DESIGN OF BEAMS FOR MOMENTS

Transcription

1 CHAPTER Stuctual Steel Design RFD ethod Thid Edition DESIGN OF BEAS FOR OENTS A. J. Clak School of Engineeing Deatment of Civil and Envionmental Engineeing Pat II Stuctual Steel Design and Analysis 9 FA 2002 By D. Iahim. Assakkaf ENCE Intoduction to Stuctual Design Deatment of Civil and Envionmental Engineeing Univesity of ayland, College Pak CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 1 ateal Suot of Beams ost eams ae designed in such a way that thei flanges ae estained against lateal uckling. The ue flanges of eams used to suot concete uilding and idge floos ae often incooated in these concete floos. Theefoe, these tye of eams fall into Zone 1. 1

2 CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 2 ateal Suot of Beams If the comession flange of a eam is without lateal suot fo some distance, it will have a stess distiution simila to that of columns. When the comession flange of a eam is long enough and slende enough, it may uckle unless lateal suot is ovided CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 3 ateal Suot of Beams Twisting o Tosion of Beams When the comession flange egin to uckle, twisting o tosion will occu, and the smalle the tosional stength of the eam the moe aid will e the failue. Standad shaes such as W, S, and channels used fo eam sections do not have a geat deal of esistance to lateal uckling and the esulting tosion. Some othe shaes, notaly the uilt-u ox shaes ae temendously stonge. 2

3 CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 4 ateal Suot of Beams ateal Suot not Povided y Sla Should lateal suot of the comession flange not e ovided y a floo sla, it is ossile that such suot my e ovided with connecting eams o with secial memes inseted fo that uose. Deending on the sacing of the suot, the eam will fall into Zones 1, 2, o 3. CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 5 Intoduction to Inelastic Inelastic uckling can occu when the acing is insufficient to emit the meme to develo and each a full lastic stain distiution efoe uckling occus. Because of the esence of esidual stesses, yielding will egin in a section at alied stesses equal to Fy F (1) 3

4 CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 6 Intoduction to Inelastic In Eq. 1, F y yield stess of the we, and F comessive esidual stess, and assumed equal to 10 ksi fo olled shaes and 16.5 ksi fo welded shaes. When a constant moment occus along the unaced length of a comact I- o C-shaed section and is lage than, the eam will fail inelastically unless is geate than a distance. CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 7 Intoduction to Inelastic ateal Buckling of Beams Fig. 1 shows that eams have thee distinct anges o zones of ehavio deending on thei lateal acing situation: Zone 1: closely saced lateal acing, eams fail lastically. Zone 2: modeate unaced lengths, eams fail inelastically. Zone 3: age unaced lengths, eams fail elastically 4

5 CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 8 Intoduction to Inelastic Plastic Behavio-full Plastic moment (Zone 1) Inelastic uckling (Zone 2) Elastic uckling (Zone 3) Figue 1. n as a function of n d (lateally unaced length of comession flange) CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 9 Intoduction to Inelastic Bending Coefficients A moment coefficient, designated y C, is included in design fomulas to account fo the effect of diffeent moment gadients on lateal-tosional uckling. The use of this coefficient is to take into account the effect of the end estaint and loading condition of the meme on lateal uckling. 5

6 CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 10 Intoduction to Inelastic Bending Coefficients (cont d) In Fig 2a, the moment in the unaced eam causes a wose comession flange situation than does the moment in the unaced eam of Fig. 2. Fo one eason, the ue flange in Fig. 2a is in comession fo its entie length, while in Fig. 2 the length of the column, that is the length of the ue flange that is in comession is much less (shote column). CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 11 Intoduction to Inelastic Bending Coefficients (cont d) w u w u C C 2 w u 8 2 w u 24 ength of ue Flange column (a) Single cuvatue Figue 2 2 w u 12 ength of ue Flange column () Doule cuvatue 2 w u 12 6

7 CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 12 Intoduction to Inelastic Bending Coefficients (cont d) Values of C : Fo the simly suoted eam of Fig.2a: C 1.14 Fo the fixed-end eam of Fig. 2: C 2.38 The asic moment caacity equations fo Zones 2 and 3 wee develoed fo lateally unaced eams sujected to single cuvatue with C 1.0 CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 13 Intoduction to Inelastic RFD Secification RFD Secification ovides moment o C coefficients lage than 1.0 which ae to e multilied y the comuted n values. The esults ae highe moment caacities. The value of C 1.0 is a consevative value. In should e noted that that value otained y multilying n y C may not e lage than the lastic moment of Zone 1, which is equal to F y Z. 7

8 CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 14 Intoduction to Inelastic RFD Secification The anual ovides an equation fo calculating the coefficient C as follows: C 2.5 max 12.5 max (2) A B + 3 C max lagest moment in unaced segment of a eam A moment at the ¼ oint B moment at the ½ oint C moment at the ¾ oint CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 15 Intoduction to Inelastic RFD Secification C 1.0 fo cantileves o ovehangs whee the fee end is unaced. Some secial values of C calculated with Eq. 2 ae shown in Fig. 3 fo vaious eam moment situations. ost of these values ae also given in Tale 5.1 of the RFD anual 8

9 CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 16 Intoduction to Inelastic Figue 3a w u ( k/ft) w u ( k/ft) C 1.14 C 1.30 C 1.32 C 1.67 CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 17 Intoduction to Inelastic Figue 3 w u ( k/ft) C vaies / 3 / 3 idsection C End section C / / 4 / 4 Fo two cente sections C Fo two end sections C / 4 / C 1.0 9

10 CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 18 Intoduction to Inelastic Figue 3c 1 1 w u ( k/ft) C 2.27 C 2.38 w u ( k/ft) C 2.38 C 1.92 CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 19 Intoduction to Inelastic Figue 3d Pu C 2.27 C

11 CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 20 oment Caacity, Zone 2 If the distance etween oints of tosional acing is inceased eyond (see Fig. 1), the moment caacity of the section will ecome smalle and smalle. Finally, at an unaced length, the section will uckle elastically as soon as the yield stess is eached. CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 21 oment Caacity, Zone 2 Effect of Residual Stesses Due to the olling oeation on steel shaes, thee is esidual stess in the section equal to F. Thus, the elastically comuted stess caused y ending can only each F y F as given y Eq. 1. Assuming C 1, the design moment fo a comact I- o C shaed section may e detemined as follows if : x ( F F ) S (3) y 11

12 CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 22 oment Caacity, Zone 2 Povisions fo Zone 2 Design y the RFD If decease the unaced length fom to, uckling does not not occu when the yield stess is fist eached. This ange etween and is called Zone 2 and is illustated in Fig. 1. Fo these cases, when the unaced length falls etween and, the design moment stength will fall aoximately on a staight line etween n F Z at and y S x ( Fy F ) at CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 23 oment Caacity, Zone 2 Povisions fo Zone 2 Design y the RFD Fo intemediate values of the unaced length, the moment caacity may e detemined y ootions o y sustituting into exessions. If C is lage than 1.0, the section will esist additional moment ut not moe than FyZ (4) 12

13 CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 24 oment Caacity, Zone 2 Povisions fo Zone 2 Design y the RFD The moment caacity can detemined y the following two exessions: o nx C [ x BF( )] x (5) n C ( ) (6) CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 25 oment Caacity, Zone 2 Povisions fo Zone 2 Design y the RFD In Eq. 5, BF is a facto given in RFD Tale 5-3 fo each section, which enales us to do the ootioning with simle fomula. Note that in Eq. 6, afte the moment n has een comuted, it should e multilied y to otain n. 13

14 CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 26 oment Caacity, Zone 2 Examle 1 Detemine the moment caacity of a W24 62 with F y 50 ksi if 8.0 ft and C 1.0. Fo F y 50 ksi, the RFD Tale 5-3 (P. 5-46) gives the following fo W24 62: 4.84 ft, 13.3 ft, 396 ki - ft 578 ki - ft, BF 21.6 kis and CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 27 oment Caacity, Zone 2 Examle 1 (cont d) ( 4.84 ) < ( 8.0 ) < ( 13.3 ) Since The moment caacity fallsin Zone 2, nx n n C C [ x BF( )] [ BF( )] and [ 21.6( ) ] Theefoe, The moment caacity ft - ki x Eq. 5 14

15 CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 28 oment Caacity, Zone 2 Examle 1 (cont d) Plastic Behavio-full Inelastic Plastic moment uckling (Zone 1) (Zone 2) Elastic uckling (Zone 3) Figue 1. n as a function of n 8.0 ft d 4.84 ft 13.3 ft (lateally unaced length of comession flange) CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 29 oment Caacity, Zone 2 Examle 2 Select the lightest availale section fo a factoed moment of 290 ft-kis if 10.0 ft. Use 50 ksi steel and assume C 1.0. Ente RFD Tale 5-3 (P. 5-47) and notice that fo W18 40 is 294 ft-ki. Fo this section: 4.49 ft, 205 ki - ft 294 ki - ft, BF 11.7 kis 12.0 ft, and 15

16 CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 30 oment Caacity, Zone 2 Examle 2 (cont d) ( 4.49 ) < ( 10.0 ) < ( 12.0 ) Since The moment caacity fallsin Zone 2, nx n n C C [ x BF( )] [ BF( )] and [ ( ) ] x Theefoe, The moment caacity 229 ft - ki < u 290 ft - ki NG CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 31 oment Caacity, Zone 2 Examle 2 (cont d) oving u in the tale and afte seveal tials, ty a W21 48 that has the following oeties: 6.09 ft, 15.4 ft, 279 ki - ft 401ki - ft, BF 13.2 kis and ( 6.09 ) < ( 10.0 ) < ( 15.4 ) Since The moment caacity fallsin Zone 2, and 16

17 CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 32 oment Caacity, Zone 2 Examle 2 (cont d) nx n n C C [ x BF( )] [ BF( )] [ ( ) ] x Theefoe, The moment caacity 349 ft - ki > u 290 ft - ki OK Hence, USE W21 48 CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 33 Elastic Buckling, Zone 3 When a eam is not fully aced, it may fail due to uckling of the comession otion of the coss section lateally aout the weak axis. This will e accomanied also with twisting of the entie coss section aout the eam s longitudinal axis etween oints of lateal acing. Fo the moment caacity to fall into Zone 3, 17

18 CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 34 Elastic Buckling, Zone 3 Plastic Behavio-full Plastic moment (Zone 1) Inelastic uckling (Zone 2) Elastic uckling (Zone 3) Figue 1. n as a function of n d (lateally unaced length of comession flange) CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 35 Elastic Buckling, Zone 3 RFD Secifications fo Zone 3 The classic equation fo detemining the flexual-tosional uckling moment is given y 2 π πe c C EI ygj + I ycw G shea modulus of steel 11,200 ksi J tosional constant (in 4 ) C w waing constant (in 6 ) NOTE: These oeties ae ovided in Tales 1.25 to 1.35 Of the RFD anual fo olled sections (7) 18

19 CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 36 Elastic Buckling, Zone 3 Examle 3 Comute c fo a W18 97 consisting of 50 ksi steel if the the unaced length is 38 ft. Assume C 1.0. Fo W18 97, Tale 5-3 (P. 5-46) of the anual gives 9.36 ft, 27.5 ft, 564 ki - ft 791ki - ft, BF 12.8 kis and CHAPTER 9. DESIGN OF BEAS FOR OENTS Slide No. 37 Elastic Buckling, Zone 3 Fom P.1-17 Examle 3 (cont d) ( 9.36 ) < ( 27.5 ) < ( 38 ) Since The moment caacity fallsin Zone 3, and Fom Pat 1 of the anual, tales fo tosion oeties (P. 1-91): I 4 4 y 201 in, J 5.86 in, and Cw Theefoe, using Eq.7 c π ,800 in 3 () ( 201)( 11,200)( 5.86) + ( 201)( 15,800) 4,916.9 in - ki 410 ft - ki 3 π Theefoe, c 0.9 (410) 369 ft-ki