Chapter 3 Prestressing with Post-Tensioning
9.1.4 Local Shear and Flexure in Webs
The webs of concrete box girder bridges must be verified with regard to local shear pull-out and flexure of cover concrete in the vicinity of ducts and duct banks. Figure 9.11 shows one interior web of a post-tensioned box girder bridge that contains three post-tensioning tendons arranged in a duct stack. The detail to the right in figure 9.11 is an enlargement of the duct stack. The dimensions shown are defined as:
dduct = outside diameter of the post-tensioning tendon duct (in) dc = cover of the duct on towards the direction of curvature (in) sduct = clear distance between tendon ducts in the vertical direction (in) hds = height of the duct stack (in)
tw = web width (in)
Ducts are considered stacked for flexure of the concrete cover when the clear distance between ducts (sduct) is less than 1.5 inches. For shear resistance calculations, duct stacks include clear spacings greater than the diameter of the ducts.
Figure 9.11 – Parameters for Local Shear and Flexure Design
Chapter 9 – Other Design Considerations 217 of 369 9.1.4.1 Shear Resistance to Pull-out
In-plane tendon forces may cause tendons to pull-out of the webs towards the center of box girder curvature unless they are sufficiently embedded in web concrete or tied back to transverse reinforcing within the web. Resistance to tendon pull-out is provided by the shear capacity of the concrete in the webs on the side of the ducts towards the center of box girder curvature. The LRFD Specifications define the shear resistance as:
(Eqn. 9.4) Vr =fVn
In which the nominal capacity is defined by:
(Eqn. 9.5) Vn = 0.15 deff f 'ci
Where, Vn = nominal shear resistance of two shear planes per unit length (kips/in) ϕ = resistance factor for shear = 0.75
f’ci = concrete strength at time of prestressing (ksi)
deff = one-half of the effective length of the failure plane in shear and tension for a curved element (in)
The length of the shear failure plane (deff) varies as a function of the clear spacing (sduct) between the tendons in a duct stack. For duct clear spacings less than the diameter of the duct (sduct< dduct), the effective length of the failure plane is defined by equation 9.6. The effective length of the failure plane for this case is shown in Figure 9.12. The detail to the left in figure 9.12 shows the special case of zero duct spacing. The detail to the right shows the general case for equation 9.6.
(Eqn. 9.6)
4
eff c dduct
d =d +
Figure 9.12 – Effective Length of Failure Plane for Equation 9.6
Chapter 9 – Other Design Considerations 218 of 369 When the duct spacing is greater than or equal to the duct diameter, the definition of the effective length of the failure plane changes. In this case, the lesser value of equation 9.7 and 9.8 is used for deff. These two equations are shown graphically in figure 9.13.
(Eqn. 9.7)
2
eff w dduct
d = −t
(Eqn. 9.8)
4 2
duct duct
eff c
d s
d =d + +∑
Figure 9.13 – Effective Length of Failure Plane for Equations 9.7 and 9.8
Procedure for verifying resistance to shear pull-out:
• Compute the factored in-plane force for all tendons in a web (equation 9.1).
• Compute the effective length of the failure plane for shear by equation 9.6, 9.7 or 9.8.
• Compute the shear resistance using equations 9.3 and 9.4.
• If the factored in-plane force is greater than the shear resistance, provide duct ties and stirrup similar to those shown in LRFD Figure C5.10.4.3.1b-1.
• As web radii vary, preform for each web in the cross section.
9.1.4.2 Cracking of Concrete Cover
Excessive in-plane force in a duct stack can cause cracking of the concrete cover. This cracking can negatively impact the long-term durability of the web and post-tensioning system.
The LRFD Specifications requires a calculation of flexural stresses in an idealized beam of concrete cover that spans the duct stack as shown in figure 9.14. This calculation is required when the clear spacing between ducts is less than 1.5 inches. These flexural stresses are combined with flexural stresses from regional bending moments for evaluation.
Chapter 9 – Other Design Considerations 219 of 369 Figure 9.14 – Local Bending Moments for Evaluating Cracking of Concrete Cover
The bending moments in the beam of concrete cover are those of a fixed end beam subjected to a uniform load over its length. The magnitude of the uniform load is equal to the summation of the factored in-plane forces divided by the duct stack height. The resulting end and mid-point bending moments are:
(Eqn. 9.9)
2
12
u in ds end ds
F h
M h
−
=
∑
(Eqn. 9.10)
2
24
u in ds mid ds
F h
M h
−
=
∑
The stresses from the local and regional bending in the web are computed and compared to a permissible cracking stress given as:
(Eqn. 9.11) fcr =ffr =0.85 0.24 ( f 'ci)
Procedure for verifying cracking of concrete cover:
• Compute the factored in-plane force for all tendons in a web (equation 9.1).
• Compute local bending moments from equations 9.9 and 9.10.
• Compute the stresses in the beam of concrete cover.
• Combine stresses resulting from regional bending with those from local bending.
• Compare the stresses to the permissible cracking stress from equation 9.11
• If the stresses are greater than the permissible cracking stress, provide duct ties and stirrup similar to those shown in LRFD Figure C5.10.4.3.1b-1 (Caltrans recommends duct ties when the bridge horizontal radius is less than 800’).
• As web radii vary, preform for each web in the cross section.