Attachment K. Supplemental Shear Test Information

Transcription

1 Attachment K. Supplemental Shear Test Information Computer Program Due to the iterative nature of shear design following the General Procedure in Article of the AASHTO LRFD Bridge Specifications a computer program was developed in order to help speed up the calculation process. The program was created through the use of the software, Mathcad, v. 14.0, copyrighted by Parametric Technology Corporation. The main program appears in Figure K-1, with the supplemental subroutines shown in Figure K-2. There was a third program, also developed using Mathcad, that helped to interpolate the tabularized values for and found in Appendix B5 of the AASHTO LRFD Bridge Specifications. This program is not presented here due to the size of the program rendering the program impractical to display. Although the equations used in Article are certainly easier than interpolating the tabularized values, the initial research started prior to having access to the latest design code. Furthermore, the results using the and equations proved to be even more conservative than the original LRFD method. Although not nearly as iterative in nature, a computer program was also developed for calculating the shear strength following the Simplified Procedure for Prestressed and Nonprestressed Sections in Article of the AASHTO LRFD Bridge Specifications. That program is given in K-1

2 calc ShearStrengthCheckFinal "NG" iterations 0 iterations.tot 0 for i 1 rows P assume 28 i while iterations iterations 1 for ShearStrengthCheckFinal x_ x critical.assume if i 1 1 for 1 rows P assume rows i 1 rows "NG" P assume.old P i assume i if rows P assume rows old i i for i 1 rows P assume.old P assume P assume.old i rows P assume old old i rows P assume P assume rows P assume P assume.old rows Passume P assume rows P assume if rows P assume rows old LiveLoad LoadSpacing x bearing x bearing.roller DeadLoad DeadLoad 9 DeadLoad 8 Load Live P assume ShearSpan L span DeadLoad Dead Load L span ShearSpan w beam DC 1 DC 2 DW P diaphragm V assume 1.00 DeadLoad LiveLoad 3 kip M assume DeadLoad DeadLoad LiveLoad 4 kipft V p V p. cg harp cg harp. f c'c f c'deck if f c'deck 0 f c' 1 1. f c'c otherwise for i 1 rows Num straight cg straight Num harp cg harp i cg ps i Num strand d pi h c cg ps i c c calc A ps f pu A s.bot A s.top f y f c'c 1 b eff b v h flange Figure K-1. Mathcad program used to calculate the design shear strength of a girder. k d p K-2

3 c c calc A ps f pu A s.bot A s.top f y f c'c 1 b eff b v h flange k d p for i 1 rows c i f ps.maxi f pu 1 k d pi 2 l d.psi f ps.maxi 3 f pe f ps f ps. l d.ps f ps.max f y s.yield E s Num strand.bot Num strand.bot. cg harp cg ps.bot cg ps.bot. cg straight cg harp h c Num straight for i 1 rows d p.boti h c cg ps.bot i d s.bot h i c cg s.bot i A ps.bot Num i strand.bot A i strand d ei A ps.bot f i psi d p.boti A s.bot f i y d s.bot i A ps.bot f i psi si d c ei c i i A s.bot f i y i return "Tension steel does not yield at x.dist (inches)" in if si s.yield a i 1 c i a i d vi max d ei 0.9d 2 ei 0.72h c M assumei max M assume1i V assumei V pi d vi v u v u. V assume V p d v MinReinforcingCheck l d.mild l d.mild. d long.rebar f c' f y IsTopBarEffectApplicableIsLongSteelEpoxyCoatedLongSteelCoverLongSteelCl for i 1 rows i x bearing RedFactor x.psi if x l disti x bearing l d.psi d.psi A ps.bot. xi 1 if i x bearing l d.psi Num strand.bot A i strand RedFactor x.psi Figure K-1 (cont.) Mathcad program used K-3 to calculate the design shear strength

4 A ps.bot. xi Num strand.bot A i strand RedFactor x.ps i x i bearing RedFactor x.s if x i l dist x d.mild i bearing l d.mild A s.bot. xi 1 if x i bearing l d.mild A s.bot RedFactor i x.s i iterate A ps.bot. x A s.bot. x f ps c d p M assume d v V assume V p cg harp v u MinReinforcingCheck 1 2 iterations.tot iterations.tot if method "General" 3 V r V r. d v s v.design V p for i 1 rows ShearStrengthCheck "OK" if V i ri V assume 0.1kip V i ri V assume 0kip i if V ri V assume 0.1kip V i ri V assume 0kip i ShearStrengthCheck "NG!!!" i ShearStrengthSum ShearStrengthSum 1 1 V assume i 2 V r V i assume 0.01kip i 5 V assume 1.00DeadLoad i i DeadLoad 7 i P assume i 2L span ShearSpan LoadSpacing V assume 1.00DeadLoad 5 i i DeadLoad 7 i L span 2ShearSpan LoadSpacing V assume 1.00 DeadLoad 5 i i DeadLoad 7 i 2ShearSpan LoadSpacing x_x critical.assume 2 vect_elemnt x.critical for ShearStrengthCheckFinal if if i 1 1 d v.critical d v vect_elemntx.critical1 critical vect_elemntx.critical1 in 1 rows vect_elemnt x.critical 1 rows vect_elemnt x.critical d v.critical critical "OK" if ShearStrengthSum 0 iterations 500 d v vect_elemnt x.critical vect_elemntx.critical K-4 i 1.00 DeadLoad 9 i 1.00 DeadLoad 9 i 1.00 DeadLoad 9 kip L span kip L span kip L span Figure K-1 (cont.) Mathcad program used to calculate the design shear strength if if if x d Sh x d

5 critical vect_elemntx.critical 1 x critical.assume max d v.critical 2 d v.criticalcot critical 180 V n.critical x critical 0 0 for if ( ) i 1 rows vect_elemnt x.critical V n.criticali V r vect_elemnt x.criticali x criticali vect_elemnt x.criticali 1 rows vect_elemnt x.critical V n.critical V rvect_elemntx.critical if rowsvect_elemnt x.critical 1 x critical vect_elemnt x.critical for i 1 rows calc "Simple" if method "Simple" N i u 0 MinReinforcingCheck "OK" h i c 16in calc "General" if method "General" i "error" otherwise V r P assume kip kip in x critical in V n.critical kip iterations iterations.tot calc MinReinforcingCheck s Figure K-1 (cont.) Mathcad program used to calculate the design shear strength K-5

6 in 2 kip 1.6 if h 24in 1.0 if h 24in "error" otherwise f c' f r.v 0.20 ksi ksi if FineAggType "sand" 0.17 "error" f c' ksi ksi otherwise if FineAggType "lightweight" cg harp. cg harp 0 for i 1 rows cg harpi cg harp.end i x bearing tan harp if i HarpDist cg harpi cg harp.ms if i HarpDist cg harpi "error" otherwise cg harp A rebar ( rrebarsize) for i 1 r A rebari if RebarSize 1.9 i A rebari if RebarSize 2 i A rebari if RebarSize 2.5 i A rebari 0.11 if RebarSize 3 i A rebari 0.20 if RebarSize 4 i A rebari 0.31 if RebarSize 5 i A rebari 0.44 if RebarSize 6 i A rebari 0.60 if RebarSize 7 i A rebari 0.79 if RebarSize 8 i A rebari 1.00 if RebarSize 9 i A rebari 1.27 if RebarSize 10 i A rebari 1.56 if RebarSize 11 i A rebari 0 if RebarSize 0 i in 2 A rebar K-6 Figure K-2. Subroutines used in the main calculation program given in Figure K-1.

7 v 1 if ConcreteType "NWC" ConcreteType "LWC" Value_f ct "specified" 4.7 f ct f c' ksi 4.7 f ct f c' ksi if ConcreteType "LWC" Value_f ct "specified" 4.7 f ct f c' ksi 0.85 if ConcreteType "LWC" Value_f ct "not specified" FineAggType "sand" 0.75 if ConcreteType "LWC" Value_f ct "not specified" FineAggType "lightweight" "error" otherwise MinReinforcingCheck "OK" if r v f c' ( ksi) "NG!!!" if v f c' ( ksi) b v s v.design r f yv b v s v.design r f yv A vr A vr "error" otherwise 0.85 if f c'deck 4ksi 1. f c'deck x x m o x_ x critical.assume f c'deck max if f ksi c'deck 4ksi x 0 x 0 1 m 1 for i 1 12 ( ) if 0.1ShearSpan( i 1) x critical.assume x critical.assume 0.1ShearSpan( i) x x i 1 critical.assume n i 1 if 0.1ShearSpan( i 1) ShearSpan x critical.assume ShearSpan x critical.assume 0.1ShearSpan for x ShearSpan x i 2 critical.assume o i 2 i 2 13 m 2 if i n m 3 if i o ( break) if i 13 1 x critical.assume 2 ShearSpan x 0.1ShearSpan( i m) otherwise i X x 1 X 2 n o in if o 0 ( nin ) if o 0 "error" otherwise K-7 X Figure K-2 (cont.) Subroutines used in the main calculation program given in Figure K-1.

8 Load Live ft V assume V assume kip ShearSpan ShearSpan ft LoadSpacing LoadSpacing ft L span L span ft for i 1 rows i x% i ShearSpan if i ShearSpan V LLi M LLi V assumei 2L L span ShearSpan LoadSpacing span V assumei 2L L span ShearSpan span LoadSpacing x disti if ShearSpan i i ShearSpan LoadSpacing V LLi V assumei L L span 2ShearSpan span LoadSpacing if M LLi V assumei 2L L span ShearSpan span i ShearSpan LoadSpacing LoadSpacing x disti V assumei i ShearSpan V assumei V LLi ( 2ShearSpan LoadSpacing) L span M LLi V assumei ( 2ShearSpan LoadSpacing) L L span i span ans augment x% V LL M LL ans Figure K-2 (cont.) Subroutines used in the main calculation program given in Figure K-1. K-8

9 Dead Load x dist ft L span L span ft x bearing x bearing ft x bearing.roller x bearing.roller ft ShearSpan ShearSpan ft w beam w beam klf DC 1 DC 1 klf DC 2 DC 2 klf DW DW klf P diaphragm P diaphragm kip for i 1 rows i x% i ShearSpan w beam V beami x 2L bearing L span x bearing.roller 2 w beam x bearing.roller x span L bearing L span x bearing.roller w beam i x bearing span w beam M beami x 2L bearing L span x bearing.roller 2 w beam x bearing.roller 1 x span L bearing L span x bearing.roller i span 2 w beam x i bea L span V diaphragmi P diaphragm if i 3 K-9 Figure K-2 (cont.) Subroutines used in the main calculation program given in Figure K-1.

10 L span V diaphragmi P diaphragm if i 3 V diaphragmi 0 otherwise L span M diaphragmi P diaphragm i if i 3 M diaphragmi L span P diaphragm 3 otherwise DC 1 V DC1i x 2L bearing L span x bearing.roller 2 DC 1 x bearing.roller x span L bearing L span x bearing.roller DC 1 i x bearing V span diaphragmi DC 1 M DC1i 2L x bearing L span x bearing.roller 2 DC 1 x bearing.roller 1 x span L bearing L span x bearing.roller i span 2 DC 1 x 2 dist x i bearing M diaphragmi DC 2 V DC2i x 2L bearing L span x bearing.roller 2 DC 2 x bearing.roller x span L bearing L span x bearing.roller DC 2 i x bearing span DC 2 M DC2i 2L x bearing L span x bearing.roller 2 DC 2 x bearing.roller 1 x span L bearing L span x bearing.roller i span 2 DC 2 x 2 dist x i bearing DW V DWi x 2L bearing L span x bearing.roller 2 DWx bearing.roller x span L bearing L span x bearing.roller DW i x bearing span DW M DWi 2L x bearing L span x bearing.roller 2 DWx bearing.roller 1 x span L bearing L span x bearing.roller i span 2 DW x 2 dist x i bearing ans augment x% V beam M beam V DC1 M DC1 V DC2 M DC2 V DW M DW ans Figure K-2 (cont.) Subroutines used in the main calculation program given in Figure K-1. K-10

11 V p 0 V p. for i 1 rows f pe i x bearing V pi Num l harp A strand sin harp if i x bearing l t i L span HarpDistRatio t V pi f pe Num harp A strand sin harp if i x bearing l t i L span HarpDistRatio V pi 0 if i L span HarpDistRatio "error" otherwise V p f ps 0 f ps. l d f ps.max for i 1 rows f pe i x bearing f psi if x l disti x bearing l t t i x bearing l t f psi f pe f l di l ps.maxi f pe if x t disti x bearing f psi f ps.maxi if i x bearing l di l t i x bearing l di f psi "error" otherwise f ps c calc for c i 1 rows d pi 10 6 in if d pi 0 c i A ps f pu A s.bot f i y 0.85f c'c 1 b eff ka ps A s.top f y f pu d pi A ps f pu A s.bot f i y A s.top f y 0.85 f c'c b eff b v h flange c if c h i f i flange pu 0.85f c'c 1 b v ka ps d pi Figure K-2 (cont.) Subroutines used in the main calculation program given in Figure K-1 K-11

12 A long.rebar A rebar rowsd long.rebar for i l d.mild. d long.rebar d long.rebar 1.25A long.rebari f y d long.rebari f y l dbi max 0.4 in f c' ksi in 8 ksi 1i 1.0 if IsTopBarEffectApplicable "No" 1.4 if IsTopBarEffectApplicable "Yes" "error" otherwise 2i 1.0 if IsLongSteelEpoxyCoated "No" d long.rebari d long.rebari 1.5 if IsLongSteelEpoxyCoated "Yes" LongSteelCover 3 in LongSteelClearSpacing 6 in 8 8 d long.rebari d long.rebari 1.2 if IsLongSteelEpoxyCoated "Yes" LongSteelCover 3 in LongSteelClearSpacing 6. in 8 8 "error" otherwise 12i min f ct 10 6 ksi if f ct 0 Value_f ct "not specified" 3i 1 if ConcreteType "NWC" f c' ksi max if ConcreteType "LWC" Value_f f ct ct "specified" 1.3 if ConcreteType "LWC" Value_f ct "not specified" FineAggType "lightweight" 123i 1 rows d long.rebar l d.mildi 1.2 if ConcreteType "LWC" Value_f ct "not specified" FineAggType "sand" "error" 12i 3i l dbi 123i otherwise l d.mild1 Figure K-2 (cont.) Subroutines used in the K-12 main calculation program given in Figure K-1.

13 Num strand.bot. cg harp Num strand.bot 0 for i 1 rows Num strand.bot Num i strand Num harp cg harpi 2in 1 if 2 h c Num strand.bot Num i strand Num harp 2 cg harpi 2in 1 2 h 1 if c cg harpi 2 h c 1 Num strand.bot Num i strand Num harp 4 cg harpi 2 h 1 if c cg harpi 2in 2 h c 1 Num strand.bot Num i strand if cg harpi 2in 2 h c Num strand.bot "error" otherwise i Num strand.bot x1 x2 x3 x x. 180 for i 1 rows ( ) x1i x2i M assume i d vi M assume i d vi 0.5 N u 0.5 V assume V pi i cot i 2 E s A s.bot. xi E p A ps.bot. xi 0.5 N u 0.5 V assume V pi i E s A s.bot. xi cot i E p A ps.bot. xi A ps.bot. xi f po A ps.bot. xi f po x3i M assume i d vi 0.5 N u 0.5 V assume V pi i 2 E beam A ct E s A s.bot. xi cot i xi x1i if MinReinforcingCheck "OK" i x1i 0 E p A ps.bot. xi x2i if MinReinforcingCheck "NG!!!" i x2i 0 x3i if x1i 0 x2i 0 A ps.bot. xi f po x Figure K-2 (cont.) Subroutines used in the main calculation program given in Figure K-1. K-13

14 cg ps.bot. cg straight cg harp h c Num straight for i 1 rows 1 cg ps.bot cg i straight if cg harpi 2in 2 h c Num straight cg straight 2 cg harpi 2in 1 cg ps.bot cg i Num straight 2 harpi 2in 2 h 1 if c cg harpi 2 h c Num straight cg straight 2 cg harpi 2in cg harpi 1 cg ps.bot cg i Num straight 4 harpi 2 h 1 if c cg harpi 2in 2 h c Num straight cg straight 6cg harpi cg ps.bot if cg harpi 2in i cg ps.bot Num straight h c v 1 s x s xe v for i v u. V assume V p d v MinReinforcingCheck ( ) V assume i v V pi v 1i v b v d vi s xi d vi 1.38in s xei min s xi 80in a g 0.63in v i 1 rows v 1i if MinReinforcingCheck "OK" f i c' v s xei in if MinReinforcingCheck "NG!!!" i Figure K-2 (cont.) Subroutines used in the main calculation program given in Figure K-1. K-14

15 L R r L R u rows cg harp x v u MinReinforcingCheck ( ) Num strand.bot Num strand.bot. cg harp for i 1 1 v u1. v u1 v u x u MinReinforcingCheck v u2. v u2 v u x u MinReinforcingCheck xl. xl v u x u MinReinforcingCheck xr. xr v u x u MinReinforcingCheck for i 1 1 MinOK1 1L. MinOK1 1L v u1. xl. u MinReinforcingCheck MinOK2 1L. MinOK2 1L v u1. xl. u MinReinforcingCheck MinNG1 1L. MinNG1 1L v u1. xl. u MinReinforcingCheck MinNG2 1L. MinNG2 1L v u1. xl. u MinReinforcingCheck for i 1 1 MinOK1 1R. MinOK1 1R v u1. xr. u MinReinforcingCheck MinOK2 1R. MinOK2 1R v u1. xr. u MinReinforcingCheck MinNG1 1R. MinNG1 1R v u1. xr. u MinReinforcingCheck MinNG2 1R. MinNG2 1R v u1. xr. u MinReinforcingCheck for i 1 1 MinOK1 1L. MinOK1 1L v u1. xl. u MinReinforcingCheck MinOK2 1L. MinOK2 1L v u1. xl. u MinReinforcingCheck MinNG1 1L. MinNG1 1L v u1. xl. u MinReinforcingCheck MinNG2 1L. MinNG2 1L v u1. xl. u MinReinforcingCheck for i 1 1 MinOK1 1R. MinOK1 1R v u1. xr. u MinReinforcingCheck MinOK2 1R. MinOK2 1R v u1. xr. u MinReinforcingCheck MinNG1 1R. MinNG1 1R v u1. xr. u MinReinforcingCheck MinNG2 1R. MinNG2 1R v u1. xr. u MinReinforcingCheck for i 1 1 MinOK1 2L. MinOK1 2L v u2. xl. u MinReinforcingCheck MinOK2 2L. MinOK2 2L v u2. xl. u MinReinforcingCheck MinNG1 2L. MinNG1 2L v u2. xl. u MinReinforcingCheck MinNG2 2L. MinNG2 2L v u2. xl. u MinReinforcingCheck for i 1 1 Figure K-2 (cont.) Subroutines used in the main calculation program given in Error! K-15

16 K-16

17 for i 1 1 MinOK1 2R. MinOK1 2R v u2. xr. u MinReinforcingCheck MinOK2 2R. MinOK2 2R v u2. xr. u MinReinforcingCheck MinNG1 2R. MinNG1 2R v u2. xr. u MinReinforcingCheck MinNG2 2R. MinNG2 2R v u2. xr. u MinReinforcingCheck for i 1 1 MinOK1 2L. MinOK1 2L v u2. xl. u MinReinforcingCheck MinOK2 2L. MinOK2 2L v u2. xl. u MinReinforcingCheck MinNG1 2L. MinNG1 2L v u2. xl. u MinReinforcingCheck MinNG2 2L. MinNG2 2L v u2. xl. u MinReinforcingCheck for i 1 1 MinOK1 2R. MinOK1 2R v u2. xr. u MinReinforcingCheck MinOK2 2R. MinOK2 2R v u2. xr. u MinReinforcingCheck MinNG1 2R. MinNG1 2R v u2. xr. u MinReinforcingCheck MinNG2 2R. MinNG2 2R v u2. xr. u MinReinforcingCheck for i 1 1 1L MinOK1 1L. MinOK2 1L. MinNG1 1L. MinNG2 1L. 1R MinOK1 1R. MinOK2 1R. MinNG1 1R. MinNG2 1R. 1L MinOK1 1L. MinOK2 1L. MinNG1 1L. MinNG2 1L. 1R MinOK1 1R. MinOK2 1R. MinNG1 1R. MinNG2 1R. 2L MinOK1 2L. MinOK2 2L. MinNG1 2L. MinNG2 2L. 2R MinOK1 2R. MinOK2 2R. MinNG1 2R. MinNG2 2R. 2L MinOK1 2L. MinOK2 2L. MinNG1 2L. MinNG2 2L. 2R MinOK1 2R. MinOK2 2R. MinNG1 2R. MinNG2 2R. for i 1 rows Li 2Li if v u2.i v u1.i 0 v u2.i v ui 1Li 2Li v u2.i v u1.i Ri 2Ri if v u2.i v u1.i 0 v u2.i v ui 1Ri 2Ri v u2.i v u1.i ri Ri if xr.i xl.i 0 2Li 2Ri otherwise otherwise Li Ri xr.i xi xr.i xl.i Ri otherwise Figure K-2 (cont.) Subroutines used in the K-17 main calculation program given in Figure K-1.

18 Li Ri xr.i xi xr.i xl.i Li 2Li if v u2.i v u1.i 0 Ri otherwise v u2.i v ui 1Li 2Li v u2.i v u1.i Ri 2Ri if v u2.i v u1.i 0 v u2.i v ui 1Ri 2Ri v u2.i v u1.i i Ri if xr.i xl.i 0 2Li 2Ri otherwise otherwise Li Ri xr.i xi xr.i xl.i Ri otherwise r V c V s V n V r V r. d v s v.design V p 180 for i 1 rows V ci i v V si A vi f yv ( ) f c' ksib v d vi d vi cot i V n1 V c V s V p V n f c' b v d v V p for i 1 rows V ni min V n1i V n2i V r V r v V n cot( ) s v.design i sin( ) Figure K-2 (cont.) Subroutines used in the main calculation program given in Figure K-1. K-18

19 iterate for i 1 rows if method "Simple" N u 0 MinReinforcingCheck "OK" h 1 c 16in ri 45 2 i otherwise AngleCheck "NG" iterations 0 Num strand.bot while AngleCheck "NG" iterations iterations 1 Num strand.bot. cg harp iterations.tot iterations.tot 1 sum 0 x x1 x2 x3 x 180 for i 1 rows ( ) x1i x2i M assume i d vi M assume i d vi 0.5 N u 0.5 V assume V i pi cot i 2 E s A s.bot. xi E p A ps.bot. xi 0.5 N u 0.5 V assume V pi i E s A s.bot. xi cot i E p A ps.bot. xi A ps.bot. xi f po A ps.bot. xi f po x3i M assume i d vi 0.5 N u 0.5 V assume V pi i 2 E beam A ct E s A s.bot. xi cot i xi x1i if MinReinforcingCheck "OK" i x1i 0 E p A ps.bot. xi x2i if MinReinforcingCheck "NG!!!" i x2i 0 x3i if x1i 0 x2i 0 A ps.bot. xi f po x 1 r cg harp x v u MinReinforcingCheck 2 cg harp x v u MinReinforcingCheck 180 for i 1 rows Figure K-2 (cont.) Subroutines used in the K-19 main calculation program given in Figure K-1.

20 for if i 1 rows ri.0005 i sum sum 1 1 i 2 r i i AngleCheck "OK" if sum 0 iterations 50 r iterations x Figure K-2 (cont.) Subroutines used in the main calculation program given in Figure K-1. K-20

21 calc ShearStrengthCheckFinal "NG" iterations 0 while iterations iterations 1 for iterations 1 rowsp assume i x_ x critical.assume 1 if i 1 1 rows for 1 rows P assume.oldi P assumei for i 1 rows P assume.old P assume P assume.oldi rows Passume if rowsp assume rows rowsp assume P assumerows Passume K-21 if rowsp assume rows P assume P assume.old LiveLoad Load Live P assume ShearSpanLoadSpacingL span DeadLoad Dead Load L span x bearing x bearing.roller ShearSpanw beam DC 1 DC 2 DWP diaphragm LiveLoad if LiveLoad V assume 1.00 DeadLoad 5 DeadLoad DeadLoad LiveLoad 3 kip M assume 1.00 DeadLoad 6 DeadLoad DeadLoad LiveLoad 4 kip ft M DC1 DeadLoad 6 kipft V d DeadLoad 5 DeadLoad 7 DeadLoad 9 kip Figure K-3. Mathcad program used to calculate the shear strength following the Simplified Procedure.

22 K-22

23 V d DeadLoad 5 DeadLoad 7 DeadLoad 9 kip V i V assume V d M max M assume DeadLoad 6 DeadLoad 8 DeadLoad 10 kip ft V p. V p cg harp cg harp. f c'c f c'deck if f c'deck 0 1 f c' otherwise 1. f c'c f y s.yield E s Num strand.bot Num strand.bot. cg harp cg ps.bot cg ps.bot. cg straight.bot cg harp h c Num straight.bot for i 1 rows cg psi d pi h c cg psi Num straight cg straight Num strand Num harp cg harpi c d i pi 10 6 in if d pi 0 c i A ps f pu A s.boti f y 0.85f c'c 1 b eff A s.top f y ka ps f pu d pi A ps f pu A s.boti f y A s.top f y 0.85 f c'c b eff b v h flange c if c h i f i flange pu 0.85f c'c 1 b v ka ps d pi c i f ps.maxi f pu 1 k d pi 2 l d.psi f ps.maxi 3 f pe f ps f ps. l d.ps f ps.max for i 1 rows A ps.bot Num i strand.bot A i strand d p.boti h c cg ps.bot i d s.bot h i c cg s.bot i Figure K-3 (cont.) Mathcad program used to calculate the shear strength following the Simplified Procedure. K-23

24 d s.bot h i c cg s.bot i d ei A ps.bot f psi d p.boti A i s.bot f i y d s.boti A ps.bot f psi i si d c ei c i i A s.bot f i y i return "Tension steel does not yield at x.dist" in if si s.yield a i 1 c i a i d vi max d ei 0.9d 2 ei 0.72h c f c' f r.v 0.20 ksi ksi if FineAggType "sand" 0.17 "error" f c' ksi ksi e strand y b cg ps l t 60 for i 1 rows P f.cpei otherwise f pe A strand f pe A strand "error" if FineAggType "lightweight" i x bearing Num strand if x l disti x bearing l t t Num strand if i x bearing l t otherwise P f.cpei P f.cpei e strand i f cpei A beam S b DeadLoad 6 kip ft i M crei S bc f r.v f cpei S b f V c' ii M crei V cii max 0.02 v ksib ksi v d vi V di 0.06 M v maxi P f.pci f pci f psi A strand V cwi 0.06 v Num strand f c' ksi ksi b vd vi P f.pci P f.pci e strand y i b y bc M DC1i y b y bc if y A beam I beam I bc h top.flange beam P f.pci P f.pci e strand i y b h top.flange M DC1i y b h top.flange if y A beam I beam I bc h top.flange beam f c' ksi 0.30 f ksi pci b v d vi V pi K-24 Figure K-3 (cont.) Mathcad program used to calculate the shear strength following the Simplified Procedure.

25 f c' V cwi 0.06 v ksi 0.30 f ksi pci b v d vi V pi V ci min V cii V cwi cot 1 if V i cii V cwi V si f pci min 1 3 v f c' ksi if A vi f yv d vi cot cot( ) sin( ) i s v.designi V cii V cwi V n1 V c V s V n f c' b v d v V p for i 1 rows ShearStrengthCheck i V ri v min V n1i V n2i "OK" if V ri V assumei 0.1kip V ri V assumei 0kip if V ri V assumei 0.1kip V ri V assumei 0kip ShearStrengthCheck "NG!!!" i ShearStrengthSum ShearStrengthSum 1 1 V assumei 2 V r V assumei 0.01kip i P assume.ri P assumei i DeadLoad 7 i i V assumei DeadLoad 1.00 DeadLoad 9 kip L span P assumei 2L span ShearSpan LoadSpacing V assumei 1.00 DeadLoad 5 i DeadLoad 7 i 1.00 DeadLoad 9 i kip L span L span 2ShearSpan LoadSpacing V assumei 1.00 DeadLoad 5 i DeadLoad 7 i 1.00DeadLoad 9 i kip L span 2ShearSpan LoadSpacing vect_elemnt x.critical for i 1 1 x_x critical.assume 2 in ShearStrengthCheckFinal "OK" if ShearStrengthSum 0 iterations 500 d v.critical d v if rows vect_elemntx.critical1 vect_elemnt x.critical 1 d v.critical d v if vect_elemnt x.critical rowsvect_elemnt x.critical 1 if if if i ShearSpan ShearSpan i ShearSpan 1 x critical.assume max d v.critical 2 d v.critical cot vect_elemntx.critical1 if rowsvect_elemnt x.critical 1 1 K-25 Figure K-3 (cont.) Mathcad program used to calculate the shear strength following the Simplified Procedure.

26 1 x critical.assume max d v.critical 2 d v.critical cot vect_elemntx.critical1 if 1 x critical.assume max d v.critical 2 d v.critical cot vect_elemntx.critical if V n.critical x critical 0 0 for i ( ) 1 rows vect_elemnt x.critical V n.criticali V r vect_elemnt x.criticali x criticali vect_elemnt x.criticali otherwise V n.critical V r vect_elemnt x.critical x critical vect_elemntx.critical if rowsvect_elemnt x.critical 1 rowsvect_elemnt x.critical 1 rowsvect_elemnt x.critical 1 Figure K-3 (cont.) Mathcad program used to calculate the shear strength following the Simplified Procedure. Beam Example Following the General Procedure using Appendix B5 The following calculations are for a simply-supported, Type II prestressed girder with a cast-inplace composite deck, containing a typical amount of shear reinforcement found in such a girder spanning 60 ft. As discussed in Section , for the actual beam in this example, a portion of the girder was removed from the original 60-ft design such that that girder was shortened by some distance between the harping points. The drawing details for this example beam, T2.8.Typ, is shown in Figure A-2. The parameters that are necessary to perform the calculations are given the section below. Inputs Programming mechanism used for setting the lightweight concrete modifier. In this example, the assumption is that the lightweight concrete should not have a modifier; hence the concrete is assumed to be normal weight. K-26

27 the splitting tensile strength was assumed to be unspecified height of the beam web width distance from bottom of beam to beam centroid distance from top of beam to beam centroid girder spacing, which in this case, is the width of the deck thickness of the haunch structural depth of the deck area of concrete in the flexural tension zone, as defined in A distance from the extreme compression flange to the webflange intersection number of unharped tendons number of unharped tendons near the bottom of the beam number of harped tendons centroid of the unharped tendons centroid of the harped tendons at the end of the beam centroid of the harped tendons at the harping location K-27 centroid of the unharped tendons near the bottom of the beam

28 strand diameter distance from the support to the harping location yield strength of the stirrups stirrup size area of a double-legged stirrup angle of the stirrups relative to the longitudinal axis of the beam area of the mild steel reinforcement at the top of the girder distance from the extreme compression fiber to the steel in the top of the girder distance from the girder bottom to the centroid of the mild steel distance from the bottom of the girder to the centroid of the mild steel used for tension reinforcement size of the longitudinal mild reinforcement stirrup spacing, listed at distinct points along the shear span; see calculation for variable below area of mild reinforcing steel in the bottom of the beam, listed at distinct points along the shear span; see explanation for variable below. area of the mild steel in the longitudinal direction K-28

29 additional parameters for dead load that were all set to equal zero since the beam being tested was a stand-alone structure with a cast-in-place composite deck distance from the end of the beam to the center of bearing of the pin support distance from the end of the beam to the center of bearing of the roller support assumed distance from the support to the critical section axial force, assumed to be equal to zero resistance factor for shear, assumed to be equal to zero for analysis purposes programming mechanism used to calculate the shear strength following A (if applicable) or A Note that individual values for P assume are listed at distinct locations along the shear span (see the calculations for below. Each value is the assumed load applied by each of the actuators, whose locations are specified by the parameters ShearSpan and LoadSpacing, that would result in a shear failure at the corresponding section indicated in. Ordinarily, the calculation would start with an arbitrary value for all of the section locations, say 150 kips or 250 kips. However, for the purposes of this example, the given P assume values are used to mitigate the number of iterations necessary to reach convergence. Subsequent calculations will require knowing the location of the sections for which the shear strength is being calculated. Those locations are defined by the term, given in this example in terms of the distance from the support closest to the point loads: K-29

30 where each location is a tenth point along the length of the shear span, with the exception of the critical section, which in this case, is assumed to be 36.5 in. K-30

31 Basic Geometry Calculations Area of unharped prestessing tendons along the bottom of the girder. Angle of the harped tendon, relative to the longitudinal axis of the beam K-31 depth from the top of the composite section to the centroid of the mild steel reinforcement at the bottom of the beam.

32 centroid of the harped tedons Note that the centroid of the harped tendons changes along the length of the shear span. Thus, the example above and all other example calculations that are dependent on the location of the section give a detailed calculation at a specific location, = 5.73 in., which is the second location given in the parameter. The results for all sections are then given at the end of each specific calculation. K-32

33 Basic Material Calculations modululs of rupture for shear calculations As discussed earlier, although the concrete in this beam was actually lightweight concrete, the assumption was that the lightweight modifier was not necessary. Hence, this example assumes that lightweight concrete behaves similarly to normal weight concrete. K-33

34 Load Effects Live Loads For a simply-supported beam subjected to two point loads, the shear at any point along the beam within the shear span, V LL, can be calculated as K-34

35 The moment at the same locations, M LL, can be calculated as Dead Loads The uniformly distributed forces due to the weight of the beam, the haunch and the deck are calculated as: K-35

36 where DC 1 is the dead load of the composite structure. There were no other dead loads present. These results can be used to calculate the shear and moment due to composite dead loads, V DC1 and M DC1, respectively, as: K-36

37 Total Load Effects Given the assumed live loads, the total assumed shear and moment in the girder, V assume and M assume1 are the sum of the live and dead loads given in Sections 0 and 0 above, such that: K-37

38 Calculate the Shear Resistance Determine and Calculate the Effective Shear Depth of the Beam, d v Calculate the Depth of the Compression Block, c Assuming that the depth of the compression block is less than the thickness of the deck, and thus the compressive strength of the concrete, f c'c, is that of the deck concrete, f c'deck, then K-38

39 Calculate the Stress in the Prestressing Strands, f ps Let f ps.max be the stress in the steel according to Eq in the AASHTO LRFD Bridge Specifications, and l d.ps be the development length of the prestress. Then, K-39

40 K-40

41 Find the Effective Depth, d e The centroid of the prestressing steel in the flexural tension region of the beam is calculated as: K-41

42 Therefore, d p.bot and d s.bot, the depths from the top of the composite section to the centroid of the prestressing steel and mild steel, respectively, that are on the flexural tension side of the beam, are calculated as: In order to calculate the area of strand in the flexural tension side of the beam, the number of strands is determined as: K-42

43 Consequently, the area of strand in the flexural tension side of the beam is: Using the information above, the effective depth, d e, is calculated as: K-43

44 Ensure that the mild steel in the flexural tension side yields: K-44

45 Since the strain in the steel is greater than the yield strain, all assumptions used in calculating the depth of the compression block, c, are validated. Find the effective shear depth, d v Find the Factored Moment, M assume According to Article of the AASHTO LRFD Bridge Specifications, the factored moment used in determining and should not exceed K-45

46 M u V u -V p d v ( K - 1 ) where V p is calculated as: Therefore, the assumed moment in the girder at the time of failure, M assume, is the maximum of M assume1 found in ( K - 1 ): K-46

47 Find The Average Factored Shear Stress on the Concrete, v u Check for Minimum Reinforcement Requirements Calculate v u K-47

48 Determine the Reduced Area of Steel, A ps.bot. x and A s.bot. x According the Article , if the full development length has not been reached at a given section, then the area of steel must be reduced proportionately to the ratio of the length of the steel at that section versus the development length. K-48

49 Calculate the Development Length of the Mild Reinforcement K-49

50 K-50

51 Determine the Area Reduction Factors Calculate the Reduced Area of Steel, A s.bot. x and A ps.bot. x K-51

52 Determine and Even within each iteration for calculating the shear strength of a girder, the process for finding itself can be iterative. The reason is because, the predicted crack angle relative to the horizontal axis, depends on the longitudinal strain at mid-depth of the beam, x, which in turn, depends on, according to Article B5.2. If following the General Procedures in Article , then this process is not as iterative because the cracking angle was removed from the equation for longitudinal strain as a conservative simplification. If using Appendix B5, one must assume an initial value for the cracking angle and then repeat the calculation until a value for converges. In the case of this example, which again, follows the tabularized method in Appendix B5 for finding and, the assumed values for (in radians) at specific sections along the length of the shear span are: K-52

53 Having assumed, and knowing all of the other necessary parameters, one can calculate x as: K-53

54 Using the results for x and for v u, foundpreviously, one can go to the tables in Appendix B5 of the AASHTO LRFD Bridge Specifications to obtain and. Note that Article B5.2 states that if x is greater than zero, then the initial value for x should not be greater than if at least the minimum transverse reinforcement is used or if the beam has less than the minimum transverse reinforcement. Hence, in this example, most of the values for x along the length of the shear span were taken as As discussed earlier, a separate computer program was developed to facilitate linear interpolation of these values using the ranges given in the aforementioned tables. That program is not discussed here, but is a relatively basic routine. Using the values for x and v u, the resulting values for and are: Note that the results for are quite similar to the assumed cracking angle, assume. In terms of degrees, is: Calculate the Concrete Contribution to the Shear Resistance, V c K-54

55 Calculate the Steel Contribution to the Shear Resistance, V s K-55

56 Calculate the Nominal Shear Resistance Reiterate with New Value for P assume In order to refine the calculations for the nominal shear strength, the engineer might need to assume a different value for the externally applied load to the beam, P assume, based on a revised K-56

57 input for the shear in the beam, V assume. The new V assume can be taken as the midpoint between the original V assume and V n found in Section 0 above. In the case of this example, Therefore, the new assumed load from each of the two actuators located at the distances indicated by the parameters ShearSpan and LoadSpacing that would result in a shear failure However, note that P assume.new is practically the same as P assume given in Section 0. In this case, the calculations for the shear strength have converged, resulting in the shear strengths at the various sections along the shear span, indicated by V n in Section 0. The lowest value in V n is taken as the beam s shear strength, provided that the distance from the support to the corresponding section is not less than the critical section. The critical section, x critical, depends on K-57

58 the effective shear depth and the theoretical cracking angle, which in turn, are indirectly dependent on the critical section. Hence, finding x critical is also iterative in nature, and this iteration has been incorporated into the program indicated in Figure K-1. Using x critical.assume = in., which is the eighth element in the parameter, the shear depth at the critical section, d v.critical is the eighth element in d v. Likewise, the corresponding cracking angle at the critical section is the eighth element of given in radians as Thus, x critical can be calculated as: Again, note that the calculated value for x critical is the same as x critical.assume, so no further iterations are required. So, for this example, the smallest shear strength given in V n is kip at = in. However, since the distance to this section is less than the critical section, the nominal shear strength for the beam should be taken as kip, which is rounded to 222 kip. K-58

59 Inputs and Detailed Results for Shear Calculations Following the General Procedure Using Appendix B5 Beam T2.8.Typ.1 Inputs c (pcf) t struct 8.5 f y (ksi) 67.3 f c ' (psi) 8865 A ct (in 2 ) 243 f yv (ksi) 67.3 f ci ' (psi) 6090 h flange 8.5 E s (ksi) f c.deck (psi) 5829 Num strand c.deck (pcf) Num straight 18 A s.top (in 2 ) 0 E beam (ksi) 3605 Num straight.bot 16 d top 9.5 E beam.i (ksi) 3585 Num harp 6 d long.rebar 5 E slab (ksi) 3566 cg straight 7.00 IsLongSteelEpoxyCoated No h 36 cg harp.end 29 LongSteelClearSpacing 3.38 A beam (in 2 ) 369 cg harp.ms 4 LongSteelCover 1.75 I beam (in 4 ) cg straight.bot 3.75 IsTopBarEffectApplicable No L beam (ft) 41 A strand (in 2 ) 0.15 A long.rebar (in2) 0.31 b v ShearSpan y b f pu (ksi) 270 x bearing 6 y t f pe (ksi) x bearing.roller 6 S (ft) 7 E p (ksi) x critical.assume t haunch 1 HarpDist (ft) 17.5 A ct (in 2 ) cg s.bot cg s s v.design A v (in 2 ) P assume assume (rad) K-59

60 Basic Geometry and Material Calculations d s A ps.bot cg harp (in 2 ) h c L span (ft) A ps (in 2 ) harp (rad) f po (ksi) f r.v (psi) 1 K-60

61 Load Effect Calculations V LL M LL V DC1 M DC1 V assume M assume1 M assume V p K-61

62 Calculations for and cg ps d p c a f ps.max (ksi) l d.ps f ps (ksi) cg ps.bot d p.bot d s.bot Num strand.bot RedFactor x.s RedFactor x.ps A ps.bot (in 2 ) d e s d v v u A s.bot.x (in 2 ) A ps.bot.x (in 2 ) K-62 x (rad) MinReinfCheck OK OK OK OK OK OK OK OK OK OK OK

63 OK Final Results for Shear Strength V c V s V n1 V n2 V n K-63

64 Beam T2.8.Typ.2 Inputs c (pcf) t struct 8.5 f y (ksi) 67.3 f c ' (psi) 8890 A ct (in 2 ) 243 f yv (ksi) 67.3 f ci ' (psi) 6090 h flange 8.5 E s (ksi) f c.deck (psi) 6088 Num strand c.deck (pcf) Num straight 18 A s.top (in 2 ) 0 E beam (ksi) 3605 Num straight.bot 16 d top 9.5 E beam.i (ksi) 3585 Num harp 6 d long.rebar 5 E slab (ksi) 3210 cg straight 7.00 IsLongSteelEpoxyCoated No h 36 cg harp.end 29 LongSteelClearSpacing 3.38 A beam (in 2 ) 369 cg harp.ms 4 LongSteelCover 1.75 I beam (in 4 ) cg straight.bot 3.75 IsTopBarEffectApplicable No L beam (ft) 41 A strand (in 2 ) 0.15 A long.rebar (in2) 0.31 b v ShearSpan y b f pu (ksi) 270 x bearing 6 y t f pe (ksi) x bearing.roller 64.5 S (ft) 7 E p (ksi) x critical.assume t haunch 1 HarpDist (ft) 17.5 A ct (in 2 ) cg s.bot cg s s v.design A v (in 2 ) P assume assume (rad) K-64

65 Basic Geometry and Material Calculations d s A ps.bot (in 2 ) cg harp h c L span A ps harp f po (ft) (in 2 ) (rad) (ksi) f r.v (psi) K-65

66 Load Effect Calculations V LL M LL V DC1 M DC1 V assume M assume1 M assume V p K-66

67 Calculations for and cg ps d p c a f ps.max (ksi) l d.ps f ps (ksi) cg ps.bot d p.bot d s.bot Num strand.bot RedFactor x.s RedFactor x.ps A ps.bot (in 2 ) d e s d v v u A s.bot.x (in 2 ) A ps.bot.x (in 2 ) K-67 x (rad) MinReinfCheck OK OK OK OK OK OK OK OK OK OK

68 OK OK OK Final Results for Shear Strength V c V s V n1 V n2 V n K-68

69 Beam T2.8.Min.1 Inputs c (pcf) t struct 8 f y (ksi) 67.3 f c ' (psi) 8890 A ct (in 2 ) 243 f yv (ksi) 67.3 f ci ' (psi) 6090 h flange 8.5 E s (ksi) f c.deck (psi) 5360 Num strand c.deck (pcf) 125 Num straight 18 A s.top (in 2 ) 0 E beam (ksi) 3605 Num straight.bot 16 d top 9.5 E beam.i (ksi) 3585 Num harp 6 d long.rebar 5 E slab (ksi) 3240 cg straight 7.00 IsLongSteelEpoxyCoated No h 36 cg harp.end 29 LongSteelClearSpacing 3.38 A beam (in 2 ) 369 cg harp.ms 4 LongSteelCover 1.75 I beam (in 4 ) cg straight.bot 3.75 IsTopBarEffectApplicable No L beam (ft) A strand (in 2 ) 0.15 A long.rebar (in2) 0.31 b v ShearSpan y b f pu (ksi) 270 x bearing 6 y t f pe (ksi) x bearing.roller 6 S (ft) 7 E p (ksi) x critical.assume t haunch HarpDist (ft) 17.5 A ct (in 2 ) cg s.bot cg s s v.design A v (in 2 ) P assume assume (rad) K-69

70 Basic Geometry and Material Calculations d s A ps.bot cg harp (in 2 ) h c L span (ft) A ps (in 2 ) harp (rad) f po (ksi) f r.v (psi) 1 K-70

71 Load Effect Calculations V LL M LL V DC1 M DC1 V assume M assume1 M assume V p K-71

72 Calculations for and cg ps d p c a f ps.max (ksi) l d.ps f ps (ksi) cg ps.bot d p.bot d s.bot Num strand.bot RedFactor x.s RedFactor x.ps A ps.bot (in 2 ) d e s d v v u A s.bot.x (in 2 ) A ps.bot.x (in 2 ) K-72 x (rad) MinReinfCheck OK OK OK OK OK OK OK OK OK OK OK

73 OK Final Results for Shear Strength V c V s V n1 V n2 V n K-73

74 Beam T2.8.Min.2 Inputs c (pcf) t struct 8.5 f y (ksi) 67.3 f c ' (psi) 8890 A ct (in 2 ) 243 f yv (ksi) 67.3 f ci ' (psi) 6090 h flange 8.5 E s (ksi) f c.deck (psi) 5360 Num strand c.deck (pcf) 125 Num straight 18 A s.top (in 2 ) 0 E beam (ksi) 3605 Num straight.bot 16 d top 9.5 E beam.i (ksi) 3580 Num harp 6 d long.rebar 5 E slab (ksi) 3240 cg straight 7.00 IsLongSteelEpoxyCoated No h 36 cg harp.end 29 LongSteelClearSpacing 3.38 A beam (in 2 ) 369 cg harp.ms 4 LongSteelCover 1.75 I beam (in 4 ) cg straight.bot 3.75 IsTopBarEffectApplicable No L beam (ft) 41 A strand (in 2 ) 0.15 A long.rebar (in2) 0.31 b v ShearSpan y b f pu (ksi) 270 x bearing 6 y t f pe (ksi) x bearing.roller 69 S (ft) 7 E p (ksi) x critical.assume t haunch 1 HarpDist (ft) 17.5 A ct (in 2 ) cg s.bot cg s s v.design A v (in 2 ) P assume assume (rad) K-74

75 Basic Geometry and Material Calculations d s A ps.bot cg harp (in 2 ) h c L span (ft) A ps (in 2 ) harp (rad) f po (ksi) f r.v (psi) 1 K-75

76 Load Effect Calculations V LL M LL V DC1 M DC1 V assume M assume1 M assume V p K-76

77 Calculations for and cg ps d p c a f ps.max (ksi) l d.ps f ps (ksi) cg ps.bot d p.bot d s.bot Num strand.bot RedFactor x.s RedFactor x.ps A ps.bot (in 2 ) d e s d v v u A s.bot.x (in 2 ) A ps.bot.x (in 2 ) K-77 x (rad) MinReinfCheck OK OK OK OK OK OK OK OK OK OK

78 OK OK OK Final Results for Shear Strength V c V s V n1 V n2 V n K-78

79 Beam BT.8N.Typ.1 Inputs c (pcf) 150 t struct 8.5 f y (ksi) 69.8 f c ' (psi) 8860 A ct (in 2 ) f yv (ksi) 69.8 f ci ' (psi) 6183 h flange 8.5 E s (ksi) f c.deck (psi) 4890 Num strand c.deck (pcf) 127 Num straight 28 A s.top (in 2 ) 0 E beam (ksi) 4823 Num straight.bot 28 d top 9.5 E beam.i (ksi) 4670 Num harp 6 d long.rebar 5 E slab (ksi) 2940 cg straight 3.68 IsLongSteelEpoxyCoated No h 45 cg harp.end 41 LongSteelClearSpacing 3.75 A beam (in 2 ) cg harp.ms 4.25 LongSteelCover 2 I beam (in 4 ) cg straight.bot 3.68 IsTopBarEffectApplicable No L beam (ft) A strand (in 2 ) A long.rebar (in2) 0.31 b v ShearSpan y b f pu (ksi) 270 x bearing 6 y t f pe (ksi) x bearing.roller 6 S (ft) 7 E p (ksi) x critical.assume t haunch 1 HarpDist (ft) 26.5 A ct (in 2 ) cg s.bot cg s s v.design A v (in 2 ) P assume assume (rad) K-79

80 Basic Geometry and Material Calculations d s A ps.bot (in 2 ) cg harp h c L span (ft) A ps (in 2 ) harp (rad) f po (ksi) f r.v (psi) 1 K-80

81 Load Effect Calculations V LL M LL V DC1 M DC1 V assume M assume1 M assume V p K-81

82 Calculations for and cg ps d p c a f ps.max (ksi) l d.ps f ps (ksi) cg ps.bot d p.bot d s.bot Num strand.bot RedFactor x.s RedFactor x.ps A ps.bot (in 2 ) d e s d v v u A s.bot.x (in 2 ) A ps.bot.x (in 2 ) K-82 x (rad) MinReinfCheck OK OK OK OK OK OK OK OK OK OK

83 OK OK OK Final Results for Shear Strength V c V s V n1 V n2 V n K-83

84 Beam BT.8N.Typ.2 Inputs c (pcf) 150 t struct 8.5 f y (ksi) 69.8 f c ' (psi) 8570 A ct (in 2 ) f yv (ksi) 69.8 f ci ' (psi) 6183 h flange 8.5 E s (ksi) f c.deck (psi) 4990 Num strand c.deck (pcf) 127 Num straight 28 A s.top (in 2 ) 0 E beam (ksi) 4950 Num straight.bot 28 d top 9.5 E beam.i (ksi) 4670 Num harp 6 d long.rebar 5 E slab (ksi) 3110 cg straight 3.68 IsLongSteelEpoxyCoated No h 45 cg harp.end 41 LongSteelClearSpacing 3.75 A beam (in 2 ) cg harp.ms 4.25 LongSteelCover 2 I beam (in 4 ) cg straight.bot 3.68 IsTopBarEffectApplicable No L beam (ft) A strand (in 2 ) 0.15 A long.rebar (in2) 0.31 b v ShearSpan y b f pu (ksi) 270 x bearing 6 y t f pe (ksi) x bearing.roller 113 S (ft) 7 E p (ksi) x critical.assume t haunch 1 HarpDist (ft) 26.5 A ct (in 2 ) cg s.bot cg s s v.design A v (in 2 ) P assume assume (rad) K-84

85 Basic Geometry and Material Calculations d s A ps.bot cg harp (in 2 ) h c L span (ft) A ps (in 2 ) harp (rad) f po (ksi) f r.v (psi) 1 K-85

86 Load Effect Calculations V LL M LL V DC1 M DC1 V assume M assume1 M assume V p K-86

87 Calculations for and cg ps d p c a f ps.max (ksi) l d.ps f ps (ksi) cg ps.bot d p.bot d s.bot Num strand.bot RedFactor x.s RedFactor x.ps A ps.bot (in 2 ) d e s d v v u A s.bot.x (in 2 ) A ps.bot.x (in 2 ) K-87 x (rad) MinReinfCheck OK OK OK OK OK OK OK OK OK OK

88 OK OK OK K-88

89 Final Results for Shear Strength V c V s V n1 V n2 V n K-89

90 Beam BT.8.Typ.1 Inputs c (pcf) t struct 8.5 f y (ksi) 67.3 f c ' (psi) 9080 A ct (in 2 ) f yv (ksi) 67.3 f ci ' (psi) 6330 h flange 8.5 E s (ksi) f c.deck (psi) 5580 Num strand c.deck (pcf) Num straight 28 A s.top (in 2 ) 0 E beam (ksi) 3590 Num straight.bot 28 d top 9.5 E beam.i (ksi) 3790 Num harp 6 d long.rebar 5 E slab (ksi) 3600 cg straight 3.68 IsLongSteelEpoxyCoated No h 45 cg harp.end 41 LongSteelClearSpacing 3.75 A beam (in 2 ) cg harp.ms 4.25 LongSteelCover 2 I beam (in 4 ) cg straight.bot 3.68 IsTopBarEffectApplicable No L beam (ft) 59 A strand (in 2 ) 0.15 A long.rebar (in2) 0.31 b v ShearSpan y b f pu (ksi) 270 x bearing 6 y t f pe (ksi) x bearing.roller 6 S (ft) 7 E p (ksi) x critical.assume t haunch 1 HarpDist (ft) 26.5 A ct (in 2 ) cg s.bot cg s s v.design A v (in 2 ) P assume assume (rad) K-90

91 Basic Geometry and Material Calculations d s A ps.bot cg harp (in 2 ) h c L span (ft) A ps (in 2 ) harp (rad) f po (ksi) f r.v (psi) 1 K-91

92 Load Effect Calculations V LL M LL V DC1 M DC1 V assume M assume1 M assume V p K-92

93 Calculations for and cg ps d p c a f ps.max (ksi) l d.ps f ps (ksi) cg ps.bot d p.bot d s.bot Num strand.bot RedFactor x.s RedFactor x.ps A ps.bot (in 2 ) d e s d v v u A s.bot.x (in 2 ) A ps.bot.x (in 2 ) K-93 x (rad) MinReinfCheck OK OK OK OK OK OK OK OK OK OK

94 OK OK OK K-94

95 Final Results for Shear Strength V c V s V n1 V n2 V n K-95

96 Beam BT.8.Typ.2 Inputs c (pcf) t struct 8.5 f y (ksi) 67.3 f c ' (psi) 9080 A ct (in 2 ) f yv (ksi) 67.3 f ci ' (psi) 6330 h flange 8.5 E s (ksi) f c.deck (psi) 6690 Num strand c.deck (pcf) Num straight 28 A s.top (in 2 ) 0 E beam (ksi) 3590 Num straight.bot 28 d top 9.5 E beam.i (ksi) 3790 Num harp 6 d long.rebar 5 E slab (ksi) 4050 cg straight 3.68 IsLongSteelEpoxyCoated No h 45 cg harp.end 41 LongSteelClearSpacing 3.75 A beam (in 2 ) cg harp.ms 4.25 LongSteelCover 2 I beam (in 4 ) cg straight.bot 3.68 IsTopBarEffectApplicable No L beam (ft) 59 A strand (in 2 ) 0.15 A long.rebar (in2) 0.31 b v ShearSpan y b f pu (ksi) 270 x bearing 6 y t f pe (ksi) x bearing.roller S (ft) 7 E p (ksi) x critical.assume t haunch 1 HarpDist (ft) 26.5 A ct (in 2 ) cg s.bot cg s s v.design A v (in 2 ) P assume assume (rad) K-96

97 Basic Geometry and Material Calculations d s A ps.bot cg harp (in 2 ) h c L span (ft) A ps (in 2 ) harp (rad) f po (ksi) f r.v (psi) 1 K-97

98 Load Effect Calculations V LL M LL V DC1 M DC1 V assume M assume1 M assume V p K-98

99 Calculations for and cg ps d p c a f ps.max (ksi) l d.ps f ps (ksi) cg ps.bot d p.bot d s.bot Num strand.bot RedFactor x.s RedFactor x.ps A ps.bot (in 2 ) d e s d v v u A s.bot.x (in 2 ) A ps.bot.x (in 2 ) K-99 x (rad) MinReinfCheck OK OK OK OK OK OK OK OK OK OK

100 OK OK OK K-100

101 Final Results for Shear Strength V c V s V n1 V n2 V n K-101

102 Beam BT.10.Typ.1 Inputs c (pcf) t struct 8.5 f y (ksi) 67.3 f c ' (psi) 8890 A ct (in 2 ) f yv (ksi) 67.3 f ci ' (psi) 6120 h flange 8.5 E s (ksi) f c.deck (psi) 4130 Num strand c.deck (pcf) Num straight 28 A s.top (in 2 ) 0 E beam (ksi) 3910 Num straight.bot 28 d top 9.5 E beam.i (ksi) 4230 Num harp 6 d long.rebar 6 E slab (ksi) 3160 cg straight 3.68 IsLongSteelEpoxyCoated No h 45 cg harp.end 41 LongSteelClearSpacing 1 A beam (in 2 ) cg harp.ms 4.25 LongSteelCover 2 I beam (in 4 ) cg straight.bot 3.68 IsTopBarEffectApplicable No L beam (ft) A strand (in 2 ) 0.15 A long.rebar (in2) 0.31 b v ShearSpan y b f pu (ksi) 270 x bearing 6 y t f pe (ksi) x bearing.roller 6 S (ft) 7 E p (ksi) x critical.assume t haunch 1 HarpDist (ft) 26.5 A ct (in 2 ) cg s.bot cg s s v.design A v (in 2 ) P assume assume (rad) K-102

103 Basic Geometry and Material Calculations d s A ps.bot cg harp (in 2 ) h c L span (ft) A ps (in 2 ) harp (rad) f po (ksi) f r.v (psi) 1 K-103

104 Load Effect Calculations V LL M LL V DC1 M DC1 V assume M assume1 M assume V p K-104

105 Calculations for and cg ps d p c a f ps.max (ksi) l d.ps f ps (ksi) cg ps.bot d p.bot d s.bot Num strand.bot RedFactor x.s RedFactor x.ps A ps.bot (in 2 ) d e s d v v u A s.bot.x (in 2 ) A ps.bot.x (in 2 ) K-105 x (rad) MinReinfCheck OK OK OK OK OK OK OK OK OK OK

106 OK OK OK K-106

107 Final Results for Shear Strength V c V s V n1 V n2 V n K-107

108 Beam BT.10.Typ.2 Inputs c (pcf) 129 t struct 8.5 f y (ksi) 67.3 f c ' (psi) 9730 A ct (in 2 ) f yv (ksi) 67.3 f ci ' (psi) 6123 h flange 8.5 E s (ksi) f c.deck (psi) 4930 Num strand c.deck (pcf) 129 Num straight 28 A s.top (in 2 ) 0 E beam (ksi) 4060 Num straight.bot 28 d top 9.5 E beam.i (ksi) 4230 Num harp 6 d long.rebar 6 E slab (ksi) 3270 cg straight 3.68 IsLongSteelEpoxyCoated No h 45 cg harp.end 41 LongSteelClearSpacing 1 A beam (in 2 ) cg harp.ms 4.25 LongSteelCover 2 I beam (in 4 ) cg straight.bot 3.68 IsTopBarEffectApplicable No L beam (ft) A strand (in 2 ) 0.15 A long.rebar (in2) 0.31 b v ShearSpan y b f pu (ksi) 270 x bearing 6 y t f pe (ksi) x bearing.roller S (ft) 7 E p (ksi) x critical.assume t haunch 1 HarpDist (ft) 26.5 A ct (in 2 ) cg s.bot cg s s v.design A v (in 2 ) P assume assume (rad) K-108

109 Basic Geometry and Material Calculations d s A ps.bot cg harp (in 2 ) h c L span (ft) A ps (in 2 ) harp (rad) f po (ksi) f r.v (psi) 1 K-109

110 Load Effect Calculations V LL M LL V DC1 M DC1 V assume M assume1 M assume V p K-110

111 Calculations for and cg ps d p c a f ps.max (ksi) l d.ps f ps (ksi) cg ps.bot d p.bot d s.bot Num strand.bot RedFactor x.s RedFactor x.ps A ps.bot (in 2 ) d e s d v v u A s.bot.x (in 2 ) A ps.bot.x (in 2 ) K-111 x (rad) MinReinfCheck OK OK OK OK OK OK OK OK OK OK

112 OK OK OK K-112

113 Final Results for Shear Strength V c V s V n1 V n2 V n K-113

114 Beam BT.10.Min.1 Inputs c (pcf) 129 t struct 8.5 f y (ksi) 67.3 f c ' (psi) 9730 A ct (in 2 ) f yv (ksi) 67.3 f ci ' (psi) 6120 h flange 8.5 E s (ksi) f c.deck (psi) 5170 Num strand c.deck (pcf) Num straight 28 A s.top (in 2 ) 0 E beam (ksi) 4060 Num straight.bot 28 d top 9.5 E beam.i (ksi) 4230 Num harp 6 d long.rebar 6 E slab (ksi) 3200 cg straight 3.68 IsLongSteelEpoxyCoated No h 45 cg harp.end 41 LongSteelClearSpacing 1 A beam (in 2 ) cg harp.ms 4.25 LongSteelCover 2 I beam (in 4 ) cg straight.bot 3.68 IsTopBarEffectApplicable No L beam (ft) 59 A strand (in 2 ) 0.15 A long.rebar (in2) 0.31 b v ShearSpan y b f pu (ksi) 270 x bearing 6 y t f pe (ksi) x bearing.roller 6 S (ft) 7 E p (ksi) x critical.assume t haunch 1 HarpDist (ft) 26.5 A ct (in 2 ) cg s.bot cg s s v.design A v (in 2 ) P assume assume (rad) K-114

115 Basic Geometry and Material Calculations d s A ps.bot cg harp (in 2 ) h c L span (ft) A ps (in 2 ) harp (rad) f po (ksi) f r.v (psi) 1 K-115

116 Load Effect Calculations V LL M LL V DC1 M DC1 V assume M assume1 M assume V p K-116

117 Calculations for and cg ps d p c a f ps.max (ksi) l d.ps f ps (ksi) cg ps.bot d p.bot d s.bot Num strand.bot RedFactor x.s RedFactor x.ps A ps.bot (in 2 ) d e s d v v u A s.bot.x (in 2 ) A ps.bot.x (in 2 ) K-117 x (rad) MinReinfCheck OK OK OK OK OK OK OK OK OK OK

118 OK OK OK K-118

119 Final Results for Shear Strength V c V s V n1 V n2 V n K-119

120 Beam BT.10.Min.2 Inputs c (pcf) t struct 8.5 f y (ksi) 67.3 f c ' (psi) A ct (in 2 ) f yv (ksi) 67.3 f ci ' (psi) 8940 h flange 8.5 E s (ksi) f c.deck (psi) 5860 Num strand c.deck (pcf) Num straight 28 A s.top (in 2 ) 0 E beam (ksi) 4140 Num straight.bot 28 d top 9.5 E beam.i (ksi) 4230 Num harp 6 d long.rebar 6 E slab (ksi) 3410 cg straight 3.68 IsLongSteelEpoxyCoated No h 45 cg harp.end 41 LongSteelClearSpacing 1 A beam (in 2 ) cg harp.ms 4.25 LongSteelCover 2 I beam (in 4 ) cg straight.bot 3.68 IsTopBarEffectApplicable No L beam (ft) 59 A strand (in 2 ) 0.15 A long.rebar (in2) 0.31 b v ShearSpan y b f pu (ksi) 270 x bearing 6 y t f pe (ksi) x bearing.roller S (ft) 7 E p (ksi) x critical.assume t haunch 1 HarpDist (ft) 26.5 A ct (in 2 ) cg s.bot cg s s v.design A v (in 2 ) P assume assume (rad) K-120

121 Basic Geometry and Material Calculations d s A ps.bot cg harp (in 2 ) h c L span (ft) A ps (in 2 ) harp (rad) f po (ksi) f r.v (psi) 1 K-121

122 Load Effect Calculations V LL M LL V DC1 M DC1 V assume M assume1 M assume V p K-122

123 Calculations for and cg ps d p c a f ps.max (ksi) l d.ps f ps (ksi) cg ps.bot d p.bot d s.bot Num strand.bot RedFactor x.s RedFactor x.ps A ps.bot (in 2 ) d e s d v v u A s.bot.x (in 2 ) A ps.bot.x (in 2 ) K-123 x (rad) MinReinfCheck OK OK OK OK OK OK OK OK OK OK

124 OK OK OK K-124

125 Final Results for Shear Strength V c V s V n1 V n2 V n K-125

126 Beam Example Following the Simplified Procedure for Prestressed and Nonprestressed Sections Additional Inputs and Basic Calculations The same scenario provided in Section 0 is used for example calculations following Article in the AASHTO LRFD Bridge Specifications. The same information and calculations presented previously apply here, with the exception that the values in P assume are: Additional required inputs and basic calculations for using this procedure include: Load Effects Since the amount of assumed live load changed from the original example, the shear and moment due to that assumed live load are calculated as: K-126

127 Using the dead load calculations from previously, the total shear and moment assumed to be in the beam are calculated as: Additionally, the maximum shear and moment due to externally applied loads, V i and M max are: K-127

128 where, in the case of this example, the terms V d and M d are equivalent to V DC1 and M DC1. Calculate the Shear Resistance Determine the Concrete Contribution to the Shear Resistance Calculate V ci Calculate f cpe, the Compressive Stress in the Concrete Due to Effective Prestress Forces K-128

129 Calculate the Moment Due to Externally Applied Loads Causing Cracking, M cre In the case of this example, M dnc, the total unfactored dead load moment acting on the noncomposite section is simply M DC1. K-129

130 Calculate V ci Calculate V cw Determine the Compressive Stress at the Composite Centroid, f pc K-130

131 Determine V cw K-131

132 Calculate V c Calculate the Steel Contribution to the Shear Resistance Determine cot K-132

133 Find V s Determine the Nominal Shear Resistance Note that for V n1, V p is taken as zero. Since V n2 is the same as what was calculated previously, the nominal shear resistance is calculated as: K-133