Chapter 4 Pure Bending

Transcription

1 Chapter Pure endg INTRODUCTION endg tress W W L endg of embers made of everal aterials 0 5 lumum 0.5 TYP rass teel rass lumum Cross-section, Cross-section, tress Concentrations r r D d D d 2r Introduction -1

2 Eccentric ial Loadg a Plane of mmetr P Unsmmetric endg -2 Introduction

3 ENDING TRE L endg tress -

4 ENDING TRE- contued z z σ = I - endg tress

5 ECTION ODULU σ 2 c2 c1 z σ 1 h b σ = = I ection odulus -5

6 Eample Fd the maimum bendg stress at section a-a ( m from ) of the W beam. Units: N, m a 000 a W mm 6mm mm 2 mm rea, = 790 mm Depth, d = 157 mm Flange Width, bf = 15 Flange Thickness, tf = 9. Web Thickness, t = 6.6 I = I = = = mm w mm mm -6

7 Eample Fd the bendg stresses at the wall at pots and for a 6" pipe with a wall thickness of 0.125". Units: lb, ft Cross-section -7

8 Eample Fd the bendg stresses at the wall at pots and for the W620 beam. Units: lb, ft W620 Cross-section 2 rea, = 5.87 Depth, d = 6.20 Flange Width, bf = 6.02 Flange Thickness, tf = 0.65 Web Thickness, t = I = 1. I = 1. = 1. =.1 w -8

9 Eample Fd the bendg stresses at the wall at pots and for the C61 beam. Units: lb, ft. 200 C61 Cross-section 2 rea, =.8 Depth, d = 6.00 Flange Width, bf = 2.16 Flange Thickness, tf = 0. Web Thickness, t = 0.7 I = 17. I = 1.05 = 5.80 = 0.62 = 0.51 w -9

10 Eample Fd the maimum bendg stress at section a-a ( m from ) of the W beam. Units: N/m, m a a 6 W mm 6mm mm 2 mm rea, = 790 mm Depth, d = 157 mm Flange Width, bf = 15 Flange Thickness, tf = 9. Web Thickness, t = 6.6 I = I = = = mm w mm mm -10

11 Eample The sign is subjected to a wd force of 250 lb at it s centroid 7' from the center of the column. The column has an outside diameter of 10" and a wall thickness of 0.25". Considerg onl this force, determe the maimum bendg stress. Units: ft. 7 ENGINEERING WY NEXT EXIT 0 Front View ide View -11

12 PRLLEL-XI THEORE OENT OF INERTI PRLLEL-XI THEORE I I d 2 = ( + ) I I d 2 = ( + ) EXPLE t (uniform thk.) a/2 a/2 b -12 Parallel-is Theorem

13 Eample The simpl-supported beam below has a cross-sectional area as shown. Determe the bendg stress that acts at pots c and d, located at section a-a ( m from ). Units: N/m, mm (uno) a a 6 m 20 c d -1

14 Eample The member is designed to resist a moment of 5 kip about the horizontal ais. Determe the maimum normal stress the member for the two similar cross-sections. Units:

15 Eample Compare the bendg stresses between the two cases for a moment about the horizontal ais. Case 1 is a simple solid cross-section, whereas case 2 is made up of identical boards. The boards aren t connected together and simpl rest on one another. Units:. 1.5 Case Case 2-15

16 Eample The two beams are connected b a th rigid plate on the top and bottom side of the flanges. Fd the bendg stresses at the wall at pots, and C for the W620 beam. Units: lb, ft, C 2000 C W620 2 rea, = 5.87 Depth, d = 6.20 Flange Width, bf = 6.02 Flange Thickness, tf = 0.65 Web Thickness, t = I = 1. I = 1. Cross-section = 1. =.1 w -16

17 Eample The two beams are connected b a th rigid plate on the top and bottom side of the flanges. Fd the bendg stresses at the wall at pots, and C for the W620 beam. Units: lb, ft, C 2000 C 2000 W620 Cross-section 2 rea, = 5.87 Depth, d = 6.20 Flange Width, bf = 6.02 Flange Thickness, tf = 0.65 Web Thickness, t = I = 1. I = 1. = 1. =.1 w -17

18 Eample The two beams are connected b bolts through the flanges. Fd the bendg stresses at the wall at pots and for the W620 beam. Units: lb, ft 2000 W620 Cross-section 2 rea, = 5.87 Depth, d = 6.20 Flange Width, bf = 6.02 Flange Thickness, tf = 0.65 Web Thickness, t = I = 1. I = 1. = 1. =.1 w -18

19 Eample The two beams are connected b bolts through the flanges. Fd the bendg stresses at the wall at pots and for the W620 beam. Units: lb, ft W620 Cross-section 2 rea, = 5.87 Depth, d = 6.20 Flange Width, bf = 6.02 Flange Thickness, tf = 0.65 Web Thickness, t = I = 1. I = 1. = 1. =.1 w -19

20 Eample The simpl-supported beam below has a cross-sectional area as shown. Determe the bendg stress that acts at pots c and d, located at section a-a ( m from ). Units: N/m, mm (UNO) a a 6 m 150 c d 2 Cross-section -20

21 ENDING OF EER DE OF EVERL TERIL N.. ε σ σ = n I endg of embers ade of everal aterials -21

22 Eample Fd the stress each of the three metals if a moment of 12 k- is applied about the horizontal ais. E (alumum)= 10E6, E(steel)= 0E6, E (brass)= 15E6 psi. Units:. lumum 0.5 TYP rass teel 2.5 rass lumum

23 Eample W wide flange beam is reforced with wood planks that are securel connected to the flanges. Esteel/Ewood= 20. If the allowable stresses the wood and steel are.5 Pa and 52 Pa, respectivel, determe the allowable distributed load w based on section a-a ( m from ). Units: N/m, mm (UNO) w a w a 6 m Cross-section W mm 6mm mm 2 mm rea, = 790 mm Depth, d = 157 mm Flange Width, bf = 15 Flange Thickness, tf = 9. Web Thickness, t = 6.6 I = I = = = mm w mm mm -2

24 Eample Two steel plates are securel fastened to a 6"10" wood beam. Esteel/Ewood= 20. Knowg that the beam is bent about the horizontal ais b a 125 kip- moment, determe the maimum stress (a) the wood, (b) the steel. Units:. 2 TYP 0.75 TYP Cross-section -2

25 Eample Determe the stress the concrete and steel if a moment of 1500 kip- is applied about the horizontal ais. rea of steel=.1 sq.. E (steel)= 0E6 psi, E (concrete)=.75e6 psi. Units: Cross-section 0 Cross-section -25

26 Eample Determe the required steel area for the beam to be balanced. llowable stress the steel and concrete are,000 and,000 psi respectivel. E (steel)= 29E6 psi, E (concrete)=.5e6 psi. Units: Cross-section Cross-section -26

27 TRE CONCENTRTION r r D d D d σ ma = k I 2r k D/d= D 1.1 r d k D 1.1 2r d r r/d tress-concentration factors for flat bars with fillets under pure bendg r/d tress-concentration factors for flat bars with groves under pure bendg Ref.: W.D. Pilke, Peterson s tress Concentration Factors, 2nd ed., John Wile and ons, New York, 1997 tress Concentrations -27

28 Eample For the 1 mm thick plate, determe the largest bendg moment that can be applied if the allowable bendg stress is 90 Pa. Units: mm. R k D/d= r/d tress-concentration factors for flat bars with fillets under pure bendg D r d Ref.: W.D. Pilke, Peterson s tress Concentration Factors, 2nd ed., John Wile and ons, New York,

29 Eample For the 7/8" thick plate, determe the largest bendg moment that can be applied if the allowable bendg stress is 2,000 psi. Units:. R k r/d tress-concentration factors for flat bars with groves under pure bendg Ref.: W.D. Pilke, Peterson s tress Concentration Factors, 2nd ed., John Wile and ons, New York, 1997 D 2r d r -29

30 ECCENTRIC XIL LODING IN PLNE OF YETRY P P In general, P σ = + I -0 Eccentric ial Loadg a Plane of mmetr

31 Eample For the solid rectangular bar, determe the largest load P that can be applied based on a maimum normal stress of 10 Pa. Ignore an stress concentrations. Units: mm. P P a P b a b ection a-a ection b-b P -1

32 Eample The three loads are applied at the end of the W620 beam. Fd the normal stress at the wall at pot for the beam, (a) if all three loads are applied, (b) the bottom load is removed. Units: lb, ft W620 2 rea, = 5.87 Depth, d = 6.20 Flange Width, bf = 6.02 Flange Thickness, tf = 0.65 Web Thickness, t = I = 1. I = 1. = 1. =.1 Cross-section w -2

33 Eample The 100 mm diameter solid circular bar has an eccentric load P applied. Determe the maimum location that the load can be placed without ducg an tensile stresses. Units: mm. P -

34 Eample Compute the maimum tension and compression stresses located at section a-a. Units: N, mm (UNO). 2 m a 6 m a (uniform) 60 ection a-a a m a -

35 UNYETRIC ENDING z z σ = + Iz z I P z I I z σ = + z z z z Unsmmetric endg -5

36 Eample For the W620 section, determe the normal stresses at and. Units: kip W620 2 rea, = 5.87 Depth, d = 6.20 Flange Width, bf = 6.02 Flange Thickness, tf = 0.65 Web Thickness, t = I = 1. I = 1. = 1. =.1 w -6

37 Eample For the WT18150 section, determe the normal stresses at, and C. Units: k WT18150 C 2 rea, =.10 Depth, d = 18. Flange Width, bf = 16.7 Flange Thickness, tf = 1.68 Web Thickness, t = 0.95 I = 120 I = 68 = 86.1 = 77.8 =.1 w -7

38 Eample For the channel section, determe the normal stresses at and. Units: kn m, mm TYP -8

39 Eample For the L angle section, determe the normal stresses at and. Units: N m. z 2 mm rea, = 1770 mm d= b = 76 mm = = 2.6 Thickness, t = 12.7 I = I = r = 1.8 mm mm 6mm 00-9

40 URY endg tress W L σ = = I endg of embers made of everal aterials 0 5 lumum 0.5 TYP rass teel rass 2.5 σ = n I 2 lumum Cross-section, Cross-section, tress Concentrations r r D d D d.0 r 2.8 D/d= 2r 2.6 D d k r/d tress-concentration factors for flat bars with fillets under pure bendg -0 ummar

41 URY Eccentric ial Loadg a Plane of mmetr P σ = P I Unsmmetric endg z z z P z I I z σ = + z ummar -1