M.sc. Concrete Formulae

8/12/2019 M.sc. Concrete Formulae

1/17

SELECTED FORMULAE AND CURVES

1. l d = K

c

et y

f

f

d b where the constant K is taken as follows:

No. 20 anddeformed wires

> No. 20deformed bars

Given bar spacingcriteria satisfied 25

12 = 0.48

53

= 0.60

Other cases2518

= 0.72109

= 0.90

2. l d =109

c

y

f

f

b

b

tr b

set d

d K c

+

where

b

tr b

d K c +

> 2.5

3. K tr =n s

f A yt tr 10

4. If 0.2 < fm 2.0 , h min = ( )2.05361500

8.0

+

+

fm

yn

f

l

120 mm

5. If fm > 2.0, hmin =

936

15008.0

+

+ yn

f l

90 mm

6. = y

c

f f 85.0

, R = 2d b M u , =

c f R614.2

11 when t 0.005

7. Transverse Distribution of Moments Let l 2 / l 1 = A 0.5 A 2.0

t = B If t > 2.5, B = 2.5

f11

2

l

l = D If f1

1

2

l

l> 1.0, D = 1.0

Interior negative moment (%age): 75 + 30(1 A) D Exterior negative moment (%age): 100 10 B + 2 BD (1 A)Positive moment (%age): 60 + 15(3 2 A) D

8. V c for punching shear = lesser of c f

+

2

161

bod , co

s f b

d

+2

121

bod , and

c f 31

bod


2/17

9. For interior columns: Ac = 2( b1 + b2)d , J c =266

212

31

31 bdbdbd b ++

For edge columns: Ac = (2 b1 + b2)d , x = b12/(2b1 + b2)

J c = 22

21

1

31

31

212122 dxb x

bd b

dbd b +

++

For corner columns: Ac = ( b1 + b2)d , x = b12d / (2 Ac)

J c = 22

21

1

31

31

21212dxb x

bd b

dbd b +

++

10. Slabs in EFM: m = 0.09 015.0

2

2

1

1

ll

cc 0.24 0.083

k = 5.3 05.0

2

2

1

1

ll

cc 0.9 4.0

COF = 0.5702.0

2

2

1

1

ll

cc 0.37 0.5

where =slabof depth paneldropatdepth

Columns in EFM: k a = 4.008.0

b

a

t t

7.2

u

c

l

l 4.0

for t a / t b = 0.4 to 2.2 and l c / l u up to 1.2

COF a = 0.5( )( ) 08.0/

/

ba

uc

t t

ll 0.5

11. Ag(trial) = yc

y xu

f f

M M P

008.043.0

22

+++

, yc

y xu

f f

M M P

010.050.0

22

+++

12. a n+1 = an ( )( )nn

a F a F

13. P n =

85.02

+

d eh

hb f c +

d d e

f A y st

+2

1 where = 0.408 0.00021 f y

14. If e x / b e y / h, e ox = e x + h

e y b

for 4.0 g cu

A f P

, 6.0720

3005.0

+

+=

y

g c

u f

A f P

for 4.0> g cu

A f P

, 5.0720

3003.1

+

=

y

g c

u f

A f P

15. 0.1=

+

uyo

uy

uxo

ux

M

M

M M

where = log 0.5 / log


3/17

16. E c = 4700 c f for normal weight concrete

17. C m = 0.6 + 0.4 2

1

M M

0.4 for members without transverse loads between supports

18. T cr = 0.33 c f

cp

cp

P

A2

; T n = s

f A A yvt 02cot ;

19. Al = 2cot

l y

yvh

t

f

f

s A

20.2

2

2

7.1

+

oh

hu

w

u

AT

d bV

+ c

w

c f d b

V 66.0

21. V c for one-way shear 1/6 c f bw d

22. Al ,min = y

yt h

t

y

cpc

f

f p

s A

f

A f

42.0

where At / s must not be less than yt

w

f b

175.0

23. ( Av + 2 At)min = larger of yt

wc f

sb f 062.0 and

yt

w

f sb

35.0

24. k = ( ) nnn +22 & k = sc

c

f nf nf +

25. A s,min. = y

c

f

f

4

bwd

y f 4.1

bwd

26. b = 0.85 1 y

c

f f

600

600+ y f

, b = b + y

s

f f

, wb = b + f

27. d min. =b f

M c

u

205.0 , max = 0.85 1 yc

f f

83

28. R = y

x

l

l, slab width for interior longer beam = x

Rl

31

2

29. sp = 0.45 y

c

c

g

f f

A

A

1

30. smax. = [ ]145.02

c g cc y sp

A A f D

f d

31. k =18

321 2

y

x

y

yn

q

m

l

l

l

32. myp = + k kq

y

x y 3832

2

l

ll


4/17

33. k 2 =)2(

/2)1( 221

y xn qmk l

34. A s,min. = y

c

f

f

4

bwd

y f 4.1

bwd

35. b = 0.85 1 y

c

f f 600600+ y f

, b = b + y

s

f f

, wb = b + f

36. V c = +u

uwc M

d V f 2.17158.0 bwd > 0.29 c f bwd ,

u

u

M d V

> 1.0

37. V n =

+ yt cccv f f A 6

ccv f A 32

ccw f A 65

where c = 2 when hw / l w 2.0 (high-rise wall)= 3 when hw / l w 1.5 (low-rise wall)= 6 hw / l w 2 when hw / l w is between 1.5 and 2.

38. Boundary element is to be provided if c ( )wuw

h/600

l

39. The minimum ratio of spiral or circular reinforcement in boundary element is:

s = larger of yt

c

f f

12.0 and yt

c

ch

g

f f

A

A

145.0

40. The minimum ratio of rectangular hoop reinforcement in boundary element is:

Ash = yt

cc f

f b s

09.0

41. l dc = F 1 F 2 bc

y d f

f

41

F 1 F 2 0.04 f y d b 200 mm

42. The minimum lap length for splices of deformed bars in compression is as under:

l = b y d f F 07.0 300 mm for f y 420 MPa = ) b y d F f 2413.0 300 mm for f y > 420 MPa

where F = 1 when f c > 20 MPa = 4/3 when f c 20 MPa

43. l dh = F 1 F 2 bc

ye d

f

f

41

8 d b 150 mm

44. l d K u

n

V M

+ l a

45. a) If fm < 0.20, the provisions for slabs without interior beams must beapplied.

b) For panel with one or more discontinuous edges having edge beam with f 0.29 c f bwd ,u

u

M d V > 1.0

68.

+++= A AC W mkg A

c

c f f 101000[)/(

3

69. Table: Moment and shear values using ACI coefficients.

1. Positive Moment:End spans:

If discontinuous end is unrestrained11

1wul n

2

If discontinuous end is integral with the support141

w ul n2

Interior spans:161

w ul n2

2. Negative moment at exterior face of first interior support:

Two spans91

wul n2

More than two spans101

w ul n2

3. Negative moment at other faces of interior supports 11

1

wul

n

2

(l n in no. 3 is the average of clear spans of the two adjacent panels.)

4. Negative moment at face of all supports for (1) slabs with spans notexceeding 3m and (2) beams and girders where ratio of sum of columnstiffness to beam stiffness exceeds 8 at each end of the span.

121

wul n2

5. Negative moment at interior faces of exterior supports for members builtIntegrally with their supports:

Where the support is a spandrel beam or girder241

wul n2

Where the support is a column161 wul n2

6. Shear in end members at first interior support 1.152

nuw l

7. Shear at all other supports2

nuw l


10/17

70. Table: Minimum Depth of Beams and One-Way Slabs

Member SteelGradeSimply

SupportedOne End

ContinuousBoth EndsContinuous Cantilever

300 l /25 l /30 l /35 l /12Solid One-Way Slabs 420 l /20 l /24 l /28 l /10

300l /20

l /23

l /26

l /10Beams 420 l /16 l /18.5 l /21 l /8

515 l /14 l /16 l /18 l /7

71. Table: BS 8110 Requirements For Durability.

Enviro-nment

Exposure Condition MaximumFree Water/CementitiousMaterial

Minimum Content ofCementitious Material(kg/m 3) for Following Max.Size 0f Aggregate (mm)

AverageCylinderStrength

Ratio 40 20 14 10 (MPa)Mild Concrete surfaces protected

against weather or aggressive

conditions.

0.80 150 180 200 220 15

Moderate Concrete surfaces shelteredfrom severe rain or freezingwhilst wet. Concrete surfacecontinuously under water or incontact with non-aggressivesoil

0.65 245 275 295 315 24

Severe Concrete surfaces exposed tosevere rain alternating wettingand drying or occasionalfreezing.

0.60 270 300 320 340 27

Verysevere

Concrete surface exposed to seawater spray, de-icing salts,corrosive fumes or severe

freezing conditions.

0.55 295 325 345 365 30 withentrained

airExtreme Concrete surfaces exposed toabrasive action by sea watercarrying solids or flowing waterwith pH 4.5 or machinery orvehicles.

0.50 320 350 370 390 36

72. Table: ACI Requirements Against Sulphate Attack.

SulphateExposure

Water-SolubleSulphate (SO 4)in Soil(percent bymass)

Sulpahte (SO 4)in Water

(ppm)

Type of CementMaximum FreeWater / CementRatio For NormalWeight AggregateConcrete

Negligible 0.00 0.10 0 150 No limitation No limitationModerate(sea water)

0.10 0.20 150 1500 Modified (Type II)Portland-pozzolan,Portland blast furnace

0.50

Severe 0.20 2.00 1500 10000 Sulphate-resisting Portland(Type V)

0.45

Very severe over 2.00 over 10,000 Sulphate-resisting Portland plus fly ash or other pozzolan

0.45


11/17

73. Table: Relation Between Water / Cement Ratio and

Average Compressive Strength of Concrete.

Average Compressive Strengthat 28 days

(MPa)

Effective Water / CementRatio by Mass for Non-Air-

Entrained Concrete45 0.3840 0.4335 0.4830 0.5525 0.6220 0.7015 0.80

74. Table: Approximate Water requirements.

Workability(slump)

Water Content (kg/m 3) of Concrete for Maximum AggregateSize (mm) for Non-Air-Entrained Concrete

10 12.5 20 25 4030 50 205 200 185 180 160

80 100 225 215 200 195 175150 180 240 230 210 205 185

Approximate EntrappedAir Content, percent

3 2.5 2 1.5 1

Recommended AverageAir Content Percent for:

MildExposure

4.5 4.0 3.5 3.0 2.5

Moderate Exposure 6.0 5.5 5.0 4.5 4.5Extreme Exposure 7.5 7.0 6.0 6.0 5.5

75. Table: Required Amount of Coarse Aggregate, ACI 211.1 Modified For LessFineness Modulus.

Maximum Sizeof Aggregate

(mm)

Dry Bulk Volume of Rodded Coarse Aggregate Per Unit Volumeof Concrete for Fineness Modulus of Sand of:

2.0 2.2 2.4 2.6 2.8 3.010 0.54 0.52 0.50 0.48 0.46 0.44

12.5 0.63 0.61 0.59 0.57 0.55 0.5320 0.70 0.68 0.66 0.64 0.62 0.6025 0.75 0.73 0.71 0.69 0.67 0.6540 0.79 0.77 0.75 0.73 0.71 0.6950 0.82 0.80 0.78 0.76 0.74 0.7270 0.85 0.84 0.82 0.80 0.78 0.76

150 0.89 0.88 0.87 0.85 0.83 0.81


12/17

76. Longitudinal Distribution of MomentsFactored M at supports = 0.65 M o Factored M + at mid-span = 0.35 M o

Exterior edgeunrestrained, withno beams

Slab with beams between allsupports

Slab with edge beams, but withoutinterior beams

(1) (2) (3)

Int. M 0.75 0.70 0.70M+ 0.63 0.57 0.52

Ext. M 0 0.16 0.26

77. P u =

bb

o

o

ee

P P

P

+ 11

78 - Table. ACI 1963 Coefficients For Dead Load Positive Moments InSlabs Increased by 25%.

Ratiom

Case1

Case2

Case3

Case4

Case5

Case6

Case7

Case8

Case9

Cx 0.045 0.023 0.023 0.034 0.034 0.041 0.034 0.025 0.0291.00Cy 0.045 0.023 0.034 0.034 0.023 0.034 0.038 0.029 0.025Cx 0.050 0.025 0.026 0.038 0.035 0.045 0.039 0.028 0.0300.95Cy 0.041 0.020 0.031 0.030 0.019 0.030 0.039 0.026 0.021Cx 0.056 0.028 0.031 0.041 0.036 0.049 0.044 0.031 0.0330.90Cy 0.036 0.018 0.030 0.028 0.016 0.026 0.035 0.024 0.019Cx 0.063 0.030 0.036 0.045 0.039 0.053 0.050 0.036 0.0350.85Cy 0.033 0.015 0.028 0.024 0.014 0.021 0.031 0.021 0.016Cx 0.070 0.033 0.043 0.049 0.040 0.056 0.056 0.040 0.0360.80Cy 0.029 0.014 0.025 0.020 0.011 0.019 0.028 0.019 0.013Cx 0.076 0.035 0.050 0.054 0.041 0.060 0.064 0.045 0.0390.75Cy 0.024 0.011 0.023 0.016 0.009 0.015 0.025 0.016 0.009Cx 0.085 0.038 0.058 0.058 0.044 0.064 0.073 0.050 0.0410.70Cy 0.020 0.009 0.020 0.014 0.006 0.011 0.021 0.014 0.008Cx 0.093 0.040 0.068 0.063 0.045 0.068 0.081 0.055 0.0430.65Cy 0.016 0.008 0.018 0.011 0.005 0.009 0.018 0.011 0.006Cx 0.101 0.043 0.078 0.066 0.046 0.070 0.091 0.060 0.0450.60Cy 0.013 0.005 0.014 0.009 0.004 0.008 0.015 0.009 0.005Cx 0.110 0.044 0.089 0.070 0.048 0.073 0.101 0.065 0.0460.55Cy 0.010 0.004 0.011 0.006 0.003 0.005 0.011 0.006 0.004Cx 0.119 0.046 0.100 0.074 0.049 0.076 0.111 0.070 0.0480.5Cy 0.008 0.003 0.009 0.005 0.001 0.004 0.009 0.005 0.003


13/17

79.

Table. ACI 1963 Coefficients For Live Load Positive Moments In SlabsIncreased by 25%.

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9

Cx 0.045 0.034 0.034 0.040 0.040 0.044 0.040 0.035 0.0381.00Cy 0.045 0.034 0.040 0.040 0.034 0.040 0.044 0.038 0.035Cx 0.050 0.038 0.039 0.044 0.043 0.048 0.045 0.039 0.0400.95Cy 0.041 0.031 0.036 0.036 0.030 0.036 0.040 0.034 0.031Cx 0.056 0.043 0.044 0.049 0.046 0.053 0.050 0.044 0.0450.90Cy 0.036 0.028 0.034 0.033 0.026 0.031 0.036 0.030 0.028Cx 0.063 0.046 0.050 0.054 0.051 0.058 0.056 0.050 0.0490.85Cy 0.033 0.024 0.030 0.029 0.024 0.028 0.033 0.028 0.025Cx 0.070 0.051 0.056 0.060 0.055 0.064 0.064 0.055 0.0530.80Cy 0.029 0.021 0.028 0.025 0.020 0.024 0.029 0.024 0.021Cx 0.076 0.056 0.064 0.065 0.059 0.069 0.070 0.061 0.0580.75Cy 0.024 0.018 0.024 0.020 0.016 0.020 0.025 0.020 0.016

80.

Table. ACI 1963 Coefficients For Live Load Positive Moments In SlabsIncreased by 25%.

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9

Cx 0.085 0.061 0.071 0.071 0.064 0.075 0.079 0.068 0.0630.70Cy 0.020 0.015 0.020 0.018 0.014 0.016 0.021 0.018 0.014Cx 0.093 0.066 0.080 0.078 0.069 0.080 0.088 0.074 0.068

0.65 Cy 0.016 0.013 0.018 0.014 0.011 0.013 0.018 0.014 0.011Cx 0.101 0.073 0.089 0.084 0.074 0.085 0.096 0.081 0.0740.60Cy 0.013 0.009 0.014 0.011 0.009 0.010 0.014 0.011 0.009Cx 0.110 0.078 0.100 0.090 0.079 0.091 0.106 0.088 0.0790.55Cy 0.010 0.008 0.011 0.009 0.006 0.008 0.011 0.009 0.008Cx 0.119 0.083 0.110 0.096 0.084 0.098 0.115 0.095 0.0840.5Cy 0.008 0.005 0.009 0.006 0.005 0.006 0.009 0.063 0.005


14/17

81.

Table. ACI 1963 Coefficients For Negative Moments In SlabsDecreased by 10%.

Case

1

Case

2

Case

3

Case

4

Case

5

Case

6

Case

7

Case

8

Case

9

Cx 0.041 0.045 0.068 0.064 0.030 0.0551.00Cy 0.041 0.068 0.045 0.064 0.055 0.030Cx 0.045 0.050 0.071 0.068 0.034 0.0590.95Cy 0.037 0.065 0.041 0.060 0.050 0.026Cx 0.050 0.054 0.072 0.071 0.039 0.0610.90Cy 0.033 0.063 0.036 0.056 0.047 0.023Cx 0.054 0.059 0.074 0.075 0.044 0.0650.85Cy 0.028 0.059 0.031 0.051 0.041 0.019Cx 0.059 0.064 0.075 0.077 0.050 0.0680.80Cy 0.024 0.055 0.026 0.046 0.037 0.015Cx 0.062 0.068 0.077 0.079 0.055 0.0700.75Cy 0.020 0.050 0.022 0.040 0.032 0.013Cx 0.067 0.073 0.077 0.082 0.061 0.0730.70Cy 0.015 0.045 0.017 0.034 0.026 0.010Cx 0.069 0.077 0.078 0.084 0.067 0.0750.65Cy 0.013 0.039 0.014 0.028 0.022 0.007Cx 0.073 0.080 0.079 0.086 0.072 0.0770.60Cy 0.009 0.032 0.010 0.022 0.016 0.005Cx 0.076 0.083 0.080 0.086 0.077 0.0770.55Cy 0.006 0.025 0.007 0.017 0.013 0.005Cx 0.077 0.085 0.081 0.087 0.080 0.0790.5Cy 0.005 0.020 0.005 0.013 0.009 0.003


15/17

82. f c = 20 MPa : f y = 300 MPa ; = 0.9

0

2

4

6

8

10

12

14

16

18

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

M u /Ag h , MPa

P u

/ A g ,

M P a

= 0.015

= 0.020

= 0.025

= 0.030

= 0.010


16/17

83. f c = 20 MPa : f y = 300 MPa ; = 0.75

0

2

4

6

8

10

12

14

16

18

0 0.5 1 1.5 2 2.5 3 3.5 4

M u /Ag h , MPa

P u

/ A g ,

M P a

= 0.015

= 0.020

= 0.025

= 0.030

= 0.010


17/17

84. f c = 20 MPa : f y = 300 MPa ; = 0.6

0

2

4

6

8

10

12

14

16

18

0 0.5 1 1.5 2 2.5 3

M u /Ag h , MPa

P u

/ A g ,

M P a

= 0.015

= 0.020

= 0.025

= 0.030

= 0.010

Documents

M.sc. Concrete Formulae