Summary
This paper proposes s new approach to the design of prestressed continuous beams, one-way slabs, and two-way slabs. The essence of this method is the control of the secondary moments produced by prestress forces and adjusting them to the specific characteristics of the designed prestressed element. The Secondary Moment Control method is described by the following design characteristics:
The benefits of this design method are a more economical prestressed design. The required prestressing forces in the members are reduced, labor required is decreased due to considerably simpler on site cable placement, and fewer “chairs” are needed to support the tendon in the member prior to casting.
The Secondary Moment Control Method
The method is based on particular detailing during the design process to exploit the magnitude of secondary moments produced by the prestressing force and by that, manipulate the total moment distribution produced by the prestress force.
Secondary moments can be viewed as the moments resulting from interior support reactions, caused by forces created at these supports to overcome the deflection Δ induced by the prestress force. That is the deflection that would have been caused by the primary moment if interior supports were absent.
The deflection Δ is the function of the second integral of prestress force multiplied by the eccentricity as a function of distance along the beam ∫∫F×e_{x}dx. In general, as more as the tendon is below the center of gravity of the section (e_{x} is positive) Δ increases (Δ is positive when the element deflects upward) and correspondingly, the positive secondary moment at the support is increased. The reverse, improved secondary negative moment at midspan, would result from tendon profile with tendons primarily above the c.g. line.
The total prestress moment difference between the average moment at an adjacent support and the moment at middle are constant and equal to the prestress force multiplied by the sag.
Figure (1) depicts tendon profiles for four tendon layouts. The first (figure 1b) is a profile of a tendon designed using the common load balancing method. This method utilizes a tendon profile that creates a transverse load equal and opposite to the external loads. Figures (1c), (1d), and (1e) are typical tendons designed by the secondary moment control method. The profile of all of these tendons is made from straight line sections and from second order double curvature lines to shift the tendons from the top to the bottom of the beam. By varying the length of the straight line between the curvatures we change the ratio of the areas between the tendon and the center of gravity of the section above and below the c.g. line, -F×e_{x} and +F×e_{x}, and thus the magnitude of the secondary moment. Profiles no. 2 & 3 in figure (1a) show the different moment distribution lines produced by such tendons. Moving the location of the double curvatures change the location of the moment sign change as is shown by profile no 4.
Advantages of the secondary moments control method
Unlike the load balancing method that accounts only for the shape of the external load, using the secondary moment control method the designer can account for additional parameters such as the relation between LL and DL, differences between the allowable stresses at different locations on the element, different sections of beams (for instance T sections that usually benefit from increased positive moment produced by prestress forces at the support), and construction sequences.
Benefits of the method
Design for Service Conditions
The better control of the prestress force moment with the proposed method often results with need of less prestress force to keep the stresses in the structure within the allowable range.
Design for Ultimate Condition
The secondary moment with a safety factor 1.0 can be added to the factored external load moment at ultimate condition. By controlling the secondary moment magnitude the designer can reduce the required ultimate strength at the support or at midspan.
During Construction:
Less “chairs”: | Using the proposed method the need for “chairs” to support the tendons prior to casting is significantly reduced. At the straight line sections the tendon can be tied to the mild reinforcement. |
Less “weaving”: | In two-way construction the placement of the tendons is much simpler and less or no weaving is required, while keeping good distribution of the tendons between column and middle strips. See figure (2). |
Figure 1a: Moment distribution lines produced by the different tendon profiles. | |
Figure 1b: 750 kN tendon load balancing method layout and equivalent load. | |
Figure 1c: 750 kN tendon secondary moment control method layout and equivalent load for tendon with minimized negative F×e_{x} zone. | |
Figure 1d: 750 kN tendon secondary moment control method layout and equivalent load for tendon with increased negative F×e_{x} zone. | |
Figure 1e: 750 kN tendon secondary moment control method layout and equivalent load for tendon with shifted double curvature lines. | |
Section along middle strip | Section along column strip |
Figure 2: Tendon placement design by secondary moment control in two-way construction. |
Example 1:
Bridge Composed of Simple Span Precast Prestressed Girders Made Continuous with Cast in Place Concrete Over Supports and Continuous Tendons
Figure (3a) depicts a bridge of three equal spans made of AASHTO VI precast girders.
Figure (3b) shows the typical moment envelope due to gravity loads on the bridge. The positive moment due to the gravity loads is resisted by low-cost pretension strands in the precast beams. The continuous post tensioned tendon is utilized mainly to resist the negative moments over the supports. Using the proposed method increases the effectiveness of the continuous tendon.
Figure (3c) and figure (3d) show the moment produced by the load balancing method, and by tendon designed by the Secondary Moment Control method. The second tendon produces positive moment at the support 27% larger than the first. The effective eccentricity (the moment produced by the prestress force divided by the prestress force) of this cable is 1.4 meter, 3.6 times larger than the geometric eccentricity.
Figure 3a: Bridge Geometry | |
Figure 3b: Gravity load moment envelope | |
Figure 3c: Equivalent load and moment produced by 200kN cable designed by the load balancing method. | |
Figure 3d: Equivalent load and moment produced by 200kN cable designed by thesecondary moment control method. |
Example 2:
Bridge Composed of Cantilever Precast Girder made Continuous with Cast in Place Joint in Low Moment Zone
The bridge and its construction sequence are described in figure (4a).
The prestressing of the bridge girders is made of pretension strands, “pre-post” tendon at the cantilevered girders (no 1), and two continuous tendons through the entire length of the continuous girders (no 2 & 3).
Tendon no 3 was required to produce maximum positive moment at the support. However, geometrically the tendon had to be kept within the precast section underneath tendons no 1 and no 2, thus dictating a geometric eccentricity of only 0.15m. Mobilizing favorable positive secondary moments the 2000 kN tendon provided a 2500 kN×m positive moment, equivalent to 1.25m effective eccentricity, 8.3 times the geometric one. Figure (4b) shows the bending moment diagram resulting from the tendon induced moments.
The maximum ultimate strength of the composite AASHTO VI girder for negative moment are 12500 kN×m, factored gravity load moment at the support is 14100 kN×m. The secondary moment produced by tendon no 3, 2200 kN×m (2500-300) reduced the external ultimate moment to be within the moment capacity of the beam, otherwise modification of the concrete section would have been needed and a non standard girder would have been required.
Figure 4a: Bridge Geometry |
Final Construction and Stressing Sequence |
Figure 4b: Tendon no 3 Induced Bending Moment |
Example 3:
One-Way or Two-Way Solid Slab
Figure (5a) describes a one-way 30cm thick solid slab or a strip of a two-way plate. The moment envelope per meter width, due to self weight, additional S.D.L. of 2.5 kN/m2 and LL of 5 kN/m2 is shown in figure (5b). The maximum allowable tensile stress at service condition is 3 MPa.
Using the Secondary Moment Control method, it was found that with 600kN tendon force and 0.5m length of straight line above supports, both the maximum concrete tensile stress at the support and at midspan reach the maximum allowable stress. At these conditions the secondary moments are ultimately shifted such that required prestress force is optimized.
Superimposing the top and bottom fiber stress envelope due to service load together with the top and bottom fiber stresses due to prestress forces, figures (5c) and (5d), verifies that the tensile stresses do not exceeds 3 MPa at any section along the slab.
The utilization of the Secondary Moment Control method results in tendon material savings, however, the benefits of the tendon arrangement using this method are important as well. The labor costs are reduced as bi-directional tendon weaving often occurring in two-way slabs is minimized. In addition the multilevel chairs required for tendon placement are reduced in both numbers and variety as the tendon in its straight level sections can be tied to the mild reinforcements.
Figure 5a: Slab Geometry |
Figure 5b: Service Load Moment Envelope |
Figure 5c: Superimposed top fiber stresses envelope due to service load with top fiber stresses due to prestress forces (stresses above line are tensile). |
Figure 5d: Superimposed bottom fiber stresses envelope due to service load with bottom fiber stresses due to prestress forces (stresses above line are tensile). |
Conclusion
The proposed secondary moment control is a method to optimize the design of prestressed continuous elements. The method provides a system to look for the tendon layout that is the optimum for the specific characteristics and restrictions of the element. It verifies the real minimum prestress force required while it facilitates construction. In total the results are savings in material and labor.