Aircraft Structures Structural & Loading Discontinuities

Transcription

1 Universit of Liège Aerospace & Mechanical Engineering Aircraft Structures Structural & Loading Discontinuities Ludovic Noels Computational & Multiscale Mechanics of Materials CM3 Chemin des Chevreuils 1, B4000 Liège Aircraft Structures - Structural & Loading Discontinuities

2 Balance of bod B Momenta balance Linear Angular Boundar conditions Neumann Dirichlet Elasticit 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 Aircraft Structures - Structural & Loading Discontinuities 2

3 General epression for unsmmetrical beams Stress Pure bending: linear elasticit summar q With a Curvature M In the principal aes I = 0 Euler-Bernoulli equation in the principal ais for in [0 L] BCs f() T M u =0 du /d =0 M>0 L Similar equations for u Aircraft Structures - Structural & Loading Discontinuities 3

4 Beam shearing: linear elasticit summar General relationships f() T M u =0 du /d =0 M>0 L Two problems considered L Thick smmetrical section h Shear stresses are small compared to bending stresses if h/l << 1 Thin-walled (unsmmetrical) sections Shear stresses are not small compared to bending stresses Deflection mainl results from bending stresses h 2 cases Open thin-walled sections» Shear = shearing through the shear center + torque Closed thin-walled sections» Twist due to shear has the same epression as torsion t L Aircraft Structures - Structural & Loading Discontinuities 4

5 Beam shearing: linear elasticit summar Shearing of smmetrical thick-section beams Stress With b() t h h t Accurate onl if h > b Energeticall consistent averaged shear strain t A * T + T d with g g d Shear center on smmetr aes T g ma Timoshenko equations d & q On [0 L]: q g Aircraft Structures - Structural & Loading Discontinuities 5

6 Beam shearing: linear elasticit summar Shearing of open thin-walled section beams Shear flow In the principal aes T S C q T s T T T Shear center S On smmetr aes At walls intersection T Determined b momentum balance Shear loads correspond to Shear loads passing through the shear center & Torque h C t t S b t Aircraft Structures - Structural & Loading Discontinuities 6

7 Shearing of closed thin-walled section beams Shear flow Beam shearing: linear elasticit summar T T Open part (for anticlockwise of q, s) T q C p s Constant twist part T The q(0) is related to the closed part of the section, but there is a q o (s) in the open part which should be T T considered for the shear torque q C p da h ds s Aircraft Structures - Structural & Loading Discontinuities 7

8 Beam shearing: linear elasticit summar Shearing of closed thin-walled section beams Warping around twist center R T T T With u (0)=0 for smmetrical section if origin on q C p da h ds s Shear center S the smmetr ais T Compute q for shear passing thought S Use S With point S=T q C p ds s Aircraft Structures - Structural & Loading Discontinuities 8

9 Beam torsion: linear elasticit summar Torsion of smmetrical thick-section beams Circular section h t M t ma M Rectangular section C r C If h >> b & b h/b a /3 b / Aircraft Structures - Structural & Loading Discontinuities 9

10 Beam torsion: linear elasticit summar Torsion of open thin-walled section beams Approimated solution for twist rate Thin curved section Rectangles t 3 R p R t q C u s M t s t n l 2 t 2 l 3 t t 1 l 1 Warping of s-ais Aircraft Structures - Structural & Loading Discontinuities 10

11 Beam torsion: linear elasticit summar Torsion of closed thin-walled section beams Shear flow due to torsion Rate of twist Torsion rigidit for constant m M q C p da h ds s Warping due to torsion A Rp from twist center R p R q u s Y C p Aircraft Structures - Structural & Loading Discontinuities 11

12 Structure idealiation summar Panel idealiation Booms area depending on loading For linear direct stress distribution b b t D A 1 A 2 s 1 s 2 s 1 s Aircraft Structures - Structural & Loading Discontinuities 12

13 Structure idealiation summar Consequence on bending If Direct stress due to bending is carried b booms onl The position of the neutral ais, and thus the second moments of area Refer to the direct stress carring area onl Depend on the loading case onl Consequence on shearing Open part of the shear flu Shear flu for open sections T T d Consequence on torsion If no aial constraint Torsion analsis does not involve aial stress So torsion is unaffected b the structural idealiation Aircraft Structures - Structural & Loading Discontinuities 13

14 Deflection of open and closed section beams summar Virtual displacement In linear elasticit the general formula of virtual displacement reads s (1) is the stress distribution corresponding to a (unit) load P (1) D P is the energeticall conjugated displacement to P in the direction of P (1) that corresponds to the strain distribution e Eample bending of semi cantilever beam In the principal aes u =0 du /d =0 M>0 L T Eample shearing of semi-cantilever beam Aircraft Structures - Structural & Loading Discontinuities 14

15 Limitations of these theories Previousl developed equations Stresses & displacements produced b Aial loads Shear forces Bending moments Torsion No allowance for constrained warping Due to structural or loading discontinuities Eample torsion of a built-in beam No warping allowed at clamping Coupling shearing-bending neglected Effect of shear strains on the direct stress Shear strains prevent cross section to g T + T d g d remain plane T g ma Direct stress predicted b pure bending theor not correct anmore For wing bo, shear strains can be important d Aircraft Structures - Structural & Loading Discontinuities 15

16 Limitations of these theories These effects can be analed on simple problems Problem of aial constraint divided in two parts Shear stress distribution calculated at the built-in section Stress distribution calculated on the beam length for the separate loading cases of bending & torsion Problem related to instabilities as buckling See later For more comple problems Finite element simulations required Aircraft Structures - Structural & Loading Discontinuities 16

17 Shear stress distribution at a built-in end Idealied or not cross-sections Assume a beam with closed cross-section Center of twist R Undistorted section of the beam Shear flow, displacements and rotation of the section were found to be q C R p R p q u s Y With At built-in this relation simplifies into Aircraft Structures - Structural & Loading Discontinuities 17

18 Shear stress distribution at a built-in end (2) T At built-in shear flu is written B equilibrium C T R p R p T q u s Y q After substitution of shear flu Aircraft Structures - Structural & Loading Discontinuities 18

19 Shear stress distribution at a built-in end (3) New sstem of 3 equations and 3 unknowns Solution of the sstem:, & This solution is then substituted into T Shear flow and shear stress are then defined Remains true for an choice of C as long as p is computed from there C T R p R p T q u s Y q Aircraft Structures - Structural & Loading Discontinuities 19

20 Eample Built-in end Section with constant shear modulus Shear stress distribution? Center of twist? C D B T = 22 kn T = 0.1 m A Wall Length (m) Thickness (mm) AB BC CD DA Aircraft Structures - Structural & Loading Discontinuities 20

21 Deformation Sign convention: >0 anticlockwise Angle a: sin a = 0.25/0.5 a = 30 Coefficients B T = 22 kn C D a q,s,q,y T = 0.1 m A Wall Length (m) Thickness (mm) AB BC CD DA Aircraft Structures - Structural & Loading Discontinuities 21

22 Deformation (2) Coefficients (2) B T = 22 kn C D a q,s,q,y T = 0.1 m A Wall Length (m) Thickness (mm) AB BC CD DA Aircraft Structures - Structural & Loading Discontinuities 22

23 Deformation (3) Coefficients (3) B T = 22 kn C D a q,s,q,y T = 0.1 m A Wall Length (m) Thickness (mm) AB BC CD DA Aircraft Structures - Structural & Loading Discontinuities 23

24 Deformation (4) Coefficients (4) B T = 22 kn C D a q,s,q,y T = 0.1 m A Wall Length (m) Thickness (mm) AB BC CD DA Aircraft Structures - Structural & Loading Discontinuities 24

25 Deformation (5) Coefficients (5) B T = 22 kn C D a q,s,q,y T = 0.1 m A Wall Length (m) Thickness (mm) AB BC CD DA Aircraft Structures - Structural & Loading Discontinuities 25

26 Deformation (6) Coefficients (6) B T = 22 kn C D a q,s,q,y T = 0.1 m A Wall Length (m) Thickness (mm) AB BC CD DA Aircraft Structures - Structural & Loading Discontinuities 26

27 Deformation (7) Sstem with origin of the ais at point A (C A) Aircraft Structures - Structural & Loading Discontinuities 27

28 Deformation (8) Sstem (2) Aircraft Structures - Structural & Loading Discontinuities 28

29 Shear flu Wall AB C D a q,s,q,y B T = 22 kn T = 0.1 m A Wall DA Wall Length (m) Thickness (mm) AB BC CD DA Aircraft Structures - Structural & Loading Discontinuities 29

30 Shear flu (2) B T = 22 kn Wall BC C D a q,s,q,y T = 0.1 m A Wall Length (m) Thickness (mm) AB BC CD DA Aircraft Structures - Structural & Loading Discontinuities 30

31 Shear flu (3) B T = 22 kn Wall CD C D a q,s,q,y T = 0.1 m A Wall Length (m) Thickness (mm) AB BC CD DA Aircraft Structures - Structural & Loading Discontinuities 31

32 Center of twist Sstem linked to point A B T = 22 kn Remarks The center of twist Depends on loading ( T and T) Does not correspond to the center of shear Due to the warping constrain Shear flu discontinuiti at corners Requires booms in order of avoiding stress concentrations C a T = q,s,q,y 0.1 m D A Wall Length (m) Thickness (mm) AB BC CD DA Aircraft Structures - Structural & Loading Discontinuities 32

33 Thin walled rectangular-section beam subjected to torsion In the case of free warping, we found B A t h C M s C t b b D h If warping is constrained (built-in end) Direct stress are introduced Different shear stress distribution C M Aircraft Structures - Structural & Loading Discontinuities 33

34 Thin walled rectangular-section beam subjected to torsion (2) Idealiation Warping to be suppressed is linear & smmetrical Direct stress also linear & smmetrical Idealiation b t D t h B C M s A s 1 s 2 b C t b Four identical booms carring direct stress onl b D h s 1 A 1 A 2 s 2 Panels carr shear flu onl Aircraft Structures - Structural & Loading Discontinuities 34

35 Thin walled rectangular-section beam subjected to torsion (3) Warping at a given section Shearing (see beam lecture) du ds s d n du s Warping If u m is the maimum warping On webs d u m On covers q b qh h L M b Aircraft Structures - Structural & Loading Discontinuities 35

36 Thin walled rectangular-section beam subjected to torsion (4) Warping of a given section (2) Kinematics See lecture on beams As twist center is at section center (b smmetr) On webs C R p R q u s On covers Combining results On webs d u m q b On covers h qh L M b Aircraft Structures - Structural & Loading Discontinuities 36

37 Thin walled rectangular-section beam subjected to torsion (5) Torque From shear flow q h & q b d q b qh Using h M b L Twist rate is directl obtained Aircraft Structures - Structural & Loading Discontinuities 37

38 Thin walled rectangular-section beam subjected to torsion (6) Shear flows From shear flow q h & q b d q b qh Using h L M b Missing balance equation is obtained from boom balance Aircraft Structures - Structural & Loading Discontinuities 38

39 Thin walled rectangular-section beam subjected to torsion (7) Boom (of section A) balance equation As boom carries direct stress onl s q b With s + s d q q i h d Aircraft Structures - Structural & Loading Discontinuities 39

40 Thin walled rectangular-section beam subjected to torsion (8) Differential equation with Solution General form Boundar conditions at = 0 (constraint warping) Boundar conditions at = L (free edge) Final form Aircraft Structures - Structural & Loading Discontinuities 40

41 Thin walled rectangular-section beam subjected to torsion (9) Warping At free end: To be compared with the warping of the free-free beam Same for L d u m q b qh h L M b Aircraft Structures - Structural & Loading Discontinuities 41

42 Thin walled rectangular-section beam subjected to torsion (10) Direct stress in booms Direct load in booms d u m q b qh h L M b Aircraft Structures - Structural & Loading Discontinuities 42

43 Thin walled rectangular-section beam subjected to torsion (11) Shear flow Using d u m The shear flows becomes q b qh h L M b Aircraft Structures - Structural & Loading Discontinuities 43

44 Thin walled rectangular-section beam subjected to torsion (12) Shear stress d u m q b h qh L covers M b webs /L Aircraft Structures - Structural & Loading Discontinuities 44

45 Thin walled rectangular-section beam subjected to torsion (13) Rate of twist Using The rate of twist becomes d u m q b To be compared with the unconstraint theor h qh L Constraint reduces the twist rate M b Aircraft Structures - Structural & Loading Discontinuities 45

46 Problem of aial constraint In previous eample the twist center was known b smmetr In the general case Twist center differs from shear center due to aial constraint Proceed b increment of DL Shear stress distribution calculated at the built-in section» As in first eample» Allows determination of the twist center AT THAT SECTION Use the previousl developed theor on DL New stress distribution on the new section» New twist center» Aircraft Structures - Structural & Loading Discontinuities 46

47 Shear lag Beam shearing Shear strain in cross-section Deformation of cross-section Elementar theor of bending For pure bending Not valid anmore due to cross section deformation New distribution of direct stress For wings Wide & thin walled beam Shear distortion of upper and lower skins causes redistribution of stress in the stringers Aircraft Structures - Structural & Loading Discontinuities 47

48 Eample Assumptions Doubl smmetrical 6-boom beam Shear load trough shear center No twist No warping due to twist Uniform panel thickness t Shear loads applied at corner booms q d A 1 A 2 A 1 q h h L T /2 d d T / Aircraft Structures - Structural & Loading Discontinuities 48

49 Shear lag (2) For a given section Uniform shear flow between booms Shear flow in web should balance the shear load q d A 1 A 2 A 1 q h Corner booms subjected to opposite loads P 1, with, b equilibrium h L T /2 d d T /2 Equilibrium of central boom Due to smmetric distribution of q d q d q d P 1 P 1 q h P 1 P 1 P 2 q h q d h T / Aircraft Structures - Structural & Loading Discontinuities 49 d

50 Shear lag (3) For a given section (2) Equilibrium of the cover At the free end q d A 1 A 2 A 1 q h Summar h L T /2 d d T /2 q d P 2 P 1 Third equation is the integration of the first two 3 unknowns so one equation is missing Compatibilit P 1 q h L Aircraft Structures - Structural & Loading Discontinuities 50

51 Shear lag (4) Deformations of top cover q d A 1 A 2 A 1 q h As h T /2 d d T /2 L q h (1+e 1 )d d g g + g d d (1+e 2 )d q h d Aircraft Structures - Structural & Loading Discontinuities 51

52 Shear lag (5) Equations q d A 1 A 2 A 1 q h h L T /2 d d T /2 General solution with Aircraft Structures - Structural & Loading Discontinuities 52

53 Shear lag (6) General solution Boundar conditions Zero aial load at = L C 1 = 0 Zero shear deformation at = 0 As & Booms direct loadings Aircraft Structures - Structural & Loading Discontinuities 53

54 Shear lag (7) Direct load in top cover q d A 1 A 2 A 1 q h h L T /2 d d T /2 Pure bending theor leads to q d P 2 P 1 Compared to pure bending theor Compression in central boom is lower Compression in corner boom is higher P 1 q h L Aircraft Structures - Structural & Loading Discontinuities 54

55 Shear lag (8) Shearing of top cover As q d A 1 A 2 A 1 q h h L Deformation of top cover T /2 d d T /2 d d Aircraft Structures - Structural & Loading Discontinuities 55

56 Shear lag (9) Remark The solution depends on BCs For a realistic wing structure, intermediate stringers have different BCs d d Aircraft Structures - Structural & Loading Discontinuities 56

57 Open-section beam I-section beam subjected to torsion without built-in end Reminder Shear Or R p R q t C M t s t n Warping Particular case of the I-Section beam u s t 3 t There is no shear stress at mid plane of flanges The remain rectangular after torsion t t 2 l 2 M l 3 t 1 l 1 M M Aircraft Structures - Structural & Loading Discontinuities 57

58 Open-section beam I-section beam subjected to torsion with built-in end Contraril to the free/free beam M M Presence of the built-end leads to deformation of the flanges M The beam still twists but with a non-constant twist rate Method of solving: Combination of Saint-Venant shear stress Bending of flanges Aircraft Structures - Structural & Loading Discontinuities 58

59 Open-section beam I-section beam subjected to torsion with built-in end (2) Saint-Venant shear stress Where is not constant M Aircraft Structures - Structural & Loading Discontinuities 59

60 Open-section beam I-section beam subjected to torsion with built-in end (3) Bending of the flanges For a given section Angle of torsion q Lateral displacement of lower flange» q d Bending moment in lower flange» h L» With M b f» It has been assumed that displacement of the flange results T f from bending onl Shearing in the lower flange» h t f q T f M f b f Aircraft Structures - Structural & Loading Discontinuities 60

61 Open-section beam I-section beam subjected to torsion with built-in end (4) Bending of the flanges (2) For a given section (2) Shearing in the lower flange» q d As shearing in top flange is in opposite direction, moment due to bending of the flange becomes h b f L M Total torque on the beam T f h q t f T f M f b f Aircraft Structures - Structural & Loading Discontinuities 61

62 Open-section beam Arbitrar-section beam subjected to torsion with built-in end Wagner torsion theor Assumptions Length >> sectional dimensions Undistorted cross-section C Shear stress at midsection s s M negligible» But shear load not negligible Under these assumptions, we can use the primar warping (of mid section) epression developed for torsion of free/free open-section beams R p R t C M t s t n q As twist rate is not constant u s t There is a direct induced stress Aircraft Structures - Structural & Loading Discontinuities 62

63 Open-section beam Arbitrar-section beam subjected to torsion with built-in end (2) Wagner torsion theor (2) Direct stress resulting from primar warping s As onl a torsion couple is applied M Integrating on the whole section C t should lead to Aircraft Structures - Structural & Loading Discontinuities 63

64 Open-section beam Arbitrar-section beam subjected to torsion with built-in end (3) Wagner torsion theor (3) Direct stress resulting from primar warping (2) s As onl a torsion couple is applied (2) M Direct stress is equilibrated b shear flow s s + s s s ds See lecture on beams s In this case s q + s q ds s q + q d + s d ds d q q s s Aircraft Structures - Structural & Loading Discontinuities 64

65 Open-section beam Arbitrar-section beam subjected to torsion with built-in end (4) Wagner torsion theor (4) Equations s M As for s = 0 (free edge) q(0) = Aircraft Structures - Structural & Loading Discontinuities 65

66 Open-section beam Arbitrar-section beam subjected to torsion with built-in end (5) Wagner torsion theor (5) Torque With R p R t C M t s t n q u s t Aircraft Structures - Structural & Loading Discontinuities 66

67 Open-section beam Arbitrar-section beam subjected to torsion with built-in end (6) Wagner torsion theor (6) Torque (2) Using the second term becomes For s = 0, A Rp = 0 For s = L, as the edge is free, there is no shear flu Using these two boundar conditions, second term is rewritten Aircraft Structures - Structural & Loading Discontinuities 67

68 Open-section beam Arbitrar-section beam subjected to torsion with built-in end (7) Wagner torsion theor (7) Torque (3) Using the integral of first term becomes As for s = 0, A Rp = 0, and using The final epression reads Aircraft Structures - Structural & Loading Discontinuities 68

69 Open-section beam Arbitrar-section beam subjected to torsion with built-in end (8) General epression for torque With Case of the I-section beam Center of twist is the center of smmetr C For the web: A Rp (s) = 0 no contribution to C G For lower flange For the I-section h t f C M s b f Aircraft Structures - Structural & Loading Discontinuities 69

70 Open-section beam Arbitrar-section beam subjected to torsion with built-in end (9) Case of the I-section beam (2) Epression With To be compared with T f h t f q T f M f b f Aircraft Structures - Structural & Loading Discontinuities 70

71 Open-section beam Idealied beam subjected to torsion with built-in end For idealied sections with booms In epression The direct stress is carried out b t direct & Booms of section A i d Aircraft Structures - Structural & Loading Discontinuities 71

72 Open-section beam Applications of beam subjected to torsion with built-in end Solution for pure torque with s Solution M Boundar conditions At built-in end = 0: No warping, and as At free end = L: no direct load, and as Aircraft Structures - Structural & Loading Discontinuities 72

73 Open-section beam Applications of beam subjected to torsion with built-in end (2) Solution for pure torque (2) Twist rate s M Free/free end 1 Built-end /L Aircraft Structures - Structural & Loading Discontinuities 73

74 Open-section beam Applications of beam subjected to torsion with built-in end (3) Solution for pure torque (3) Angle of twist As s M Boundar condition at built end = 0: No twist At free end Reduction compared to freefree case Aircraft Structures - Structural & Loading Discontinuities 74

75 Open-section beam Applications of beam subjected to torsion with built-in end (4) Distributed torque loading m Two contributions to torque Balance equation M m M + M d d As To be solved with adequate boundar conditions Built-in end: q = 0 & q, = 0 (no warping) Free end: q, = 0 (no direct stress) & No torque at free end Aircraft Structures - Structural & Loading Discontinuities 75

76 Open-section beam Remark We have studied Aial loading resulting from torsion A similar theor can be derived to deduce torsion resulting from aial loading Aircraft Structures - Structural & Loading Discontinuities 76

77 Lecture notes References Aircraft Structures for engineering students, T. H. G. Megson, Butterworth- Heinemann, An imprint of Elsevier Science, 2003, ISBN Other references Books Mécanique des matériau, C. Massonet & S. Cescotto, De boek Université, 1994, ISBN X Aircraft Structures - Structural & Loading Discontinuities 77