APPENDIX C Slab Calculation 2 - · PDF fileAPPENDIX C Slab Calculation 2 Now try calculating...

APPENDIX CSlab Calculation 2

Now try calculating the same slab, with the minimum recommended balanced load of 50% of self-weight. (As opposed to 50% of DLLL used in the first calculation). The same thickness of slab will be used, with the same layout

Given:-- 4 bays each way 8000 x 7000 Ult. Load factors1.0 kPa finish1.8 kPa partitions2.5 kPa live loadSlab will be protected from weatherInternal columns 500 x 500External columns 300 x 500Wind loads taken bv shear walls

DL 1.2 and 1.5 as SABS 0160-1989LL 1.6

Floor to floor 3.0mAssume 15.2mm Dia cables, low relaxation

Breaking Strength 260.5 KNArea 140 sq. mm. E= 198 GPaRelaxation 1.5% at 1000 hrs

CALCULATIONAssume a 210 thick slab.

Balance 50% of Permanent Load 8.05kPa /2for internal spansFor external 8M spans, balance same proportion of load as previously, i.e 80/66x 0.5 =.606 Permanent loadFrom the formula, the slab should be somewhat thicker than 230mm, but take 210 as minimum practicable.

COMMENTSee diagrams infirst part ofcalculation

Formula wouldgive 230 slab

then self DLTotal perm =Service (1.1)

Live load

TOTAL

5.25 kPa8.058.85

2.50

11.35

Factored9.66

4.0

13.66

revised DL

12 factor toallow for extrashear in first int.

Check for shear as before, 210 is a bit thin, but acceptable with capitals

As contractor has column shutters with capitals, use capitals:For analysis assume E of slab = code value: i.e. (20 + 0.2 x 30) = 26.0 GPa

From the cross-sections, take the effective top cover as 47mm to C/L cable for 7m span, and (63 + 31)/2 = 47mmfor 8 m span. As bottom cover is 41 for 8 m span, effective drape - 210 - 88mm = 122 mm.For 7m span, drape is 210 - 98 =112 mm. (See Diag. Calculation 1)

From the diagrams of the cable shapes in the end spans (the cables in centre spans will have the same positions asat the ends of outer spans), the properties of the parabolas may be deduced, and then the losses due to curvature.Referring to the formula in Appendix A, the following table may be calculated, as before.

Span

8.0 end

8.0 cent

7.0 end

7.0 cent

al0.4

0.4

0.35

0.35

a2

0.4

0.4

0.35

0.35

0.1.,

0.163

0.13

0.163

b2.041

.041

.051

.051

b30.163

0.163

0.163

0.163

L

8.0

8.0

7.0

7.0

1

0.0330

0.0

.0330

0.0

m

1.34

1.854

1.039

149

n

-5.411

-7.418

-3.677

-5.214

X

3.7018

4.0

3.2114

3.5

cl.0096

.0122

.0086

.0112

c2

.0114

.0122

.0103

.0112

Drape

.0944

.1098

.0853

.1008

CALCULATIONLoads to be Balanced:—

For 8m span, balance 0.606 x permanent load in external span:For normal external bay 7 x 8.05 x .606 = 34.15 kN/mFor 8 m internal span, balance 0.5 of permanent loadi.e. 7 x 8.05 x 5 = 28.17 kN/m

For 7m span, balance 0.50x permanent load in external & internal span.For internal and external bay, 8 x 8.05 x .5 = 32.20kN

COMMENTSee page 1

For first internalbay, increasecables to allowfor increasedload at first

internal support

SAMPLE CALCULATION 2

CALCULATION

~rom the loads to be balanced, the required prestress force =H > I 2

8xDrapeThis is the final force after

COMMENTinternal spanswould havesmaller prestressforce for same

losses, and assuming a loss of say 18% for 8m spans, and 16% for 7m spans, the initial prestress can be calculated, load balanced.The angle which the cable 'rotates' through is also required for the friction calculation. They are calculated from the because offormula for a parabola. larger drape.e.g. for the 8m end span O = Arctan (2 x 78.4/3268.4) x 2 + Arctan( 2 x 113.5/3931.6) x 2 = .2113 Radians

The following table shows the calculation results.

Span Load Drape Eff. Length Angle to C/L Angle C/L to Total angle(rad) end (Cumul)

8 mend span 34.15 .0944 7.2 .09615 .1135 .20965

8m 2nd span 28.17 .1098 7.2 .122 .122 .45365

8m 3rd span -do- -do- -do- -do- -do- .69765

8m last span as 1st .9073 rad

7m 1st span 32.2 .0853 6.3 .09838 .1182 .21658

7m 2nd span 32.2 .1008 6.3 .128 .128 .4726

7m 3rd span -do- -do- .7286

7m 4th span as 1st span .9452 rad

factor = e-0025 * 32-° - -06 * -9073 = .8742 i.e. 12.58 % losses

7m spans

The total loss due to friction and wobble is

i.e. 11.9 % losses.

factor = e-'0025 x 28-° - -06 x a9452 = .8810

CALCULATION8 m spans: end spanThe prestress required to balance 59 kN/m with a drape of .0951, and an effective length of parabola of 8 - 0.8

34.15 x 122

7.2 m is given by = 2344 kN8 x .0944

allowing an estimated 18% losses, P—, = 2859 kNIf cables of 15.2mm dia (ult. strength 260 kN) are stressed to 80% of ult. the no. of cables required

28590.8 x 260

= 13.7 say 14 cables

COMMENTAlthough oneshould calculatethe losses fromthe exponentialequation, it isquite acceptableto calculate theloss at the end,and interpolate.An even numberis better ifstressed fromboth ends.

8m spans: 2nd spanThe prestress required to balance 28.17 kN with drape of .1122 =

28.17 x 7.22

= 1627 kN8 x .1122

Allowing the same losses, Pinit( = 1984 kN. and no of cables = ,

1984208

= 9.5 cables

say 10.Assume that the cables in the end span are taken to 1.5 m past the 1st support, and that half the cables are stressedfrom each end, the diagrams on the next page may be drawn. Although strictly one should calculate the losses fromthe exponential equation it is quite acceptable to use a linear method.An even number is better if stressed from both ends.

Losses:8 m spans

The total loss due to friction and wobble is


.33 kN


7 m Spans Effects of frictionStressing forces in cables before other losses


CALCULATION

For 7m external span, force to balance 32.20 kN/m, with drape of .0853, and effective length of 6.3m.=1873 kNWith 16% losses , no of cables = 10.8 say 12For internal span, force to balance34.15 kN/m with drape of .1008, Pfma] = 1585 kN -> 1887 kN before lossesNo. of cables = 9.0 say 10

Take losses due to friction and pull-in as before (see first calculation)The results are diagrammed on pages 5 and 6

COMMENT

Diagrams suchas shown makeit easy to knowwhat the force atany section is.

Other Losses:Elastic Shortening8m spansAverage stress in end 9.5 m =(2590+2699)7(7 x .21 x 2)= 1.80 MPaIn Centre part ((1927+1949)/2 x3.52)+ (1949)x2.98)/6.5( 7 x .21) = 1.32 MPaLosses in end span =1.80 x 1987(26 x 2) = 6.85 MPa

Average loss over whole span = (1.80 x 9.5 x 2 + 1.32 x 13)/(2 x 32m)x 198726 = 6.11 MPaLoss per cable = 140 x 10"6 x 6.85 x 1000 kN = .96kN in end spans (x 4 = 3.8kN)

= " x6.11 x " = 0.86 kN in centre spans (x 10=8.6kN)Total in end spans 12.4 kN

Because thecables arestressed one at atime, theaverage loss is0.5 times theRatios of Emoduli x stress

7m spansAverage stress whole length =((2223 + 2313.6)x 8.5+(1925+1956)/2 x 2.87 +1956 x 1.63) /(14x8x.21)= 1.19 MPaIn end average force =((2223 + 2313.6)/2=2268.2—>1.35MPa

Losses 10 cables per cable = 1.19x 1987(26 x 2) x 140 x 10-6 x 1000 = .63 kN

Loss in 2 cables = 1.35x 140x 10'6x 198726/2=0.72 kN10 x .63 + 2 x .72 = 7.8kN totalTotal in centre span = 6.3 kNThe resultant forces after friction and elastic losses are shown in the diagram below.


CALCULATIONLong term lossesRelaxation 1.5% x 2 = 3%For 8m spans: end span 3% x (2578+2687)/2 kN = 79 kN

centre 3% x( (1918+1940)/2x3.52 +(1940 x 2.98/6.5))=1934kN x 3%=58kNFor 7m spans: end span 3% x 2252 = 67.6kN

centre 3% x 1950 = 58.5 kNCreep. (Same creep factor as 1st calc)If stressed at 3 days (common for prestressed flat slabs) and with a humidity ol 45%, the creep for a 210 slab,interpolating between 150 and 300 thick slabs given in the code (BS8110) gives a creep factor of 3.7Then loss = stress in concrete x ratio of moduli x 3.78m spans: End Section conc, stress = (2.578 MN + 2.687)/(2*7*.21) = 1.79MPa

Loss of steel stress = 1.79 * 3.7 * 198/26 = 50.5MPaPer cable 50.5 x .140(Area) = 7.06 kNCentre Section( (1.918+1.94)x 3.52+1.94x 2.98)/6.5/(7 x .21) x 198/26 x 3.7 = 37.1 MPaPer cable 37. 1x .14 = 5.19 kN Therefore average loss in long cables=(5.19x6.5+7.06x9.5)/16m = 6.30 kN

Total creep loss: End 7.06 x 4 + 6.3 x 10 = 91.2 kNCentre 6.3x10 = 63kN

7m spans End Section Conc stress after elastic losses(2215.1 + 2305.8)/(2x 8 x.21)= 1.345MPaCentre ((1918.7 + 1947.7)/2 x 2.87 + 1949.7x 1.63) /(4.5 x 8 x 2.1)= 1.154 MPaAverage(1.154 x 5.5 + 1.345 x 8.5)/14 = 1.27MPaLoss of steel stress : end 1.345 x 3.7 x 198/26= 37.9MPaPer cable 37.9 x .14 = 5.306 kNCentre loss of steel stress 1.27 x 3.7 x 198/26 = 35.78MPa

loss/cable = 35.7 x .14 = 5.0kNLosses in centre 10 x 5.0 = 50kNLosses in end section 12 x 5.306 = 63.7 kN

7m spans Relaxation end span 2 cables 3% x (373.4 + 388.6V2 = 11.43kNTotal 10 cables (1956x 11.37 + 1950 x 2.63)/14--> 1912.8 kN1912.8 x 3% = 57.4 kNTotal for 12 cables = 68.8 kN

Shrinkage as before: for 1 cable 10.26 kN

8m spans 14 cables —>143.6 kN10 cables —> 102.6 kN

7m spans 12 cables —> 123.1 kN10 cables —>102.6 kN

COMMENTThe code givesa factor of 2.0on the 1000hour value forclass 1 BS5896

Varies withinitial prestress,temp etc.Should strictlybe taken asstress at thecentroid of thecables.

Total loss

Cause

No of cables

Initial force(208kN/cable)

Elastic

Relaxation

Shrinkage

Creep

TOTAL

Percentage of init. forceloss after friction loss

%Loss due to friction

Total % loss of initial

8m End span

14

2912 kN

12.4 kN

79 kN

143.6

91.2

321.2

11.1

11.4

22.5

8m Centre

10

2080

8.6

58

102.6

63

232

11.1

7.45

18.5

7m End span

12

2496

7.8

67.6

123.1

63.7

262

10.5

11.08

21.6

7m Center

10

2080

6.3

58.5

102.6

50

217.4

10.4

8.31

18.7

The losses are somewhat lower than with the higher prestress but still appreciably higher than the 16% (or less) assumed by some commercialdesigners.From the figures for long-term loss, the final prestress can be calculated


Position

8m end span

8m 2nd span

7m end span

7m 2nd span

Initial prestress (after elastic losses)

2578 kN

2669

2687

1918

1940

1940

2215

2289.5

2306

1919

1948

Loss

311

311

223

223

255.6

-do-

-do-

210

210

Final prestress

2267 kN

2358

2376

1695

1717

1717

1959

2034

2050

1709

1738

ANALYSISThe slabs are analysed by the equivalent frame method, using Long's method to calculate the equivalent stiffness of the columns.Three loading cases are needed: Dead load on all spans, Line load on even spans, and live load on odd spans.These may be combined together with the load factors to give the desired bending moments diagrams.Any method of analysis may be used: moment distribution if a computer is not available, or a frame analysis if one is available.The hogging moments for the ultimate limit state may then be reduced by 15%, and the sagging moments increased accordingly to maintainequilibrium.The design moment is at the face of the column or capital, but the the total statically required moment is: W(L-2D/3) /8If a computer program is used, there is an advantage in arranging a node at 1/3 of the column or capital dimension from the centreline ofthe column, as the moment at that point will not be less than the moment given for statically required moment.Loads (As for previous calculation)

LOAD

Finishes

Partitions

Self. Wt

Total DL

Live load

Service(1.1DL + LL)

1.l0kPa

1.98

5.77

8.85

2.50

Ultimate(1.2DL + 1.6LL)

1.20kPa

2.16

6.30

9.66

4.00

Ultimate Dead Load(1.5 DL)

l.SkPa

2.7 kPa

7.87kPa

12.07kPa

Total loads on spans

8m Spans DL

8m spans LL

7m spans DL

7m spans LL

Service

61.95kn/m

17.5

70.8

20.0

Ultimate

67.62kN/m

28.0

77.28

32.0

Self only

36.75

36.75

42.0

42.0

NOTE: As the Ultimate Deadload case of 1.5 DL at 12.07 kPa is less than the Ultimate DL + LL of 13.66 kPa, it may be effectively ignored,although there may be some places where the moments could be fractionally higher. A construction loading at initial prestress may needto be calculated if the propping is not adequately arranged.

Loads due to prestressFor a parabola, the equivalent uniform load caused by a tension in the cable is given by w L2 /8 = P h where h is the drapefor the cables. The drapes can be read from the table on page 1The loads will not be uniform, but will be trapezoidal if the variation of prestress along the beam is taken into accountIn addition the moments due to eccentricity at the ends must be taken into account in the analysis if the cables are not exactly central. Inour case we have assumed a 20mm eccentricity upwards at the end'The moment at the end of the 8m spans will be 2267 x .02 =56.7 kNmThe moment at the end of the 7m spans will be 1959 x .02 = 49.0kNmThe loads given are for final prestress. Initial prestress (after friction and elastic losses) combined with deadload only may be a critical state,but it is probably sufficient to take the stresses due to final prestress and multiply them by an average factor.


Note: The forces at intermediate points are interpolated

Position

8m 1st. span L end

8m 1st start of sag

8m 1st . end of sag

8m 1st span RH end

8m 2nd span L end

-do- end of cable

-do- beyond end

8m 2nd RH end

7m 1st span LH

7m 1st: start of sag

7m 1st end of sag

7m 1st RH

7m 2nd LH end

7m 2nd end of cable

7m 2nd beyond end cable

7m 2nd Rh end

Prestress

2267

2272

2353

2358

2354

2376

1717

1717

1959

1966

2026

2034

2034

2050

1709

1738

Drape

.0096

.0944

.0944

.0114

.0122

.1098

.1098

.0122

.0086

.0853

.0853

.0103

.0122

.1008

.1008

.0122

Length

0.8

7.2

7.2

0.8

0.8

7.2

7.2

0.8

0.7

6.3

6.3

0.7

0.7

6.3

6.3

0.7

Lateral load

272.0 kN/m down

33.1kN/m up

34.28 -do-

336.0 kN/m down

359.6 down

40.26 up

29.1 up

261.8 kN/m down

275.1 down

33.80 up

34.83 up

342.0 down

405.1 down

41.65 up

34.72 up

346.2 kN/m down

272 kN/m 336 359.6 261.6

0.4

Loads on 8m spans due to final prestress

275.1 kN/m 342 405.1 346.2

Loads on 7m spans due to f i n a l prestress

Columns: Stiffness by Long's method (See Sample Calculation 1)7m spans Exterior: E (30 MPa)=26GPa

fe =.5 x .33 /12 = 1.125 x 10-3 m unitsKc = 4EI/LC

Kc = Kc /(1+.1272KC L/EhJc)Ke = Kc /(I + .1272 (4IcL/Lch

3c)


where L=8m, LC =3, h=.21. and c= .3Then equivalent stiffness = stiffness x .645orequiv. I = 725 x 10"6

8m spans interior : Ic = .54/12 = 5.208 x 10-3 , k = .0564c= 0.5Then equiv. stiff = stiff x .596 Equiv I = 3.10 x 10-3

7m exterior Ic = .3 x -53/12 = :- .125 x 10-3

c=0.5 k=.1272Then equiv stiff = stiff x .555 Equiv I = 1.73 x 10-3

7m interior I = 5.20?, x 10-3 . c=.5. k= .0564Then equiv. stiff = stiff x .628 Equiv I = 3.27 x 1-3

Slab Stiffness The moment of Inertia of the 8m spans is 7 x .213 /12 = .0054 mThe moment of inertia of the 7m spans is 8 x .213 /12 = .00617 m

The E is taken as the same as the columns, and the creep factor (for calculation deflections due to long term loads) as 3.5The results of the analysis are given below: Because the structure is symmetrical, only the first two spans are shown.

It should be realised that as the loads are calculated for an interior span, the first interior column band should be reinforced forapproximately a 7% greater load.

i.e. for this design, 1 extra prestressing cable, and some additional reinforcement over the columns.

MOMENTS

Position

8m 1st span Left

8m 1st span centre

8m 1st span right

8m 2nd span left

8m 2nd span centre

8m 2nd span right

7m 1st span left

7m 1 st span centre

7m 1st span right

7m 2nd span left

7m 2nd span centre

7m 2nd span right

ULT(1.2DL +1.6L1)

-22.9 kNm

367.0

-557.3

-476.8

215.5

-343.1

-40.8

293.7

-445.4

-400.9

185.9

-287.3

ULT(adjusted 15%)

-14.1

403.1

-441.7

-378.4

264.1

-264.7

-25.6

326.5

-352.0

-314.1

225.9

-217.5

Service(1.1DL +LL)

-17.0

301.4

-457.9

-396.2

170.9

-285.1

-31.1

240.8

-369.9

-333.1

146.6

-238.7

DL self only

- 6.0

134.4

-211.6

-183.3

67.5

-131.9

-10.5

106.8

-170.8

-154.1

56.8

-110.4

Final Prestress

+63.4

-99.6

167.9

148.1

-54.1

105.7

48.8

-69.7

125.1

118.6

-49.7

98.0

DEFLECTIONS (allowing 3.5 creep factor on permanent loads)Assuming an uncracked section (see later for correction)

Position

8m 1st span

8m 2nd span

7m 1st span

7m 2nd span

Service Load (1.1 DL +LL)

28.7 mm

9.0 mm

14.9 mm

5.0 mm

Final Prestress

-10.7 mm

- 3.7 mm

- 4.6 mm

- 2.0 mm

Net Deflection

18.0 mm

6.3 mm

10.3 mm

3.0 mm

Serviceability Limit State

For controlling cracking, it has been traditional to limit tensile stresses.For the sake of completeness, this will be done, but the incremental stressmethod is considered better.. (Use the program described in Appendix E)The permissible tensile stress is given as 0.45Ö fcor 2.46 MPa

4.O4 Mfti

\^

\T = 4.O4x75.2/2 \=151.9 kN ^

4.P4 \ /1 x 210 mm/

4.04 -i- 7.24 / /

= 75.2 mm

// \

t

752. mm

»

h = 21O mm

\•

V^ 4.0

7.24 MRi

Elastic interacted etraeeee- equivalent tension


Serviceability Tensile stress

Position

8m 1st L

8m 1st C

8m 1st R

8m 2nd L

8m 2nd C

8m 2nd R

7m 1st L

7m 1st C

7m 1st R

7m 2nd L

7m 2nd C

7m 2nd R

Net moment(service-prestress)

46.4

201.8

-290.0

-248.1

116.8

179.4

17.7

172.1

-244.8

-214.5

96.9

-140.7

Comp. kN

2267

2358

2376

2376

1717

1717

1959

2034

2050

2050

1709

1738

Stress(moment)

±0.90

3.93

5.64

4.83

2.27

3.49

0.30

2.93

4.16

3.65

1.65

2.39

Stress (comp)

1.54

1.57

1.60

1.60

1.17

1.17

1.17

1.19

1.21

1.21

1.02

1.03

Net tensilestress (MPa)

2.36

4.04*

3.23*

1.10

2.32

-

1.74

2.95*

2. 44

0.63

1.36

Z of section(M units)

.0514

.0588

Slab Area(M units)

1.47

1.68

It may be seen that 3 of the stresses are more than the Report 25 stresses, and reinforcement is required. The tension to be taken byreinforcement is calculated by simply taking the force from the tensile stress diagram as in the sketch at a stress of 0.58 fv It should benoted that it is assumed that the moments are evenly distributed across the section. This is clearly incorrect.151.9kN /(.58 x 450) ->582sq.mm/m ->Y12 @ 180

To calculate the steel required to control crackwidth by a more logical method, the formula in BS8007 is used. Moments are taken aboutthe level of the tension steel reinforcement, i.e.The required area of steel is given by calculating the tension required for a moment of M + P(d-h/2), and then reducing the tension by P.i.e. transferring the compressive force to the reinforcement level.. 75% of the total hogging moment and 55% of the total sagging momentshould be taken in the column band for the purpose of calculating crackwidths

The following table was calculated using a computer program for crackwidths to BS8007 It may be seen that areas are similar to the areasrequired for ultimate load, which supports the Report 25 stresses. SABS 0100 seems to give slightly higher areas if the tensile stress in theconcrete is taken into account. Hogging moments are taken as 75% of total in column bands, and 55% for sagging momentsThe result is slightly more reinforcement in the position which requires more than nominal reinforcement

Required area of reinforcement for cracking (0.15% = 315 sq.mm/m)

Position

8m 1st L

8m 1st C

8m 1st R

8m 2nd L

8m 2nd C

8m 2nd R

7m 1st L

7m 1st C

7m 1st R

7m 2nd L

7m 2nd C

7m 2nd R

Net. Moment(service-prestress)

46.4 kNm

201.8

-290.0

-248.1

116.8

-179.4

-17.7

172.1

-244.8

-214.5

96.9

-140.7

Moment /min band

9.9

31.7

62.!

-18.3

38.4

-

23.6

45.9

-13.3

26.4

PrestressMPa

1.54

1.57

1.60

1.17

1.17

1.17

1.19

1.21

1.21

1.02

1.03

Steel stressMPa

-

238

246

360

271

Bar Dia.

10@300

16 @ 200

12@ 120

nom

10@250

12@180

nom

d

180

166

Area(sq.mm/m)

nil

nil

992

946

nil

Nominal(297)

nil

Nominal

625

Nominal

Nominal

(Calculated using SABS0100 with 0.2mm crackwidth. See Appendix E) The areas are slightly greater than for the Report 25 Calculation(e.g Y16 at 200 or Y12 @ 1?0 as compared with Y12 at 180)

Ultimate Load Limit State.15% reduction of hogging moments ( with corresponding increase of sagging moments) is allowed and has been taken.The decompression moment, equal to the prestress x Z, is the moment required to reduce the compression on the extreme fibre to zero.If the applied moment (Ultimate dead + Live + prestress) is less than the decompression moment, no reinf. is required. If it is greater, anequivalent moment M' is calculated, by displacing the prestress to the level of the reinforcement, and adding a moment of P(d-h/2). Thetension calculated from this moment then has the prestress force deducted, to obtain the net tension on the reinforcement M7Jd-P. Therequired steel area, is then (M;/Jd-P)/fs

The ultimate M.R. of concrete = 4651 bd2 = 126 kNm for the 7m spans and 154 kNm for the 8m spans. None of the design moments


exceeds this. The lever arm as a proportion of effective depth is given approximately by

d-h/2 = 65mm for the 8m spans, and 49mm for the 7m spans.

The approximate moment /metre taken by Nominal reinforcement is 391 MPa x 0.75 d x .15% x .21m=

Required Area of reinforcement for Ultimate Limit State

16.8 kNmford=.182 mm15.3kNm/m for d=. 166

Position

8m 1st L

8m 1st C

8m 1st R

8m 2nd L

8m 2nd C

8m 2nd R

7m 1st L

7m 1st C

7m 1st R

7m 2nd L

7m 2nd C

7m 2nd R

Net. Moment(Ult-prestress)

49.3

303.5

-273.8

-230.3

210.0

-159.0

-23.2

256.8

-226.9

176.2

-119.5

Net Moment/m in band

(A)

10.6

47.7

-58.7

33.0

-34.1

4.3

35.3

-42.5

24.2

-22.4

Mom. /mP(d-h/2)

(B)

21.0

21.4

21.8

16.0

16.0

'2.0

12.2

12.4

10.5

10.6

DecomMom/mPrestress x Z

11.3

8.6

DesignMoment/m

(A+B)

—69.1

80.5

49.0

50.1

—

47.5

54.9

34.7

33.0

NetTension

kN/m

—

116.6

155.4

47.1

105.4

Bar Dia& spacing

Y10@250

Y10@250

Y10@200

Y10@250

Area(sq.mm/m)

Nom

298

398

Nom

121

Nom

Nom

270

Nom

Nom

The reinforcement areas required for ultimate load by this method are a bit smaller than those required for crack controlCompare the above calculation with the Report 25 method of calculating the reinforcement required for ultimate moments:-

This method does not take account of the moments due to prestress, but effectively assumes the cables are bonded, and that the momentsare taken over the full width of the section.At the first interior support, of the 8m spansf, = 2.358lMN/( 14 x 140. x 10-6) =1203.1 MPa (Effectivepeprestress)f pu = 260.5 MN/140. x 10-6=1860.7 MPa

-.Aps = 14x 140 x10-6 =l.96x 10-6

Effective depth of cables =.163m (see page 3 of calcs)From equation 52 of BS8110

for l/d =16/0.163 =98, f pb = 1203.1 + 61.2= 1264 MPa0.7 f = 1302 MPa therefore use 1264MPaFor 7m end spans , 1=28/2=14, and for 7m internal spans, 1=28/3=9.33

For prestress only, Tult = 2.478 MNThen N.A. depth = 2.478/(7 x .45 x 30) = 26.2 mm andlever arm = .163 -.026/2 = .150 mM.R. Prestress = .150x 2.478= 371.4 kNmActual ultimate moment 441.7kNm. Therefore additional steel is required.

d re inf=.182and dcable = .163Say approximately Tensionreinff= (441.7- 371.4)7371.4 x 2478kN = 469At 391 MPa (450/1.15), the area required = 1392 sq.mm over 3.5 m width = 343 sq. mm/m (Y10 @ 220 )Try Y10 @ 200 : Area = 1374 sq.mm Tensile capacity =537 kNSee diagram, Tension = 2478+537 kN = 3015kN. N.A depth = 3015/C7 - .45 x 30)=31.9mmMR =537x .16 6+ 2478 x .147 = 453kNm O.K.The other positions are checked in the table below0.15% =315sqmm/m-->Y10 @ 250

For end 8m spans (SABS 0100) 1=32/2=16For internal spans, 1= 32/3 = 10.67

(for 8m int 1295MPa, for 7m end 1274.for 7m internal f b = 1302MPa)

Position

8m 1st L

8m 1st C

8m 1st R

8m 2nd C

Ult. M

-14.1

403.1

-441.5

264.1

fpb x!40. x 10-6

x No. cables

2.478 MN

-do-

-do-

1.813 MN

Eff.d

.130

.163

-do-

-do-

N.A. depth

27mm

-do-

19.3mm

Lever arm

116

149

153

M.R.

287

371

371

279

DM

0

32

70.5

0

As reqd.

0(.15%

min)

Nom

Y10 @200

Nominal

SAMPLE CALCULATION 2 Pase 13

Position

8m 2nd R

7m 1st L

7m 1st C

7m 1st R

7m 2nd C

7m 2nd R

Ult. M

-264.7

- 25.6

326.5

-352.0

225.9

-217.5

fpb x!40. x 10-6

x No. cables

1.813 MN

2.138 MN

-do-

-do-

1.813MN

-do-

Eff.d

-do-

.130

.163

-do-

-do-

-do-

N.A. depth

19.8mm

-do-

16.8mm

Lever arm

120

153

154.6

M.R.

268

257

327.3

-do-

282

-do-

DM

0

25

0

As reqd.

Y10 @250

Y10@250

Y10@250

The reinforcement required for ultimate moments by this method are slightly less than required for cracking, and slightly less than the reinforcementrequired by the suggested method..

Ultimate MR of section = 537x .166 + 2473 x .147= 453 kNm

It is also desirable to check the ultimate load case with half of the cables in the end span removed, with DL + .25 LL. in the end spans only. TheMR due to half the number of cables may be taken as half the M.R. in the table above if Report 25 method is used, or the moments calculated forprestress may be halved and deducted from the moments for DL + .25LLThe second method will be used

Position

8m 1st L

8m 1st C

8m 1st R

7m 1st L

7m 1st C

7m 1st R

Ult. M(DL + .25LL)

- 8.2 kNm

172.8

-267.9

- 15.4

137.6

-190.0

PSMom/2

36.7

-49.8

83.9

27.0

-24.4

62.6

Moment/mPS (d-h/2)/2

10.7

10.7

4.3

4.3

4.3

UltMom/m

27.1

57.4

2.9

18.9

35.6

Net. M/mfor tension

37.8

68.1

7.2

23.2

39.9

Required Area(per m col band)

-

169. sq mm

622. sq mm

-

-

355. sq mm

Norn

Y12 at 180

Nom

It may be seen that the reinforcement for the 1.2DL + 1.6LL ultimate load case is adequate everywhere except the first interior support of the 8mspans

Deflections:The deflections in the table above were calculated for a non-cracked slabIf the sections are cracked, the Moment of Inertia would be 2.56 x 10-3' instead of the uncracked value of 5.4 x 10-3

Using the ACI formula , and taking moment at first cracking as the moment at which the tensile stress reaches 2MPa, the

cracking moment in the 8m 1st span is (1.54+ 2) x 7 x .212/6 = 182 kNm. Taking the design moments as the serviceability moments, and takingthe moments due to prestress into account, the moment at midspan of the 8 m 1st span is 201.8 kNm. i.e. cracked, and at the first internal support290 kNmThen Ie midspan = 4.13 xl0-3

I Support = 3.26 x 10-3 using the service load moments calculated aboveThe actual moment over the support is more concentrated, and there may be more cracking.The net equivalent M. of I. is then.85 x 4.13 + .15 x 3.26 =4.0The calculated deflection will then be 5.4/4.0 x the calculated one.Now the calculated deflection assumes that the slab acts as a band spanning in one direction, and neglects the span in the other direction.If one adds half the calculated deflection in the short direction to the calculated deflection in the long direction, this will probably be a reasonableestimate.


The total deflection is then (18.0 + 5.15) x 5.4/4.0 = 31mmThis is 1/223 of the short span, and may be acceptable if there are no rigid partitions. It should be noted that if there are reasons the slab might bemore cracked, e.g. temperature stresses or shrinkage, the deflections could be considerably greater.

Shear: The calculated ultimate shear at the first interior support ( at the face of the supports) is (from the computer calculations) 841.95 kN (ascompared to the simply supported load of 765.9 kN) . The shear assumed in the preliminary calculations was 860 kNThe design shear is 1.15 times this, or 968. kNAlternatively, from the code, Veff = Vt (1 + 1.5M/Vt x)

Mt, the moment transmitted to the column, is 226.0 kNm, and x is 1.5 + .18 x 1.5 (The column capital +1.5 slab depths)

then Veff = 1.227 x 841.95 =1033.4 kNFrom this may be deducted the vertical component of the prestressing cablesAt 1.06 m from the c/1 of column, the slope of the cables is .0529 from the properties of the parabola in the 8m spanvertical component = 4/14x 2267 x .0529 =34 kN x 2 = 68 kNIn the 7m span the slope is .0575Component = 5/12 x .0525 x 2056 kN =44.9 kN x 2 =89.8 kNTotal vert. component on 4 sides =157.8 kN and net design shear = 1033.4 - 157.8=875 kN

Area of reinforcement (Y10@200) in one direction, and Y12 @ 180 in the other. Average is 510 sq mm/mAverage % = 100(1860 x 4,5 x 140.x 10"6 + 510x1.71 x 10-6 x 450)/(1.71 x .21x 450) = 0.97%Permissible shear stress = .0.70 MPaActual shear stress = 851/(4 x 2.04 x .167) = .63 MPa. Therefore no shear reinf. reqdAt other columns, shear is less, and moment transferred to column is smaller. No shear reinforcement required.

One should also calculate that the width of slab at the external columns is adequate to transfer the moment, but with Long's method, the momenttransferred is quite small, and with the column capitals, it is not necessary.

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