Chapter 2 -- Derivation of Basic Equations

Transcription

1 Chapter -- Derivatin f Basic Equatins. Overview In structural analysis, plates and shells are defined as bdies which are bunded by tw surfaces, where the distance between the bunding surfaces is small cmpared t their lateral dimensins. The set f all material pints equidistant frm the tw bunding surfaces is termed the middle surface (MS. If the MS is curved, then the bdy is a shell; therwise it is a plate. The thinness f the bdy justifies use f a specialized structural thery, in which the shell r plate is treated essentially like a surface surrunded by a small thickness envelpe. The thery f shells is a cmplicated subject, and there are numerus bks devted slely t its expsitin, including the texts by Flügge (973 and Nvzhilv (959 which cver the linear thery, and Librescu (975 and Mushtari and Galimv (96 which als cver the case f gemetrically nnlinear respnse. In this chapter, we attempt t prvide nly thse details required fr the wrk at hand. The interested reader is directed t any f the abve referenced texts fr further study. We begin with an intrductin t the thery f surfaces, then prceed t derive a shell thery by descent frm the 3D thery f elasticity, in lines f curvature crdinates. The derived thery will be presented in bth linear and nnlinear frms, fr cmpsite laminate shells.. Intrductry thery f surfaces Cnsider a surface in three-dimensinal space. Psitins f pints within the surface may be described three-dimensinally, with reference t a glbal system, r tw-dimensinally, with reference t a crdinate system which is lcal t the surface. Such a D system is knwn as a gaussian crdinate system, and must be defined s that there is a unique ne-t-ne crrespndence between material pints f the surface and gaussian crdinate pairs. Let us take a rectangular cartesian crdinate system with crdinates (x,y,z as ur fixed glbal system, and take crdinates (ξ,ξ as ur gaussian crdinates, as shwn in Fig...

2 ξ ξ z ξ -curve ξ -curve y x ξ is cnstant alng a ξ -curve; ξ is cnstant alng a ξ -curve Fig.. Surface in 3-Dimensinal Space ξ z ξ r, r, B dr r A y x Fig.. Additinal Gemetry f the Surface Althugh there are any number f chices which culd be made fr the gaussian crdinate system (ξ,ξ, in the develpment which fllws we will use nly crdinates wherein the ξ and ξ curves passing thrugh any pint are mutually rthgnal. Let r be a vectr which describes the psitin f a pint A within the surface. See Fig... We may write r as 3

3 ξ curves. The terms and A are knwn as the metrics f the surface. They relate infinitesir x( ξ, ξ î + y( ξ, ξ ĵ + z( ξ, ξ kˆ (. where î, ĵ, kˆ are unit vectrs in the x-,y-,and z- directins, respectively. Dente by r, the deriva- tive vectr ( r ; similarly, r, ( r. Clearly, in view f the definitin f a derivative, ξ ξ r, is tangent t the ξ curve, and r, is tangent t the ξ curve. Nw, cnsider a secnd pint B als within the surface, separated frm A by a vectr given by dr. The distance ds between A and B is ds dr dr (. In view f equatin (., we find dr ξ ( rdξ + ( rdξ ξ (.3 The equatin (.3 is used in equatin (. t get ds dξ + A dξ (.4 where A r, r, r, r, r, r, (.5 r r, A r, (.6 and we have used the relatin r, r, 0, which arises frm the rthgnality f the ξ and 4

4 We will require knwledge f the derivatives f the basis vectrs with respect t the gaus- mal distances within the surface t infinitesimal changes in the values f the gaussian crdinates. Fr example, if dr is directed alng a ξ curve, then ds dξ. In general terms, we may write dr dξ tˆ + A dξ tˆ (.7 where tˆ, tˆ are unit vectrs in the directins f r,, r,, respectively. Unit vectrs tangent t and nrmal t the surface (i.e., the basis vectrs f the surface may be fund by nrmalizing r, and r, : tˆ r, tˆ r, A (.8 nˆ ± ( r, r, ( A where tˆ, tˆ are tangent t the ξ - and ξ -curves, and nˆ is nrmal t the surface. The sign f nˆ may be chsen fr cnvenience t place the nrmal as either inwardly f utwardly directed, as desired. The unit basis vectrs f the surface change directin frm pint t pint within the surface as a result f curvature. nˆ nˆ nˆ + nˆ tˆ ξ -curve Fig..3 Derivatives f the Nrmal Vectr 5

5 sian crdinates. These derivatives appear in their simplest frm when expressed in lines f curvature crdinates. The crdinates ξ, ξ are lines f curvature crdinates if, in additin t rthgnality, they pass the fllwing test: fr every ξ i -curve n the surface (i,, a plane which cntains that curve must als cntain the nrmal t the surface at every pint n that curve. These cnditins ensure that adjacent nrmals t the surface, taken alng a crdinate curve, intersect in the tˆi - nˆ plane. That is, nˆ, tˆ 0 and nˆ, tˆ 0. Fr lines f curvature crdinates, we may make the fllwing argument (See Nvzhilv (959, Chapter t find the derivatives f the nrmal vectrs. Tw neighbring pints alng a ξ -curve are separated by a small crdinate value ξ, and have nrmal vectrs as shwn in Fig..3. The difference between the tw nrmal vectrs is dented by nˆ, and the length f the arc which jins the tw pints is ξ. The radius f curvature f the ξ -curve is. As ξ becmes small, the arc which jins pints A and B becmes a straight line, directed alng tˆ, and the vectr nˆ als becmes parallel t tˆ. Then, using similar triangles, we see nˆ nˆ ξ tˆ The vectr nˆ has unit magnitude. In the limit as ξ 0, we get ξ ( nˆ tˆ (.9 Likewise, ξ ( nˆ A tˆ R (.0 where R is the radius f curvature f a ξ -curve. 6

6 nˆ Fig..4 Tangent and Nrmal Vectrs f a Gaussian Crdinate Curve tˆ ξ -curve Calculatin f the derivatives f the tangent vectrs is as fllws. First, it will be seen that the derivative f the tangent vectr is perpendicular t the tangent vectr. Then, scalar prducts f basis vectrs with the derivative will be taken t get the cmpnents f the derivative vectrs. Cnsider a small arc f a ξ -curve, with basis and nrmal vectrs tˆ, nˆ as shwn in Fig..4. Because the magnitude f tˆ is unity, we may write tˆ tˆ tˆ Taking the derivative f the abve expressin, we get tˆ ξ tˆ tˆ ( 0 ξ ξ, and by similar reasning, tˆ ξ tˆ tˆ ( 0 ξ ξ which prves that the derivatives f the tangent vectrs are perpendicular t the tangent vectrs. Next, cnsider the mixed derivative f the psitin vectr r with respect t the gaussian crdinates. If Thus, r is cntinuus and single-valued, then the rder f differentiatin is immaterial. r ξ ξ r ξ, r, ξ r ξ ξ r, using equatin (.8, 7

7 ξ ( A tˆ ξ ( A tˆ whence tˆ ξ A ξ ( tˆ A ξ tˆ Nw, tˆ tˆ ξ tˆ ( ξ tˆ tˆ tˆ ξ tˆ tˆ ξ, with the last step arising frm the rthgnality f the ξ - and ξ -curves. Thus, tˆ tˆ ξ A tˆ ( A A ξ tˆ ( ξ tˆ tˆ tˆ A A ξ tˆ ( tˆ A ξ tˆ Using the rthgnality f the tangent vectr with its derivative, we finally get tˆ tˆ ξ A A ξ Similar manipulatins may be perfrmed t get the remaining derivatives and prducts. If this apprach is taken, we get the Gauss-Weingarten relatins: 8

8 0, A tˆ, tˆ, tˆ, tˆ, nˆ, nˆ, 0, A A, A , 0 A R tˆ tˆ nˆ (. 0 A R In equatin (. and hencefrth, the ntatin (*, will represent the derivative f (* with respect t ξ. Similarly, (*, represents the derivative f (* with respect t ξ. It may be nted frm equatin (. that nˆ r, nˆ r, 0. This is the mathematical cnditin which, alng with their mutual rthgnality, define the crdinate curves as lines f curvature. There are, finally, tw ther useful relatins t be derived frm cnsideratin f the derivatives f the basis vectrs. These tw relatins are knwn as the Gauss-Cdazzi relatins, and stem frm the cntinuus, single-valued nature f the basis vectrs. First, nting that nˆ, nˆ,, and using equatin (., we may derive the relatins f Cdazzi: A A ξ R ξ A A ξ R ξ (. Then by asserting that r alng with equatin (., we get tˆ, tˆ, tˆ, tˆ, 9

9 A ξ ξ A ξ A ξ A R, (.3 which is knwn as the Gauss relatin. z r η y θ x Fig..5 Gemetry f a Surface f Revlutin Fr a general surface f revlutin with the z-axis as the axis f rtatin, we may find expressins fr the radii f curvature as fllws. Chse fr (ξ,ξ the crdinates (η,θ as shwn in Fig..5. where η crrespnds t the latitude f a pint and θ crrespnds t the lngitude. Given this chice f crdinates, we may write the vectr r as r ( ηθ, x( ηθ, î + y( ηθ, ĵ + z( ηθ, kˆ r r( η, θ R( η csηcsθî + R( η csηsinθĵ + R( η sinηkˆ (.4 where, due t axisymmetry, r R( η. Nw, with equatin (.4, we may find the metrics and A by equatin (.6; we find the unit basis vectrs by equatin (.8, using the negative sign n nˆ in rder t have a psitive utward nrmal. By the Gauss-Weingarten relatins, (., we see that 0

10 ----- nˆ nˆ R tˆ η tˆ θ A which leads us (with sme algebra t R ( R RR [ R +( R ] R csη Rsinη (.5 in which primes indicate differentiatin with respect t η. As previusly stated, the terms, R are the radii f curvature f the lines f curvature; the reciprcals f these radii are knwn as the principal curvatures f the surface. Finally, cnsider that there exists a secnd surface, parallel t and separated frm the first by a small distance ζ, measured nrmal t the first (reference surface. If the vectr r dentes the lcatin f a pint n the reference surface, then a crrespndent pint n the parallel surface is lcated by See Fig..6. R r + ζnˆ ζnˆ r R Fig..6 Parallel Surface Psitin Vectr

11 Tangent vectrs t the parallel surface are fund by R, r, + ζnˆ, and s n. By using equatins (.8 and (., we get, R, H tˆ R, H tˆ (.6 R,ζ H 3 nˆ where ζ H A ζ H A R H 3 H ζ (.7 H, H, and H 3 are the metrics f the parallel surface, s that dr H dξ t ˆ + H dξ t ˆ + dζnˆ (.8 The magnitude f dr is given by ds, where ds H dξ + H dξ +dζ (.9 which cncludes the intrductry thery f surfaces..3 Kinematics f defrmatin & definitin f strain measures Recall frm the intrductry sectin f this chapter that it is ur intentin t treat a shell as a small thickness envelpe surrunding a reference surface. Fr ur purpses, we will take the middle surface (MS t be the reference surface. The shell will have a cnstant thickness f h, with the bunding surfaces thus lcated at ζ h and ζ h. Furthermre, we are specifically interested here in thin shells, wherein h R«, where R is the lesser f and R.

12 The system f crdinates ξ, ξ and ζ, alng with the tangent vectrs tˆ, tˆ and nˆ frm an rthnrmal basis fr a three-dimensinal bdy. undefrmed defrmed nˆ A tˆ B g 3 g A * dr g tˆ R R * dr * Origin Fig..7 Defrmed Basis Vectrs Cnsider that a material pint A is lcated by the vectr R in the undefrmed bdy. A secnd pint B is lcated by dr given by equatin (.8 relative t A. The pints A and B, alng with the basis vectrs, define a rectangular parallelepiped. Under defrmatin, the pint A mves thrugh a displacement vectr U t A * lcated by R * R+ U (.0 In the same defrmatin, the basis vectrs are translated, rtated and stretched int new basis vectrs g, g, g 3 fr the defrmed system, as illustrated in Fig..7, and the defrmed parallelepiped is nt generally rectangular. That is, the defrmed basis vectrs d nt frm an rthnrmal set. Under defrmatin, the line element dr becmes 3

13 dr * H dξ g + H dξ g + dζg 3. (. We may als write dr * as dr * * R,dξ R *, dξ R * + + dζ,ζ (. Cmparisn f (. and (., in view f (.0 and (.6 yields g tˆ U H, g tˆ U H, (.3 g 3 nˆ + U,ζ The displacement gradients within (.3 may be written in terms f the basis vectrs f the undefrmed system as U H, ε tˆ + ε tˆ + ε 3 nˆ U H, ε tˆ + ε tˆ + ε 3 nˆ (.4 whence, U,ζ ε 3 tˆ + ε 3 tˆ + ε 33 nˆ g ( + ε tˆ + ε tˆ + ε 3 nˆ g ε tˆ + ( + ε tˆ + ε 3 nˆ (.5 g 3 ε 3 tˆ + ε 3 tˆ + ( + ε 33 nˆ We nw make an assumptin: straight line elements initially nrmal t the MS remain straight after defrmatin. This assumptin implies that the basis vectr g3 is independent f the thickness crdinate ζ. Then by the third f equatins (.3, U,ζ is als independent f ζ. We 4

14 further assume vectr U,ζ is parallel t the tangent plane in the undefrmed shell, r ε See equatin (.4. These assumptins limit us t the First-rder Transverse Shear Defrmatin Thery. Fr the displacement vectr U given by U U tˆ + U tˆ + U ζ nˆ (.6 the first-rder transverse shear defrmatin assumptins result in U ( ξ, ξ, ζ u( ξ, ξ + ζφ ( ξ, ξ U ( ξ, ξ, ζ v( ξ, ξ + ζφ ( ξ, ξ U 3 ( ξ, ξ, ζ w( ξ, ξ (.7 with u, v, w, φ, φ as graphically depicted in Fig..8; u, v, w are translatins, φ, φ are rtatins f the nrmal element. z φ w φ u v ξ -curve ξ -curve Fig..8 Middle Surface Displacements and Rtatins Use f (.6, (.7 in (.4, alng with the Gauss-Weingarten relatins (. yields, after sme algebra, 5

15 ε ( ζ ε ( + ζκ ε ( ζ R ε ( + ζκ ε ( ζ ε ( + ζκ ε ( ζ R ε ( + ζκ ε 3 ( ζ ε ζ φ 3 ε 3 ( ζ R ε ζ φ 3 R ε 3 φ ε 3 φ ε 33 0 (.8 with A -----, w u, v ε A ε A A -----, u, v A ε A -----, v, u ε A ε 3 u w, ε 3 A, w u v A A, A v w, R R (.9 A κ -----, A φ, φ A κ -----, φ, φ A A κ -----, A φ A, , φ κ φ φ A, Equatins (.8, (.9 allw us t write the displacement gradients f any pint within the shell as a linear functin f the thickness crdinate, ζ. The superscript n the terms abve indicates ε ij a MS value; terms represent MS displacement gradients, and terms are gradients f the rtatins, cmmnly referred t as curvatures. Green s strain measure is defined by the equatin κ ij ( ds * ( ds E g ( ds (.30 where ds * dr * dr * dr *, and ds dr dr dr. We make the fllwing H dξ H definitins: l dξ d ζ, l,. Here, define the directin csines f ds l ds l ds, l, l 3 6

16 the line element dr; equatin (.9 shws that l + l + l 3. Then, using equatins (.8, (., (.5 in (.30, we get E g E l + Γ l l + Γ 3 l l 3 + E l + Γ 3 l l 3 + E 33 l 3 (.3 where E ε + -- ( ε + ε + ε 3 Γ 3 ε 3 + φ + ε φ + ε φ E ε + -- ( ε + ε + ε 3 Γ 3 ε 3 + φ + ε φ + ε φ E ( φ + φ Γ ε + ε + ε ε + ε ε + ε 3 ε 3 (.3 The strain measures E ij, Γ ij represent the full nnlinear Green s strains cnsistent with the firstrder transverse shear defrmatin thery. We may further simplify the strain-displacement relatins, under the assumptin f small strains and mderate rtatins. ε 3 ε 3 nˆ shell nrmal g3 ε 33 assumed t be zer g tˆ ξ -curve ε ε 3 ε tˆ ε 3 g ε ε ξ -curve Fig..9 Shell Strains and Rtatins Cnsider three infinitesimal line elements ( tˆdξ, tˆdξ, nˆdζ in the undefrmed shell which displace, stretch and rtate t vectrs ( gdξ, gdξ, g3dζ, respectively, in the defrmed shell. Omitting the displacement, equatins (.5 are used t depict the basis vectrs in the 7

17 law. We may write the strain measures (.34 explicitly in ζ by the fllwing prcess: use equa- defrmed shell. See Fig..9. The displacement gradients ε 3, ε 3 represent the ut-f-plane rtatins f the elements riginally in the tangent plane f the undefrmed shell. Fr the purpse f this argument, we will refer t ε 3, ε 3, and the rtatinal displacements φ, φ as rtatins, and the remaining gradients as strains. Nte that by equatin (.8, ε 3 φ, and ε 3 φ. Under the assumptin that all strains and all rtatins are small, we are able t simplify the strain measures f equatins (.3 by simply retaining nly the lwest-rdered terms f each strain measure. This leads us t the linear strain measures: L Γ L E L ε + ε Γ 3 alng with a nnlinear equatin fr E 33 : L ε E ε L ε 3 + φ Γ 3 E ( φ + φ ε 3 + φ (.33 The linearized strain-displacement relatins may be made explicit in the thickness crdinate by use f equatins (.8, (.9. In rder t accunt fr sme gemetric nnlinearity, we may assume the strains t be small, and rtatins t be f mderate size. That is, we assume the strains t be n the rder f µ, 0 < µ «, and assume rtatins ε 3, ε 3, φ, φ t be n the rder f µ. These assumptins are justified because we are interested in develping a shell thery, wherein we expect the shell t have large stiffness in-plane, relative t ut-f-plane stiffness. Keeping terms t the lwest rder nly frm equatin (.3 thus leads t the strain measures E ε + -- ε 3 E ε + -- ε 3 E ( φ + φ Γ 3 γ 3 ε 3 + φ Γ 3 γ 3 ε 3 + φ Γ γ ε + ε + ε 3 ε 3 (.34 Nte that the nrmal strain in the thickness directin, E 33, is nn-zer, and is a nnlinear functin f the rtatins. This issue f nn-zer E 33 will be dealt with in the discussin f the material 8

18 tins (.8 t substitute fr ε, etc.; factr ut a cmmn denminatr frm each strain measure; in the numeratr, apprximate the terms ( + ζ R by the first term f their series expansins:, fr, which is an apprpriate apprximatin fr thin shells; lin- ( ζ R ζ -- R R R, R ζ ζ earize the numeratrs with respect t -----, We finally arrive at the terms R with E ( ζ E ( + ζχ E ( ζ R E ( + ζχ E ( φ + φ Γ ( ζ ( + ζ R Γ ( + ζχ Γ 3 ( ζ Γ Γ 3 3 ( ζ R Γ 3 Γ ε E -- ( ε 3 E ε ε + ε 3ε3 + + Γ 3 ε 3φ ε 3 ε + -- ( ε 3 + φ Γ 3 ε 3 ε χ κ χ R κ R ε χ κ + κ R ε ε 3φ R ε 3φ ε 3 ε 3φ R + φ (.35 (.36.4 Material law We will assume linear elastic behavir, with the shell cnstructed f a number f laminae f mnclinic materials, each lamina having a plane f symmetry parallel t the shell MS. In such a case, the generalized Hke s Law fr each lamina is given by 9

19 S C C C C 6 S C C C C 6 S 33 C 3 C 3 C C 36 S C 44 C 45 0 S C 45 C 55 0 S C 6 C 6 C C 66 E E E 33 Γ 3 Γ 3 Γ where S ij are stresses f the secnd Pila-Kirchhff type. That is, they measure frce n a small element in the defrmed state, divided by the undefrmed area f the element. A reduced material law is fund by assuming S 33 t be negligible cmpared t the in-plane stresses S, S, and S. This allws us t take S We then slve fr E 33 in terms f the ther stresses, substitute back int the material law and get where S Q Q 0 0 Q6 S Q Q 0 0 Q6 S Q44 Q45 0 S Q45 Q55 0 S Q6 Q6 0 0 Q66 E E Γ 3 Γ 3 Γ (.37 Qij C ij ( C i3 C j3 C 33 ij, 6,, Qij C ij i, j 45, Nte that the reductin has effectively remved nrmal strain E 33 frm the prblem. The trublesme cnditin f having a nnlinear expressin fr E 33 in the small strain, mderate rtatin thery is thus circumvented. See the third f equatins (

20 .5 Equilibrium.5. Internal virtual wrk Frm the thery f elasticity, we knw that the internal virtual wrk f a bdy is given by δw int S δg Vl [ + S δg + S 3 δg 3 ]H H dζdξ dξ (.38 where the vlume element dv is equal t H H dξ dξ dζ. The integral expressin in (.38 represents the applicatin f a small virtual displacement t each bdy element in the defrmed state. The stress vectrs S, S, S 3 may be chsen as referring t the undefrmed bdy axes and the undefrmed areas f the element (first Pila-Kirchhff stresses r P-K-, t the defrmed bdy axes and undefrmed areas (secnd Pila-Kirchhff stresses r P-K- r t the defrmed bdy axes and defrmed area (Cauchy-Lagrange stresses. In terms f P-K-, we write S T tˆ + T tˆ + T 3 nˆ S T tˆ + T tˆ + T 3 nˆ S 3 T 3 tˆ + T 3 tˆ + T 33 nˆ We get variatins f the lattice vectrs frm equatin (.5 as δg ε δ tˆ + δε tˆ + δε 3 nˆ δg δε tˆ + δε tˆ + δε 3 nˆ (.39 δg 3 φ δ t ˆ +δφ tˆ Then we get frm (.38 W int δ [ T δε + T δε + T 3 δε 3 + T δε + T δε Vl + T 3 δε 3 + T 3 δφ + T 3 δφ ]H H dζdξ dξ (.40 The displacement gradients may be made explicit in ζ by equatins (.8, (.9, and integratin 3

21 thrugh the thickness may be perfrmed t get δw int [ N ε Area δ + N δε + N δε + N δε + M δκ + M δκ + M δκ + M δκ + ( Q δε 3 + S δφ (.4 + ( Q δε 3 + S δφ ] A dξ dξ where the implicit definitins have been made: h h ( N, M T, ζ h h ( N, M T, ζ h h ( N, M T, ζ h h ( N, M T, ζ h h ( ( + ζ R dζ ( ( + ζ R dζ ( ( + ζ dζ ( ( + ζ dζ Q T 3 ( + ζ R dζ h h Q T 3 ( + ζ dζ h S [ T 3 ( + ζ ( + ζ R T 3 ( + ζ R ( ζ ] dζ h h S [ T 3 ( + ζ ( + ζ R T 3 ( + ζ ( ζ R ] dζ h (.4 Here we have lked at the case f gemetric nnlinearity by cnsidering virtual wrk n the defrmed bdy. Fr a gemetrically linear analysis, we allw equilibrium stresses t exist n an undefrmed bdy. Mment equilibrium then leads t the cnclusins ( T 3 T 3, ( T 3 T 3 and ( T T. This symmetry f stresses leads t a different expressin fr δw int : 3

22 L W int δ [ N δε + N δε + N δε + N δε + M δκ Area + M δκ + M δκ + M δκ + Q δγ 3 + Q δγ 3 ] A dξ dξ (.43 Nte that in this latest expressin, the shear stress cuple terms S, S d nt appear, and the shear stress resultants Q and Q multiply different strain measures than in the nnlinear case. These new strain measures are defined by the equatins ε 3 + φ, γ 3 ε 3 + φ. We als γ ε ε 3 define +. The internal virtual wrk may als be written in terms f P-K- stresses, wherein γ 3 S S g + S g + S 3 g 3 S S g + S g + S 3 g 3 (.44 These expansins fr the stress vectrs are substituted int equatin (.38 alng with (.5 and (.39 and the multiplicatin is carried ut, recalling that we have assumed in the material law that S 33 0 S 3 S 3 g + S 3 g + S 33 g 3. Mment equilibrium n the defrmed bdy leads t the cnclusin that S ij is symmetric. In view f equatins (.3 we are thus able t write δw int (.45 Vl [ S δe + S δγ + S 3 δγ 3 + S δe + S 3 δγ 3 ]H H ζ d dξ dξ which, in terms f the small strain, mderate rtatin apprximatins (.34 becmes 33

23 It may be smewhat interesting t nte that symmetry f the P-K- stresses leads t symδw int [ S δε + S δε + ( S 3 + S ε 3 + S ε 3 δε 3 Vl + S δε + S δε + ( S 3 + S ε 3 + S ε 3 δε 3 + S 3 δφ + S 3 δφ ] (.46 H H dζdξ dξ Cmparing equatins (.46 and (.40, we see a crrespndence between stresses in the P-K- frm and the P-K- frm: T S T S T 3 S 3 + S ε 3 + S ε 3 T S T S T 3 S 3 + S ε 3 + S ε 3 T 3 S 3 T 3 S 3 (.47 We nte frm (.33 and (.40 that the P-K- stresses are cnjugate t the displacement gradients, and frm (.45 that the P-K- stresses are cnjugate t the Green s strains. Use f the equatins (.35 in (.45 and integratin thrugh the thickness results in the expressin fr internal virtual wrk δw int [ NδE Area + Mδχ + QδΓ NδΓ + NδE + Mδχ + Mδχ QδΓ 3 ] A dξ dξ (.48 where h ( N, M S (, ζ ( + ζ R dζ h h ( N, M S (, ζ ( + ζ dζ N, M h h ( S (, ζ dζ h h Q S 3 ( + ζ R dζ h h Q S 3 ( + ζ dζ h (.49 34

24 metry f the stress resultants ( and, regardless f the gemetry f the shell. N such symmetry exists when P-K- stresses are used. As a result, the gemetrically linear analysis has 0 stress resultants and 0 strain measures, where the gemetrically nnlinear analysis has nly eight. Further, use f P-K- stresses in the gemetrically nnlinear analysis prduces stress resultants and cuples. In cnsideratin f the crrespndence between P-K- stresses and P-K- stresses shwn in equatins (.47, we may cmpare the equatins (.4 and (.49 t get a crrespndence between stress resultants and stress cuples in the tw systems. Beginning, fr example, with the first f (.4 using the first f equatins (.47 and finally cmparing directly t the first f (.49, leads t the cnclusin Similar cmparisns may be made amng all the stress resultants and stress cuples, using equatins (.8 as necessary t yield N N M M h ( N, M T, ζ h h ( N, M S, ζ h ( ( + ζ R dζ ( ( + ζ R dζ ( N, M ( N, M ( N, M ( N, M ( N, M ( N, M Q Q+ ε 3 + Q Q+ ε 3 + N N ε γ 3 3 N N ε φ 3 N N M N N φ M M R γ M M R M (.50 M M M M 35

25 S ε 3 ε Q M M S R Q ε 3 R ε M M R (.50, cnt d The crrelatins f equatin (.50 are apprximate -- in additin t the thin shell assumptin, higher-rder resultants have been discarded in the relatins fr M, M, S and S. Fr example, in the exact crrelatin fr M, we have M M L R with L h h S ζ dζ We have chsen t discard the higher-rder resultants like because their retentin cmplicates the equilibrium equatins, t be intrduced in the next sectin. We ratinalize this decisin n the basis f smallness: magnitudes f the resultants tend t diminish as the rder f the thickness crdinate increases within the integrals which define the resultants. We nte, hwever, that the simplificatin leads t the errneus cnclusin, cntrary t the definitins f (.4. We thus accept a certain lss f accuracy in exchange fr a set f equatins which will be mre readily slvable. L ( M M.5. External virtual wrk We cnsider that the shell is laded under hydrstatic pressure nly, except at the edges; the lading remains nrmal t the shell middle surface under defrmatin. It is assumed that the pressure remains cnstant as the structure respnds. On the defrmed MS, the frce due t pressure lading n an infinitesimal area is given by f pda * nˆ* The term da * nˆ* is fund by 36

26 nˆ *da * g A dξ g A dξ with g, g given by (.5, applied at the MS. External virtual wrk is fund by applying a small (virtual displacement and integrating ver the defrmed area f the shell. This gives δw ext p g g δu A A dξ dξ Area If the multiplicatin is carried ut, the expressin becmes δw ext p { ε ε3 Area ε3 [ ( + ε ε 3 ] δu + [ ε ( + ε ε 3 ] δv + [( + ε ( + ε ε ] δw ε } A dξ dξ Nw applying the small strain, mderate rtatin assumptins and keeping terms t rder µ, we get δw ext p { ε 3δu ε3δv + ( + ε + ε δw} A dξ dξ Area Finally, if the displacement gradients are written in terms f displacements accrding t equatins (.9 and the necessary integratins are perfrmed, we get δw ext pδ w R A ξ b + p ua δw d ξ a C ξ u v w u w R A, v w A, da (.5 where A is the area f the surface; da A dξ dξ and C is the bunding curve defining the shell edge. Fr the gemetrically linear analysis, we assume applicatin f the lad n an undefrmed surface, and simply drp all f the nnlinear terms f equatin (.5 t get 37

27 L δw ext pδ w A dξ d ξ + p ua δw d ξ a Area C ξ b ξ. ( Principle f virtual wrk The principle f virtual wrk states that the shell is in a state f equilibrium if and nly if the virtual wrk f internal frces and mments exactly balances the virtual wrk f external frces and mments, fr all kinematically admissible virtual displacements. Mathematically, if δw int δw ext fr every kinematically admissible displacement, then the bdy is in equilibrium. We have internal virtual wrk in terms f P-K- stresses frm (.4 and external virtual wrk frm (.5. We write the variatins f the displacement gradients in terms f displacements with the aid f (.8, (.9 and integrate by parts as necessary t get Area +, A, ( A A N, ( A N, N N A Q δ u A ( A A N , A, ( A N ,, N A N Q R δ v C ( A A Q, ( A Q, N N R δw, A, ( A A M, ( A M, M M A + S δφ A,, ( A A M, ( A M, M A M + S δφ da b A N δu ξ a + A N δv + ξ a b b A Q δw ξ a + b A M δφ ξ a b + A M δφ ξ dξ p u w a A, δu p v w A, R δv Area + A p u A, , v A w -----, v A A, u δw A da R 38

28 Nting that the virtual displacements are arbitrary and independent, we see that the fllwing equilibrium cnditins must hld: ( A A N , ( A N +, A ,, N N A Q R p u A w, ( A A N , ( A N + A, A ,, N A N A Q R p v A w, R ( ( A A Q, ( A Q, N N R A p u A, , v A w -----, v A A, u A R and A ( A A M , A, ( A M , +, + M M A S 0 A ( A A M , A, ( A M , +, + M A M S 0 Bundary cnditins are such that ne element frm each f the fllwing pairs must be prescribed at the edges a, ξ b: ξ (N, u, (N, v, (Q, w, (M, φ, (M, φ. (.54 The entire set f equilibrium equatins and bundary cnditins may be recast in terms f P-K- stress resultants and stress cuples by use f equatins (.50. It is in this cnversin that we justify the decisin t neglect higher-rder resultants such as. L.6 Cnstitutive law A relatinship between the stress resultants and the middle surface strains is fund by cmbining the definitins (.49 and the material law (.37. Specifically, fr a laminate f plies f mnclinic material, the material law is applicable t each lamina; it is assumed that each ply is hmgeneus. 39

29 k th ply Ply #3 ζ ζ k- ζ ζ k ζ N h/ ζ 0 h/ t k MS Ply # Ply # Fig..0 Laminate Nmenclature The integrals defining the stress resultants and stress cuples may be viewed as sums f integrals ver the ply thicknesses s that, fr example, h N S ( + ζ R dζ h N k ζ k ζ k ( ( + ζ R dζ S k (.55 where there are N plies in the laminate, ζ 0 h, ζ k ζ k + t k and t k is the thickness f the k th ply. See Fig..0. The superscript (k indicates the k th ply. Use f the material law (.37 in (.55 leads t N N k ζ k ζ k ( k ( k ( k Q E + Q E + Q6 Γ [ ]( + ζ R dζ Use equatins (.35 t get relatins f the frm N E + E + 6 Γ + B χ + B χ + B 6 χ fr all f the stress resultants and stress cuples. The prcess yields the cnstitutive law: 40

30 N N Q Q N M M M B B B 6 A 0 0 A 6 B B B A 44 A A 45 A A A 66 B 6 B 6 B 66 B B 0 0 B 6 D D D 6 B B 0 0 B 6 D D D 6 B 6 B B 66 D 6 D 6 D 66 E E Γ 3 Γ 3 Γ χ χ χ (.56 The cefficients f equatin (.56 are defined by ζ k ( k (, B, D Q (, ζζ, + ζ R dζ + ζ ζ k ( k (, B, D Q (, ζζ, dζ ζ k ζ k ζ k ( k ( A, B, D Q (, ζζ, + ζ ζ R dζ ζ k ζ k ( 6, B 6, D 6 Q6 ζ k ζ k (,, ζ dζ ( k ζζ ( k ( A 6, B 6, D 6 Q6 (, ζζ, dζ + ζ R ζ k (.57 ζ ( k ζ ζ k (,, ( A 66, B 66, D 66 Q ( + ζ ( + ζ R dζ ζ k A 44 ζ k ζ k ( + ζ R ζ R dζ k Q44 A 55 A 45 ζ k ζ k ζ k ζ k Q55 ( k Q45 dζ + ζ R ζ dζ where the sums are taken ver the plies: k N. 4

31 .7 Summary f equatins The preceding sectins f this chapter give a smewhat detailed develpment f the thery f shells t be used in this wrk. As such, there are many equatins presented which are imprtant fr the derivatin, but nt fr the analysis. This sectin recaps the relevant equatins fr stress analysis. These equatins are fr a thin shell f revlutin, in lines f curvature crdinates, and it is assumed that the shape f the meridian is knwn; that is, the vectr r which describes the psitin f the surface relative t a fixed glbal crdinate system is knwn as a functin f the glbal crdinates and/r the Gaussian crdinates. Gemetry: The metrics f the middle surface are fund by equatins (.5, and metrics f a parallel surface within the shell are fund by equatins (.7. The radii f curvature f the middle surface are given by equatins (.5. Displacement gradients: Displacement gradients which vary thrugh the thickness are described in terms f their values n the middle surface, and in terms f MS displacements by equatins (.8, (.9. These displacement gradients frm the building blcks f the strain measures, which are derived depending upn the assumptins used. Strain measures: In the preceding derivatin, tw sets f strain measures were develped -- ne fr small strains and rtatins (i.e., a linear thery, and ne set fr small strains and mderate ut-f-plane rtatins. Fr the linear thery, the strain measures are as given in equatin (.33, using the displacement gradients fund previusly. Nte that fr the linear thery, the displacement gradients are the same as the strains, with the exceptin f the (negligible strain strains in terms f MS values are fund frm equatins (.8, (.9.. Thus, the linear Fr the nnlinear (i.e., small strain, mderate rtatins thery, the strain measures are fund by equatin (.34. In terms f MS values, the nnlinear strains are expressed as in equatins (.35, (.36. Internal virtual wrk: The integral expressin fr internal virtual wrk is given in equatin (.43 fr the linear thery, in terms f P-K- stresses. The stress resultants in terms f P-K- ε 33 4

32 are as given by equatins (.4. Fr the nnlinear thery, there are tw expressins fr internal virtual wrk: in terms f P-K- resultants in equatin (.4 with (.4, and in terms f P-K- in equatin (.48 with (.49. Crrelatin f P-K- stresses with P-K- stresses is shwn in equatin (.47, and crrelatin f resultants between the tw systems is as given in equatin (.50. External virtual wrk: The integral expressin fr incremental wrk f external lading is given in equatin (.5 fr the linear respnse case. Fr the gemetrically nnlinear respnse, the external virtual wrk is given by equatin (.5. Equilibrium: Equilibrium equatins are here shwn nly in terms f P-K- stress resultants and fr the gemetrically nnlinear respnse (equatins (.53, with bundary cnditins (.54. If the equatins f equilibrium are desired in their linear frm, these may be fund by use f the virtual wrk principle, using the linear equatins fr internal and external virtual wrk already discussed, and using the linear strain-displacement relatins. If the equatins are desired in terms f P-K- resultants, they may be fund by transfrmatin f (.53 and (.54 using the crrelatin (.50. Cnstitutive law: The cnstitutive law fr a cmpsite laminate shell in terms f P-K- stress resultants is given by equatins (.55, (.56, with the lamina material prperties given in equatin (.37. If a thery is desired fr linear respnse, it is cmmn t use the material law (.37, disregarding the differences between P-K- and P-K- stresses, and als disregarding the difference between the linear and nnlinear strains. That is, we simply replace the P-K- stress terms S ij with the P-K- stress terms T ij, and replace the strains E ij, Γ ij with the linear strains ε ij, and γ ij, all f which is cnsistent with the assumptin f small strains and small rtatins. 43