Chapter 5—Preliminary Design 105 of 355 Multiplying the components of equation 5.29, the loss of prestress stress resulting from the shrinkage of concrete is:
0.000454(28500)(1.054) 13.6
fpSD ksi
D = =
5.7.4 Losses from Concrete Creep
The loss of stress in the prestressing steel resulting from the creep of concrete is:
28500
( , ) (.66)(1.226)(1.054) 6.3 3834
p
pCD cgp b f i df
ci
f E f Ψ t t K ksi
D = E = =
5.7.5 Losses from Steel Relaxation
Using the AASHTO LRFD permissible value for low relaxation steel, the loss of prestressing stress is resulting from relaxation of the prestressing steel is:
pR
2.4
f ksi
D =
5.7.6 Total of Losses and Tendon Sizing
The total loss of prestressing force after jacking is the sum of elastic shortening, shrinkage, creep and relaxation:
2.6 13.6 6.3 2.4 24.9
fp ksi
D = + + + =
The resulting stress in the prestressing steel at the center of the middle span is then:
171.62 24.9 146.72
fp = − = ksi
The resulting force in a 0.6” diameter strand would be 31.9 kips. The number of strands in the 15 tendons, based on the force requirement at the center of Span 2 of 8,172 kips, would be 17.2. Therefore, use 18 strand tendons in final design. The jacking stressed assumed in the loss calculations was 75 percent of the ultimate strength of the strand. The resulting jacking force (Pjack) for each tendon would be 791 kips.
Chapter 5—Preliminary Design 106 of 355 reinforcing in the precompressed tensile zone to permit tension during this phase. Applied bending moments are taken from table 5.1.
Three post-tensioning tendons in each of the five webs of the box girder are stressed. Each of the tendons contains 18, 0.6” diameter strands. Tendon forces at the three study sections are based on stresses shown in figure 5.27, reduced for elastic shortening losses of 2.6 ksi as computed in section 5.7.2.
The allowable concrete stresses in the concrete before losses are presented in AASHTO LRFD Article 5.9.4.1. For a 28-day concrete strength of 5 ksi the allowable stresses are:
0.60 ' 0.60(4) 2.4 345.6 ( )
a ci
f = f = = ksi= ksf compression
0.0 ( )
fa = ksf no tension At the point of maximum eccentricity in Span 1 and 3
10,819 10,819(2.5)(2.732) 19,126(2.732)
99.45 643.7 643.7 75.2
fTOP = − + = ksf
10,819 10,819(2.5)(3.768) 19,126(3.768)
155.2
99.45 643.7 643.7
fBOT = + − = ksf
Over Piers 2 and 3
10, 556 10, 556( 1.25)(2.732) 20,879(2.732)
99.45 643.7 643.7 73.5
fTOP = − − +− = ksf
10, 556 10, 556( 1.25)(3.768) 20,879(3.768)
151.1
99.45 643.7 643.7
fBOT = + − −− = ksf
At the center of the middle Span
9, 903 9, 903(2.5)(2.732) 29, 613(2.732)
120.2
99.45 643.7 643.7
fTOP = − + = ksf
9, 903 9, 903(2.5)(3.768) 29, 613(3.768)
99.45 643.7 643.7 71.2
fBOT = + − = ksf
The concrete stresses in the box girder superstructure are within permissible AASHTO LRFD limits. Stresses over Piers 2 and 3 could have been checked at the face of the support.
*From the first row of AASHTO LRFD Table 5.9.4.1.2-1, where a N/A is given as the stress limit.
Chapter 5—Preliminary Design 107 of 355 5.8.2 Service Limit State III Flexure Before Long-Term Losses
Longitudinal stresses in the concrete superstructure are verified at three locations along the bridge. The stresses are being verified when the bridge is first open to traffic. The assumption for this example is that there is insufficient bonded mild reinforcing in the precompressed tensile zone to permit tension during this phase. Applied bending moments are taken from table 5.1.
Three post-tensioning tendons in each of the five webs of the box girder are stressed. Each of the tendons contains 18, 0.6” diameter strands. Tendon forces at the three study sections are based on stresses shown in figure 5.27, reduced for elastic shortening losses of 2.6 ksi as computed in section 5.7.2.
The allowable concrete stresses in the concrete before losses are presented in AASHTO LRFD Article 5.9.4.2. For a 28-day concrete strength of 5 ksi the allowable stresses are:
0.6
'0.6(1)(5) 3.0 432.0
a w c
f = f f = = ksi= ksf (compression)
0.19
'0.19 5 0.425 61.2
a c
f = − f = − = − ksi= − ksf (tension)
At the point of maximum eccentricity in Span 1 and 3
10,819 10,819(2.5)(2.732) 32,867(2.732)
133.5
99.45 643.7 643.7
fTOP = − + = ksf
10,819 10,819(2.5)(3.768) 32,867(3.768)
99.45 643.7 643.7 74.7
fBOT = + − = ksf
Over Piers 2 and 3
10, 556 10, 556( 1.25)(2.732) 36, 952(2.732)
99.45 643.7 643.7 5.3
fTOP − − ksf
= − + =
10, 556 10, 556( 1.25)(3.768) 36, 952(3.768)
245.2
99.45 643.7 643.7
fBOT = + − −− = ksf
At the center of the middle Span
9, 903 9, 903(2.5)(2.732) 46,850(2.732)
193.3
99.45 643.7 643.7
fTOP = − + = ksf
9, 903 9, 903(2.5)(3.768) 46,850(3.768)
99.45 643.7 643.7 29.7
fBOT = + − = − ksf
The concrete stresses in the box girder superstructure are within permissible AASHTO LRFD limits. Stresses over Piers 2 and 3 could have been checked at the face of the support.
Chapter 5—Preliminary Design 108 of 355 5.8.3 Service Limit State III Flexure After Long-Term Losses
Longitudinal stresses in the concrete superstructure are verified at three locations along the bridge. The stresses are being verified when the bridge is first open to traffic. The assumption for this example is that there is insufficient bonded mild reinforcing in the precompressed tensile zone to permit tension during this phase. Applied bending moments are taken from table 5.1.
Three post-tensioning tendons in each of the five webs of the box girder are stressed. Each of the tendons contains 18, 0.6” diameter strands. Tendon forces at the three study sections are based on stresses shown in figure 5.27, reduced by long-term losses of 24.9 ksi as computed in section 5.7.6.
The allowable concrete stresses in the concrete before losses are presented in AASHTO LRFD Article 5.9.4.2. For a 28-day concrete strength of 5 ksi the allowable stresses are:
0.6
'0.6(1)(5) 3.0 432.0
a w c
f = f f = = ksi= ksf (compression)
0.19
'0.19 5 0.425 61.2
a c
f = − f = − = − ksi= − ksf (tension)
At the point of maximum eccentricity in Span 1 and 3
9, 513 9, 513(2.5)(2.732) 32, 357(2.732)
132.0
99.45 643.7 643.7
fTOP = − + = ksf
9, 513 9, 513(2.5)(3.768) 32, 357(3.768)
99.45 643.7 643.7 45.5
fBOT = + − = ksf
Over Piers 2 and 3
9, 249 9, 249( 1.25)(2.732) 38, 372(2.732)
99.45 643.7 643.7 20.8
fTOP − − ksf
= − + = −
9, 249 9, 249( 1.25)(3.768) 38, 372(3.768)
249.9
99.45 643.7 643.7
fBOT = + − −− = ksf
At the center of the middle Span
8, 596 8, 596(2.5)(2.732) 45, 429(2.732)
188.0
99.45 643.7 643.7
fTOP = − + = ksf
8, 596 8, 596(2.5)(3.768) 45, 429(3.768)
99.45 643.7 643.7 53.7
fBOT = + − = − ksf
The concrete stresses in the box girder superstructure are within permissible AASHTO LRFD limits. (Stresses checked over Piers 2 and 3 could have been checked at the face of the supporting pier.)
Chapter 5—Preliminary Design 109 of 355 5.8.4 Principal Tension in Webs after Losses
AASHTO LRFD does not require principal tension verifications at service limit states for concrete box girders other than those built segmentally. These verifications are, however, useful for preliminary design, as an indicator whether the webs are sized appropriately. It is strongly recommended that principal tension be verified in cast-in-place concrete box girder construction when the box girder has only two webs.
Consider the cross section 0.72h (4.68’) to the left of Pier 2 in Span 2. The shear forces acting are:
16(80 4.68) 1, 205
VDC = − = kips
1.4(80 4.68) 105
VDW = − = kips
(1 )
53
LL lane
V = kips
(1 )
68
LL truck
V = kips
1.33(68) 53 143
VLL I+ = + = kipsThe live load distribution for shear is found by AASHTO LRFD Table 4.6.2.2.3a-1 for cross section type d and two or more design lanes loaded:
0.1 0.1
0.9 0.9
12.25 78
1.157
7.3 12( ) 7.3 12(160)
S d
DF L
= = =
5(1.157) 5.785
L w
N =N ⋅DF = =
(0.8)5.785(143) 662
VLL I+ = = kips
( )
( ) ( )
2 2
2 0.9375
2 4.68 0.0219 1.26
20
hx rad deg
θ b −
= = = − = −
( )
9, 249 sin 1.26 203
VP = − = − kips
( )
1205 105 662 203 1769
V = + + + − = kips
∑
The stresses acting on an element at the neutral axis are:
9, 249
93.00 99.45
x ksf
s
= =y 0.0 s =
( ) 1, 769(118.1)
64.91 ( ) 643.7(5)
xy
V Q ksf
t
= I B = =Chapter 5—Preliminary Design 110 of 355 The features of Mohr’s Circle with these stresses are:
( )
2 2
2
93.00
264.91 79.85
2 2
x
R=
s
+t
xy = + = ksfmax
126.35
2
x R ksf
s
=s
+ =' '
min
33.35 231.6 3.28 ( )
2
x
c c
R ksf psi f f in psi
s =s − = − = =
The Mohr’s Circle representation of stress at this location is then:
Figure 5.28 – Mohr Circle for Location of Maximum Shear in Middle Span
The maximum principal tension of -44 ksf is greater than what would be allowable for a segmental box girder, but does reflect a level of stress that can be adequately reinforced during final design.