LA ESTRUCTURA OCUPACIONAL

limit states

During the application of a displacement in the top of a precast post-tensioned concrete rocking wall or frame, several main points can be defined to draw a force-displacement diagram. After the initial linear-elastic response, decompression is reached and a gap is opened in the base connection. At this stage there is no significant modification to the stiffness of the structure, what happens when the neutral axis is much deeper into the section or when the concrete has a nonlinear response under compression. After the change to the stiffness there are three options: the yielding of the energy dissipater (if present); reaching the limit of proportionality of the post-tensioning steel and excess of compression on the concrete. This last stage corresponds to the ultimate limit state and the three options can be achieved sequentially.

Figure 2.33: Fundamentals of Direct Displacement-Based Design (extracted from Priestley et al. (2007)).

Priestley and Tao (1993), developed a bilinear idealisation, considering the limit of proportionality of the prestress as the ultimate state. In this study, a precast prestressed frame with partially debonded tendons without dissipaters was considered, as can be seen in Figure 2.34a. To define the bilinear idealisation (Fig. 2.34b), three points are needed, the first is the origin, point one, then the end of the initial stiffness, point two and the limit of proportionality of the prestress, point three. Point two is obtained considering that the stiffness changes when the gap opening has propagated to the centroidal axis, unless the average initial stress in the concrete due to prestress is higher than 0.25fcm.

This point corresponds to a moment twice the one related to the decompression of the cross section. The author considers that the prestress should remain elastic, thus point three is when the limit of proportionality of the prestress is reached. In this example no dissipation device is considered, what means that only an excess of compression stress in the concrete can change this bilinear idealisation.

unbonded length prestress tendons hb hc lb lc

(a) Beam-column connection with partially debonded tendons F ∆ ∆y ∆u Fu Fy bilinear actual 1 2 3 (b) Bilinear force-displacement

Figure 2.34: Beam-column connection with partially debonded tendons (adapted from Priestley and Tao (1993)).

Perez (2004) made an approach similar to Priestley and Tao (1993), using some of their conclusions and added the possibility of the prestress steel to reach stresses beyond the limit of proportionality. Based on Kurama (1997), Perez (2004) presented seven wall stages for the behaviour of an unbonded post-tensioned precast prestressed wall:

1. decompression at the base of the wall (DEC);

2. elastic limit of the linear-elastic response of the wall (ELL); 3. limit of proportionality of the prestress steel (LLP);

4. base shear capacity;

5. loss of prestress under cyclic lateral load; 6. crushing of confined concrete (CCC); 7. rupture of prestressing steel.

1. DEC is when the wall starts to lift on one side and opens a gap in the connection. As already mentioned, at this stage there is no significant change to the wall stiffness. The initial compressive force is due to the prestressing force and the other vertical loads.

2. ELL is when the wall changes stiffness, this occurs by the growing of the opening of the gap or by a nonlinear behaviour of the concrete in compression. According to El- Sheikh et al. (2000) the value of the bending moment that corresponds to this stage should be calculated as the minimum value considering two possible situations: the concrete nonlinear response in compression, assuming an extreme fiber (including the concrete cover) strain of the concrete equal to 0.003 and a prestressing force equal to the initial one; the geometric softening, that is considered to be reached when the gap opening length propagates beyond 75% of the cross section depth, corresponding to 2.5 times the

decompression bending moment. This last situation, that corresponds to the geometric softening, is different from the one previously suggested by (Priestley and Tao, 1993), where the effect of the gap opening on the stiffness is considered small until the neutral axis reaches the section centroid.

3. LLP is when the limit of proportionality of the prestress steel is reached. Beyond this limit the prestress force begins to have losses and the structural initial elastic stiffness is reduced. The prestress can be applied with several bars or tendons, which means that the first bar or tendon to reach the limit of proportionality may not impose a significant loss of stiffness due to prestress. Either way, LLP is considered to be reached when the first bar or tendon reaches the limit of proportionality.

4. The base shear capacity is the maximum base shear that can be reached. Consid- ering that there is no sliding, the base shear is controlled by the overturning capacity of the wall and is achieved when the prestress reaches the limit of proportionality (LLP), neglecting the strain-hardening effects in the steel.

5. As mentioned above, the loss of prestress occurs when the wall is unloaded from a drift after the prestress had went beyond the limit of proportionality. Prestress losses can also be associated with local damage or system adjustments during loading, especially in short length bars or tendons.

6/7. CCC is a limit state that should be avoided. To delay this limit state beyond the design drift ratio, the concrete should be conveniently confined. Priestley and Tao (1993) recommend interlocking spirals as confinement for rectangular beams (Figure 2.5), that can also be used in the walls. As well as CCC, rupture of post-tensioning steel should never occur within the design drift ratio.

Using ELL, LLP and CCC limit states, Perez (2004) presented the tri-linear ideal- isation, base shear vs drift ratio, that is shown in Figure 2.35. The difference for the idealisation presented by Priestley and Tao (1993) is essentially by considering the struc- ture beyond LLP.