mδw ( ) + Fδ w( ), whee wx ( ) is the tansvese displacement and w dw / dx. Detemine the undelying diffeential equation and ounday conditions. Assume that the solution is sought fom the set of functions having continuous deivatives of all odes and δw() = δ w () =. Paametes M and F ae given. The given functions EI, k and f have continuous deivatives of all odes ut they ae not constants. 5. Conside a cuved eam foming ¾ of a full cicle of adius in the hoizontal plane. Given toque of magnitude P is acting on the fee end as shown. Wite down the equiliium equations and ounday conditions fo the stess esultants and solve the equations fo Ns, () Qn () s, Q () s, T() s, Mn() s, and M() s. s x P z y

2 Mapping (, φ, z) = cosφi + sinφ j + zk detail) the geneic fomula e e eφ= ( [ F])[ F] eφ α α ez ez to deive the deivatives of the asis vectos. Solution defines the cylindical φ z coodinate system. Use (in, whee α {, φ, z} 3p In tems of the asis vectos of the Catesian system, expessions of the asis vectos of the cylindical φ z coodinate system ae e, /, cosφ sinφ i i eφ=, φ /, φ = sinφ cosφ j= [ F] j ez, z /, z k k in which T =. [ F] [ F] 3p Diect use of the definition gives (just take the deivatives on oth sides of the elationship aove and use invese of the same elationship to eplace the asis vectos of the Catesian system y the asis vectos of the φ z system) e cosφ sinφ e eφ= sinφ cosφ eφ=, ez ez e sinφ cosφ cosφ sinφ e e eφ eφ= cosφ sinφ sinφ cosφ eφ= eφ= e, φ ez ez ez e cosφ sinφ e eφ= sinφ cosφ eφ=. z ez ez eaning outcome/ geomety: poo -, satisfactoy -4, good 4-6

3 Deive the expessions of linea stain components ε, ε φ, εφ and ε φφ of the pola coodinate system. Use the displacement epesentation u= ue + ue φ φ, asis vecto deivative expessions e / φ = e φ, eφ / φ = e and definition ε = + Solution [ u ( u) ] c, whee = e + eφ. φ 3p Bute foce type method woks always: u= e ( ue ) + e ( ue ) e ( ue ) e ( ue ) φ φ + φ + φ φ φ φ φ, u= e ( ue ) + e ( ue ) e ( ue ) e ( ue ) φ φ + φ + φ φ φ φ φ, u= eu e + eue + eu e + eue + e u e + e ue + e u e + e ue,, φ, φ φ φ, φ, φ φ, φ φ φ, φ φ φ φ φ, φ u = eu, e+ euφ, eφ + eφ u, φe+ eφ ueφ + eφ uφ, φeφ eφ uφe, u= eeu, + eeu φ φ, + ee φ ( u, φ uφ) + ee φ φ ( u+ uφφ, ). p Conjugate is otained y changing the odes of the asis vectos in each tem, ( u) c = eeu, + ee φ ( u, φ uφ) + eeu φ φ, + ee φ φ ( u+ uφφ, ). p Definition of the small stain ε = [ u+ ( u) ] c gives ε = [ eeu, + eeu φ φ, + ee φ ( u, φ uφ) + ee φ φ ( u+ uφφ, ) + eeu, + ee φ ( u, φ uφ) + eeu φ φ, + ee φ φ ( u+ uφφ, )] ε = eeu, + ee φ ( uφ, + u, φ uφ) + ee φ ( u, φ uφ + uφ, ) + ee φ φ ( u+ uφ, φ). p Finally, collecting the components ε = u, and ε φ = ε φ = ( u,, ) φ + u u φ φ and ε φφ = ( u + uφ, φ ). eaning outcome/ kinematics: poo -, satisfactoy -4, good 4-6

4 Conside the simply suppoted xy plane eam of length shown. Mateial popeties E and G, coss-section popeties A, S =, I, and loading ae constants. Wite down the equiliium equations, constitutive equations, and ounday conditions accoding to the Benoulli eam model. Afte that, solve the equations fo the tansvese displacement. y x Solution p In xy plane polem Timoshenko eam equations of the fomulae collection simplify to (the non-zeo displacements and otations ae u, v, and ψ and S = ) N + x Q y + y = Mz + Qy + cz N EAu Qy = GA( v ψ ) M z EIψ p The additional kinematic constaint of the Benoulli eam model v ψ = follows fom the constitutive equation of the shea foce. Shea foce Q y ecomes a constaints foce to e otained fom the moment equation ( Q y = M z c z ). Theefoe the equiliium and constitutive equations and ounday condition (otained fom the figue) take the foms N + x = Mz cz + y and N EAu = M EI v z yy x ], [, u v = M z N x = and v = M z x =. p Bounday value polem fo the tansvese displacement is (moment is eliminated using the constitutive equation) (4) EI v + = x ], [ and EI v = x {, } and v = x {, }. yy yy p epeated integations of the diffeential equation taking into account the ounday conditions give (3) v = x+ a EI yy () v = xx ( ) EI yy () 3 v = ( x x ) + a EI 6 4 yy v = ( x x + x ). EI 4 yy eaning outcome/ eam model: poo -, satisfactoy -4, good 4-6

5 Vitual wok expession of cetain eam model is given y δw = ( δw EIw + δwkw + δwf ) dx Mδw ( ) + Fδw( ), whee wx ( ) is the tansvese displacement and w dw / dx. Detemine the undelying diffeential equation and ounday conditions. Assume that the solution is sought fom the set of functions having continuous deivatives of all odes and δw() = δ w () =. Paametes M and F ae given. The given functions EI zz, k and f have continuous deivatives of all odes ut they ae not constants. Solution 3p Integation y pats gives equivalent foms (the aim is to emove the deivatives fom vaiations in the integal ove the domain) δw = ( δw EIw + δwkw + δwf ) dx Mδw ( ) + Fδw( ) δw = ( δw ( EIw ) + δwkw+ δwf) dx δw ( )( EIw ) x= Mδw ( ) + Fδw( ) δw = [ ( EIw ) + kw + f ] δwdx + δw( )[( EIw ) + F] δw ( )( EIw + M ) x= x= p Accoding to the pincipal of vitual wok δ W = δ w. et us conside fist vaiations satisfying δ w ( ) = and δ w ( ) = so that the ounday tems vanish. The fundamental lemma of vaiation calculus implies that ( EIw ) + kw + f = in Ω= ], [.. p Knowing the equations aove and consideing vaiations with δ w ( ) = gives EIw + M = x=. Afte that, vaiation satisfying δ w ( ) = gives ( EIw ) + F = x=. p As δw() = δ w () = y assumption, tansvese displacement and it s deivative ae known at x = and theefoe w w= and w + θ = x =. in which w and θ ae the given ounday values. eaning outcome/ pinciple of vitual wok: poo -, satisfactoy -4, good 4-6

6 Conside a cuved eam foming ¾ of a full cicle of adius in the hoizontal plane. Toque of magnitude P is acting on the fee end as shown. Wite down the ounday value polem fo stess esultants and solve the equations fo Ns, () Qn () s, Q () s, T() s, Mn() s, and M() s. s x P z y Solution p In the geomety of the figue κ, κ = /. Extenal distiuted foces and s = moments vanish. Theefoe the cuved eam equiliium equations of the fomulae collection simplify to N Qn / Qn + N / = Q and T Mn / Mn + T / Q= M + Qn s ], [ whee 3 = π. p Bounday conditions at s = ae (notice the unit outwad nomal to the solution domain n =, moment on s e s e is negative) is pointing to the diection of s, and the component of the given N Qn = Q T + P and Mn = M s =. 3p Solution to the ounday values polem fo Q Q = s ], [ and Q = s = Q () s =. Solution to the connected ounday values polem fo Q n and N N Q n =, Qn + N = s ], [, Q n = and N = s = N + N = s ], [ and N =, N = s = s s N = asin cos + s ], [, N = and N = s = N() s = and Q () s =. n Solution to the ounday values polem fo M M = s ], [ and M = s = M () s =.

7 Solution to the connected ounday values polem fo M n and T T M n = and Mn + T = s ], [, T = P and M n = s = T + T = s ], [, T = P and T = T( s) Pcos s s = and Mn( s) = Psin. eaning outcome/ cuved eam model: poo -, satisfactoy -4, good 4-6

8 Kul Models fo eam, plate and shell stuctues INDEX NOTATION (Othonomal asis) a i i= a i i= a + a+ + ann i I ai/ xj aij, δ ij ei ej {,} ( e i e j = δ ij ) ε e ( e e ) {,,} ( e i e j = ε ijk e k ) ijk i j k εijkεimn = δ jmδkn δ jnδ km ε det( a) = ε a a a ijk lmn il jm kn GENEA a = ae i i a= a ij ee i j a = aijklee i je ke l... I a = a I = a a ( I = ii + jj + kk ) I : a = a: I = a a ( I = iiii + jjjj + kkkk + ijji + jiij + ikki + kiik + kjjk + jkkj ) a= aijee i j ac = aee ij j i a = a c a = a IDENTITIES a ( c) = ( a ) c a ( c) = ac ( ) ca ( ) a:( ) = ( a ) ( a) c CYINDICA φ z SYSTEM = cosφi + sinφ j + zk e cφ sφ i e e eφ = sφ cφ j eφ= eφ φ ez k ez ez = e + eφ + ez φ z SPHEICA θφ SYSTEM ( θφ,, ) = (s θ c φ i + s θ s φ j + c θ k)

9 eθ cθφ c cθφ s sθ i eφ = sφ cφ j e sθφ c sθφ s cθ k eθ cθ eφ eφ= sθe cθeθ φ e sθeφ eθ e eφ =, θ e eθ = eθ + eφ + e θ sinθ φ THIN BODY sn SYSTEM FO PANA BEAMS (, s n) = () s + ne () s n es, s /, s, s es en / = = = e n ess, / ess, ess, s en es / = es + en n s n OTHONOMA CUVIINEA COODINATES eα eα eα i e = ( i[ F])[ F] e = [ D] () i e e = D e en en en β β β i j ijk k T T eα α eα α = e F H = e D e e T β [ ] [ ] β β [ ] β = ed i ij j = ed i i n n n n Γ = e e e = e = ( e e ) D D ( e e ) ijk i j k k i s s jl l k a= ( dae ) i a= ( da + a Γ ) ee i i j k ikj i j a= da +Γ a i i iji j a= ( da +Γ a +Γ a ) e i ij kik ij ikj ik j Γ ijk = D i D jk a= ( a) = dda i i +Γjijda i PATE GEOMETY ( φ n) (, φ, n) = [ i cosφ+ j sin φ ] + nen

10 e cosφ sinφ i eφ = sinφ cosφ j en k e eφ eφ = e φ e n d = d = d = φ φ n n Γ = Γ = φφ φφ dv = dndω BEAM GEOMETY ( sn ) ( s, n, ) = [ ( s)] + ne n + e es, s es κ es κen en= ess, / ess, en= κ κs en= κse κes s e es en e κs e κsen d s = n ) ( s + s n sn ( κ κ κ ) d n = n d = ssn sns ( n ) Γ = Γ = κ κ dv = ( nκ ) dads sn Γ sn = ( nκ ) κs Γ = CYINDICA SHE GEOMETY ( zφ n) ( z, φ, n) = [ i cosφ+ jsin φ + kz] + nen ez i ez eφ = sinφ cosφ j eφ = en φ en cos φ sinφ k en eφ d = z z φ = ( ) φ d n = n d n Γ φφn = Γ φnφ = ( n) dv = ( n ) dn( dφ ) dz = ( n ) dndω INEA ISOTOPIC EASTICITY σ = E: ε = E: u (mino and majo symmeties of the elasticity dyad assumed) ε = [ u + ( u )] c

11 T T ii ν ν ii ij + ji G ij + ji E = jj E ν ν jj + jk + kj G jk + kj kk ν ν kk ki + ik G ki + ik T T ii ν ii ij + ji G ij + ji E E = jj ν jj + jk + kj jk + kj (plane stess) ν kk kk ki + ik ki + ik T T ii E ii ij + ji G ij + ji E = jj jj + jk + kj G jk + kj (eam) kk kk ki + ik G ki + ik T T ii E ii ij + ji ij + ji E = jj jj + jk + kj jk + kj (uni-axial) kk kk ki + ik ki + ik E G = ( +ν ) 3 Et D = ( ν ) PINCIPE OF VITUA WOK ext int δw = δw + δw = δ u U (a function set) δw = ( σ : δε ) dv + ( f δu) dv + ( t δ u) da V c V A BEAM EQUATIONS F + F σ = = da M + i F + c M ρ σ F σ E E ρ u + i θ = da = da M ρ σ ρ E ρ E ρ θ E = Eii + Gjj + Gkk TIMOSHENKO BEAM ( xyz ) N + x Q y + y= Qz + z T + cx M y Qz + cy= Mz + Qy + cz

12 N EAu ESzψ + ES yθ Qy= GA( v ψ) GS yφ Q z GA( w + θ) + GSzφ T GS y( v ψ) + GSz( w + θ) + GIφ M y = ES yu EIzyψ + EI yyθ M z ESzu + EIzzψ EI yzθ TIMOSHENKO BEAM ( sn ) N Qnκ + s Qn + Nκ Qκs + n= Q + Qnκ s + T Mnκ + cs Mn + Tκ Mκs Q + cn= M + Mnκ s + Qn + c N EA( u vκ ) + ESn( θ + φκ ψκ s) ES( ψ + θκ s) Qn= GA( v + uκ wκ s ψ ) GSn( φ θκ) Q GA( w + vκ s + θ ) + GS( φ θκ) T GS( w + vκ s + θ ) + GI( φ θκ) GSn( v + uκ wκ s ψ ) Mn = ESn( u vκ ) + EInn( θ + φκ ψκ s) EIn( ψ + θκ s) M ES( u vκ ) EIn( θ + φκ ψκ s) + EI( ψ + θκ s) PATE EQUATIONS F + = ( M Q+ c) k = F = σ dz = iinxx + ijnxy + jin yx + jjn yy + ( ki + ik ) Qx + ( kj + jk ) Qy M = σ zdz = iim + ijm + jim + jjm + ( ki + ik ) + ( kj + jk ) xx xy yx yy x y EISSNE-MINDIN PATE ( xyz ) Nxx, x + Nyx, y + x = Nyy, y + Nxy, x + y Qxx, + Qyy, + z Mxx, x + Myx, y Qx + cx = Myy, y + Mxy, x Qy + cy Qx w, x + θ = Gtk Q w φ y, y Nxx u, x + ν v, y Et Nyy = v, y + νu, x ν N ( ν )( u + v ) / xy, y, x M xx θ, x νφ, y Myy = D φ, y + νθ, x M ( ν)( θ φ ) / xy, y, x Qn Q o w w n Nnn Nn o un un = M ns M s o θn θn = N ns Ns o us u s M nn M n o θs θs KICHHOFF PATE ( xyz )

13 Nxx, x + Nyx, y + x = Nyy, y + Nxy, x + y Mxx, xx + Mxy, xy + Myy, yy + z ( Mxx, x + Myx, y Qx + cx ) = ( Myy, y + Mxy, x Qy + cy ) Nxx u, x + ν v, y Et Nyy = v, y + νu, x ν N ( ν )( u + v ) / xy, y, x Mxx w, xx + ν w, yy Myy = D w, yy + ν w, xx M ( ν ) w xy, xy Nnn Nn o un un = N ns Ns o us us Q + M Q M o w w M nn M n o w, n + θ s n nss, ss, = EISSNE-MINDIN PATE ( φ z) [( N ) + N N ] / + [( Nφ ), + Nφφ, φ + Nφ] / + φ, φ, φ φφ = N u, + ν ( u + uφφ, )/ Et Nφφ = u ν, + ( u+ uφ, φ )/ ν N ( ν )[( u u ) / + u ] / φ, φ φ φ, [( Q), + Qφφ, ] / + z [( M ), + Mφ, φ Mφφ ] / Q + c = [( Mφ ), + Mφφ, φ + Mφ] / Qφ + cφ Q w, + θφ = Gt Qφ w, φ / θ M θφ, + νθ ( φ θ, φ)/ Mφφ = D νθφ, + ( θφ θ, φ )/ M ( ν)[( θ + θ ) / θ ] / φ φφ,, OTATION SYMMETIC KICHHOFF PATE D w+ z = d d = ( ) d d MEMBANE EQUATIONS IN CYINDICA GEOMETY ( zφ n) Nφz, φ + Nzz, z z Nzφ, z + Nφφ, φ + φ = n Nφφ te [ u zz, + ν ( u φφ, u n)] Nzz ν te Nφφ = [ ( u φ, φ un) + νuzz, ] ν Nzφ tg( uz, φ + uφ, z) MEMBANE EQUATIONS IN SPHEICA GEOMETY ( φθ n )

14 cscθnφφ, φ + Nθφ, θ + cot θ( Nθφ + Nφθ ) φ csc θnφθ, φ + Nθθ, θ + cot θ ( Nθθ Nφφ ) + θ = Nφφ + Nθθ n te [ csc θ(cosθu θ + ν sin θuθθ, + uφφ, ) ( + ν) un] N φφ ν te Nθθ = [ csc θ ( ν cosθu sin u θ + θ θθ, + νuφφ, ) ( + ν) un] ν Nφθ tg( cscθuθφ, co tθuφ + uφθ, ) SHE EQUATIONS IN CYINDICA GEOMETY ( zφ n) κ Nφz, φ + Nzz, z + z Nzφ, z + κnφφ, φ κqφ + φ = κqφ, φ + Qzz, + κnφφ + n Mzφ, z + κmφφ, φ κmφn Qφ + cφ M + κm Q + c = zz, z φz, φ z z Nzz uz, z + νκ( uφφ, un) Et Nφφ = u ν z, z + κ( uφφ, un) ν Nzφ ( ν)( uφ, z + κuz, φ) / Mzz ωzz, + κνωφφ, κuzz, Mφφ νω zz, + κωφφ, + κ ( uφφ, un) Mzφ= D ( ν )( ωφ, z + κωz, φ κuφ, z) / Mφz ( ν)( ωφ, z + κωz, φ + κ uz, φ) / M ( νκκ ) ( u + κu + ω) / φn n, φ φ φ Qz unz, + ωz = tg Q ω + κ( u + u ) φ φ n, φ φ ωz θ φ = ωφ θz