Εθνικό Μετσόβιο Πολυτεχνείο National Technical Universit of thens erodnamics & eroelasticit: Beam Theor Σπύρος Βουτσινάς / Spros Voutsinas
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simple eample: the beam Beams are defined as slender bodies, i.e. solid bodies with one dominant dimension (the other two are assumed much smaller). Throughout the our analsis, the ais of the beam will coincide with ais of the sstem defining the beam. Beams are in theor considered as one-dimensional structures. This means that all detail over its cross sections will be neglected and cross sections are behaving as rigid surfaces. s we are going to see this is done b integration. lthough cross sections will not deform, the can displace b following the ais of the beam. Beams can deform in three was: in fleion, in bending and in torsion. These three modes of deformation are described b 3 displacements: u,v,w and 3 rotations:,, erodnamics & eroelasticit Beam Theor 3
simple eample: the beam Beam is a slender structure with one of its dimensions dominant Tension, bendings and torsion can be defined: O Radial fleion v O Bending w θ O In classical beam theor the 3 displacements and the torsion rotation are independent while the bending rotations are given as: Torsion θ θ w θ u erodnamics & eroelasticit Beam Theor 4
simple eample: the beam Definition of displacements & strains P 0 (u,w) P u v U u 1 0 0 0 0 w V v 0 1 0 0 W w 0 0 1 0 0 (U,V,W) defines the displacements of an arbitrar point P 0 of a beam section. u u 1 0 0 0 0 0 0 v v o 0 1 r ro 0 1 0 0 o 0 o 0 o w r S u Su w 0 0 1 o 0 0 1 0 Usuall S 1 is eliminated in which case we are left onl with S 0 erodnamics & eroelasticit Beam Theor 5
simple eample: the beam Definition of displacements & strains P 0 (u,w) P u v U u 1 0 0 0 0 w V v 0 1 0 0 W w 0 0 1 0 0 (U,V,W) defines the displacements of an arbitrar point P 0 of a beam section. V 1U V W U V U U V V W W W V U U V V W W v u w v u w erodnamics & eroelasticit Beam Theor 6
simple eample: the beam Definition of stresses & internal loads In beam theor stresses, strains and displacements are related as follows: σ Eε V E v E E u E w Ev Eu Ew τ τ Gγ Gγ U V G W V G θ G θ G Gθ Gθ τ τ Where E is Young s modulus and G the torsion rigidit or stiffness. Primes will denote from now on differentiation with respect to. erodnamics & eroelasticit Beam Theor 7
simple eample: the beam Ev Eu Ew E G G F d Evd Ev M ( - )d G( ) d GI t v M w u F M M M - d - (-E u - E w )d EI u EI w M d (-E u- E w )d -EI u- EI w erodnamics & eroelasticit Beam Theor 8 GI G( ) d EI EI EI t Ed E d E d E d
simple eample: the beam Equilibrium equations The balance of forces and moments over a differential length d of the beam result in the equations of motion. In classical beam theor there will be 4 eq s: the 3 force equations and the torsion moment equation. In case shear is added then 6 eq s will be derived F +df M +dm δl M F +df M +dm F F (a) dr r a (b) F +df M F M +dm M erodnamics & eroelasticit Beam Theor 9
simple eample: the beam Balance of forces along the df U u d( u ) dg L d In order to determine the derivative of F we use the moment balance in. 0 dm Fd Fdu O( d ) df d FM (F u ) F (EI u ) (EI w ) (Fu ) ( ) u ( ) * ( EI u) ( EI w ) ( Tu) ( ) g L Where T=F and * denotes the location of the center of gravit. erodnamics & eroelasticit Beam Theor 10
simple eample: the beam Balance of forces along the ( - df W w d w ) dg L d In order to determine the derivative of F we use the moment balance in. 0 dm Fd Fdw O( d ) df d FM (F w ) F (EI u ) (EI w ) (F w ) * ( )w ( ) (EI u ) (EI w ) (Tw) ( )g L erodnamics & eroelasticit Beam Theor 11
simple eample: the beam Balance of forces along the -ais (tension) B neglecting and introducing surface properties we obtain: ρd(u - w ) d (v u - w ) (( E) v) ( )v d g (( E) v) ( )g L L Balance of moments in the _ais (torsion) d( )θ ρu ρw)d (GI θ t ( a ) δl ρ(g a δl ) g )d * * * * ( I )θ ( ) u (ρ) w (GIθ ) ( ) g ( ) g t t ( δl δl ) a a erodnamics & eroelasticit Beam Theor 1
simple eample: the beam In matri form: 0 1 dii1s u diis u = K u K u K u K u dii g II L 11 1 1 1 1 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 II 1 II 0 0 1 ΙI 0 0 1 a 0 0 1 0 0 0 a a flap tension torsion edge F 0 0 0 EI 0 0 0 0 0 EI 0 0 E 0 0 0 E 0 0 0 0 0 0 E 0 E 0 0 0 0 0 K Κ K Κ 11 0 0 F 0 EI 0 0 0 0 0 EI 0 1 1 0 E 0 0 0 0 0 GI 0 0 0 0 t 0 0 0 0 0 0 0 0 erodnamics & eroelasticit Beam Theor 13
How can we generate structural data? Beams are approimations of otherwise 3D structures. In order to appl beam theor we need to know the averaged properties of the section. To this end we need as input the sectional geometr as well as the distribution of the basic material properties: mass densit and the two elastic module E and G. Then for most of the properties simple averaging is required. For eample: Cell 1 Cell Cell 3 L e (X 1e,Y 1e ) (X e,y e ) E= el e = N e=1 N e=1 L t E e e e e E(X X ) / e e 1e e E N ρ = ρe e=1 e erodnamics & eroelasticit Beam Theor 14
How can we generate structural data? From averaging we get: E, G, ρ, EI, EI, EI, CM, CM, el, el However, GI t and the shear center which defines the ais of a beam, requires a more elaborate procedure. For a section in pure torsion, the St Venant torsional constant I t is defined as: M t It= G We need to calculate the twist for e.g. M t =1. The equations we introduce shear flows along the section and appl: shear flow continuit, balance of moments, and finall that all cells should undergo the same twist. e q 6 M t q 1 q 5 q 3 q q 4 e erodnamics & eroelasticit Beam Theor 15
How can we generate structural data? The shear center is the point wherefrom the resultant load must pass in order to prevent the development of twisting moments. For multi cell sections, the procedure is as follows: 1. In ever cell we introduce a slit. n etra shear flow is added so that all slits close, We appl again shear flow continuit and compatibilit at nodes. The shear flows will generate a moment which must counter the moment generated b the eternal force V which defines the shear center. 1 a 4 6 e 4 3 1 1 b a b 3 a 3 b Node 1 5 q 1 4 e q 1 1b q 1 b erodnamics & eroelasticit Beam Theor 16
End of presentation erodnamics & eroelasticit Beam Theor 17
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