Appendix XVI Cracked Section Properties of the Pier Cap Beams of the Steel Girder Bridge using the Moment Curvature Method and ACI Equation
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1 ppndix XV rakd Stion Proprti o th Pir ap Bam o th Stl Girdr Bridg ug th omnt urvatur thod and Equation Wt Bound Pir ap Bam Figur XV- Th atual pir ap bam ro tion [Brown, 99] Th ¾ - al i no longr orrt 5 8 Figur XV- Th impliid pir ap bam ro tion Thi igur wa not drawn to al 6
2 omnt urvatur thod T S + b ' S d + ' ' ' b d d ( ( ( 5 + ( 7( ( 5 ( ( + ( 7( 5 ' ( ( ( 7( um th omprion tl will not ild: d d ' d' ' d' d d 6ki 7 E 9,ki 5 ' 7 5 ' ( ki 9 E 57 ki ' E ' b ' d ( ( ( d B pluggg th ollowg valu to th quation abov, 6
3 ( #9bar 6ki ' 889 ( 7#bar E 9,ki 5 ' 7 5 b 5 ' ki 7 9 d 5 thn a third-dgr quation trm o i obtad Th olution o th third-dgr quation ar or 5 or i obvioul ruld out, and i ruld out bau th rorg ara at th bottom i l than that at th top, thror i uppod to b l than hal o 5, whih i qual to 55 Thu 5 i th onl poibl olution Now th itial aumption that th omprion tl will not ild mut b hkd: d' 5 5 ' 7 5 < 7 d 5 5 Thu th itial aumption i orrt Now max mut b hkd to i it i l than it i, thn th tr blok will hav a paraboli hap 5 max 7 75 < 9 d 5 5 6
4 Thu th tr blok will b paraboli Th urvatur at th ild pot i φ max / Now th momnt at ild an b obtad ug Figur V- max max max ( ( max ( ( ( 5 ( 5 ( 5 ( b ' max ( ( ( 5 ( ki ( 5 ( 5 559k 6k b d max ( 889 ( 9,ki( 7 ( d + ( ( 559k + ( 6k ( 5 5 5,8kip ( d d'
5 E g 5,8kip 9,,kip / mm φ 9,,kip 57 ki 6, ( 5 ( 8 5 6, Equation thod ' ki r 75 r r ' 75 pi pi r 7,kip ( ki ( 5 ( 8 Thi rakg momnt mut b ompard with th maximum poitiv and ngativ momnt th pir ap bam ( a, whih ar givn Tabl XV- Tabl XV- Th alulation or th maximum poitiv and ngativ momnt th pir ap bam Eat Bound Lan Wt Bound Lan aximum Poitiv aximum Ngativ aximum Poitiv aximum Ngativ Dign Truk Dign Tandm Two Dign Truk Lan ontrollg Load Liv Load Et Dad Load Et LL+DL Et Not Load ar kip 6
6 r 7,kip > 679kip or th maximum poitiv momnt a r 7,kip > 7kip or th maximum ngativ momnt Thror aordg to th Equation mthod, g 5, ( 5 ( 8 a Eat Bound Pir ap Bam Figur XV- Th atual pir ap bam ro tion [Brown, 99] Th /' - al i no longr orrt 65
7 5 8 Figur XV- Th impliid pir ap bam ro tion Thi igur wa not drawn to al omnt urvatur thod T S + b ' S d + ' ' ' b d d (( 5 ( 5 + ( 7( ( 5 8 ' ( ( (( 5 + ( 7( ((
8 um th omprion tl will not ild: d d ' d' ' d' d d 6ki 7 E 9,ki 5 ' 7 8 ' ( ki 9 E 57 ki ' E ' b ' d ( ( ( d B pluggg th ollowg valu to th quation abov, ( #9bar 6ki ' ( 9#bar E 9,ki 5 ' 7 8 b 5 ' ki 7 9 d 8 thn a third-dgr quation trm o i obtad Th olution o th third-dgr quation ar or 59 or -9 67
9 -9 i obvioul ruld out, and i ruld out bau th rorg ara at th bottom i l than that at th top, thror i uppod to b l than hal o 8, whih i qual to 5 Thu 5 i th onl poibl olution Now th itial aumption that th omprion tl will not ild mut b hkd: d' 59 5 ' 7 56 < 7 d 8 59 Thu th itial aumption i orrt Now max mut b hkd to i it i l than it i, thn th tr blok will hav a paraboli hap < d 8 59 max 9 Thu th tr blok will b paraboli Th urvatur at th ild pot i φ max / Now th momnt at ild an b obtad ug Figur V- max max max ( ( max ( ( ( 59 ( 59 ( 59 (
10 69 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( , 6, 57,,,, / 66 8, 8, ' ( , ' 5 max max ki kip kip mm kip E kip k k d d d k ki k ki b d b g φ
11 Equation thod ' ki r 75 r r ' 75 pi pi r 7,kip ( ki ( 5 ( 8 Thi rakg momnt mut b ompard with th maximum poitiv and ngativ momnt th pir ap bam ( a, whih ar givn Tabl XV- r 7,kip > 6,5kip or th maximum poitiv momnt a r 7,kip >,78kip or th maximum ngativ momnt Thror aordg to th Equation mthod, g 5, ( 5 ( 8 a 7
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