County: Any Hwy: Any Design: BRG Date: 6/2005

LRFD Bent Cap Design Example

Design Parameters

Span 1100' Type IV Beams (0.821k/ft)5 Beams Spaced @ 8.50' with 3' overhangs

Span 2120' Type IV Beams (0.821k/ft)6 Beams Spaced @ 6.80' with 3' overhangs

All SpansType T501 Rail (0.33k/ft)8" Thick Slab (0.100 ksf)Assume 2" Overlay @ 140 pcf (0.023 ksf)Use Class "C" Concrete

f'c=3.60 ksiGrade 60 Reinforcing

Fy=60 ksi

Assume3'-3" X 3'-3" Cap3 Columns Spaced @ 15'-0"Cap will be modeled as a continuous beam with simple supports using TxDOT's CAP18 program.

TxDOT does not consider frame action for typical multi-column bents.

LRFD Cap Design Example

Back Span Ahead Span

Span1 100 ft Span2 120 ft RailWt 0.376 klf

BmSpa1 8.5 ft BmSpa2 6.8 ft SlabThk 8 in

BmNo1 5 BmNo2 6 OverlayThk 2 in

IM 0.33 Dynamic load allowance, LRFD 3.6.2BmWt1 0.821 klf BmWt2 0.821 klf

Cap Properties:

CapWidth 3.25 ft

CapDepth 3.25 ft

station 0.5ft

staStation increment for CAP18.

fc 3.60 ksi Ec 33000 0.1451.5 fc Ec 3457.14 ksi

Es 29000 ksi

page 1

20.0 33.0 47.0 60.0

Dead Load

SPAN 1

Rail12 RailWt

Span12

BmNo1Rail1 7.52

Increase slab DL by 5% to accountfor haunch and thickened ends.Slab1 0.150BmSpa1

SlabThk12

Span12

1.05 Slab1 44.62

Beam1 BmWt1Span1

2Beam1 41.05

Overlay1 0.140 BmSpa1OverlayThk

12Span1

2Overlay1 9.92 Design for future overlay.

DLRxn1 Rail1 Slab1 Beam1 Overlay1

DLRxn1 103.11 kip/beam

For bents with different beamspacings forward and back, TXDOTstandard design procedure requirestwo CAP18 problems as follows:

Problem 1, Table 3 describesstringers for SPAN 1 only.

Problem 2, same as problem 1except hold envelopes fromproblem 1 and on Table 3 describestringers for SPAN 2 only. Useproblem 2 results.

SPAN 2

Rail22 RailWt

Span22

BmNo2Rail2 7.52

Slab2 0.150 BmSpa2SlabThk

12Span2

21.05 Slab2 42.84

Beam2 BmWt2Span2

2Beam2 49.26

Overlay2 0.140 BmSpa2OverlayThk

12Span2

2Overlay2 9.52

DLRxn2 Rail2 Slab2 Beam2 Overlay2

DLRxn2 109.14 kip/beam

Cap 0.150 CapWidth CapDepth station Cap 0.79 kip/station

page 2

Cap 18 Live Load Model

Live Load LRFD 3.6.1.2.2 and 3.6.1.2.4

LongSpan if Span1 Span2 Span1 Span2( )

ShortSpan if Span1 Span2 Span2 Span1( )

LongSpan 120.00 ft

ShortSpan 100.00 ft

IM 0.33

Lane 0.64LongSpan ShortSpan

2

Lane 70.40 kip

For maximum reaction at interiorbents, "Design Truck" will alwaysgovern over "Design Tandem".

Truck 32.0 32.0LongSpan 14

LongSpan8.0

ShortSpan 14ShortSpan

Truck 67.15 kip

Combine "Design Truck" and"Design Lane" loadings. LRFD 3.6.1.3

Dynamic load allowance, IM,does not apply to "Design Lane."LRFD 3.6.1.2.4

LLRxn Lane Truck 1 IM( )

LLRxn 159.71 kip/lane

P 16.0 1 IM( )

P 21.28 kipTXDOT practice has been tomodel live load as two 16kipwheel loads increased bymaximum impact with theremainder of the live load to bedistributed over a 10ft designlane width. It is reasonable tocontinue this for LRFD.

kftft

sta

WLLRxn 2 P( )

10

station

W 5.86 kip/station

page 3

Cap 18 Data Input

Input "Multiple Presence Factors"into Cap18 as "Load ReductionFactors".

Multiple Presence Factors, m LRFD 3.6.1.1.2

No. of Lanes Factor "m"

1 2 3>3

1.201.000.850.65

Note: When using CAP18 for LRFD,an additional analysis will need to beperformed that defines one large laneas the clear width of the bridge. Thisstep corrects the unconservativeproblem of CAP18 not using the 1.2factor for a single lane in the randomload calculation.

Limit States LRFD 3.4.1

Strength IFor cap design, need only toconsider Strength I and Service I.Live Load and Dynamic Load Allowance LL + IM = 1.75

Dead Load Components DC = 1.25For Strength I, LRFD specifies adead load factor of 1.50 for"Wearing Surface". CAP18 will onlyallow input of one dead load factor.Therefore, input 1.25 for ALL deadloads. Wearing surface is only afraction of the dead weight, thus,this approximation will be closeenough.

Dead Load Wearing Surface (Overlay) DW = 1.50

Use Dead Load factor = 1.25 and Live Load factor = 1.75.

Service I

Live Load and Dynamic Load Allowance LL + IM = 1.00

Dead Load and Wearing Surface DC & DW = 1.00Summary of CAP18 Input problemcards:

Problem 1- Bridge defined with Span1 beam spacing.

Problem 2- Same as Problem 1 with1 large lane defined and envelopesheld from previous problem.

Problem 3- Bridge defined with Span2 beam spacing and envelopes heldfrom previous problem..

Problem 4- Same as Problem 3 with1 large lane defined and envelopesheld from previous problem.

Use Dead Load factor = 1.00 and Live Load factor = 1.00.

Dead Load

TXDOT also considers Service level Dead Load only.

Cap 18 Output( kip ft)

Max +M(kip ft)Max -M

Dead Load posDL 390.8 negDL 599.1

Service posServ 1026.4 negServ 985.9

Ultimate posUlt 1630.4 negUlt 1425.7

Mdl if posDL negDL posDL negDL( ) Mdl 599.10 kip ft

Ms if posServ negServ posServ negServ( ) Ms 1026.40 kip ft

Mu if posUlt negUlt posUlt negUlt( ) Mu 1630.40 kip ft

page 4

Minimum Flexural Reinforcement LRFD 5.7.3.3.2

Mr must be greater than or equal to the lesser of 1.2 Mcr or 1.33Mu

where

IgCapWidth 12( ) CapDepth 12( )3

12 Ig 192786.75 in4

fr 0.24 fc fr 0.455 ksi

ytCapDepth 12

2yt 19.50 in

McrIg fryt

Mcr 4502.00 kip in

Therefore,Design for the lesser of 1.2Mcr or1.33Mu when determining minimumarea of steel required. TXDOT'stypical practice is to provide at least1.2 Mcr in bent caps.

Mcr11.2 Mcr

12Mcr1 450.20 kip ft

Mcr2 1.33 Mu Mcr2 2168.43 kip ft

Mf if Mcr1 Mcr2 Mcr1 Mcr2( )

Thus, Mr must be greater than Mf 450.20 kip ft

Moment Capacity Design LRFD 5.7.3.2

Try, BarNo 8 (#11's Top & Bottom)

As BarNo( ) 1.56 in2 As 12.48 in2

d CapDepth 12( ) 258

1.412

d 35.67 in

b CapWidth 12 b 39.00 in

Beta 0.85 fc 3.60 ksi LRFD 5.7.2.2MPhi 0.9 fy 60 ksi

To find a:c

As fy0.85 fc Beta b

c 7.38 in LRFD 5.7.3.1.2-4

a c Beta a 6.27 in

Thus,Mn As fy d

a2

Mn 24360.52 kip in LRFD 5.7.3.2.2-1

MrMPhi Mn

12Mr 1827.04 kip ft

Mu 1630.40 kip ft

MinReinfChk if Mr Mf( ) OK NG[ ] MinReinfChk "OK!"

UltimateMom if Mr Mu( ) OK NG[ ] UltimateMom "OK!"

page 5

Check Servicability "Control of cracking by Distribution of Reinforcement", LRFD 5.7.3.4

Check allowable Ms, given: dc 258

1.412

dc 3.33 in

Adc 2 CapWidth 12( )

BarNoA 32.47 in2

To find fs: z 170kipin

fs1z

3 dc Afs1 35.68 ksi

fs2 0.6 fy fs2 36.00 ksi

fs if fs1 fs2 fs1 fs2( ) fs 35.68 ksi

To find j:n

EsEc

n 8.39

pAsb d p 0.0090

k 2 p n( ) p n( )2 p n( ) k 0.320

Thus,For simplicity one can take j=0.9 for"typical" bent caps.j 1

k3

j 0.89

AllowMsAs d j fs

12AllowMs 1182.62 kip ft

Ms 1026.40 kip ft

ServiceabilityMom if AllowMs Ms( ) OK NG[ ] ServiceabilityMom "OK!"

Check Dead LoadTXDOT limits dead load stress to22 ksi. This is due to observedcracking under dead load.

Check allowable Mdl: fdl 22 ksi

AllowMdlAs d j fdl

12AllowMdl 729.09 kip ft

Mdl 599.10 kip ft

DeadLoadMom if AllowMdl Mdl( ) OK NG[ ] DeadLoadMom "OK!"

TXDOT typically uses the samereinforcement top and bottom forsimplicity.

Flexural Steel Summary: Use BarNo 8 (Top & Bottom)

page 6

Shear Design LRFD 5.8 (For flow chart of shear design procedure see Figure C5.8.3.4.2-5)

Vn must be the lesser of Vc+Vs+Vp or 0.25fcbvdv LRFD 5.8.3.3-1&2

LRFD 5.8.3.3-3Where, Vc = 0.0316 Beta fc b d

LRFD C5.8.3.3-1Vs =

Av fy dv cot theta( )s

Since shear is dependent on location, let's look at STA 18:

Vu 438 kips

Sta12Mu 692 kip ft

Mn 24360.52 kip in w/ BarNo 8 #11's (Top & Bottom) Refer to page 5.

Find dv:

dv1 must not be less than the greater of dv2 and dv3:

dv1Mn

As fydv1 32.53 in LRFD C5.8.2.9-1

dv2 0.9 d dv2 32.10 in

dv3 0.72 CapDepth 12( ) dv3 28.08 in

tempdv if dv2 dv3( ) dv2 dv3[ ]

dv if dv1 tempdv dv1 tempdv( )

dv 32.5 in

Find Maximum Vn: Vn1 0.25 fc b dv Vn1 1141.90 kips

Maximum Vr1 must be greater than Vn

VPhi 0.9 Vu 438.00 kips LRFD 5.5.4.2.1

Vp 0 kips

Thus, Vr1 VPhi Vn1 Vr1 1027.71 kips LRFD 5.8.2.1-2

MaxVr1 if Vr1 Vu( ) OK NG[ ] If Vrmax < Vumax, then use aLARGER cap depth in order tosatisfy shear requirements. LRFD 5.8.2.9-1MaxVr1 "OK!"

page 7

Shear Design

To find Vc+Vs we need to determine and LRFD 5.8.2.9

vVu VPhi Vp( )

VPhi b dvv 0.38 ksi

ratiovfc

ratio 0.107

Using Table 5.8.3.4.2-1: with ratio 0.107 and x1 0.001 Determining and is aninterative process, therefore,assume initial value of x1 =0.001per LRFD 5.8.3.4.2 and then verifythat the assumption was valid.

Theta 37.0 and BetaShr 2.13

Verify assumed value of x:

Recall,Nu 0 kips Vu 438 kips

Vp 0 kips Sta12Mu 692 kip ft

dv 32.5 in However, Mu must be greater than Vu*dv so:

Es 29000 ksi newMu if Sta12Mu Vudv12

Sta12Mu Vudv12

Theta 37.00 deg

newMu 1187.45 kip ftAs 12.48 in2

x

newMu 12dv

0.5 Nu 0.5 Vu Vp( ) cot Theta180

2 Es As If x < 0, then use equation5.8.3.4.2-3 and re-solve for x.x 0.0010

The table values for and canbe applied over a range, thus, nointerpolation is required. (*Note: Shear spreadsheet willautomatically interpolate andvalues so results will slightly varyfrom hand calculations.)

Ex if 0.75 x 1000( ) OK Recalculate[ ] Ex "OK!"

Theta 37.00 deg and BetaShr 2.13

Recall,fc 3.60 ksi

b 39.00 in

dv 32.53 in

Find Vc: LRFD 5.8.3.3-3

Vc 0.0316 BetaShr fc b dv Vc 162.03 kips

page 8

Shear Design (Continued)

Assuming #5 stirrups at in spacing,s 9.5

Av 4 0.31( ) fy 60.00 ksi Theta 37.00 deg

Av 1.24 in2 dv 32.53 in

Find Vs:

LRFD 5.8.3.3-1

VsAv fy dv cot Theta

180s

Vs 338.11 kips

Find Vn:Vp 0.00

Vn1 1141.90 (Refer to page 7)

Vn2 Vc Vs Vp Vn2 500.14 kips LRFD 5.8.3.3-1

Thus,

Vn if Vn1 Vn2 Vn1 Vn2( )

Vn 500.14 kips

Shear Resistance: Vphi 0.9 LRFD 5.8.2.1-2

Vr Vphi Vn

Vr 450.13 kips and Vu 438.00 kips

ShearResistance if Vr Vu OK NG( ) ShearResistance "OK!"

Therefore, use double #5 stirrups @ s 9.50 in spacing

This process should be repeated at ALL points of critical shear. The ConcreteSection Shear Capacity spreadsheet can be used in lieu of hand calculations.

page 9

County: Descrip:

Highway:

C-S-J: Design: RCR Ck Dsn:

Bridge Division Rev: 7/6/04 Date: Nov-04

Resistance Factors: Units: US

V = 0.9

M = 0.9

N = 1

Concrete: Mild Steel: Prestressed Steel:f'c = 3.6 ksi fy = 60 ksi fps = 270 ksiEc = 3453 ksi Es = 29000 ksi Ep = 28500 ksi

SECTIONSUnits Sta 8 Sta 12 Sta 18 Sta 25 Sta 31 Sta 35 Sta 45 Sta 50

Input DataWeb width, bv in 39.00 39.00 39.00 39.00 39.00 39.00 39.00 39.00

Shear depth, dv in 32.53 32.53 32.53 32.53 32.53 32.53 32.53 32.53Bending moment, Mu kip-ft 491 692 1199 1286 655 473 585 593

Shear force, Vu kip 488 438 432 241 247 522 454 202Prestress shear, Vp kip 0 0 0 0 0 0 0 0

Axial force, Nu kip 0 0 0 0 0 0 0 0Mild steel reinf. area, As in^2 12.48 12.48 12.48 12.48 12.48 12.48 12.48 12.48

Prestressed steel area, Aps in^2Area of beam below h/2, Ac in^2 760.5 760.5 760.5 760.5 760.5 760.5 760.5 760.5

Area of stirrups, Av in^2 0.62 1.24 1.24 0.62 0.62 1.24 1.24 0.62Stirrup spacing, s in 4 10 10 12 12 7.5 9 12

Effective prestress fpe ksiTorsional Moment, Tu kip-ft

Shear flow area, Ao in^2Area of one leg of stirrup, At in^2

Perimeter of stirrup, Ph inArea enclosed by stirrup, Aoh in^2

Calculated ValuesVc kip 162.8 165.1 165.1 175.0 189.4 162.0 164.3 194.7Vs kip 403.0 323.5 323.5 143.0 161.4 428.3 359.5 168.4Vn kip 509 440 440 286 316 531 471 327

x 1.00E+00 1.00E+00 1.00E+00 8.91E-01 6.14E-01 1.00E+00 1.00E+00 5.35E-01

deg 36.90 36.80 36.80 35.20 32.00 37.00 36.80 30.922.140 2.170 2.170 2.300 2.490 2.130 2.160 2.560

Req'd Shear reinf. Av/S in^2/in 0.146 0.123 0.121 0.034 0.027 0.161 0.130 0.009Req'd Torsion reinf. At/S in^2/in 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Maximum stirrup spacing, Smax in 4.2 10.1 10.3 15.9 15.9 7.7 9.5 15.9Conclusion

Shear Reinforcing OK OK OK OK OK OK OK OKLongitudinal Reinforcing OK OK FAILED FAILED OK OK OK OK

Note: Longitudinal Reinforcing check can be ignored for typical multi-column bent caps. For straddle bents with no overhangs, this check must be considered. Refer to LRFD 5.8.3.5 for further information. May use 6" spa. forSpreadsheet columns must be used in consecutive order to work properly. DO NOT skip a column. cantilevered regions.If torsion is not being considered, leave last five rows of input data blank. TXDOT max. spa. = 12"

CONCRETE SECTION SHEAR CAPACITY BY AASHTO LRFD 2004 BRIDGE DESIGN SPECIFICATIONS

Any

Any

Any

LRFD Design for Shear: Rectangular Bent Cap Design

Designed withdouble stirrups

page 10

BRG

County: Descrip:

Highway:

C-S-J: Design: RCR Ck Dsn:

Bridge Division Rev: 7/6/04 Date: Nov-04

Resistance Factors: Units: US

V = 0.9

M = 0.9

N = 1

Concrete: Mild Steel: Prestressed Steel:f'c = 3.6 ksi fy = 60 ksi fps = 270 ksiEc = 3453 ksi Es = 29000 ksi Ep = 28500 ksi

SECTIONSUnits Sta 56 Sta 61 Sta 68 Sta72

Input DataWeb width, bv in 39.00 39.00 39.00 39.00

Shear depth, dv in 32.53 32.53 32.53 32.53Bending moment, Mu kip-ft 1072 878 692 491

Shear force, Vu kip 196 339 346 488Prestress shear, Vp kip 0 0 0 0

Axial force, Nu kip 0 0 0 0Mild steel reinf. area, As in^2 12.48 12.48 12.48 12.48

Prestressed steel area, Aps in^2Area of beam below h/2, Ac in^2 760.5 760.5 760.5 760.5

Area of stirrups, Av in^2 0.62 1.24 1.24 0.62Stirrup spacing, s in 12 12 12 4

Effective prestress fpe ksiTorsional Moment, Tu kip-ft

Shear flow area, Ao in^2Area of one leg of stirrup, At in^2

Perimeter of stirrup, Ph inArea enclosed by stirrup, Aoh in^2

Calculated ValuesVc kip 181.1 177.2 175.7 162.8 189.4 161.3 164.3 194.7Vs kip 151.2 294.6 292.4 403.0 161.4 428.3 359.5 168.4Vn kip 299 425 421 509 316 531 471 327

x 7.49E-01 8.10E-01 8.25E-01 1.00E+00 6.14E-01 1.00E+00 1.00E+00 5.35E-01

deg 33.70 34.40 34.60 36.90 32.00 37.00 36.80 30.922.380 2.330 2.310 2.140 2.490 2.120 2.160 2.560

Req'd Shear reinf. Av/S in^2/in 0.013 0.070 0.074 0.146 0.027 0.163 0.132 0.009Req'd Torsion reinf. At/S in^2/in 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Maximum stirrup spacing, Smax in 15.9 17.7 16.8 4.2 15.9 7.6 9.4 15.9Conclusion

Shear Reinforcing OK OK OK OK OK OK OK OKLongitudinal Reinforcing OK OK OK OK OK FAILED FAILED OK

Note: Longitudinal Reinforcing check can be ignored for typical multi-column bent caps. For straddle bents with no overhangs, this check must be considered. Refer to LRFD 5.8.3.5 for further information. May use 6" spa. forSpreadsheet columns must be used in consecutive order to work properly. DO NOT skip a column. cantilevered regions.If torsion is not being considered, leave last five rows of input data blank. TXDOT max. spa. = 12"

CONCRETE SECTION SHEAR CAPACITY BY AASHTO LRFD 2004 BRIDGE DESIGN SPECIFICATIONS

Any

Any

Any

LRFD Design for Shear: Rectangular Bent Cap Design

Designed withdouble stirrups for symmetry

page 11

BRG

County: Descrip:

Highway:

C-S-J: Design: RCR Ck Dsn:

Bridge Division Rev: 7/6/04 Date: Nov-04

Capacity Reduction Factors: Units: USV = 0.9 Section: Sta 12M = 0.9

N = 1

Concrete: Mild Steel: Prestressed Steel: Stirrups:f'c = 3.6 ksi fy = 60 ksi fps = 270 ksi Av = 1.24 in^2Ec = 3453 ksi Es = 29000 ksi fpe = ksi s = 9.5 inAc = 760.5 in^2 As = 12.48 in^2 Ep = 28500 ksi At = in^2

Aps = in^2 Ph = in

Section Geometry: Loads: Concrete Stresses:bv = 39 in Mu = 692 kip-ft fpo =0.7fps 189.0 ksi

dv = 32.5 in Vu = 438 kip v = 0.3836 ksi (LRFD 5.8.3.6.2-4)Aoh = in^2 Vp = 0 kip x1 = 1.0000 (1,000's) (LRFD 5.8.3.4.2-1)

Ao = in^2 Nu = 0 kip x2 = NA (1,000's) (LRFD 5.8.3.4.2-2)

Tu = kip-ft x3 = NA (1,000's) (LRFD 5.8.3.4.2-3)

Shear Capacity Calculations: Determination of and :

Vc = 165 kip (LRFD 5.8.3.3-3) x = 1.0000 (1,000's) (LRFD 5.8.3.4.2-1,2,3)

Vs = 341 kip (LRFD 5.8.3.3-4) v/fc = 0.107Vp = 0 kip = 36.8 ° (Trial value)Vn = 506 kip (LRFD 5.8.3.3-1) 1 = 36.8 ° (LRFD Table 5.8.3.4.2-1)

Vn(max)= 1142 kip (LRFD 5.8.3.3-2) = 0.0Vn = 455 kip = 2.17 (LRFD Table 5.8.3.4.2-1)

Shear Stirrup Requirements: Torsion Stirrup Requirements:Av/S (min) = 0.039 in^2/in (LRFD 5.8.2.5-1) Vueq = 438 kip (LRFD 5.8.3.6.2-2)

Req'd Vs = 322 kip (Vu/ v-Vc-Vp) Tn = 0 kip-ft (LRFD 5.8.3.6.2-1)Req'd Av/S = 0.123 in^2/in (Vs/(bv*dv*cot )) Req'd At/S = 0.000 in^2/in

Av = 1.24 in^2 Torsion, S <= 1000.0 inShear, S <= 10.1 in OK

OK

Longitudinal Reinforcing Requirements: Combined Shear and Torsion Requirements:As*fy+Aps*fps = 749 kip Smax <= 24 in (LRFD 5.8.2.7-1,2)Req'd Force = 707 kip (LRFD 5.8.3.6.3-1) Req'd Spacing = 10.1 in

OK OK

Final Shear Design Statement: OK

CONCRETE SECTION SHEAR CAPACITY BY AASHTO LRFD 2004 BRIDGE DESIGN SPECIFICATIONS

LRFD Design for Shear: Rectangular Bent Cap Design

page 12

BRG

BRG

Formulas: Items UnitsForce kip

Determine unit shear stress: Moment kip-ft v = [((Vu- Vp)/( bv*dv))^2 + ((Tu*Ph)/( Aoh^2))^2]^0.5 (5.8.3.6.2-4) Stress ksi

Distance inDetermine strain in reinforcement on tension side of the member: (Mu not less than Vu*dv) Area in^2

x1 = (Mu/dv + 0.5Nu +0.5(Vueq-Vp)*cot - Aps*fpo)/2(EsAs+EpAps) <= 0.001 (5.8.3.4.2-1) Area/Distance in^2/in

If combined shear and torsion: (5.8.3.6.2)

x2 = (Mu/dv + 0.5Nu +0.5(Vueq-Vp)*cot - Aps*fpo)/(EsAs+EpAps) (5.8.3.4.2-2)

If x1 < 0 (compression) then:

x3 = (Mu/dv + 0.5Nu +0.5(Vueq-Vp)*cot - Aps*fpo)/2(EcAc+EsAs+EpAps) (5.8.3.4.2-3)

Equivalent ultimate shear for combined shear and torsion Vueq = [Vu^2 + ((0.9*Ph*Tu)/(2*Ao))^2]^0.5 (5.8.3.6.2-2)

Find values of and as a function of concrete shear stress v and strain x from Figure 5.8.3.4.2-1

Calculate shear capacity of concrete VcVc = 0.0316 *sqrt(f'c)bv*dv (5.8.3.3-3)

Calculate stirrups shear capacity Vs Vs = Av*fy*dv*cot /S (5.8.3.3-4)

Calculate total shear capacity Vn, as Lesser of: Vn = Vc+Vs+Vp (5.8.3.3-1) Vn = 0.25f'c*bv*dv+Vp (5.8.3.3-2)

Vn =min(Vn1, Vn2)*

Maximum Spacing of Stirrups, S (5.8.2.7) if Vu < 0.125 f'c*bv*dv then S<=0.8dv<=24 in if Vu >= 0.125 f'c*bv*dv then S<=0.4dv<=12 in

Minimum shear reinforcing Av/S >=0.0316*Sqrt(f'c)*bv/fy (5.8.2.5-1)

Torsional Resistance Tn = (2*Ao*At*fy*cot )/s (5.8.3.6.2-1)

Minimum torsional reinforcing At/S >= Tu/( 2*Ao*fy*cot )

Minimum transverse reinforcement Smax = At/((Av/S)min*(At/Av) + (At/S)min)

Longitudinal reinforcement requirements (5.8.3.6.3-1)

As*fy+Aps*fps >= Mu/(dv M)+0.5Nu/ N+cot [(Vu/ V-0.5Vs-Vp)^2+((0.45PhTu)/(2Ao v))^2]^0.5

with Vs not to be taken as greater than Vu/ V (5.8.3.5-1)

page 13

Notations:

bv effective web width (5.8.2.9)dv effective shear depth taken as a distance between resultants of the tensile and compressive

forces due to flexure, but need not be taken less than greater of 0.9de or 0.72h (5.8.2.9)Mu factored moment, taken as a positive quantity, input as kip-ft, used in equations as kip-in (5.8.3.4.2)(5.8.3.5-1)Vu factored shear force (kip), taken as a positive quantity (5.8.3.4.2)Nu factored axial force (kip), positive if tensile, negative if compressive (5.8.3.4.2)As area of nonprestressed steel (in^2), on the flexural tension side of the member (5.8.3.4.2)

Aps area of prestressing steel (in^2), on the flexural tension side of the member (5.8.3.4.2)Ac area of concrete (in^2), on the flexural tension side of the member (5.8.3.4.2)Av area of shear reinforcing (in^2), within distance S (5.8.2.5)s spacing of transverse reinforcement (in) (5.8.2.5)

fpe effective stress in prestressing steel (ksi), after losses (5.6.3.4.1)Tu factored torsional moment, input as kip-ft, used in formulas as kip-in (5.8.2.1)

Aoh area enclosed by centerline of exterior closed transverse torsion reinforcement (in^2), including area of holes (5.8.3.6.2Ao area enclosed by the shear flow path (in^2), including area of holes therin, if any (5.8.3.6.2)At area of one leg of transverse torsion reinforcement (in^2) (5.8.3.6.2)Ph perimeter of the centerline of the closed transverse torsion reinforcement (in) (5.8.3.6.2)f'c compressive strength of concrete (ksi)Ec modulus of elasticity of concrete (ksi)fy yield strength of mild reinforcement (ksi)Es modulus of elasticity of mild reinforcement (ksi)fps ultimate strength of prestressing strands (ksi)Ep modulus of elasticity of prestressing strands (ksi)

Vc nominal shear resistance (kip), provided by tensile stresses in concrete (5.8.2.4)Vs shear resistance provided by shear reinforcement (kip) (5.8.3.3)Vn factored nominal shear resistance of the section considered (kip) (Vn - 5.8.2.1) ( - 5.5.4.2.1)

x longitudinal strain in the web reinforcement (in/in) on the flexural side of the member (5.8.3.4.2)angle of inclination of diagonal compressive stresses, (5.8.3.4)factor indicating ability of diagonally cracked concrete to transmit tension (5.8.3.4)

Av/S minimum required shear reinforcement (in^2/in) (5.8.2.5-1)At/S minimum required torsional reinforcement (in^2/in) (5.8.3.6.2)Smax maximum stirrup spacing for combined shear and torsion effects (in)

v unit shear stress, (ksi) (5.8.3.6.2-4)Vp

Vueq equivalent factored shear for combined shear and torsion (kip), for calculating x, , and (5.8.3.6.2-2)

fpo stress in reinforcment when stress in concrete is zero (ksi), typically 0.7*fpu (5.8.3.4.2)Tn factored nominal torsional resistance of the section considered (kip-in) (Tn - 5.8.3.6.2-1) ( - 5.5.4.2.1)

fps average stress in prestressing steel at the time for shich the nominal resistance of member is required (ksi) (C5.6.3.3.3)

Inpu

tO

utpu

t

component in the direction of the applied shear of the effective prestressing force (kip), positive if resisting the applied shear (C5.8.2.3)

Inte

rmed

iate

page 14

LRFD CAP18 Example 0000-00-000 RCR 11-01-2004 INTERIOR BENTS 1 E HL-93; Span 1 (Beam Spa= 8.5') TABLE 1: 2 8 15 -2 0.000E+00 TABLE 2: 80 5.000E-01 20 2 58 1 3 1.250E+00 1.750E+00 3 1.200E+00 1.000E+00 8.500E-01TABLE 3: 3 5 3 11 3 (Left) 2 26 54(Right) 26 54 78 (Stringers) 6.0 23.0 40.0 57.0 74.0 (Supports) 10 40 70 (Moments) 6 19 23 33 40 47 57 60 74 10 70 (Shears) 8 12 38TABLE 4: 2 78 1.000E+09 -7.920E-01 6 6 -1.031E+02 23 23 -1.031E+02 40 40 -1.031E+02 57 57 -1.031E+02 74 74 -1.031E+02 6 6 -1.091E+02 20 20 -1.091E+02 33 33 -1.091E+02 47 47 -1.091E+02 60 60 -1.091E+02 74 74 -1.091E+02 0 20 -5.860E+00 4 4 -2.130E+01 16 16 -2.130E+01 2 E Span 1 (1 lane loading)- envelopes held TABLE 1: 1 1 1 8 -2 0.000E+00 TABLE 3: 1 5 3 11 3 (Left) 2(Right) 78 (Stringers) 6.0 23.0 40.0 57.0 74.0 (Supports) 10 40 70 (Moments) 6 19 23 33 40 47 57 60 74 10 70 (Shears) 8 12 38 3 E Span 2 (Beam spa= 6.8')- Span 1 envelopes held TABLE 1: 1 1 1 8 -2 0.000E+00 TABLE 3: 3 6 3 11 3 (Left) 2 26 54(Right) 26 54 78 (Stringers) 6.0 19.6 33.2 46.8 60.4 74.0 (Supports) 10 40 70 (Moments) 6 19 23 33 40 47 57 60 74 10 70

(Shears) 8 12 38 4 E Span 2 (1 lane loading)- Span 1 envelopes held TABLE 1: 1 1 1 8 -2 0.000E+00 TABLE 3: 1 6 3 11 3 (Left) 2 (Right) 78 (Stringers) 6.0 19.6 33.2 46.8 60.4 74.0 (Supports) 10 40 70 (Moments) 6 19 23 33 40 47 57 60 74 10 70 (Shears) 8 12 38CEASE

CAP18 Input

page 15

See page 4 for description of CAP18 problem cards.

20.0 33.0 47.0 60.0

20.0 33.0 47.0 60.0

CAP18 Output

NOV 7, 2004 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 39 CAP18 BENT CAP ANALYSIS Win32 Ver 5.1 Aug 2001

PROB 4 Span 2 (1 lane loading)- Span 1 envelopes held (CONTINUED)

TABLE 4A. DEAD LOAD RESULTS ( WORKING STRESS )

-------------------------------------------------------------------- STA DIST X (FT) DEFLECTION (FT) MOMENT (K-FT) SHEAR (K) -------------------------------------------------------------------- -1 -.50 .000000 .0 .0 0 .00 .000000 .0 .0 1 .50 .000000 .0 .0 2 1.00 .000000 .0 -.2 3 1.50 .000000 -.2 -.8 4 2.00 .000000 -.8 -1.6 5 2.50 .000000 -1.8 -2.4 6 3.00 .000000 -3.2 -109.3 7 3.50 .000000 -111.1 -216.2 8 4.00 .000000 -219.3 -217.0 9 4.50 .000000 -328.0 -217.7 10 5.00 .000000 -437.1 -30.4 11 5.50 .000000 -358.4 156.9 12 6.00 -.000001 -280.2 156.1 13 6.50 -.000001 -202.3 155.3 14 7.00 -.000002 -124.9 154.5 15 7.50 -.000002 -47.8 153.7 16 8.00 -.000003 28.9 152.9 17 8.50 -.000003 105.1 152.1 18 9.00 -.000004 181.0 151.4 19 9.50 -.000004 256.5 150.6 20 10.00 -.000004 331.6 95.2 21 10.50 -.000005 351.7 39.9 22 11.00 -.000005 371.4 39.1 23 11.50 -.000005 << 390.8 >> -13.3 24 12.00 -.000005 358.2 -65.6 25 12.50 -.000005 325.2 -66.4 26 13.00 -.000005 291.8 -67.2 27 13.50 -.000005 258.0 -68.0 28 14.00 -.000004 223.8 -68.8 29 14.50 -.000004 189.3 -69.5 30 15.00 -.000004 154.3 -70.3 31 15.50 -.000003 118.9 -71.1 32 16.00 -.000003 83.1 -71.9 33 16.50 -.000002 47.0 -127.3 34 17.00 -.000002 -44.1 -182.6 35 17.50 -.000001 -135.6 -183.4 36 18.00 -.000001 -227.5 -184.2 37 18.50 -.000001 -319.8 -185.0 38 19.00 .000000 -412.5 -185.8 39 19.50 .000000 -505.6 -186.6 40 20.00 .000000 << -599.1 >> .0 41 20.50 .000000 -505.6 186.6 42 21.00 .000000 -412.5 185.8 43 21.50 -.000001 -319.8 185.0 44 22.00 -.000001 -227.5 184.2

page 16

CAP18 Output (Continued)

NOV 7, 2004 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 42 CAP18 BENT CAP ANALYSIS Win32 Ver 5.1 Aug 2001

PROB 4 Span 2 (1 lane loading)- Span 1 envelopes held (CONTINUED)

TABLE 6. ENVELOPES OF MAXIMUM VALUES ( WORKING STRESS )

------------------------------------------------------------------------ STA DIST X MAX + MOM MAX - MOM MAX + SHEAR MAX - SHEAR ( FT ) ( FT-K ) ( FT-K ) ( K ) ( K ) ------------------------------------------------------------------------ -1 -.50 .0 .0 .0 .0 0 .00 .0 .0 .0 .0 1 .50 .0 .0 .0 .0 2 1.00 .0 .0 -.2 -.2 3 1.50 -.2 -.2 -.8 -.8 4 2.00 -.8 -.8 -1.6 -1.6 5 2.50 -1.8 -1.8 -2.4 -2.4 -->6 3.00 -3.2 -3.2 -109.3 -171.3 7 3.50 -111.0 -173.1 -216.2 -340.2 8 4.00 -219.3 -343.4 -217.0 -341.0 9 4.50 -328.0 -514.1 -217.7 -341.8 10 5.00 -437.1 -685.2 39.1 -65.9 11 5.50 -289.0 -580.0 295.9 143.2 12 6.00 -141.2 -475.3 295.1 142.4 13 6.50 6.1 -370.9 294.3 141.6 14 7.00 153.0 -266.9 293.5 140.8 15 7.50 299.6 -163.3 292.7 140.0 16 8.00 445.8 -62.5 291.9 139.3 17 8.50 591.5 33.0 291.1 138.5 18 9.00 736.9 126.3 290.3 137.7 19 9.50 881.8 194.9 289.5 136.9 -->20 10.00 << 1026.4 >> 263.2 171.3 81.5 21 10.50 1001.3 276.4 116.0 -50.5 22 11.00 980.4 289.3 115.2 -51.3 -->23 11.50 964.2 301.8 6.0 -103.7 24 12.00 896.1 262.4 -59.5 -156.0 25 12.50 827.5 222.6 -60.3 -156.8 26 13.00 758.6 182.3 -61.1 -157.6 27 13.50 689.3 141.7 -61.9 -158.4 28 14.00 619.6 100.7 -62.7 -159.1 29 14.50 549.5 59.2 -63.5 -159.9 30 15.00 479.0 17.4 -64.3 -160.7 31 15.50 408.1 -24.8 -65.1 -161.5 32 16.00 336.7 -67.4 -65.8 -162.3 -->33 16.50 265.7 -110.4 -127.3 -246.2 34 17.00 96.2 -209.5 -182.6 -349.8 35 17.50 -55.6 -309.0 -183.4 -350.6 36 18.00 -207.8 -409.3 -184.2 -351.4 37 18.50 -319.8 -511.7 -185.0 -352.2 38 19.00 -412.5 -667.2 -185.8 -352.9 39 19.50 -505.6 -826.3 -186.6 -353.7 -->40 20.00 -599.1 << -985.9 >> 50.3 -57.8 41 20.50 -505.6 -845.6 314.9 186.6 42 21.00 -412.5 -726.8 314.1 185.8 43 21.50 -319.8 -608.5 313.4 185.0

page 17

CAP18 Output (Continued)

NOV 7, 2004 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 43 CAP18 BENT CAP ANALYSIS Win32 Ver 5.1 Aug 2001

PROB 4 Span 2 (1 lane loading)- Span 1 envelopes held (CONTINUED)

TABLE 6. ENVELOPES OF MAXIMUM VALUES ( WORKING STRESS )

------------------------------------------------------------------------ STA DIST X MAX + MOM MAX - MOM MAX + SHEAR MAX - SHEAR ( FT ) ( FT-K ) ( FT-K ) ( K ) ( K ) ------------------------------------------------------------------------ 44 22.00 -207.8 -490.6 312.6 184.2 45 22.50 -55.6 -373.1 311.8 183.4 46 23.00 96.2 -256.0 311.0 182.6 -->47 23.50 247.6 -161.4 202.5 127.3 48 24.00 290.7 -124.6 136.9 65.9 49 24.50 336.6 -88.2 136.1 65.1 50 25.00 382.4 -52.2 135.3 64.3 51 25.50 427.8 -16.6 134.5 63.5 52 26.00 472.8 18.7 133.7 62.7 53 26.50 519.6 53.5 132.9 61.9 54 27.00 583.6 87.9 132.1 61.1 55 27.50 649.5 122.0 131.3 60.3 56 28.00 714.9 155.6 130.5 59.5 -->57 28.50 780.0 188.9 43.4 -6.0 58 29.00 727.7 170.2 -8.9 -115.3 59 29.50 693.6 151.1 -9.7 -116.1 -->60 30.00 688.5 131.6 -74.6 -171.4 61 30.50 575.2 76.5 -110.6 -236.6 62 31.00 462.0 21.0 -111.4 -237.4 63 31.50 348.9 -34.8 -112.1 -238.1 64 32.00 236.2 -91.1 -112.9 -238.9 65 32.50 123.6 -177.8 -113.7 -239.7 66 33.00 10.7 -266.8 -114.5 -240.5 67 33.50 -102.7 -370.8 -115.3 -241.3 68 34.00 -216.5 -475.2 -116.1 -242.1 69 34.50 -329.3 -580.0 -116.9 -242.9 70 35.00 -437.0 -685.2 65.9 -3.5 71 35.50 -328.0 -514.1 341.8 217.7 72 36.00 -219.3 -343.4 341.0 216.9 73 36.50 -111.0 -173.1 340.2 216.1 -->74 37.00 -3.2 -3.2 171.3 109.3 75 37.50 -1.8 -1.8 2.4 2.4 76 38.00 -.8 -.8 1.6 1.6 77 38.50 -.2 -.2 .8 .8 78 39.00 .0 .0 .2 .2 79 39.50 .0 .0 .0 .0 80 40.00 .0 .0 .0 .0 81 40.50 .0 .0 .0 .0

page 18

CAP18 Output (Continued)

NOV 7, 2004 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 46 CAP18 BENT CAP ANALYSIS Win32 Ver 5.1 Aug 2001

PROB 4 Span 2 (1 lane loading)- Span 1 envelopes held (CONTINUED)

TABLE 6. ENVELOPES OF MAXIMUM VALUES ( LOAD FACTOR )

------------------------------------------------------------------------ STA DIST X MAX + MOM MAX - MOM MAX + SHEAR MAX - SHEAR ( FT ) ( FT-K ) ( FT-K ) ( K ) ( K ) ------------------------------------------------------------------------ -1 -.50 .0 .0 .0 .0 0 .00 .0 .0 .0 .0 1 .50 .0 .0 .0 .0 2 1.00 .0 .0 -.2 -.3 3 1.50 -.2 -.3 -1.0 -1.0 4 2.00 -1.0 -1.0 -2.0 -2.0 5 2.50 -2.2 -2.3 -3.0 -3.0 -->6 3.00 -4.0 -4.0 -136.6 -245.2 7 3.50 -138.8 -247.4 -270.2 -487.3 8 4.00 -274.1 -491.3 -271.2 -488.3 9 4.50 -410.0 -735.7 -272.2 -489.3 10 5.00 -546.3 -980.6 83.6 -100.2 11 5.50 -326.5 -835.9 439.3 172.2 12 6.00 -107.1 -691.6 438.3 171.2 13 6.50 111.8 -547.9 437.3 170.2 14 7.00 330.3 -404.6 436.4 169.2 15 7.50 548.2 -261.8 435.4 168.2 16 8.00 765.6 -123.8 434.4 167.2 17 8.50 982.6 5.2 433.4 166.2 18 9.00 1199.0 130.5 432.4 165.2 19 9.50 1415.0 212.9 431.4 164.3 -->20 10.00 << 1630.4 >> 294.7 252.3 95.1 21 10.50 1576.5 307.9 183.1 -108.4 22 11.00 1530.0 320.6 182.1 -109.4 -->23 11.50 1492.0 332.8 17.1 -174.8 24 12.00 1389.0 280.1 -71.4 -240.3 25 12.50 1285.6 226.9 -72.4 -241.2 26 13.00 1181.6 173.2 -73.3 -242.2 27 13.50 1077.2 119.0 -74.3 -243.1 28 14.00 972.3 64.3 -75.3 -244.1 29 14.50 866.9 9.1 -76.3 -245.1 30 15.00 761.0 -46.6 -77.3 -246.1 31 15.50 654.6 -102.8 -78.3 -247.1 32 16.00 547.7 -159.5 -79.3 -248.1 -->33 16.50 441.5 -216.7 -159.1 -367.2 34 17.00 190.4 -344.5 -228.3 -520.8 35 17.50 -29.5 -472.9 -229.3 -521.8 36 18.00 -249.9 -602.5 -230.2 -522.8 37 18.50 -399.8 -735.6 -231.2 -523.8 38 19.00 -515.6 -961.3 -232.2 -524.8 39 19.50 -632.0 -1193.3 -233.2 -525.8 -->40 20.00 -748.9 << -1425.7 >> 88.1 -101.1 41 20.50 -632.0 -1226.9 457.9 233.2 42 21.00 -515.6 -1065.7 456.9 232.2 43 21.50 -399.8 -905.0 455.9 231.3 44 22.00 -249.8 -744.8 454.9 230.3

page 19

CAP18 Output (Continued)

NOV 7, 2004 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 47 CAP18 BENT CAP ANALYSIS Win32 Ver 5.1 Aug 2001

PROB 4 Span 2 (1 lane loading)- Span 1 envelopes held (CONTINUED)

TABLE 6. ENVELOPES OF MAXIMUM VALUES ( LOAD FACTOR )

------------------------------------------------------------------------ STA DIST X MAX + MOM MAX - MOM MAX + SHEAR MAX - SHEAR ( FT ) ( FT-K ) ( FT-K ) ( K ) ( K ) ------------------------------------------------------------------------ 45 22.50 -29.5 -585.1 453.9 229.3 46 23.00 190.4 -425.9 452.9 228.3 -->47 23.50 409.8 -306.0 290.8 159.1 48 24.00 467.2 -259.7 203.6 79.3 49 24.50 529.5 -213.8 202.6 78.3 50 25.00 592.0 -168.5 201.6 77.3 51 25.50 653.9 -123.6 200.6 76.3 52 26.00 715.4 -79.3 199.6 75.3 53 26.50 780.3 -35.4 198.6 74.3 54 27.00 875.4 7.9 197.6 73.3 55 27.50 973.9 50.8 196.6 72.3 56 28.00 1072.0 93.2 195.6 71.3 -->57 28.50 1169.5 135.1 69.4 -17.1 58 29.00 1087.8 112.1 4.0 -182.2 59 29.50 1037.8 88.5 3.0 -183.2 -->60 30.00 1039.1 64.5 -82.9 -252.3 61 30.50 878.3 5.6 -118.2 -338.7 62 31.00 717.9 -53.7 -119.2 -339.7 63 31.50 558.1 -113.5 -120.2 -340.7 64 32.00 398.9 -173.9 -121.2 -341.7 65 32.50 240.2 -287.2 -122.2 -342.6 66 33.00 81.1 -404.5 -123.1 -343.6 67 33.50 -78.6 -547.8 -124.1 -344.6 68 34.00 -238.7 -691.6 -125.1 -345.6 69 34.50 -397.1 -835.8 -126.1 -346.6 70 35.00 -546.3 -980.6 100.1 -21.3 71 35.50 -409.9 -735.7 489.3 272.2 72 36.00 -274.1 -491.3 488.3 271.2 73 36.50 -138.8 -247.4 487.3 270.2 -->74 37.00 -3.9 -4.0 245.1 136.6 75 37.50 -2.2 -2.2 3.0 3.0 76 38.00 -1.0 -1.0 2.0 2.0 77 38.50 -.2 -.2 1.0 1.0 78 39.00 .0 .0 .2 .2 79 39.50 .0 .0 .0 .0 80 40.00 .0 .0 .0 .0 81 40.50 .0 .0 .0 .0

page 20