CAPÍTULO 1: MARCO TEÓRICO REFERENCIAL DE LA INVESTIGACIÓN
1.5. Procedimiento diagnóstico para la comunicación institucional
The yield (in some references also called critical) horizontal acceleration, g K. y, is a
key parameter informing practitioners of the level of seismic acceleration for which a given slope, stable under static conditions, becomes unstable. Also, it is needed to calculate earthquake induced permanent displacements via the Newmark’s approach (Newmark, 1965).
The global minimum of fy
, , , , , /c H,
over the three geometrical variables , , (see Eq.(3.5)) Provides the least upper bound on the yield seismic coefficient,Ky assuming that the most unfavourable crack for the slope is present. InFigure 3.8, the Ky values obtained are plotted for slopes of various characteristics , ,
and c/H together with the values of Kyobtained for intact slopes. In
Figure 3.9 the difference in percent between the obtained yield seismic coefficients and the corresponding coefficients for a slope of the same characteristics but intact is plotted. It can be seen that the presence of cracks causes substantial reduction of the yield seismic coefficient, especially for steep slopes of low . This result is in agreement with the trend observed in Figure 3.4 for the reduction of the stability factor under a prescribed .g Kh. Figure 3.9 is useful to investigate the relative influence between the two strength parameters (c and ) on the yield seismic coefficient. Looking at the charts for =60° and =75° (see Figure 3.9b and c respectively) it can be noticed that the reduction of Kydue to the presence of cracks becomes less significant for c increasing. However, in case of gentle slopes (see Figure 3.9a), there
Chapter 3: On the Stability of Fissured Slopes Subject to Seismic Action (3)
38
is an inversion of the trend at =30°: for slopes with >30° the reduction in Kydue to the presence of cracks becomes more significant for c increasing.
Figure 3.8 Coefficient of yield acceleration versus slope inclination for intact slopes (solid lines) and for fissured slopes for the most unfavourable crack scenario (dotted lines). Vertical acceleration is absent (=0): a)) =20º; b)) =30º; c)) =40º . Grey lines indicate a translational failure mechanism. Dashed and dashed-dotted lines indicate a below the slope toe mechanism occurring for intact and fissured slopes respectively.
Chapter 3: On the Stability of Fissured Slopes Subject to Seismic Action (3)
39
Figure 3.9 Percentage of reduction in the yield acceleration due to the presence of the most unfavourable crack for the stability of the slope with =0. a) =45º , b) =60º , and c) =75º. As noted in the investigation of the stability factor under prescribed seismic excitation, assuming the presence of the most unfavourable crack can be overly conservative. When the maximum depth of cracks in a slope can be inferred by either a stress analysis or in-situ measurements, this information can be included in the
(a) 𝜷=45˚ c: increasing c: increasing (b) 𝜷=60˚ c: increasing (c) 𝜷=75˚ c: increasing
Chapter 3: On the Stability of Fissured Slopes Subject to Seismic Action (3)
40
search for the least upper bound on Ky (problem ii) listed in the Introduction). Mathematically, this is done by imposing the following constraint (Utili, 2013):
exp tan . sin exp tan . sin 1 h h exp tan . sin
H H
(3.6)
into the maximisation of fy
, , , , , /c H,
in Eq.(3.5). In Figure 3.10, the function Khy
h obtained from the maximisation of fy
, , , , , /c H,
constrained by Eq. (3.6), is plotted against the prescribed values for =0, =0.5 and =1. Khy
h gradually decreases for h increasing until a minimum at hhmin is reached and then increases for h increasing (see the grey curves in Figure 3.10). Note that the results represented by the grey curves are obtained assuming the log-spiral failure surface C-D constrained to depart from the crack bottom end (see Eq. (3.6)). When h H/ 1, the function Khy
h tends to infinity because the volume of thewedge E-B-C-D sliding away becomes infinitesimal. However, physics dictates that the failure mechanism taking place may involve only one part of the total crack depth,
i.e. the log-spiral C-D may depart from the crack above its bottom end. This possibility is not reflected by the mathematical function Khy
h since Eq. (3.6)constrains the failure log-spiral C-D to depart from the crack bottom end. Forhhmin, the least upper bound on the yield acceleration coefficient is provided byKhy
hhmin
which is represented by black horizontal lines in Figure 3.10.Finally from Figure 3.10 emerges that for steep slopes (Figure 3.10d and e), the presence of a vertical downward acceleration reduces the yield seismic coefficient (hence it is detrimental to slope stability) whereas for gentle slopes with high
Chapter 3: On the Stability of Fissured Slopes Subject to Seismic Action (3)
41
(Figure 3.10c) the opposite is true. This trend is in agreement with the results of the investigation, carried out in the previous section, on the influence of Kv on the stability factor for prescribed values of Kh.
Figure 3.10 a) Visualisation of a slope subject to cracks of known depth but unspecified location. In figures b), c), d) and e) K is plotted against the prescribed crack depth for slopes of variousy , and values with c/H =0.15: b) =20º, =45º; c) =40º, =45º; d) =20º, =70º; e) =40º, =70º. The grey lines represent the mathematical functionKhy h , whilst the black lines represent the yield seismic coefficient of the slope.
(a) H (a) h 𝝓=20˚, 𝛽=45˚ (b) (c) 𝝓=40˚, 𝛽=45˚ (d) 𝝓=20˚, 𝛽=70˚ (e) 𝝓=40˚, 𝛽=70˚ h/H h/H h/H h/H
Chapter 3: On the Stability of Fissured Slopes Subject to Seismic Action (3)
42