Objective - Analysis and design of beams. Beams - structural members supporting loads at various points along the member

Transcription

1 EHANIS OF ATERIALS Introduction eer Johnston DeWolf zurek Objective - Anlsis nd design of bes es - structurl ebers supporting lods t vrious points long the eber Trnsverse lodings of bes re clssified s concentrted lods or distributed lods Applied lods result in internl forces consisting of sher force (fro the sher stress distribution) nd bending couple (fro the norl stress distribution) Norl stress is often the criticl design criteri I c I S Requires deterintion of the loction nd gnitude of lrgest bending oent 9 The cgrw-hill opnies, Inc. All rights reserved. 5-

2 EHANIS OF ATERIALS Introduction eer Johnston DeWolf zurek lssifiction of e Supports 9 The cgrw-hill opnies, Inc. All rights reserved. 5-

3 EHANIS OF ATERIALS Sher nd ending oent Digrs eer Johnston DeWolf zurek Deterintion of iu norl nd shering stresses requires identifiction of iu internl sher force nd bending couple. Sher force nd bending couple t point re deterined b pssing section through the be nd ppling n equilibriu nlsis on the be portions on either side of the section. Sign conventions for sher forces nd nd bending couples nd 9 The cgrw-hill opnies, Inc. All rights reserved. 5-

4 EHANIS OF ATERIALS eer Johnston DeWolf zurek Reltions Aong Lod, Sher, nd ending oent Reltionship between lod nd sher: : w F : w d d D w D w d Reltionship between sher nd bending oent: w w d d D D d 9 The cgrw-hill opnies, Inc. All rights reserved. 5-4

5 EHANIS OF ATERIALS Design of Pristic es for ending The lrgest norl stress is found t the surfce where the iu bending oent occurs. I c S A sfe design requires tht the iu norl stress be less thn the llowble stress for the teril used. This criteri leds to the deterintion of the iniu cceptble section odulus. S in ll ll Aong be section choices which hve n cceptble section odulus, the one with the sllest weight per unit length or cross sectionl re will be the lest epensive nd the best choice. eer Johnston DeWolf zurek 9 The cgrw-hill opnies, Inc. All rights reserved. 5-5

6 EHANIS OF ATERIALS Sple Proble 5. eer Johnston DeWolf zurek SOLUTION: Treting the entire be s rigid bod, deterine the rection forces For the tiber be nd loding shown, drw the sher nd bendoent digrs nd deterine the iu norl stress due to bending. Section the be t points ner supports nd lod ppliction points. Appl equilibriu nlses on resulting free-bodies to deterine internl sher forces nd bending couples Identif the iu sher nd bending-oent fro plots of their distributions. Appl the elstic fleure foruls to deterine the corresponding iu norl stress. 9 The cgrw-hill opnies, Inc. All rights reserved. 5-6

7 EHANIS OF ATERIALS Sple Proble 5. eer Johnston DeWolf zurek SOLUTION: Treting the entire be s rigid bod, deterine the rection forces fro F : R 46kN R 4kN Section the be nd ppl equilibriu nlses on resulting free-bodies F kn kn D kn F kn kn kn.5 5kN kN 6kN 4kN 4kN kN 8kN 8kN 9 The cgrw-hill opnies, Inc. All rights reserved. 5-7

8 EHANIS OF ATERIALS Sple Proble 5. eer Johnston DeWolf zurek Identif the iu sher nd bendingoent fro plots of their distributions. 6kN 5kN Appl the elstic fleure foruls to deterine the corresponding iu norl stress. S 6 bh S 5 8. N P 9 The cgrw-hill opnies, Inc. All rights reserved. 5-8

9 EHANIS OF ATERIALS Sple Proble 5. SOLUTION: eer Johnston DeWolf zurek Replce the 45 kn lod with n equivlent force-couple sste t D. Find the rections t b considering the be s rigid bod. The structure shown is constructed of W 567 rolled-steel be. () Drw the sher nd bending-oent digrs for the be nd the given loding. (b) deterine norl stress in sections just to the right nd left of point D. Section the be t points ner the support nd lod ppliction points. Appl equilibriu nlses on resulting free-bodies to deterine internl sher forces nd bending couples. Appl the elstic fleure foruls to deterine the iu norl stress to the left nd right of point D. 9 The cgrw-hill opnies, Inc. All rights reserved. 5-9

10 EHANIS OF ATERIALS Sple Proble 5. SOLUTION: eer Johnston DeWolf zurek Replce the 45 kn lod with equivlent forcecouple sste t D. Find rections t. Section the be nd ppl equilibriu nlses on resulting free-bodies. Fro A to : F kn Fro to D : F 8 45 kn 8 kn kn 8 Fro D to : 5kN 5. 5kN 9 The cgrw-hill opnies, Inc. All rights reserved. 5-

11 EHANIS OF ATERIALS Sple Proble 5. eer Johnston DeWolf zurek Appl the elstic fleure foruls to deterine the iu norl stress to the left nd right of point D. Fro Appendi for W567 rolled steel shpe, S =.8 - bout the X-X is. To the left of D : To the right of D : S S N N 9 P 96P 9 The cgrw-hill opnies, Inc. All rights reserved. 5-

12 EHANIS OF ATERIALS Sple Proble 5. eer Johnston DeWolf zurek SOLUTION: Tking the entire be s free bod, deterine the rections t A nd D. Appl the reltionship between sher nd lod to develop the sher digr. Drw the sher nd bending oent digrs for the be nd loding shown. Appl the reltionship between bending oent nd sher to develop the bending oent digr. 9 The cgrw-hill opnies, Inc. All rights reserved. 5-

13 EHANIS OF ATERIALS Sple Proble 5. SOLUTION: eer Johnston DeWolf zurek Tking the entire be s free bod, deterine the rections t A nd D. D F 7. 9 kn.8 54 kn4. 5.8kN8.4 D 5.6 kn A A A 9 kn 54 kn 5.6 kn 5.8kN 8. kn Appl the reltionship between sher nd lod to develop the sher digr. d d w d w d - zero slope between concentrted lods - liner vrition over unifor lod segent 9 The cgrw-hill opnies, Inc. All rights reserved. 5-

14 EHANIS OF ATERIALS Sple Proble 5. eer Johnston DeWolf zurek Appl the reltionship between bending oent nd sher to develop the bending oent digr. d d d d - bending oent t A nd E is zero - bending oent vrition between A,, nd D is liner - bending oent vrition between D nd E is qudrtic - net chnge in bending oent is equl to res under sher distribution segents - totl of ll bending oent chnges cross the be should be zero 9 The cgrw-hill opnies, Inc. All rights reserved. 5-4

15 EHANIS OF ATERIALS Sple Proble 5.5 eer Johnston DeWolf zurek SOLUTION: Tking the entire be s free bod, deterine the rections t. Appl the reltionship between sher nd lod to develop the sher digr. Drw the sher nd bending oent digrs for the be nd loding shown. Appl the reltionship between bending oent nd sher to develop the bending oent digr. 9 The cgrw-hill opnies, Inc. All rights reserved. 5-5

16 EHANIS OF ATERIALS Sple Proble 5.5 SOLUTION: eer Johnston DeWolf zurek Tking the entire be s free bod, deterine the rections t. F w R w L R w w L Results fro integrtion of the lod nd sher distributions should be equivlent. Appl the reltionship between sher nd lod to develop the sher digr. A w w d w re under lod curve - No chnge in sher between nd. - optible with free bod nlsis 9 The cgrw-hill opnies, Inc. All rights reserved. 5-6

17 9 The cgrw-hill opnies, Inc. All rights reserved. EHANIS OF ATERIALS ifth eer Johnston DeWolf zurek 5-7 Sple Proble 5.5 Appl the reltionship between bending oent nd sher to develop the bending oent digr. 6 w w d w A 6 L w L w L w d w L Results t re coptible with free-bod nlsis

18 EHANIS OF ATERIALS Sple Proble 5.8 eer Johnston DeWolf zurek SOLUTION: onsidering the entire be s freebod, deterine the rections t A nd D. A sipl supported steel be is to crr the distributed nd concentrted lods shown. Knowing tht the llowble norl stress for the grde of steel to be used is 6 P, select the wide-flnge shpe tht should be used. Develop the sher digr for the be nd lod distribution. Fro the digr, deterine the iu bending oent. Deterine the iniu cceptble be section odulus. hoose the best stndrd section which eets this criteri. 9 The cgrw-hill opnies, Inc. All rights reserved. 5-8

19 EHANIS OF ATERIALS Sple Proble 5.8 eer Johnston DeWolf zurek onsidering the entire be s free-bod, deterine the rections t A nd D. D5 6kN.5 5kN4 F A D 58.kN A A 5.kN 58.kN 6kN 5kN Develop the sher digr nd deterine the iu bending oent. A A A 8kN 5.kN re under lod curve 6kN iu bending oent occurs t = or =.6. re under sher curve, AtoE 67.6kN 9 The cgrw-hill opnies, Inc. All rights reserved. 5-9

20 EHANIS OF ATERIALS Sple Proble 5.8 eer Johnston DeWolf zurek Deterine the iniu cceptble be section odulus. S in ll kN 6P Shpe W4 8.8 W6.9 W 8.7 W W 46. S hoose the best stndrd section which eets this criteri. W6.9 9 The cgrw-hill opnies, Inc. All rights reserved. 5-