Chapter 3 Prestressing with Post-Tensioning
3.4 Selection of Prestressing Force for a Given Eccentricity
Where service limit state flexural verification of prestressed members involves summing of stresses due to applied forces, it is often more convenient to work with internal forces and moments when designing prestressed members. This section develops expressions for prestressing based on force and eccentricity.
Stresses are determined at the extreme top and bottom fibers by the familiar equations:
(Eqn. 3.1)
(Eqn. 3.2)
Chapter 3 – Prestressing with Post-Tensioning 40 of 389 These equations can be rearranged to express the required prestressing force as a function of the other equation variables.
Consider first the prestressing requirements for the bottom stress. The limiting prestressing force would be that which satisfies equation 3.2 when the bottom stress fb is set to a permissible concrete stress fa. Making this substitution and multiplying equation 3.2 through by the cross section inertia and dividing by the distance from the neutral axis to the extreme bottom fiber the equation becomes:
(Eqn. 3.3)
This equation can be reduced further by noting that the left hand side of the equation is a bending moment that produces the permissible stress in the concrete at the bottom of the girder:
(Eqn. 3.4)
2 ab f Ia
M = c
Equation 3.3 now becomes:
(Eqn. 3.5)
2
ab FI
M Fe M
= Ac + −
Further simplification is made by defining the dimensionless parameter:
(Eqn. 3.6)
The parameter ρ is termed the Efficiency of the cross section with regard to prestressing. Cross section efficiencies for three typical shapes are shown in figure 3.6. As seen in this figure, cross section efficiency increases as material is moved away from the neutral axis and is located in top and bottom flanges.
Figure 3.6 – Efficiencies of Various Cross Sections
Chapter 3 – Prestressing with Post-Tensioning 41 of 389 Recognizing that:
(Eqn. 3.7)
Equation 3.5 is now simplified to:
(Eqn. 3.8) Mab =F c Fe Mρ 1+ −
Solving for the prestressing force:
(Eqn. 3.9)
1
M Mab
F ρc e
= +
+
The numerator of this equation is the bending moment at the cross section under study, adjusted by the moment causing allowable stress. The sign of Ma is established by the sign of the allowable stress at the section. A permissible tension would cause Ma to be negative, reducing the required prestressing force. A requirement for a residual compressive stress would cause Ma to be positive, increasing the required prestressing force.
When Ma is established by the minimum allowable stress, equation 3.9 becomes the expression for minimum required prestress force. When maximum permissible compressive stress is controlling, equation 3.9 becomes the expression for maximum permissible prestress force.
This exercise can be repeated for limiting stress control at the top of a cross section. In this case, equation 3.1 can be rearranged to find:
(Eqn. 3.10)
2
M Mat
F e ρc
= −
−
It is interesting to study equations 3.9 and 3.10 for additional implications. Figure 3.7 shows the internal equilibrium expressed by equation 3.9 for a positive bending moment. Figure 3.8 shows a similar diagram for the equilibrium expressed by equation 3.10.
Figure 3.7 – Internal Equilibrium for Positive Bending.
Chapter 3 – Prestressing with Post-Tensioning 42 of 389 Figure 3.8 – Internal Equilibrium for Negative Bending.
Now consider the case of zero bending moment acting on the cross section. Equation 3.9 becomes:
(Eqn. 3.11)
This equation is satisfied in one of two conditions, either the prestressing force is equal to zero or:
(Eqn. 3.12)
Equation 3.12 shows that any prestressing force can be applied at a distance of ρc1 above the neutral axis with the result being zero stress at the bottom of the cross section. Likewise, from equation 3.10:
(Eqn. 3.13)
Equation 3.13 shows that any prestressing force can be applied at a distance of ρc2 below the neutral axis with the result being zero stress at the top of the cross section. Graphically, these two limiting eccentricities are seen in figure 3.9.
Together, these limiting eccentricities define the upper and lower locations of the cross section kern—that portion of the cross section through which no tension occurs if the resultant compressive force is located therein. Figure 3.10 shows the layout of the upper and lower kern for a cross section with bending about the horizontal axis. Further analysis of the cross section could be undertaken to define the kern limits in any direction.
Chapter 3 – Prestressing with Post-Tensioning 43 of 389 Figure 3.9 – Limiting Eccentricities for Zero Tension Under Axial Force Only
Figure 3.10 – Kern of a Cross Section for Bending About the Horizontal Axis.
Chapter 3 – Prestressing with Post-Tensioning 44 of 389 Example: Consider the line girder segment of a concrete box girder bridge shown in
figure 3.11. The girder is simply supported, with a span length of 120’. In addition to the girder self weight, the girder is subjected to a uniformly distributed load of 1.0 kip/ft. The concrete strength of the girder is 5000 psi.
Determine the limiting values of prestressing for maximum and minimum stress at the top and bottom of the girder. The maximum permissible compressive stress is 0.6f’c. The minimum permissible tensile stress is 3√f’c.
The section properties of the girder are:
Figure 3.11 –Example Concrete I-Girders The efficiency of the cross section is:
The moment required to produce the permissible tensile stress at the bottom- most fiber of the girder is:
The moment required to produce the permissible compressive stress at the bottom-most fiber of the girder with no axial force is:
The moment required to produce the permissible tensile stress at the top- most fiber of the girder is:
149.55
0.6617 21.5(3.0247)(3.4753)
ρ = =
( ) ( ) ( )
( )
1
3 ' 3 5000 149.55
0.144 1, 510.3 3.0247
at T f c I
M ft kips
= − c = − = − −
( ) ( ) ( )
( )
2
3 ' 3 5000 149.55
0.144 1, 314.5 3.4753
ab T f c I
M ft kips
= − c = − = − −
( ) ( ) ( )
( )
2
0.6( ' ) 0.6(5000) 149.55
0.144 18, 590 3.4753
ab C f c I
M ft kips
= c = = −
Chapter 3 – Prestressing with Post-Tensioning 45 of 389 The moment required to produce the permissible compressive stress at the top-most fiber of the girder is:
The mid-span bending moment due to the applied load is:
The minimum prestressing required to limit the bottom girder tension is:
The maximum prestressing permissible to not overstress the bottom of the girder is:
The maximum prestressing permissible to not exceed the minimum tension in the top of the girder is:
The minimum prestressing permissible to not over compress the top of the girder is:
Though the four limiting forces can be computed, the two limiting forces for top stress are not useful for this example. For the case of the maximum top tension, the limiting prestress force causes the bottom of the beam to exceed the maximum compressive stress. To reach the maximum top compression, the post-tensioning force would need to be negative. As a result, for this example, the prestressing force can vary between 1397.4 kips and 5819.2 kips without exceeding allowable stresses.
( )
( ) 1
7, 605 ( 1, 314.5)
1, 397.4 0.6617 3.0247 2.5
M Mab T
F kips
ρc e
+ + −
≥ = =
+ +
( )2
2 4.225 120
7, 605
8 8
M = wL = = ft kips−
( )
( ) 1
7, 605 18, 590
5,819.2 0.6617 3.0247 2.5
M Mab C
F kips
c e ρ
+ +
≤ = =
+ +
( )
( ) 2
7, 605 21, 359
68, 635 2.5 .6617 3.4753
M Mat C
F kips
e ρc
− −
≥ = = −
− −
( )
( ) 2
7, 605 ( 1, 510.3)
45, 487 2.5 .6617 3.4753
M Mat T
F kips
e ρc
− − −
≤ = =
− −
( )
0.15 21.5 1 4.225 /
w= + = k ft
( ) ( ) ( )
( )
1
0.6( ' ) 0.6(5000) 149.55
0.144 21, 359 3.0247
at C f c I
M ft kips
= c = = −