8jan_latest final bridge formula (1)

8/12/2019 8Jan_Latest Final Bridge Formula (1)

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SUMMARY OF EQUATIONS

AND TABLES

EN 1992-2 Eurocode 2:

Design of Concrete Structures,

Part 2: Concrete Bridges


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Prestressed Concrete Beam (Additional Sub-section)

Prestressed concreteM beam

- Characteristic value of maximum force =As x

(kN)

-

Characteristic value of 0.1% proof force =Maximum force 90% (kN)- (MPa)- - Stressing and losses, initial stressing = 75% , Losses at transfer = 10% =

0.9

Long-term losses = 20% = 0.8

- - Find the following values:

1. (MPa)2. (MPa)3.

a) Consider horizontal top branch and a neutral axis depth, dneutral- Determine the strain profile (interpolate)- Determine the value for total strains at ULS in the four layers of strands,

including prestrain - Compared between total strains and - If total strains > , therefore all stresses (MPa)- Total steel force, (kN)- Calculate splitting compression zone into the following three sections and

taking account of the different concrete strengths

(MPa) ; (MPa) {

}

= 0.0035; 0.0020


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i) Rectangular part of stress block in top slab, Fc1 (mm)

(mm)

(kN)ii) Parabolic part of stress block in top slab, Fc2 (mm) (mm) * + (kN)

iii) Parabolic part of stress block in the top flange to neutral axis, F c3 (mm) (mm) * +

(kN)

- Fc = Fc1 + Fc2+ Fc3 ; compare with Fs (check the balance of section)- Find moments about the neutral axis level, M

[ ] [ ] (kNm)N1 = Number of strand at second layer.

N2 = Number of strand at bottom layer.

As = Cross-sectional area of strands (mm2)

X1

X2

X3

X4


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tf

Prestressed Sections Uncracked In Flexure Shear Tension

Post-tensioned concrete box girder, un-cracked in flexure

Global prestressing foce= (no.tendon x As x fpk x initial stressing% x (100-totallosses%)) (kN) where ; ; (MPa)

(mm)

( )

= (MPa)

Where hc=height to centroid, zna=neutral axis height , hu=height of underneath= H - tf

(kN)Therefore, resistance for inner web = (bw inner0.5) / (kN)resistance for outer web = (bw outer0.5) / (kN)

Strands at 2ndlayer

Strands at bottom


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Shear At Points of ContraflexureSection Cracked in Flexure

Post-tensioned concrete box girder without tendon drape.

a) Consider vertical links (=90o) and try a compression strut angle, =45o-

(MPa) where

= 1.15

- (mm2/mm)- Using two legs of 20 bars, (mm2)

i.e: can use any size, bars

- Find spacing, s and make conclusion.b) Using vertical links (=90o) and try a compression strut angle, =41.5o(for centre

web)

- (mm2/mm) , refer table 2 to determine spacing- Maximum allowable shear, (kN)- - - -

- Compared with VEd, VEd,so the arrangement is adequate.c) Using link inclined at =45o, =29.5o

- - - - (kN)-

(kN)

- Take the lowest VRd


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Torsion in Slabs (Additional Sub-section)

Box girder bridge

Enclosed area,

(mm2

)

where Area of reinforcement required, (mm2/mm)

Where is the truss angle Equivalent amount of longitudinal reinforcement,

(mm2/mm)

Checking crushing resistance for web, > total applied torque (MN)

- (MPa) ; - - - = thickness of web, tweb (mm)

Lweb


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PunchingPile Cap (Additional Sub-section)

Reinforced concrete pile cap

This is sufficient to resist the applied bending moment at the face of the pier, taken as the

maximum of: (kNm/m)And

(kNm/m)1) The flexure shear plane across the face of the support is checked (1 face side = 2

piles)

Load per pile = NEd/4 (kN) , pile cap width= (B+A) x 2 (m)

(kN/m)

* +(kN), if < VEd ,the shear reinforcement isrequired.

Shear force required to reduces by (kN) if < VEd,new ,theshear reinforcement is still required.


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(kN)

(mm2)

, therefore, conclude theno. of links adopted.

Checking the max.shear force; , if > VEdtherefore the provided shearreinforcement is okay.

2) Face shear at corner pile and column is checked (using uoperimeters). (MPa)

; if therefore, its ok as required. Size of column is relatively small and checked punching at face of column

against shear maximum shear stress limit.

- -

- ; if therefore, its ok asrequired.


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3) Flexural shear across the corner pile is checked.

(mm) (kN/m) (kN/m)

(

)

(kN) ; if

its ok as required.


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Verification of Concrete Under Compression or Shear

Concrete fatigue verification for a concrete road bridge

Maximum and minimum fibre stresses under frequent load combination

- - -

( ) t = 7 days

- s = 0.20 for rapid hardening high strength cements = 0.25 for normal and rapid hardening cement

s = 0.38 for slow hardening cement

- , for fatigue ; (MPa) Final checks for

o Sagging section Compared between maximum and minimum fibre stresses

, fatigue resistance adequate.o Hogging section

Compared between maximum and minimum fibre stresses , fatigue resistance adequate.

0.9 for fck 50 MPa or 0.8 for fck >50 Mpa


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Stress Limitation

Reinforced concrete deck slab

a. Check section whether cracked or un-cracked. Depth to neutral axis, x=h/2 (mm) Second moment of area, I = bh3/12 (mm4/m) ; b=1000 mm Check the un-cracked section for compressive and tensile stress at top and bottom

of section ; section is uncracked or vice versa.Where My= sagging moment (Nmm) ; x = depth to neutral axis ; from table

b. Check for performed at an age when bridge first opens, assuming minimal creep hasoccurred.

Es= 200 GPa (GPa) Depth of concrete in compression, (mm) Cracked second moment of area in steel, (mm4) Concrete stress at top of section, , (mm

3

)

Compression limit Compared between concrete stress at top section and compression limit. Reinforcement stress, , (mm3) Tensile limit

c. Check for performed after all the creep has taken place. Creep factor,

Effective modulus of elasticity, ()(GPa)Mqp = % moment of permanent actions ,Mst= % moment of transient actions from traffic.

Repeated calculation process in (b) above (from dccalculation)


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APPENDIX

Summary of Tables and Graphs


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Table 1 Sectional areas per meter width for various bar spacing (mm)

Bar size Spacing of bars

(mm) 50 75 100 125 150 175 200 250 300

6 566 377 283 226 189 162 142 113 94

8 1010 671 503 402 335 287 252 201 168

10 1570 1050 785 628 523 449 393 314 262

12 2260 1510 1130 905 754 646 566 452 377

16 4020 2680 2010 1610 1340 1150 1010 804 670

20 6280 4190 3140 2510 2090 1800 1570 1260 1050

25 9820 6550 4910 3930 3270 2810 2450 1960 1640

32 16100 10700 8040 6430 5360 4600 4020 3220 2680

40 25100 16800 12600 10100 8380 7180 6280 5030 4190

Table 2 Asv/ sv for varying stirrup diameter and spacing

Stirrup

diameter

(mm)

Stirrup spacing (mm)

85 90 100 125 150 175 200 225 250 275 300

8 1.183 1.118 1.006 0.805 0.671 0.575 0.503 0.447 0.402 0.366 0.335

10 1.847 1.744 1.57 1.256 1.047 0.897 0.785 0.698 0.628 0.571 0.523

12 2.659 2.511 2.26 1.808 1.507 1.291 1.13 1.004 0.904 0.822 0.753

16 4.729 4.467 4.02 3.216 2.68 2.297 2.01 1.787 1.608 1.462 1.34

20 7.392 6.981 6.283 5.027 4.189 3.590 3.142 2.793 2.513 2.285 2.094

25 11.550 10.908 9.817 7.854 6.545 5.610 4.909 4.363 3.927 3.570 3.272

32 18.923 17.872 16.085 12.868 10.723 9.191 8.042 7.149 6.434 5.849 5.362

40 29.568 27.925 25.133 20.106 16.755 14.362 12.566 11.170 10.053 9.139 8.378


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Table 3 Sectional areas of groups of bars (mm)

Bar

sizeNumber of bars

(mm) 1 2 3 4 5 6 7 8 9 10

6 28.3 56.6 84.9 113.2 141.5 169.8 198.1 226.4 254.7 283

8 50.3 100.6 150.9 201.2 251.5 301.8 352.1 402.4 452.7 503

10 78.5 157 235.5 314 392.5 471 549.5 628 706.5 785

12 113 226 339 452 565 678 791 904 1017 1130

16 201 402 603 804 1005 1206 1407 1608 1809 2010

20 314 628 942 1256 1570 1884 2198 2512 2826 3140

25 491 982 1473 1964 2455 2946 3437 3928 4419 4910

32 804 1608 2412 3216 4020 4824 5628 6432 7236 8040

40 1260 2520 3780 5040 6300 7560 8820 10080 11340 12600

8jan_latest final bridge formula (1)

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