8.2.1 Cross section model used to consider post-tensioning
The model of the cross section used to consider the effects of the post-tensioning is very similar to that used to model a normally reinforced concrete circular hollow section. The normally reinforced cross section model is modified so that the reduction of concrete, due to the inclusion of the post- tension ducts, is taken into account. The modified cross section is then analysed as discussed in Section 4.3.
After the completion of the cross section analysis, the normal forces required to ensure that the cross section remains entirely in compression are calculated. The process used to obtain the post-tension force is iterative, due to the nonlinear material properties present in the cross section. This force is found by performing multiple cross section analyses and incrementing the post-tension force after each analysis. This is done until the post-tension force is large enough to ensure that the entire cross section is in compression.
The post tension force is then applied to the cross section as a concentric normal force. In the deformed state, the restoring moment from the tendons is calculated and subtracted from the moments caused by lateral forces and P-delta effects. The process in finding the prestressing restoring moment is also iterative, due to the restoring moment being dependent on the existing moment acting on the cross section. The effect of the post-tensioning is thus modelled as an externally applied normal force and restoring moment, which act together with the original axial force and moment.
Due to the accurate results obtained from the cross section analysis performed in Section 4.6, it is deemed sensible to apply the “smeared area”-concept that is used to model the reinforcing steel, to the prestressing steel as well. This is also deemed sensible because the prestressing steel model is treated independently from the cross section model. Unlike the reinforcing, the prestressing steel is assumed not to contribute directly to the composite cross section properties.
8.2.2 Algorithm developed to consider post-tensioning
The overall structural analysis process developed to consider the effects of post-tensioning is summarised below.
Step 1: Perform a structural analysis as discussed in Chapter 7, but with the modified cross section discussed in Section 8.2.1.
Step 2: Find the minimum post-tension force required to ensure that the tower segments remain entirely in compression, eliminating all cracks.
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Step 3: Apply the post-tension force as an external axial force, along with the original internal forces (axial force and moment).
Step 4: Perform another structural analysis to calculate the post-tension restoring moments and subtract it from the existing internal moments. Perform Step 4 until the restoring moment converges.
Step 4 above is repeated until the restoring moment converges because the restoring moment is dependent on the curvature of the cross section. The curvature, however, is dependent on the total applied moment on the cross section. By subtracting the restoring moment from the existing internal moment in Step 4, the curvature is reduced. If Step 4 is repeated, the calculation of the restoring moment will yield a lower value to what is obtained during the first iteration. The third iteration would yield a slightly bigger restoring moment to what is obtained in the second iteration. By repeating this process, the true restoring moment is obtained.
It should be noted that the calculation of losses associated with the post-tensioning is ignored for this study. The minimum post-tension force required to eliminate cracking is calculated with the assumption that friction losses are taken into account. It is thus left to the engineer to design the post-tensioning so that the final post-tension force (including losses) equals the post-tension force obtained from the developed software.
8.2.3 Implemented material model for prestressing steel
In order to calculate the post-tension restoring moment, a material model is implemented, similar to the other materials present in the cross section. The force present in each tendon is dependent on its strain state. Therefore, tendons located further away from the bending axis have a bigger contribution to the restoring moment compared to tendons located near or on the bending axis.
Figure 8.2: Idealized and design stress-strain diagrams for prestressing steel. Obtained from Eurocode 2 (EN 1992-1- 1:2004, 2004)
The prestressing steel material model is obtained from Eurocode 2: Design of concrete structures. This is illustrated in Figure 8.2, where:
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𝑓𝑝𝑘 is the characteristic tensile strength of prestressing steel;
𝑓𝑝0.1𝑘 is the characteristic 0.1% proof-stress of prestressing steel;
𝛾𝑠 is the partial factor for reinforcing and prestressing steel, which can be taken as 1.15 (EN
1992-1-1:2004, 2004);
𝜀𝑢𝑑 and 𝜀𝑢𝑘 are the design and characteristic strains of reinforcing or prestressing steel at
maximum load, respectively; and
𝐸𝑝 is the design value of modulus of elasticity of prestressing steel, which can be taken as 205 GPa (EN 1992-1-1:2004, 2004).
For the prestressing tendons to achieve the required tension force, the tendons are tensioned to a specific strain. This strain can be interpreted as the initial strain of the tendons, before the tower progresses into the deformed state. The actual strain in each tendon in the deformed state is calculated as the initial strain plus the strain resulting from the curvature at the height where the cross section is analysed. The total strain within a tendon is thus composed of an initial normal strain and the strain resulting from bending.