Box Culvert by B.C.punmia Example 30.2

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DESIGN OF BOX TYPE CULVERT DESIGN OF BOX TYPE CULVERT 1

1 In In siside de didiamamenentitiononss

3.50

3.50 m

m x

x

3.50

mm 2

2 SuSupeper imr impoposesed lod loadad

12000

3

3 LLiivve e llooaadd

45000

4

4 WiWiegeght ht oof sf sooilil

18000

5

5 AnAnglgle oe of rf repepososee

30

DegreeDegree 6

6 NoNomiminanal l cocovever r totop p / b/ botottotomm

50

mm mm Nominal Nominal cover cover sideside

50

mmmm 6

6 CCooccrreettee MM--

20

wt. of concretewt. of concrete

25000

7

mm

13

7

7 SStteeeell

4

1

5

1

5

0

1

9

0

8

8 ThThicickekess ss of of siside de wawallll

₃₃₀

mmmm thickness of side wall is OKthickness of side wall is OK Thickness of top slab

Thickness of top slab

₃₂₀

mmmm O.K.O.K.

Thickness of bottom slab

₃₅₀

mmmm 9 9 Reinforcement Reinforcement T Toop p ssllaabb MaMaiinn

20

130

mm c/cmm c/c Distribution Distribution

8

130

mm c/cmm c/c At supports At supports

₈

₂₀₀

mm c/cmm c/c B

Boottttoom m ssllaabb MaMaiinn

₂₀

₁₂₀

mm c/cmm c/c Distribution

Distribution

₈

₁₂₀

mm c/cmm c/c At supports

At supports

₈

₃₀₀

mm c/cmm c/c Through out slab at bottomThrough out slab at bottom S

Siidde e vveerrttiiccaal wl waallll VVeerrttiiccaall

₂₀

₃₀₀

mm c/cmm c/c Both sideBoth side O.K.O.K. Distribution Distribution

8

130

mm c/cmm c/c 2 200 226600 mmmmcc//cc 8 8 220000 mmmmCC//CC 8 8 113300 mmmmCC//cc 320 320 7 70000 2200 2 200 113300 mm C/Cmm C/C 3 30000 mmm m CC//CC 3.50 3.50 8 8 1 13300 mmm m CC//CC 20 20 1 12200 mmm m cc//cc 350 350 20 20 88 88 2 24400 mmm m cc//cc ₂₂₀₀₀₀ _C_C//C_C ₁₁₃₃₀₀ _C_C// N/m N/m33 N/m N/m22 N/m N/m22 kg/m kg/m33 σ σ cbc cbc N/mN/m22 ter side ter side σσ st st N/mN/m 2 2 _σ_σ st st N/mN/m22 mm mmΦΦ_@_@ mm mmΦΦ_@_@ mm mmΦΦ_@_@ mm mmΦΦ_@_@ mm mmΦΦ_@_@ mm mmΦΦ_@_@ mm mmΦΦ_@_@ mm mmΦΦ_@_@ mm mmΦΦ_@_@ mm mmΦΦ_@_@ mm mmΦΦ_@_@ mm mmΦΦ_@_@ mm mmΦΦ_@_@ mm mmΦΦ_@_@ mm mmΦΦ_@_@ mm mmΦΦ_@_@ _mm_mmΦΦ_@_@ _mm_mmΦΦ_@_@

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[email protected]

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DESIGN OF BOX TYPE CULVERT DESIGN OF BOX TYPE CULVERT 1

1 IIn n ssiidde e ddiiaammeennttiioonnss 33..55 xx 33..55 mm 2 S

2 Suuppeer r iimmppoosseed d llooaadd 1122000000 3

3 LivLive e loaload d 4504500000 4

4 WWiieegghht t oof f ssooiil l 1188000000 wwtt. . oof f wwaatteer r 99880000 5

5 AAnngglle e oof f rreeppoossee 3300 DDeeggrreeee 6

6 NNoommiinnaal l ccoovveer r ttoopp//bboottttoom m 5500 mmmm NNoommiinnaal l ccoovveer r SSiidde e 5500 mmmm 6

6 CCooccrreettee M M -- 2020 wwtt. . oof f ccoonnccrreettee 2255000000 7 7 mm 1133 7 7 SStteeeell FFyy 441155 119900 150 150 1

1 SolSolutiution on GeGenranrall

For the purpose of design ,

For the purpose of design , one metre length of the box is one metre length of the box is considered.considered. The analysis is done for the f

The analysis is done for the following cases.ollowing cases. (I) Live load, dead load and earth

(I) Live load, dead load and earth prssure acting , with no water prssure acting , with no water pressure from inside.pressure from inside. (II) Live and dead load on

(II) Live and dead load on top and earth pressure acting from out side, top and earth pressure acting from out side, and water pressure acting from insand water pressure acting from ins with no live load on sides

with no live load on sides

(III) Dead load and earth pressure acting from

(III) Dead load and earth pressure acting from out side and water pressure from in out side and water pressure from in side.side.

L

Leet t tthhe e tthhiiccnneesss s oof f HHoorriizzoonnttaal l ssllaabb 333300 mmmm == 00..3333 mm V

Veerrttiiccaal l wwaalll l tthhiiccnneesss s 332200 mmmm == 00..3322 mm E

Effffeeccttiivve e ssllaab b ssppaan n 3..535 ++ 00..3333 == 33..8833 mm E

Effffeeccttiivve e HHeeiigghht t oof f wwaallll 33..55 ++ 00..3322 == 33..8822 mm

2

2 Case 1 : Dead and live Case 1 : Dead and live load from out side of while no water pressure from inside.load from out side of while no water pressure from inside. S

Seellf f wweeiigghht t oog g ttoop p ssllaab b == 00..3333 xx 11 xx 11 xx 2255000000 == 88225500 L

Liivve e llooaad d aannd d ddeeaad d llooaad d == 4455000000 ++ 1122000000 == 5577000000 T

Toottaal l llooaad d oon n ttoop =p = 6655225500 W

Weeiigghht t oof f ssiidde e wwaalll l == 33..8822 xx 00..3322 xx 2255000000 == 3300556600 NN//mm 6 65522550 x0 x 33..8833 ))++(( 22 xx 3300556600 )=)= 81208.22 81208.22 3.83 3.83 Ka Ka == 1 1 - - ssiinn 3300 == 11 -- 0..505 == 0.50.5 == 11 == 00..3333 1 1++ssiinn 3300 1 +1+ 00..55 1.51.5 33 p p == 5577000000 xx 0..30333 == 1199000000 L

Laattrraal l pprreessssuurre e dduue e tto o ssooiil l KKa a x x w w x x hh == 00..333 x3 x 181800000 h 0 h == 60006000 hh H

Heenncce e ttoottaal l pprreessssuurre e == 1199000000 ++ 66000000 hh L

Laattrraal l pprreessuurre e iinntteenncciitty y aat t ttoopp == 1199000000 La

Latrtral al ppreressssuure re iintnteencnciity ty aat t bbotottotom m == 11909000+00+ 66000000 xx 3.823.82 == 4411992200 w w == 6655225500 1 199000000 1199000000 A A EE BB h h 33..8833 1 199000000 33..8822 60 600000 hh D D FF CC 4 411992200 1199000000 2222992200 # ##### N/m N/m33 N/m N/m22 N/m N/m22 _N/m_N/m33 N/m N/m33 σ σ cbc cbc N/mN/m22 Out side Out side σσ st st N/mN/m22

water side side water side side σσ

st st N/mN/m22 N/m N/m22 N/m N/m22 N/m N/m22 ∴

∴ _{Upward soil}_{Upward soil reaction at}_{reaction at base =}_{base = ((}

N/m N/m22

∴

∴ _{Latral pressure due to dead load and live load =}_{Latral pressure due to dead load and live load = Pv x Ka}_{Pv x Ka}

N/m N/m22 N/m N/m22 N/m N/m22

Fig 1 show the box culvert Fig 1 show the box culvert frame

frame ABC ABC _{D, along with the external}_{D, along with the external} loads, Due to symmetry, half of loads, Due to symmetry, half of frame (i.e.

frame (i.e. AEFD AEFD_{) of box culvert is}_{) of box culvert is} considered for moment distribution. considered for moment distribution. Since all the members have uniform Since all the members have uniform thickness, and

thickness, and uniform diamentions,uniform diamentions, the relative stiffness K for

the relative stiffness K for ADAD_{will be}_{will be} equal to 1 while the relative stiffness equal to 1 while the relative stiffness for

for AE AE _and_andDF DF _{will be 1/2.}_{will be 1/2.}

N/m N/m22

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1

= 2/3 1/2 = 1/3

1+1/2 1+1/2

Fix end moments will be as under : = 65250 x 3.82 ### N - m

12 12

+ = ### x 3.82 98751.91 N - m

12 12

+ + WL Where W is the total tringular earth pressure.

12 15 + 19000 x 3.82 22920 x 3.82 x 3.82 = 34254 N-m 12 2 15 - - WL 12 15 - 19000 x 3.82 22920 x 3.82 x 3.82 = -23105 -16723 = -39 12 2 10

The Moment distribution is carried out as illustrate in table

Fixed End Moments

Member DC DA AD AB 55075 65250 98751.91 -39828 34254 -79346 46852 Joint D A 124627.5 1246 Member DC DA AD AB 46852 55075 Distribution factore 0.33 0.67 0.67 0.33 19000

Fix end moment 98751.91 -39828 34254 -79346 A A

Balance -19641 -39283 30061 15031 55075 1.91 Carry over 15031 -19641 7197 balance -5010 -10020 13094 6547 3.82 m Carry over 6547 -5010 balance -2182 -4365 3340 1670 71023 1.91 Carry over 1670 -2182 22920 D D balance -557 -1113 1455 727 41920 69810 71023 Carry over 727 -557 balance -242 -485 371 186 155514 155 Carry over 186 -242 69810 balance -62 -124 162 81 Carry over 81 -62 81208 balance -27 -54 41 21 Fig 2 Carry over 21 -27 balance -7 -14 18 9 Final moment 71023 -71023 55075 -55075

For horizontal slab AB, carrying UDL @ 65250

Vertical reactionat a and B = 0.5 x 65250 x 3.82 = 124627.5 N/m2 Similarly, for the Bottom slab DC carrying U.D.L.loads @ ###

Vertical reaction at D and C = 0.5 x 81208.22 x 3.83 = ### N

The body diagram for various members, including loading, B.M. And reactions are shown in fig.2 For the vertical member AD, the horizontal reaction at A is found by taking moments at D.Thus

( -ha x 3.83 ) + 55075 - 71023 + 19000 x 3.83 x 3.83 x 1/2

+ 1/2 22920 3.83 3 83 1/3

[email protected]

Distribution factore for AD_andDA= Distribution factore for AB_andDC=

M_FAB= wL 2 2₌ M_fdc= wL 2 2₌ M_FAD= pL 2 M_FAD= 2₊ M_FDA= pL 2 M_FDA= 2-

-The moment distribution carried out as per table 1 for

case 1

N/m2_.

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Hence , hd =( 19000 + 41920 )x 3.83 - 46852 = 69810 N 2

Free B.M. at mid point E = 65250 x 3.83 119644 N-m 8

Net B.M. at E = 119644 - 55075 = 64569 N-m Similarly, free B.M. at F = 81208.22 x 3.83 ### N -m

8

Net B.M. at F = 148904.42 - 71023 = 77881 N-m For vertical member AD , Simply supported B.M. At mid span

imply supporetd at mid sapn 19000 x 3.83 1/16 x 22920 x 3.83 ### 8

Net B.M. = 71023 + 55075 = 63049 - ### = 7197 N-m

2

3 Case 2 : Dead load and live load from out side and water pressure from inside.

In this case , water pressure having an intensity of zero at A and 9800 x 3.82 = 37436

w = 65250 19000 19000 190 Itensity = 19000 A E B 14516 And = 41920 - 37436 = 4484 3.83 3.82 D F C 41920 41920 4484 w = ### Fig 3

12 12

= ### x 3.83 99269.61 N - m

12 12

+ + WL Where W is the total tringular earth pressure.

12 10 + 4484 x 3.83 14516 x 3.83 x 3.83 = 16128 N-m 12 2 10 - - WL 12 15 - 4484 x 3.83 14516 x 3.83 x 3.83 = -12579 N -m 12 2 15

The moment distribution is carrired out as illustred in table.

Fixed End Moments

Member DC DA AD AB 45069 65250 99269.61 -12579 16128 -79762 23451 Joint D A 124627.5 1246 Member DC DA AD AB 45069 Distribution factore 0.33 0.67 0.67 0.33 19000 23451

Fix end moment 99269.61 -12579 16128 -79762 A A

[email protected] 2₌ 2₌ 2₊ 2₌ N/m2

At D, is acting, in addition to the pressure considered in case 1. The various pressures are marked in fig 3 .The vertical walls will thus be subjected to a net latral pressure of

N/m2 N/m2_{At the Top} N/m2_{at the bottom} N/m2 M_FAB= wL 2 2₌ M_fdc= wL 2 2₌ M_FAD= pL 2 M_FAD= 2₊ M_FDA= pL 2 M_FDA= 2

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balance -7070 -14141 19265 9632 3.82 Carry over 9632 -7070 balance -3211 -6422 4714 2357 58813 1.91 Carry over 2357 -3211 4484 D D balance -786 -1571 2141 1070 21404 58813 Carry over 1070 -786 balance -357 -714 524 262 155108 1551 Carry over 262 -357 21404 balance -87 -175 238 119 Carry over 119 -87 81208 balance -40 -79 58 29 Fig 4 Carry over 29 -40 balance -10 -19 26 13 Final moment 58813 -58813 45069 -45069

Vertical reactionat a and B = 0.5 x 65250 x 3.82 = 124627.5 N/m2 Similarly, for the Bottom slab DC carrying U.D.L.loads @ 81208

Vertical reaction at D and C = 0.5 x 81208 x 3.82 = ### N

The body diagram for various members, including loading, B.M. And reactions are shown in fig.3 For the vertical member AD, the horizontal reaction at A is found by taking moments at D.Thus

( -ha x 3.82 ) + 45069 - 58813 + 4484 x 3.82 x 3.82 x 1/2 + 1/2 x 14516 x 3.82 x 3.82 x 2/3 -ha x 3.82 + -13744 + 32716.16 + 70607.76 From which, ha = 23451 Hence , hd =( 4484 + 19000 )x 3.82 - 23451 = 21404 N 2

Net B.M. at E = 119020 - 45069 = 73951 N-m Similarly, free B.M. at F = 81208 x 3.82 ### N -m

8

imply supporetd at mid sapn 4484 x 3.82 1/16 x 14516 x 3.82 21418 8

Net B.M. = 58813 + 45069 = 51941 - 21418 = 30523 N-m

2

4 Case 3 : Dead load and live load on top water pressure from inside no live load on side. in this case, it is assume that there is no latral oressure due to live load . As before .

The top slab is subjected to a load of '= 65250

and the bottom slab is subjected to a load w = 65250

Itensity = 4000 4000

Lateral pressure due to dead load = A E B 4000

1/3 x 12000 = 4000

Lateral pressure due to soil = 3.83

1/3 x 18000 = 6000 3.82

Hence earth pressure at depth h is =

N/m2_. N/m2 2₌ 2₌ 2₊ 2₌ N/m2 N/m2 81208. N/m2 N/m2 N/m2

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Earth pressure intensity at top = 37436w= ### 37436 Fig 5

Earth pressure intensity at Bottom= ### + 6000 x 3.82 = 26920

In addition to these, the vertical wall lslab subjectednto water pressure of intensity ZERO at top and 374 N/m2 at Bottom, acting from inside . The lateral pressure on vertical walls Is shown in fig 5 and 6

12 12

= ### x 3.83 99269.61 N - m

12 12

+ - WL Where W is the total tringular earth pressure.

12 15 + 4000 x 3.83 14516 x 3.83 x 3.83 = -2209 N-m 12 2 15 - + WL ### - ### 12 10 - 4000 x 3.83 14516 x 3.83 x 3.83 = 5757 N -m 12 2 10

The moment distribution is carrired out as illustred in table.

Fixed End Moments

Member DC DA AD AB 35902 65250 3 99269.61 5757 -2209 -79762 = Joint D A 124627.5 1246 Member DC DA AD AB 35902 Distribution factore 0.33 0.67 0.67 0.33 4000 8

Fix end moment 99269.61 5757 -2209 -79762 A A

Balance -35009 -70018 54647 27324 35902 1.91 Carry over 27324 -35009 48748 balance -9108 -18216 23339 11670 3.82 Carry over 11670 -9108 balance -3890 -7780 6072 3036 49646 1.91 Carry over 3036 -3890 0 D D balance -1012 -2024 2593 1297 14516 49646 9 Carry over 1297 -1012 balance -432 -864 675 337 155108 155 Carry over 337 -432 5200 balance -112 -225 288 144 Carry over 144 -112 81208 balance -48 -96 75 37 Fig 4 Carry over 37 -48 balance -12 -25 32 16 Final moment 49646 -49646 35902 -35902

Vertical reactionat a and B = 0.5 x 65250 x 3.82 = 124627.5 N Similarly, for the Bottom slab DC carrying U.D.L.loads @ 81208

Vertical reaction at D and C = 0.5 x 81208 x 3.82 = ### N

The body diagram for various members, including loading, B.M. And reactions are shown in fig.6

4000 N/m2 _N/ N/m2 [email protected] M_FAB= wL 2 2₌ M_fdc= wL 2 2₌ M_FAD= pL 2 M_FAD= 2 -M_FDA= pL 2 M_FDA= 2

-The moment distribution carried out as per table 1 for

case 1

N/m2_.

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- 1/2 x 14516 x 3.82 x 3.82 x 1/3

-ha x 3.82 + -13744 + 29184.8 - 35304

From which, ha = 5200

Hence , hd =( 14516 x 3.82 )- 4000 x 3.82 - 5200 = 7245.56 2

Net B.M. at E = 119644 - 35902 = 83742 N-m Similarly, free B.M. at F = 81208 x 3.83 ### N -m

8

Simply supporetd at mid sapn = 4000 x 3.83 1/16 x 14516 x 3.83 5973.91

8

Net B.M. = 49646 + 35902 = 42774 + ### = 48748 N-m 2

5 Design of top slab :

Mid section

The top slab is subjected to following values of B.M. and direct force Case B.M. at Center (E) B.M. at ends (A) Direct force (ha)

(i) 64569 55075 46852

(II) 73951 45069 23451

(II) 83742 35902 5200

The section will be design for maximum B.M. = 83742 N -m for water side force

= 150 wt. of concrete = 25000

= 7 wt of water = 9800

m

= 13 for waterside f

m*c

= 13 x 7 =

0.378

K = 0.3

13 x 7 + 150

= 1 - 0.378 / 3 =

0.874

J = 0.8

= 0.5 x 7 x 0.87 x 0.378 =

1.155

R = 1.1

Provide over all thickness =

320

mm so effective thicknesss =

270

mm

= 1.155 x 1000 x 270 84216794 > 83742000 O.K.

Ast = 83742000 = 2365

150 x 0.874 x 270

using 20 A = = 3.14 x 20 x 20 = 314

4x100 4

Spacing of Bars = x1000/Ast 314 x 1000 / 2365 = 133 say =

130

mm

Hence Provided

20

130

mm c/c

Acual Ast provided 1000 x 314 / 130 = 2415

Bend half bars up near support at distance of L/5 = 3.83 / 5 = 0.80 m Area of distributionn steel = 0.3 - 0.1 x( 320 - 100 = 0.24 %

450 - 100 = 0.24 x 320 x 10 = 759 # using 8 A = = 3.14 x 8 x 8 = 50 4 100 4 2₌ 2₌ 2₊ 2₌ σ

_{st =}

N/mm2 _N/m3 σ

_{cbc =}

N/mm2 _N/mm2 k=

m*c+

σ

_st

j=1-k/3 R=1/2xc x j x k Mr = R . B .D2 2₌ BMx100/σ_stxjxD= mm 2 mmΦ_bars 3.14xdia 2 mmΦ _{Bars @} mm2

A_st mm2 _{area on each face=}

mmΦ_bars 3.14xdia

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Section at supports :-

Maximum B.M.= 55075 N-m. There is direct compression of 46852 N also. But it effect is not considered because the slab is actually reinforced both at top and bottom .

Since steel is at top = 190 concrete M 20

k = 0.32 J = 0.89 R = 1.01

= 55075000 = 1204

190 x 0.89 x 270

Area available from the bars bentup from the middle section = 2415 / 2 = ### 1204 < ###

6 Design of bottom slab:

The bottom slab has the following value of B.M. and direct force. Case B.M. at Center (F) B.M. at ends (D) Direct force (ha)

(i) 77881 71023 69810

(II) 89315 58813 21404

(II) 99258 49646 7246

The section will be design for maximum B.M. = 99258 N -m for water side force

= 150 wt. of concrete = 25000

= 7 wt of water = 9800

m

= 13 for waterside f

m*c = 13 x 7 =

0.378

K = 0.3 13 x 7 + 150 = 1 - 0.378 / 3 =

0.874

J = 0.8 = 0.5 x 7 x 0.87 x 0.378 =

1.155

R = 1.1

=

99258416 = 294 mm

D

=

344 mm 1000 x 1.155

Provide thickness of bottom slab D=

350

mm so that d =

300

mm

Ast = 99258416 = 2523

150 x 0.874 x 300

using 20 mm bars A = = 3.14 x 20 x 20 = 314

4x100 4

Spacing of Bars = x1000/Ast 314 x 1000 / 2523 = 124 say =

120

mm

Hence Provided

20

120

mm c/c

Acual Ast provided 1000 x 314 / 120 = 2617

Bend half bars up near support at distance of L/5 = 3.83 / 5 = 0.80 m Area of distributionn steel = 0.3 - 0.1 x( 350 - 100 = 0.23 %

450 - 100

= 0.23 x 350 x 10 = 800 400

using 8 mm bars A = = 3.14 x 8 x 8 = 50

4x100 4

Spacing of Bars = Ax1000/Ast = 50 x 1000 / 400 = 126 say =

120

mm

Hence Provided

8

120

mm c/c on each face

Section at supports :-

Maximum B.M.= 71023 N-m. There is direct compression of 69810 N also. But it effect is not considered because the slab is actually reinforced both at top and bottom .

Si t l i t t 190 t M 20 σ st N/mm2 ∴ _Ast _mm2 [email protected] mm2

Hence these bars will serve the purpose. However, provide 8 mm dia. Additional bars @ 200 mm c/c σ

_{st =}

N/mm2 _N/m3 σ

_{cbc =}

N/mm2 _N/mm2 k=

m*c+

σ

_st

j=1-k/3 R=1/2xc x j x k d BMx100/σ_stxjxD= mm 2 3.14xdia2 mmΦ _{Bars @} mm2

A_st mm2 _{area on each face=}

3.14xdia2

mmΦ _{Bars @}

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190 x 0.89 x 300

Area available from the bars bentup from the middle section = 2617 / 2 = ### 1397 > ### Fail , hence additional reinforcement will provided.

Additional reinforcemet required = 88.67

using 8 mm bars A = = 3.14 x 8 x 8 = 50

4x100 4

Spacing of Bars = Ax1000/Ast = 50 x 1000 / 89 = 567 say =

560

mm Hence Provided

8

300

mm c/c throught out the slab, at its bottom.

7 Design of side wall:

The side wall has the following value of B.M. and direct force. Case B.M. at Center (F) B.M. at ends (D) Direct force (ha)

(i) 7197 71023 155514

(II) 30523 58813 155108

(II) 48748 49646 155108

The section will be design for maximum B.M. = 71023 N -m, and direct force = 155514

Eccentricity = 71023 x 1000 = 457 mm

155514

proposed thickness of side wall '= 330 mm ∴ _e_/_D _{457 /} _{330 =} _{1.38 <} _1.5

thickness of side wall is OK

Let us reinforce the section with

20

300

mm c/c provided on both faces, as sho in fig xxx . With cover of 50 mm and D = 330 mm

Asc = Ast = 1000 x 3.14 x 20 x 20 = 1047

300 4

The depth of N.A. is computed from following expression: n 3 3 n = e + D - dt b n + (m -1) Asc n - dc - m Ast D- dt- n 2 n n or 1000 n 330 - 50 - n + 12 x ### x n - 50 x 2 3 n 1000 n+ 12 x ### x n - 50 - 13 x ### x 330 - 50 2 n n 500 n 280 - n + n - 50 x -1256000 3 n = 457 + 115 500 n+ 12560 x n - 50 - ### x 280 - n n n 140000 n - ### + -1256000 - -62800000 n = 572 500 n + 12560 - 628000 - ### + ### n n multiply by n 140000 n2 - ### n3 + -1256000 n - -62800000 = 572 500 2 + 12560 628000 ### + ### mm2 mm2 3.14xdia2 mmΦ_{Bars @} mmΦ_{bars @} mm2 b n D - dt - + (m - 1)Asc 1 (n - dc)(D - dt- dc) n2

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286000 n2 + 14967333 n - 2538459733

-146000 n2 - 13711333 n - -2475659733 = 167

-876 n2 + 82268 n - 14853958 =

Solwing this trial and error we get, n =

91.47

mm

= ( 500 x 91.47 + 12 x ### ( 91.47 - 50 ) - 13 x ###

91.47 91.47

x ( 330 - 50 - 91.47 )

or ### + 137.32 x 41.47 - ### x 188.53 = 23383

= 155514 =

6.65

<

7

Stress is less than permissi 23383

Also stress in steel t = m c' (D-dc-n) = 13 x 6.65 x ( 330 - 50 - 91.47 )

n 91.47

=

178.21

N/mm2 <

190

N/mm2 O.K.

Stress in steel is less than permissiable Hence section is O.K.

n3

∴ _c'

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28 55075 27.5 64569 77881 14

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00 N e t l a t r a l p r e s s u r e d i a g r a m 45069 27.5 73951

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89315 08 a t r a l p r e s s u r e d i a g r a m

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36 5902 27.5 3742 9258 08

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orce 78 74 55 mm2 mm2 mm2

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orce 78 74 55 mm2 mm2 mm2

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N OK n -100 - n mm2

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Box culverts

20 260 mm c/c 8 200 mmC/C 8 130 mm C/c 320 700 20 20 130 mm C/C 300 mm C/C 3.50 8 130 mm C/C 20 120 mm c/c 350 20 8 8 240 mm c/c _{200 mm C/C} _{130 mm C/c} 330 3.50 330 mΦ_@ mmΦ_@ mmΦ_@ mmΦ_@ mmΦ_@ mmΦ_@ mmΦ_@ mΦ_@ _mmΦ_@ _mmΦ_@

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Grade of co M-10 M-15 M-20 M-25 M-30 M-35 M-40 b 1.2 2.0 2.8 3.2 3.6 4.0 4.4 0. 0. 0. 1. 1. (N/mm2) (N/mm2) (N/mm2) 1. M 10 3.0 300 2.5 250 -- -- 1. M 15 5.0 500 4.0 400 0.6 60 2. M 20 7.0 700 5.0 500 0.8 80 2. M 25 8.5 850 6.0 600 0.9 90 2. M 30 10.0 1000 8.0 800 1.0 100 2. M 35 11.5 1150 9.0 900 1.1 110 3.00 an M 40 13.0 1300 10.0 1000 1.2 120 M 45 14.5 1450 11.0 1100 1.3 130 M 50 16.0 1600 12.0 1200 1.4 140 Over all de Grade of co M-10 M-15 M-20 M-25 M-30 M-35 M-40 Modular ra Grade of Grade of concrete M-15 M-20 M-25 M-30 M-35 M-40

Modular Ratio 18.67 13.33 10.98 9.33 8.11 7.18 Grade of concre

5 7 8.5 10 11.5 13 93.33 93.33 93.33 93.33 93.33 93.33 0.4 0.4 0.4 0.4 0.4 0.4 0.87 0.87 0.87 0.87 0.87 0.87 0.87 1.21 1.47 1.73 1.99 2.25 0.71 1 1.21 1.43 1.64 1.86 0.33 0.33 0.33 0.33 0.33 0.33 M 15 0.89 0.89 0.89 0.89 0.89 0.89 M 20 0.73 1.03 1.24 1.46 1.68 1.9 M 25 0.43 0.61 0.74 0.87 1 1.13 M30 0.29 0.29 0.29 0.29 0.29 0.29 M 35 0.9 0.9 0.9 0.9 0.9 0.9 M40 0.65 0.91 1.11 1.31 1.5 1.7 M 45

Table 1.15. PERMISSIBLE DIRECT TENSILE STRESS

Table 3.1

10 Tensile

stress N/mm2

< 0

Table 1.16.. Permissible stress in concrete (IS : 456-2000)

Grade of concrete

Permission stress in compression (N/mm2₎

Permissible stress in bond (Average) for plain bars in tention (N/mm2₎ Bendingα

cbc Direct (α_cc)

Kg/m2 _Kg/m2 _{in kg/m}2

Table 1.18. MODULAR RATIO

Table

31 (31.11) 19 (18.67) 13 (13.33) 11 (10.98) 9 (9.33) 8 (8.11) 7 (7.18) τ c.

Table 2.1. VALUES OF DESIGN CONSTANTS

σ cbcN/mm 2 _τ bd (N / mm 2 mσ cbc (a)σ st= 140 N/mm2 (Fe 250) k_c j_c R_c _{Grade of} concrete P_c(%) (b)σ st= 190 N/mm2 k_c j_c R_c P_c(%) (c )σ st = 230 N/mm2 k_c j_c R_c

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0.31 0.44 0.53 0.63 0.72 0.82 M 50

Reiforcement % Value of angle

M-20 M-20 Degree sin cos tan

bd bd 10 0.17 0.98 0.18 0.15 0.18 0.18 0.15 11 0.19 0.98 0.19 0.16 0.18 0.19 0.18 12 0.21 0.98 0.21 0.17 0.18 0.2 0.21 13 0.23 0.97 0.23 0.18 0.19 0.21 0.24 14 0.24 0.97 0.25 0.19 0.19 0.22 0.27 15 0.26 0.97 0.27 0.2 0.19 0.23 0.3 16 0.28 0.96 0.29 0.21 0.2 0.24 0.32 17 0.29 0.96 0.31 0.22 0.2 0.25 0.35 18 0.31 0.95 0.32 0.23 0.2 0.26 0.38 19 0.33 0.95 0.34 0.24 0.21 0.27 0.41 20 0.34 0.94 0.36 0.25 0.21 0.28 0.44 21 0.36 0.93 0.38 0.26 0.21 0.29 0.47 22 0.37 0.93 0.40 0.27 0.22 0.30 0.5 23 0.39 0.92 0.42 0.28 0.22 0.31 0.55 24 0.41 0.92 0.45 0.29 0.22 0.32 0.6 25 0.42 0.91 0.47 0.3 0.23 0.33 0.65 30 0.50 0.87 0.58 0.31 0.23 0.34 0.7 35 0.57 0.82 0.70 0.32 0.24 0.35 0.75 40 0.64 0.77 0.84 0.33 0.24 0.36 0.82 45 0.71 0.71 1.00 0.34 0.24 0.37 0.88 50 0.77 0.64 1.19 0.35 0.25 0.38 0.94 55 0.82 0.57 1.43 0.36 0.25 0.39 1.00 60 0.87 0.50 1.73 0.37 0.25 0.4 1.08 65 0.91 0.42 2.14 0.38 0.26 0.41 1.16 0.39 0.26 0.42 1.25 0.4 0.26 0.43 1.33 0.41 0.27 0.44 1.41 0.42 0.27 0.45 1.50 0.43 0.27 0.46 1.63 0.44 0.28 0.46 1.64 0.45 0.28 0.47 1.75 0.46 0.28 0.48 1.88 0.47 0.29 0.49 2.00 0.48 0.29 0.50 2.13 0.49 0.29 0.51 2.25 0.5 0.30 0.51 0.30 0.52 0.30 e _P c(%) Shear stress t_c 100A_s 100A_s

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0.53 0.30 0.54 0.30 0.55 0.31 0.56 0.31 0.57 0.31 0.58 0.31 0.59 0.31 0.6 0.32 0.61 0.32 0.62 0.32 0.63 0.32 0.64 0.32 0.65 0.33 0.66 0.33 0.67 0.33 0.68 0.33 0.69 0.33 0.7 0.34 0.71 0.34 0.72 0.34 0.73 0.34 0.74 0.34 0.75 0.35 0.76 0.35 0.77 0.35 0.78 0.35 0.79 0.35 0.8 0.35 0.81 0.35 0.82 0.36 0.83 0.36 0.84 0.36 0.85 0.36 0.86 0.36 0.87 0.36 0.88 0.37 0.89 0.37 0.9 0.37 0.91 0.37 0.92 0.37 0.93 0.37 0.94 0.38 0.95 0.38 0.96 0.38 0.97 0.38 0.98 0.38 0.99 0.38 1.00 0.39 1.01 0.39 1.02 0.39

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1.03 0.39 1.04 0.39 1.05 0.39 1.06 0.39 1.07 0.39 1.08 0.4 1.09 0.4 1.10 0.4 1.11 0.4 1.12 0.4 1.13 0.4 1.14 0.4 1.15 0.4 1.16 0.41 1.17 0.41 1.18 0.41 1.19 0.41 1.20 0.41 1.21 0.41 1.22 0.41 1.23 0.41 1.24 0.41 1.25 0.42 1.26 0.42 1.27 0.42 1.28 0.42 1.29 0.42 1.30 0.42 1.31 0.42 1.32 0.42 1.33 0.43 1.34 0.43 1.35 0.43 1.36 0.43 1.37 0.43 1.38 0.43 1.39 0.43 1.40 0.43 1.41 0.44 1.42 0.44 1.43 0.44 1.44 0.44 1.45 0.44 1.46 0.44 1.47 0.44 1.48 0.44 1.49 0.44 1.50 0.45 1.51 0.45 1.52 0.45

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1.53 0.45 1.54 0.45 1.55 0.45 1.56 0.45 1.57 0.45 1.58 0.45 1.59 0.45 1.60 0.45 1.61 0.45 1.62 0.45 1.63 0.46 1.64 0.46 1.65 0.46 1.66 0.46 1.67 0.46 1.68 0.46 1.69 0.46 1.70 0.46 1.71 0.46 1.72 0.46 1.73 0.46 1.74 0.46 1.75 0.47 1.76 0.47 1.77 0.47 1.78 0.47 1.79 0.47 1.80 0.47 1.81 0.47 1.82 0.47 1.83 0.47 1.84 0.47 1.85 0.47 1.86 0.47 1.87 0.47 1.88 0.48 1.89 0.48 1.90 0.48 1.91 0.48 1.92 0.48 1.93 0.48 1.94 0.48 1.95 0.48 1.96 0.48 1.97 0.48 1.98 0.48 1.99 0.48 2.00 0.49 2.01 0.49 2.02 0.49

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2.03 0.49 2.04 0.49 2.05 0.49 2.06 0.49 2.07 0.49 2.08 0.49 2.09 0.49 2.10 0.49 2.11 0.49 2.12 0.49 2.13 0.50 2.14 0.50 2.15 0.50 2.16 0.50 2.17 0.50 2.18 0.50 2.19 0.50 2.20 0.50 2.21 0.50 2.22 0.50 2.23 0.50 2.24 0.50 2.25 0.51 2.26 0.51 2.27 0.51 2.28 0.51 2.29 0.51 2.30 0.51 2.31 0.51 2.32 0.51 2.33 0.51 2.34 0.51 2.35 0.51 2.36 0.51 2.37 0.51 2.38 0.51 2.39 0.51 2.40 0.51 2.41 0.51 2.42 0.51 2.43 0.51 2.44 0.51 2.45 0.51 2.46 0.51 2.47 0.51 2.48 0.51 2.49 0.51 2.50 0.51 2.51 0.51 2.52 0.51

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2.53 0.51 2.54 0.51 2.55 0.51 2.56 0.51 2.57 0.51 2.58 0.51 2.59 0.51 2.60 0.51 2.61 0.51 2.62 0.51 2.63 0.51 2.64 0.51 2.65 0.51 2.66 0.51 2.67 0.51 2.68 0.51 2.69 0.51 2.70 0.51 2.71 0.51 2.72 0.51 2.73 0.51 2.74 0.51 2.75 0.51 2.76 0.51 2.77 0.51 2.78 0.51 2.79 0.51 2.80 0.51 2.81 0.51 2.82 0.51 2.83 0.51 2.84 0.51 2.85 0.51 2.86 0.51 2.87 0.51 2.88 0.51 2.89 0.51 2.90 0.51 2.91 0.51 2.92 0.51 2.93 0.51 2.94 0.51 2.95 0.51 2.96 0.51 2.97 0.51 2.98 0.51 2.99 0.51 3.00 0.51 3.01 0.51 3.02 0.51

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3.03 0.51 3.04 0.51 3.05 0.51 3.06 0.51 3.07 0.51 3.08 0.51 3.09 0.51 3.10 0.51 3.11 0.51 3.12 0.51 3.13 0.51 3.14 0.51 3.15 0.51

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d M-15 M-20 M-25 M-30 M-35 M-40 0.18 0.18 0.19 0.2 0.2 0.2 5 0.22 0.22 0.23 0.23 0.23 0.23 0 0.29 0.30 0.31 0.31 0.31 0.32 5 0.34 0.35 0.36 0.37 0.37 0.38 0 0.37 0.39 0.40 0.41 0.42 0.42 5 0.40 0.42 0.44 0.45 0.45 0.46 0 0.42 0.45 0.46 0.48 0.49 0.49 5 0.44 0.47 0.49 0.50 0.52 0.52 0 0.44 0.49 0.51 0.53 0.54 0.55 5 0.44 0.51 0.53 0.55 0.56 0.57 0 0.44 0.51 0.55 0.57 0.58 0.60 5 0.44 0.51 0.56 0.58 0.60 0.62 above _0.44 _0.51 _0.57 _0.6 _0.62 _0.63

pth of slab 300 or more ₂₇₅ ₂₅₀ ₂₂₅ ₂₀₀ ₁₇₅ 150 or less

1.00 1.05 1.10 1.15 1.20 1.25 1.30

concrete M-15 M-20 M-25 M-30 M-35 M-40

1.6 1.8 1.9 2.2 2.3 2.5

10 15 20 25 30 35 40 45 50

-- 0.6 0.8 0.9 1 1.1 1.2 1.3 1.4

Plain M.S. Bars H.Y.S.D. Bars

0.6 58 0.96 60 0.8 44 1.28 45 0.9 39 1.44 40 1 35 1.6 36 1.1 32 1.76 33 1.2 29 1.92 30 1.3 27 2.08 28

. Permissible shear stress Table

_c

in concrete (IS : 456-2000)

A_s Permissible shear stress in concrete t_c N/mm2

.15

Table 3.2. Facor k

.3. Maximum shear stress

τ

c.max

in concrete (IS : 456-2000)

ax

able 3.4. Permissible Bond stress Table

_bd

in concrete (IS : 456-2000)

Table 3.5. Development Length in tension

τ

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1.4 25 2.24 26

Value of angle

tan Degree sin cos

0.18 10 0.17 0.98 0.19 11 0.19 0.98 0.21 12 0.21 0.98 0.23 13 0.23 0.97 0.25 14 0.24 0.97 0.27 15 0.26 0.97 0.29 16 0.28 0.96 0.31 17 0.29 0.96 0.32 18 0.31 0.95 0.34 19 0.33 0.95 0.36 20 0.34 0.94 0.38 21 0.36 0.93 0.40 22 0.37 0.93 0.42 23 0.39 0.92 0.45 24 0.41 0.92 0.47 25 0.42 0.91 0.58 30 0.50 0.87 0.70 35 0.57 0.82 0.84 40 0.64 0.77 1.00 45 0.71 0.71 1.19 50 0.77 0.64 1.43 55 0.82 0.57 1.73 60 0.87 0.50 2.14 65 0.91 0.42

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