Chapter 3 Prestressing with Post-Tensioning
5.8 Service Limit State Stress Verifications
This Section presents stress summaries for important load cases at service limit states.
Flexural stresses are verified after the tendons are stressed and before grouting, after the bridge is open to traffic and before long-term losses, and the bridge in operation after long-term
Chapter 5 – Preliminary Design 108 of 369 losses have occurred. A verification of principal stresses in the webs is made with the bridge in operation and after long-term losses have occurred.
5.8.1 Service Flexure—Temporary Stresses (DC and PT Only)
Longitudinal stresses in the concrete superstructure are verified at three locations along the bridge. The stresses are being verified after all post-tensioning tendons are stressed, but before grouting. 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 5.9.4.1. 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
a 0.0
f = ksf (no tension when not considering bonded reinforcing) 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
Chapter 5 – Preliminary Design 109 of 369 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.
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 110 of 369 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.)
5.8.3 Service Limit State III Flexure After Long-Term Losses
Chapter 5 – Preliminary Design 111 of 369 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+ = + = kips
The 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
h x 20 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 = =
( )
2 2
2 93.00 2
64.91 79.85
2 2
x xy
R= s +t = + = ksf
max 126.35
2
x R ksf
s =s + =
' '
min 33.35 231.6 3.28 ( )
2
x R ksf psi fc f in psic
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.