Slab Bridges

SLAB BRIDGES

ORIENTATION

Straight

Skew

Types of Concrete Deck Sections

1. Solid slab

2. Composite of in situ concrete in-filling precast beams

3. Voided slab

4. Voided of precast box beams post tensioned transversely

Behaviour, Modeling and Analysis Behaviour

- Understood by the deformation profile of bridge under loading

Cell Distortion in multi cellular decks

Differential deflection of precast beams

Behaviour, Modelling and Analysis

Modelling Simple Mathematical

Numerical Rigorous

Mathematical Models Analysis is done after making a mathematical model of the bridge deck depending on the force transfer and force flow.

Typical mathematical models 1.Simple beam 2.Plate 3.Grillage

Mathematical Models – Beam models • Applicable for narrow slab bridges – Predominant one way action • Adopted mainly for pedestrian bridges Governing equation

∂ 4 wx EI = p( x) 4 ∂x

Mathematical Models- Plate model

Governing equation

 ∂4w ∂4w ∂4w  D 4 + 2 2 2 + 4  = q ∂x ∂y ∂y   ∂x D - flexural rigidity of plate q- intensity of load

Mathematical Models- Orthotropic Plate Theory Applicability- Flexural stiffness differ in two directions

Mathematical Models- Orthotropic Plate Theory Governing equation -

∂4w ∂4w ∂4w Dx 4 + 2 B 2 2 + Dy 4 = q ( x, y ) ∂x ∂x ∂y ∂y

2 Bυ= Dy

where Ex h3 Dx = 12 (1 − υx υy )

(

Dy =

;

2 Dυ = t υ 1 − D xDy

)

E y h3

υ+Dx

x

y

+D4

t

12 (1 − υx υy )

flexural rigidities of plate

x

torsional rigidity of plate

y

Mathematical Models- Grillage Analogy

Original solid slab

Corresponding grillage mesh

Mathematical Models- Grillage Analogy   





A computer-aided method for analysis of bridge decks The deck is idealized as a series of ‘grid’ elements (or grillages), connected and restrained at their joints. Each element is given an equivalent bending and torsional inertia to represent the portion of the deck which it replaces. Bending and torsional stiffness in every region of the slab are assumed to be concentrated in the nearest equivalent grid element. Restraints, load and supports may be applied at the joints between the members, and members framing into a joint may be at any angle.

Mathematical Models- Grillage Analogy

Solid Slab

Mathematical Models- Grillage Analogy Voided slabs

d

Numerical Methods (rigorous) Finite element method: •Plate considered as an assemblage of elements •Displacement at the nodes evaluated using energy theorems •Stress field in the elements obtained from the displacements. Finite strip method : •Structure discretized in one direction(width) and is assumed to be continuous in other direction (longitudinal). •Displacement function will be continuous in longitudinal direction and discrete in width direction.

01

Analysis of slabs under point loads Cl. 305.16, IRC 21 (2000) Annexure B-3, IRC 112 (2011) “The effect of concentrated loads on slabs spanning in one or two directions or on cantilever slabs may be calculated from the influence fields of such loads or by any other rational method. A value of 0.15 may be assumed for Poisson’s ratio.” For solid slabs spanning in one direction, the ‘effective width’ method may be used.

Effective width method for estimating longitudinal moments in slabs under point loads

02

Slabs simply supported on two opposite edges effective width = b

b What does b/B depend on ?

Effective width method for estimating bending moments in slabs under point loads Slabs simply supported on two opposite edges x = 0.1 L B = 0.1 L

L

b/B depends on x/L ?

03

04

b/B increases as x/L increases, up to mid-span

Effective width method for estimating bending moments in slabs under point loads Slabs simply supported on two opposite edges

b/B depends on B/L ?

05

06

b/B increases as x/L increases b/B decreases as B/L increases Effective width be = b + bw = k (1 - x/L) + bw function of B/L

b/L = k (x/L) (1 - x/L)

07

Cl. 305.16.2, IRC 21 - 2000 Cl. B3.2, IRC 112 - 2011

B/L

k

x/L

b/B

0.1

0.40

0.1

0.36

0.25

0.75

0.50

1.00

0.1

0.31

0.25

0.645

0.50

0.86

0.1

0.22

0.25

0.46

0.50

0.62

0.1

0.135

0.25

0.25

0.50

0.375

0.5

1.0

2.0

1.72

2.48

3.00

08

b/L = k (x/L) (1 - x/L) 0.30

0.25

c = (x/L)(1 - x/L)

0.20

0.15

0.10

0.05

0.00 0.00

0.20

0.40

0.60

x/L

0.80

1.00

b/L = k (x/L) (1 - x/L)

09

Cl. 305.16.2, IRC 21 - 2000 Cl. B3.2, IRC 112 - 2011

B/L

k s.s.

k cont.

0.1

0.40

0.40

0.2

0.80

0.80

0.3

1.16

1.16

0.4

1.48

1.44

0.5

1.72

1.68

0.6

1.96

1.84

0.7

2.12

1.96

0.8

2.24

2.08

0.9

2.36

2.16

1.0

2.48

2.24

b/L = k (x/L) (1 - x/L)

10

Cl. 305.16.2, IRC 21 - 2000 Cl. B3.2, IRC 112 - 2011

B/L

k s.s.

k cont.

1.1

2.60

2.28

1.2

2.64

2.36

1.3

2.72

2.40

1.4

2.80

2.48

1.5

2.84

2.48

1.6

2.88

2.52

1.7

2.92

2.56

1.8

2.96

2.60

1.9

3.00

2.60

>=2.0

3.00

2.60

2.50 ?

11

b/L = k (x/L) (1 - x/L) 3.0 2.8 2.6 2.4 2.2 2.0

Simply supported slabs

oo

Continuous slabs

k

1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4

Cl. 305.16.2, IRC 21 -2000 Cl. B3.2, IRC 112 - 2011

0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

B/L

Effective width method for estimating bending moments in slabs under point loads

12

Cl. 305.16, IRC 21 Cl. B3.2, IRC 112 Effective width be = b + bw = k x (1 - x/L) + bw bw is the width of the concentrated area of the load, i.e., the dimension of the tyre or track contact area over the road surface of the slab in a direction normal to the span plus twice the thickness of the wearing coat or surface finish above the structural slab. wearing coat

bw

RC slab

Effective width method for estimating bending moments in slabs under point loads

13

Cl. 305.16, IRC 21 Cl. B3.2, IRC 112 Effective width be = b + bw = k x (1 - x/L) + bw The calculated effective width (1) should not exceed the actual width of the slab cast insitu or width of precast slab unit. (2) should not exceed half the above value plus distance of the load from the unsupported edge in case of loads positioned near the edge. (3) in the case of two or more adjacent loads, should be taken as the sum of the calculated effective widths for each load minus the width of the overlap, provided the slab is also analysed for the two loads acting separately.

14 Cl. 305.16.3, IRC 21 Cl. B3.3, IRC 112 DISPERSION OF LOADS ALONG SPAN The effect of contact of wheel or track load in the direction of the span shall be taken as equal to the dimension of the tyre contact area plus twice the overall depth of the slab inclusive of the thickness of the wearing surface.

wearing coat RC slab contact length

Modelling of RC Culvert Slab Bridge in SAP2000

span = 5.4m, width = 12m plate-shell element (300×300×thickness)

2

3

Longitudinal bending moments due to DL

Simplified Analysis results (kNm/m): 1) at centre: 43.7 ; 2) at edge: 32.4

4

Transverse bending moments due to DL

Simplified Analysis results (20%) (kNm/m): 1) at centre: 8.74 ; 2) at edge:

5

Twisting moments due to DL

Simplified Analysis results (kNm/m): 1) at centre: 0.0 ; 2) at edge:

6

Arrangement of IRC Class AA tracked vehicle loads

7

Longitudinal bending moments due to LL

Simplified Analysis results (kNm/m): 1) at centre: 101.5 ; 2) at edge:

8

Transverse bending moments due to LL

Simplified Analysis results (30%) (kNm/m): 1) at centre: 30.35 ; 2) at edge: 30.35

9

Twisting moments due to LL

Simplified Analysis results (kNm/m): 1) at centre: 0.0 ; 2) at edge:

Deformed Shape under combined DL + LL

10

11

Longitudinal bending moments due to DL+LL

Simplified Analysis results (kNm/m): 1) at centre: 145.2 ; 2) at edge:

12

Transverse bending moments due to DL+LL

Simplified Analysis results (kNm/m): 1) at centre: 39.2 ; 2) at edge:

13

Twisting moments due to DL+LL

Simplified Analysis results (kNm/m): 1) at centre: 0.0 ; 2) at edge:

14

Twisting moment (at location of max. long. moment)

DESIGN OF RC SLAB CULVERTS • •For culverts and minor bridges of total length less than 60m, the width between the outermost faces of the bridge should be the full formation width of the approaches, but not less than 12m for roads on National Highways (Cl. 112.1, IRC 5). • Minimum width of kerb of 500 mm and height of 300 mm above the top of the deck are desirable. • Raised footpaths need not be provided for culverts and minor bridges having total length less than 30m, unless such footpaths exist on the approaches.

DESIGN OF RC SLAB CULVERTS • The deck slab for 2-lane carriageway should be designed for the worst effect of (a) one lane of IRC Class AA / 70R tracked vehicle (b) one lane of IRC Class AA / 70R wheeled vehicle (c) two lanes of Class A loads • In general, it is seen that Class AA wheeled vehicle will give max. bending moment due to live load for spans up to 4m, and Class AA tracked vehicle for larger spans.

DESIGN OF RC SLAB CULVERTS • For maximum shear, it is seen that Class AA wheeled vehicle governs the design for all spans. •The design transverse bending moment (distribution steel) should be taken as 0.3 times the longitudinal moment due to live load plus 0.2 times that due to other loads (such as dead load, shrinkage, temperature, etc). (Cl. 305.18, IRC 21). Revised as : • The transverse reinforcement (distribution steel) should be atleast 20 percent of the main reinforcement. (Cl. 16.6.1.1, IRC 112). Reference: “Standard plans for 3.0 to 10.0m RCC solid slab superstructure with and without footpaths for highways”, MORTH (Roads Wing), 1991.

GENERAL ARRANGEMENT DRAWING

wearing course: either 56mm thick asphalt concrete or 75mm thick cement concrete

DETAILS OF DECK AND APPROACH SLABS

DETAILS OF ABUTMENTS AND WING WALLS

Design Data Clear Span

5.0 m

Width of bridge (without footpath) Exposure condition Width of bearing Wearing course

12.0 m

‘Moderate’ (Table 14.1, IRC 112) 400 mm 56 mm thick asphalt

Use M25 concrete & Fe 415 steel (As per durability requirements in table no. 14.2, IRC 112)

Design and detail of RC deck slab Preliminary dimension •Assume slab thickness ≈ span / 11 = 550 mm (at centre) & sloping to 400 mm (at kerb). •Effective depth (assuming 20φ bars & clear cover 50 mm) d= 490 mm at centre = 340 mm at edge. •Effective span = c/c span = 5000 + 400 = 5400 mm (smaller) = clear span + effective depth = 5490 mm (Cl. 7.6.1.1, IRC 112)

Geometry of deck slab Deck slab

Approach slab

5.4 m

12 m

D2 =400

D1 = 550

Cross section

Dead load effects 11.88 (1.23) kN/m2 Self wt. of RC slab (wearing coat)

L =5.4 m 340 mm

S.F.D

  5.4 11.88 − 0.34  = 28.1 kN/m   2

(2.91 kN/m)

 5.4 2 11.88  8 

B.M.D

  = 43.28 kNm/m   (4.49 kN/m)



Live Load – Class AA tracked

Class AA Load effects •Impact factor fraction •Load position for maximum effects • for maximum moment for maximum shear

Load position for maximum B.M 550 1200

850

1200

505

454 2175

850

2050

2506 454 + 505   3600 + 2 ×  56 +  = 4672 2  

864.5 kN Including impact

L = 5400 mm

Load position for maximum S.F

340

x = 340 +

4672

4672 = 2676 2

864.5 kN

Live load - Class AA (wheeled)

Dispersion width- for class AA wheeled

1900

300

300

1200 300

600

1000

300

600

Load position for maximum effects – class AA wheeled Maximum B.M

x = 2100

Position of wheel load

500 kN (Including impact)

2412

Load position for maximum effects – class AA wheeled Shear Force

x

= 946

340

Position of wheel load

2412

500 kN

Class A & Class B Loading

Class A & Class B Loading

Dispersion width for class A two lane loading 1200

500

500

500

500

150

950

1800

1700

1800

Load position for maximum effects – class A two lane loading Maximum B.M

x

= 2100

Position of wheel load

2542

636 kN (Including impact)

Load position for maximum effects – class A two lane loading Shear Force

x

= 1011

340

Position of wheel load

2542

636 kN

Summary of analysis results

Loading

Max. B.M (kNm/m)

Max. S.F (kN/m)

Class AA - tracked

98.4

64.8

Class AA - wheeled

84.1

72.1

Class A (2 lanes)

77.4

55.9

Grillage Analysis

Bending moment diagram (Class AA tracked)

Results from Grillage Analysis (Bending moment)

Max. Bending Moment (kNm/m) Torsionless grillage

Considering torsional modifier 0.2

Considering full torsion

Effective width method

Class AA - tracked

123.6

114.6

97.5

98.4

Class AA - wheeled

110.4

102

85.8

84.1

Class A (2 lanes)

104.3

95.2

79.1

77.4

Loading



Results from Grillage Analysis (Shear force)

Max. Shear force (kNm/m) Torsion less grillage

Considering torsional modifier 0.2

Considering full torsion

Class AA - tracked

81.8

81.8

82.5

64.8

Class AA - wheeled

82.7

83.5

86.4

72.1

Class A (2 lanes)

67.0

69.3

72.0

55.9

Loading

Effective width method

Plate element model

(Longitudinal bending moment – Class AA tracked)

Results from plate element model

Loading

Longitudianal B.M. (kNm/m)

Transverse B. M. (kNm/m)

Class AA - tracked

103.7

35.9

Class AA wheeled

92.1

38.3

Class A (2 lanes)

88.5

33.5

Reference: “Standard plans for 3.0 to 10.0m RCC solid slab superstructure with and without footpaths for highways”, MORTH (Roads Wing), 1991.

Reinforcement detailing for 5.0m span

Reinforcement detailing for 5.0m span

Working stress method IRC 21: 2000 & Annexure A4 of IRC 112: 2011

Load combinations & permissible stresses (Table 1, IRC 6: 2010)

Permissible stresses in concrete Table A4.2, IRC 112 Table 9, IRC 21 Properties / permissible stresses

M15

M20

M25

M30

M35

M40

M45

M50

1. Modulus of elasticity Ec – design value (GPa)

26

27.5

29

30.5

31.5

32.5

33.5

35

36

37

5

6.25

7.5

8.75

10

11.25

12.5

13.75

15

6.67 8.33

10

11.67 13.33

15

16.67

18.3

20

2. Permissible direct comp. stress 3.75 σcbc (MPa) 3. Permissible flexural comp. stress σ cbc (MPa)

5

M55 M60

Permissible stresses in steel Table A4.4, IRC 112 Table 10, IRC 21 Type of stress in steel reinforcement

Permissible stress in MPa 125

Fe 415

Tension in flexure, shear of combined bending

Fe 500

Tension in flexure or combined bending

240

Tension in shear

200

Bar grade Fe 240

Fe 240 Fe 415

200

115 Direct compression

170

Fe 500

205

Fe 240

95

Fe 415 Fe 500

Tension in helical reinforcement

95 95

Permissible shear stresses in concrete Table A4.6, IRC 112 Table 12B, IRC 21

Design constants For singly reinforced sections,

1 K k b = σ st 1+ mσ cbc

M = Rσ k= j2 0.5 b bd M A st = σ st jb d

cbc b b

Where, m = 10

Where, jb = 1 − kb / 3

Longitudinal Bending moment, M = MDL + MLL = 146 kNm/m kb = 0.294 jb = 0.902 Ast required = 1954 mm2 (20Φ @ 160 mm) Transverse Bending moment, M = 0.2 MDL + 0.3 MLL = 39 kNm/m Ast required = 515 mm2 (10Φ @ 125 mm)

Check for shear Design shear force = VDL + VLL =103.1 kN/m Nominal shear stress ζv= 0.30 MPa For M25 concrete, ζc = 0.31 MPa

ζv < ζc Hence OK

Limit states method IRC 112: 2011

Ultimate limit states





Limit state of equilibrium: When subjected to various design combinations of ultimate loads the bridge or any of its components, considered as rigid body, shall not become unstable. Limit state of strength: The bridge or any of its components shall not lose its capacity to sustain various ultimate load combinations by excessive deformation, transformation into a mechanism, rupture, crushing or buckling.

Serviceability limit states 





Stress limitation: Concrete and steel stresses should be within permissible stresses. (Cl. 12.2) Limit state of cracking: The cracking is kept within acceptable limits of crack widths (Cl. 12.3.4). Allowable crack width depends on exposure conditions. Limit state of deformation: The deformation of bridge shall not affect the proper functioning of its elements, appurtenances and riding quality. Deformations during construction shall be controlled to achieve proper geometry of the structure.

Load factors

(Annexure B, IRC 6: 2010) 







Partial safety factor for verification of equilibrium (Table 3.1) Partial safety factor for verification of structural strength (Table 3.2) Partial safety factor for verification of serviceability limit state (Table 3.3) Combination for base pressure and design of foundation (Table 3.4)

Annexure B, IRC 6: 2010

(Annexure B, IRC 6: 2010)

Annexure B, IRC 6: 2010

Annexure B, IRC 6: 2010

Annexure B, IRC 6: 2010

Serviceability combinations Annexure B, IRC 6: 2010

Materials (Section 7, IRC 112)

Stress strain relationship for concrete in compression for design

Design based on strength Design bending moment

A st f ck = bd 2f y

=1.35 DL + 1.75 Surfacing + 1.5 LL = 214.2 kNm/m

 R 1 − 1 − 4.598 f ck 

  

R=

Mu bd 2

(Parabolic stress block)

Ast Required = 1522 mm2 20 mm Φ @ 200 mm spacing (A st = 1570 mm2) Parabolic stress Bilinear stress block block Ast

A

req.

(mm2)

1522

= 589 mm2 (Cl. 16.5.1.1)

1528

A

Uniform stress block 1518

= 8500 mm2 (Cl. 16.5.1.1)

Transverse reinforcement A st

= 20% of Longitudinal reinforcement (Cl. 16.6.1.1) required = 349 mm2 10mm Φ @150 mm spacing

Minimum top reinforcement = 25% of main reinforcement. (Cl. 16.6.1.3) A st required = 357.7 mm2 10 mm Φ @ 200 mm spacing Maximum spacing as per Cl, 16.6.1.1

Check for shear (Cl. 10.3.2)

Design shear force Ved = 1.35 VDL + 1.75 VSurfacing + 1.5 VLL = 151 kN K = 1.694 ρ1 = 0.00373 Design shear resistance, VRd.c = 164 kN

VRd.c ≥ VEd Hence shear reinforcement not required

Serviceability Section 12, IRC 112

Stress check (Cl 12.2) Permissible stresses  Concrete = 0.48 fck = 12 MPa  Steel = 0.8 fy = 332 MPa

Crack width (Cl. 12.3.4) Allowable crack width = 0.3 mm (Table 12.1)

Deflection (Cl. 12.4.1) Allowable vehicular deflection = span/800 = 6.75 mm

h−x 3

Method

Main Crack reinforcem width ent (DL+ LL)

IRC 21 (WSM)

20 Φ @ 160

IRC 112 strength

20 Φ @ 200

IRC 112 serviceability

20 Φ @ 160

0.27 0.32 0.27

Max. Stress in concrete (MPa)

Max. stress in steel (MPa)

Deflectio Transverse n due to reinforcem live load ent (mm)

7.06

197

1.9

10 Φ @ 125

9.07

244

1.9

10 Φ @ 150

8.28

195

1.9

10 Φ @ 125