Buckling Buckling of a rucure mean failure due o exceive diplacemen (lo of rucural iffne), and/or lo of abiliy of an equilibrium configuraion of he rucure The rule of humb i ha buckling i conidered a mode of failure for lender member in compreion, or for hin panel in compreion or hear. Abou 50% of an airplane deign may be limied by he buckling of hin kin. AOE 2104 Inro. o Aero Engineering. Lec. 6: 1 of 19
Concep of abiliy of equilibrium Sabiliy of equilibrium mean ha he repone of he rucure due o a mall diurbance from i equilibrium configuraion remain mall; he maller he diurbance he maller he reuling magniude of he diplacemen in he repone.if a mall diurbance caue large diplacemen, perhap even heoreically infinie, hen he equilibrium ae i unable. able equilibrium g unable equilibrium neural equilibrium AOE 2104 Inro. o Aero Engineering. Lec. 6: 2 of 19
Sabiliy of equilibrium wih repec o load racical rucure are able a no load. Now conider increaing he load lowly. We are inereed in he value of he load, called he criical load, a which buckling occur. Tha i, we are inereed in when a equence of equilibrium ae a a funcion of he load, one ae for each value of he load, ceae o be able. 0 unloaded configuraion able equilibrium 0 < < ure compreion repone able equilibrium > urely compreive configuraion loe abiliy o a combined bending compreion configuraion AOE 2104 Inro. o Aero Engineering. Lec. 6: 3 of 19
Load-horening curve for compreion erfecly raigh, elaic column cr 0 compreive axial force axial diplacemen criical load of perfec rucure diplacemen a he criical load < < cr > > cr 1 0 pure compreion 1 u u fla plae column cylindrical hell able u unable cr Afer buckling he column canno rei much of an increae in load. I pobuckling iffne i near zero. I i aid he column i neural in pobuckling AOE 2104 Inro. o Aero Engineering. Lec. 6: 4 of 19
Load-horening curve (coninued) A perfecly fla recangular plae All four edge uppored; elaic buckling 1 pure compreion u fla plae column > unloaded edge 0 1 u cylindrical hell cr able u unable unloaded edge > The plae can rei increaed load afer buckling becaue he unloaded edge are uppored. I iffne i reduced in pobuckling. The plae i aid o have pobuckling rengh. AOE 2104 Inro. o Aero Engineering. Lec. 6: 5 of 19
Load-horening curve (coninued) A circular cylindrical hell > cr R 1 0 pure compreion cr able u unable The hell canno rei increaed load afer buckling. The load and diplacemen decreae on he iniial, unable pobuckling equilibrium pah. The hell ha no pobuckling rengh. Deigner have o knockdown he value of obained from he heory of he perfec hell by a ubanial amoun. 1 u u fla plae column cylindrical hell AOE 2104 Inro. o Aero Engineering. Lec. 6: 6 of 19
Euler load for a pinned-pinned column π ----------- 2 EI b 2 The criical load increae wih increaed bending iffne EI, and decreae wih increaing column lengh b. N.B., Eq. 7.24, p. 263, in he ex i incorrec. b 0 < < cr > > cr For deign of elaic column he criical load, or Euler load, i ued o deermine failure by buckling. Alo ue he minimum I for he cro-ecional area. AOE 2104 Inro. o Aero Engineering. Lec. 6: 7 of 19
For column deign ue minimum I h h > I min I max ------- 3 h 12 h ------- 3 12 h w b f f I min I max b 3 -------- f 6 h 2 b ------------ f 2 for mo I-ecion h 3 + ---------- w 12 r I min I max πr 3 hin-walled ube AOE 2104 Inro. o Aero Engineering. Lec. 6: 8 of 19
Deign buckling load for an elaic plae For purpoe of deign, he compreive buckling load of a recangular, hin plae wih all four edge uppored by hinge i ν 4π 2 cr -------- ------------------------- E 3 b 12( 1 ν 2 ) where i oion raio, a dimenionle maerial propery. ν 0.3 For mo aluminum alloy. all four edge wih hinge uppor a 0 < «a and b b Acually a b i a funcion of he plae apec raio. The formula given above i good lower bound eimae of for a b > 1.0. AOE 2104 Inro. o Aero Engineering. Lec. 6: 9 of 19
Buckling load for a circular cylindrical hell Theory give he formula for he criical compreive axial normal re,, a σ cr σ 1 cr -------------------------- E ----- 31 ( ν 2 ) R The correponding compreive axial normal force,, a he buckling i obained from σ cr ( 2πR) hin hell R ---» 1 R L AOE 2104 Inro. o Aero Engineering. Lec. 6: 10 of 19
Deign buckling load for he hell The formula for σ cr above i valid for elaic buckling of a hin, circular cylindrical hell if he hell i moderaely long. Moderaely long i characerize by parameer Z > 2.85, where hi parameer (called he Badorf parameer) i defined by Z L ----- 2 1 ν 2 R If he hell i oo long i will buckle a a column raher han hell, and we ue Euler formula o eimae ha criical load. For deign we ue a knockdown facor, which accoun for he fac ha experimenal value of he buckling load of axially compreed circular cylindrical hell are ubanially le han he heoreical predicion. Tha i, he deign buckling load i relaed o he heoreical value by σ cr deign γ γσ cr heory AOE 2104 Inro. o Aero Engineering. Lec. 6: 11 of 19
Deign buckling load for he hell (concluded) The knockdown facor i a funcion of he radiu o hickne R γ R raio,. Facor decreae for hinner hell, i.e., a increae. For example, a deign recommendaion i R/ γ 10 0.84 100 0.58 500 0.32 1000 0.23 4000 0.11 R If i mall, hen he hell will no buckle in he elaic maerial range. AOE 2104 Inro. o Aero Engineering. Lec. 6: 12 of 19
Srucural erformance Indice For aeropace applicaion weigh i an exremely imporan meaure of he performance of a rucure. In comparing differen maerial one need o ake ino accoun heir relaive efficiency in erm of rengh o weigh raio and iffne o weigh raio. However, hee raio depend on he ype of loading. We will compare hee performance indice for a panel of lengh a, widh b, and hickne. a b AOE 2104 Inro. o Aero Engineering. Lec. 6: 13 of 19
Tenion panel The weigh of he panel i w ( ρg)ab where g ρ ma deniy acceleraion due o graviy Aume he lengh a and widh b are given. For he panel ubjeced o enion, he axial force N i N pecified. Le σ f denoe he failure rengh of he maerial. The value of σ could be he a f enile rengh, or he yield rengh, or an allowable re for he maerial. b N AOE 2104 Inro. o Aero Engineering. Lec. 6: 14 of 19
Tenion panel (coninued) The axial normal re in he panel i, and we e hi equal o he failure re σ f of he maerial and olve for he hickne o ge N ( bσ f ). Now eliminae he hickne in he weigh equaion o find N ( b) w N a -------------------- 1 ( ρg) σ f Here he axial force N i a pecified funcional requiremen, lengh a i pecified geomery, and σ f ( ρg) i a maerial propery. A maerial wih a large value of σ f ( ρg) will reul in a lower weigh panel of pecified dimenion a by b ha ha o carry he pecified enile force N. AOE 2104 Inro. o Aero Engineering. Lec. 6: 15 of 19
Tenion panel (coninued) From able 7.3 in he ex, we li value of rengh o pecific weigh raio, ( ρg), for eleced maerial. σ f Maerial, lug/in 3, in 10 3 in 4340 Seel 0.00879 636 2024-T4 Aluminum 0.00311 570 7075-T6 Aluminum 0.00314 772 Tianium 0.00497 981 Graphie/Epoxy 0.00174 3034 a Spruce wood 0.00048 608 a a. correced from he value lied in Table 7.3 ρ σ u ( ρg) AOE 2104 Inro. o Aero Engineering. Lec. 6: 16 of 19
Tenion panel pecified iffne The panel iffne i pecified, where iffne K i defined a K N and i he elongaion under enile load N. σ From Hooke law, bu and. So, N Eε σ N ( b) ε a N be( a) a b N Noe ha K N and we ee ha K b ---- E a. AOE 2104 Inro. o Aero Engineering. Lec. 6: 17 of 19
Tenion panel pecified iffne (con.) Now olve for o ge Ka ------- Eb Eliminae hickne in he weigh equaion and wrie i a w Ka 2 ------------------ 1 E ( ρg) where K i he funcional requiremen (pecified iffne) a 2 i he pecified geomery and E ( ρg) i a maerial propery AOE 2104 Inro. o Aero Engineering. Lec. 6: 18 of 19
Tenion panel pecified iffne (concl.) A maerial wih a large value of pecific modulu reul in a lower weigh panel. The pecific modulu i Young modulu divided by he pecific weigh of he maerial. E ( ρg) Maerial, lug/in 3, ρ in 10 6 in 4340 Seel 0.00879 102 a 2024-T4 Aluminum 0.00311 107 7075-T6 Aluminum 0.00314 102 a Tianium 0.00497 100 Commen The meal panel all have abou he ame pecific iffne. Graphie/Epoxy 0.00174 393 impreive Spruce wood 0.00048 84 a a. Correced from he value lied in Table 7.3 of he ex AOE 2104 Inro. o Aero Engineering. Lec. 6: 19 of 19