Aircraft Structures Kirchhoff-Love Plates
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1 University of Liège erospace & Mechanical Engineering ircraft Structures Kirchhoff-Love Plates Ludovic Noels Computational & Multiscale Mechanics of Materials CM3 Chemin des Chevreuils 1, B4000 Liège ircraft Structures - Kirchhoff-Love Plates
2 Balance of body B Momenta balance Linear ngular Boundary conditions Neumann Dirichlet Elasticity T Small deformations with linear elastic, homogeneous & isotropic material n b (Small) Strain tensor, or Hooke s law, or with Inverse law l = K - 2m/3 2m with ircraft Structures - Kirchhoff-Love Plates 2
3 Reissner-Mindlin equations summary Deformations (small transformations) In plane membrane Curvature Out-of-plane sliding z q y = Dt 1 Dt Dt t t x q y = Dt 1 -u z,1 g 1 1/k 11 1/Dt, ircraft Structures - Kirchhoff-Love Plates 3
4 Reissner-Mindlin equations summary Resultant stresses in linear elasticity Membrane stress Bending stress ñ 22 Out-of-plane shear stress ñ 12 z q y = Dt 1 ñ 11 ñ 21 Dt Dt t t q 3 x ~ m 22 -u z,1 q y = Dt 1 g 1 q 3 ~ m 11 ~ m 21 ~ m 12 1/k ircraft Structures - Kirchhoff-Love Plates 4
5 Resultant Hooke tensor in linear elasticity Membrane mode Bending mode Shear mode Reissner-Mindlin equations summary ircraft Structures - Kirchhoff-Love Plates 5
6 Resultant equations Membrane mode Reissner-Mindlin equations summary with with Clearly, the solution can be directly computed in plane Oxy (constant H n ) Boundary conditions» Dirichlet D N n» Neumann n 0 = n a E a Remaining equation along : ircraft Structures - Kirchhoff-Love Plates 6
7 Reissner-Mindlin equations summary Resultant equations (2) Bending mode & with with Solution is obtained by projecting into the plane Oxy (constant H q, H m ) 2 equations (a=1, 2) with 3 unknowns (Dt 1, Dt 2, u 3 ) Use remaining equation ircraft Structures - Kirchhoff-Love Plates 7
8 Reissner-Mindlin equations summary Resultant equations (3) Bending mode (2) 3 equations with 3 unknowns To be completed by BCs Low order constrains D p N T» Displacement or» Shearing n 0 = n a E a High order» Rotation or T p M M» Bending n 0 = n a E a ircraft Structures - Kirchhoff-Love Plates 8
9 Reissner-Mindlin equations summary Resultant equations (4) Remarks Compare bending equations With Timoshenko beam equations z q y q y g x Membrane and bending equations are uncoupled No initial curvature Small deformations (equilibrium on non curved configuration) ircraft Structures - Kirchhoff-Love Plates 9
10 Shear effect Kirchhoff-Love assumption Beam analogy z q y Timoshenko beam equations q y g x Euler-Bernoulli equations It has been assumed that the cross-section remains planar and perpendicular to the neutral axis u z,x = -q y Validity: (EI)/(L 2 m) << 1 The same assumption can be made for plates z f(x) T z M xx u z =0 x du z /dx =0 M>0 L ircraft Structures - Kirchhoff-Love Plates 10
11 Shear effect Kirchhoff-Love assumption (2) s D z q y q y g x Kirchhoff-Love assumption requires D 1 + ν L 2 Eh 0 = h L 2 1 ν 1 Where L is a characteristic distance ircraft Structures - Kirchhoff-Love Plates 11
12 Plate kinematics Description In the reference frame E i The plate is defined by with Mapping of the (deformed) plate Neutral plane x 1 x 2 F = j(x 1, x 2 )+x 3 t(x 1, x 2 ) S a =1 or 2, I = 1, 2 or 3 x 1 =cst x 2 =cst t Cross section with t 0 = Initial plate S 0 S 0 = x [-h 0 /2 h 0 /2], for a plate of initial thickness h 0 Deformed plate S ircraft Structures - Kirchhoff-Love Plates 12
13 Plate kinematics Kirchhoff-Love plate ssumptions Kirchhoff (cross section remains plane) & Small deformations,, Displacement field with Kinematics x 2 x ircraft Structures - Kirchhoff-Love Plates 13
14 Kirchhoff-Love theory Deformations (small transformations) Reissner-Mindlin Using Dt Dt t t 1/k 11 1/Dt, ircraft Structures - Kirchhoff-Love Plates 14
15 Kirchhoff-Love theory Resultant stresses Reissner-Mindlin ñ 22 ñ 12 Using ñ 11 ñ 21 Dt Dt t t ~ m 22 ~ m 11 ~ m 21 ~ m 12 1/k ircraft Structures - Kirchhoff-Love Plates 15
16 Kirchhoff-Love theory Resultant equations Membrane mode with Clearly, the solution can be directly computed in plane Oxy (constant H n ) Boundary conditions» Neumann D N n» Dirichlet Same membrane problem as for Reissner-Mindlin Remark: n 0 = n a E a ircraft Structures - Kirchhoff-Love Plates 16
17 Kirchhoff-Love theory Resultant equations (2) Bending mode & with Solution is obtained by projecting into the plane Oxy (constant H m ) 1 higher-order equation with 1 unknown (u 3 ) With Compare to Euler-Bernoulli beams How to include pressure effect? ircraft Structures - Kirchhoff-Love Plates 17
18 Kirchhoff-Love theory pplied pressure 1 higher-order equation with 1 unknown D p N T In case of pressure loading p ssume static problem (and constant H m ) n 0 = n a E a has no component along a in case T p M M of pressure loading Use energy conservation n 0 = n a E a Work of external forces Bending part of internal energy (as problems are uncoupled)? ircraft Structures - Kirchhoff-Love Plates 18
19 Kirchhoff-Love theory pplied pressure (2) Internal energy Last lecture we considered the decomposition g a /2 with Bending contribution to internal energy s Remark other contributions (for the uncoupled cases) follow For Kirchhoff plates» ircraft Structures - Kirchhoff-Love Plates 19
20 Kirchhoff-Love theory pplied pressure (3) Energy conservation s ircraft Structures - Kirchhoff-Love Plates 20
21 pplied pressure (4) Resulting bending equations Kirchhoff-Love theory Double integration by parts D p N T T p M M n 0 = n a E a n 0 = n a E a ircraft Structures - Kirchhoff-Love Plates 21
22 pplied pressure (5) Resulting bending equations (2) Kirchhoff-Love theory Using, and essential boundary conditions On : On N : On M : D p N T T p M M n 0 = n a E a n 0 = n a E a ircraft Structures - Kirchhoff-Love Plates 22
23 Kirchhoff-Love theory pplied pressure (6) Comparison with Euler-Bernoulli beams Plates Beams On : On N : On M : D p N T T p M M n 0 = n a E a n 0 = n a E a ircraft Structures - Kirchhoff-Love Plates 23
24 Kirchhoff-Love example Example: simply supported plate Equations D p with a Bending couple can be rewritten b General equation: BCs for x = 0 & x = a No torque s on these edges ircraft Structures - Kirchhoff-Love Plates 24
25 Kirchhoff-Love example Simply supported plate: Methodology Using BCs for x= 0, a & for y = 0, b p Solution can be represented as a Fourier series a b Coefficients mn such that Solution for p(x, y)? p(x,y) = p 0? ircraft Structures - Kirchhoff-Love Plates 25
26 Kirchhoff-Love example Simply supported plate: General solution pplied pressure can also be expended as a Fourier series Coefficient a mn can be evaluated using p a b ircraft Structures - Kirchhoff-Love Plates 26
27 Simply supported plate: General solution (2) Equations Kirchhoff-Love example p a What remains to be defined are the mn (as a mn are known) Balance equation yields b Should remain true for any x, y ircraft Structures - Kirchhoff-Love Plates 27
28 Kirchhoff-Love example Simply supported plate: Constant pressure p 0 Coefficients a mn p 0 a For m or n even: a mn = 0 b For m & n odd: Coefficients mn for m & n odd: Solution of the problem ircraft Structures - Kirchhoff-Love Plates 28
29 Kirchhoff example Simply supported plate: Constant pressure p 0 (2) Solution of the problem p 0 For a square plate b a m=1 m=3 m=5 m=7 n= n=3 Sym n=5 Sym. Sym n=7 Sym. Sym. Sym ircraft Structures - Kirchhoff-Love Plates 29
30 Kirchhoff example Simply supported plate: Constant pressure p 0 (3) For a square plate (2) Maximum deflection at x=y=a/2 4 x 10-3 u 3 D/(p 0 a 4 ) y/a 0 0 x/a ircraft Structures - Kirchhoff-Love Plates 30
31 Kirchhoff example Simply supported plate: Constant pressure p 0 (4) Resultant stress couple From & Bending Twisting ircraft Structures - Kirchhoff-Love Plates 31
32 m 11 /(p 0 a 2 ) m 22 /(p 0 a 2 ) Kirchhoff example Simply supported plate: Constant pressure p 0 (5) Resultant stress couple (2) For a square plate and n=0.3 u 3 D/(p 0 a 4 ) 4 2 x y/a 0 0 x/a ~ m aa /p 0 a ~ m 22 ~ m 11 m 12 /(p 0 a 2 ) ~ m ab /p 0 a ~ m ~ m y/a 0 0 x/a y/a 0 0 x/a ircraft Structures - Kirchhoff-Love Plates 32
33 Kirchhoff example Simply supported plate: Constant pressure p 0 (6) Stresses Using Direct stresses due to bending m 11 /(p 0 a 2 ) m 22 /(p 0 a 2 ) ~ m aa /p 0 a m ~ 22 y/a 0 0 x/a m ~ Shear stress due to torsion For a square plate (n=0.3) m 12 /(p 0 a 2 ) ~ m ab /p 0 a m ~ 21 m ~ y/a 0 0 x/a ircraft Structures - Kirchhoff-Love Plates 33
34 Tension effect on bending Uncoupled theory of Kirchhoff-Love plates We found for tension D N n With n 0 = n a E a Completed by appropriate BCs We found in bending D p N T With n 0 = n a E a Completed by appropriate BCs Low order T p M M High order n 0 = n a E a ircraft Structures - Kirchhoff-Love Plates 34
35 Tension effect on bending Uncoupled theory of Kirchhoff-Love plates (2) Uncoupled theory Results from the fact that Equilibrium is written on the initial configuration Small deformations assumption This initial configuration is planar s equilibrium should be written In deformed configuration, which has a curvature Tension will affect bending Second order theory D D T p p N n 0 = n a E a N n 0 = n a E a M n T M n 0 = n a E a ircraft Structures - Kirchhoff-Love Plates 35
36 Tension effect on bending Second order theory: Tension without external loading Equilibrium in x direction Cosines lead to terms in 1+O(u 3,x ) 2 Second order theory becomes Similarly Resulting equations s for first order theory Lead to 3rd order dy ñ 11 ñ 21 ñ 11 u 3,1 ñ 11 dx dx ñ 22 + y ñ 22 dy ircraft Structures - Kirchhoff-Love Plates 36 ñ 12 ñ 22 ñ 21 ñ 12 ñ 12 + y ñ 12 dy ñ 12 + x ñ 12 dx ñ 11 + x ñ 11 dx ñ 11 + x ñ 11 dx u 3,1 + x u 3,1 dx ñ 22
37 Second order theory: Bending Tension and shearing have a resultant loading in the z direction Due to x tension Tension effect on bending Second order theory: sin (x) x dy ñ 11 ñ 21 ñ 22 + y ñ 22 dy ñ 12 + y ñ 12 dy ñ 12 + x ñ 12 dx ñ 11 + x ñ 11 dx ñ 12 ñ 22 fter simplification (and removing d 3 terms) dx ñ 11 + x ñ 11 dx u 3,1 u 3,1 + x u 3,1 dx Due to y tension ñ 11 Similarly dx Due to yx shearing? ircraft Structures - Kirchhoff-Love Plates 37
38 Tension effect on bending Second order theory: Bending (2) Tension and shearing have a resultant loading in the z direction (2) Due to yx shearing ñ 22 + y ñ 22 dy Second order theory: sin (x) x dy ñ 11 ñ 12 + y ñ 12 dy ñ 12 + x ñ 12 dx ñ 21 ñ 11 + x ñ 11 dx fter simplification (and removing d 3 terms) ñ 12 ñ 22 dx dy Due to xy shearing Similarly dx ñ 21 ñ 21 + y ñ 21 dy u 3,1 u 3,1 + y u 3,1 dy ircraft Structures - Kirchhoff-Love Plates 38
39 Tension effect on bending Second order theory: Bending (3) Tension and shearing have a resultant loading in the z direction Value on surface dxdy: But tension equilibrium reads Final expression Using tension equilibrium twice more ircraft Structures - Kirchhoff-Love Plates 39
40 Tension effect on bending Second order theory: Bending (4) First order theory D p N T Second order theory Tension & shearing lead to a force n 0 = n a E a along z on a surface dxdy This corresponds to a pressure T p M M New resulting equation n 0 = n a E a ircraft Structures - Kirchhoff-Love Plates 40
41 Second order theory: Energy Equation Tension effect on bending Bending energy We saw that with s But part of the bending results from tension ñ 11 + x ñ 11 dx u 3,1 u 3,1 + x u 3,1 dx ñ 11 dx ircraft Structures - Kirchhoff-Love Plates 41
42 Tension effect on bending Second order theory: Energy (2) Bending energy (2) Part of the bending resulting from tension This corresponds to a pressure ñ 11 u 3,1 ñ 11 + x ñ 11 dx u 3,1 + x u 3,1 dx dx s ssuming BCs are such that there is no contribution to energy (u 3 =0 or u 3,a = 0), and if ñ is constant with u 3, Total bending energy ircraft Structures - Kirchhoff-Love Plates 42
43 Tension effect on bending Example: simply supported plate with tension BCs for x= 0, a & for y = 0, b (no torque) p ñ 11 Constant tension along Ox Equations b a Can be developed using ñ 11 with Final equation Solution for p(x,y) = p 0? ircraft Structures - Kirchhoff-Love Plates 43
44 Tension effect on bending Simply supported plate with tension & constant pressure p 0 Pressure can be expended in a Fourier series ñ 11 With Coefficients a mn in case of constant pressure p For m or n even: a mn = 0 For m & n odd: b a ñ 11 Final equation to be solved ircraft Structures - Kirchhoff-Love Plates 44
45 Tension effect on bending Simply supported plate with tension & constant pressure p 0 (2) Governing equation to be solved Considering BCs ñ 11 for x= 0, a & for y = 0, b p The solution can take the form b a ñ 11 The governing equation leads to a system of equations with mn as unknowns ircraft Structures - Kirchhoff-Love Plates 45
46 Tension effect on bending Simply supported plate with tension & constant pressure p 0 (3) System of equations By identification For m or n even: mn = 0 For m & n odd: With the solution stated as ircraft Structures - Kirchhoff-Love Plates 46
47 Tension effect on bending Simply supported plate with tension & constant pressure p 0 (4) Solution For a square plate a=b u 3 D p 0 a 4 u 3 D/(p 0 a 4 ) 6 x Tension increases the bending stiffness n 11 a 2 /D n 11 a 2 D ircraft Structures - Kirchhoff-Love Plates 47
48 Tension effect on bending Simply supported plate with tension & constant pressure p 0 (5) For a square plate a=b (2) Central displacement: If we increase the compression range u 3 D p 0 a 4 u 3 D/(p 0 a 4 ) Instability in compression See next lecture n 11 a 2 /D n 11 a 2 D ircraft Structures - Kirchhoff-Love Plates 48
49 Simply supported plate with tension & arbitrary pressure p For constant pressure we found Tension effect on bending For an arbitrary pressure we cannot particularize the a mn with General solution From Equation leads to ircraft Structures - Kirchhoff-Love Plates 49
50 Small initial curvature Plate with initial curvature This corresponds to n initial mapping along : j 03 Deflection u 3 is measured from this mapping Final mapping along : j 3 = j 03 + u 3 ssumption Initial mapping j 03 is small ñ 11 u 3 ñ 21 j 03 ñ 12 ñ 22 compared to the shell thickness momentum is still computed from u 3,ab ñ 11 + x ñ 11 dx There is a tension resultant along Previous analysis ñ 11 j 3,1 j 3 u 3 j 03 j 3,1 + x j 3,1 dx But here this effect is proportional to j 3,ab dx ircraft Structures - Kirchhoff-Love Plates 50
51 Example: simply supported plate with initial curvature BCs for x= 0, a & for y = 0, b (no torque) Constant tension along Ox Equations Can be developed using with & Final equation Solution for p(x,y) =0? Small initial curvature ñ 11 u 3 j ircraft Structures - Kirchhoff-Love Plates 51
52 Small initial curvature Simply supported plate with initial curvature We can expend the initial mapping in a Fourier series u 3 j 03 ñ 11 Considering BCs: for x= 0, a & for y = 0, b The solution can take the form ircraft Structures - Kirchhoff-Love Plates 52
53 Small initial curvature Simply supported plate with initial curvature (2) s we have u 3 j 03 ñ 11 The governing equation becomes Due to the initial curvature, a tension produces a z-deflection ircraft Structures - Kirchhoff-Love Plates 53
54 Kirchhoff-Love plate summary Membrane mode On : With D N n n 0 = n a E a Completed by appropriate BCs Dirichlet Neumann ircraft Structures - Kirchhoff-Love Plates 54
55 Kirchhoff-Love plate summary Bending mode On : With D Completed by appropriate BCs Low order On N : D p N T On D : n 0 = n a E a High order On T : with T p M M On M : n 0 = n a E a ircraft Structures - Kirchhoff-Love Plates 55
56 Kirchhoff-Love plate summary Membrane-bending coupling The first order theory is uncoupled For second order theory On : ñ 22 Tension increases the bending stiffness of the plate In case of small initial curvature (k >>) ñ 11 ñ 21 ñ 12 On : Tension induces bending effect u 3 ñ 22 General theories For not small initial curvature: Linear shells ñ 11 ñ 21 j 03 ñ 12 To fully account for tension effect Non-Linear shells ircraft Structures - Kirchhoff-Love Plates 56
57 Kirchhoff-Love plate summary FE implementation High order equation Requires C 1 finite-elements Shape functions too complex Can be enforced weakly Not common, usually Reissner-Mindlin plates are implemented ñ 11 u 3 j 03 ñ 21 ñ 12 ñ ircraft Structures - Kirchhoff-Love Plates 57
58 Introduction to shells Shell kinematics In the reference frame E i The shell is described by with x 1 =cst t 0 x 2 =cst S0 c = F o F 0-1 S x 1 =cst x 2 =cst t Initial configuration S 0 mapping Neutral plane Cross section Thin body F 0 = j 0 (x 1, x 2 )+x 3 t 0 (x 1, x 2 ) x 1 x 2 F = j(x 1, x 2 )+x 3 t(x 1, x 2 ) Deformed configuration S mapping Thin body Two-point deformation mapping ircraft Structures - Kirchhoff-Love Plates 58
59 Shell kinematics (2) Deformation gradient Two-point deformation mapping Two-point deformation gradient Introduction to shells Small strain deformation gradient It is more convenient to evaluate these tensors in a convected basis Example g 0I basis convected to S 0 s S 0 is described by One has with the convected basis j 0,1 (x 1, x 2 ) x 2 =cst t 0 The picture shows the basis for x 3 = 0 x 1 =cst j 0,2 (x 1, x 2 ) S 0» g 0a (x 3 =0) = j 0,a ircraft Structures - Kirchhoff-Love Plates 59
60 Shell kinematics (3) Convected basis g 0I to S 0 Introduction to shells j 0,1 (x 1, x 2 ) x 2 =cst t 0 x 1 =cst j 0,2 (x 1, x 2 ) S 0 If t 0,a 0 (there is a curvature) Tension and bending are coupled This leads to locking with a FE integration: stiffness of the FE Locking can be solved by» dding internal degrees of freedom (ES) more expensive» Using only one Gauss point (reduced integration) hourglass modes (zero energy spurious deformation modes)» See next lecture Both methods lead to complex computational schemes ircraft Structures - Kirchhoff-Love Plates 60
61 Shell kinematics (4) Convected basis g 0I to S 0 (2) Introduction to shells j 0,1 (x 1, x 2 ) x 2 =cst t 0 x 1 =cst j 0,2 (x 1, x 2 ) S 0 The basis is not orthonormal vector component is still defined as So can be written?» If so which is not consistent d IJ ircraft Structures - Kirchhoff-Love Plates 61
62 Shell kinematics (5) Convected basis g 0I to S 0 (3) Introduction to shells j 0,1 (x 1, x 2 ) x 2 =cst t 0 x 1 =cst j 0,2 (x 1, x 2 ) S 0 The basis is not orthonormal (2) conjugate basis g 0 I has to be defined» Such that g 02» So vector components are defined by & a 2 g 01» The vector can be represented by g 0 2 a 1 as For plates g ircraft Structures - Kirchhoff-Love Plates 62
63 Lecture notes References ircraft Structures for engineering students, T. H. G. Megson, Butterworth- Heinemann, n imprint of Elsevier Science, 2003, ISBN Other references Papers Simo JC, Fox DD. On a stress resultant geometrically exact shell model. Part I: formulation and optimal parametrization. Computer Methods in pplied Mechanics and Engineering 1989; 72: Simo JC, Fox DD, Rifai MS. On a stress resultant geometrically exact shell model. Part II: the linear theory, computational aspects. Computer Methods in pplied Mechanics and Engineering 1989; 73: ircraft Structures - Kirchhoff-Love Plates 63
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