MECHANICS OF MATERIALS

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1 00 Th McGraw-Hill Companis, Inc. ll rights rsrvd. T Edition CHTER MECHNICS OF MTERIS Frdinand. Br E. Russll Johnston, Jr. John T. DWolf Columns ctur Nots: J. Walt Olr Txas Tch Univrsit

2 00 Th McGraw-Hill Companis, Inc. ll rights rsrvd Columns Stabilit of Structurs Eulr s Formula for in-endd Bams Extnsion of Eulr s Formula Sampl roblm 10.1 Eccntric oading; Th Scant Formula Sampl roblm 10. Dsign of Columns Undr Cntric oad Sampl roblm 10.4 Dsign of Columns Undr an Eccntric oad

- allowabl strss is not xcdd all - dformation falls within spcifications δ E δ spc ftr

3 00 Th McGraw-Hill Companis, Inc. ll rights rsrvd Stabilit of Structurs In th dsign of columns, oss-sctional ara is slctd such that - allowabl strss is not xcdd all - dformation falls within spcifications δ E δ spc ftr ths dsign calculations, ma discovr that th column is unstabl undr loading and that it suddnl bcoms sharpl curvd or buckls.

4 00 Th McGraw-Hill Companis, Inc. ll rights rsrvd Stabilit of Structurs Considr modl with two rods and torsional spring. ftr a small prturbation, K ( θ ) sin θ rstoring momnt θ dstabilizing momnt Column is stabl (tnds to rturn to alignd orintation) if θ < < K ( θ ) 4K

5 00 Th McGraw-Hill Companis, Inc. ll rights rsrvd Stabilit of Structurs ssum that a load is applid. ftr a prturbation, th sstm sttls to a nw quilibrium configuration at a finit dflction angl. sinθ K 4K ( θ ) θ sinθ Noting that sinθ < θ, th assumd configuration is onl possibl if >.

6 00 Th McGraw-Hill Companis, Inc. ll rights rsrvd Eulr s Formula for in-endd Bams Considr an axiall loadd bam. ftr a small prturbation, th sstm rachs an quilibrium configuration such that d d dx dx + M EI EI EI 0 Solution with assumd configuration can onl b obtaind if > π EI > π E ( ) r π E ( r)

7 00 Th McGraw-Hill Companis, Inc. ll rights rsrvd Eulr s Formula for in-endd Bams Th valu of strss corrsponding to th itical load, r > π π > E ( r) ( ) r E π EI itical strss slndrnss ratio rcding analsis is limitd to cntric loadings.

8 00 Th McGraw-Hill Companis, Inc. ll rights rsrvd Extnsion of Eulr s Formula column with on fixd and on fr nd, will bhav as th uppr-half of a pin-connctd column. Th itical loading is calculatd from Eulr s formula, π EI π ( r) E quivalnt lngth

9 00 Th McGraw-Hill Companis, Inc. ll rights rsrvd Extnsion of Eulr s Formula

10 00 Th McGraw-Hill Companis, Inc. ll rights rsrvd Sampl roblm 10.1 n aluminum column of lngth and rctangular oss-sction has a fixd nd at B and supports a cntric load at. Two smooth and roundd fixd plats rstrain nd from moving in on of th vrtical plans of smmtr but allow it to mov in th othr plan. a) Dtrmin th ratio a/b of th two sids of th oss-sction corrsponding to th most fficint dsign against buckling. 0 in. E 10.1 x 10 6 psi 5 kips FS.5 b) Dsign th most fficint oss-sction for th column.

11 00 Th McGraw-Hill Companis, Inc. ll rights rsrvd Sampl roblm 10.1 SOUTION: Th most fficint dsign occurs whn th rsistanc to buckling is qual in both plans of smmtr. This occurs whn th slndrnss ratios ar qual. Buckling in x lan: r z, z r z I z 1 1 ba ab 0.7 a 1 3 a 1 Buckling in xz lan: r, r I 1 1 ab ab b / 1 3 b 1 r r z a 1 b 1 Most fficint dsign:, z, r r z 0.7 a 1 a b b / a b 0.35

00 Th McGraw-Hill Companis, Inc. ll rights rsrvd. 10-1 Sampl roblm 10.1 0 in. E 10.1 x 10 6 psi 5 kips FS.5 a/b 0.35 Dsign: r b 1 ( 0.35b) ( 0 in) ( FS ) (.

12 00 Th McGraw-Hill Companis, Inc. ll rights rsrvd Sampl roblm in. E 10.1 x 10 6 psi 5 kips FS.5 a/b 0.35 Dsign: r b 1 ( 0.35b) ( 0 in) ( FS ) (.5)( 5 kips) π ( r) 1500 lbs b E ( 0.35b) 1500 lbs π b b 1 π ( 6 ) psi ( b) ( 6 ) psi ( b) b 1.60 in. a 0.35b in b 1.5 kips

13 00 Th McGraw-Hill Companis, Inc. ll rights rsrvd Eccntric oading; Th Scant Formula Eccntric loading is quivalnt to a cntric load and a coupl. Bnding occurs for an nonzro ccntricit. Qustion of buckling bcoms whthr th rsulting dflction is xcssiv. Th dflction bcom infinit whn d dx EI max π sc Maximum strss + 1+ max max r ( ) c sc r 1 c E r π EI

14 00 Th McGraw-Hill Companis, Inc. ll rights rsrvd Eccntric oading; Th Scant Formula max Y 1 + c sc r 1 E r

00 Th McGraw-Hill Companis, Inc. ll rights rsrvd. 10-15 Sampl roblm 10. Th uniform column consists of an 8-ft sction of structural tubing having th oss-sction shown.

15 00 Th McGraw-Hill Companis, Inc. ll rights rsrvd Sampl roblm 10. Th uniform column consists of an 8-ft sction of structural tubing having th oss-sction shown. a) Using Eulr s formula and a factor of saft of two, dtrmin th allowabl cntric load for th column and th corrsponding normal strss. E psi. b) ssuming that th allowabl load, found in part a, is applid at a point 0.75 in. from th gomtric axis of th column, dtrmin th horizontal dflction of th top of th column and th maximum normal strss in th column.

16 Sampl roblm 10. SOUTION: Maximum allowabl cntric load: - Effctiv lngth, ( 8 ft) 16 ft 19 in. - Critical load, π EI π 6.1 kips ( 6 )( 4 ) 9 10 psi 8.0 in ( 19 in) 00 Th McGraw-Hill Companis, Inc. ll rights rsrvd. - llowabl load, all FS all 6.1 kips 31.1 kips 3.54 in all 31.1kips 8.79 ksi 10-16

17 00 Th McGraw-Hill Companis, Inc. ll rights rsrvd Sampl roblm 10. Eccntric load: - End dflction, m m π sc ( in) sc in. 1 π - Maximum normal strss, c π m 1 + sc r 31.1 kips in ( 0.75 in)( in) ( 1.50 in) π sc m.0 ksi

18 00 Th McGraw-Hill Companis, Inc. ll rights rsrvd Dsign of Columns Undr Cntric oad rvious analss assumd strsss blow th proportional limit and initiall straight, homognous columns Exprimntal data dmonstrat - for larg /r, follows Eulr s formula and dpnds upon E but not Y. - for small /r, is dtrmind b th ild strngth Y and not E. - for intrmdiat /r, dpnds on both Y and E.

19 00 Th McGraw-Hill Companis, Inc. ll rights rsrvd Dsign of Columns Undr Cntric oad Structural Stl mrican Inst. of Stl Construction For /r > C c π E / r ( ) all FS FS 1.9 For /r > C c ( / r) FS 5 3 Y / r C c C c 1 / r 8 Cc 3 all FS t /r C c 1 Y C c π E Y

20 00 Th McGraw-Hill Companis, Inc. ll rights rsrvd Dsign of Columns Undr Cntric oad luminum luminum ssociation, Inc. llo 6061-T6 /r < 66: all [ ( / r) ] ksi [ ( / r) ]Ma /r > 66: all llo 014-T6 /r < 55: all /r > 66: all ksi ( / r) ( / r) [ ( / r) ] ( / r) Ma ksi [ ]Ma ksi ( / r) ( / r) Ma

21 00 Th McGraw-Hill Companis, Inc. ll rights rsrvd Sampl roblm 10.4 SOUTION: With th diamtr unknown, th slndrnss ration can not b valuatd. Must mak an assumption on which slndrnss ratio rgim to utiliz. Calculat rquird diamtr for assumd slndrnss ratio rgim. Evaluat slndrnss ratio and vrif initial assumption. Rpat if ncssar. Using th aluminum allo014-t6, dtrmin th smallst diamtr rod which can b usd to support th cntric load 60 kn if a) 750 mm, b) 300 mm

22 00 Th McGraw-Hill Companis, Inc. ll rights rsrvd Sampl roblm 10.4 For 750 mm, assum /r > 55 Dtrmin clindr radius: all πc N ( r) Ma Ma m c/ c mm c r clindr radius radius of I πc gration 4 πc 4 c Chck slndrnss ratio assumption: r c / 750mm mm ( ) assumption was corrct d c 36.9 mm 81.3 > 55

00 Th McGraw-Hill Companis, Inc. ll rights rsrvd. 10-3 Sampl roblm 10.4 For 300 mm, assum /r < 55 Dtrmin clindr radius: all 60 10 πc 3 c 1.

23 00 Th McGraw-Hill Companis, Inc. ll rights rsrvd Sampl roblm 10.4 For 300 mm, assum /r < 55 Dtrmin clindr radius: all πc 3 c 1.00 mm N r Ma 0.3 m / c Chck slndrnss ratio assumption: r c / 300 mm 1.00 mm ( ) 50 < 55 6 a assumption was corrct d c 4.0 mm

24 00 Th McGraw-Hill Companis, Inc. ll rights rsrvd Dsign of Columns Undr an Eccntric oad n ccntric load can b rplacd b a cntric load and a coupl M. Normal strsss can b found from suprposing th strsss du to th cntric load and coupl, + max cntric + Mc I bnding llowabl strss mthod: Mc + all I Intraction mthod: Mc + ( ) ( ) all cntric all I bnding 1

Strength of Materials

Strength of Materials Strngth of Matrials Sssion Column 08 ctur not : ramudiyanto, M.Eng. Strngth of Matrials STBIITY OF STRUCTURE In th dsign of columns, oss-sctional ara is slctd such that - allowabl strss is not xcdd all