Universit of Liège Aerospace & Mechanical Engineering Aircraft Structures Structural & Loading Discontinuities 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 Aircraft Structures - Structural & Loading Discontinuities
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 2
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 3
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 4
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 5
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 6
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 2013-2104 Aircraft Structures - Structural & Loading Discontinuities 7
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 8
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 1 1.5 2 4 a 0.208 0.231 0.246 0.282 1/3 b 0.141 0.196 0.229 0.281 1/3 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 9
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 10
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 11
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 2 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 12
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 13
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 14
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 15
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 16
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 17
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 18
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 19
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 0.375 1.6 BC 0.500 1.0 CD 0.125 1.2 DA 1.0 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 20
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 0.375 1.6 BC 0.500 1.0 CD 0.125 1.2 DA 1.0 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 21
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 0.375 1.6 BC 0.500 1.0 CD 0.125 1.2 DA 1.0 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 22
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 0.375 1.6 BC 0.500 1.0 CD 0.125 1.2 DA 1.0 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 23
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 0.375 1.6 BC 0.500 1.0 CD 0.125 1.2 DA 1.0 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 24
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 0.375 1.6 BC 0.500 1.0 CD 0.125 1.2 DA 1.0 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 25
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 0.375 1.6 BC 0.500 1.0 CD 0.125 1.2 DA 1.0 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 26
Deformation (7) Sstem with origin of the ais at point A (C A) 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 27
Deformation (8) Sstem (2) 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 28
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 0.375 1.6 BC 0.500 1.0 CD 0.125 1.2 DA 1.0 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 29
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 0.375 1.6 BC 0.500 1.0 CD 0.125 1.2 DA 1.0 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 30
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 0.375 1.6 BC 0.500 1.0 CD 0.125 1.2 DA 1.0 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 31
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 0.375 1.6 BC 0.500 1.0 CD 0.125 1.2 DA 1.0 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 32
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 33
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 34
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 35
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 36
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 37
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 38
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 39
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 40
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 41
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 42
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 43
Thin walled rectangular-section beam subjected to torsion (12) Shear stress d u m q b h qh L covers M b webs /L 1 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 44
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 45
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» 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 46
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 47
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 /2 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 48
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 /2 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 49 d
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- 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 50
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 51
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 52
Shear lag (6) General solution Boundar conditions Zero aial load at = L C 1 = 0 Zero shear deformation at = 0 As & Booms direct loadings 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 53
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- 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 54
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 55
Shear lag (9) Remark The solution depends on BCs For a realistic wing structure, intermediate stringers have different BCs d d 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 56
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 57
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 58
Open-section beam I-section beam subjected to torsion with built-in end (2) Saint-Venant shear stress Where is not constant M 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 59
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 60
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 61
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 62
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 0 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 63
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 64
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) = 0 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 65
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 66
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 67
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 68
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 69
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 70
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 71
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 72
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 1 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 73
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 74
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 75
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 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 76
Lecture notes References Aircraft Structures for engineering students, T. H. G. Megson, Butterworth- Heinemann, An imprint of Elsevier Science, 2003, ISBN 0 340 70588 4 Other references Books Mécanique des matériau, C. Massonet & S. Cescotto, De boek Université, 1994, ISBN 2-8041-2021-X 2013-2014 Aircraft Structures - Structural & Loading Discontinuities 77