University of Liège erospace & Mechanical Engineering ircraft Structures Kirchhoff-Love Plates Ludovic Noels Computational & Multiscale Mechanics of Materials CM3 http://www.ltas-cm3.ulg.ac.be/ Chemin des Chevreuils 1, B4000 Liège L.Noels@ulg.ac.be ircraft Structures - Kirchhoff-Love Plates
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 2
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,1 2013-2014 ircraft Structures - Kirchhoff-Love Plates 3
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 11 2013-2014 ircraft Structures - Kirchhoff-Love Plates 4
Resultant Hooke tensor in linear elasticity Membrane mode Bending mode Shear mode Reissner-Mindlin equations summary 2013-2014 ircraft Structures - Kirchhoff-Love Plates 5
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 : 2013-2014 ircraft Structures - Kirchhoff-Love Plates 6
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 7
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 8
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) 2013-2014 ircraft Structures - Kirchhoff-Love Plates 9
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 10
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 0 2 12L 2 1 ν 1 Where L is a characteristic distance 2013-2014 ircraft Structures - Kirchhoff-Love Plates 11
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 12
Plate kinematics Kirchhoff-Love plate ssumptions Kirchhoff (cross section remains plane) & Small deformations,, Displacement field with Kinematics x 2 x 1 2013-2014 ircraft Structures - Kirchhoff-Love Plates 13
Kirchhoff-Love theory Deformations (small transformations) Reissner-Mindlin Using Dt Dt t t 1/k 11 1/Dt,1 2013-2014 ircraft Structures - Kirchhoff-Love Plates 14
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 11 2013-2014 ircraft Structures - Kirchhoff-Love Plates 15
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 16
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? 2013-2014 ircraft Structures - Kirchhoff-Love Plates 17
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)? 2013-2014 ircraft Structures - Kirchhoff-Love Plates 18
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» 2013-2014 ircraft Structures - Kirchhoff-Love Plates 19
Kirchhoff-Love theory pplied pressure (3) Energy conservation s 2013-2014 ircraft Structures - Kirchhoff-Love Plates 20
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 21
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 22
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 23
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 24
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? 2013-2014 ircraft Structures - Kirchhoff-Love Plates 25
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 26
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 27
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 28
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=1 4.161 10-3 0.055 10-3 0.005 10-3 0.001 10-3 n=3 Sym. 0.006 10-3 0.001 10-3 0.002 10-4 n=5 Sym. Sym. 0.003 10-4 0.001 10-4 n=7 Sym. Sym. Sym. 0.004 10-5 2013-2014 ircraft Structures - Kirchhoff-Love Plates 29
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 ) 2 1 0 0.5 y/a 0 0 x/a 0.5 1 2013-2014 ircraft Structures - Kirchhoff-Love Plates 30
Kirchhoff example Simply supported plate: Constant pressure p 0 (4) Resultant stress couple From & Bending Twisting 2013-2014 ircraft Structures - Kirchhoff-Love Plates 31
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 10-3 1 0 0.5 y/a 0 0 x/a 0.5 1 ~ m aa /p 0 a 2 0.04 0.02 ~ m 22 ~ m 11 m 12 /(p 0 a 2 ) ~ m ab /p 0 a 2 0.05 0 ~ m 21 21 ~ m 12 12 1 0 0.5 y/a 0 0 x/a 0.5 1-0.05 1 0.5 y/a 0 0 x/a 0.5 1 2013-2014 ircraft Structures - Kirchhoff-Love Plates 32
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 2 0.04 0.02 1 0 0.5 m ~ 22 y/a 0 0 x/a m ~ 11 0.5 1 Shear stress due to torsion For a square plate (n=0.3) m 12 /(p 0 a 2 ) ~ m ab /p 0 a 2 0.05 0 m ~ 21 m ~ 12-0.05 1 0.5 y/a 0 0 x/a 0.5 1 2013-2014 ircraft Structures - Kirchhoff-Love Plates 33
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 34
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 35
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 2013-2014 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
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? 2013-2014 ircraft Structures - Kirchhoff-Love Plates 37
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 38
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 39
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 40
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 41
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 42
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? 2013-2014 ircraft Structures - Kirchhoff-Love Plates 43
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 44
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 45
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 46
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 10-3 5 4 3 2 1 Tension increases the bending stiffness 0-10 -5 0 5 10 n 11 a 2 /D n 11 a 2 D 2013-2014 ircraft Structures - Kirchhoff-Love Plates 47
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 ) 1 0.5 0-0.5 Instability in compression See next lecture -1-50 0 50 n 11 a 2 /D n 11 a 2 D 2013-2014 ircraft Structures - Kirchhoff-Love Plates 48
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 49
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 50
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 03 2013-2014 ircraft Structures - Kirchhoff-Love Plates 51
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 52
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 53
Kirchhoff-Love plate summary Membrane mode On : With D N n n 0 = n a E a Completed by appropriate BCs Dirichlet Neumann 2013-2014 ircraft Structures - Kirchhoff-Love Plates 54
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 55
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 56
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 22 2013-2014 ircraft Structures - Kirchhoff-Love Plates 57
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 58
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 59
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 60
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 2013-2014 ircraft Structures - Kirchhoff-Love Plates 61
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 0 1 2013-2014 ircraft Structures - Kirchhoff-Love Plates 62
Lecture notes References ircraft Structures for engineering students, T. H. G. Megson, Butterworth- Heinemann, n imprint of Elsevier Science, 2003, ISBN 0 340 70588 4 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; 72:267 304. 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; 73:53 92. 2013-2014 ircraft Structures - Kirchhoff-Love Plates 63