III. Itinerario político
III.9. Gobierno de la mayoría y discurso de la incapacidad
Classical impedance function solutions, such as those presented in Table 2-2 and Table 2-3, strictly apply for rigid foundations. As illustrated in Figure 2-1, soil-foundation interaction for rigid soil-foundations can be represented by individual springs for each foundation degree of freedom. Actual foundation slabs and basement walls, however, are non-rigid structural elements. The few theoretical solutions that exist apply to circular foundations supporting a rigid core (Iguchi and Luco, 1982),
flexible perimeter walls (Liou and Huang, 1994), or rigid concentric walls (Riggs and Waas, 1985). Figure 2-7 shows the effect of flexible foundation elements on
rotational stiffness, krr, and rotational radiation damping ratio, rr, for the cases of a circular foundation supporting a rigid core or flexible perimeter wall.
Figure 2-7 Effect of flexible foundation elements on rotational stiffness (krr) and rotational radiation damping ratio (rr) for circular foundations supporting a rigid core (Iguchi and Luco, 1982) and flexible perimeter walls (Liou and Huang, 1994).
The flexibility of the foundation is represented by a relative soil-to-foundation stiffness ratio, , taken from plate theory as:
0 f / s
a r V a0
Young’s modulus and Poisson’s ratio of the foundation concrete. The case of = 0 corresponds to a rigid foundation slab.Liou and Huang (1994) showed that foundation flexibility does not significantly affect translational stiffness and damping terms for the case of flexible perimeter walls. For rotational stiffness and radiation damping, Figure 2-7 shows that foundation flexibility effects are relatively modest for the case of flexible perimeter walls, and most significant for the case of a rigid core.
Typical practice does not adjust the impedance function for non-rigid foundations as shown in Figure 2-7. Instead, foundations springs are distributed across the extent of the foundation, as illustrated in Figure 1-2c. Distributed springs allow the foundation to deform in a natural manner given the loads imposed by the superstructure and the spring reactions. For vertical springs, this can be accomplished by calculating the vertical translational impedance, as described above, and normalizing it by the foundation area to compute stiffness intensity, k (also known as coefficient of zi subgrade reaction), with dimensions of force per cubic length:
4
i z
z
k k
BL (2-20a)
A dashpot intensity can be similarly calculated as:
4
i z
z
c c
BL (2-20b)
As illustrated in Figure 2-8, the stiffness of an individual vertical spring in the interior portion of the foundation can be taken as the product of k and the spring’s zi tributary area dA. If this approach were used across the entire length, the vertical stiffness of the foundation would be reproduced, but the rotational stiffness would generally be underestimated. This occurs because the vertical soil reaction is not uniform, and tends to increase near the edges of the foundation. Using a similar process with c would overestimate radiation damping from rocking. This occurs iz because translational vibration modes (including vertical translation) are much more effective radiation damping sources than rocking modes.
To correct for underestimation of rotational stiffness, strips along the foundation edge (of length ReL) are assigned stiffer springs. When combined with springs in the interior, the total rotational stiffness of the foundation is reproduced. Harden and Hutchinson (2009) present expressions for end length ratios and spring stiffness increases as a function of L/B using static stiffnesses from Gazetas (1991).
Figure 2-8 Vertical spring distribution used to reproduce total rotational stiffness kyy. A comparable geometry can be shown in the y-z plane (using foundation dimension 2B) to reproduce kxx.
More generally, the increase in spring stiffness, Rk, can be calculated as a function of foundation end length ratio, Re, as:
Equations 2-21 were derived by matching the moment produced by the springs for a unit foundation rotation to the rotational stiffness kyy or kxx. In these equations, a value of Re can be selected (typically in the range of 0.3 to 0.5), which then provides a unique Rk. This correction for rotational stiffness, however, does not preserve the original vertical stiffness kz. This is considered an acceptable approximation, in general, because rocking is the more critical foundation vibration mode in most structures.
To correct for overestimation of rotational damping, the relative stiffness intensities and distribution are used (based on the stiffness factor Rk and end length ratio Re), but dashpot intensities over the full length and width of the foundation are scaled down by a factor, Rc, computed as:
Use of the above procedures for modifying vertical spring impedances will reproduce the theoretical rotational stiffness and damping through distributed vertical springs and dashpots. While this allows foundation flexibility to be accounted for, in the sense that foundation structural elements connected to springs and dashpots are non-rigid, a question that remains is whether or not the rotational impedance computed using a rigid foundation impedance function is an appropriate target for calibration.
For the case of a rigid core illustrated in Figure 2-7 it is not, but solutions for more practical situations are not available.
In the horizontal direction, the use of a vertical distribution of horizontal springs depends largely on whether the analysis is two-dimensional or three-dimensional, and whether or not the foundation is embedded. Current recommendations are as
follows:
For two-dimensional analysis of a foundation on the ground surface, the horizontal spring from the impedance function is directly applied to the foundation, as shown in Figure 2-8 (i.e., no distributed springs).
For two-dimensional analysis of an embedded foundation, the component of the embedded stiffness attributable to the base slab (i.e., the stiffness without the embedment modifier, kx/x) can be applied to the spring at the base slab level. Distributed springs are then positioned along the height of the basement walls with a cumulative stiffness equal to kx (1–1/x).
For three-dimensional analysis, springs are distributed in both horizontal directions uniformly around the perimeter of the foundation. The sum of the spring stiffnesses in a given direction should match the total stiffness from the impedance function.