Beam Data mp := 10. Beam length (ft) = length := 100. Composite slab strength (ksi) = fc := 4. Concrete unit weight (kcf) = γc := 0.

Transcription

1 LRFD pre-stressed beam.mcd 7//3 of 7 Beam Data mp := Beam length (ft) = length := Composite slab strength (ksi) = fc := 4 Concrete unit weight (kcf) = γc :=.5 Initial strength of concrete (ksi) = fci := 6 Final Strength of concrete (ksi) = fcf := 8 Modulus of beam concrete based on final (ksi) = Ec := 33 γc.5 fcf Ec = Modulus of slab concrete (ksi) = Esl:= 33 γc.5 fc Esl = Number of Spa = spa := n :=.. spa n :=.. Which span is used in design = comp := Length of all spa (ft) = L := n Should the haunch depth be used in calculatio (yes or no) = ha_dec := "yes" Depress point to use for draped strands = depress :=.4 Number of span points calculatio shall be done to = (Please choose only an even number of points) sp := :=.. Interior or Exterior beam used in design (intput "int" or "ext") = aa := "int"

2 LRFD pre-stressed beam.mcd 7//3 of 7 Beam type to use = AASHTO TYPE I = AASHTO TYPE II 3 = AASHTO TYPE III 4 = AASHTO TYPE IV 5 = BT54 6 = BT63 7 = BT7 type := 4 8 = IDOT 36 INCH 9 = IDOT 4 INCH = IDOT 48 INCH = IDOT 54 INCH = Box Box Beam dimeio (if no box set to zero) Width (in) = Depth (in) = Top flange (in) = Bottom Flange (in) = a := a := a3 := a4 := Web (in) = a5 := Beam area (in^) = Area = 789 Web thickness (in) = web = 8 Distance from bottom to cg (in) = yb = 4.73 Total beam depth (in) = h = 54 Section inertia (in^) = Inc = 673 Width of top flange (in) = fwt = Beam weight (k/ft) = bwt =.8

3 LRFD pre-stressed beam.mcd 7//3 3 of 7 Strand pattern Data strand := PICK Description DIAMETER AREA WEIGHT PER LENGTH Fpu STEEL TYPE TYPE english in in^ lb/ft ksi 6/-7k SR 6/-7k-LL LL 9/6-7k SR 3 9/6-7k-LL LL 4 /-7k SR 5 /-7k-LL LL 6 /-7k-SP LL 7 7/6-7k SR 8 7/6-7k-LL LL 9 3/8-7k SR 3/8-7k-LL LL Strand Type to use s_type := Strand_description := strand s_type, Strand_diameter := strand s_type, Strand_area := strand s_type, Strand_weight := strand s_type, 3 Strand_strength := strand s_type, 4 Strand_type := strand s_type, 5 Strand_description = "6/-7k-LL" Strand_diameter =.6 Strand_area =.7 Strand_weight =.745 Strand_strength = 7 Strand_type = "LL" Trafer length = 6*bd trafer := 6 Strand_diameter trafer = 36

4 LRFD pre-stressed beam.mcd 7//3 4 of 7 Calculatio of Dead Loads, non-composite and composite General Information Out to out width (ft) = oto := 4.5 Beam spacing (ft) = bs := 8 Slab thickness (ft) = slab := 8.5 ts := slab Wearing surface (ksf) = wear :=.5 Number of beams = beams := 5 Width of one lane (ft) = lane_width := Multiple presence factor = RF :=. Top slab to top beam (in) = tstw :=.75 Haunch Selection haunch := tstw slab haunch = 4.5 ha := if( ha_dec = "yes", haunch, ) ha = 4.5 Beam weight per foot (k/ft) = bwt =.8 Max span length (ft) = (for ETFW) Width of top flange of beam (in) = max_span := length fwt = max_span =

5 LRFD pre-stressed beam.mcd 7//3 5 of 7 RAIL OR PARAPET DATA Rail width on outside (ft) = outside :=. Rail weight per foot (k/ft) = railwt :=.5 Number of parapet's = npar := MEDIAN BARRIER DATA Median barrier width (ft) = med_width := Median barrier weight (k/ft) = median := Number of barriers = nmed := Diaphragm Data Weight of Diaphragms (k) = Number of Diaphragms (k) = Optional Loads wdia :=.664 ndia := Note: Program assumes diaphragms are point loads at equal spaces over the length of the beam. If you do not wish to use any of the optional loads then simply set the values to zero. If SIP metal forms will be used then the first three should probably be used. However, it is most certanly not necessary to adjust for the deck grooving. SIP form weight (psf) = sipw := 3 Depth of valley in SIP form (in) = vald := Amount of deflection in SIP form (in) = sipd :=.5 If the user so desires, you may adjust the deck weight for the deck grooving, just enter the depth of grooving. Enter a positive value for an increased thickness, and enter a negative value for an decreased thickness. This adjustment in really not necessary at all, and the user may set the value equal to. gt :=.5

6 LRFD pre-stressed beam.mcd 7//3 6 of 7 Filler weight (k/ft) = filler := fwt haunch 44 γc filler =.94 SIP form (k/ft) = say (3 psf) SIP := bs fwt sipw SIP =.9 Concrete in valley of SIP form (k/ft) = (say each inch of valley is equal to /" of concrete depth) valley := bs fwt vald γc 4 valley =.79 Weight from deflectio (k/ft) = (this assumes that the SIP form will deflect, adding about /" depth for every " of deflection) wdefl := bs fwt sipd γc 4 wdefl =. Deck grooving (k/ft) = (Say that the deck grooving adds /4" in depth) groov := bs gt γc 4 groov =.5 Total optional loads (k/ft) = optional := filler + SIP + valley + wdefl optional =. Final Composite and Non-Composite Loads NON COMPOSITE DL (excluding beam weight) (DLnc) (DC) DLnc := max oto slab beams bs slab γc γc + optional DLnc =.47 COMPOSITE DL (DW) Roadway width (ft) = roadway := oto npar outside med_width roadway = 38.5 DLc := roadway wear + railwt npar + median nmed beams + groov DLc =.47

7 LRFD pre-stressed beam.mcd 7//3 7 of 7 Unit Load for Diaphragm, to be used only for Deflectio (the actual point loads will be used for shear and moment) wdia ndia dwt := dwt =.33 length Unit weight to be used in in the calculation of Non-Composite DL Deflection railwt npar + median nmed w_defl := DLnc + + dwt beams

8 LRFD pre-stressed beam.mcd 7//3 8 of 7 Effective flange width (LRFD ) (use the smaller of interior or exterior) Interior - smaller of the following. /4 span length. center to center beams 3. *T+B ; B = larger of the web thickness or / top flange width etfw := length 4 etfw = 3 etfw := bs etfw = 96 etfw3 := slab + fwt etfw3 = 9 ETFW_int := min etfw etfw etfw3 ETFW_int = 96 Exterior - / effective width of adjacent interior beam plus the smaller of the following. /8 Effective Span. 6*ts + B ; B = largter of the web thickness or / top flange width 3. overhang etfw := length 8 etfw = 5 etfw := 6 slab + fwt etfw = 59.5 etfw3 := oto ( beams ) bs etfw3 = 5 ETFW_ext := min etfw etfw etfw3 ETFW_int = 96 Effective flange width used in design ETFW := ETFW_ext if aa = "ext" ETFW_int ETFW = 96

9 LRFD pre-stressed beam.mcd 7//3 9 of 7 Section Diagram 7 Section 6 5 beam xa, xh xhn, xe xhn, beam xa,, xh xhn,, xe xhn,

10 LRFD pre-stressed beam.mcd 7//3 of 7 Composite moment of Inertia Effective compression slab width (in) = ETFW = 96 Modular ratio = η := fc η =.77 fcf Traformed slab width (in) = b := ETFW η b = Slab thickness (in) = ts = 8.5 Composite distance from bottom to c.g. (in) = ts b ts h + ha + + Area yb ybc := ybc = 4.46 b ts + Area Composite N.A. to top beam (in) = ytb := h ybc ytb = Composite N.A. to top slab (in) = yts := h + ts + ha ybc yts = 6.88 b ts 3 Composite moment of inertia (in^t) = Ic := Inc + + Area ( yb ybc) + b ts yts ts Ic = Composite Section Modulus Section modulus bottom of beam (in^3) = Section modulus top beam (in^3) = Section modulus top concrete (in^3) = Ic Sbc := ybc Sbc = Ic Stb := ytb Stb = Ic Stc := yts η Stc = Non-Composite Section Modulus Section modulus bottom of beam (in^3) = Section modulus top beam (in^3) = Inc Sb := yb Sb = Inc St := h yb St =

11 LRFD pre-stressed beam.mcd 7//3 of 7 Live Load Distribution Factors LRFD Number of design lanes lanes := floor roadway lanes = 3 Table b- - Interior beam distribution factor Range of applicability ; 3.5 <= S <= <= ts <= <= L <= 4 Nb >= 4, <= Kg <= 7,, Distance from N.A. non composite beam and CL. deck (in) = eg := h + ts yb eg = kg := fcf ( Inc + Area eg ) fc kg = DFM I :=.75 + bs bs length. kg length slab 3. DFM I =.669 Table d- - Exterior beam distribution factor for Moment Range of applicibility -. <= de <= 5.5 de := oto ( beams ) bs de = 3.5 "OK" if (. de) ( de 5.5) "NG" = "OK" e := max de e =.7 DFM E := DFM I e DFM E =.754

12 LRFD pre-stressed beam.mcd 7//3 of 7 Distribution Factor for Moment Used in Design LLDFM := DFM I if aa = "int" LLDFM =.669 DFM E if aa = "ext" Table a- - Interior beam distribution factor for shear Range of applicibility: 3.5 <= S <= 6 <= L <= <= ts <= <= kg <= 7,, Nb >= 4. DFV I :=. + bs bs 35 DFV I =.84 Table b- - Exterior beam distribution factor for shear Range of applicibility - <= de <= 5.5 e := max de.6 +. e = DFV E := DFV I e DFV E =.84 Distribution Factor for Shear Used in Design LLDFV := DFV I if aa = "int" LLDFV =.84 DFV E if aa = "ext" If the user wants to overide the distribution factors that have been calculated, simply enable the two numbers below and imput the desired factor. Live Load distribution factor for moment LLDFM :=.66 currently disabled Live Load distribution factor for shear LLDFV :=.84 currently disabled

13 LRFD pre-stressed beam.mcd 7//3 3 of 7 ALLOWABLE STRESS IN CONCRETE At release Compression (ksi) = fic :=.6 fci fic = 3.6 Teiion (ksi) = fit :=. fci fit =.539 At final conditio Case I full PS + DL + LL Compression (ksi) = fc :=.6 fcf fc = 4.8 Teion (ksi) = fft :=.9 fcf fft =.537 Case II PS + DL Compression (ksi) = fc :=.45 fcf fc = 3.6 Teion (ksi) = fft =.537 Case III 5%PS + 5%DL + LL Compression (ksi) = fc3 :=.4 fcf fc3 = 3. Teion (ksi) = fft =.537

14 LRFD pre-stressed beam.mcd 7//3 4 of 7 Simple Span Shear and Moment Span length (ft) = length = Data range (ft) = rg := for j.. j length j j j Beam Weight (k/ft) = bwt =.8 bwt rg Self weight Moment at tenth points (k*ft) = Mself := ( ) length rg Self weight Shear (k) = Vself := bwt length rg Non composite moment (k*ft) = Mnonc := DLnc rg ( ) length rg Non composite shear (k) = Vnonc := DLnc length rg

15 LRFD pre-stressed beam.mcd 7//3 5 of 7 Shear and Moment from Diaphragm Load for diaphragm (k) = P := wdia :=.. ndia if ndia P =.664 Range for variable "x" = rg Definition of variable "a" = ad := length ndia + ( + ) ad = Definition of variable "b" = bd := length ad bd = Moment at point of load (k*ft) = P bd rg Md :=, length if rg < ad P bd rg length P rg ad ( ) Md := Md, Shear at point of load (k) = P bd Vd :=, length P bd length if rg < ad P Vd := Vd, Md = Moment from Diaphragm (k*ft) Vd = Shear from Diaphragm (k)

16 LRFD pre-stressed beam.mcd 7//3 6 of 7 Moment and Shear, Generated by DL on the Composite Section. This generator is capable of handling from to spa, and is capable of returning values for continuous sectio. This is done by moment distribution. The values returned are SL. Use a unit load "w" =. unit := DLc unit =.47 disp = mc n8 5 vc n8 column = span point column = moment column = shear Based on continuous section, cotant inertia n8 sp n8 sp

17 LRFD pre-stressed beam.mcd 7//3 7 of 7 Notes on Live Load: The HL-93 LL shall be used as described in (LRFD) The Design Lane: The design lane shall coist of a load of.64 k/ft uniformaly distributed in the longitudinal direction. Traversley the load shall be assumed to be ft wide. DO NOT apply the dynamic load allowance (Impact) to the lane. The design lane shall accompany the design truck and tandem. The Design Truck Design truck axal spacing from rear The Design Tandem: The design tandem coists of a pair of 5k axles spaced 4ft apart. Apply the dynamic load allowance to the tandem

18 LRFD pre-stressed beam.mcd 7//3 8 of 7 Load Combinatio Combination : The effect of the design tandem combined with the effect of the design lane. Combination : The effect of the design truck combined with the effect of the design lane. Combination 3: For both the negative moment between points of contraflexure under a uniform load on all spa, and reaction at interior piers only. 9% of two design trucks spaced a minimum of 5 ft between the lead axle of truck and the rear axle of truck. 9% the design Lane The distance between 3 k axles shall be 4 ft.

19 LRFD pre-stressed beam.mcd 7//3 9 of 7 Moment, SL, LLDF =. wheels, Impact included, input to tenth points ldm := DC LOADS (non-comp) DW Loads LL + I LOCATION self wt other (comp) M (+) M (-) the other loads include slab, diaphragms (if there are any) and any other non-composite loads. ( Mself Mnonc mca Md ) Shear Load, SL, LLDF = wheels, Impact included, input to tenth points ldv := DC LOADS (non-comp) DW Loads LL + I LOCATION self wt other (comp) V (+) V (-) the other loads include slab, diaphragms (if there are any) and any other non-composite loads. ( Vself Vnonc vca Vd )

20 LRFD pre-stressed beam.mcd 7//3 of 7 Expand area for moment and shear iteratio, Also LLDF is applied here Service I loads (moment) full SI SI SI3 SI4 SI5 SI6 SI7 := SI SI SI3 SI4 SI5 SI5 SI7 DC LOADS (non-comp) DW Loads LL + I TOTAL LOADS LOCATION self wt other (slab) (comp) M (+) M (-) M (+) M (-) ldm_f

21 LRFD pre-stressed beam.mcd 7//3 of 7 Service III loads (moment) SIII SIII SIII3 SIII4 SIII5 SIII6 SIII7 := SIII SIII SIII3 SIII4 SIII5 SIII5 SIII7 DC LOADS (non-comp) DW Loads LL + I TOTAL LOADS LOCATION self wt other (slab) (comp) M (+) M (-) M (+) M (-) full

22 LRFD pre-stressed beam.mcd 7//3 of 7 Strength I loads (moment) Maximum.5*DW +.5*DW +.75*(LL + IM) Minimum.9*DC +.65*DW +.75*(LL + IM) The loads shown in the DL colum reflect the values from Service I. The appropriate load combination (max or min) is shown in the total loads colum. The minimum load factors for dead load are used when dead load and future wearing survace stresses are of opposite sign to that of the live load. STI STI STI3 STI4 STI5 STI6 STI7 := STI STI STI3 STI4 STI5 STI6 STI7 DC LOADS (non-comp) DW Loads LL + I TOTAL LOADS LOCATION self wt other (slab) (comp) M (+) M (-) M (+) M (-) full

23 LRFD pre-stressed beam.mcd 7//3 3 of 7 Service I loads (shear) fullv SIv SIv SI3v SI4v SI5v SI6v SI7v := SI SI SI3 SI4 SI5 SI5 SI7 DC LOADS (non-comp) DW Loads LL + I TOTAL LOADS LOCATION self wt other (slab) (comp) V (+) V (-) V (+) V (-) ldv_f

24 LRFD pre-stressed beam.mcd 7//3 4 of 7 Service III loads (shear) SIIIv SIIIv SIII3v SIII4v SIII5v SIII6v SIII7v := SIII SIII SIII3 SIII4 SIII5 SIII5 SIII7 DC LOADS (non-comp) DW Loads LL + I TOTAL LOADS LOCATION self wt other (slab) (comp) V (+) V (-) V (+) V (-) fullv

25 LRFD pre-stressed beam.mcd 7//3 5 of 7 Strength I loads (shear) Maximum.5*DC +.5*DW +.75*(LL + IM) Minimum.9*DC +.65*DW +.75*(LL + IM) The loads shown in the DL colum reflect the values from Service I. The appropriate load combination (max or min) is shown in the total loads colum. The minimum load factors for dead load are used when dead load and future wearing surface stresses are of opposite sign to that of the live load. STIv STIv STI3v STI4v STI5v STI6v STI7v := STI STI STI3 STI4 STI5 STI6 STI7 DC LOADS (non-comp) DW Loads LL + I TOTAL LOADS LOCATION self wt other (slab) (comp) V (+) V (-) V (+) V (-) fullv

26 LRFD pre-stressed beam.mcd 7//3 6 of 7 Estimate number of Required Strands Final stress in bottom (no pre-stress) (ksi) = (I will use the moments at mid point) SI SI fa max ( + ) ( SI3 + SI4) := + fa = 4.36 Sb Sbc Required stress from pre-stress (ksi) = f_reqd := fa.9 fcf f_reqd = Approximate force per strand (k) = (estimate 4 ksi loss) F_est := Strand_area (.75 Strand_strength 4) F_est = Approximate number of strands required = f_reqd N := N = yb 4 F_est + Area Sb The area below gives the proper data for strand pattern for the proper beam.

27 LRFD pre-stressed beam.mcd 7//3 7 of 7 End Strand pattern Input Strand pattern at end (only fill in the colum in red) If the user wants to cut strands in the middle (break bond in middle) input a "y" in the column middle break and enter the number of strands broken, and the distance from center on each side. Will harped strands be used ("y" or "n") = harped := "n" mb br act sn stt := max Row actual middle BREAK NO. BREAK NO. BREAK NO. 3 harp. per row Height strands break strands dist strands dist strands dist y 5 5 y 5 ( x_s x_s)

28 LRFD pre-stressed beam.mcd 7//3 8 of 7 Middle Strand pattern Input Strand pattern at middle (only fill in the colum in red), do not input middle break here. x_u_m br_m act_m sn_m := max strands Row actual Harped per row Height strands "y" or "n" n 4 n 6 n n 5 n ( x_s x_s)

29 LRFD pre-stressed beam.mcd 7//3 9 of 7 6 End Pattern beam xa, x_s xa4, k xa4, 5 4 Do not worry about if the "x" coordinate of a strand or a strand break is not correct. The critical thing is that the "y" coordinate is correct. k_b n8, 3 k_b n8, k_b3 n8, beam xa,, x_s xa4,, k xa4,, k_b n8,, k_b n8,, k_b3 n8, 6 Middle Pattern beam xa, x_s xa4, k_m xa4, 5 4 Do not worry about if the "x" coordinate of a strand or a strand break is not correct. The critical thing is that the "y" coordinate is correct. k_b_m n8, k_b_m n8, k_b3_m n8, beam xa,, x_s xa4,, k_m xa4,, k_b_m n8,, k_b_m n8,, k_b3_m n8,

30 LRFD pre-stressed beam.mcd 7//3 3 of 7 Calculate Eccentricity Eccentricity From bottom of beam ecc x The graph indicates values from the bottom of beam ecc = 3.88 mp Eccentricity for non-composite section enc := yb ecc enc =.848 mp disp := x disp := ecc disp := enc,,, disp = column = span point column = eccentricity from bottom of beam column = eccentricity of non-comp section

31 LRFD pre-stressed beam.mcd 7//3 3 of 7 Prestress Losses LRFD The total loses shall be the sum of elastis shortening (ES) + shrinkage (SR) + Creep (CR) + Relaxation (R) Elastic Shortening LRFD a Initial modulus of elasticity for concrete Eci := fci Eci = Modulus of elasticity for the strands Ep := 85 Number of strands at middle Ns_middle := ye4 Ns_middle = 34 sp ceil Area of pre-stress strands Aps_ := Ns_middle Strand_area Aps_ = Total force in strands Fs := Strand_strength Aps_.75 Fs = Moment from beam alone (k*ft) M_ := max( SI) M_ = 7.5 Area of concrete (in^) = Ac := Area Ac = 789 Eccentricity of strands for N.A. non-composite e_ := yb ecc e_ =.848 sp ceil fcgp c.g. strands from prestress and beam weight only) at bottom Fs Fs e_ M_ fbi := + fbi = Ac Sb Sb at top Fs Fs e_ M_ fti := + fti =.9 Ac St St ( fti fbi) ( yb e_ ) fcgp := fbi + fcgp = h Elastic shortenting (ksi) = Ep ES := Eci fcgp ES =.64

32 LRFD pre-stressed beam.mcd 7//3 3 of 7 Shrinkage LRFD Preteioned members shrinkage = 7-.5*H H can be abtained from figure H := 7 SR := 7.5 H SR = 6.5 Creep of concrete LRFD For pre-teioned members creep = *fcgp - 7*fcdp >= fcdp = stress at c.g. strands from all permanent loads, except the loads used in fcgp at bottom fb := SI 7 Sb ( SI3 + SI4 7 7 ) Sbc fb =.869 at top ft := SI 7 St + ( SI3 + SI4 7 7 ) Stb ft =.49 fcdp := fb + ( ft fb) ( yb e_ ) h fcdp =.58 Creep (ksi) = CR := fcgp 7 fcdp CR = 3.9 Relaxation at Trafer (fpr) LRFD b Time of trafer (8 hours) =.75 days time :=.75 Yield strength of tendo (ksi) = fpy := Strand_strength fpy = 7 Initial stress in tendon (ksi) = fpj :=.75 fpy Relaxation at trafer = R := log( 4 time) 4 fpj fpy.55 fpj R =.7

33 LRFD pre-stressed beam.mcd 7//3 33 of 7 Relaxation after Trafer LRFD c fpr =.3 * (the value from formula c-) for Low Lax strands R :=.3 [.4 ES. ( SR + CR) ] R =.74 Total Initial Loses fi := R + ES fi =.895 Total Final Loses ft := ES + SR + CR + R ft = 5.85 Total Initial stress in the strands with Initial Losses fi :=.75 Strand_strength fi fi = 8.65 Total Final stress in the strands with Final Loses ff :=.75 Strand_strength ft ff = 5.45

34 LRFD pre-stressed beam.mcd 7//3 34 of 7 Design variables Eccentricity for non-compite section enc := yb ecc enc =.848 mp Effective strand area (including bond break) in^ = Aps := ye4 Strand_area Aps = mp Number of effective strends = ye4 = 34 mp Initial Strand force (k) = Fi := Aps fi Fi = mp Final Strand force (k) = Ff := Aps ff Ff = mp disp := disp := enc, disp := ye4, 3 disp := SI, 5 disp := Ff, 7 disp := x, disp :=, disp := Aps, 4 disp := Fi, 6 column = span point column = eccentricity of strands on non-compostie section column = blank column 3 = number of effective strands column 4 = Area of prestressing strands at a given point column 5 = Service I moment column 6 = initial force column 7 = final force disp =

35 LRFD pre-stressed beam.mcd 7//3 35 of Stress at initial conditio This includes the beam weight and the pre-stress force only. Initial strand force (lb) = Fi := Aps fi Fi = mp Initial stress top (psi) = top_i := Fi Ac Fi enc St + SI St top_i =.46 mp Initial stress in bottom (psi) = bot_i := Fi Ac + Fi enc Sb SI Sb bot_i = 3.54 mp Pass fail condition = check_i := x, check_i := "top fail" if top_i <, 3 ( ) top_i fit "top fail" if top_i > ( < ) ( ) top_i fic ( > ) "top OK" check_i := "bot fail" if bot_i <, 6 ( ) bot_i fit "bot fail" if bot_i > ( < ) ( ) bot_i fic ( > ) "bot OK" check_i := top_i, check_i := fit if top_i <, fic check_i := bot_i, 4 check_i := fit if bot_i <, 5 fic

36 LRFD pre-stressed beam.mcd 7//3 36 of check_i = "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" column = span point column = top stress column = top allowable column 3 = top check column 4 = bottom stress column 5 = bottom allowable column 6 = bottom check

37 LRFD pre-stressed beam.mcd 7//3 37 of "top OK" "bot OK" 4 Initial Stress 3 check_i, fic check_i, 4 fit x Positive moment envelope at Service I ( ) ( ) Ff Ff enc SI + SI SI3 + SI4 Final stress top (psi) = SI_pt := + + SI_pt =.568 Ac St St Stb mp Final stress in bottom (psi) = ( ) ( ) Ff Ff enc SI + SI SI3 + SI4 SI_pb := + SI_pb =.75 Ac Sb Sb Sbc mp Pass fail condition = From LRFD under final conditio it is only necessary to check teion under service III load combinatio. If the flag "see SIII" is shown in the tables below see the section with service III loads for teion checks. check_ := x,

38 LRFD pre-stressed beam.mcd 7//3 38 of 7 ( ) check_ := "see SIII" if SI_pt <, 3 "top fail" if SI_pt > "top OK" ( ) SI_pt fc ( > ) ( ) check_ := "see SIII" if SI_pb <, 6 "bot fail" if SI_pb > "bot OK" ( ) SI_pb fc ( > ) check_ := SI_pt check_ := SI_pb,, 4 check_ := fft if SI_pt <, fc check_ := fft if SI_pb <, 5 fc check_ = "see SIII" "bot OK" "top OK" "bot OK" "top OK". 4.8 "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK". 4.8 "bot OK" "top OK" "see SIII" "top OK" "see SIII" "top OK" "see SIII" "top OK" "see SIII" "top OK" "see SIII" "top OK" "see SIII" "top OK" "see SIII" "top OK" "see SIII" "top OK" "see SIII" "top OK". 4.8 "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK". 4.8 "bot OK" "top OK" "bot OK" "see SIII" "bot OK" column = span point column = top stress column = top allowable column 3 = top check column 4 = bottom stress column 5 = bottom allowable column 6 = bottom check

39 LRFD pre-stressed beam.mcd 7//3 39 of "see SIII" "bot OK" 5 Service I Positive Moment Envelope 4 check_, 3 fc check_, 4 fft x Negative moment envelope at Service I ( ) ( ) Ff Ff enc SI + SI SI3 + SI5 Final stress top (psi) = SI_nt := + + SI_nt =.47 Ac St St Stb mp Final stress in bottom (psi) = ( ) ( ) Ff Ff enc SI + SI SI3 + SI5 SI_nb := + SI_nb =.534 Ac Sb Sb Sbc mp Pass fail condition = From LRFD under final conditio it is only necessary to check teion under service III load combinatio. If the flag "see SIII" is shown in the tables below see the section with service III loads for teion checks. check_ := x,

40 LRFD pre-stressed beam.mcd 7//3 4 of 7 ( ) check_ := "see SIII" if SI_nt <, 3 "top fail" if SI_nt > "top OK" ( ) SI_nt fc ( > ) ( ) check_ := "see SIII" if SI_nb <, 6 "bot fail" if SI_nb > "bot OK" ( ) SI_nb fc ( > ) check_ := SI_nt check_ := SI_nb,, 4 check_ := fft if SI_nt <, fc check_ := fft if SI_nb <, 5 fc check_ = "see SIII" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "see SIII" "bot OK" column = span point column = top stress column = top allowable column 3 = top check column 4 = bottom stress column 5 = bottom allowable column 6 = bottom check

41 LRFD pre-stressed beam.mcd 7//3 4 of "see SIII" "bot OK" 5 Service I Negative Moment Envelope 4 check_, 3 fc check_, 4 fft x Positive moment envelope at Service III ( ) ( ) Ff Ff enc SIII + SIII SIII3 + SIII4 Final stress top (psi) = SIII_pt := + + SIII_pt =.484 Ac St St Stb mp Final stress in bottom (psi) = ( ) ( ) Ff Ff enc SIII + SIII SIII3 + SIII4 SIII_pb := + SIII_pb =.474 Ac Sb Sb Sbc mp Pass fail condition = From LRFD under final conditio it is only necessary to check teion under service III load combinatio. If the flag "see SIII" is shown in the tables below see the section with service III loads for teion checks. check_ := x,

42 LRFD pre-stressed beam.mcd 7//3 4 of 7 check_ := "top fail" if SIII_pt <, 3 ( ) SIII_pt fft "top fail" if SIII_pt > "top OK" ( < ) ( ) SIII_pt fc3 ( > ) check_ := "bot fail" if SIII_pb <, 6 ( ) SIII_pb fft "bot fail" if SIII_pb > "bot OK" ( < ) ( ) SIII_pb fc3 ( > ) check_ := SIII_pt check_ := SIII_pb,, 4 check_ := fft if SIII_pt <, fc3 check_ := fft if SIII_pb <, 5 fc check_ = "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK".4 3. "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK".4 3. "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" column = span point column = top stress column = top allowable column 3 = top check column 4 = bottom stress column 5 = bottom allowable column 6 = bottom check

43 LRFD pre-stressed beam.mcd 7//3 43 of "top OK" "bot OK" 4 Service III Positive Moment Envelope 3 check_, fc3 check_, 4 fft x Negative moment envelope at Service III ( ) ( ) Ff Ff enc SIII + SIII SIII3 + SIII5 Final stress top (psi) = SIII_nt := + + SIII_nt =.47 Ac St St Stb mp Final stress in bottom (psi) = ( ) ( ) Ff Ff enc SIII + SIII SIII3 + SIII5 SIII_nb := + SIII_nb =.534 Ac Sb Sb Sbc mp Pass fail condition = From LRFD under final conditio it is only necessary to check teion under service III load combinatio. If the flag "see SIII" is shown in the tables below see the section with service III loads for teion checks. check_ := x,

44 LRFD pre-stressed beam.mcd 7//3 44 of 7 check_ := "top fail" if SIII_nt <, 3 ( ) SIII_nt fft "top fail" if SIII_nt > "top OK" ( < ) ( ) SIII_nt fc3 ( > ) ( ) check_ := "see SIII" if SI_nb <, 6 "bot fail" if SI_nb > "bot OK" ( ) SI_nb fc3 ( > ) check_ := SIII_nt check_ := SI_nb,, 4 check_ := fft if SIII_nt <, fc3 check_ := fft if SIII_nb <, 5 fc check_ = "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" "top OK" "bot OK" column = span point column = top stress column = top allowable column 3 = top check column 4 = bottom stress column 5 = bottom allowable column 6 = bottom check

45 LRFD pre-stressed beam.mcd 7//3 45 of "top OK" "bot OK" 5 Service III Negative Moment Envelope 4 check_, 3 fc check_, 4 fft x Flexural Strength - LRFD stress in prestressing tendo I will coider the bonded case only. Use Strength I. fps = fpu k c dp fpy k =.4 for values of k see Table C fpu Aps fpu + As fy A's f'y.85 β fc ( b bw) hf c = for "T" section behavior (ts<c) fpu.85 fc β bw + k Aps dp Aps fpu + As fy A's f'y c = for rectangular section behavior (ts>c) fpu.85 fc β b + k Aps dp fpu = ultimate strength of tendio

46 LRFD pre-stressed beam.mcd 7//3 46 of 7 fps = actual stress in strand c = compression depth dp = distance from extreme compression fiber to centroid of tendio Aps = area of prestressing strands fy = yield of teion reinforcing As = area of teion reinforcing A's = area of compression reinforcing f'y = yield of compression reinforcing β = b = width of compression flange (un-traformed) bw = width of web of beam hf = depth of compression flange Calculatio k :=.8 A's := β :=.85 fpu := Strand_strength fpu = 7 Aps = f'y := 6 bw = 6 As := fy := 6 mp dp := h + ts ecc dp = hf := ts mp

47 LRFD pre-stressed beam.mcd 7//3 47 of 7 Aps fpu + As fy A's f'y c := j fpu.85 fc β ETFW + k Aps dp c = 6.94 mp Aps + As fy A's f'y.85 β fc ( ETFW bw) hf j fpu.85 fc β bw + k Aps dp j if j < ts j c fps := fpu k dp fps = 6. mp depth of compressive stress block a := c β a = 5.9 mp Allowable flexural strength (k*ft) = a Mn := Aps fps dp Mn = mp Check pass or fail condition check_m := "OK" if Mn > STI6 "FAIL" Prepare for output display disp_m := x, disp_m := c, 3 disp_m := check_m, 6 disp_m := Aps, disp_m := STI6, 4 disp_m := dp, disp_m := Mn, 5 Untimate Moment Capacity Output column = span point "OK" column = Aps "OK" column = dp "OK" column 3 = c column 4 = applied loads "OK" (Strength I) "OK" column 5 = capacity Mn "OK" column 6 = pass or fail "OK" "OK" "OK"

48 LRFD pre-stressed beam.mcd 7//3 48 of 7 disp_m = "OK" "OK" "OK" "OK" "OK" "OK" "OK" "OK" "OK" "OK" "OK" "OK" "OK". 4 Ultimate Moment Capacity 8 Mn 6 STI x Maximum reinforcing LRFD c de.4 where c = the distance fro the extreme compression fiber to the N.A. de = Aps fps dp + As fy ds Aps fps + As fy sinze As = then de = dp

49 LRFD pre-stressed beam.mcd 7//3 49 of 7 de := dp ratio := c dp check := "OK" if ratio <.4 "NG" disp := disp := x disp :=.4,, disp := ratio disp := check,, 3 disp = "OK" "OK"..5.4 "OK" "OK" "OK" "OK" "OK" "OK" "OK" "OK" "OK" "OK" "OK" "OK" "OK" "OK" column = span point column = actual ratio column = allowable ratio column 3 = check Minimum Steel Requirement LRFD Unless othewise specified, at any section of a flexural component, the amount of prestressed and nonprestressed teile reinforcement shall be adequate to develop a factored flexural resistance, Mr, at least equal to the lesse of.. times the cracking strength determined on the basis of elastic stress distribution and the modulus of rupture, fr, of the concrete as specified in Article times the factored moment required by the applicable strength load combinatio specified in Table I shall basically ignore this one because it makes no see. You see the allowable in the section is by default greater that the Strength I combination, so I shall use this as my minimum. If they wanted the allowable to be larger, to the

50 LRFD pre-stressed beam.mcd 7//3 5 of 7.33 value, they would have increased it by.33 already. The LRFD code does not specify a method for calculation Mcr, I shall therefore use the method specified in the AASHTO standard spec. AASHTO 9.8..; The total amount of prestressed and non-prestressed reinforcement shall be adequate to develop an ultimate moment at the critical section at least.*mcr φ Mn. Mcr Mcr = Sc ( fr + fpe) Mdnc Sbc Sb Sbc = composite section modulus at extreme teion fiber Sb = non-composite section modulus at extreme teion fiber fr = Modulus of rupture of concrete LRFD fpe = compressive stress in concrete due to effective prestress forces only (after allowance for all loses) at extreme fiber of section where teile stress is caused by externally applied loads. Composite section modulus (in^3) = Sbc = Non compoiste section modulus (in^3) = Sb = Modulus of rupture (psi) = fr :=.4 fcf fr =.679 fpe (psi) = fpe := Ff Ac + Ff enc Sb fpe = 3.6 mp Non-composite DL moment (lb*in) = ( ) Mdnc := SI + SI Mdnc = 87.9 mp Cracking moment (k*ft) = ( ) Sbc Mcr := Sbc fr + fpe Mdnc Sb Mcr = mp Check Minimum check_min := "Min OK" if Mn >. Mcr check_min = "Min OK" mp "Min NG" disp := disp := x, disp := Mn, disp := Mcr, disp := check_min, "Min OK" column = span point

51 LRFD pre-stressed beam.mcd 7//3 5 of 7 disp = "Min OK" "Min OK" "Min OK" "Min OK" "Min OK" "Min OK" "Min OK" "Min OK" "Min OK" "Min OK" "Min OK" "Min OK" "Min OK" "Min OK" "Min OK" "Min OK" "Min OK" "Min OK" "Min OK" "Min OK" "Min OK" column = capacity of section column = cracking moment column 3 = minimum check Check Deflectio n :=.. 4 Deflection due to pre-stress force (in) I will use the M/(E*I) method Length of each section = length int := int = Preliminary moment curve (all points) M := enc Fi

52 LRFD pre-stressed beam.mcd 7//3 5 of 7 Define actual moment curve = M3a := M M 3 M 4 M 5 M 6 M3a = M3 := ( M3a + M3a ).5 ( M3a + M3a ).5 ( M3a + M3a 3 ).5 ( ).5 M3a + M3a 3 4 M3a M3a n. 4 M3 n n Define range for each point on moment curve Area along each block Determine reaction area Area of each block * range xr := M4 := n M5 := M4 n n M6 := n 4.5 int 3.5 int.5 int.5 int.5 int M3 int n M4 xr n n xr = M4 = M5 = Final Pre-Stress Deflection defl_p := ( M5 length.5) Method two for pre-stress camber using a little different take on the M/EI method Eci Inc M6 n n defl_p = 3.45 lc := length xr := Length of each section = int := lc sp int = 6

53 LRFD pre-stressed beam.mcd 7//3 53 of 7 Range (in) = :=.. sp xr := for j j j.. sp int j j Preliminary moment curve (all points) M := enc Fi M := M sp+ sp Block area (moment*length) (kip*in) = M := M int Reaction = M3 M := M3 = Sum at points M5 := for j.. sp for j3.. j j int j j j int j3 j3 int + int j4 j3 M j j3 j3 j j5 j j6 = j4 j6 j5 Deflection at points defl := M3 xr Eci Inc M5

54 LRFD pre-stressed beam.mcd 7//3 54 of defl disp := disp := x, disp := defl, disp = column = span point column = deflection (in)

55 LRFD pre-stressed beam.mcd 7//3 55 of 7 Deflectio under Live Load (SL, max Positive), These will be based on simple span Live Load Moments I will use the M/(E*I) method Length of section for calculatio (ft) = lc := length lc = Length of each section = int := length sp int = 6 Define actual moment curve = M3 := disp3aa LLDFM M Define range for each point on moment curve xr := for j.. sp Area along each block M4 := M3 length sp lc j j j int j int Determine reaction area M5 := M4 Area of each block * range M6 := M4 xr Final Deflection defl_p := M5 lc.5 sp.5 Ec Ic j = M6 j defl_p =.948 Positive value indicates an downward deflection Max allowable deflection max_d := length 8 max_d =.5

56 LRFD pre-stressed beam.mcd 7//3 56 of 7 Deflectio due to non-composite Dead Loads (DC) I will calculate the deflection due to beam weight based on the initial strength (and modulus) of the beam. The slab weight shall be applied to the final concrete strength. n :=.. range for tenth points (in) = w bwt DLnc := w =.69 ws := range :=.5 x := range length n n ws =.87 w x n Deflection from self weight at tenth points (k*in) = Db := n 4 Inc Eci ( length ) 3 length ( ) x n + ( ) 3 x n Deflection from Non-Composite at tenth points (k*in) = ws x n Ds := n 4 Inc Ec ( length ) 3 length ( ) x n + ( ) 3 x n

57 LRFD pre-stressed beam.mcd 7//3 57 of 7 column = span point column = self wt column = non-comp Defl column 3 = rail deflection column 4 = Total for Oklahoma curve Db n Ds n disp = range n

58 LRFD pre-stressed beam.mcd 7//3 58 of 7 Shear Design, pre-stress method LRFD Except for slabs footings and culverts, traverse reinforcement shall be provided where Vu >.5 φ ( Vc + Vp) Vu = factored shear force Vc = nominal resistance of concrete Vp = component of the prestressing force in direction of the shear force LRFD : Minimum traverse reinforcing bv S Av =.36 fcf fy Av = area of traverse reinforcing within distance S bv = width of web adjusted for the presence of ducts as specified in S = spacing of traverse reinforcing fy = yeild strength of traverse reinforcing fcf = final concrete strength LRFD : Maximum spacing of traverse reinforcing. If Vu<.5*fcf then: Smax =.8*dv<= 4 in If Vu>=.5*fcf then: Smax =.4*dv<= in Vu = the shear stress calculated in accordance with dv = effective shear depth as defined in LRFD : Shear stress in concrete Vu φ Vp V = φ bv dv bv = effective web width dv = effective shear depth dv = As fy φ = resistance factor for shear Mn + Aps fps LRFD : The nominal shear resistance Vn shall be determined as the lesser of Vn = Vc + Vs + Vp Vn =.5 fcf bv dv + Vp for which Vc =.36 β fcf bv dv ( ) Av fy dv cot θ Vs = this is as per commentary EQ C S β = Factor as defined in article θ = angle of inclination of diagonal compressive stresses as determined in α = angle of inclination of traverse reinforcement to longitudinal axis

59 LRFD pre-stressed beam.mcd 7//3 59 of 7 LRFD : Use for the calculation of β and θ εt = Mu dv + Vu Vp Ep Aps Aps fpo εx = εt Ac = area of concrete on the flexural teion side of the member (see fig ) Aps = area of prestressing steel on the flexural teion side of the member (see fig ) As = area of nonprestressed steel on the flexural teion side of the member at the section (fig ) fpo = for the usual levels of prestressing.7*fpu will be appropriate Mu = factored moment taken as positive, but not taken less than Vu*dv Vu = factored shear force taken as positive mp := 3 Width of web (in) = bv := bw bv = 6 Distance from top slab to CL. prestressing (in) = de := h + ts ecc de = mp Effective shear depth (in) = ( ) dv := j.9 de if.9 de >.7 ( h + ts) dv = mp.7 ( h + ts) j de a.5 j if j > j j Define factored shear (k) = Vu := max STI6v STI7v Vu = mp Define factored moment (k*in) = Mu := max STI6 STI7 Mu = mp Mu shall not be less than Vu*dv (k*in) = Mu := max Mu Vu dv Mu = mp

60 LRFD pre-stressed beam.mcd 7//3 6 of 7 Resistance factor for concrete (5.5.4.) = φ :=.9 Calculate force "fpo" (ksi) = fpo :=.7 Strand_strength fpo = 89 Use stress for vertical force (k) = ( ) Aps Fp := fpo ft Fp = mp Component of the prestressing force in direction of the shear force Vp (k) = angle := if harped = "n" atan enc enc ceil( sp.5) length depress 8 π π Vp := Ff sin angle 8 Vp = mp Shear stress on the concrete (ksi) = Vu φ Vp V := φ bv dv V =.94 mp Ratio of V/fcf = r := V fcf r =.4 mp Maximum spacing of traverse reinforcing (in) = Smax := min min.8 dv 4..4 dv. if V <.5 fcf Smax = 4 mp disp := disp := x, disp := de, disp := Vu, 4 disp := Vp, 6 disp := r, 8 disp := bv, disp := dv, 3 disp := Mu, 5 disp := V, 7 disp := Smax, 9

61 LRFD pre-stressed beam.mcd 7//3 6 of 7 column = span point column = "bv" web width column = "de" column 3 = "dv" column 4 = Vu, Strength I shear column 5 = Mu, Strength I moment column 6 = Vp, vertical component of shear column 7 = applied shear stress column 8 = ration of shear stress to concrete strength column 9 = Maximum spacing of traverse reinforcing disp =

62 LRFD pre-stressed beam.mcd 7//3 6 of 7 Compute the strain in the reinforcement Modulus of prestressing strands (ksi) = Ep = 85 εt := Mu dv + Vu Vp Ep Aps Aps fpo εt mp =.985 εx := min. εt εx mp =.493 Enter Table and indicate values of θ and β expand area for value determination disp := disp := x, disp = disp := εx, disp := r, disp := θ, 3 column = span point column = strain column = ratio of V/fcf column 3 = angle θ column 4 = factor β disp := β, 4