Torsion, Thermal Effects and Indeterminacy

Transcription

1 ENDS Note Set 7 F007bn orson, herml Effects nd Indetermncy Deformton n orsonlly Loded Members Ax-symmetrc cross sectons subjected to xl moment or torque wll remn plne nd undstorted. At secton, nternl torque (resstng ppled torque) s mde up of sher forces prllel to the re nd n the drecton of the torque. he dstrbuton of the sherng stresses depends on the ngle of twst, φ. he cross secton remns plne nd undstored. Sherng Strn Sherng strn s the ngle chnge of strght lne segment long the xs. where γ ρ s the rdl dstnce from the centrod to the pont under strn. he mum strn s t the surfce, dstnce c from the centrod: ρφ L γ cφ L G s the Sher Modulus or Modulus of Rgdty: τ G γ Sherng Strn nd Stress In the lner elstc rnge: the torque s the summton of torson stresses over the re: τj ρ gves: ρ τ J Mxmum torsonl stress, τ, occurs t the outer dmeter (or permeter). Polr Moment of Inert For x-symmetrc shpes, there s only one vlue for polr moment of nert, J, determned by the rdus, c: sold secton: J πc hollow secton: J ( ) c o c π

2 ENDS Note Set 7 F007bn Combned orson nd Axl Lodng Just s wth combned xl lod nd sher, combned torson nd xl lodng result n mum sher stress t 5 oblque plne of twst. Sherng Strn In the lner elstc rnge: L JG nd for composte shfts: L Σ J G orson n Noncrculr Shpes J s no longer the sme long the lterl xes. Plne sectons do not remn plne, but dstort. τ s stll t the furthest dstnce wy from the centrod. For rectngulr shpes: τ For /b > 5: c b L c b G ( 0. b ) c c 60 b Open Sectons For long nrrow shpes where /b s very lrge (/b ) c c / nd: τ b L b G > b Sher Flow of Closed hn Wlled Sectons q s the nternl sherng force per unt length, nd s constnt on cross secton even though the thckness of the wll my very. s the re bounded by the centerlne of the wll secton; s, s length segment of the wll nd t s the correspondng thckness of the length segment. τ t L t s t

3 ENDS Note Set 7 F007bn Sher Flow n Open Sectons he sher flow must wrp round t ll edges, nd the totl torque s dstrbuted mong the res mkng up the cross secton n proporton to the torsonl rgdty of ech rectngle (b /). he totl ngle of twst s the sum of the φ vlues from ech rectngle. t s the thckness of ech rectngle nd b s the length of ech rectngle. t b t L τ Σ GΣb t Exmple τj ( ks )( 9. 7n ) ft 87. 5k ft ρ 5. 5n n π( c c ) (( 5. 5n ) (. 75n ) ) o π J 9. 7n τ t ft τ t ( ks ) ( 0. 5n )( 7n ) 8k ft n ( n )( 6n ) 7n τ t bt τj ( ks )(. 08n ) ft. 8k ft t n n [ 0n( 0. 5n ) + ( 5. 5n )( n ) + ( 5. 5n )( n ) ]. n J 08 herml Strns Physcl restrnts lmt deformtons to be the sme, or sum to zero, or be proportonl wth respect to the rotton of rgd body. We know xl stress reltes to xl strn: δ PL whch reltes δ to P

4 ENDS Note Set 7 F007bn Deformtons cn be cused by the mterl rectng to chnge n energy wth temperture. In generl (there re some exceptons): Sold mterls cn contrct wth decrese n temperture. Sold mterls cn expnd wth n ncrese n temperture. he chnge n length per unt temperture chnge s the coeffcent of therml expnson, α. It hs unts of F or C nd the deformton s relted by: δ α ( Δ )L herml Strn: ε αδ here s no stress ssocted wth the length chnge wth free movement, BU f there re restrnts, therml deformtons or strns cn cuse nternl forces nd stresses. How A Restrned Br Feels wth herml Strn. Br pushes on supports becuse the mterl needs to expnd wth n ncrese n temperture.. Supports push bck.. Br s restrned, cn t move nd the recton cuses nternl stress. Superposton Method If we wnt to solve sttclly ndetermnte problem tht hs extr support forces: We cn remove support or supports tht mkes the problem look sttclly determnte Replce t wth recton nd tret t lke t s n ppled force Impose geometry restrctons tht the support mposes For Exmple: δ α ( Δ )L δ P + δ 0 δ p PL PL + α ( Δ ) L 0 P α α L/ ( Δ ) L/ ( Δ ) f P α ( Δ )E A

5 ENDS Note Set 7 F007bn Exmple (pg 8) ALSO: If the bem s nchored to concrete slb, nd the steel sees temperture chnge of 50 F whle the concrete only sees chnge of 0 F, determne the compressve stress n the bem. α c 5.5 x 0-6 / F α s 6.5 x 0-6 / F E c x 0 6 ps E s 9 x 0 6 ps 5

6 ENDS Note Set 7 F007bn Exmple 6