Chapter 3 Prestressing with Post-Tensioning
9.1.3 Regional Effects—Transverse (Regional) Bending
Consider the hypothetical case of a tendon following a circular arc within a concrete member whose area is concentrated as to be completely coincident with the tendon. Figure 9.6 shows this hypothetical case with the tendon shown separated from the concrete for clarity. The curved stressed tendon produces an inward radial force. The concrete is placed in compression at the tendon anchorages and its curvature induces an outwardly resisting radial load equal and opposite to the tendon force. The radial forces negate each other and the curved concrete member only shortens under the prestressing force.
Figure 9.6 – Hypothetical Concrete Member Completely Coincident with a Tendon
Figure 9.7 shows a tendon following a circularly curving plate with a height of hc. In this case, the post-tensioning tendon and area of the concrete member being prestressed are not coincident. The radial force produced by the post-tensioning tendon is equal to the tendon force divided by the radius of the curved member. Circumferential compression is produced in the curved plate that is in equilibrium with the prestressing force. This circumferential compression is distributed over the height of the plate and produces a distributed pressure in the plate radial
Chapter 9 – Other Design Considerations 214 of 369 to the plate curvature. The member will shorten under the action of the prestressing force, but as the radial distributed force of the tendon is not coincident with the radial pressure produced in the concrete, the plate will bend radially.
Figure 9.7 – Post-Tensioning a Curved Plate
The curved plate shown in figure 9.7 is free to deflect radially under the applied post-tensioning.
The top and bottom of the plate will tend to deflect radially outward from the center of curvature, while deflections would tend to be inward at the elevation of the post-tensioning tendon. In concrete box girder construction the tops and bottoms of webs are restrained by their monolithic connections with the top and bottom slabs, producing transverse restraining moments as shown in figure 9.8.
Figure 9.8 – Web Flexure Restrained by Top and Bottom Slabs
Chapter 9 – Other Design Considerations 215 of 369 Transverse moments (Regional Bending in LRFD Article 5.10.4.3.1d) must be evaluated and combined with other transverse moments and shear requirements to determine the appropriate amount and placement of transverse reinforcing. The transverse moments from the in-plane radial forces could be determined by either a 2-dimensional analysis of a unit length of transverse cross section (as in chapter 8) or a more complex 3-dimensional finite element analysis. The LRFD Specifications offers a simplified approach to evaluating these transverse moments in the webs of box girder bridges. This approach first considers the web to be simply supported at its connection with the top and bottom slab, and loaded at mid-height of the web.
A continuity factor is then applied to express end moments as a percentage of the maximum positive transverse bending moment in the simply supported beam.
Figure 9.9 shows a comparison of the LRFD simplified approach to moments developed by more detailed analysis performed on a transverse cross section of a box girder bridge. In the detailed analysis, the bending moments can be seen as the superposition of the concentrated post-tensioning force and the resisting radial force in the compressed concrete web. Figure 9.9 shows these component loads and their summation. The LRFD simplified approach is shown at the left of figure 9.9. The effects of the radial force produced in the compressed concrete web are not considered in the LRFD simplified approach.
Figure 9.9 – Web Transverse (Regional) Bending Moments
The factored moment in using the LRFD simplified approach is:
(Eqn. 9.3)
4
cont u in c
u F h
M f −
=
Where, ϕcont = continuity factor = 0.6 for interior webs and 0.7 for exterior webs
hc = span of the web between the top and bottom slabs, measured along the axis of the web
The span of the web for equation 9.3 is shown in figure 9.10. Only components of forces acting to produce bending in the direction transverse to the axis of the web needs to be considered.
Though the height of the inclined outer web is taller than the vertical interior web in figure 9.10
Chapter 9 – Other Design Considerations 216 of 369 by one over the cosine of the web slope, the force transverse to the web axis is the in-plane force multiplied by the cosine of the web slope. These two negate each other such that the regional bending in the outer web is only different from the inner web by the difference of the radii of the two webs.
Figure 9.10 – Web Height for Equation 9.3