Lecture 5 Buckling Buckling of a structure means failure due to excessive displacements (loss of structural stiffness), and/or
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1 AOE 204 Inroducion o Aeropace Engineering Lecure 5 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. Lec. 5: of 2
2 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 Lec. 5: 2 of 2
3 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 Lec. 5: 3 of 2
4 Load-horening curve for compreion erfecly raigh, elaic column 0 compreive axial force axial diplacemen criical load of perfec rucure diplacemen a he criical load < < > > 0 pure compreion u u fla plae column cylindrical hell able u unable 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 Lec. 5: 4 of 2
5 Load-horening curve (coninued) A perfecly fla recangular plae All four edge uppored; elaic buckling pure compreion u fla plae column > unloaded edge 0 u cylindrical hell 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. Lec. 5: 5 of 2
6 Load-horening curve (coninued) A circular cylindrical hell > R 0 pure compreion 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. u u fla plae column cylindrical hell Lec. 5: 6 of 2
7 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 < < > > 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. Lec. 5: 7 of 2
8 For column deign ue minimum I h h > I min I max 3 h h h w b f I max f I min b f 6 h 2 b f 2 for mo I-ecion h w 2 r I min I max πr 3 hin-walled ube Lec. 5: 8 of 2
9 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 E cr b 2( 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 a b Acually i a funcion of he plae apec raio. The formula given above i good lower bound eimae of for a b >.0. Lec. 5: 9 of 2
10 Buckling load for a circular cylindrical hell Theory give he formula for he criical compreive axial normal re,, a σ cr σ cr E 3 ( ν 2 ) R The correponding compreive axial normal force,, a he buckling i obained from σ cr ( 2πR) hin hell R --» R L Lec. 5: 0 of 2
11 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 ν R 2 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 Lec. 5: of 2
12 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 R If i mall, hen he hell will no buckle in he elaic maerial range. Lec. 5: 2 of 2
Buckling of a structure means failure due to excessive displacements (loss of structural stiffness), and/or
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
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