PSC Girder Design

PROPERTIES FOR GRILLAGE ANALYSIS

SPAN 18.571 OR say 19 m Length of each segment = 1.1875

Transverse Members

End Crossgirder 0.794

0.240

A Yt Ayt Ayt2

Iself

slab 0.1905 0.1200 0.0229 0.0027 0.0009 1.500

diaph 0.6000 0.9900 0.5940 0.5881 0.1125

0.7905 0.6169 0.5908 0.1134

0.400

Yt = 0.6169 = 0.7803 m

0.7905

Iz = 0.1134 + 0.5908 - 0.6169 x 0.7803

= 0.2229 m4

Ix = 0.002 slab + 0.027 diaph =

= 0.0284 m4

(half for slab & full for diaph)

Intermediate Crossgirder 1.188

0.240

A Yt Ayt Ayt2

Iself

slab 0.2850 0.1200 0.0342 0.0041 0.0014 1.500

diaph 0.4500 0.9900 0.4455 0.4410 0.0844

0.7350 0.4797 0.4451 0.0857

0.300

Yt = 0.4797 = 0.6527 m

0.7350

Iz = 0.0857 + 0.4451 - 0.4797 x 0.6527

= 0.2178 m4

Ix = 0.002 slab + 0.012 diaph =

= 0.0140 m4

(half for slab & full for diaph)

Slab members Transverse

1.188

A = 0.285 m2

0.240

Iz = 1.188 x 0.2403

= 0.00137 m4

12

Ix = 0.00237 m4

(Half )

Effects of Curvature

Torsion due to the effects of curvature has been calculated as per the formulae in Raina.

t = applied torque per unit length = w X e

R = Radius of element at CL of section

t' = any locally applied torque (udl or distributed over a dispersion width)

T = Torsional moment as a result of M, w and t'

M = Moment externally applied (including parasitic moment)

W = shear

w = UDL weight (self wt+SDL etc)

e = r.Iyy / (R.w) ecc of the CG of the uniformly distributed self wt measured from the CL of the section

r = density of the material

Iyy = MI about vertical axis through CG of section.

a = distance from CL of torque span to point under consideration

torque span = Length of structure between points of torsional restraint.

DL (deck slab only) SIDL Total

w = 75.6 + 57.5 = 133.1 kN

Iyy = 40.3 m4

length = 20.5 m

depth = 1.7 m

web = 0.3 m midspan 0.6 m at supp

width = 12.6 m

CG bot = 1.19 m

r = 25.0 kN/cum

e = 0.084 m (outward)

R = 90 m

V = 25 kmph

camber = 3.1%

Centrifugal force = 0.055 x 1000 kN / 20.5 m = 2.67 kN/m

Ecc of CF = 1.200 m (LL) + 0.5 m (cg top)= 1.7 m

1--1 2--2 3--3 4--4 5--5 6--6

Support 0.9D L/8 L/4 3L/8 L/2

Distance 0 1.35 2.321375 4.64275 6.964125 9.2855

Effects of Curvature

Torsion due to DL & SIDL ecc due to curvature = 0.084 m

Shear due to DL + SIDL 1364.3 1184.6 1055.3 746.3 437.3 128.4

Torsion = total shear x ecc 114.6 99.5 88.7 62.7 36.7 10.8

(ecc due to curvature x shear force)

Torsion due to centrifugal force

Coeff of centrifugal force = V.V/(127R) 0.055

Distance betn CG of girder & CG of LL = 1.7 m

torsion due to centrifugal force = 47.8 41.5 37.0 26.1 15.3 4.5

(LL shear x coeff of centrifugal force x ecc)

Torsion due to Bending Moment effect = M/R Radius of Curvature = infinity for the straight girders

Total Additional torsion in superstr due to curvature of the deckslab

Torsion in the total superstr. 162.4 141.0 125.6 88.8 52.1 15.3

Torsion per girder 5 girders = 32.48 28.20 25.12 17.77 10.41 3.06

Additional Moment due to torsion (kNm)

Eq. Moment =T/1.7*(1+d/b) 74.5 64.7 100.5 71.1 41.6 12.2

Design of Post Tensioned Beam Outer GirdersALL DISTANCES ARE IN M, STRESSES IN MPA, FORCES IN KN AND MOMENTS IN KNM

Tension (-) Compression (+), anti-CW (+), CW (-)

0.18

0.10

Girder Length 19.2 m

Span of girder c/c of bearing 18.571 m

L/D 10.67 0.300

Girder overhang on bearing 0.314 m

0.20

0.24

Mid span Support

Width of top flange(PSC) 1.000 1.000 m. Midspan End(supports)

Width of bot flange(PSC) 0.600 0.600 m.

Thk of top flange 0.180 0.180 m.

Thk of bot flange 0.240 0.000 m.

Web thickness 0.300 0.600 m.

Thk of top fl at web 0.280 0.237 m.

Thk of bot fl at web 0.440 0.000 m.

Depth of girder 1.500 1.500 m. 1.8 1.5

Thickness of top slab 0.240 0.240 m. Web Taper profile

Beff of top slab 2.650 2.650 m. =MIN(spacing,12*web thk))*Eslab/Egird

Spacing of girders 2.650

Actual width of slab 2.325 Outer Girder slab

Mid span Supports

SECTION PROPERTIES web top flange bot flange web top flange bot flange Slab

Area 0.450 0.126 0.035 0.072 0.030 0.900 0.072 0.011 0.000 0.00 0.636

x from bottom 0.750 1.410 1.287 0.120 0.307 0.750 1.410 1.301 0.000 0.000 1.620

distance from cg 0.061 0.599 0.476 0.691 0.504 0.055 0.605 0.496 0.000 0.000 0.428

Moment 0.084 0.000 0.000 0.000 0.000 0.169 0.000 0.000 0.000 0.000 0.003

Total Moment 0.182 0.201

Moment I yy 0.003 0.002 0.039 0.000191 0.00459 0.027 0.00031 0.007 0 0

Total Moment Iyy 0.049 0.034

PRE-CAST GIRDER mid span Support Torsional Property (for use in staad)

Area 0.713 0.983 m2 mid end

Area * x 0.578 0.791 slab 0.010 0.010

cg. Of girder 0.811 0.805 m web 0.008 0.071

Mom of inertia 0.182 0.201 m4 top fl 0.002 0.002

Top cg 0.689 0.695 m bot fl 0.002 0.000

Ztop 0.264 0.289 m3 Ixx = 0.022 0.083

Zbot 0.224 0.250 m3

Composite girder mid span Support

Area 1.349 1.619

Area * x 1.608 1.822 m2

cg. Of girder 1.192 1.125 m

Mom of inertia 0.402 0.458 m4

Top cg 0.548 0.615 m

Ztop 0.734 0.744 m3

Zbot 0.337 0.407 m3

Ztop girder 1.307 1.220 m3

(Ay)p 0.045 0.065 m3

(Ay)c 0.317 0.380 m3

Density Concrete (kn/cu.m) 25 Girder fck 40 Mpa Ec 31622.8 Mpa

slab fck 40 Mpa Ec 31622.8 Mpa

1.00

0.60 0.60

Sectional Properties

Section No. 1--1 2--2 3--3 4--4 5--5 6--6

Section at Support 0.9D L/8 L/4 3L/8 L/2

Distance (x) 0.000 1.350 2.321 4.643 6.964 9.286

Area of Girder m2

0.983 0.983 0.889 0.713 0.713 0.713

Moment of Inertia m4

0.201 0.201 0.194 0.182 0.182 0.182

CG of Section (bot) m 0.805 0.805 0.807 0.811 0.811 0.811

Z bot of section m4

0.250 0.250 0.241 0.224 0.224 0.224

Z top of section m4

0.289 0.289 0.280 0.264 0.264 0.264

Width of Web m 0.600 0.600 0.496 0.300 0.300 0.300

Area of CompositeGirder m2

1.619 1.619 1.525 1.349 1.349 1.349

Moment of Inertia m4

0.458 0.458 0.438 0.402 0.402 0.402

CG of Section (bot) m 1.125 1.125 1.148 1.192 1.192 1.192

Z bot of section m4

0.407 0.407 0.382 0.337 0.337 0.337

Z top of girder. m4

1.220 1.220 1.246 1.307 1.307 1.307

Z top of Composite m4

0.744 0.744 0.741 0.734 0.734 0.734

Calculation of Bending Moments due to the following at Various Xn

Dead Load of the PSC Girder only

C/S Area of Girder = 0.7 m2

. . . U.D.L due to Dead Load = 0.7 x 25

= 17.8 kN/m

Additional Area at End Section = 0.983 - 0.713 m2

. . . U.D.L due to Dead Load = 0.270 x 25

= 6.8 kN/m

24.6 kN/m

17.8 kN/m

0.3145 1.8 1.5 5.9855

support Centreline

support reaction of girder only = 2.1145 x 24.6 + 1.5 x 21.21

+ 17.8 x 5.9855

= 190.486 KN

Section No. 1--1 2--2 3--3 4--4 5--5 6--6


Distance (x) 0 1.35 2.32 4.64 6.96 9.29

Shear (kN) 182.8 149.6 131.5 82.8 41.4 0.0

Moment(kNm) -1.2 233.5 435.2 633.6 795.7 861.6

Dead Load of the Deck Slab

UDL due to Deck slab = 2.325 m x 0.240 m x 25 T/cum = 13.95 KN/m

Reaction at support 13.950 KN/m x 9.6 m = 133.92 KN

Section No. 1--1 2--2 3--3 4--4 5--5 6--6


Distance (x) 0 1.35 2.32138 4.64275 6.96413 9.2855

Shear (kN) 133.9 115.1 101.5 69.2 36.8 4.4

Moment(kNm) 0.0 162.2 263.1 451.0 563.8 601.4

Weight of Interm. Diaphragm = 1.260 X 0.3 x 2.350 x 25 = 22.21 KN

Number of intermediate diaphragms = 1 => support reaction = 11.1038 KN

Section No. 1--1 2--2 3--3 4--4 5--5 6--6

Shear (kN) 11.1 11.1 11.1 11.1 11.1 0.0

Moment(kNm) 0.0 15.0 25.8 51.6 77.3 103.1

Dead Load of the Deck Slab+Diaphragms only

Section No. 1--1 2--2 3--3 4--4 5--5 6--6


Distance (x) 0 1.35 2.32 4.64 6.96 9.29

Shear (kN) 145.0 126.2 112.6 80.3 47.9 4.4

Moment(kNm) 0.0 177.1 288.9 502.6 641.1 704.5

Torsion(kNm) 0.0 0.0 0.0 0.0 0.0 0.0

Super Imposed Dead Load : (Shear & moment from staad)

Section No. 1--1 2--2 3--3 4--4 5--5 6--6


Shear (kN) 280.5 232.1 184.0 133.2 32.5 81.9

Moment(kNm) 322.1 618.8 850.4 1016.1 1145.1 1108.1

Torsion(kNm) 99.4 73.5 48.6 30.4 16.0 37.0

Due to Curvature Effect

Section No. 1--1 2--2 3--3 4--4 5--5 6--6


Shear (kN) 0.0 0.0 0.0 0.0 0.0 0.0

Moment(kNm) 74.5 64.7 100.5 71.1 41.6 12.2

Torsion(kNm) 0.0 0.0 0.0 0.0 0.0 0.0

The Effect of Live Load is taken from STAAD Results.

Governing Loads due to LL with impact (From STAAD Summary)

Section No. 1--1 2--2 3--3 4--4 5--5 6--6


Shear (kN) 135.6 134.0 131.7 129.7 127.7 126.5

Moment(kNm) 195.6 373.2 545.9 715.1 1212.8 1378.8

Torsion (kNm) 30.6 24.9 19.4 14.5 12.5 16.5

Summary of Shear & Moments at Various Xns

Moment / Shear due to . 1--1 2--2 3--3 4--4 5--5 6--6

Support 0.9d L/8 L/4 3L/8 L/2

1) Dead Load of PSC Girder M -1.2 233.5 435.2 633.6 795.7 861.6

S 182.8 149.6 131.5 82.8 41.4 0.0

T 0.0 0.0 0.0 0.0 0.0 0.0

2) Dead Load of Deckslab M 0.0 177.1 288.9 502.6 641.1 704.5

S 145.0 126.2 112.6 80.3 47.9 4.4

T 0.0 0.0 0.0 0.0 0.0 0.0

3) S.I.D.Load M 322.1 618.8 850.4 1016.1 1145.1 1108.1

S 280.5 232.1 184.0 133.2 32.5 81.9

T 99.4 73.5 48.6 30.4 16.0 37.0

4)Vehicular Live Load

+Curvature Effects M 270.1 437.9 646.4 786.2 1254.5 1391.0

S 135.6 134.0 131.7 129.7 127.7 126.5

T 30.6 24.9 19.4 14.5 12.5 16.5

M Moment in kNm S Shear in kN T Torsion in kNm

CHECK FOR PRESTRESSING AND STRESSES

Check for stresses (Stage I- stressing all cables to full design force)

Horizontal Component of Cable forces at various Xn.(from prestress calculations)

Cable 1--1 2--2 3--3 4--4 5--5 6--6

Nos Support 0.9D L/8 L/4 3L/8 L/2

1 2169.2 2181.1 2195.6 2228.4 2258.4 2223.6

2 1602.9 1610.4 1618.7 1637.7 1655.6 1600.7

3 2217.6 2225.9 2233.9 2252.6 2270.7 2192.6

ΣΣΣΣ Force 5989.7 6017.5 6048.3 6118.7 6184.7 6016.9

Vertical Component of Cable forces at various Xn.

Cable 1--1 2--2 3--3 4--4 5--5 6--6

Nos Support 0.9D L/8 L/4 3L/8 L/2

1 283.0 266.8 234.9 157.2 77.4 0.0

2 147.2 138.1 120.3 77.1 32.9 0.0

3 109.4 102.1 87.9 53.4 18.3 0.0

ΣΣΣΣ Force 539.6 507.0 443.1 287.7 128.7 0.0

Effect of Prestressing :- (Stage I)

Sections 1--1 2--2 3--3 4--4 5--5 6--6 Check for 0% loss in force

Force, F( KN ) 5989.7 6017.5 6048.3 6118.7 6184.7 6016.9

Area, A(m2) 0.98 0.98 0.89 0.71 0.71 0.71

F/A (KN/m2) 6090.66 6118.9 6800.15 8581.66 8674.18 8438.79

Yb (m) 0.80 0.80 0.81 0.81 0.81 0.81

Cable y (m) 0.75 0.61 0.54 0.40 0.32 0.30

e = Yb-y (m) 0.06 0.19 0.27 0.41 0.49 0.51

Zt (m3) 0.29 0.29 0.28 0.26 0.26 0.26

Zb (m3) 0.25 0.25 0.24 0.22 0.22 0.22

F x e / Zt 1201.77 4002.93 5829.89 9566.49 11503 11619.4 kN/m2

Fxe / Zb 1390.94 4633.05 6785.33 11251.5 13529.1 13666 kN/m2

F/A -(Fxe / Zt) 4.89 2.12 0.97 -0.98 -2.83 -3.18 Mpa

F/A +(Fxe / Zt) 7.48 10.75 13.59 19.83 22.20 22.10 Mpa

Time Dependent Losses consisting of the following ;-

Losses 1

Elastic Shortening of Wires: (Vide Cl:11.1 of I.R.C:-18-2000)

Loss = 0.5 x m x Stress at C.G of the Cables at that Xn.

It is proposed to stress the cables after 28 Days when the concrete attains 40.0 MPa.

The effect of Prestress & Dead Load acts together.

Descriptions

1--1 2--2 3--3 4--4 5--5 6--6

Prestress Stage I σ t 4.89 2.12 0.97 -0.98 -2.83 -3.18

σ b 7.48 10.75 13.59 19.83 22.20 22.10

Dead Load σ t 0.00 0.81 1.55 2.40 3.01 3.26

σ b 0.00 -0.94 -1.81 -2.82 -3.55 -3.84

Resultant σσσσ t 4.88 2.92 2.52 1.42 0.19 0.08

σσσσ b 7.49 9.82 11.78 17.01 18.66 18.27

Sections 1--1 2--2 3--3 4--4 5--5 6--6

4.88 2.92 2.52 1.42 0.19 0.08

Stress at 0.8 0.9 1.0 1.1 1.2 1.2

the C.G of

the Cables 6.19 7.00 8.47 12.87 14.72 14.62

0.75 0.61 0.54 0.40 0.32 0.30

7.49 9.82 11.78 17.01 18.66 18.27

Average stress at C.G of the Xn. = 11.37

∴ Loss for stage I cables = 0.5 x 10.0 x 11.4

= 56.87 = 4.77 % Loss 3

Initial Stress in cables = 1191.49

Losses 2

Losses from Stage I prestress to addition of Deckslab I.e., 28 days to 42 days

Creep of Concrete:- (Vide Cl:11.1 of I.R.C:-18-2000)

1--1 2--2 3--3 4--4 5--5 6--6


σ b 7.12 10.24 12.94 18.89 21.14 21.05

Dead Load σ t 0.00 0.81 1.55 2.40 3.01 3.26

σ b 0.00 -0.94 -1.81 -2.82 -3.55 -3.84

Resultant σσσσ t 4.7 2.8 2.5 1.5 0.3 0.2

σσσσ b 7.1 9.3 11.1 16.1 17.6 17.2

Sections 1--1 2--2 3--3 4--4 5--5 6--6

4.65 2.82 2.48 1.46 0.32 0.24

Stress at 0.75 0.89 0.96 1.10 1.18 1.20

the C.G of

the Cables 5.90 6.66 8.03 12.19 13.92 13.81

0.75 0.61 0.54 0.40 0.32 0.30

7.13 9.30 11.13 16.06 17.60 17.21


Concrete maturity at 28 days = 40.0 / 40 = 100 % coeff = 0.0004

Concrete maturity at 42 days = 42.3 / 40 = 106 % coeff = 0.0004

Creep strain during this period = 0.00002 / 10 MPa.

N/mm2

MPa.

N/mm2

N/mm2

Es = 195000

∴ Loss = 0.00002 x 10.08 x 195000 = 4.65

10

Shrinkage of Concrete:- (Vide Cl:11.1 of I.R.C:-18-2000)

Shrinkage coeff at age 28 days = 0.00019


Shrinkage Strain during this period = 0.00001

∴ Loss = 0.00001 x 195000 = 1.8

Relaxation of H.T.Steel (Vide Cl:11.1 of I.R.C:-18-2000)

Average stress in HTS = 0.6399 x UTS when stressed (after friction and slip) for stage 2

Relaxation loss for HTS for this stress = 1.7487 %

Relaxation loss for HTS on 28 days means 0 hrs after stressing

Relaxation loss for HTS for this period = 0.000 % of initial stress



Relaxation loss from 28 days to 42 days

for stage I =( 1.295 - 0.000 ) x 1191 = 15.43 MPa.

Total Time dependent Losses 4 =

for stage I cables = 4.65 + 1.8 + 15.43 = 21.85 MPa.

% loss (of initial stress) = 21.85 / 1191.49 = 1.834 % Loss 2 stage I

Losses 3

Losses from addition of Deckslab till addition of SIDL I.e., 42 days to 60 days


1--1 2--2 3--3 4--4 5--5 6--6


σ b 6.99 10.04 12.69 18.52 20.74 20.64

Dead Load σ t 0.00 0.81 1.55 2.40 3.01 3.26

σ b 0.00 -0.94 -1.81 -2.82 -3.55 -3.84

Deck Slab + Diaphragm σ t 0.00 0.61 1.03 1.90 2.43 2.67

σ b 0.00 -0.71 -1.20 -2.24 -2.86 -3.14

Resultant σσσσ t 4.6 3.4 3.5 3.4 2.8 3.0

σσσσ b 7.0 8.4 9.7 13.5 14.3 13.7

Sections 1--1 2--2 3--3 4--4 5--5 6--6

4.56 3.40 3.49 3.38 2.80 2.96

Stress at 0.75 0.89 0.96 1.10 1.18 1.20

the C.G of

the Cables 5.78 6.35 7.47 10.79 11.88 11.52

0.75 0.61 0.54 0.40 0.32 0.30

6.99 8.40 9.68 13.46 14.33 13.67


Concrete maturity at 42 days = 42 / 40 = 106 % coeff = 0.0004

Concrete maturity at 60 days = 43.89 / 40 = 109.7 % coeff = 0.0004

Creep strain during this period = 0.00002 / 10 MPa.

Es = 195000

∴ Loss = 0.00002 x 8.96 x 195000 = 2.73

10




Shrinkage Strain during this period = 0.00001

∴ Loss = 0.00001 x 195000 = 2


Average stress in HTS = 0.6399 x UTS when stressed (after friction and slip) for stage 1

Relaxation loss for HTS for this stress = 1.7487 %

Relaxation loss for HTS on 42 days means 336 hrs stressing I




Relaxation loss from 42 days to 60 days

for stage I =( 1.627 - 1.295 ) x 1191 = 3.95 MPa.

Total Time dependent Losses 5 =

MPa.

MPa.

MPa.

MPa.

MPa.

for stage I cables = 2.7 + 2.3 + 3.95 = 8.94 MPa.

% loss (of initial stress) = 8.94 / 1191.49 = 0.751 % Loss 3 stage I

Losses 4

Losses in prestressing from addition of SIDL to infinity 60 days to infinity


1--1 2--2 3--3 4--4 5--5 6--6


σ b 6.93 9.96 12.59 18.37 20.57 20.48

Dead Load σ t 0.00 0.81 1.55 2.40 3.01 3.26

σ b 0.00 -0.94 -1.81 -2.82 -3.55 -3.84

Deckslab+diaphragm σ t 0.00 0.61 1.03 1.90 2.43 2.67

σ b 0.00 -0.71 -1.20 -2.24 -2.86 -3.14

SIDL σ t 0.26 0.51 0.68 0.78 0.88 0.85

σ b -0.79 -1.52 -2.23 -3.01 -3.40 -3.29

Resultant σ t 4.79 3.89 4.16 4.17 3.70 3.83

σ b 6.14 6.79 7.35 10.30 10.77 10.22

Sections 1--1 2--2 3--3 4--4 5--5 6--6

4.79 3.89 4.16 4.17 3.70 3.83

Stress at 0.75 0.89 0.96 1.10 1.18 1.20

the C.G of

the Cables 5.47 5.61 6.21 8.67 9.27 8.94

0.75 0.61 0.54 0.40 0.32 0.30

6.14 6.79 7.35 10.30 10.77 10.22


Concrete maturity at 60 days = 43.89 / 40.00 = 109.7 % coeff = 0.0004

Creep strain till infinity = 0.00036 / 10 MPa.

Es = 195000

∴ Loss = 0.00036 x 7.36 x 195000 = 51.8

10



Shrinkage Strain till infinity = 0.00015

∴ Loss = 0.00018 x 195000 = 35.3




Relaxation loss for HTS for 500000 hrs 5.246 % of initial stress

Relaxation loss from 60 days to 500000 hrs after stressing

Stage I cables =( 5.246 - 1.627 ) x 1191 = 43.1 MPa.

Total Time dependent Losses 4 (I)= 51.82 + 35.3 + 43.12 = 130.2 MPa.

% loss (of initial stress) for stage I= 130.23 / 1191 = 10.930 % Loss 4

Summary of Losses at mid span

Values in % of initial stress Stage I

Instantaneous Losses cables

Friction Loss 3.36

Slip Loss 5.21

Elastic Shortening 4.77

13.34

Time Dependent Loss

Steel relaxation 5.25

Shrinkage 3.30

Creep 3.58

Total Losses 12.13

20% extra time dep losses 2.4

Total Losses 14.6

MPa.

MPa.

MPa.

Recapitulation of Stresses at Various Xn.

1--1 2--2 3--3 4--4 5--5 6--6


σ b 5.93 8.52 10.77 15.73 17.60 17.53

Dead Load σ t 0.00 0.81 1.55 2.40 3.01 3.26

σ b 0.00 -0.94 -1.81 -2.82 -3.55 -3.84

Deck slab + Diaphragm σ t 0.00 0.61 1.03 1.90 2.43 2.67

σ b 0.00 -0.71 -1.20 -2.24 -2.86 -3.14

SIDL Load σ t 0.26 0.51 0.68 0.78 0.88 0.85

σ b -0.79 -1.52 -2.23 -3.01 -3.40 -3.29

Resultant σ t 4.14 3.61 4.04 4.30 4.08 4.26

σ b 5.15 5.36 5.54 7.65 7.81 7.26

Live Load σ t 0.22 0.36 0.52 0.60 0.96 1.06

σ b -0.66 -1.08 -1.69 -2.33 -3.72 -4.12

Final stress σ t 4.36 3.97 4.55 4.90 5.04 5.32

σ b 4.48 4.28 3.84 5.32 4.09 3.14

Final stress (with 50% LL) σ t 4.25 3.79 4.29 4.60 4.56 4.79

(for temperature check) σ b 4.81 4.82 4.69 6.48 5.95 5.20

Remarks about Stresses at various Conditions.

(Vide Cl: 7.1 to 7.1.4 of IRC:-18-2000 )

Permissible Stress in Concrete at Stage I Prestress

Maximum Compressive Stress immediately after Prestressing shall not exceed minimum of the following

0.5 Fcj

Fcj = Expected Concrete Strength at the time of Prestressing.

= 0.5 x 40 = 20

Max Compressive Stress developed = 17.60 Hence O.K

Temporary Tensile Stress in the extreme fibre immediately after Prestressingshall not exceed,

= 1 of Maximum Compressive Stress immediately after Prestressing

10

= -1 x 17.60 = -1.76

10

Minimum Stress developed = 0.08 Hence O.K

Permissible Stress in Concrete at Service Condition

Maximum Compressive Stress allowed during Service Condition

= 0.33 Fck

= 0.33 x 40 = 13

Maximum Compressive Stress attained at Service

= 5.3 Hence O.K

Minimum Stress attained at Service

= 3.139

No Tension is developed .The Stresses are Compressive only.Hence O.K

Check for Stress in top slab

DL PS I SL+diaph PS II SIDL LL Total allowable

σ t(slab) midspan 0 0 1.0 0.0 1.5 1.9 4.4 13.33 Safe

σ t(slab) support 0 0 0.0 0.0 0.4 0.4 0.8 13.33 Safe

Check for Deflection at Midspan.

Downward deflection is given by δ

= 5 x M x L2

48 E x I

M = Moment = 4065 kNm

L = Span = 18.571 m

E = Modulus of Elasticity of Concrete = 31622.8

I = Moment of Inertia = 0.402

∴ δ 1 = 5 x 4.06523 x 18.572

48 x 31622.8 x 0.402

= 0.01148 m

= 11.48 mm

N/mm2

N/mm2

20

Mpa

m4

N/mm2

N/mm2

N/mm2

N/mm2

N/mm2

N/mm2

or

x x

Upward deflection due to prestress

= P x e x L2

8 x E x I

P = Prestressing Force at Mid Span = 6017 KN

e = eccentricity = 0.51 m

∴δ 2 = 6.0E+00 x 0.50978 x 18.572

8 x 31622.8 x 0.402

= 0.010 m

= 10.40 mm

∴ Net δ = 11.48 - 10.40

= 1.09 Downward

Permissible Deflection = L = 18571 = 23.2 mm HenceO.K

800 800

Minimum reinforcement

IRC 18-2000: cl.15.1 atleast 10 mm dia bar at not greater than 200 mm.

IRC 18-2000: cl.15.2 Vertical dirn. 0.18% of web area

IRC 18-2000: cl.15.3 Longit. Min. 0.15% of c/s area for Upto M45; beyond M45, 0.18%.

min Longit. Steel reqd 1069.5 mm2 10 dia 14 Nos in web+top+bot flange

min Web vertical steel 540 mm2/m 10 dia 2 lg 200 mm spacing

(Note: the values given here are only as a check for min. steel. Actual steel provided is checked in the

following sections)

Ultimate load Capacity (IRC 18:2000 cl 12 & 13)

Check for Ultimate Strength at Various Xn

.

Failure by yield of steel ( Under Reinforced section )

M ult ( Steel ) = 0.9 d b A s f p + 0.87 d b A st f y (Ast and fy are for non-prestressed steel)

A s = Area of High Tensile Steel (This is neglected)

f p = The Ultimate Tensile Strength of Steel .

d b = The Depth opf the beam from the maximum compression edge to C.G of Tendons.

Failure by crushing of concrete ( over reinforced section )

M ult ( Con :) = 0.176 b d b2 f ck + (2/3) x 0.8( B f - b )(d b - t / 2)x t x fck

b = width of the Web.

B f = Overall width of the top flange of PSC Girder.or Slab Eff width

t = Average thickness of flange.

Sections Xn 1-1 X

n 2-2 X

n 3-3 X

n 4-4 X

n 5-5 X

n 6-6

A s (m2) 0.005 0.005 0.005 0.005 0.005 0.005

f p (kN/m2) 1862000 1862000 1862000 1862000 1862000 1862000

d b (m) 0.993 1.127 1.203 1.342 1.420 1.439

M ult(HT Steel ) (kNm) 8543 9697 10350 11543 12216 12377

M ult(Tot Steel ) (kNm) 8543 9697 10350 11543 12216 12377

b (m) 0.600 0.60 0.40 0.30 0.30 0.300

B f (m) 2.650 2.650 2.650 2.650 2.650 2.650

t (m) 0.240 0.240 0.240 0.240 0.240 0.240

M ult( Conc ) (kNm) 13332 15943 16577 18507 19905 20245

M ult (section) 8543 9697 10350 11543 12216 12377

1.5*DL+2*SIDL+2.5*LL (kNm) 1383 3096 4623 5987 7961 8445

Remarks Safe Safe Safe Safe Safe Safe

Check for Ultimate Shear Strength at Various Xn

. (Vide Cl:14.1 of I.R.C:-18-2000)

Sections Uncracked in flexure

V co = 0.67bd (f t +0.8 f cp* f t )

V co = Ultimate Shear Resistance of the Xn.

b = Width of Webs - (2/3 x Duct Diametre) if the Cables are grouted.

d = Overall depth

f t = Max principal stress 0.24 fck

f cp = Stress at c.g at the section due to prestress after inst: loss is accounted.

Sections Cracked in flexure

V cr = 0.037bd b f ck + (M t xV/M)

d b = Distance of extreme comp.fibre from centroid of tendons.

M t = (0.37 f ck + 0.8 f pt ) I/y

V and M = Ultimate Shear & corresponding moment at the section

V cr (min) = 0.1bd f ck

Acc.to IRC :18 - 2000 Cl. No. 14.1.5 &Table 6.

V Capacity =( 4700 x b x db )+

db = 0.8 x Overall Depth or Dist: from comp: face to C.G of Tendons,which ever is more.

Shear Design(IRC 18:2000 cl 14.1) ft = 1.52 Mpa

Section 1--1 2--2 3--3 4--4 5--5 6--6

Vult (5%extra) 1461 1274 1117 835 544 511 (1.5*DL+2*SIDL+2.5*LL)

f pt due to prestress (bottom fibre) 4.89 2.12 0.97 -0.98 -2.83 -3.18

f pc due to prestress(top fibre) 7.48 10.75 13.59 19.83 22.20 22.10

Total Prestress force 5989.7 6017.5 6048.3 6118.7 6184.7 6016.9

Area of precast section 1.0 1.0 0.9 0.7 0.7 0.7

I p of precast section 0.2 0.2 0.2 0.2 0.2 0.2

cg of cables in precast section 0.7 0.6 0.5 0.4 0.3 0.3

cg of precast section 0.8 0.8 0.8 0.8 0.8 0.8

cg of composite section 1.1 1.1 1.1 1.2 1.2 1.2

f cp (stress at composite cg ) 5.5 4.3 3.9 3.3 2.3 2.0

Section 1--1 2--2 3--3 4--4 5--5 6--6

M pc =1.5*DL moment of girder -2 350 653 950 1193 1292

f cm (stress at comp. Cg due to Mpc) 0.0 -0.6 -1.1 -2.0 -2.5 -2.7

f' cp = 0.8*f' cp +f cm 4.4 2.9 2.0 0.6 -0.7 -1.1

V c1 =1.5*DL shear of girder 274.1 224.3 197.2 124.1 62.1 0.0

I c of composite section 0.5 0.5 0.4 0.4 0.4 0.4

f s =V c1 *(A y ) p /(I p *b) 0.1 0.1 0.1 0.1 0.1 0.0

check if f s < f t Safe Safe Safe Safe Safe Safe

V c2 =(I c *b)/(A yc )*[(f t2+f cp *f t )

1/2-f s ] 2260 2057 1769 1651 1499 1470

V co = V c1 +V c2 2534 2282 1966 1775 1561 1470 IRC 18:2000 cl 14.1.2.2

(for precast girders+slab)

V co = 0.67bd(ft2+0.8f cp f t )

1/21466 1335 649 612 551 531 IRC 18:2000 cl 14.1.2.1

(superstr is cast at once)

M t =(0.37*sqrt(f ck )+.8*f pt )*Z b 2078 2731 2965 4086 4512 4494

V cr =0.037*b*db*(f ck )1/2

+(M t /M)V 2301 1248 784 647 391 356

V cr (min) = 0.1bd(fck)^0.5 660 660 330 330 330 330

Section is uncracked uncracked uncracked uncracked uncracked uncracked

V (Psin a ) due to Cables 540 507 443 288 129 0 (for uncracked only)

V co (uncracked section incl Psin α ) 3074 2789 2409 2063 1690 1470

V cr (Cracked section) 2301 1248 784 647 391 356

V c (section) 2301 1248 784 647 391 356

V c (section capacity) 4072 4040 3294 1858 1731 1623 IRC 18:2000 cl 14.1.5

web width to be reduced by (2/3 of 1 1 1 1 1 1 cable/s

Is V ult <= V capacity Safe Safe Safe Safe Safe Safe IRC 18:2000 cl 14.1.5

Is V ult <= 0.5*V c section No No No No No No

Shear Reinft required or not Min.reqd Reqd Reqd Reqd Reqd Reqd IRC 18:2000 cl 14.1.4

Assume Spacing 200 200 200 200 200 200

Min Asv Required 133 133 66 66 66 66 IRC 18:2000 cl 14.1.4

Asv required 0 8 109 62 50 51 IRC 18:2000 cl 14.1.4

Asv reqd for torsion 145 138 23 34 22 40 (See Torsion Calculations below)

dia of shear reinft 12 12 12 12 12 12

No. of legs 4 4 4 2 2 2

Asv provided 452 452 452 226 226 226

Total Asv reqd. 278 271 132 100 88 106

Safe Safe Safe Safe Safe Safe

P Sin( θθθθ) if the Xn is Uncracked.

Torsion Design(IRC 18:2000 cl 14.2) Vt= 4.70 Mpa Vc = 0.42 Mpa

Section 1--1 2--2 3--3 4--4 5--5 6--6

Tult (KNm) 275 209 146 97 63 115

sum of bd3/3 0.11 0.11 0.07 0.03 0.03 0.03

T in slab KNm 31.7 24.1 25.9 38.1 24.8 45.2

T in top flange KNm 7.8 6.0 8.6 12.6 8.2 15.0

T in web KNm 235.8 179.2 14.9 21.9 14.2 26.0

T in bottom flange KNm 0.0 0.0 16.7 24.5 15.9 29.1

Shear Stress in Slab(Mpa) due to torsion 0.43 0.33 0.35 0.51 0.33 0.61

Shear Stress in Top fl (Mpa) due to torsion 0.39 0.29 0.35 0.52 0.34 0.61

Shear Stress in Web (Mpa) due to torsion 1.17 0.89 0.10 0.40 0.26 0.47

Shear Stress in Bot fl (Mpa) due to torsion 0.00 0.00 0.59 0.87 0.57 1.03

ult Shear Stress V/bd (Mpa.) 1.40 1.22 1.29 1.60 1.04 0.98

ult Torsion shear stress (Mpa) 1.17 0.89 0.10 0.40 0.26 0.47

Total shear stress 2.57 2.11 1.40 2.00 1.30 1.45

Vtc (allowable) for torsion 0.42

Vtu (allowable) 4.70

Provide Torsional shear reinft Reqd. Reqd. No No No Reqd.

Is ultimate shear safe Safe Safe Safe Safe Safe Safe

Torsion reinft.

Asv in top slab mm2 / m 172 131 141 207 135 246

Asv in top flange mm2 / m 151 115 166 243 158 289

Asv in web mm2 / m 726 689 115 168 109 200

Asv in bot flange mm2 / m 0 0 401 589 383 699

Asl in top slab mm2 498 379 408 599 389 711

Asl in top flange mm2 178 135 195 287 187 340

Asl in web mm2 1448 1448 206 303 197 360

Asl in bot flange mm2 0 0 337 470 322 587

Design of Shear Connectors (with slab)

Section 1--1 2--2 3--3 4--4 5--5 6--6

A*y (first moment of top slab) 0.315 0.315 0.300 0.272 0.272 0.272

VL=V(sidl+ll)*A*y/I KN/m 769 680 593 481 308 329 IRC 22:cl 611.4.2.5

shear stress Mpa. 0.77 0.68 0.59 0.48 0.31 0.33 0.77 < 2.1

Qu of vert. Strps(KN) 176 176 176 88 88 88 safe

spacing reqd (mm) 458 517 594 366 571 535

stirrup spacing ok/not 1 1 1 1 1 1 Safe

min shear (12 dia 2 lg) at 200 mm spacing

Q (of slab at critical pl.) 0.17 0.17 0.17 0.17 0.17 0.17

Shear flow KN/m 0 11 140 79 64 65

0.4*L*(fck)^.5 (slab) 1214 1214 1214 1214 1214 1214 IRC 22:cl 611.5.1

0.7*As*fy+.08*L*(fck)^.5 749 749 749 749 749 749 IRC 22:cl 611.5.1

(based on slab reinft of .9%) 1 1 1 1 1 1 Safe

Ast/m (in slab) min 1157 1157 1157 1157 1157 1157 IRC 22:cl 611.5.2.3

Ast prov. In slab (16@150) 2680 2680 2680 2680 2680 2680

1 1 1 1 1 1 Safe

Min Ast reqd in Top flange 1131 1131 1131 565 565 565 mm2/ m 50% of web

Min Ast reqd in Bot flange 1131 1131 1131 589 565 699 mm2/ m 50% of web

Provide 16@ 200 200 200 200 200 200 mm

Ast provided = 1005 1005 1005 1005 1005 1005 mm2/ m

Longitudinal steel in top flange 6 Nos 12 dia 679 mm2 Reqd is 340 OK

Longitudinal steel in web (mid) 12 Nos 10 dia 942 mm2 Reqd is 360 OK

Extra Longl. steel in web (end) 12 Nos 10 dia 942 mm2 Reqd is 1448 OK

Longitudinal steel in bot flange 6 Nos 12 dia 679 mm2 Reqd is 587 OK

Refer Temperature design stress in the following section.

STAGE I PRESTRESSING

CABLES DATA

Input particulars

Effective span c to c of bearings L 18.571 m

Modulus of elasticity of steel Es 195000 N/mm2

initital stress

UTS of Ht strands(Class2) Fy 19000 Kg/cm2

13300 Kg/cm2

UTS of Ht strands(Class2) Fy 1862 N/mm2

1303.4 N/mm2

Type of strands used T 13 As 98.7 MM2

Maximum jack pressure = 0.9 X 0.85 = 0.765 of UTF is permissible

Grade of concrete 40

Ec = 31622.78 N/mm2

Details of different strands

Types of cables

Cable

capacity

No of

strands

stressed

Dia No. Area UTF (kN)

Sheathin

g Int dia

Jack

type

Jack

piston

area

Length

of cable

for jack

Stress

upto (of

UTS)

Cable 1 19 19 T 13 1 1875.3 3491 85 K-350 490 0 0.7

Cable 2 19 14 T 13 1 1381.8 2572 85 K-350 490 0 0.7

Cable 3 19 19 T 13 1 1875.3 3491 85 K-350 490 0 0.7

5132.4 9554

Coefficient fo friction k 0.002 Per m

u 0.17 per rad.

Slip accounted s 0.006 m

Each cable is to be stressed from both ends simultaneously.

CABLE PROFILES (TYPICAL)

Horizontal Splay of Cables

Bearing

L/2

X3

314.5

Elevation of a typical Cable

Typical cases of slip loss

Case 1 (X4 case)P3

P2

P1Po

P4

P1'

Po'

X4

L3L2L1

Y4

L4

CL

+X3-X3

Y3

Y2

Y1

Xa X2 X1

Case 2 (X5 case)

Case 3 (Y5 case)

CABLE PROFILE(Stressing from both ends) base len gir len diff

Cable nos 1 2 3 19.96 19.2 0.76

No. of cables 1 1 1

Total force 3491 2572 3491

Check for length 9.450 9.450 9.450 9.450 Total length - 150 mm for setting

Xa 0 0 0 0 mm extra length of cable)

X1 0.120 0.620 1.120

Y1 0.480 0.300 0.120

Y1+Y2+Y3 1.5 0.350 1.150 0.750 0.350

X3 1.000 1.000 1.000

Y4a (hor splay 1) 0.000 0.000 0.000

L4a 1.000 1.500 1.500

Y4a (hor splay 2) 0.000 0.000

L4b 1.000 1.000

X2 8.330 7.830 7.330

c=(Y2+Y3)/X2(X2+2*X3) 0.008 0.006 0.003

a1 Atan(2*c*X2)(Rad) 0.129 0.091 0.049

(degree) 7.388 5.229 2.821

Y2 = 0.540 0.358 0.181

L1 = X3/cosa1 1.008 1.004 1.001

L2 = L1+X2+2y2^2/3*X2 9.362 8.845 8.334

L3 = L2+X1 9.482 9.465 9.454

a2 =2*Atan(2*Y4/X4) 0.000 0.000 0.000

Stress before slip (UTS)

P0 0.700 0.700 0.700

P1 =Po/Exp(kL1) 0.699 0.699 0.699

P2 =Po/Exp(u(a1+a2)+kL2) 0.672 0.677 0.683

P3 =Po/Exp(u(a1+a2)+kl3) 0.672 0.676 0.681

Elongation (mm) 62.2 62.2 62.3

(Xa*Po+(Po+P1)L1+(P1+P2)*

(L2-L1)+(P2+P3)(L3-L2))Fy/2Es

Fav =((Po+P1)L1+(P1+P2)* 0.687 0.688 0.691

(L2-L1)+(P2+P3)(L3-L2))/(2xL3)

Stress after slip (UTS)

Po' 0.626 0.626 0.636

P1' 0.628 0.627 0.637

P2' 0.654 0.648 0.653

P3' 0.637 0.622 0.628

P4 0.637 0.622 0.628

Fav 0.640 0.636 0.644

X4

X5

Y5 0.018 0.027 0.027

Selected Case Y5 Y5 Y5

L3

L2

L1

P1

Po

P2 P3

P3'P2'P1'

Y5

L3

L2

L1

P0'

X5

Po

P1 P2

P4P3

Po'

P1'P2'

Slip travels upto 9.450 9.450 9.450

CABLE COORDINATES

CABLE NO 1 2 3

At support 9.2855 1.150 0.750 0.350

At 0.9D from Support7.94 0.956 0.613 0.276

L/8 6.96 0.845 0.535 0.235

L/4 4.64 0.639 0.395 0.162

3L/8 2.32 0.518 0.317 0.125

L/2 0 0.48 0.3 0.12

CABLE FORCES AND MOMENTS

Cable number 1 2 3

P (average) 2232.93 1636.79 2247.065

At midspan

Distance from cg to bottom fibre=Yb 0.811 0.811 0.811

Distance from bottom fibre to cable cg=Yc0.480 0.300 0.120

a radian 0.000 0.000 0.000

Eccentricitye=Yb-Yc 0.331 0.511 0.691 avg ecc 0.50978 m

Pi 2223.6 1600.7 2192.6

Pi cosa 2223.6 1600.7 2192.6

Pisina 0.0 0.0 0.0

Picosa*e 735.4 817.5 1514.4

At 3l/8 2.32138

Yb 0.811 0.811 0.811

Yc 0.518 0.317 0.125

a radian 0.034 0.020 0.008

e=Yb-Yc 0.293 0.494 0.686 avg ecc 0.49098 m

Pi 2259.7 1655.9 2270.8

Picosa 2258.4 1655.6 2270.7

Pisina 77.4 32.9 18.3

Picosa*e 661.6 817.5 1557.4

At quarter span4.64275

Yb 0.811 0.811 0.811

Yc 0.639 0.395 0.162

a radian 0.070 0.047 0.024

e=Yb-Yc 0.171 0.416 0.649 avg ecc 0.41272 m

Pi 2233.9 1639.5 2253.2

Picosa 2228.4 1637.7 2252.6

Pisina 157.2 77.1 53.4

Picosa*e 382.0 681.5 1461.9

At 1/8 span 6.96413

Yb 0.807 0.807 0.807

Yc 0.845 0.535 0.235

a radian 0.107 0.074 0.039

e=Yb-Yc -0.038 0.271 0.572 avg ecc 0.27014 m

Pi 2208.2 1623.2 2235.6

Picosa 2195.6 1618.7 2233.9

Pisina 234.9 120.3 87.9

Picosa*e -83.3 439.5 1277.7

At 0.9D 7.9355

Yb 0.805 0.805 0.805

Yc 0.956 0.613 0.276

a radian 0.122 0.086 0.046

e=Yb-Yc -0.151 0.192 0.529 avg ecc 0.19215 m

Pi 2197.4 1616.3 2228.3

Picosa 2181.1 1610.4 2225.9

Pisina 266.8 138.1 102.1

Picosa*e -329.1 308.9 1176.4

At support 9.2855

Yb 0.805 0.805 0.805

Yc 1.150 0.750 0.350

a radian 0.130 0.092 0.049

e=Yb-Yc -0.345 0.055 0.455 avg ecc 0.05796 m

Pi 2187.6 1609.7 2220.3

Picosa 2169.2 1602.9 2217.6

Pisina 283.0 147.2 109.4

Picosa*e -749.0 87.7 1008.4

Average stress in steel 0.64023 times UTS

Cable Cordinates Of all the Cables

CABLE 1 CABLE 2 CABLE 3

X (m) Y (mm) Y (mm) Y (mm)

0 480 300 120

1 486 301 120

2 508 311 123

3 545 333 132

4 597 367 148

5 665 412 171

6 749 469 200

7 849 538 236

8 963 618 279

8.450 1020 658 301

9.286 1129 735 342

9.450 1150 750 350

Curve Starts

at

0.120 0.620 1.120

Curve Ends

at

8.450 8.450 8.450

Cable No.

Type No of

strands

stressed

Force in

Cable

(jack)

(kN)

Duct dia

(mm)

Jack Type

(or eqv)

Cable

Length

(mm)

Elongati

on (each

end)

(mm)

angle at

anchora

ge (deg)

Stressin

g Stage

1 19-T-13 19 2443.7 85 K-350 18963 62.2 7.388 1 9482

2 19-T-13 14 1800.4 85 K-350 18930 62.2 5.229 1 9465

3 19-T-13 19 2443.7 85 K-350 18908 62.3 2.821 1 9454

Note cable length includes 0 mm extra for jacking.

DESIGN OF END BLOCK

Refer Clause 17.2 of IRC:18-2000

0.35

1

0.40

2 Anchorage plate size assumed

0.40 270 mm

3

270

Min. concrete beyond anchorage edge

0.35 = 0.165

0.165 = 0.165 > 0.05

0.60 Safe

STAGE 1 CABLES Moment from Moment from

Jacking/Stressing forces in cables (after slip& friction) bottom Rt. Edge

Force in Cable 1 1967.7 KN Cable type 19 T 13 2262.8 590.3



4047.72 1623.80

(Note: here all forces are taken as horizontal only

Check for bursting forces, case 1 vertical is neglected)

Max force = 1997.1 KN

size of eff sq. = 0.350 m from bottom

Hence, 2Yo = 0.70 m.

Size of each anchorage is 270 mm square

Along Vertical line of action, anchorage size 2Ypo= 1 x 0.27 = 0.27 m

Ypo/Yo = 0.27 / 0.70 = 0.386

For this ratio of Ypo/Yo, from Table 11 of above code,

Fbst/Pk = 0.20

Now Pk = 1997.1129 kN

Hence, Fbst = 407.98 kN

For external anchorage the design force has to be increased by 10%.

Hence, revised Fbst = 448.78 kN

This max stress occurs at 0.5*Yo = 0.5 x 0.350 = 0.175 m

Providing 12 φ mesh reinforcement , with 0.87Fy permissible stress

For Fy = 415 Mpa,

Area of steel reqd. = 448.78 x 1000 = 1242.99 mm2

0.87 x 415

Number of bars reqd. = 1242.99 = 11 Nos

113.10

(note vertical check gives more steel than the horizontal check hence only vertical check has been done)

Bearing Stress

As the anchorages are set out as per the manufacturer's specifications for embedded anchorages, bearing stress

checks are not carried out. Refer Note ii under clause 7.3 of IRC:18-2000.

0.27

Design temperatrure differences

Idealised section

2.650 2.650

0.332 0.325

1.018 0.97

0.300 0.600

0.390 0.44

0.600 0.6

1.740 Mid Section 1.740 End Section

Temperature Rise case Temperature Fall case

17.8'C 10.6'C

0.25

h1 0.15 h1

4'C 0.7'C

h2 0.25 h2

0.2

h

0.2

h3

h3 0.15

0.8'C h4

0.25

2.1'C 6.6'C

h1=0.3h (Max 0.15m) h1=h4=0.2h<=0.25m

h2=0.3h>=0.1m<=0.25m h2=h3=0.25h <=0.20m

h3=0.3h <=0.15m h1 0.25

h1 0.15 h2 0.2

h2 0.25 h3 0.2

h3 0.15 h4 0.25

CALCULATION OF THERMAL STRESSES AT MID SECTION

TEMPERATURE RISE CASE

Section Depth Width Area Y from top Ay I t At It

m m (m^2) (m) (m^3) (m^4) (C) (m^2.C) (m^3.C)

1 0.15 2.750 0.413 0.075 0.031 0.002 10.900 4.496 0.025

2 0.15 2.750 0.407 0.224 0.091 0.020 2.815 1.147 0.058

3 0.10 0.300 0.031 0.349 0.011 0.004 0.815 0.025 0.003

4 1.02 0.300 0.305 0.909 0.277 0.252 0.000 0.000 0.000

5 0.17 0.600 0.104 1.504 0.156 0.234 0.000 0.000 0.000

6 0.15 0.600 0.090 1.665 0.150 0.250 1.050 0.095 0.262

Sum 1.74 1.349 0.716 0.762 5.762 0.348

0.0000117

εo*ΣA - θ*ΣA*Y = α*A*T

εo∗ΣA*Y - θ*ΣA*Y2 = α*A*T*Y

θ 0.000083

εo 0.000094

Ec 31623 N/mm2

Coefficient of thermal expansion α =

Point Y from top y*theta t alfa*tεεεεo-yθθθθ-ααααt Fci=Ec(ε(ε(ε(εo-

yθθθθ-ααααt)

Comp /

Ten

(m) 'C MPA

1 0 0.00E+00 17.80 2.08E-04 -1.14E-04 -3.615 Compression

2 0.15 1.24E-05 4.00 4.68E-05 3.47E-05 1.098 Tension

3 0.30 2.47E-05 1.63 1.91E-05 5.02E-05 1.586 Tension

4 0.40 3.32E-05 0.00 0.00E+00 6.08E-05 1.922 Tension

5 1.42 1.18E-04 0.00 0.00E+00 -2.36E-05 -0.745 Compression

6 1.59 1.32E-04 0.00 0.00E+00 -3.79E-05 -1.197 Compression

7 1.74 1.44E-04 2.10 2.46E-05 -7.49E-05 -2.367 Compression

Temperature Fall in Mid section

Section Depth Width Area y from top Ay I t At It

m m (m^2) (m) (m^3) (m^4) (C) (m^2.C) (m^3.C)

1 0.250 2.750 0.688 0.125 0.086 0.011 5.650 3.884 0.061

2 0.048 2.750 0.132 0.274 0.036 0.010 0.616 0.081 0.006

3 0.152 0.300 0.046 0.374 0.017 0.006 0.266 0.012 0.002

4 0.840 0.300 0.252 0.870 0.219 0.191 0.000 0.000 0.000

5 0.128 0.300 0.038 1.354 0.052 0.070 0.255 0.010 0.018

6 0.073 0.600 0.044 1.454 0.063 0.092 0.655 0.028 0.060

7 0.250 0.600 0.150 1.615 0.242 0.391 3.700 0.555 1.448

Sum 1.740 1.349 0.716 0.771 4.571 1.594

0.0000117



θ 0.000025

εo 0.000053

Ec 31623 N/mm2



Comp /

Ten

(m) 'C MPA

1 0 0.00E+00 10.6 1.24E-04 -7.12E-05 -2.251 Tension

2 0.250 6.21E-06 0.700 8.19E-06 3.84E-05 1.215 Compression

3 0.298 7.41E-06 0.532 6.22E-06 3.92E-05 1.240 Compression

4 0.450 1.12E-05 0.000 0.00E+00 4.16E-05 1.317 Compression

5 1.290 3.21E-05 0.000 0.00E+00 2.08E-05 0.657 Compression

6 1.418 3.52E-05 0.446 5.22E-06 1.24E-05 0.392 Compression

7 1.490 3.70E-05 0.800 9.36E-06 6.45E-06 0.204 Compression

8 1.740 4.32E-05 6.600 7.72E-05 -6.76E-05 -2.139 Tension

END SECTION

Temperature Rise

Section Depth Width Area Y from top Ay I t At It

m m (m^2) (m) (m^3) (m^4) (C) (m^2.C) (m^3.C)

1 0.15 2.750 0.413 0.075 0.031 0.0023 10.900 4.496 0.025

2 0.129 2.750 0.356 0.215 0.076 0.0164 2.966 1.054 0.049

3 0.121 0.6 0.072 0.340 0.025 0.0084 0.966 0.070 0.008

4 0.970 0.6 0.582 0.885 0.515 0.4558 0.000 0.000 0.000

5 0.22 0.6 0.132 1.480 0.195 0.2891 0.000 0.000 0.000

6 0.15 0.6 0.090 1.665 0.150 0.2495 1.050 0.095 0.262

Sum 1.740 1.644 0.992 1.022 5.715 0.344

0.0000117



θ 0.000086

εo 0.000092

Ec 31623 N/mm2





Comp /

Ten

(m) 'C MPA

1 0 0.00E+00 10.6 1.24E-04 -3.16E-05 -0.998 Compression

2 0.150 1.29E-05 0.700 8.19E-06 7.14E-05 2.258 Tension

3 0.279 2.40E-05 0.423 4.94E-06 6.35E-05 2.009 Tension

4 0.400 3.43E-05 0.000 0.00E+00 5.81E-05 1.838 Tension

5 1.370 1.18E-04 0.000 0.00E+00 -2.52E-05 -0.796 Compression

6 1.590 1.37E-04 0.770 9.01E-06 -5.31E-05 -1.678 Compression

7 1.740 1.49E-04 0.800 9.36E-06 -6.63E-05 -2.097 Compression

Temperature Fall

Section Depth Width Area y from top Ay I t At It

m m (m^2) (m) (m^3) (m^4) (C) (m^2.C) (m^3.C)

1 0.250 2.750 0.688 0.125 0.086 0.011 5.650 3.884 0.061

2 0.029 2.750 0.080 0.265 0.021 0.006 0.649 0.052 0.004

3 0.171 0.600 0.102 0.365 0.037 0.014 0.299 0.031 0.004

4 0.840 0.600 0.504 0.870 0.438 0.381 0.000 0.000 0.000

5 0.080 0.600 0.048 1.330 0.064 0.085 0.160 0.008 0.014

6 0.120 0.600 0.072 1.430 0.103 0.147 0.560 0.040 0.082

7 0.250 0.600 0.150 1.615 0.242 0.391 3.700 0.555 1.448

Sum 1.740 1.644 0.992 1.035 4.570 1.612

0.0000117



θ 0.000031

εo 0.000051

Ec 31623 N/mm2



Comp /

Ten

(m) 'C MPA

1 0 0.00E+00 10.6 1.24E-04 -7.30E-05 -2.308 Tension 0.16868

2 0.250 7.68E-06 0.700 8.19E-06 3.52E-05 1.112 Compression

3 0.279 8.58E-06 0.598 6.99E-06 3.55E-05 1.122 Compression

4 0.450 1.38E-05 0.000 0.00E+00 3.72E-05 1.177 Compression

5 1.290 3.96E-05 0.000 0.00E+00 1.14E-05 0.361 Compression

6 1.370 4.21E-05 0.280 3.28E-06 5.69E-06 0.180 Compression

7 1.490 4.58E-05 0.800 9.36E-06 -4.08E-06 -0.129 Tension

8 1.740 5.34E-05 6.600 7.72E-05 -7.96E-05 -2.518 Tension

SUMMARY (Check with 50% LL)

stress with

Temp rise Temp Fall Max Tension 50% LL Net stress

Mid section Stress(MPA) Mpa Mpa Mpa Mpa

Top Slab 3.61 -2.25 -2.25 4.36 2.11

Top Flange -1.10 1.22 -1.10 4.79 3.69

Web -0.59 0.99 -0.59

Bottom fl 2.37 -2.14 -2.14 5.20 3.06

stress with

Temp rise Temp Fall Max Tension 50% LL Net stress

End section Stress(MPA) Mpa Mpa Mpa Mpa

Top Slab 1.00 -2.31 -2.31 0.81 -1.50

Top flange -2.26 1.11 -2.26 4.25 1.99

Web -0.52 0.77 -0.52

Bottom fl 2.10 -2.52 -2.52 4.81 2.30

At the end spans, tensile force = 126.60 KN

Provided extra 10 mm dia at 240 mm c/c at top and bottom in slab is sufficient for this


Design Of End Diaphragm

Diaphragm is designed for two Loading Condition

1 ) Selfweight Of Diaphragm + Reaction From Superstructure ( Girders ) ( Service Condition )

2 ) Replacement Of Bearing / Superstructure supported on Jacks - ( No Live Load )

Diaphragm is spanning between 4 girders . It act as continuous beam

Diaphragm is designed for Jacking end Force During Bearing Replacement

Jacks are kept at distance mentioned below away from the face of the girder as shown in figure.

Hence It acts as a continuous beam between 6 Jacks. Jack Positions are as shown in figure & indicated

by T To Y

During Bearing Replacement there will be no Live Load

Hence Only Dead Loads and SIDL are considered (in KNs)

Reactions due to G 4 G3 G 2 G 1

1) Selfweight 335.51 335.51 335.51 335.51

2) S.I.D.L. 366.57 19.1 19.91 365.53

702.08 354.61 355.42 701.04

Girder & Jack Positions

G 1 G 2 G 3 G 4 Diaphragm

702 355 355 701

2750

1500

500 1750 1000 1750 1000 1750 500.00 350

T U V W X Y

200 200 200 200 Jack distances.

200 200

Thickness of base of girder 600 mm

CG of girder loads 8713.56 / 2113.15 = 4.123 m from G1

CG of Jacks 4.125 m

T To Y are Jack positions at the time Of Bearing replacement

Note: all the jacks will have the same force as they are connected to the same hydraulic pump.

Note: the self weight of diaphragm has been considered in the reactions of girders and hence, not taken again.

Force in the jacks = 6 x P = 2113.15 KN

P = 352.192 KN

M (T) = 702 x 0.5 = 351.04 KNm

351 KNM

M(U) = 702 x 2.25 - 352.192 x 1.75 = 963.344 KNm

963 KNM

M(G2)= 702 x 2.75 - 352.192 x 2.25 - 352.192 x 0.5

962.193 KNM

M(V)= 702 x 3.25 - 352.192 x 2.75 - 352.192 x 1 +

355 x 0.2

1031.96 KNM

M(W) = 702 x 5 - 352.192 x 4.5 - 352.192 x 2.75 +

355 x 1.95 - 352.192 x 1.75

1032.16 KNM

M(G3)= 702 x 5.5 - 352.192 x 5 - 352.192 x 3.25 +

355 x 2.45 - 352.192 x 2.25

1032.22 KNM

Design Moment is 1032 kNm

Shear is taken on right side.

S(G1) = 702

S(T) = 350

S(U) = -2

S(G2)= 352

S(V)= 0

S(W) = -352

S(G3)= 3

S(x)= -349

S(Y)= -701

Max Design Shear is 702 kN (Note: this is at the jack face or the CL of the girder. Actual shear

will be little less)

Check For Moment Of Resistant

Grade Of Concrete M 40 L/D min = 1750 / 1500 = 1.167 < 2.5

Grade Of Steel Fe 415 Deep Beam

Acc.to IRC : 21 -2000 Cl. No. 303.1

Permissible Bending Stress in Concrete σ cbc = 13.3 Mpa

Permissible Bending Stress in bending for H.Y.S.D. bars σ st = 200 Mpa

Modular Ratio = 10

Lever Arm for Deep beam= 0.2*(l+1.5D) for 1<L/D<2.5 = 800 mm

0.5L for L/D <1

0.40

m x σ cbc σ st 133.333 200

J = 1 - K / 3 = 0.867 But here J = 800 / 1500 = 0.53333

Q = 1 / 2 * σ cbc * j * k = 1.42

Moment Of Resistant = Q . B . D2

= 1120 kN - mtr > Max Moment = 1031.96 kN - mtr

Hence Safe

Clear Cover 50 mm

Bar Dia 32 mm

deff provided = 1500 - 50 - 96 = 1354.0

Steel Required =

0.2D = 300

= -Ast Ast1 (0.5(L/D-.5)= 3 bars

200 0.533 1354.0

= 7145.24 mm2

Mid ht 0.3D = 450

Ast2 = 6 bars

Using 32 mm dia bar ∴ ∴ ∴ ∴ No of Bars = 9

+Ast 0.25D-0.05L = 288 mm

Ast = 9 bars

Side face reinft

Side face reinft reqd as per IS 456:2000 is 0.002 X 350 x 1500

= 1050 mm2

4825.49 mm2

Provide 16 dia 12 Nos. on each face

Km x σ cbc 133.3333333

M

σ st x j x d

1031963075.27

+

==

+

=

∴∴∴∴

xx

a) Check for Max Shear

Total Shear = 702.08 kN

Shear stress , τ = V ( Vide cl - 304.7.1.1 of I.R.C:-21-1987 )

B x d

V = The design shear across the section

d = Effective depth of the section

B = Breadth of slab

. .

. τ = 702.1 x 1000 = 1.48 N/mm2

350 x 1354.00

Maximum Permissible Shear Stress :- ( Vide cl - 304.7.2 of I.R.C:-21-2000 )

b = 350 mm

d revised = 1354 mm

p = 1.53 %

f ck = 40.00 N/mm2

τ c = 0.49 N/mm2

τ max = 2.5 N/mm2

= V / bd = 1.481 > 0.49 N/mm2

Shear reinft. reqd

= 1.481 < 2.5 N/mm2

SAFE

Vsteel = V −τ c*b*d 46.987 T Shear reinft. Reqd.

Spacing = σ sc X A SV X d

Vst

Assuming 2 legged 12 dia stirrups

A SV = 226.08 mm2

σ sc = 200 N/mm2

Spacing of stirrups = 130.3 mm

Provide 2 legged 12 dia stirrups at 125.0 c/c

DESIGN LOADS During Service (KN, KNm)

SHEAR DL SIDL LIVE LOAD Total Shear with Torsion

End Diaph- 77.12 104.3 181.467 368.9

Mid Diaph - 211.74 236.0 447.725 476.9

TORSION DL SIDL LIVE LOAD Total

End Diaph- 15.78 25.2 41.0

Mid Diaph - 0 5.5 5.5

MOMENTSDL SIDL LIVE LOAD Total Moment with Torsion

End Diaph- 189.07 55.6 244.69 274.4

Mid Diaph - -757.62 610.7 757.62 760.8

Check For Moment Of Resistant for Intermediate Diaphragm

Grade Of Concrete M 40 L/D min = 2750 / 1500 = 1.833 < 2.5

Grade Of Steel Fe 415 Deep Beam

Acc.to IRC : 21 -2000 Cl. No. 303.1

Permissible Bending Stress in Concrete σ cbc = 13.3 Mpa

Permissible Bending Stress in bending for H.Y.S.D. bars σ st = 200 Mpa

Modular Ratio = 10

Lever Arm for Deep beam= 0.2*(l+1.5D) for 1<L/D<2.5 = 1000 mm

0.5L for L/D <1

0.40

m x σ cbc σ st 133.333 200

J = 1 - K / 3 = 0.867 But here J = 1000 / 1500 = 0.66667

Q = 1 / 2 * σ cbc * j * k = 1.78

Moment Of Resistant = Q . B . D2

= 1400 kN - mtr > Max Moment = 760.84 kN - mtr

Hence Safe

Clear Cover 50 mm

Bar Dia 32 mm

deff provided = 1500 - 50 - 96 = 1354.0

Km x σ cbc 133.3333333

+

==

+

=

∴∴∴∴

Steel Required =

0.2D = 300

= -Ast Ast1 (0.5(L/D-.5)= 4 bars

200 0.667 1354.0

= 4214.38 mm2

Mid ht 0.3D = 450

Ast2 = 2 bars

Using 32 mm dia bar ∴ ∴ ∴ ∴ No of Bars = 6

+Ast 0.25D-0.05L = 238 mm

(Note: MOST standard drawings have same reinf at Ast = 6 bars

top and bottom)

Side face reinft

Side face reinft reqd as per IS 456:2000 is 0.002 X 1500 x 350

= 1050 mm2

1608.5 mm2

Provide 10 dia 11 Nos. on each face

a) Check for Max Shear

Total Shear = 476.87 kN

Shear stress , τ = V ( Vide cl - 304.7.1.1 of I.R.C:-21-1987 )

B x d

V = The design shear across the section

d = Effective depth of the section

B = Breadth of slab

. .

. τ = 476.9 x 1000 = 1.01 N/mm2

350 x 1354.00

Maximum Permissible Shear Stress :- ( Vide cl - 304.7.2 of I.R.C:-21-2000 )

b = 350 mm

d revised = 1354 mm

p = 1.02 %

f ck = 40.00 N/mm2

τ c = 0.42 N/mm2

τ max = 2.5 N/mm2

= V / bd = 1.006 > 0.42 N/mm2

Shear reinft. reqd

= 1.006 < 2.5 N/mm2

SAFE

Vsteel = V −τ c*b*d 27.784 T Shear reinft. Reqd.

Spacing = σ sc X A SV X d

Vst

Assuming 2 legged 12 dia stirrups

A SV = 226.08 mm2

σ sc = 200 N/mm2

Spacing of stirrups = 220.4 mm

Provide 2 legged 12 dia stirrups at 200.0 c/c

All these loads are less than or equal to the loads experienced during the jacking operations.

M

σ st x j x d

760835523.24

xx

Documents

PSC Girder Design