Design Parameters. Span 1. Span 2. Span 3. All Spans. Bents

Transcription

1 County: Any Hwy: Any Design: BRG Date: 6/2010 Inverted Tee Bent Cap Design Example Design example is in accordance with the AASHTO LRFD Bridge Design Specifications, 5th Ed. (2010) as prescribed by TxDOT Bridge Design Manual - LRFD (May 2009). Design Parameters Span 1 54' Type TX54 Girders (0.851 k / ft ) 6 Girders 8.00' with 3' overhangs 2" Haunch Span 2 112' Type TX54 Girders (0.851 k / ft ) 6 Girders 8.00' with 3' overhangs 3.75" Haunch Span 3 54' Type TX54 Girders (0.851 k / ft ) 6 Girders 8.00' with 3' overhangs 2" Haunch All Spans Deck is 46ft wide Type T551 Rail (0.382k/ft) 8" Thick Slab (0.100 ksf) Assume 2" 140 pcf (0.023 ksf) Use Class "C" Concrete f' c =3.60 ksi w c =150 pcf (for weight) w c =145 pcf (for Modulus of Elasticity calculation) Grade 60 Reinforcing F y =60 ksi "AASHTO LRFD" refers to the AASHTO LRFD Bridge Design Specification, 5th Ed. (2010) "BDM-LRFD" refers to the TxDOT Bridge Design Manual - LRFD (May 2009) "TxSP" refers to TxDOT guidance, recommendations, and standard practice. "Furlong & Mirza" refers to "Strength and Servicability of Inverted T-Beam Bent Caps Subject to Combined Flexure, Shear, and Torsion", Center for Highway Research Research Report No F, The University of Texas at Austin, August 1974 The basic bridge geometry can be found on the Bridge Layout located in the Appendices. (TxSP) (BDM-LRFD, Ch. 4, Sect. 5, Materials) (BDM-LRFD, Ch. 4, Sect. 5, Materials) Bents Use 36" Diameter Columns (Typical for Type TX54 Girders) LRFD Inverted Tee Bent Cap Design Example 1 June 2010

2 Design Parameters Define Variables (Con't) Back Span Span1 GdrSpa1 Forward Span 54ft Span2 112ft 8ft GdrSpa2 8ft GdrNo1 6 GdrNo2 6 GdrWt1 Haunch klf GdrWt klf 2in Haunch2 3.75in Span Length Girder Spacing Number of Girders in Span Weight of Girder Size of Haunch Bridge Skew 0deg BridgeW 46ft RdwyW 44ft GirderD 54in BrgSeat 1.5in BrgPad 2.75in SlabThk 8in OverlayThk 2in RailWt 0.382klf w c 0.150kcf Skew of Bents Width of Bridge Deck Width of Roadway Depth of Type TX54 Girder Bearing Seat Buildup Bearing Pad Thickness Thickness of Bridge Slab Thickness of Overlay Weight of Rail Unit Weight of Concrete for Loads W Olay 0.140kcf Unit Weight of Overlay Bents f c 3.60ksi Concrete Strength w ce 0.145kcf Unit Weight of Concrete for E c 1.5 E c = w ce f c E c 3457ksi Modulus of Elasticity of Concrete, (AASHTO LRFD Eq ) f y 60ksi Yield Strength of reinforcement E s 29000ksi Modulus of Elasticity of Steel D column 36in Diameter of Columns Other Variables IM 33% Dynamic load allowance, (AASHTO LRFD Table ) LRFD Inverted Tee Bent Cap Design Example 2 June 2010

3 Determine Cap Dimensions Stem Width b stem D column 3in b stem 39in Stem Height The stem is typically at least 3" wider than the Diameter of the Column (36") to allow for the extension of the column reinforcement into the Cap. (TxSP) Distance From Top of Slab to Top of Ledge D Slab_to_Ledge SlabThk Haunch2 GirderD BrgPad BrgSeat Haunch 2 is the larger of the two haunches. D Slab_to_Ledge 70.00in StemHaunch 3.75in The top of the stem must be 2.5" below the bottom of the slab. (BDM-LRFD, Ch. 4, Sect. 5, Geometric Constraints) Accounting for the 1/2" of bituminus fiber, the top of the stem must have at least 2" of haunch on it, but the haunch should not be less than either of the haunches of the adjacent spans. d stem D Slab_to_Ledge SlabThk StemHaunch 0.5in Use: d stem 57in d stem 57.75in The stem must accommodate 1/2" of bituminous fiber. Round the Stem Height down to the nearest 1". (TxSP) LRFD Inverted Tee Bent Cap Design Example 3 June 2010

4 Determine Cap Dimensions Ledge Width (Con't) The Ledge Width must be adequate for Bar M to develop fully. 26" is typically used for TX54 girders, as it is adequate to develop a # 6 bar with the typical 2.5" cover. If the cover is increased to 3", allowing for a modification factor of 0.7, a 24" Ledge is adequate to develop a # 7 bar. I-Beams have 9.5" from the face of the cap to the CL of Bearing. The typical ledge width for these bridges is 23". "L dh,prov " must be greater than or equal to "L dh,req " for bar M. cover 2.5in "cover" is measured from the center of the transverse bars. L 8in Determine the Required Development Length of Bar M: "L" is the length of the Bearing Pad along the girder. A typical type TX54 bearing pad is 8"x21" as shown in the IGEB standard. Try # 6 Bar for Bar M. d bar_m 0.750in A bar_m 0.44in 2 Basic Development Length L dh = 38.0d bar_m L dh 15.02in (AASHTO LRFD Eq ) f c Modification Factors for L dh: (AASHTO LRFD ) Is Top Cover greater than or equal to 2.5", and Side Cover greater than or equal to 2"? SideCover cover d bar_m 2 SideCover 2.13in TopCover cover d bar_m 2 TopCover 2.13in "Side Cover" and "Top Cover" are the clear cover on the side and top of the hook respectively. The dimension "cover" is measured from the center of Bar M. No, Factor = 1.0 LRFD Inverted Tee Bent Cap Design Example 4 June 2010

5 Determine Cap Dimensions (Con't) The Required Development Length is the larger of the following: (AASHTO LRFD ) L dh Factor 8d bar_m 6in Therefore, L dh_req 15.02in 6.00in 15.02in L b ledge_min L dh_req cover 12in b 2 ledge_min 25.52in The distance from the face of the stem to the center of bearing is 12" for TxGirders. (IGEB) Use: b ledge 26in Width of Bottom Flange b f 2b ledge b stem b f 91in Ledge Depth Use a Ledge Depth of 28" d ledge 28in As a general rule of thumb Ledge Depth is greater than or equal to 2'-3". This is the depth at which a bent from a typical bridge will pass the punching shear check (calculations found on Pg. 21). Total Depth of Cap h cap d stem d ledge h cap 85in Summary of Cross Sectional Dimensions b stem 39in From Pg. 3 d stem 57in From Pg. 3 b ledge 26in d ledge 28in h cap 85in LRFD Inverted Tee Bent Cap Design Example 5 June 2010

6 Determine Cap Dimensions (Con't) Length of Cap First define Girder Spacing and End Distance: S 8ft c 2ft L Cap S( GdrNo1 1) 2c L Cap 44 ft Girder Spacing "c" is the distance from the Center Line of the Exterior Girder to the Edge of the Cap measured along the Cap. Length of Cap TxDOT policy is as follows, "The edge distance between the exterior bearing pad and the end of the inverted T-beam shall not be less than 12in." (BDM-LRFD, Ch. 4, Sect. 5, Design Criteria) replacing the statement in AASHTO LRFD stating it shall not be less than d f. Preferably, the stem should extend at least 3" beyond the edge of the bearing seat. Bearing Pad Dimensions L 8in (IGEB standard) Length of Bearing Pad W 21in Width of Bearing Pad Cross Sectional Properties of Cap A g d ledge b f d stem b stem A g 4771in 2 ybar d ledge b f 1 2 d ledge 1 d stem b stem d ledge 2 d stem A g Distance from bottom of cap to the center of gravity of the cap ybar 33.80in 3 b f d ledge I g 12 3 b stem d stem 12 b f d ledge 2 1 ybar 2 d ledge I g in 4 1 b stem d stem ybar d ledge 2 d stem 2 LRFD Inverted Tee Bent Cap Design Example 6 June 2010

7 Cap Analysis Cap Model Assume: 4 Columns 12'-0" The 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. (BDM-LRFD, Ch. 4, Sect. 5, Structural Analysis) Cap 18 Model Station = 0.5' The circled numbers are the stations that will be used in the CAP 18 input file. One station is 0.5ft in the direction perpendicular to the pgl, not parallel to the bent. station 0.5ft Station increment for CAP18 Recall: E c ksi (Pg. 2) I g in 4 (Pg.6) E c I g kipin 2 / 12 in ft 2 = E c I g kipft 2 LRFD Inverted Tee Bent Cap Design Example 7 June 2010

8 Cap Analysis Dead Load SPAN 1 Rail1 (Con't) Span1 2RailWt 2 kip Rail min( GdrNo1 6) girder Values used in the following equations can be found on Pg. 2. Rail weight is distributed evenly among stringers, up to 3 stringers per rail. (TxSP) Slab1 Span1 kip w c GdrSpa1SlabThk 1.10 Slab girder Increase slab DL by 10% to account for haunch and thickened slab ends. Girder1 GdrWt1 Span1 kip Girder girder kip DLRxn1 Rail1 Slab1 Girder1 DLRxn girder Overlay is calculated separately, because it has a different load factor than the rest of the dead loads. Overlay1 Span1 kip W Olay GdrSpa1OverlayThk Overlay girder Design for future overlay. SPAN 2 Rail2 Slab2 Span2 2RailWt 2 kip Rail min( GdrNo2 6) girder Span2 kip w c GdrSpa2SlabThk 1.10 Slab girder Girder2 GdrWt2 Span2 kip Girder girder kip DLRxn2 Rail2 Slab2 Girder2 DLRxn girder Overlay2 Span2 kip W Olay GdrSpa2OverlayThk Overlay girder CAP Cap w c A g kip * ft 0.5ft station kip = Cap station LRFD Inverted Tee Bent Cap Design Example 8 June 2010

9 Cap Analysis (Con't) Live Load (AASHTO LRFD and ) LongSpan ShortSpan max( Span1 Span2) min( Span1 Span2) LongSpan ft ShortSpan ft IM 0.33 Lane 0.64klf Lane kip lane LongSpan ShortSpan LongSpan 14ft Truck 32kip 32kip 8kip LongSpan Truck kip lane LLRxn Lane Truck( 1 IM) LLRxn kip lane P 16.0kip( 1 IM) P 21.28kip LLRxn ( 2P) w 10ft w 9.83 kip * ft 0.5ft station kip w 4.92 station 2 LongSpan 28ft LongSpan Use HL-93 Live Load. For maximum reaction at interior bents, "Design Truck" will always govern over "Design Tandem". For the maximum reaction when the long span is more than twice as long as the short span, place the rear (32 kip) axle over the support and the middle (32 kip) and front (8 kip) axles on the long span. For the maximum reaction when the long span is less than twice as long as the short span, place the middle (32 kip) axle over the support, the front (8 kip) axle on the short span and the rear (32 kip) axle on the long span. Combine "Design Truck" and "Design Lane" loadings. (AASHTO LRFD ) Dynamic load allowance, IM, does not apply to "Design Lane." (AASHTO LRFD ) The Live Load is applied to the slab by two 16 kip wheel loads increased by the dynamic load allowance with the remainder of the live load distributed over a 10 ft (AASHTO LRFD ) design lane width. (TxSP) The Live Load applied to the slab is distributed to the beams assuming the slab is hinged at each beam except the outside beam. (BDM-LRFD, Ch. 4, Sect. 5, Structural Analysis) LRFD Inverted Tee Bent Cap Design Example 9 June 2010

10 Cap Analysis Cap 18 Input Multiple Presence Factors, m (AASHTO LRFD Table ) Input "Multiple Presence Factors" into Cap18 as "Load Reduction Factors". No. of Lanes Factor "m" >3 (Con't) Limit States (AASHTO LRFD 3.4.1) The cap design need only consider Strength I Live Load and Dynamic Load Allowance LL + IM = 1.75 Dead Load Components DC = 1.25 Strength I, Service I, and Service I with DL. (TxSP) Dead Load Wearing Surface (Overlay) Service I Live Load and Dynamic Load Allowance DW = 1.50 LL + IM = 1.00 TxDOT allows the Overlay Factor to be reduced to 1.25 (TxSP), since overlay is typically used in design only to increase the safety factor, but in this example we will use DW = Dead Load and Wearing Surface DC & DW = 1.00 Dead Load TxDOT considers Service level Dead Load only with a limit reinforcement stress of 22 ksi to minimize cracking. (BDM-LRFD, Chapter 4, Section 5, Design Criteria) Cap 18 Output Max +M Max -M Dead Load: M posdl 250.1kipft M negdl 379.4kipft Service Load: M posserv 492.5kipft M negserv 590.9kipft These loads are the maximum loads from the Cap 18 Output File located in the Appendices. Factored Load: M posult 741.7kipft M negult 852.1kipft LRFD Inverted Tee Bent Cap Design Example 10 June 2010

11 Cap Analysis (Con't) Girder Reactions on Ledge: Dead Load kip DLSpan1 Rail1 Slab1 Girder1 DLSpan girder kip Overlay girder kip DLSpan2 Rail2 Slab2 Girder2 DLSpan girder kip Overlay girder For calculations of these loads see Pg. 8. Live Load (AASHTO LRFD and ) Loads per Lane: Use HL-93 Live Load. For maximum reaction at interior bents, "Design Truck" will always govern over "Design Tandem" for Spans greater than 26ft. For the maximum reaction, place the back (32 kip) axle over the support. LaneSpan1 LaneSpan2 Span1 0.64klf LaneSpan kip 2 lane Span2 0.64klf LaneSpan kip 2 lane Span1 14ft TruckSpan1 32.0kip 32.0kip Span1 28ft 8.0kip TruckSpan kip Span1 Span1 lane Span2 14ft TruckSpan2 32.0kip 32.0kip Span2 28ft 8.0kip TruckSpan kip Span2 Span2 lane LRFD Inverted Tee Bent Cap Design Example 11 June 2010

12 Cap Analysis (Con't) Girder Reactions on Ledge (Con't) : Live Load (Con't) Loads per Lane (Con't) : IM 0.33 LLRxnSpan1 LaneSpan1 TruckSpan1 ( 1 IM) LLRxnSpan kip lane LLRxnSpan2 LaneSpan2 TruckSpan2 ( 1 IM) LLRxnSpan kip lane gv Span1_Int gv Span1_Ext gv Span2_Int gv Span2_Ext Combine "Design Truck" and "Design Lane" loadings. (AASHTO LRFD ) Dynamic load allowance, "IM", does not apply to "Design Lane." (AASHTO LRFD ) The Live Load Reactions are assumed to be the Shear Live Load Distribution Factor multiplied by the Live Load Reaction per Lane. The Shear Live Load Distribution Factor was calculated using the "LRFD Live Load Distribution Factors" Spreadsheet found in the Appendices. The Exterior Girders must have a Live Load Distribution Factor equal to or greater than the Interior Girders. This is to accommodate a possible future bridge widening. Widening the bridge would cause the exterior girders to become interior girders. LLSpan1Int gv Span1_Int LLRxnSpan1 kip LLSpan1Int girder LLSpan1Ext gv Span1_Ext LLRxnSpan1 kip LLSpan1Ext girder LLSpan2Int gv Span2_Int LLRxnSpan2 kip LLSpan2Int girder LLSpan2Ext gv Span2_Ext LLRxnSpan2 kip LLSpan2Ext girder LRFD Inverted Tee Bent Cap Design Example 12 June 2010

13 Cap Analysis (Con't) Girder Reactions on Ledge (Con't) : Span 1 Interior Girder Service Load Service I Limit State, (AASHTO LRFD 3.4.1) V s_span1int DLSpan1 Overlay1 LLSpan1Int V s_span1int 134kip Factored Load Strength I Limit State, (AASHTO LRFD 3.4.1) V u_span1int 1.25DLSpan1 1.5Overlay1 1.75LLSpan1Int V u_span1int 208kip Exterior Girder Service Load Service I Limit State, (AASHTO LRFD 3.4.1) V s_span1ext DLSpan1 Overlay1 LLSpan1Ext V s_span1ext 134kip Factored Load Strength I Limit State, (AASHTO LRFD 3.4.1) V u_span1ext 1.25DLSpan1 1.5Overlay1 1.75LLSpan1Ext V u_span1ext 208kip Span 2 Interior Girder Service Load Service I Limit State, (AASHTO LRFD 3.4.1) V s_span2int DLSpan2 Overlay2 LLSpan2Int V s_span2int 215kip Factored Load Strength I Limit State, (AASHTO LRFD 3.4.1) V u_span2int 1.25DLSpan2 1.5Overlay2 1.75LLSpan2Int V u_span2int 322kip Exterior Girder Service Load Service I Limit State, (AASHTO LRFD 3.4.1) V s_span2ext DLSpan2 Overlay2 LLSpan2Ext V s_span2ext 215kip Factored Load Strength I Limit State, (AASHTO LRFD 3.4.1) V u_span2ext 1.25DLSpan2 1.5Overlay2 1.75LLSpan2Ext V u_span2ext 322kip LRFD Inverted Tee Bent Cap Design Example 13 June 2010

14 Cap Analysis (Con't) Torsional Loads Strength I Limit State, (AASHTO LRFD 3.4.1) To maximize the torsion, the live load only acts on the longer span in the configuration shown. The loads are applied to the cap as depicted in the following picture: a v 12in "a v " is the value for the distance from the face of the stem to the center of bearing for the girders. 12" is the typical value for TxGirders on Inverted Tee Bents (IGEB). 9" is the typical value for I-Beams (IBEB). The lever arm for the torsional loads is the distance from the center line of bearing to the centerline of the cap ( 1 / 2 b stem + a v ). b stem 39in From Pg. 3 1 LeverArm a v 2 b stem LeverArm 31.5in LRFD Inverted Tee Bent Cap Design Example 14 June 2010

15 Cap Analysis (Con't) Torsional Loads (Con't) Interior Girders Girder Reactions R u_span1 1.25DLSpan1 1.5Overlay1 R u_span1 70kip R u_span2 Torsional Load 1.25DLSpan2 1.5Overlay2 1.75gV Span2_Int [ LaneSpan2 TruckSpan2 ( 1 IM) ] R u_span2 322kip T u_int R u_span1 R u_span2 LeverArm T u_int 660kipft Exterior Girders Girder Reactions R u_span1 1.25DLSpan1 1.5Overlay1 R u_span1 70kip R u_span2 Torsional Load 1.25DLSpan2 1.5Overlay2 1.75gV Span2_Ext [ LaneSpan2 TruckSpan2 ( 1 IM) ] R u_span2 322kip T u_ext R u_span1 R u_span2 LeverArm T u_ext 660kipft Torsion on Cap Analyzed assuming Bents are torsionally rigid at Effective Face of Columns. T u 660kipft Maximum Torsion on Cap LRFD Inverted Tee Bent Cap Design Example 15 June 2010

16 Cap Analysis (Con't) Load Summary Ledge Loads Interior Girder Service Load V s_int max V s_span1int V s_span2int V s_int kip Factored Load V u_int max V u_span1int V u_span2int V u_int kip Exterior Girder Service Load V s_ext Factored Load max V s_span1ext V s_span2ext V s_ext kip V u_ext max V u_span1ext V u_span2ext V u_ext kip Cap Loads Positive Moment (From CAP 18) Dead Load: M posdl kipft Service Load: M posserv kipft Factored Load: M posult kipft Negative Moment (From CAP 18) Dead Load: M negdl kipft Service Load: M negserv kipft Factored Load: M negult kipft Maximum Torsion and Concurrent Shear and Moment (Strength I) T u 660kipft V u 448.1kip M u 335.6kipft Located two stations away from centerline of column. V u and M u values are from CAP 18 In this example the maximum Torsion and the maximum Shear are concurrent with each other. If they are not, it becomes necessary to check the location of the maximum Torsion with its concurrent Shear and the location of the maximum Shear with its concurrent Torsion. LRFD Inverted Tee Bent Cap Design Example 16 June 2010

17 Locate and Describe Reinforcement Recall: b stem 39in From Pg. 3 d stem 57in From Pg. 3 b ledge 26in From Pg. 5 d ledge 28in From Pg. 5 b f 91in From Pg. 5 h cap 85in From Pg. 5 cover 2.50in From Pg. 4 Measured from Center of bar LRFD Inverted Tee Bent Cap Design Example 17 June 2010

18 Locate and Describe Reinforcement Describe Reinforcing Bars (Con't) Use # 11 bars for Bar A A bar_a 1.56in 2 d bar_a 1.410in Use # 11 bars for Bar B A bar_b 1.56in 2 d bar_b 1.410in Use # 6 bars for Bar M A bar_m 0.44in 2 d bar_m 0.75in Bar M must be a # 6 bar or smaller to allow it to fully develop, as stated on Pg 4. Use # 6 bars for Bar N A bar_n 0.44in 2 d bar_n 0.75in To prevent confusion, use the same bar size for Bar N as Bar M. Use # 6 bars for Bar S A bar_s 0.44in 2 d bar_s 0.75in Use # 6 bars for Bar T A bar_t 0.44in 2 d bar_t 0.750in Calculate Dimensions d s_neg h cap cover 1 2 d bar_s 1 2 d bar_a d s_neg 81.42in d s_pos h cap cover 1 2 max d bar_s d bar_m 1 2 d bar_b d s_pos 81.42in a v 12in Typical for TX Girders on Inverted Tee Bent Caps (IGEB standard) a f a v cover a f 14.50in d e d ledge cover d e 25.50in 1 d f d ledge cover 2 d 1 bar_m 2 d bar_b d f 24.42in h d ledge BrgSeat h 29.50in "BrgSeat" is the height of the Bearing Seat Buildup. This value is defined on Pg. 2. LRFD Inverted Tee Bent Cap Design Example 18 June 2010

19 Locate and Describe Reinforcement Calculate Dimensions (Con't) (Con't) α 90deg Angle of Bars S Recall: L 8in From Pg. 4 W 21in From Pg. 6 LRFD Inverted Tee Bent Cap Design Example 19 June 2010

20 Check Bearing (AASHTO LRFD 5.7.5) The load on the bearing pad propagates along a truncated pyramid whose top has the area A 1 and whose base has the area A 2. A 1 is the loaded area (the bearing pad area: LxW ). A 2 is the area of the lowest rectangle contained wholly within the support (the Inverted Tee Cap). A 2 must not overlap the truncated pyramid of another load in either direction, nor can it extend beyond the edges of the cap in any direction. Elevation View Plan View ϕ 0.7 A 1 WL A 1 168in 2 (AASHTO LRFD ) Area under Bearing Pad Interior Girders 1 B min b ledge a v 2 L a 1 v 2 b stem 1 2 L 2d 1 ledge 2 S 1 2 W B 10.00in L 2 L 2B L in "B" is the distance from the perimeter of A 1 to the perimeter of A 2, as seen in the above figures. W 2 W 2B W in A 2 L 2 W 2 A in 2 m = the minimum of: Modification Factor A & 2 m 2.00 (AASHTO LRFD Eq ) A 1 ϕv n ϕ0.85f c A 1 m ϕv n 720kip (AASHTO LRFD Eq & AASHTO LRFD Eq ) V u_int 322kip < ϕv n BearingChk "OK!" V u_int From Pg. 16 Exterior Girders B min b ledge a v 1 2 L a v 1 2 b stem 1 2 L 2d 1 ledge 2 S 1 2 W c 1 2 W B 10.00in L 2 L 2B L in W 2 W 2B W in "B" is the distance from the perimeter of A 1 to the perimeter of A 2. A 2 L 2 W 2 A in 2 m = the minimum of: Modification Factor A A 1 & 2 m 2.00 (AASHTO LRFD Eq ) ϕv n ϕ0.85f c A 1 m ϕv n 720kip V u_ext 322kip < ϕv n BearingChk "OK!" (AASHTO LRFD Eq & AASHTO LRFD Eq ) V u_ext From Pg. 16 LRFD Inverted Tee Bent Cap Design Example 20 June 2010

21 Check Punching Shear (AASHTO LRFD with modifications from BDM-LRFD, Ch. 4, Sect. 5, Design Criteria) ϕ 0.9 (AASHTO LRFD ) Determine if the Shear Cones Intersect Is 1 2 S 1 2 W d f? Yes. Therefore Shear Cones do not intersect in the longitudinal direction of the Cap. Is 1 2 S 1 d f 2 W 24.42in 1 2 b stem a v in 1 2 L TxDOT uses "d f " instead of "d e " for Punching Shear (BDM-LRFD, Ch. 4, Sect. 5, Design Criteria). This is because "d f " has traditionally been used for inverted tee bents and was used in the Inverted Tee Research (Furlong & Mirza pg. 58). d f? Yes. Therefore Shear Cones do not intersect in the transverse direction of the Cap. 1 2 b stem a v d f 24.42in 1 2 L in Interior Girders V n f c W 2L 2d f d f V n 497kip (BDM-LRFD, Ch. 4, Sect. 5, Design Criteria) ϕv n 447kip V u_int 322kip < ϕv n PunchingShearChk "OK!" V u_int From Pg. 16 Exterior Girders V n = minimum of: (BDM-LRFD, Ch. 4, Sect. 5, Design Criteria) f c 2 W L d f c d f 388kip V n f c W 2L 2d f 388kip d f 497kip ϕv n 349kip V u_ext 322kip < ϕv n PunchingShearChk "OK!" V u_ext From Pg. 16 LRFD Inverted Tee Bent Cap Design Example 21 June 2010

22 Check Shear Friction (AASHTO LRFD ) Checks are for concrete only see ϕ 0.9 (AASHTO LRFD ) Determine the Distribution Width (AASHTO LRFD ) Interior Girders b s_int = minimum of: W 4 a v 69.00in "Ledge Reinforcement" for reinforcement checks for Bars M and N. b s_int S 69.00in 96.00in "S" is the girder spacing. Exterior Girders b s_ext = minimum of: W 4 a v 69.00in b s_ext S 96.00in 2 c 48.00in 48.00in "S" is the girder spacing. Interior Girders A cv d e b s_int A cv 1759in 2 V n = minimum of: 0.2 f c A cv 1267kip (AASHTO LRFD Eq ) 0.8 ksia cv 1408kip (AASHTO LRFD Eq ) V n ϕv n V u_int 1267kip 1140kip 322kip < ϕv n ShearFrictionChk "OK!" V u_int From Pg. 16 Exterior Girders A cv d e b s_ext A cv 1224in 2 V n = minimum of: 0.2 f c A cv 0.8 ksia cv 881kip 979kip (AASHTO LRFD Eq ) (AASHTO LRFD Eq ) V n ϕv n V u_ext 881kip 793kip 322kip < ϕv n ShearFrictionChk "OK!" V u_ext From Pg. 16 LRFD Inverted Tee Bent Cap Design Example 22 June 2010

23 Flexural Reinforcement for Negative Bending (Bars A) (Tension in Top) M dl M negdl M dl 379.4kipft M s M negserv M s 590.9kipft From Cap 18 Output. See Pg. 10 M u M negult M u 852.1kipft Minimum Flexural Reinforcement (AASHTO LRFD ) Factored Flexural Resistance, M r, must be greater than or equal to the lesser of 1.2 M cr (Cracking Moment) or 1.33 Mu (Ultimate Moment) I g in 4 Gross Moment of Inertia (From Pg. 6) h cap 85in Depth of Cap (From Pg. 5) ybar 33.80in f r = 0.24 f c f r 0.455ksi y t h cap ybar y t 51.20in Distance to the Center of Gravity of the Cap from the bottom of the Cap (From Pg. 6) Modulus of Rupture (BDM-LRFD, Ch. 4, Sect. 5, Design Criteria) Distance from Center of Gravity to extreme tension fiber S I g S in 3 y t Section Modulus for the extreme tension fiber M cr 1ft Sf r M 12in cr kipft Cracking Moment (AASHTO LRFD Eq ) M f = minimum of: 1.2M cr 1.33M u kipft kipft Design for the lesser of 1.2M cr or 1.33Mu when determining minimum area of steel required. Thus, M r must be greater than M f kipft LRFD Inverted Tee Bent Cap Design Example 23 June 2010

24 Flexural Reinforcement for Negative Bending (Con't) (Bars A) Moment Capacity Design (AASHTO LRFD ) Try, 5 ~ #11's Top BarANo 5 d bar_a 1.41in Number of bars in tension Diameter of main reinforcing bars A bar_a 1.56in 2 Area of one main reinforcing bar A s ( BarANo) A bar_a A s 7.80in 2 Area of steel in tension d stirrup d bar_s d stirrup 0.75in Diameter of shear reinforcing bars From Pg. 18 d d s_neg d 81.42in See Pg. 18 for the calculation of "d s_neg, " b b f b 91in See Pg. 5 for the calculation of "b f." f c 3.60ksi Compressive Strength of Concrete f y 60ksi Yield Strength of Rebar β 1 = f c 4ksi (AASHTO LRFD ) Bounded by: 0.65 β β c A s f y c 1.98in 0.85f c β 1 b Depth of Cross Section under Compression under Ultimate Load (AASHTO LRFD Eq ) This "c" is the distance from the extreme compression fiber to the neutral axis, not the distance from the center of bearing of the last girder to the end of the cap. a cβ 1 a 1.68in Depth of Equivalent Stress Block (AASHTO LRFD ) Note: "a" is less than "d ledge " therefore the equivalent stress block acts over a rectangular area. If "a" was greater than "d ledge " it would act over a Tee shaped area. a 1ft M n A s f y d M 2 12in n kipft d c ε s c ε s Nominal Flexural Resistance (AASHTO LRFD Eq ) Strain in Reinforcing at Ultimate ε s > FlexureBehavior "Tension Controlled" (AASHTO LRFD ) ϕ M 0.90 (AASHTO LRFD ) M r ϕ M M n M r kipft Factored Flexural Resistance (AASHTO LRFD Eq ) M f kipft < M r MinReinfChk "OK!" M u 852.1kipft < M r UltimateMom "OK!" LRFD Inverted Tee Bent Cap Design Example 24 June 2010

25 Flexural Reinforcement for Negative Bending (Con't) (Bars A) Check Serviceability (AASHTO LRFD ) To find s max : Modular Ratio: E s n n 8.39 E c For service loads, the stress on the cross-section is located as drawn: Tension Reinforcement Ratio: A s ρ bd ρ k dk ( 2ρn) ( ρn) 2 ( ρn) k in < d ledge 28.00in k j 1 j Therefore, the compression force acts over a rectangular area. If the compression force does not act over a rectangular area, j will not be 1-k/3. f ss M s 12in f A s j d 1ft ss 11.65ksi f a 0.6f y f a 36.00ksi f ss < f a ServiceStress "OK!" 1 d c cover 2 d 1 stirrup 2 d bar_a d c 3.58in Exposure Condition Factor: γ e 1.00 d c β s 1 β 0.7 h cap d s 1.06 c s max = minimum of: Service Load Bending Stress in outer layer of the reinforcing Allowable Bending Stress in the outer layer of the reinforcing (BDM-LRFD Ch. 4, Sect. 5, Design Criteria) "cover" is measured to center of shear reinforcement. For class 1 exposure conditions. For areas where deicing chenicals are frequently used, design for Class 2 Exposure ( e = 0.75). (BDM-LRFD, Ch. 4, Sect. 5, Design Criteria) 700γ e β s f ss 2d c 49.38in (AASHTO LRFD Eq ) & 12in A good practice is to place a bar every 12in along each surface of the s max 12.00in bent. (TxSP) 1 b stem 2 cover 2 d 1 stirrup 2 d bar_a s Actual BarANo 1 s Actual 7.96in < s max ServiceabilityCheck "OK!" Check Dead Load Check allowable M dl : f dl 22ksi BDM-LRFD, Chapter 4, Section 5, Design Criteria 1ft M a A s dj f dl M 12in a kipft TxDOT limits dead load stress to 22 ksi. This is due to observed cracking under dead load. Allowable Dead Load Moment M dl kipft < M a DeadLoadMom "OK!" LRFD Inverted Tee Bent Cap Design Example 25 June 2010

26 Flexural Reinforcement for Positive Bending (Bars B) (Tension in Bottom) M dl M posdl M dl 250.1kipft M s M posserv M s 492.5kipft From Cap 18 Output. See Pg. 10 M u M posult M u 741.7kipft Minimum Flexural Reinforcement (AASHTO LRFD ) Factored Flexural Resistance, M r, must be greater than or equal to the lesser of 1.2 M cr (Cracking Moment) or 1.33 Mu (Ultimate Moment) y t S M cr ybar y t 33.80in I g S in 3 y t 1ft Sf r M 12in cr kipft M f = minimum of: 1.2M cr 1.33M u kipft 986.5kipft Distance to the Center of Gravity of the Cap from the top of the Cap See Pg. 6 for calculations of "ybar" Section Modulus for the extreme tension fiber Cracking Moment (AASHTO LRFD Eq ) Design for the lesser of 1.2M cr or 1.33Mu when determining minimum area of steel required. Thus, M r must be greater than M f 986.5kipft Moment Capacity Design (AASHTO LRFD ) Try, 11 ~ #11's Bottom BarBNo 11 d bar_b 1.41in A bar_b 1.56in 2 Number of bars in tension Diameter of main reinforcing bars Area of one main reinforcing bar A s ( BarBNo) A bar_b A s 17.16in 2 Area of steel in tension d d s_pos d 81.42in See Pg. 18 for the calculation of "d s_pos." b b stem b 39in See Pg. 3 for the calculation of "b stem." c A s f y c 10.15in 0.85f c β 1 b Depth of Cross Section under Compression under Ultimate Load (AASHTO LRFD Eq ) This "c" is the distance from the extreme compression fiber to the neutral axis, not the distance from the center of bearing of the last girder to the end of the cap. a cβ 1 a 8.63in Depth of Equivalent Stress Block (AASHTO LRFD ) Note: "a" is less than "d stem " therefore the equivalent stress block acts over a rectangular area. If "a" was greater than "d stem " it would act over a Tee shaped area. M n a 1ft A s f y d M 2 12in n kipft Nominal Flexural Resistance (AASHTO LRFD Eq ) LRFD Inverted Tee Bent Cap Design Example 26 June 2010

27 Flexural Reinforcement for Positive Bending (Con't) (Bars B) Moment Capacity Design (Con't) d c ε s c ε s > FlexureBehavior ε s Strain in Reinforcing at Ultimate "Tension Controlled" (AASHTO LRFD ) ϕ M 0.90 (AASHTO LRFD ) M r ϕ M M n M r kipft Factored Flexural Resistance (AASHTO LRFD Eq ) M u 741.7kipft < M r MinReinfChk "OK!" M f 986.5kipft < M r UltimateMom "OK!" Check Serviceability (AASHTO LRFD ) To find s max : 1 d c cover 2 d 1 stirrup 2 d bar_b d c 3.58in "cover" is measured to center of shear reinforcement. Tension Reinforcement Ratio: A s ρ ρ bd For service loads, the stress on the cross-section is located as drawn: k dk ( 2ρn) ( ρn) 2 ( ρn) k in < d stem 57.00in Therefore, the compression force acts over a rectangular area. k j 1 j f ss M s 12in f A s j d 1ft ss 4.63ksi f a 0.6f y f a 36.00ksi f ss < f a ServiceStress "OK!" Exposure Condition Factor: γ e 1.00 d c β s 1 β 0.7 h cap d s 1.06 c s max = minimum of: & 12in s max 12.00in 700γ e 2d β s f c ss in Service Load Bending Stress in outer layer of the reinforcing Allowable Bending Stress in the outer layer of the reinforcing (BDM-LRFD Ch. 4, Sect. 5, Design Criteria) For class 1 exposure conditions. For areas where deicing chenicals are frequently used, design for Class 2 Exposure ( e = 0.75). (BDM-LRFD, Ch. 4, Sect. 5, Design Criteria) (AASHTO LRFD Eq ) A good practice is to place a bar every 12in along each surface of the bent. (TxSP) LRFD Inverted Tee Bent Cap Design Example 27 June 2010

28 Flexural Reinforcement for Positive Bending (Con't) (Bars B) Check Serviceability (Con't) Bars Inside Stirrup Bar S Try: BarBInsideSNo 5 1 b stem 2cover 2 d bar_s s Actual BarBInsideSNo d bar_b Number of bars B that are inside Stirrup bar S. s Actual 7.96in < s max ServiceabilityCheck "OK!" Bars Outside Stirrup Bar S BarBOutsideSNo BarBNo BarBInsideSNo BarBOutsideSNo 6 1 2b ledge 2 cover 2 d 1 bar_s 2 d bar_b cover s Actual BarBOutsideSNo Number of bars B that are Outside Stirrup bar S. 1 2 d 1 bar_m 2 d bar_b s Actual 8.67in < s max ServiceabilityCheck "OK!" Check Dead Load BDM-LRFD, Chapter 4, Section 5, Design Criteria Check allowable M dl : f dl 22ksi 1ft M a A s dj f dl M 12in a kipft TxDOT limits dead load stress to 22 ksi. This is due to observed cracking under dead load. Allowable Dead Load Moment M dl kipft < M a DeadLoadMom "OK!" Flexural Steel Summary: Use 5~#11 Bars on Top & 11~#11 Bars on Bottom LRFD Inverted Tee Bent Cap Design Example 28 June 2010

29 Ledge Reinforcement (Bars M&N) Try Bars M at a 4.95" spacing, and Bars N at a 9.90" spacing. s bar_m s bar_n 4.95in 9.90in Determine Distribution Widths Use trial and error to determine the spacing needed for the ledge reinforcing. Bars 5" and Bars 10" failed one of the checks by a small margin (1%). 4.95" spacing was chosen because it shoud pass all the checks. It is typical for Bars N to be at every other Bar M. These distribution widths will be used on the following three pages to determine the required ledge reinforcement per foot of cap. Distribution Width for Shear (AASHTO LRFD ) Note: These are the same distribution widths used for the Shear Friction Interior Girders check on Pg. 22. b s_int = minimum of: W 4 a v 69.00in b s_int S 69.00in 96.00in "S" is the girder spacing. (From Pg. 6) Exterior Girders b s_ext = minimum of: W 4 a v 69.00in b s_ext S 96.00in 2 c 48.00in 48.00in "c" is the distance from the center of bearing of the outside beam to the end of the ledge. (From Pg. 6) Distribution Width for Bending and Axial Loads (AASHTO LRFD ) Interior Girders b m_int = minimum of: b m_int W 5 a f S Exterior Girders 93.50in 96.00in b m_ext = minimum of: b m_ext W 5 a f S 96.00in 2 c 48.00in 48.00in 93.50in 93.50in LRFD Inverted Tee Bent Cap Design Example 29 June 2010

30 Ledge Reinforcement (Con't) (Bars M&N) The reinforcing required for shear friction, flexure, and axial tension will be calculated on Pgs The reinforcing provided by Bars M and Bars N must exceed the checks that are found on the bottom of Pg. 32 (These checked can be found on pages 33 and 34). These checks combine the reinforcing requirements for shear friction, flexure, and axial tension, as per AASHTO LRFD Reinforcing Required for Shear Friction (AASHTO LRFD ) ϕ 0.9 (AASHTO LRFD ) μ 1.4 c 1 0ksi P c 0kip Recall: d e 25.5in (From Pg. 18) Minimum Reinforcing (AASHTO LRFD ) A vf_min A cv d e b s a vf_min Interior Girders 0.05 ksia cv f y and 0.05 ksid e f y A vf a vf b s A cv d e b s_int A cv 1759in 2 V u_int "" is 1.4 for monolithically placed concrete. (AASHTO LRFD ) For clarity, the cohesion factor is labeled "c 1 ". This is to prevent confusion with "c", the distance from the last girder to the edge of the cap. c 1 is 0ksi for corbels and ledges. (AASHTO LRFD ) "P c " is zero as there is no axial compression. 322kip From Pg. 16 a vf_min 0.26 in2 ft Minimum Reinforcing required for Shear Friction V n c 1 A cv μ A vf f y P c (AASHTO LRFD Eq ) ϕv n Vu ϕ c 1 A cv μ A vf f y P c μ V u A vf V u_int ϕ c 1 A cv P c f y A vf 4.26in 2 (AASHTO LRFD Eq & AASHTO LRFD Eq ) Required Reinforcing for Shear Friction a vf_int A vf a b vf_int 0.74 in2 s_int ft Required Reinforcing for Shear Friction per foot length of cap Exterior Girders A cv V u_ext d e b s_ext A cv 1224in 2 322kip From Pg. 16 V n c 1 A cv μ A vf f y P c (AASHTO LRFD Eq ) ϕv n Vu ϕ c 1 A cv μ A vf f y P c μ V u A vf V u_ext ϕ c 1 A cv P c f y A vf 4.26in 2 (AASHTO LRFD Eq & AASHTO LRFD Eq ) Required Reinforcing for Shear Friction a vf_ext A vf b s_ext a vf_ext 1.06 in2 ft Required Reinforcing for Shear Friction per foot length of cap LRFD Inverted Tee Bent Cap Design Example 30 June 2010

31 Ledge Reinforcement (Con't) (Bars M&N) Reinforcing Required for Flexure (AASHTO LRFD ) Recall: h 29.50in d e 25.5in a v 12.00in From Pg. 18 Interior Girders V u_int N uc_int M u_int 322kip From Pg V u_int N uc_int 64.37kip (AASHTO LRFD ) V u_int a v N uc_int h d e M u_int 343.3kipft (AASHTO LRFD Eq ) Use the below equations to solve for A f : ϕm n M u_int (AASHTO LRFD Eq ) a M n A f f y d e (AASHTO LRFD Eq ) 2 (AASHTO LRFD Eq ) A f f y This "c" is the distance from the c 0.85f extreme compression fiber to the c β 1 b m_int neutral axis, not the distance from the β center of bearing of the last girder to the end of the cap. a cβ 1 (AASHTO LRFD Eq ) d e ϕ = Bounded by: 0.75 ϕ 0.9 (AASHTO LRFD ) c Solve for A f : A f 3.03in 2 Required Reinforcing for Flexure A f a f_int a b f_int 0.39 in2 m_int ft Exterior Girders Required Reinforcing for Flexure per foot length of cap V u_ext 322kip From Pg. 16 N uc_ext 0.2V u_ext N uc_ext 64.37kip (AASHTO LRFD ) M u_ext V u_ext a v N uc_ext h d e M u_ext 343.3kipft (AASHTO LRFD Eq ) Use the below equations to solve for A f : ϕm n M u_ext (AASHTO LRFD Eq ) a M n A f f y d e (AASHTO LRFD Eq ) 2 (AASHTO LRFD Eq ) A f f y This "c" is the distance from the c 0.85f extreme compression fiber to the c β 1 b m_ext neutral axis, not the distance from the β center of bearing of the last girder to the end of the cap. a cβ 1 (AASHTO LRFD Eq ) d e ϕ = Bounded by: 0.75 ϕ 0.9 (AASHTO LRFD ) c Solve for A f : A f 3.07in 2 Required Reinforcing for Flexure a f_ext A f a b f_ext 0.77 in2 m_ext ft Required Reinforcing for Flexure per foot length of cap LRFD Inverted Tee Bent Cap Design Example 31 June 2010

32 Ledge Reinforcement (Con't) (Bars M&N) Reinforcing Required for Axial Tension (AASHTO LRFD ) ϕ 0.9 (AASHTO LRFD ) Interior Girders N uc_int 64.37kip From Pg. 31 A n N uc_int A ϕf n 1.19in 2 y Required Reinforcing for Axial Tension (AASHTO LRFD Eq ) a n_int A n a b n_int 0.15 in2 m_int ft Required Reinforcing for Axial Tension per foot length of cap Exterior Girders N uc_ext 64.37kip From Pg. 31 A n N uc_ext A ϕf n 1.19in 2 y Required Reinforcing for Axial Tension (AASHTO LRFD Eq ) a n_ext A n a b n_ext 0.30 in2 m_ext ft Required Reinforcing for Axial Tension per foot length of cap Minimum Reinforcing (AASHTO LRFD ) f c a s_min 0.04 d f e a s_min 0.73 in2 y ft Minimum Required Reinforcing (AASHTO LRFD ) Check Required Reinforcing (AASHTO LRFD ) Actual Reinforcing: a s A bar_m a s s 1.07 in2 bar_m ft Primary Ledge Reinforcing Provided A bar_n a h a s h 0.53 in2 bar_n ft Checks: A s Auxiliary Ledge Reinforcing Provided The area of one bar M and N can be found on Pg. 18. The spacing of bars M and N can be found on Pg. 29. A s_min (AASHTO LRFD ) A s A f A n (AASHTO LRFD ) 2A vf A s A 3 n (AASHTO LRFD Eq ) A h 0.5 A s A n (AASHTO LRFD Eq ) LRFD Inverted Tee Bent Cap Design Example 32 June 2010

33 Ledge Reinforcement (Con't) (Bars M&N) Check Required Reinforcing (Con't) The following checks bars M and N. These are the checks found on Pg. 32, modified to check the steel per unit length instead of the total steel. Check Interior Girders Bar M: Check if: a s a s_min (AASHTO LRFD ) a s a f_int a n_int (AASHTO LRFD ) 2a vf_int a s a 3 n_int (AASHTO LRFD Eq ) a s 1.07 in2 ft a s_min 0.73 in2 < a ft s a f_int a n_int 0.54 in2 < a ft s 2a vf_int 3 a n_int 0.65 in2 < a ft s BarMCheck "OK!" Bar N: Check if: a h 0.5 a s a n_int (AASHTO LRFD Eq ) a s = The maximum of: a s a f_int a n_int 2a vf_int in2 ft a n_int "a s " in this equation is the steel required for Bar M, based on the requirements for Bar M in AASHTO LRFD This is derived from the suggestion that A h should not be less than A f /2 nor less than A vf /3 (Furlong & Mirza pg. 73 & 74) a h 0.53 in2 ft 0.5 a s a n_int 0.19 in2 < a ft h BarNCheck "OK!" LRFD Inverted Tee Bent Cap Design Example 33 June 2010

34 Ledge Reinforcement (Con't) Check Required Reinforcing (Con't) Check Exterior Girders Bar M (Bars M&N) Check if: a s a s_min (AASHTO LRFD ) a s a f_ext a n_ext (AASHTO LRFD ) 2a vf_ext a s a 3 n_ext (AASHTO LRFD Eq ) a s 1.07 in2 ft a s_min 0.73 in2 < a ft s a f_ext a n_ext 1.06 in2 < a ft s 2a vf_ext 3 a n_ext 1.01 in2 < a ft s BarMCheck "OK!" Bar N Check if: a h 0.5 a s a n_int a s = The maximum of: a s a f_int a n_int 2a vf_int in2 ft a n_int (AASHTO LRFD Eq ) "a s " in this equation is the steel required for Bar M, based on the requirements for Bar M in AASHTO LRFD This is derived from the suggestion that A h should not be less than A f /2 nor less than A vf /3 (Furlong & Mirza pg. 73 & 74) a h 0.53 in2 ft 0.5 a s a n_int 0.46 in2 < a ft h BarNCheck "OK!" Ledge Reinforcement Summary: Use #6 primary ledge 4.95" maximum spacing & #6 auxiliary ledge 9.90" maximum spacing LRFD Inverted Tee Bent Cap Design Example 34 June 2010

35 Hanger Reinforcement (Bars S) Try Double # 6 Stirrups at a 8" spacing. s bar_s A hr 8.5in 2stirrupsA bar_s A hr 0.88in 2 A v 2legsA hr A v 1.76in 2 Check Minimum Transverse Reinforcement b v b stem b v 39.00in Use trial and error to determine the spacing needed for the hanger reinforcing. A hr is the area of one leg of hanger reinforcement. In this example the area of the bar is multiplied by 2 because the Inverted Tee has double stirrups. A v_min = f c b v s bar_s A f v_min 0.33in 2 (AASHTO LRFD Eq ) y A v 1.76in 2 > A v_min MinimumSteelCheck "OK!" Check Service Limit State (AASHTO LRFD with modifications from BDM-LRFD Ch.4, Sect. 5, Design Criteria) Interior Girders V all V all V s_int = minimum of: 2 A hr 3 f y W 3a s v bar_s 2 A hr 3 f y s bar_s 236kip Exterior Girders V all V all V s_ext S 398kip 236kip 215kip < V all ServiceCheck "OK!" = minimum of: V all 2 A hr 3 f y s bar_s 2 A hr 3 f y s bar_s 217kip for the interior girder W S 2 2 c 3a v c 298kip 217kip 215kip < V all ServiceCheck "OK!" TxDOT uses " 2 / 3 f y " from the original research (Furlong & Mirza Eq. 5.4) instead of "0.5 f y " from AASHTO LRFD Eq (BDM-LRFD Ch.4, Sect. 5, Design Criteria) (BDM-LRFD Ch.4, Sect. 5, Design Criteria ~ Modified to limit the distribution width to the girder spacing. This will prevent distribution widths from overlapping) (BDM-LRFD Ch.4, Sect. 5, Design Criteria ~ Modified to limit the distribution width to the edge of the cap. This will prevent distribution widths from extending over the edge of the cap.) (BDM-LRFD Ch.4, Sect. 5, Design Criteria ~ Modified to limit the distribution width to half the girder spacing and the distance to the edge of the cap. This will prevent distribution widths from overlapping or extending over the edge of the cap.) LRFD Inverted Tee Bent Cap Design Example 35 June 2010

36 Hanger Reinforcement (Con't) (Bars S) Check Strength Limit State ϕ 0.9 (AASHTO LRFD ) (AASHTO LRFD ) Interior Girders V n = minimum of: A hr f y S 596kip (AASHTO LRFD Eq ) s bar_s f c b f d f A hr f y W 2d s f bar_s 699kip (AASHTO LRFD Eq ) V n ϕv n V u_int 596kip 537kip 322kip < ϕv n UltimateCheck "OK!" Exterior Girders V n = minimum of: V n for the interior girder V n ϕv n A hr f y s bar_s V u_ext S 2 c f c b f d f 447kip 403kip 447kip (AASHTO LRFD Eq ) Modified to limit the distribution width to the edge of the cap. A hr f y W 2d f s bar_s 2 c 322kip < ϕv n UltimateCheck "OK!" 632kip (AASHTO LRFD Eq ) Modified to limit the distribution width to the edge of the cap. LRFD Inverted Tee Bent Cap Design Example 36 June 2010

37 Hanger Reinforcement (Con't) (Bars S) Combined Shear and Torsion The following calculations are for Station 36. All critical locations must be checked. See the the Concrete Section Shear Capacity spreadsheet in the appendices for calculations at other locations. Shear and Moments were calculated using the CAP 18 program. M u 335.6kipft V u 448.1kip N u 0kip T u 660kipft These loads can be found on Pg. 16 Recall: β f y 60ksi f c 3.6ksi E s 29000ksi b f 91in h cap 85in b stem 39in b v b stem b v 39.00in Find d v : A s A bar_a BarANo A s 7.80in 2 The Moment calculations to the left reiterate those on Pg. 24. c A s f y c 1.98in 0.85f c β 1 b f a cβ 1 a 1.68in d s d s_neg d 81.42in Shears are maximum near the column faces. In these regions the cap is in negative bending with tension in the top of the cap. Therefore, the calculations are based on the steel in the top of the bent cap. a M n A s f y d M 2 n kipft A ps 0in 2 d e A ps f ps d p A s f y d s d A ps f ps A s f e 81.42in (AASHTO LRFD Eq ) y d v need not be less than the greater of 0.9d e and 0.72h: (AASHTO LRFD ) d v = maximum of: A s f y M n A ps f ps 80.58in (AASHTO LRFD Eq. C ) d v 0.9d e 0.72h 80.58in 73.28in 21.24in The method for calculating and used in this design example are from AASHTO LRFD Appendix B5. The method from AASHTO LRFD may be used instead. The method from is based on the method from Appendix B5; however, it is less accurate and more conservative (often excessively conservative). The method from Apendix B5 is preferred because it is more accurate, but it requires iterating to a solution. The method from can be used when doing calculations by hand. LRFD Inverted Tee Bent Cap Design Example 37 June 2010

38 Hanger Reinforcement (Con't) (Bars S) Check Combined Shear and Torsion (Con't) Determine and : ε x ϕ v 0.9 (AASHTO LRFD ) V u ϕ v V p v u v ϕ v b v d u 0.16ksi v v u 0.04 f c Using Table B5.2-1 with θ M u d v where, M u 36.4 deg and β N u 0.5 V u V p 2 E s A s E p A ps v u 0.04 and ε f x c cot( θ) kipft Must be > V u V p ε x in > in in in A ps f po d v kipft Shear Stress on the Concrete (AASHTO LRFD Eq ) Determining and is an iterative process, therefore, assume initial shear strain value x of per LRFD B5.2 and then verify that the assumption was valid. Strain halfway between the compressive and tensile resultants (AASHTO LRFD Eq. B5.2-1) If x < 0, then use equation B5.2-3 and re-solve for x. use ε x in For values of x greater than 0.001, in the tensile strain in the reinforcing, t is greater than ( t = 2 x - c, where c is < 0) Grade 60 steel yields at a strain of 60 ksi / 29,000 ksi = By limiting the tensile strain in the steel to the yield strain and using the Modulus of Elasticity of the steel prior to yield, this limits the tensile stress of the steel to the yield stress. x has not changed from the assumed value, therefore no iterations are required. V p 0kip h cap A c b stem A 2 c in 2 s s bar_s s 8.50in "V p " is zero as there is no prestressing. (AASHTO LRFD B5.2) "A c " is the area of concrete on the flexural tension side of the cap, from the extreme tension fiber to one half the cap depth. "A c " is needed if AASHTO LRFD Eq. B5.2-1 is negative. LRFD Inverted Tee Bent Cap Design Example 38 June 2010

39 Hanger Reinforcement (Con't) (Bars S) Check Combined Shear and Torsion (Con't) A v 2 legs2 stirrupsa bar_s A v 1.76in 2 The transverse reinforcement, "A v ", is double closed stirrups. The failure surface intersects four stirrup legs, therefore the area of the shear steel is four times the stirrup bar's area (0.44in 2 ). See the sketch of the failure plane to the left. A t A oh 1legA bar_s A t 0.44in 2 b f 2cover d stem b stem 2cover d ledge 2 cover A oh 3916in 2 "A t " is the area of the outer stirrup. A t is not independent of A v, but it is one leg of the outer stirrup of A v. "A oh " is the area inside the centerline of the exterior stirrup. A o 0.85A oh A o 3329in 2 2b ledge p h b stem 2cover p h 332.0in b f 2cover 2 h cap 2 cover (AASHTO LRFD C ) A o = 0.85 * the area inside the centerline of the exterior stirrup. "p h " is the perimeter of the centerline of Bar S. Equivalent Shear Force 2 V u_eq V u 0.9p h 2A o T u 2 V u_eq 572.0kip (AASHTO LRFD Eq ) Shear Steel Required V n = the lesser of: V c V s V p (AASHTO LRFD Eq ) 0.25f c b v d v check maximum ϕv n for section: V p (AASHTO LRFD Eq ) ϕv n_max ϕ 0.25f c b v d v V p ϕv n_max 2546kip V u 448.1kip < ϕv n_max MaxShearCheck "OK!" LRFD Inverted Tee Bent Cap Design Example 39 June 2010

40 Hanger Reinforcement (Con't) (Bars S) Check Combined Shear and Torsion (Con't) Shear Steel Required (Con't) calculate required shear steel: V u ϕv n (AASHTO LRFD Eq ) V c = β V u ϕ V c V s V p f c b v d v V c 420kip (AASHTO LRFD Eq ) V s = a v_req A v f y d v ( cot( θ) cot( α) ) sin( α) (AASHTO LRFD Eq ) s req V u V ϕ c V p v a f y d v ( cot( θ) cot( α) ) sin( α) v_req 0.14 in2 ft Torsional Steel Required ϕ T 0.9 (AASHTO LRFD ) T u ϕ T T n (AASHTO LRFD Eq ) 2A o A t f y cot( θ) T n (AASHTO LRFD Eq ) s bar_s a t_req T u a ϕ T 2A o f y cot( θ) t_req 0.19 in2 ft Total Required Transverse Steel (AASHTO LRFD ) a req a v_req 2 sides a t_req a req 0.53 in2 ft A v a prov a s prov 2.48 in2 bar_s ft a prov > a req TransverseSteelCheck "OK" The transverse reinforcement is designed for the side of the section where the effects of shear and torsion are additive. (AASHTO LRFD C ) Both sides of the section need this reinforcing because the torsion can act on either side of the section. LRFD Inverted Tee Bent Cap Design Example 40 June 2010

41 Hanger Reinforcement (Con't) (Bars S) Maximum Spacing of Transverse Reinforcement (AASHTO LRFD ) Shear Stress V u ϕ v V p v u v ϕ v b v d u 0.158ksi (AASHTO LRFD Eq ) v 0.125f c 0.450ksi if v u 0.125f c, s max = minimum of: (AASHTO LRFD Eq ) 0.8d v & 24in 64.46in if v u 0.125f c, s max = minimum of: (AASHTO LRFD Eq ) 0.4d v & 12in Since v u < 0.125*f c, s max 24.00in 32.23in TxDOT limits the maximum transverse reinforcement spacing to 12", therefore: (BDM-LRFD, Ch. 4, Sect. 4, Detailing) s max 12.00in s bar_s 8.50in < s max SpacingCheck "OK!" Hanger Reinforcement Summary: Use double #6 8.5" maximum spacing Vertical End Reinforcement (Bars C) (BDM-LRFD, Ch. 4, Sect. 5, Detailing) Extra vertical reinforcing across end surfaces of the stem is provided to resist cracking which has been observed in existing bridges. (BDM-LRFD, Ch. 4, Sect. 5, Detailing) We will place #5 bars at approximately 6 in spacing. (TxSP) See the Bent Cap Details in the Appendices for detail of Bar C. Use 6~#5 bars evenly spaced across each end of the stem LRFD Inverted Tee Bent Cap Design Example 41 June 2010

42 Skin Reinforcement (Bars T) Try 7~#6 bars in stem and 3~#6 bars in Ledge on each side A bar_t 0.44in 2 NoTBarsStem 7 NoTBarsLedge 3 "a" must be within 2 / 3 of d e (AASHTO LRFD ) 2 3 d e 17.00in TxDOT typically uses: a 6in (TxSP) Required Area of Skin Reinforcement A sk_req = 0.012( d 30) (AASHTO LRFD Eq ) A sk_req 0.62 in2 ft A sk need not be greater than one quarter of the main reinforcing (A s /4) per side face within d/2 of the main reinforcing. (AASHTO LRFD Eq ) A sk_max = maximum of: A sk_max A bar_a BarANo 4 A bar_b BarBNo in2 ft d s_neg 2 d s_pos in2 ft ft in2 A skreq = minimum of: A sk_req A sk_max 0.62 in2 ft ft in2 A skreq 0.62 in2 ft LRFD Inverted Tee Bent Cap Design Example 42 June 2010

43 Skin Reinforcement (Con't) (Bars T) Required Spacing of Skin Reinforcement s req = minimum of: (AASHTO LRFD ) A bar_t A skreq d s_neg 6 d s_pos in 13.57in 13.57in & 12in s req 8.56in Actual Spacing of Skin Reinforcement Check T bars Spacing in Stem: h top d stem cover d bar_s 2 d bar_a 2 cover d bar_m 2 d bar_t 2 "cover" is measured to center of shear reinforcement. h top 56.67in s skstem h top s NoTBarsStem 1 skstem 7.08in s req 8.56in > s skstem SkinSpacing "OK!" Check T bars Spacing in Ledge: h bot d ledge cover d bar_m 2 d bar_t 2 cover d bar_s 2 d bar_b 2 "cover" is measured to center of shear reinforcement. h bot 21.17in h bot a s skledge s NoTBarsLedge 1 skledge 7.58in s req 8.56in > s skledge SkinSpacing "OK!" Check if "a" dimension is less than or equal to s req : a 6.00in < s req 8.56in Skin Reinforcement Summary: SkinSpacing "OK!" Use 7~#6 bars in stem and 3~#6 bars in Ledge on each side See the Bent Cap Detail Sheet in the Appendices for the resulting design of all the preceding calculations. LRFD Inverted Tee Bent Cap Design Example 43 June 2010

44 Appendices Bridge Layout... Pg. 45 CAP 18 Input File CAP 18 Output File Pg. 46 Pg. 47 Live Load Distribution Factor Spreadsheet... Pg. 70 Concrete Section Shear Capacity Spreadsheet... Pg. 86 Bent Cap Details... Pg. 87 LRFD Inverted Tee Bent Cap Design Example 44 June 2010

45 LRFD Inverted Tee Bent Cap Design Example 45 June 2010

46 00001 County Highwy Pro# XXXX-XX-XXX BRG Comment CAP18 Version 6.10 Inverted Tee Cap Design Example, Skew = E (Spans 54'-112'-54', Type TX54 8.0', 8" Slab, 2" O'lay) Table Table E Table (Lane Left) (Lane Right) (Stringers) (Supports) (Mom CP) (Mom CP) 86 (Shear CP) Table 4 (Cap) E (DL Span1, Bm1) (DL Span1, Bm2) (DL Span1, Bm3) (DL Span1, Bm4) (DL Span1, Bm5) (DL Span1, Bm6) (DL Span2, Bm1) (DL Span2, Bm2) (DL Span2, Bm3) (DL Span2, Bm4) (DL Span2, Bm5) (DL Span2, Bm6) (Dist. Lane Ld) (Conc. Lane Ld) (Conc. Lane Ld) LRFD Inverted Tee Bent Cap Design Example 46 June 2010

47 APR 22, 2010 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 1 CAP18 BENT CAP ANALYSIS Ver 6.1 PSF HIGHWAY PD- CONTROL- CODED NO COUNTY NO IPE SECTION-JOB BY DATE County Highwy Pro# XXXX-XX-XXX BRG APR 22, 2010 Comment CAP18 Version 6.10 Inverted Tee Cap Design Example, Skew = 0.00 PROB 1 (Spans 54'-112'-54', Type TX54 8.0', 8" Slab, 2" O'lay) ENGLISH SYSTEM UNITS TABLE 1. CONTROL DATA ENVELOPES TABLE NUMBER OF MAXIMUMS KEEP FROM PRECEDING PROBLEM (1=YES) CARDS INPUT THIS PROBLEM 16 OPTION TO CLEAR ENVELOPES BEFORE LANE LOADINGS (1=YES) 0 OPTION TO OMIT PRINT (-1=TABLE 4A, -2=TABLE 5, -3=BOTH) 0 SKEW ANGLE, DEGREES TABLE 2. CONSTANTS NUMBER OF INCREMENTS FOR SLAB AND CAP 92 INCREMENT LENGTH, FT NUMBER OF INCREMENTS FOR MOVABLE LOAD 20 START POSITION OF MOVABLE-LOAD STA ZERO 2 STOP POSITION OF MOVABLE-LOAD STA ZERO 70 NUMBER OF INCREMENTS BETWEEN EACH POSITION OF MOVABLE LOAD 1 ANALYSIS OPTION (1=WORKING STRESS, 2=LOAD FACTOR, 3=BOTH) 3 LOAD FACTOR FOR DEAD LOAD 1.25 LOAD FACTOR FOR OVERLAY LOAD 1.50 LOAD FACTOR FOR LIVE LOAD 1.75 MAXIMUM NUMBER OF LANES TO BE LOADED SIMULTANEOUSLY 3 LIST OF LOAD COEFFICIENTS CORRESPONDING TO NUMBER OF LANES LOADED LRFD Inverted Tee Bent Cap Design Example 47 June 2010

48 APR 22, 2010 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 2 CAP18 BENT CAP ANALYSIS Ver 6.1 PROB 1 (Spans 54'-112'-54', Type TX54 8.0', 8" Slab, 2" O'lay) (CONTINUED) TABLE 3. LISTS OF STATIONS NUM OF NUM OF NUM OF NUM MOM NUM SHEAR LANES STRINGERS SUPPORTS CONTR PTS CONTR PTS TOTAL LANE LEFT LANE RIGHT STRINGERS SUPPORTS MOM CONTR SHEAR CONTR TABLE 4. STIFFNESS AND LOAD DATA FIXED-OR-MOVABLE FIXED-POSITION DATA MOVABLE- STA STA CONTD CAP BENDING SIDEWALK, STRINGER, OVERLAY POSITION FROM TO IF=1 STIFFNESS SLAB LOADS CAP LOADS LOADS SLAB LOADS ( K-FT*FT ) ( K ) ( K ) ( K ) ( K ) LRFD Inverted Tee Bent Cap Design Example 48 June 2010

49 APR 22, 2010 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 3 CAP18 BENT CAP ANALYSIS Ver 6.1 PROB 1 (Spans 54'-112'-54', Type TX54 8.0', 8" Slab, 2" O'lay) (CONTINUED) TABLE 4A. DEAD LOAD RESULTS ( WORKING STRESS ) STA DIST X (FT) DEFLECTION (FT) MOMENT (K-FT) SHEAR (K) LRFD Inverted Tee Bent Cap Design Example 49 June 2010

50 APR 22, 2010 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 4 CAP18 BENT CAP ANALYSIS Ver 6.1 PROB 1 (Spans 54'-112'-54', Type TX54 8.0', 8" Slab, 2" O'lay) (CONTINUED) TABLE 4A. DEAD LOAD RESULTS ( WORKING STRESS ) STA DIST X (FT) DEFLECTION (FT) MOMENT (K-FT) SHEAR (K) LRFD Inverted Tee Bent Cap Design Example 50 June 2010

51 APR 22, 2010 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 5 CAP18 BENT CAP ANALYSIS Ver 6.1 PROB 1 (Spans 54'-112'-54', Type TX54 8.0', 8" Slab, 2" O'lay) (CONTINUED) TABLE 4A. DEAD LOAD RESULTS ( WORKING STRESS ) STA DIST X (FT) DEFLECTION (FT) MOMENT (K-FT) SHEAR (K) LRFD Inverted Tee Bent Cap Design Example 51 June 2010

52 APR 22, 2010 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 6 CAP18 BENT CAP ANALYSIS Ver 6.1 PROB 1 (Spans 54'-112'-54', Type TX54 8.0', 8" Slab, 2" O'lay) (CONTINUED) TABLE 5. MULTI-LANE LOADING SUMMARY ( WORKING STRESS ) ( *--CRITICAL NUMBER OF LANE LOADS) MOMENT ( FT-K ) AT DEAD LD LANE POSITIVE LOAD AT LANE NEGATIVE LOAD AT STA EFFECT ORDER MAXIMUM LANE STA ORDER MAXIMUM LANE STA * 0* * 0* * 0* * 2* * 0* * 2* LRFD Inverted Tee Bent Cap Design Example 52 June 2010

53 APR 22, 2010 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 7 CAP18 BENT CAP ANALYSIS Ver 6.1 PROB 1 (Spans 54'-112'-54', Type TX54 8.0', 8" Slab, 2" O'lay) (CONTINUED) MOMENT ( FT-K ) AT DEAD LD LANE POSITIVE LOAD AT LANE NEGATIVE LOAD AT STA EFFECT ORDER MAXIMUM LANE STA ORDER MAXIMUM LANE STA * 0* * 2* * 0* * 0* * 0* LRFD Inverted Tee Bent Cap Design Example 53 June 2010

54 APR 22, 2010 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 8 CAP18 BENT CAP ANALYSIS Ver 6.1 PROB 1 (Spans 54'-112'-54', Type TX54 8.0', 8" Slab, 2" O'lay) (CONTINUED) SHEAR ( K ) AT DEAD LD LANE POSITIVE LOAD AT LANE NEGATIVE LOAD AT STA EFFECT ORDER MAXIMUM LANE STA ORDER MAXIMUM LANE STA * 0* * 0* * 0* * 0* * 2* * 0* LRFD Inverted Tee Bent Cap Design Example 54 June 2010

55 APR 22, 2010 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 9 CAP18 BENT CAP ANALYSIS Ver 6.1 PROB 1 (Spans 54'-112'-54', Type TX54 8.0', 8" Slab, 2" O'lay) (CONTINUED) SHEAR ( K ) AT DEAD LD LANE POSITIVE LOAD AT LANE NEGATIVE LOAD AT STA EFFECT ORDER MAXIMUM LANE STA ORDER MAXIMUM LANE STA * 0* * 0* LRFD Inverted Tee Bent Cap Design Example 55 June 2010

56 APR 22, 2010 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 10 CAP18 BENT CAP ANALYSIS Ver 6.1 PROB 1 (Spans 54'-112'-54', Type TX54 8.0', 8" Slab, 2" O'lay) (CONTINUED) REACTION ( K ) AT DEAD LD LANE POSITIVE LOAD AT LANE NEGATIVE LOAD AT STA EFFECT ORDER MAXIMUM LANE STA ORDER MAXIMUM LANE STA * 0* * 0* * 0* * 0* LRFD Inverted Tee Bent Cap Design Example 56 June 2010

57 APR 22, 2010 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 11 CAP18 BENT CAP ANALYSIS Ver 6.1 PROB 1 (Spans 54'-112'-54', Type TX54 8.0', 8" Slab, 2" O'lay) (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 ) LRFD Inverted Tee Bent Cap Design Example 57 June 2010

58 APR 22, 2010 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 12 CAP18 BENT CAP ANALYSIS Ver 6.1 PROB 1 (Spans 54'-112'-54', Type TX54 8.0', 8" Slab, 2" O'lay) (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 ) LRFD Inverted Tee Bent Cap Design Example 58 June 2010

59 APR 22, 2010 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 13 CAP18 BENT CAP ANALYSIS Ver 6.1 PROB 1 (Spans 54'-112'-54', Type TX54 8.0', 8" Slab, 2" O'lay) (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 ) LRFD Inverted Tee Bent Cap Design Example 59 June 2010

60 APR 22, 2010 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 14 CAP18 BENT CAP ANALYSIS Ver 6.1 PROB 1 (Spans 54'-112'-54', Type TX54 8.0', 8" Slab, 2" O'lay) (CONTINUED) TABLE 7. MAXIMUM SUPPORT REACTIONS ( WORKING STRESS ) STA DIST X MAX + REACT MAX - REACT ( FT ) ( K ) ( K ) LRFD Inverted Tee Bent Cap Design Example 60 June 2010

61 APR 22, 2010 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 15 CAP18 BENT CAP ANALYSIS Ver 6.1 PROB 1 (Spans 54'-112'-54', Type TX54 8.0', 8" Slab, 2" O'lay) (CONTINUED) TABLE 5. MULTI-LANE LOADING SUMMARY ( LOAD FACTOR) ( *--CRITICAL NUMBER OF LANE LOADS) MOMENT ( FT-K ) AT DEAD LD LANE POSITIVE LOAD AT LANE NEGATIVE LOAD AT STA EFFECT ORDER MAXIMUM LANE STA ORDER MAXIMUM LANE STA * 0* * 0* * 0* * 2* * 0* * 2* LRFD Inverted Tee Bent Cap Design Example 61 June 2010

62 APR 22, 2010 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 16 CAP18 BENT CAP ANALYSIS Ver 6.1 PROB 1 (Spans 54'-112'-54', Type TX54 8.0', 8" Slab, 2" O'lay) (CONTINUED) MOMENT ( FT-K ) AT DEAD LD LANE POSITIVE LOAD AT LANE NEGATIVE LOAD AT STA EFFECT ORDER MAXIMUM LANE STA ORDER MAXIMUM LANE STA * 0* * 2* * 0* * 0* * 0* LRFD Inverted Tee Bent Cap Design Example 62 June 2010

63 APR 22, 2010 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 17 CAP18 BENT CAP ANALYSIS Ver 6.1 PROB 1 (Spans 54'-112'-54', Type TX54 8.0', 8" Slab, 2" O'lay) (CONTINUED) SHEAR ( K ) AT DEAD LD LANE POSITIVE LOAD AT LANE NEGATIVE LOAD AT STA EFFECT ORDER MAXIMUM LANE STA ORDER MAXIMUM LANE STA * 0* * 0* * 0* * 0* * 2* * 0* LRFD Inverted Tee Bent Cap Design Example 63 June 2010

64 APR 22, 2010 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 18 CAP18 BENT CAP ANALYSIS Ver 6.1 PROB 1 (Spans 54'-112'-54', Type TX54 8.0', 8" Slab, 2" O'lay) (CONTINUED) SHEAR ( K ) AT DEAD LD LANE POSITIVE LOAD AT LANE NEGATIVE LOAD AT STA EFFECT ORDER MAXIMUM LANE STA ORDER MAXIMUM LANE STA * 0* * 0* LRFD Inverted Tee Bent Cap Design Example 64 June 2010

65 APR 22, 2010 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 19 CAP18 BENT CAP ANALYSIS Ver 6.1 PROB 1 (Spans 54'-112'-54', Type TX54 8.0', 8" Slab, 2" O'lay) (CONTINUED) REACTION ( K ) AT DEAD LD LANE POSITIVE LOAD AT LANE NEGATIVE LOAD AT STA EFFECT ORDER MAXIMUM LANE STA ORDER MAXIMUM LANE STA * 0* * 0* * 0* * 0* LRFD Inverted Tee Bent Cap Design Example 65 June 2010

66 APR 22, 2010 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 20 CAP18 BENT CAP ANALYSIS Ver 6.1 PROB 1 (Spans 54'-112'-54', Type TX54 8.0', 8" Slab, 2" O'lay) (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 ) LRFD Inverted Tee Bent Cap Design Example 66 June 2010

67 APR 22, 2010 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 21 CAP18 BENT CAP ANALYSIS Ver 6.1 PROB 1 (Spans 54'-112'-54', Type TX54 8.0', 8" Slab, 2" O'lay) (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 ) LRFD Inverted Tee Bent Cap Design Example 67 June 2010

68 APR 22, 2010 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 22 CAP18 BENT CAP ANALYSIS Ver 6.1 PROB 1 (Spans 54'-112'-54', Type TX54 8.0', 8" Slab, 2" O'lay) (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 ) LRFD Inverted Tee Bent Cap Design Example 68 June 2010

69 APR 22, 2010 TEXAS DEPARTMENT OF TRANSPORTATION (TxDOT) PAGE 23 CAP18 BENT CAP ANALYSIS Ver 6.1 PROB 1 (Spans 54'-112'-54', Type TX54 8.0', 8" Slab, 2" O'lay) (CONTINUED) TABLE 7. MAXIMUM SUPPORT REACTIONS ( LOAD FACTOR ) STA DIST X MAX + REACT MAX - REACT ( FT ) ( K ) ( K ) LRFD Inverted Tee Bent Cap Design Example 69 June 2010

70 TXDOT BRIDGE DIVISION County: ANY Highway: Any Design: BRG Date: 2/2/ LRFD Specs C-S-J: XXX-XX-XXXX ID #: XXXX Ck Dsn: BRG Date: Rev. 8/08 - (No Interim) Descrip: Inverted Tee Design Example, Spans 1 & 3 File: distribution_factors_i.xls Sheet: 1 of 8 LRFD Live Load Distribution Factors* Live Load Distribution Factors are calculated according to AASHTO LRFD Bridge Design Specifications, 4th Edition (2007 with no interim revisions) as prescribed by TxDOT policies (LRFD Design Manual July 2008) and practices. The Lever Rule is used when outside the Range of Applicability. The Range of Applicability for the Skew Correction Factors is ignored. INPUT: Beam Type = Tx54 Deck Slab Beam No. Beams, N b = 6 Conc wt = k/ft 3 weight = k/ft 3 CL brg to CL brg, L = ft f' c = 4.0 ksi f' c = 8.5 ksi Beam Spacing, S = 8.00 ft E slab = 3644 ksi E beam = 5312 ksi Avg. Skew Angle, = 0.00 deg y t = in Slab Thickness, t s = 8 in A = in 2 Slab Overhang, OH = 3 ft I = in 4 Rail Width, RW = 1 ft Roadway Width, W = 44 ft Number of Lanes, N L = Longitudinal Stiffness Parameter: ( ) e g (in) = (dist. b/w cog of bm & deck) n = K g = n(i+ae 2 g ) = in 4 *For typical cross sections (a,e,i,j & k). Table RESULTS: Final LLDF Interior Shear LLDF, gv interior Interior Moment LLDF, gm interior Exterior Shear LLDF, gv exterior Exterior Moment LLDF, gm exterior The Final LLDF may be modified according to the following TxDOT policies: Exterior beams use the interior LLDF when OH S/2. When OH > S/2 the exterior beam LLDF is determined by the lever rule for a single lane with a multiple presence factor of 1.0. In no case shall the LLDF for the exterior beams be less than the LLDFs for the interior beams. When the Roadway width is less than 20ft, all beams are designed for one lane loaded only. In no case shall the LLDF be less than m N L N b. CALCULATIONS: Shear LLDF Correction for Skew (Table c-1) Check : OK Check S: 3.5' 8.0' 16.0' OK Corr. = Lts tan Check L: 20' 50.4' 240' OK K g Check N b : 6 4 OK = * [(12.0*50.4*8^3)/(1,271,611)]^0.3 * tan(0) Corr. = Moment LLDF Correction for Skew (Table e-1) Check 0 < 30 Set = Corr. = 1 - c 1 (tan )^ K g = 1-0(tan 0)^1.5 where: c 1 = 0.25 S Lt L s Corr. = c 1 = because < 30 LRFD Inverted Tee Bent Cap Design Example 70 June 2010

71 TXDOT BRIDGE DIVISION County: ANY Highway: Any Design: BRG Date: 2/2/ LRFD Specs C-S-J: XXX-XX-XXXX ID #: XXXX Ck Dsn: BRG Date: Rev. 8/08 - (No Interim) Descrip: Inverted Tee Design Example, Spans 1 & 3 File: distribution_factors_i.xls Sheet: 2 of 8 INTERIOR BEAM: Shear LL Distribution Per Lane (Table a-1): One Lane Loaded Lever Rule (Table ) mg = * 1.2 = Modify for Skew: skew correction = mg = * = Equation S g = g = (8 / 25) = Modify for Skew: skew correction = g = * = Range of Applicability (ROA) Checks Check S: 3.5' 8.0' 16.0' OK Check t s : 4.5'' 8.0'' 12.0'' OK Check L: 20' 50.4' 240' OK Check N b : 6 4 OK Use Equation from Table a-1 because all criteria is OK. gv int1 = Two or More Lanes Loaded Lever Rule (Table ) mg = Max(0.875 * 1.0, * 0.85, * 0.65) = Modify for Skew: skew correction = mg = * = Equation 2.0 S S g = g = (8 / 12) - (8 / 35)^2.0 = Modify for Skew: skew correction = g = * = Range of Applicability (ROA) Checks (same as for one lane loaded) Use Equation from Table a-1 because all criteria is OK. gv int2+ = TxDOT Policy states gv Interior must be m N L N b m N L N b = 0.85 * 3 / 6 = Is W 20ft? Yes TxDOT Policy states that if W < 20ft, gv Interior is the Maximum of: gv int1 and m N L N b. >> TxDOT Policy states that if W 20ft, gv Interior is the Maximum of: gv int1, gv int2+, m N L N b. gv interior = LRFD Inverted Tee Bent Cap Design Example 71 June 2010

72 TXDOT BRIDGE DIVISION County: ANY Highway: Any Design: BRG Date: 2/2/ LRFD Specs C-S-J: XXX-XX-XXXX ID #: XXXX Ck Dsn: BRG Date: Rev. 8/08 - (No Interim) Descrip: Inverted Tee Design Example, Spans 1 & 3 File: distribution_factors_i.xls Sheet: 3 of 8 INTERIOR BEAM: Moment LL Distribution Per Lane (Table b-1): One Lane Loaded Lever Rule (Table ) mg = * 1.2 = Modify for Skew: skew correction = mg = * = Equation S S K g g = L 12 Lt s g = (8/14)^0.4 * (8/50.4)^0.3 * (1,271,611/(12*50.4*8^3))^0.1 = Modify for Skew: skew correction = g = * = Range of Applicability (ROA) Checks Check S: 3.5' 8.0' 16.0' OK Check t s : 4.5'' 8.0'' 12.0'' OK Check L: 20' 50.4' 240' OK Check N b : 6 4 OK Check K g : 10,000 1,271,611 7,000,000 OK Use Equation from Table b-1 because all criteria is OK. gm int1 = Two or More Lanes Loaded Lever Rule (Table ) mg = Max(0.875 * 1.0, * 0.85, * 0.65) = Modify for Skew: skew correction = mg = * = Equation S S K g g = L 12 Lts g = (8/9.5)^0.6 * (8/50.4)^0.2 * (1,271,611/(12*50.4*8^3))^0.1 = Modify for Skew: skew correction = g = * = Range of Applicability (ROA) Checks (same as for one lane loaded) Use Equation from Table b-1 because all criteria is OK. gm int2+ = TxDOT Policy states gm Interior must be m N L N b m N L N b = 0.85 * 3 / 6 = Is W 20ft? Yes TxDOT Policy states that if W < 20ft, gm Interior is the Maximum of: gm int1 and m N L N b. >> TxDOT Policy states that if W 20ft, gm Interior is the Maximum of: gm int1, gm int2+, m N L N b. gm interior = LRFD Inverted Tee Bent Cap Design Example 72 June 2010

73 TXDOT BRIDGE DIVISION County: ANY Highway: Any Design: BRG Date: 2/2/ LRFD Specs C-S-J: XXX-XX-XXXX ID #: XXXX Ck Dsn: BRG Date: Rev. 8/08 - (No Interim) Descrip: Inverted Tee Design Example, Spans 1 & 3 File: distribution_factors_i.xls Sheet: 4 of 8 EXTERIOR BEAM: Shear LL Distribution Per Lane (Table b-1): One Lane Loaded Lever Rule (Table ) mg = * 1.0 = TxDOT uses a multiple presence factor of 1.0 for one lane Modify for Skew: loaded on the exterior beam. skew correction = mg = * = Use Lever Rule, as per AASHTO LRFD Table b-1. gv ext1 = Two or More Lanes Loaded Lever Rule (Table ) mg = Max(0.625 * 1.0, * 0.85, * 0.65) = Modify for Skew: skew correction = mg = * = Equation d e = dist. b/w CL web to curb d e = OH - Rail Width d e = 3ft - 1ft = 2.0 ft d e = 0.6 e 10 e = (2.0/10) = g = e*gv int2+eq g = * = Skew Correction is included in gv(interior). Range of Applicability (ROA) Checks Interior ROA is implicitly applied to the exterior beam. Check Interior Beam ROA: OK Check d e : -1.0' 2.0' 5.5' OK Check N b : 6 3 OK Use Equation from Table b-1 because all criteria is OK. gv ext2+ = TxDOT Policy states gv Exterior must be gv interior gv interior = TxDOT Policy states gv Exterior must be m N L N b m N L N b = 0.85 * 3 / 6 = Is OH S/2? Yes Is W 20ft? Yes >> TxDOT Policy states that if OH S/2, gv Exterior is gv interior. TxDOT Policy states that if OH > S/2 and W < 20ft, gv Exterior is the Maximum of: gv ext1, gv interior, and m N L N b. TxDOT Policy states that if OH > S/2 ans W 20ft, gv Exterior is the Maximum of: gv ext1, gv ext2+, gv interior, and m N L N b. gv exterior = LRFD Inverted Tee Bent Cap Design Example 73 June 2010

74 TXDOT BRIDGE DIVISION County: ANY Highway: Any Design: BRG Date: 2/2/ LRFD Specs C-S-J: XXX-XX-XXXX ID #: XXXX Ck Dsn: BRG Date: Rev. 8/08 - (No Interim) Descrip: Inverted Tee Design Example, Spans 1 & 3 File: distribution_factors_i.xls Sheet: 5 of 8 EXTERIOR BEAM: Moment LL Distribution Per Lane (Table d-1): One Lane Loaded Lever Rule mg = * 1.0 = TxDOT uses a multiple presence factor of 1.0 for one lane Modify for Skew: loaded on the exterior beam. skew correction = mg = * = Use Lever Rule as per AASHTO LRFD Table d-1. gm ext1 = Two or More Lanes Loaded Lever Rule (Table ) mg = Max(0.625 * 1.0, * 0.85, * 0.65) = Modify for Skew: skew correction = mg = * = Equation d e = 0.77 e 9.1 e = (2.0/9.1) = g = e*gm int2+eq g = 0.99 * = Skew Correction included in gm(interior). Range of Applicability (ROA) Checks Interior ROA is implicitly applied to the exterior beam. Check Interior Beam ROA: OK Check d e : -1.0' 2.0' 5.5' OK Check N b : 6 3 OK Use Equation from Table d-1 because all criteria is OK. gm ext2+ = TxDOT Policy states gm Exterior must be gm interior gm interior = TxDOT Policy states gm Exterior must be m N L N b m N L N b = 0.85 * 3 / 6 = Is OH S/2? Yes Is W 20ft? Yes >> TxDOT Policy states that if OH S/2, gm Exterior is gm interior. TxDOT Policy states that if OH > S/2 and W < 20ft, gm Exterior is the Maximum of: gm ext1, gm interior, and m N L N b. TxDOT Policy states that if OH > S/2 ans W 20ft, gm Exterior is the Maximum of: gm ext1, gm ext2+, gm interior, and m N L N b. gm exterior = LRFD Inverted Tee Bent Cap Design Example 74 June 2010

75 TXDOT BRIDGE DIVISION County: ANY Highway: Any Design: BRG Date: 2/2/ LRFD Specs C-S-J: XXX-XX-XXXX ID #: XXXX Ck Dsn: BRG Date: Rev. 8/08 - (No Interim) Descrip: Inverted Tee Design Example, Spans 1 & 3 File: distribution_factors_i.xls Sheet: 6 of 8 LEVER RULE S = 8.0 ft INTERIOR For S < 4: 16 One Lane = 32 = For 4 S < 6: 16 One Lane = 32 = Two Lanes = 16 S S = >>For 6 S < 10: 16 S 6 One Lane = 1 32 S = Two Lanes = 16 S 6 S S S = For 10 S < 12: 16 S 6 One Lane = 1 32 S = Two Lanes = 16 S 6 S 4 S S S S = For 12 S < 16: 16 S 6 One Lane = 1 32 S = Two Lanes = 16 S 6 S 4 S S S S = Three Lanes = 16 S 6 S 4 S 10 S S S S S = For 16 S < 18: 16 S 6 One Lane = 1 32 S = Two Lanes = 16 S 6 S 4 S S S S = Three Lanes = 16 S 6 S 4 S 10 S S S S S = Four Lanes = 16 S 6 S 4 S 10 S 12 S S S S S S = LRFD Inverted Tee Bent Cap Design Example 75 June 2010

76 TXDOT BRIDGE DIVISION County: ANY Highway: Any Design: BRG Date: 2/2/ LRFD Specs C-S-J: XXX-XX-XXXX ID #: XXXX Ck Dsn: BRG Date: Rev. 8/08 - (No Interim) Descrip: Inverted Tee Design Example, Spans 1 & 3 File: distribution_factors_i.xls Sheet: 7 of 8 LEVER RULE S = 8.0 ft INTERIOR (con't) For 18 S < 22: 16 S 6 One Lane = 1 32 S = Two Lanes = 16 S 6 S 4 S S S S = Three Lanes = 16 S 6 S 4 S 10 S 12 S S S S S S = Four Lanes = 16 S 6 S 4 S 10 S 12 S 18 S S S S S S S = For 22 S 24: 16 S 6 One Lane = 1 32 S = Two Lanes = 16 S 6 S 4 S S S S = Three Lanes = 16 S 6 S 4 S 10 S 12 S S S S S S = Four Lanes = 16 S 6 S 4 S 10 S 12 S 18 S 16 S S S S S S S S = EXTERIOR S = OH = Rail Width = RW = X = S+OH-RW-2ft = 8.0 ft 3.0 ft 1.0 ft 8.0 ft For X < 6: 16 X One Lane = 32 S = >>For 6 X < 12: 16 X X 6 One Lane = 32 S S = For 12 X < 18: 16 X X 6 One Lane = 32 S S = Two Lanes = 16 X X 6 X S S S = LRFD Inverted Tee Bent Cap Design Example 76 June 2010

77 TXDOT BRIDGE DIVISION County: ANY Highway: Any Design: BRG Date: 2/2/ LRFD Specs C-S-J: XXX-XX-XXXX ID #: XXXX Ck Dsn: BRG Date: Rev. 8/08 - (No Interim) Descrip: Inverted Tee Design Example, Spans 1 & 3 File: distribution_factors_i.xls Sheet: 8 of 8 LEVER RULE EXTERIOR (con't) S = 8.0 ft OH = 3.0 ft RW = 1.0 ft X = S+OH-RW-2ft = 8.0 ft For 18 X < 24: 16 X X 6 One Lane = 32 S S = Two Lanes = 16 X X 6 X 12 X S S S S = For 24 X < 30: 16 X X 6 One Lane = 32 S S = Two Lanes = 16 X X 6 X 12 X S S S S = Three Lanes = 16 X X 6 X 12 X 18 X S S S S S = For 30 X < 36: 16 X X 6 One Lane = 32 S S = Two Lanes = 16 X X 6 X 12 X S S S S = Three Lanes = 16 X X 6 X 12 X 18 X 24 X S S S S S S = For 36 X < 42: 16 X X 6 One Lane = 32 S S = Two Lanes = 16 X X 6 X 12 X S S S S = Three Lanes = 16 X X 6 X 12 X 18 X 24 X S S S S S S = Four Lanes = 16 X X 6 X 12 X 18 X 24 X 30 X S S S S S S S = For 42 X 48: 16 X X 6 One Lane = 32 S S = Two Lanes = 16 X X 6 X 12 X S S S S = Three Lanes = 16 X X 6 X 12 X 18 X 24 X S S S S S S = Four Lanes = 16 X X 6 X 12 X 18 X 24 X 30 X 36 X 42 = S S S S S S S S INTERIOR EXTERIOR One Lane Loaded = One Lane Loaded = Two Lanes Loaded = Two Lanes Loaded = Three Lanes Loaded = Three Lanes Loaded = Four Lanes Loaded = Four Lanes Loaded = LRFD Inverted Tee Bent Cap Design Example 77 June 2010

78 TXDOT BRIDGE DIVISION County: ANY Highway: Any Design: BRG Date: 2/2/ LRFD Specs C-S-J: XXX-XX-XXXX ID #: XXXX Ck Dsn: BRG Date: Rev. 8/08 - (No Interim) Descrip: Inverted Tee Design Example, Span 2 File: distribution_factors_i.xls Sheet: 1 of 8 LRFD Live Load Distribution Factors* Live Load Distribution Factors are calculated according to AASHTO LRFD Bridge Design Specifications, 4th Edition (2007 with no interim revisions) as prescribed by TxDOT policies (LRFD Design Manual July 2008) and practices. The Lever Rule is used when outside the Range of Applicability. The Range of Applicability for the Skew Correction Factors is ignored. INPUT: Beam Type = Tx54 Deck Slab Beam No. Beams, N b = 6 Conc wt = k/ft 3 weight = k/ft 3 CL brg to CL brg, L = ft f' c = 4.0 ksi f' c = 8.5 ksi Beam Spacing, S = 8.00 ft E slab = 3644 ksi E beam = 5312 ksi Avg. Skew Angle, = 0.00 deg y t = in Slab Thickness, t s = 8 in A = in 2 Slab Overhang, OH = 3 ft I = in 4 Rail Width, RW = 1 ft Roadway Width, W = 44 ft Number of Lanes, N L = Longitudinal Stiffness Parameter: ( ) e g (in) = (dist. b/w cog of bm & deck) n = K g = n(i+ae 2 g ) = in 4 *For typical cross sections (a,e,i,j & k). Table RESULTS: Final LLDF Interior Shear LLDF, gv interior Interior Moment LLDF, gm interior Exterior Shear LLDF, gv exterior Exterior Moment LLDF, gm exterior The Final LLDF may be modified according to the following TxDOT policies: Exterior beams use the interior LLDF when OH S/2. When OH > S/2 the exterior beam LLDF is determined by the lever rule for a single lane with a multiple presence factor of 1.0. In no case shall the LLDF for the exterior beams be less than the LLDFs for the interior beams. When the Roadway width is less than 20ft, all beams are designed for one lane loaded only. In no case shall the LLDF be less than m N L N b. CALCULATIONS: Shear LLDF Correction for Skew (Table c-1) Check : OK Check S: 3.5' 8.0' 16.0' OK Corr. = Lts tan Check L: 20' 106.8' 240' OK K g Check N b : 6 4 OK = * [(12.0*106.8*8^3)/(1,271,611)]^0.3 * tan(0) Corr. = Moment LLDF Correction for Skew (Table e-1) Check 0 < 30 Set = Corr. = 1 - c 1 (tan )^ K g = 1-0(tan 0)^1.5 where: c 1 = 0.25 S Lt L s Corr. = c 1 = because < 30 LRFD Inverted Tee Bent Cap Design Example 78 June 2010

79 TXDOT BRIDGE DIVISION County: ANY Highway: Any Design: BRG Date: 2/2/ LRFD Specs C-S-J: XXX-XX-XXXX ID #: XXXX Ck Dsn: BRG Date: Rev. 8/08 - (No Interim) Descrip: Inverted Tee Design Example, Span 2 File: distribution_factors_i.xls Sheet: 2 of 8 INTERIOR BEAM: Shear LL Distribution Per Lane (Table a-1): One Lane Loaded Lever Rule (Table ) mg = * 1.2 = Modify for Skew: skew correction = mg = * = Equation S g = g = (8 / 25) = Modify for Skew: skew correction = g = * = Range of Applicability (ROA) Checks Check S: 3.5' 8.0' 16.0' OK Check t s : 4.5'' 8.0'' 12.0'' OK Check L: 20' 106.8' 240' OK Check N b : 6 4 OK Use Equation from Table a-1 because all criteria is OK. gv int1 = Two or More Lanes Loaded Lever Rule (Table ) mg = Max(0.875 * 1.0, * 0.85, * 0.65) = Modify for Skew: skew correction = mg = * = Equation 2.0 S S g = g = (8 / 12) - (8 / 35)^2.0 = Modify for Skew: skew correction = g = * = Range of Applicability (ROA) Checks (same as for one lane loaded) Use Equation from Table a-1 because all criteria is OK. gv int2+ = TxDOT Policy states gv Interior must be m N L N b m N L N b = 0.85 * 3 / 6 = Is W 20ft? Yes TxDOT Policy states that if W < 20ft, gv Interior is the Maximum of: gv int1 and m N L N b. >> TxDOT Policy states that if W 20ft, gv Interior is the Maximum of: gv int1, gv int2+, m N L N b. gv interior = LRFD Inverted Tee Bent Cap Design Example 79 June 2010

80 TXDOT BRIDGE DIVISION County: ANY Highway: Any Design: BRG Date: 2/2/ LRFD Specs C-S-J: XXX-XX-XXXX ID #: XXXX Ck Dsn: BRG Date: Rev. 8/08 - (No Interim) Descrip: Inverted Tee Design Example, Span 2 File: distribution_factors_i.xls Sheet: 3 of 8 INTERIOR BEAM: Moment LL Distribution Per Lane (Table b-1): One Lane Loaded Lever Rule (Table ) mg = * 1.2 = Modify for Skew: skew correction = mg = * = Equation S S K g g = L 12 Lt s g = (8/14)^0.4 * (8/106.8)^0.3 * (1,271,611/(12*106.8*8^3))^0.1 = Modify for Skew: skew correction = g = * = Range of Applicability (ROA) Checks Check S: 3.5' 8.0' 16.0' OK Check t s : 4.5'' 8.0'' 12.0'' OK Check L: 20' 106.8' 240' OK Check N b : 6 4 OK Check K g : 10,000 1,271,611 7,000,000 OK Use Equation from Table b-1 because all criteria is OK. gm int1 = Two or More Lanes Loaded Lever Rule (Table ) mg = Max(0.875 * 1.0, * 0.85, * 0.65) = Modify for Skew: skew correction = mg = * = Equation S S K g g = L 12 Lts g = (8/9.5)^0.6 * (8/106.8)^0.2 * (1,271,611/(12*106.8*8^3))^0.1 = Modify for Skew: skew correction = g = * = Range of Applicability (ROA) Checks (same as for one lane loaded) Use Equation from Table b-1 because all criteria is OK. gm int2+ = TxDOT Policy states gm Interior must be m N L N b m N L N b = 0.85 * 3 / 6 = Is W 20ft? Yes TxDOT Policy states that if W < 20ft, gm Interior is the Maximum of: gm int1 and m N L N b. >> TxDOT Policy states that if W 20ft, gm Interior is the Maximum of: gm int1, gm int2+, m N L N b. gm interior = LRFD Inverted Tee Bent Cap Design Example 80 June 2010

81 TXDOT BRIDGE DIVISION County: ANY Highway: Any Design: BRG Date: 2/2/ LRFD Specs C-S-J: XXX-XX-XXXX ID #: XXXX Ck Dsn: BRG Date: Rev. 8/08 - (No Interim) Descrip: Inverted Tee Design Example, Span 2 File: distribution_factors_i.xls Sheet: 4 of 8 EXTERIOR BEAM: Shear LL Distribution Per Lane (Table b-1): One Lane Loaded Lever Rule (Table ) mg = * 1.0 = TxDOT uses a multiple presence factor of 1.0 for one lane Modify for Skew: loaded on the exterior beam. skew correction = mg = * = Use Lever Rule, as per AASHTO LRFD Table b-1. gv ext1 = Two or More Lanes Loaded Lever Rule (Table ) mg = Max(0.625 * 1.0, * 0.85, * 0.65) = Modify for Skew: skew correction = mg = * = Equation d e = dist. b/w CL web to curb d e = OH - Rail Width d e = 3ft - 1ft = 2.0 ft d e = 0.6 e 10 e = (2.0/10) = g = e*gv int2+eq g = * = Skew Correction is included in gv(interior). Range of Applicability (ROA) Checks Interior ROA is implicitly applied to the exterior beam. Check Interior Beam ROA: OK Check d e : -1.0' 2.0' 5.5' OK Check N b : 6 3 OK Use Equation from Table b-1 because all criteria is OK. gv ext2+ = TxDOT Policy states gv Exterior must be gv interior gv interior = TxDOT Policy states gv Exterior must be m N L N b m N L N b = 0.85 * 3 / 6 = Is OH S/2? Yes Is W 20ft? Yes >> TxDOT Policy states that if OH S/2, gv Exterior is gv interior. TxDOT Policy states that if OH > S/2 and W < 20ft, gv Exterior is the Maximum of: gv ext1, gv interior, and m N L N b. TxDOT Policy states that if OH > S/2 ans W 20ft, gv Exterior is the Maximum of: gv ext1, gv ext2+, gv interior, and m N L N b. gv exterior = LRFD Inverted Tee Bent Cap Design Example 81 June 2010

82 TXDOT BRIDGE DIVISION County: ANY Highway: Any Design: BRG Date: 2/2/ LRFD Specs C-S-J: XXX-XX-XXXX ID #: XXXX Ck Dsn: BRG Date: Rev. 8/08 - (No Interim) Descrip: Inverted Tee Design Example, Span 2 File: distribution_factors_i.xls Sheet: 5 of 8 EXTERIOR BEAM: Moment LL Distribution Per Lane (Table d-1): One Lane Loaded Lever Rule mg = * 1.0 = TxDOT uses a multiple presence factor of 1.0 for one lane Modify for Skew: loaded on the exterior beam. skew correction = mg = * = Use Lever Rule as per AASHTO LRFD Table d-1. gm ext1 = Two or More Lanes Loaded Lever Rule (Table ) mg = Max(0.625 * 1.0, * 0.85, * 0.65) = Modify for Skew: skew correction = mg = * = Equation d e = 0.77 e 9.1 e = (2.0/9.1) = g = e*gm int2+eq g = 0.99 * = Skew Correction included in gm(interior). Range of Applicability (ROA) Checks Interior ROA is implicitly applied to the exterior beam. Check Interior Beam ROA: OK Check d e : -1.0' 2.0' 5.5' OK Check N b : 6 3 OK Use Equation from Table d-1 because all criteria is OK. gm ext2+ = TxDOT Policy states gm Exterior must be gm interior gm interior = TxDOT Policy states gm Exterior must be m N L N b m N L N b = 0.85 * 3 / 6 = Is OH S/2? Yes Is W 20ft? Yes >> TxDOT Policy states that if OH S/2, gm Exterior is gm interior. TxDOT Policy states that if OH > S/2 and W < 20ft, gm Exterior is the Maximum of: gm ext1, gm interior, and m N L N b. TxDOT Policy states that if OH > S/2 ans W 20ft, gm Exterior is the Maximum of: gm ext1, gm ext2+, gm interior, and m N L N b. gm exterior = LRFD Inverted Tee Bent Cap Design Example 82 June 2010

83 TXDOT BRIDGE DIVISION County: ANY Highway: Any Design: BRG Date: 2/2/ LRFD Specs C-S-J: XXX-XX-XXXX ID #: XXXX Ck Dsn: BRG Date: Rev. 8/08 - (No Interim) Descrip: Inverted Tee Design Example, Span 2 File: distribution_factors_i.xls Sheet: 6 of 8 LEVER RULE S = 8.0 ft INTERIOR For S < 4: 16 One Lane = 32 = For 4 S < 6: 16 One Lane = 32 = Two Lanes = 16 S S = >>For 6 S < 10: 16 S 6 One Lane = 1 32 S = Two Lanes = 16 S 6 S S S = For 10 S < 12: 16 S 6 One Lane = 1 32 S = Two Lanes = 16 S 6 S 4 S S S S = For 12 S < 16: 16 S 6 One Lane = 1 32 S = Two Lanes = 16 S 6 S 4 S S S S = Three Lanes = 16 S 6 S 4 S 10 S S S S S = For 16 S < 18: 16 S 6 One Lane = 1 32 S = Two Lanes = 16 S 6 S 4 S S S S = Three Lanes = 16 S 6 S 4 S 10 S S S S S = Four Lanes = 16 S 6 S 4 S 10 S 12 S S S S S S = LRFD Inverted Tee Bent Cap Design Example 83 June 2010

84 TXDOT BRIDGE DIVISION County: ANY Highway: Any Design: BRG Date: 2/2/ LRFD Specs C-S-J: XXX-XX-XXXX ID #: XXXX Ck Dsn: BRG Date: Rev. 8/08 - (No Interim) Descrip: Inverted Tee Design Example, Span 2 File: distribution_factors_i.xls Sheet: 7 of 8 LEVER RULE S = 8.0 ft INTERIOR (con't) For 18 S < 22: 16 S 6 One Lane = 1 32 S = Two Lanes = 16 S 6 S 4 S S S S = Three Lanes = 16 S 6 S 4 S 10 S 12 S S S S S S = Four Lanes = 16 S 6 S 4 S 10 S 12 S 18 S S S S S S S = For 22 S 24: 16 S 6 One Lane = 1 32 S = Two Lanes = 16 S 6 S 4 S S S S = Three Lanes = 16 S 6 S 4 S 10 S 12 S S S S S S = Four Lanes = 16 S 6 S 4 S 10 S 12 S 18 S 16 S S S S S S S S = EXTERIOR S = OH = Rail Width = RW = X = S+OH-RW-2ft = 8.0 ft 3.0 ft 1.0 ft 8.0 ft For X < 6: 16 X One Lane = 32 S = >>For 6 X < 12: 16 X X 6 One Lane = 32 S S = For 12 X < 18: 16 X X 6 One Lane = 32 S S = Two Lanes = 16 X X 6 X S S S = LRFD Inverted Tee Bent Cap Design Example 84 June 2010

85 TXDOT BRIDGE DIVISION County: ANY Highway: Any Design: BRG Date: 2/2/ LRFD Specs C-S-J: XXX-XX-XXXX ID #: XXXX Ck Dsn: BRG Date: Rev. 8/08 - (No Interim) Descrip: Inverted Tee Design Example, Span 2 File: distribution_factors_i.xls Sheet: 8 of 8 LEVER RULE EXTERIOR (con't) S = 8.0 ft OH = 3.0 ft RW = 1.0 ft X = S+OH-RW-2ft = 8.0 ft For 18 X < 24: 16 X X 6 One Lane = 32 S S = Two Lanes = 16 X X 6 X 12 X S S S S = For 24 X < 30: 16 X X 6 One Lane = 32 S S = Two Lanes = 16 X X 6 X 12 X S S S S = Three Lanes = 16 X X 6 X 12 X 18 X S S S S S = For 30 X < 36: 16 X X 6 One Lane = 32 S S = Two Lanes = 16 X X 6 X 12 X S S S S = Three Lanes = 16 X X 6 X 12 X 18 X 24 X S S S S S S = For 36 X < 42: 16 X X 6 One Lane = 32 S S = Two Lanes = 16 X X 6 X 12 X S S S S = Three Lanes = 16 X X 6 X 12 X 18 X 24 X S S S S S S = Four Lanes = 16 X X 6 X 12 X 18 X 24 X 30 X S S S S S S S = For 42 X 48: 16 X X 6 One Lane = 32 S S = Two Lanes = 16 X X 6 X 12 X S S S S = Three Lanes = 16 X X 6 X 12 X 18 X 24 X S S S S S S = Four Lanes = 16 X X 6 X 12 X 18 X 24 X 30 X 36 X 42 = S S S S S S S S INTERIOR EXTERIOR One Lane Loaded = One Lane Loaded = Two Lanes Loaded = Two Lanes Loaded = Three Lanes Loaded = Three Lanes Loaded = Four Lanes Loaded = Four Lanes Loaded = LRFD Inverted Tee Bent Cap Design Example 85 June 2010

86 County: Descrip: Highway: Any C-S-J: XXXX-XX-XXX Design: BRG Ck Dsn: BRG Bridge Division Rev: 09/26/08 Date: Apr-10 Resistance Factors: Units: US V = 0.9 M = 0.9 N = 0.75 Concrete: Mild Steel: Prestressed Steel: Input Data f'c = 3.6 ksi fy = 60 ksi fpu = 270 ksi Ec = 3453 ksi Es = ksi Ep = ksi SECTIONS Units Bending moment, Mu kip-ft Shear force, Vu kip Axial force, Nu (+ if tensile) kip Web width, bv in Shear depth, dv in Mild steel reinf. area, As in^ Conc area on tension side, Ac in^ Area of stirrups, Av in^ Stirrup spacing, s in Prestressed steel area, Aps Prestress shear, Vp Average prestress, fps in^2 kip ksi Any LRFD Inverted Tee Cap Design Example - Bent 2 CONCRETE SECTION SHEAR CAPACITY BY AASHTO LRFD BRIDGE DESIGN SPECIFICATIONS, FOURTH EDITION, 2007 Torsional moment, Tu kip-ft Shear flow area, Ao in^ Area of one leg of stirrup, At in^ Perimeter of stirrup, Ph in Calculated Values Vc kip Vs kip Vn kip x 1.00E E E E E E E E-03 deg Req'd Shear reinf. Av/S in^2/in Req'd Torsion reinf. At/S in^2/in Maximum stirrup spacing, Smax in Conclusion Shear Reinforcing OK OK OK OK OK OK OK OK Longitudinal Reinforcing FAILED OK OK FAILED FAILED OK OK FAILED 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 for further information. If torsion is not being considered, leave last five rows of input data blank. LRFD Inverted Tee Bent Cap Design Example 86 June 2010

87 LRFD Inverted Tee Bent Cap Design Example 87 June 2010

88 LRFD Inverted Tee Bent Cap Design Example 88 June 2010