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T- Beam Bridge (Girder Bridge)

Introduction:

1- T-beam bridges are generally more economical for spans of 12 to 18 m.

2- The girder stem thickness usually varies from 35 to 55 cm and is controlled by the

required horizontal spacing of the positive moment reinforcement.

3- This type of bridges composed of slab supported by girders, and girders supported by

piers and abutments.

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Design of super structure (T – Beam Bridge):

For the above shown bridge design the super structure elements using ultimate strength design

method? Assume any dimensions you don’t know?

Steps of design:

1- Design of slab:

a- Load calculations:

Check the minimum depth of superstructure (not only slab) according to table

2.5.2.6.3-1

t= 0.07*L = 0.07 *15000 = 1050 mm (total depth (slab+girder)

Actual t = 800+300 =1100 mm >1050 mm ok;

Dead load due to weight of slab W

DD

= 0.3*24*1m width of slab =7.2 kN/m

Dead load due to wearing surface W

DW

= 0.075*18 * 1m width =1.35 kN/m

Dead load due to barrier and walk side at cantilever side =

(0.8*0.4*24*1)+(0.2*0.65*24*1) = 10.8 kN/m

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b- Design of overhanging part:

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Moment calculations:

Moment due to deck weight and any permanent objects connected to the slab =

M

DD

= 7.2(1.8

2

)/2 + (7.68*1.6)+(3.12*1.08)= 27.32 kN m

M

DW

= 1.35*(0.75

2

)/2= 0.38 kN m

Moment due to truck load, impact and pedestrian load (live load) according to article No

3.6.1.6 and 3.6.1.3.3 =

M

LL

= (1.33*72.5*0.45) + (4.1*1*0.65*1.1) = 46.3 kN m

∑η

i

γ

i

Q

i

≤ R

n

= R

r

η

i =

η

D

η

R

η

I

≥ 0.95

η

i

=0.95 * 1.05 * 1

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η

i

= 0.99 ≥ 0.95 OK

M

u

= ηi * [γ

d

M

D

+ γ

w

M

w

+ γ

l

M

l+im]

With reference to tables 3.4.1.1 and 3.4.1.2 in AASHTO then we can find the values of γ

i

M

u

= 0.99 * [1.25*(27.32) + 1.5*(0.38) + 1.75 * (46.3)] = 114.6 kN m

Shear calculations:

Shear due to deck weight and any permanent objects connected to the slab =

V

DD

= 7.2(1.8) + (7.68)+(3.12)= 23.76 kN

V

DW

= 1.35*(0.75) = 1.01 kN

Shear due to truck load, impact and pedestrian load (live load) according to article No

3.6.1.6 and 3.6.1.3.3 =

V

LL

= (1.33*72.5) + (4.1*1*0.65) = 99 kN

V

u

= 0.99 * [1.25*(23.76) + 1.5*(1.01) + 1.75 * (99)] = 202.4 kN

Flexural design of cantilever part:

d= 300-25 – 10 =265 mm

Mu< Ø Mn

Ø Mn = 0.9*0.85*f’c a b (d-(a/2)) =0.9* As fy (d-(a/2))

114.6*10

6

= 0.9*0.85*35*1000*a (265-0.5a)

Solving quadratic equation get a = 16.7 mm

114.6*10

6

= 0.9* As (420) (265-(16.7/2))

As= 1182 mm

2

Choose Ø 16 with A

b

= 200 mm

2

No of bars = 1182/200 = 6 Ø 16 / meter long of bridge deck

Spacing = 1000/5 = 200 mm Ø 16@200mm

Check minimum steel and maximum:

As min= 0.25*√35 * 1000* 265/420 = 933 <1182 ok

Check maximum steel:

As

(εs=0.005)

= (0.85*35*1000*a)/420

a =β1 c

c= (0.003/0.008)*265=99.37mm

β1=0.85- (0.05*(35-28)/7) = 0.8

As max= 0.85*35*1000*0.8*99.37/420=5630 > 1182 ok

Check shear at cantilever part:

Vc= 0.17√fc’ b d= 0.17*√35 *1000*(300-25-8) *10

-3

= 268.5 kN

ØVn= 0.75*268.5 = 201.4 kN ≈ 202 kN

Actually we should check the shear at distance d from face of support (face of girder in

this case) Vu = 146.17 kN (linear interpolation) < 201.4 kN ok;

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**** Don’t forget the shrinkage and distribution steel as per the design of Slab Bridge;

c- Live load calculations on slab:

NL = 6/3.6 = 1.67 = 2LANES according to article No. 3.6.1.1.1

The above article (3.6.1.3.3.) means that the lane load will not apply in case of design deck in

transverse direction

M

TL

≈ 0.8* P

72.5

(S+0.6)/32 = 0.8* 72.5 (1.4+0.6)/32 = 3.6 kN m

M

IM

= 0.33*3.6 = 1.18 kN m

M

DD

= 7.2*(

1.4

2

10

) = 1.4 kN m for both positive and negative moment for continuous spans

M

Dw

= 1.35*(

1.4

2

10

) = 0.26 kN m

∑η

i

γ

i

Q

i

≤ R

n

= R

r

η

i =

η

D

η

R

η

I

≥ 0.95

η

i

=0.95 * 1.05 * 1

η

i

= 0.99 ≥ 0.95 OK

M

u

= ηi * [γ

d

M

D

+ γ

w

M

w

+ γ

l

M

l+im]

With reference to tables 3.4.1.1 and 3.4.1.2 in AASHTO then we can find the values of γ

i

M

u

= 0.99 * [1.25*(1.4) + 1.5*(0.26) + 1.75 * (3.6+1.18)] = 10.4 kN m

d= 300- 25-8 = 267 mm

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Ø Mn = Mu = 10.4*10

6

= 0.9*0.85*fc’ a b (d- a/2) = 0.9*As fy (d -a/2)

10.4*10

6

= 0.9*0.85*35 a (1000) (267- a/2)

Solving quadratic equation get a=1.46mm

As =

10.4∗10

6

420∗(267−0.73)

= 93 mm

2

< As min ; use minimum reinforcement ratio

Use minimum reinforcement ratio As = 0.0025*1000*253 = 632.5 mm

2

As min= 0.25*√35 * 1000* 265/420 = 933 mm

2

choose Ø14 with Ab= 153 mm

2

No of bars = 933/ 153 = 6 bars

Spacing = 1000/ 5 = 200 mm ; choose Ø14 @200 mm top and bottom reinforcement

H W Check the shear capacity of slab in transverse direction?

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2- Design of girders in longitudinal direction (traffic direction):-

a- Interior girders:

Hints:

**

Mg= m (Mn lanes/NL)

Where m= multi presence factor

M

n lanes

= Moment due to NL No. of lanes

NL = number of lanes

**

Distribution factor:

DF is depends on the following:

a- Type of deck

b- Spacing between girders or supporting elements

c- NL

d- Support location

to find D F for moment for interior girders and with reference to table 4.6.2.2.1-1

in AASHTO specify the type of cross section (a;b;c….etc) in this case the type of

bridge cross section is type (e).

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then according to table 4.6.2.2.2b- 1 below and for number of beams (girders) >4 then

the distribution factor can be calculated from equations be low ; you should apply both

equations for one design lane loaded and two or more then take the maximum value of D.F.

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3

In this case n=1

I= 400*

800

= 1.706 x 10

10

mm

4

12

A= 800*400 = 320000 mm

2

e

g

= 400+150 = 550 mm

K

g

= 1 * (1.706 x 10

10

* + (320000 * 550

2

)) = 1.138 x10

11

Ok. Since the value of K

g

is between 4 x10

9

and 3 x10

12

Type of super structure according to table 4.6.2.2.1-1 is type (e)

Enter to table 4.6.2.2.2.b-1 to find the value of D.F

N

b

= 4 girders in our case then

From equation of one design lane loaded:

DF = 0.06+ (

1400

4300

)

0.4

(

1400

15000

)

0.3

(

1.138∗10

11

15000∗300

3

)

0.1

= 0.34

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For interior girders use D.F = 0.84

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NOTE : LEVER RULE

To find the DF using lever rule in case of number of girders = 3 as indicated from table

then we should use the following method (method of lever rule)

1- Find reactions from the deck at the first interior girder by taking the right hand side of

bridge only by assuming internal hinge at the first interior girder

R*1.4 - 72.5*(1.4) +72.5*(1.8-1.4) = 0

R = 51.78 kN

2- Divide the reactions by 72.5kN and multiply by m (in this case the right hand side of

bridge capacity = one lane only then m=1.2)

3- DF=

𝑅 𝑚

72.5

To calculate the moment due to tandem or truck load we will use the influence lines tool:

a- Truck load

M

truck

= 1.6*35+3.75*175+1.6*175 = 992.25 kN m

b- Tandem load

M

tandem

= 110*3.75+3.15*110 = 759 kN m

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Use M

truck

= 992.25 kN m

M

IM

= 0.33*992.25 =327.44 kN m

M

LL

= 9.3 * 15

2

/8 = 261.6 kN m

All of above moment values are per lane;

M

live

per girder = DF (for moment) * 0.5*m*( M

truck +

M

LL+

M

IM

)

=0.84*1*0.5*(992.25+327.44+261.6) = 664.14

kN m

To calculate the value of dead load due to wearing surface and deck weight per girder we

should find the value of flange width of T-Beam:

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Then ;

b

f

= the smallest of

150000/4= 3750 mm

or 12*300+400 = 4000mm

or 1400 mm

b

f

= 1400 mm

W

DD

= 24* (0.3*1.4)+24*(0.8*0.4) = 17.76 kN/m

W

Dw

= 18* (0.075*1.4) = 1.89 kN/m

M

DD

= 17.76*15

2

/ 8 = 500 kN.m

M

Dw

= 1.89*15

2

/ 8 = 53.15 kN.m

Mu = 0.99 * [1.25*(500) + 1.5*(53.15) + 1.75 * (664.14)] = 1848.3 kN m

Design of concrete T beam section with flange in compression and span = 15m length

Dimensions of beam (girder) :

bw= 400mm

ts=300 mm

bf= 1400 mm

h= 1100 mm

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d= 1100-40-10-15 = 1035mm (expected bar diameter for main Rft = 30 mm)

Note : flexural design of T girder will be explained in class

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To design the shear reinforcement for the interior girders:

1- Find DF from table 4.6.2.2.3.a-1 for interior beams

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1

1

V

truck

= 175 *1 + 175*0.71 + 35 *0.43 = 314.3 kN

V

tandem

= 110*1 + 110 * 0.92 =211.2 kN

Design shear according to truck load;

V

IM

= 0.33* 314.3 = 103.7 kN

V

LL

= 9.3*(15)*0.5 = 69.75 kN

V

truck

= 314.3 kN

V

live load

= 314.3+69.75+103.7= 487.75 kN/per lane

V

live load

= DF * m*

∗(V

truck+

V

LL+

V

IM

) = 0.57*1*

2 2

∗( (487.75) =139.0 kN/girder

3- Calculate the shear force per girder for dead load:

b

f

= 1400 mm

V

DD

=

17.76

2

∗ 15 = 133.2 kN

V

DW

=

1.89∗ 15

2

= 14.18 kN

Vu = 0.99 * [1.25*(133.2) + 1.5*(14.18) + 1.75 * (139)] = 426.7 kN m

2- Calculate the shear force due to live load from influence lines of shear:

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Note : shear design of T girder will be explained in class

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