Homework4 Answer1

Department of Civil Engineering, Faculty of Engineering, Mahidol University
EGCE 406: Bridge Design
Homework 4 (Due Tuesday, September 14 th 2010)
Develop an Excel worksheet to calculate the stresses in a prestressed concrete
member at initial and service stages based on AASHTO LRFD Specifications. The
worksheet must have the following features:
• Able to input material and section properties
• Able to input the prestressing force, number of strand, and eccentricity.
• Able to input (or calculate) moments due to dead load, superimposed dead
loads, and live loads on a simple-supported member.
• Able to calculate allowable stresses from the input compressive strength
according to the AASHTO specifications.
• Able to calculate stresses on the top and bottom fiber in the member for initial
(immediately after transfer) and final (service) stages.
• Able to plot a simple feasible domain for the critical section
• Able to plot prestress envelope at every 10% of the span
The example given in class may be used as a guideline and to check your results.
However, you do not need to follow it closely (you may be as creative as you want).
After successfully develop the worksheet, use this worksheet to find the number of
strands and the tendon profile for the following beam:
Assume
f’ ci = 40 MPa, f’ c = 55 MPa,
f pu = 1860 MPa, f pi = 1300 MPa,
η = 0.8, f pe = 1040 MPa
A ps (each) = 98.71 mm 2
Note that the dead load and live loads are
uniformly distributed and there are two
superimposed point loads at both ends.

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Prestressed Concrete Beam ‐ AASHTO LRFD
Materials Concrete Prestressing Steel
f' ci 40 MPa f pu 1860 MPa ƞ 0.8
f' c 55 MPa f pi 1300 MPa f pe 1040 MPa
Section Concrete Prestressing Steel
A c 131250 mm^2 Z t 1.91.E+07 mm^3 # strands 4
I c 3.04E+09 mm^4 Z b 8.91.E+06 mm^3 A ps (each) 98.71 mm^2 F i 513292
y t 158.9 mm kt ‐67.91 mm A ps 394.84 mm^2 F 410634 kN
y b 341.07 mm kb 145.76 mm e 0 266.07 mm 1/Fi 1.948E‐06 kN
Load
Span mm Midspan Moments
W D N/mm M D 7.30E+07 N‐mm 73 kN‐m
W SD N/mm M SD ‐1.00E+07 N‐mm ‐10 kN‐m
W L N/mm M L 8.17E+07 N‐mm 82 kN‐m
W T N/mm M T 1.45E+08 N‐mm 145 kN‐m
AASHTO Allowable Stress
f ti = 0.25(f' ci ) 1/2 < 1.38 MPa ‐1.38 MPa
f ci = 0.6 f' ci = 24.00 MPa
f ts = 0.50(f' c ) 1/2 = ‐3.71 MPa
f cs (P e + M D + M SD ) = 0.45f' c = 24.75 MPa
f cs (0.5(P e + M D + M SD ) + M LL ) = 0.40f' c = 22.00 MPa
f cs (P e + M T ) = 0.6f' c = 33.00 MPa
Stresses at Critical Section
Immediately after Transfer ‐ with Dead Load
Check
F i F i e 0 M D
f t = 513292 ‐ 513292 266.07 + 73000000 = 0.59 OK
131250 19131529 19131529
A c Z t Z t
F i F i e 0 M D
f b = 513292 + 513292 266.07 ‐ 73000000 = 11.04 OK
131250 8913126 8913126
A c Z b Z b
After Prestressing Losses
Top Fiber ‐ with Sustained Load
F F e 0 M D +M SD
f t = 410633.6 ‐ 410633.6 266.07 + 6.30E+07 = 0.71 OK
131250 19131529 19131529
A c Z t Z t
Top Fiber ‐ with One‐Half Prestress and Sustained Load plus Live Load
0.5F 0.5F e 0 0.5M sus +M LL
f t = 205316.8 ‐ 205316.8 266.07 + 1.13E+08 = 4.62 OK
131250 19131529 19131529
A c Z t Z t
Top Fiber ‐ with Total Load
F F e 0 M T
f t = 410633.6 ‐ 410633.6 266.07 + 144637000 = 4.98 OK
131250 19131529 19131529
A c Z t Z t
Bottom Fiber ‐ with Service III Limit State
F F e 0 M D +M SD +0.8M LL
f b = 410633.6 + 410633.6 266.07 ‐ 1.28E+08 = 0.99 OK
131250 8913126 8913126
A c Z b Z b
Check

Feasible Domain
e 0 (inch)
ƞ = 0.8 0 3.00E‐06 1/F i
M min ‐f ti Z t = 9.94E+07 146 444 I
M min +f ci Z b = 2.87E+08 ‐68 793 II
M max ‐f cs Z t = ‐4.11E+08 146 ‐1394 IIIa
M max ‐f cs Z t = ‐3.08E+08 146 ‐2162 IIIb
M max ‐f cs Z t = ‐4.87E+08 146 ‐1679 IIIc
M max +f ts Z b = 9.53E+07 ‐68 289 IV
d c,min = 75 mm 266 266 V
0.0E+00 5.0E‐07 Do 1.0E‐06 not delet 341.07 1.5E‐06341.07
2.0E‐06 2.5E‐06 3.0E‐06
‐600
1/F i
Case I
Case II
Case IIIa
Case IIIb
‐400
Case IIIc
Case IV
Case V
yb
‐200
e 0 (mm)
0
200
400
600

Prestressing Envelope
x (mm) 0 750 1500 2250 3000 3750 4500 5250 6000 6750 7500 8250 9000 9750 10500
M D 0.00E+00 ‐8.69E+05 ‐3.48E+06 ‐7.82E+06 ‐1.39E+07 2.61E+06 1.74E+07 3.04E+07 4.17E+07 5.13E+07 5.91E+07 6.52E+07 6.95E+07 7.21E+07 7.30E+07
M SD 0.00E+00 ‐2.50E+06 ‐5.01E+06 ‐7.51E+06 ‐1.00E+07 ‐1.00E+07 ‐1.00E+07 ‐1.00E+07 ‐1.00E+07 ‐1.00E+07 ‐1.00E+07 ‐1.00E+07 ‐1.00E+07 ‐1.00E+07 ‐1.00E+07
M L 0.00E+00 ‐9.72E+05 ‐3.89E+06 ‐8.75E+06 ‐1.56E+07 2.92E+06 1.94E+07 3.40E+07 4.67E+07 5.73E+07 6.61E+07 7.29E+07 7.78E+07 8.07E+07 8.16E+07
e0 case I 197.2 195.5 190.4 182.0 170.1 202.3 231.1 256.5 278.5 297.1 312.3 324.2 332.6 337.7 339.4
e0 case II 348.8 347.1 342.1 333.6 321.8 353.9 382.7 408.1 430.1 448.7 464.0 475.8 484.3 489.4 491.1
e0L 197.2 195.5 190.4 182.0 170.1 202.3 231.1 256.5 278.5 297.1 312.3 324.2 332.6 337.7 339.4
e0 case IIIa ‐1007.3 ‐1015.6 ‐1028.0 ‐1044.7 ‐1065.6 ‐1025.4 ‐989.4 ‐957.7 ‐930.1 ‐906.9 ‐887.8 ‐873.0 ‐862.4 ‐856.1 ‐854.0
e0 case IIIb ‐1904.2 ‐1917.2 ‐1943.8 ‐1984.1 ‐2038.2 ‐1908.0 ‐1791.6 ‐1688.8 ‐1599.8 ‐1524.4 ‐1462.8 ‐1414.8 ‐1380.5 ‐1360.0 ‐1353.1
e0 case IIIc ‐1391.7 ‐1402.3 ‐1421.8 ‐1450.4 ‐1487.8 ‐1402.6 ‐1326.4 ‐1259.2 ‐1200.9 ‐1151.6 ‐1111.2 ‐1079.8 ‐1057.4 ‐1044.0 ‐1039.5
e0 case IV ‐148.4 ‐158.5 ‐176.6 ‐202.8 ‐236.9 ‐160.7 ‐92.6 ‐32.4 19.7 63.8 99.9 128.0 148.0 160.1 164.1
eoU ‐148.4 ‐158.5 ‐176.6 ‐202.8 ‐236.9 ‐160.7 ‐92.6 ‐32.4 19.7 63.8 99.9 128.0 148.0 160.1 164.1
eomp 266.1 266.1 266.1 266.1 266.1 266.1 266.1 266.1 266.1 266.1 266.1 266.1 266.1 266.1 266.1
x (mm) 10500 11250 12000 12750 13500 14250 15000 15750 16500 17250 18000 18750 19500 20250 21000
M D 7.30E+07 7.21E+07 6.95E+07 6.52E+07 5.91E+07 5.13E+07 4.17E+07 3.04E+07 1.74E+07 2.61E+06 ‐1.39E+07 ‐7.82E+06 ‐3.48E+06 ‐8.69E+05 0.00E+00
M SD ‐1.00E+07 ‐1.00E+07 ‐1.00E+07 ‐1.00E+07 ‐1.00E+07 ‐1.00E+07 ‐1.00E+07 ‐1.00E+07 ‐1.00E+07 ‐1.00E+07 ‐1.00E+07 ‐7.51E+06 ‐5.01E+06 ‐2.50E+06 0.00E+00
M L 8.16E+07 8.07E+07 7.78E+07 7.29E+07 6.61E+07 5.73E+07 4.67E+07 3.40E+07 1.94E+07 2.92E+06 ‐1.56E+07 ‐8.75E+06 ‐3.89E+06 ‐9.72E+05 0.00E+00
e0 case I 339.4 337.7 332.6 324.2 312.3 297.1 278.5 256.5 231.1 202.3 170.1 182.0 190.4 195.5 197.2
e0 case II 491.1 489.4 484.3 475.8 464.0 448.7 430.1 408.1 382.7 353.9 321.8 333.6 342.1 347.1 348.8
e0L 339.4 337.7 332.6 324.2 312.3 297.1 278.5 256.5 231.1 202.3 170.1 182.0 190.4 195.5 197.2
e0 case IIIa ‐854.0 ‐856.1 ‐862.4 ‐873.0 ‐887.8 ‐906.9 ‐930.1 ‐957.7 ‐989.4 ‐1025.4 ‐1065.6 ‐1044.7 ‐1028.0 ‐1015.6 ‐1007.3
e0 case IIIb ‐1353.1 ‐1360.0 ‐1380.5 ‐1414.8 ‐1462.8 ‐1524.4 ‐1599.8 ‐1688.8 ‐1791.6 ‐1908.0 ‐2038.2 ‐1984.1 ‐1943.8 ‐1917.2 ‐1904.2
e0 case IIIc ‐1039.5 ‐1044.0 ‐1057.4 ‐1079.8 ‐1111.2 ‐1151.6 ‐1200.9 ‐1259.2 ‐1326.4 ‐1402.6 ‐1487.8 ‐1450.4 ‐1421.8 ‐1402.3 ‐1391.7
e0 case IV 164.1 160.1 148.0 128.0 99.9 63.8 19.7 ‐32.4 ‐92.6 ‐160.7 ‐236.9 ‐202.8 ‐176.6 ‐158.5 ‐148.4
eoU 164.1 160.1 148.0 128.0 99.9 63.8 19.7 ‐32.4 ‐92.6 ‐160.7 ‐236.9 ‐202.8 ‐176.6 ‐158.5 ‐148.4
eomp 266.1 266.1 266.1 266.1 266.1 266.1 266.1 266.1 266.1 266.1 266.1 266.1 266.1 266.1 266.1
‐300
Prestressing Envelope
‐200
Upper Bound
Lower Bound
‐100
Upper Row
Practical Limit
Eccentricity (mm)
0
100
e 0 Total
200
Lower Row
300
400
0 1500 3000 4500 6000 7500 9000 10500 12000 13500 15000 16500 18000 19500 21000
X (mm)