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Unformatted text preview: TutOrial 10 _ 3?;1 / The T section shown below is made of'an elasto—plastic material. The stress~strain relationship of the material is as shown . (1) Determine the ultimate moment capacity of the section. - (ii) A bending moment is applied to this Section and incresed until the entire top flange yields. ~“Calculate the magnitude of the moment at is stage Of loading. 300 fag—42...... - 160x20 + {500— 49/5 = a x' /5-'- I 2"-"56 7 mm '_ ~ ' I 5 H I F; .= cam/5. 700116 on . ”£5 ' ganja - . h‘ ' . == Mom/50x20) ' _' . -LJ“FS ‘ _ = 768 kN - 9; ~ -- * - ' a1) F2 3 @mfi- force an ad». - 7:? - :- 24“ (500—'256-7)x/5 : 5.5.? k/V Egg : fang/e yfiorce on weé =7" 24ox255-7x/5 =- €24! AN Nofe: FC 2 F7 5:. 5+5 1.5) ' “£7151de 310297sz ca/bdaiév . _ - Exam—2557) +. e x (jaw/2 . - l+g_x-d/2 ' A ‘ V /62,9/8 kAI—mm- :- Ma kmm k? 11' IE2?!” AC.- =AT I _ __,__ . mNA . N56}? emf/{rel - L” 743/5 lvfi/an e/ié/dfi', Kai/1 Md area .9719 . Ale/[a vwfltZ/Er 7484442021 yzé/aéj ' (543'_ From MLJ’EM 4/054“ 4462me and _ ‘ Candi/bun 473% , . , ~ f ffie Jamne am (A) '9 ’5"; "'240X/50x20: 75404” (6 F F; =-2x24ox454xx5— 79M " AM 7%}: S/axj-e I inanvanf'= 75f-X53w4 + (7?X§§K434%2 ‘ '+-7éo°x~--(4a4+2/ss/2) == /6o-7 Min», _ w 9 , ' " ' K4 jVMLe: mommf < mamm/é C‘s/5&5}? .' (Lo/flnflrze.) 5 7- 5.53115“- beam C=°§5'.5=?Cti°n shown is-subjected to a bending moment. The 5.11:?55 distrlbgtion due to this bending moment has the form shown w1th themaxmum tensile stress being 120 MPa. The stress} strain comes for the material of the beam in both tension an compresseon are also shown. 1 a} What is the value of the-maximum compression stress on the ‘ cross-section? - . b) What is the benfiiing moment on the crossesection? C0 MTZE'SSB!‘ ~1— T'EUSIBM "' *‘x ‘3 - ”:5 £120 E- V \J" m m . u: v, _ 350 $20} }- l- . m _I W _ swam STEAM STRESS N' _ 'm51m- m CURVE ' W . . m; compzessiou {CD—LP ' 35 7.5 - .fi 53‘ £120 '—— E vs __ V1. Em g to . m ._ h ' '0??? 01' - f 0067 ' 57mm . ' 1. ’ ' ‘5me ' [m mm; - wwwu cum/5' ' srésss _ I . x f. - m sxou . ‘ -C0MP:25_55 on _ 575‘__________/am_ “(1579‘ ’5 ”’7; f on a) Ibiue 57/ max. 125'61— — .s/ve-r: "a, “0057‘ . ($0271 07- E curl/e ln com/521231027) 3 Eon/IF 112m; s75conf 2‘ 5) From 57%)}? 01/1571f.a//c3§?rc9m by Emmi/car Ag . a, 5 .—~ -0067 -0/ £=/-'54,.. --.‘(/) From Sfress W's/r. aZ/cgjvrm_ Jmce ,L2 =-_ F], ‘-5%~5 F5 ' - _- 590x50x(/25—- 4— A)+2x000x50x4 P ‘ = 2X/20x50xz5 (2? JED/wry Cf) ‘9 C2) ”at: 402nm ._ 5“ 50 mm drvd, I25——5L—- Z7"'= 25mm . Fl: 80x50x25= MOM! ‘ _. , 2‘2-‘2 X000X50X40f- XOkN'V :9 2%: :x/20x50x560=‘= M90 kN _ 5 D "JMomenzL- /00x(40+/25) K35 ' ' +330x %X40 ' ' + APOXZ/s X50 = /4-58 kN-m 1 57-7 37- $7.é Ta \ ? guts shaws the distributinn af' handing stress a tangular grass—section. what is tha value =: bending moment an the cram-section. ' 20143:. é'o _ Crass... jgcf,‘°n 51725.5 D;.§Tr rIEU fun“. For ma 5122! cm yieiding. as weii S‘s-section shown in Hg. 1. calculate the mm as the My plastic mament Mp for the secxian. Assume that the stress-strain relationship. eiaszic-piaszic with a yield stress of 250 MFa. The cmss-saczinn of 300 MPa in the 2033”! I o i ‘ + 1:, 233m F951 . 486mm Shawn in Fig. 2 Is suhj acted to a wamemfihich induces a Compressive stress rap fibre. Determine: ‘ (a) the pcsifian of the mantra! axis. . (b) the stress a: A and (c) zha moment in the section. Tenant: ma: am M, which wiii cause the first ' . C, =_ 28%on 40 2 32000 = 32;” :59 "C; = QLX/ox 20x40 _== _4-é1\j ‘ g 77—”: éxé‘Oxfiokl4-0=3ékz\{" Mame/14f = 52 (2071/0) + 4-K %_X /0 wwzéxso” ‘ _= ['7-0‘7-4é/V-m \ 1. S 7.é The ‘igure shame the distribution a? bending strasé an a rectangular grues-eectiun. what is the value :2? the bending moment an fihe :rneeoeecflen. 4-0 mm 7 ‘ 20 ”’3 30‘ 60 _ Crass‘Stcf,‘°n Err-es: DIET/I-EU ian'- 5' 7 7 For the steel crass-section shown in fig. 1. cafcuiate the moment My which wiii cause the first yielding. as weii as the fufiy plastic moment M9 for the s'ecn'on. Assume that the stress-main _relaficnship. eies-zic-piasfic with a yield stress of 250 MPa. —.}-—.E‘"'_"_fi-._ ‘ 3'; 2:13:11 F9. ' i . ”MI mean. ?I S 7 6) The crass-section Shawn in Fig. 2 is subjected to amen-rem which induces a Camerassiva stress of 300 MPa in the rap fibre. Deiemh-ze: (a) the position of the neutral axis. . (:3) :he stress 21A and (c) the mement in the secficrz. E 15 ' G‘ 26 u r + 2!: - I A ‘ no 1:: ' ts — _ 1,- - - MA I. T 70;; A P0375027 A 07! A/A 24 ._ ‘ when C- — T . 20x/5O + 2 4(300: 0’)20X/5© H Ijézoflfl 2" . 5 fria/ 25 arm __a__ jjb/CSJ if 50mm. .juwjis 34mm lat =‘6AP-3AN ”2:30," 56%" 20>/5 A NA I: 55-7-0017 yflroM-Mfyé. 3) P/fl/K’ZL/C fjmn #37 fez/775700 9 -_.:’ gzofir/gfi; = /5(,Pm/m, 740m50é0777 k9 " ,4 //.C.F1 NIILA/n [5/3/3796 21771027,. ' 57620; 0% /}_ = 250 ME? 6) M: szo x 60 x (f 7/0) __ - 7 . :2(500' 0")020x/50(g: 75x20) 0 ._ J‘fi—20>/5xz/5(g .20) PF _ 7‘ +2xéjm5x25ox(%) /(Lf/)»’ . - 7250x115 (22021;)(2‘5Wfl220, 67)) _- == 506 kN-m‘l' 57. ‘f 57' /0 // '1. A beam has a dose-section as shown in Fig. 4. and is made iron: :1qu steei with the given stress— strain relationship- -the behaviour is the same in tension as in compression The beam :5 subieoted to 3 gradualty Increasing bending moment M until the entire flange IS piaszic' in compression. (3) is the tension zone mmietsiy elastic or partially plastic? What' IS the extent of the piastit: zone in tension? to) Where is the name! axis? to) What is the bending moment. M? Iceman I’I :n 3 Figures» A steei beam with the cross-section shown in Fig. 5. is suhieoteo to a gradually" increasing bending moment M. until the upper flange :5 plastic' to compression. The s‘ress—strain relation tor miid steel IS also given in Fig. 5 and is the same in compression as in tension. (a) Determine whether the tension zone is compieteiy elastic or partially plastic. if paniaiiy plastic. what is the extent of theplastio zone in tension? (in) Looms the, neutral axis. (c) Galatians the bending rnom‘ent. M. {d} Compute the curraiure oi the beam atthis section. Sketch the stress distribution giving important values. ' no ' . , i ""—"'T— o 2' 23: mm» “Fl 1:: E 2! ll ‘3 it - ‘Kiw' IA liq I ..-5i'€€[560-13L20fl'-I ' W W‘ 7L0 wag/aw mm W ___m0me~nl£. #77257. ujéPef’ 7%?an jg FAQ—51296 _fl . _ m Compmf/bn . _ (COM) 5 ' . . 5-7040 9 fan/b f/anje qfi 5071710247 F/anje Faye ' 53mg area . N4 15 $me af mzdd/e 0/ MeA: [ Mfé A/A dz‘ 7‘26/3/505/290/7 Fj: ’2 _ ‘5 F3 .F2 30 7%ng 00/265an a7]r _ C T AS‘ 741/ F///ed ) Kamila F/anje f; f/wF/c - '5 5) AM .fd” /oca7;‘eo{ m mm” A/aé. _ CJ F == 250x20x/00= foo/«V F5 :- 42x (0+250)x202<50 —- /25/<N Elk/my :wmanzl czéouf AM ' M= 6X00 /0J7LFX(/00 25J , -+FX2/5x50 + .Fx 5.»: 753.26 kN—mm f—= ' ‘S 21 22 For the beam cross-section show. in Eig- 1 it {a} etermine the elastic seetioh modulus (S) , and the_ plastic sectiOn modulus (2p)- Using these seation properties, calculate the shape factor of the cross—section. (b) Determine the first yield moment, MY' Indicate where the firSt yielding will otcur. (C) Determine the fully plastic moment, MD. (d) :or the case of pure bending, determine the moment when the entire top flange has just become fully plastic. You may assume that the centroid of the section is located at 212.5 mm above the bottom fibre of the section and the moment of inertia of the gettion about its centroidal x axis is 241.5 x 10 . FIGURE 1 s1 "up" 'Iuv’ \\~—l . r 30 ' ,4] :‘20;></50 = 3000 1 I 42' : BOOK/O r—-3£:u90 300 A245- '5" 302‘200-75005 I A : l2,0002z 2‘5 (ff—1250 MP6 ‘ _ _-: 244-5 x105 .m-m (flit/en) ' . . c I 0‘ =4 —é’——M 5‘ Ix , .- -_ é . . _ MT __::___1_ 2: :2502’jjf/5'5X’0 9-— 2M-JX/b‘ F, _ F = 200x59x250 I500 kN - F2 = /0X30.0x_25 = 750 FA! 1% = 750 k'Al‘. ..'. fvllb "' [‘7 7V5.) 7" ['2- X/5O '1" f3 Xj/O . ‘. ._-,22 5 + //2— —5 + 232- 5 =3é75 km ) mm - "S Z2 Cmfl-lb' V", ' , 5 4 ___ 24/ 5X/0 : /'/56X/Ogfl7md 4222-5 ' W . _ .é 0,» 2:13 .4. w. .= Wax/06,77,713 ~ 0} 250 —————-——-=—~—--————-————-—~—-— 4 6 5' .2: ”475/0 MM' 5 z 112 = .W H . * a; " I s = M (awm 74,. M W16 : .3; 30:59 xszo 7L 3-000x/50 1‘ $000505 ‘ 3: #4172005 mmj I I p a! x“ h... jg: __n ['47 '__ , .f’l’a'ln” '62:” or? ”‘2'“ 2517355 '“ ”‘75 5) M _= 2f4-/ 1W~m ‘ ' ’ gm!" j/‘e/a/I'j .‘occ_ar3;;.__a{‘ exfreme I {502?on #4725229 - = - ‘ EC) ”’5‘” = 347-5 AN—m. 5’ - -'. : 'M l-Z‘ d) W/Jen en 71/38 7‘0; 75/522373 Ara: '- Ae'come fibszl/ 17%6.‘ resf of #78 Jed/2227 wow/d A; we A5260» [15/35ng , ' xh arc/6r 7‘0 ”7;?thth C4: 7" (ghee As i (4/ +x42) - ’ 14f f/w': five/'27:“, momenf car-Him! 9/ Jae/fan ==-“ M/b . ...
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