Ejemplo: IP sobre SRT con GRE

A more rigorous form of slip circle analysis is given by Morgenstern and Price (1965). This is not a `hand calculation' method but is carried out by numerical means in a computer program. The method involves the solution of a pair of simultaneous partial differential equations ± one for force equilibrium and one for moment equilibrium ± using a two-variable Newton approximation method. It is widely regarded as the `benchmark' against which other methods are measured, particularly calculations `by hand'. This is because their method satis®es both force and moment equilibrium at the same time whereas other methods do not. Whitman and Bailey (1967) used the Morgenstern and Price method as a datum of measurement in their review and comparison of various computer methods.

The key elements of the method are: . failure surface of general shape . two-dimensionality

. force and moment equilibrium . computer analysis

. based on assumptions interrelating the slice side forces X and E. It can be seen that large errors can arise with the USBR method and we recommend therefore that use of the USBR method should be abandoned. A dif®culty with the Bishop-simpli®ed method can arise if



1‡ tantan 0

F 

Table 4.3 Factors of safety found by the USBR, simpli®ed and rigorous methods for two cases, after Bishop (1955)

Example USBR Bishop-simpli®ed Bishop-rigorous

1 1.38 1.53 1.60±1.61

2 1.5 1.84 1.92

Table 4.4 Factors of safety found by the USBR, simpli®ed and rigorous methods for four cases, after Whitman and Bailey (1967)

Example USBR Bishop-simpli®ed Bishop-rigorous

1 1.49 1.61 1.58±1.62

2 1.10 1.33 1.24±1.26

3 0.66 0.70±0.82 0.73±0.78

becomes negative or 0. This can occur if is negative and tan 0=F is large. If  1‡ tantan0 F 

is less than 0.2 the use of other methods is recommended. Stability coecients for earth slopes

Bishop and Morgenstern (1960), using the Bishop-simpli®ed method, have produced charts of stability coef®cients from which a factor of safety for an effective stress analysis can be rapidly obtained. These charts are given in Appendix 1. The general solutions are based on the assumption that the pore pressure u at any point is a simple proportion ru of the overburden pressure h. This proportion is the pore pressure ratio ruˆ u= h and is regarded as being constant throughout the cross-section. This is called a homogeneous pore pressure distribution.

For a simple soil pro®le and speci®ed shear strength parameters, the factor safety, F, varies linearly with the magnitude of the pore pressure expressed by the ratio rusuch that

Fˆ m ÿ nru

(see Fig. 4.15) where m and n are termed `stability coef®cients' for the particular slope and soil properties. The use of pore pressure ratio, ru,

3·00 2·00 1·00 0·0 0·0 0·2 0·4 0·6 0·8 1·0 ru F φ′ = 30˚ c′ γIH cotβ = 3 D = 1·0 = 0·05

Fig. 4.15 Linear relationship between factor of safety F and pore pressure ratio ru, after Bishop and Morgenstern (1960)

permits the results of stability analysis to be presented in dimensionless form.

There are seven variables (see Fig. 4.16): . the height of the slope H

. the depth to a hard stratum DH . the pore pressure ratio ruˆ u= h . the slope angle

. the cohesion intercept with respect to effective stress c0

. the angle of shearing resistance with respect to effective stress0

. the bulk unit weight .

. for a given value of c0_{= H, F depends on cot , D, r}

u and 0. Three values of c0= H have been used: 0, 0.025, 0.05

. slope inclination varies from 2:1 to 5:1 . 0_{ranges from 108 to 408}

. D ˆ 1:00, 1.25, 1.50

. If ru> rue, where rueis indicated by the dotted line on the charts, the critical circle is at greater depth

. use linear extrapolation in range c0_{= H 0.05 to 0.10 (Fig. 4.17).}

The values of the stability coef®cients have been plotted against the cotangent of the slope angle (see Appendix 1, Figs. A1.1 to A1.8) for0 varying between 108 and 408 with values of the dimensionless parameters c0= H and D speci®ed for each ®gure. The bold lines show values of m and n at intervals of 2.58. The broken lines are those of equal ru(denoted by rue). To calculate the factor of safety of a section whose c0= H lies within the range covered by these ®gures, it is necessary only to apply the equa- tion Fˆ m ÿ nru, to determine the factor of safety of the two nearest values of c0= H and then perform a linear interpolation between these values, for the speci®ed value of c0= H.

For a given set of parameters (, 0, c0= H), there is a value of the pore pressure ratio for which the factor of safety, when D equals 1.00, is the

DH h H β γw u

same as the factor of safety when D equals 1.25. If the design value of the pore pressure ratio is higher than rue for the given section and strength parameters, then the factor of safety with a depth factor Dˆ 1:25 has a lower value than with D equal to 1.00. This argument can be extended to discern whether the factor of safety with D equal to 1.50 is more critical than with D equal to 1.25.

WORKED EXAMPLES

(1) Use stability coef®cients to investigate the slope shown in Fig. 4.14. The data are cot ˆ 2:5:1, c0ˆ 18 kPa, 0_{ˆ 238, H ˆ 12 m, ˆ 19:5±}

20.0 kN/m3, say 19.8 kN/m3. Hence, referring to Fig. 4.14: ruˆ 0; 37 127; 62 138; 66 128; 49 96; 24 47 ˆ 0; 0:29; 0:45; 0:52; 0:51; 0:51

for each slice, respectively, whence the average ru (linear inter- polation)ˆ 0.38, or alternatively, the average ru(using areas)ˆ 0.39.

c0

Hˆ

1:8

19:8  12ˆ 0:076

Therefore do calculations for c0= H ˆ 0:025 and 0.050 and use linear extrapolation (see Fig. 4.17).

4·0 3·0 2·0 1·0 0·0 0·0 0·02 0·04 0·06 0·08 0·10 F c′ / γH Actual Linear extrapolation

Fig. 4.17 Factor of safety ± stability number relationship showing extrapolation in the range of c0= H 0.05 to 0.10

For dˆ 1:00, c0= H ˆ 0:05 mˆ 1:79; n ˆ 1:42; rueˆ 0:45

Therefore deeper circles are not critical since ruˆ 0:39 < 0:45 Fˆ 1:79 ÿ 0:39  1:42 ˆ 1:24

For Dˆ 1:00, c0= H ˆ 0:025 mˆ 1:52; n ˆ 1:35; rueˆ 0:8 Therefore deeper circles are not critical

Fˆ 1:52 ÿ 0:39  1:35 ˆ 0:99 for c0= H ˆ 0:076 Fˆ 1:24 ‡0:076 ÿ 0:050 0:050 ÿ 0:025 1… :24 ÿ 0:99† ˆ 1:24 ‡ 0:26 ˆ 1:50

This can be compared with Fˆ 1:43 by slip circle analysis (see Fig. 4.14).

(2) Consider the stability of an earth dam where ˆ 20 kN/m3, c0ˆ 23 kPa, 0_{ˆ 258, H ˆ 35 m, r}

uˆ 0:5, side slopes 4 horizontal to 1 vertical on a rock foundation.

Now c0= H ˆ 23=20  35 ˆ 0:033 c0= H m n 0.025 2.40 2.15 0.050 2.75 2.20 For0ˆ 258, slope 4:1 for c0= H ˆ 0:025, F ˆ 2:40 ÿ 0:5  2:15 ˆ 1:325 for c0= H ˆ 0:050, F ˆ 2:75 ÿ 0:5  2:20 ˆ 1:65

for c0= H ˆ 0:033, F ˆ 1:325 ‡ …1:65 ÿ 1:325†  8=25 ˆ 1:43 using linear interpolation.

(3) Check Lodalen (see Fig. 5.10) using Bishop and Morgenstern's stability coef®cients.

Non-homogeneous pore pressure ratio distribution

If the ruvalue varies from point to point in a slope, it is necessary to use some average value of ru when using stability coef®cients. One way of doing this is illustrated in Fig. 4.18.

For section area 1: average: ruˆ h1ru1‡ h2ru2‡ h2ru2 h1‡ h2‡ h3 overall average: ruˆ Pn 1Anrun Pn 1An where Anis the area of any slice.

Hˆ 18 m, say. c0= H ˆ 9:8=18:7  18 ˆ 0:029 ruˆ 0:4 for c0= H ˆ 0:025; D ˆ 1:0 mˆ 1:45; n ˆ 1:40 Fˆ 1:45 ÿ 0:4  1:40 ˆ 0:89 for c0= H ˆ 0:05; D ˆ 1:0 mˆ 1:75; n ˆ 1:50 Fˆ 1:75 ÿ 0:4  1:50 ˆ 1:15 for c0= H ˆ 0:029 Fˆ 1:89 ‡ 4=25…1:15 ÿ 0:89† ˆ 0:93 Sevaldson (1956) found Fˆ 1:05. ru1 h1 h2 h3 _A 2 A3 A4 ru2 ru3

Fig. 4.18 De®nition sketch for averaging non-homogeneous pore pressure ratio distribution, after Bishop and Morgenstern (1960)