Chapter 3 Prestressing with Post-Tensioning
7.2.1 Straight Bridges Supported on Bearings
The longitudinal design of straight concrete box girder superstructures can be made using two- dimensional analyses supplemented with hand calculations to estimate force effects out of the plane of the analysis. Figure 7.1 shows the elevation of a bridge similar to the one presented in design example 1 in appendix C. The bridge shown in figure 7.1 is different from the design example in that the superstructure is not integral with the piers, but rests on bearings. The cross section of this bridge, shown in figure 7.2, is the same as for design example 1.
Figure 7.1 – Example Straight Bridge on Bearings
Figure 7.2 – Box Girder Superstructure Cross Section 7.2.1.1 Nodes
Figure 7.3 shows one layout for a two-dimensional model representation of the three-span bridge in figure 7.1. Nodes are defined by coordinates in an orthogonal coordinate system, such as x-coordinates in the longitudinal direction and y-coordinates in the vertical direction.
Longitudinally, nodes were first located at support locations and at even increments along the length of the bridge. The number of nodes defined along each span is the decision of the analyst, but should be sufficient to allow the analysis software member definitions to accurately model bridge behavior. A typical nodal spacing of 10’ was used for the bridge model shown in figure 7.3. Additional nodes (3, 15, 17, 22, 35, and 47) were added for ease in using the results of the analysis. These nodes coincide with the design cross section for shear (d or h/2 from the centerline of pier, depending on the approach in finding nominal capacity). Two other nodes (1
Chapter 7 – Longitudinal Analysis & Design 132 of 369 and 49) were added behind the first and last bearing locations to facilitate the modeling of post- tensioning consistent with end anchor details
Nodes shown in figure 7.3 are defined vertically at the center of gravity of the superstructure cross section. The vertical location of the nodal coordinates changes with variations in cross section such as when a bottom slab is thickened near a pier to control compressive stresses, or in a variable depth structure used in longer spans. The vertical locations of the nodes are not adjusted for the profile of the bridge, except in the most extreme conditions. There are other bridge types, such as cable-stayed bridges, where horizontal forces are sufficiently large as to impose significant deck bending because of profile changes.
The analysis model is simply supported at all bearing locations in the vertical direction. A horizontal support is provided at Node 2 to assure model stability. Interior piers and bearing stiffnesses could be added to this model if the of long-term effects of concrete creep and shrinkage and prestressing steel relaxation are desired in the piers. Their effect is typically small on the results for the design of the superstructure (see chapter 6). If the bearings are flexible, such as for laminated elastomeric bearings, the horizontal stiffness can be input as a constraint between pier top and the superstructure. Rotational stiffnesses of these bearings should not be accounted for in the model. The very large rotational stiffnesses of these bearings rely on large vertical loads to remain active without uplift. The result is an inappropriately large attraction of bending moment to the pier if modelled, approaching that of integral piers. A rotationally free connection is more appropriate.
Figure 7.3 – Two-Dimensional Analysis Model 7.2.1.2 Elements
Nodes in the two-dimensional analysis model of the straight bridge are connected by idealized beam-column elements defined by appropriate stiffness matrices. Figure 7.4 shows a typical stiffness matrix for a beam element connecting nodes that have three degrees of displacement freedom (vertical, horizontal, and rotational).
Figure 7.4 shows that the information necessary to define the behavior of the beam-column element connecting two nodes includes:
• Cross-Sectional Characteristics—Area and Moment of Inertia about the horizontal axis passing through the centroid of the cross section. Typically the gross cross section characteristics of the concrete girder are considered. There may be instances where the cross-sectional area of the ducts is great enough to impact analysis results. When this happens, the net cross section should be used for superstructure construction and tendon stressing, and the gross properties used for loads applied after the tendons are grouted.
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• Modulus of Elasticity—For the purposes of the verification of Service and Strength Limit States, cast-in-place post-tensioned bridges are analyzed with fully elastic behavior (LRFD C4.5.1). Within these elastic analyses, however, the time-dependent behaviors of concrete and prestressing steel must be incorporated (LRFD 4.5.2.2). Time- dependent behavior is most typically captured in commercially available software by a series of time updates, where changes in concrete dimension are estimated over a time step. Typically, the modulus of elasticity is based on the 28-day strength of the concrete (see chapter 2).
Figure 7.4 –Typical Element Stiffness Matrix for a Plane Frame Member with 3DOF Nodes For the cross section shown in figure 7.2, the cross section characteristics are:
Figure 7.5 –Cross Section Properties for the Box Girder shown in Figure 7.2
End and pier diaphragms are used at the supports to transmit shear forces from the webs to the supports and to stiffen the box girder with regard to torsion. End and pier diaphragms are typically solid sections of the bridge with small openings to allow for access between the spans after construction. It is important to incorporate the weight of these diaphragms into the design of the bridge. However, members containing small length diaphragms (up to approximately the depth of the box girder superstructure) should not be represented with cross section properties including the diaphragms. The typical cross section properties should be used for these members. Longer diaphragms could warrant a change in cross section characteristics.
Chapter 7 – Longitudinal Analysis & Design 134 of 369 7.2.1.3 Post-Tensioning
The effects of the post-tensioning tendons are modeled within the software as an equivalent set of forces acting on the elements of the bridge. Most post-tensioned bridge analysis software packages use a graphical interface so that the tendons can be defined by their desired geometry. Internal to the program, the geometry is used to compute fixed end element forces which then become the post-tensioning load case within the program. When loaded with the equivalent loads, the results are the combination of primary and secondary post-tensioning moments. The secondary moments are then determined by subtracting the primary moment, from the input tendon geometry, from the total post-tensioning moments.
The effect of time-dependent deformations of the concrete and steel are incorporated through time-steps where the long-term deformations are used to change the post-tensioning forces along the length of the tendons. Iterative solutions are often used to have the assumed and final time-depended deformations converge over each time step.
General purpose software analysis packages without the capability to define post-tensioning tendons geometrically, or to track and update forces with time, can be used for the final design of cast-in-place bridge superstructures. An equivalent force approach could be taken with code predicted changes in the post-tensioning forces in order to capture appropriately long-term behavior. Design Aids 11.1.4 and 11.1.5 in chapter 11 of the Precast/Prestressed Concrete Institute Design Handbook, 7th Edition, are good resources for determining equivalent prestressing loads.