ME311 Machine Design
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1 ME311 Machine Desin Lectue 7: Columns W Donfeld 6Oct17 Faifield Univesit School of Enineein Column Bucklin We have aad discussed axiall loaded bas. Fo a shot ba, the stess /A, and the defction is L/AE. (Hamock, 4.3) If we load a lon, snde ba, howeve, it will bend and buck lon befoe it will ield in compession. The sudden natue of bucklin makes it deseve special attention so it can be avoided. This failue mode instabilit is diffeent fom the ield o fatiue failue modes. Columns ae classified b two means: 1. b thei elative nth (i.e., sndeness). b whethe o not the load is centeed on them. Hamock Section 9.3 1
2 Concenticall Loaded Columns with inned Ends You aad aned this in Bee & Johnston. The Eu column bucklin fomula [Eqn. 9.7]: cit I l Notes: Swiss mathematician Leonhad Eu (Óil e) fiued it out in ~179. His name does not hme with Feis Buel's. cit is independent of mateial stenth, S. It depends on I and not on aea, as /A does. The deivation is simp and beautiful see Yadstick Bucklin A tpical adstick is about 1/8 thick and 1 1/8 wide. What is the citical bucklin load, assumin the ends ae pinned? 3 bh Tpical E fo softwoods is 1.5x1 6 I psi. 1
3 Radius of Gation In Chap. 4, the adius of ation was defined fom I A as I A Eqn (It is simila in fom to I m fo mass moment of inetia.) Then can ewite cit Now that we have Aea, can define the citical stess: cit c A l A ( l ) ( ) This is an elastic stess, because it is < S l/ is cald the Sndeness Ratio It depends onl on eomet and Youn s Modulus, not stenth o heat teatment. Hamock ae 3 End Conditions If the ends of the column ae somethin othe than pinnedpinned, must use an effective nth in the Eu equation. Effective Lenth: Theoetical AISC Recommended L L.7L.8L.5L.65L L.1L Use These Fo othe than inned-inned, eplace the actual beam nth, L, b effective column nth, Le. 3
4 Citical Stess vs Sndeness The cuves onl depend on Modulus; Uppe limit is S. Unit Load (c / A ksi) S5. ksi S14 ksi Steel Aluminum ( l ) Sndeness Ratio (L/R) c Tansition to Yieldin Up nea this cone, it was discoveed that bucklin failues did occu below the Eu/Yield lines. 7 Based on measued esults aound 19, J.B. Johnson developed a paabolic tansition fomula fo Intemediate nth columns. [Eqn. 9.16] Unit Load (c / A ksi) S5. ksi c J S S 4 Steel Sndeness Ratio (L/R) 4
5 cj Eu-Johnson Equations Above the Tansition Sndeness atio, use Eu. Below the Tansition Sndeness atio, use Johnson. Tansition Sndeness Ratio: l e tans [Eqn. 9.18] S S Johnson Unit Load (c/a ksi) S S 51 KSI Johnson Tansition (Tanenc) oint Eu Sndeness Ratio (L/R) Eu: c ( l ) Eu-Johnson Equations As the Yield Stenth chanes, so does the tansition point, to keep the two cuves tanent. 7 Tansition Sndeness Ratio: l e tans S Note: It hits the 1 Eu cuve at about S/. Unit Load (c / A ksi) S5. ksi S37.5 ksi Sndeness Ratio (L/R) 5
6 Columns, B the Numbes 1. Usin the mateial popeties, compute the tansition sndeness atio, l e S tans. Compute the nomal stess in the component /A 3. Compute the aea moment of inetia, I 4. Compute the adius of ation, I A 5. Fom end conditions, detemine the effective nth, Le 6. Compute the component sndeness atio, Le/R 7a. If ou component sndeness atio is above the tansition, use Eu: c ( l ) e 7b. If ou component sndeness atio is below the tansition, use Johnson: S 4 c S J 8. The Facto of Safet is just the citical bucklin stess (NOT the ield stess) divided b the /A load stess on the column. Eu? o Johnson? You have a 1 ound ba of 14 steel, annead. Assumin it is loaded with its ends pinned-pinned: A. How lon would it be to be iht at the Johnson-Eu tansition point? B. What load could it take befoe bucklin? 6
7 What if the column is loaded off-cente? The eccenticit causes a moment that contibutes to bendin the beam and aidin bucklin. This will also happen if the column is initiall cooked (bent). This case is descibed b the Secant Equation [Eqn. 9.3]: ec max 1+ sec A o, eaanin EA 1 sec cos Me max A ec + 1 sec EA e Me The tem ec/ is cald the Eccenticit Ratio, and c is the distance fom neutal axis to suface (like Mc/I). This is a mess equation because /A is a function of itself. It must be solved iteativel, tpicall statin with the Johnson c. Secant Equation A max ec + sec 1 EA 7
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