ASSESSMENT OF LIQUEFACTION POTENTIAL

It is extremely important to know if a seismic site is susceptible to liquefaction and several methods have been developed to evaluate the “liquefaction potential” in free field condition. It can be easily evaluated from the factor of safety (FS) that is defined as the ratio between the capacity of the soil and the demand of the earthquake. Demand is strongly linked to the earthquake, in other words it depends on its amplitude and duration, while Capacity is the demand which generates liquefaction. If FS is higher than 1, liquefaction will not occur, conversely if FS ≤1 it does not mean that liquefaction will occur certainly, but only that the site is potentially susceptible to liquefaction.

Liquefaction potential of a soil deposit may be evaluated by using several procedures, which can be summarized in three categories: stress-based, strain-based and energy-based procedures.

43 2.3.1 STRESS-BASED METHODS

The factor of safety (FS) in stress-based methods can be defined as:

𝐹𝑆 =𝐶𝑅𝑅

𝐶𝑆𝑅 (2.20) where CRR is the capacity and CSR is the demand.

The demand CSR can be evaluated through a simple formulation proposed by Seed and Idriss (1971). During the earthquake, shear waves propagate upwards and a soil column to a depth of z is assumed to move horizontally. Defining amax as the peak horizontal acceleration on the ground surface, the maximum cyclic shear stress induced by the earthquake, τmax, acting at the bottom of soil column, is given by:

𝜏_𝑚𝑎𝑥 = 𝑟_𝑑 ∙𝑎_𝑚𝑎𝑥

𝑔 ∙ 𝛾𝑧 (2.21)

Where rd is a stress reduction coefficient to take into account the deformability of the soil column (rd<1), g is the gravitational acceleration and γ is the unit weight of the soil.

The parameter rd is a function of depth and it can be found graphically or by using empirical forms, such as (Idriss and Boulanger, 2004):

𝑟_𝑑= 𝑒𝑥𝑝 ⌊(−1.012 − 1.126 ∙ 𝑠𝑖𝑛 ( 𝑧

11.73+ 5.133)) + (0.106 + 0.118 ∙ 𝑠𝑖𝑛 ( 𝑧

11.28+ 5.142)) 𝑀⌋ (2.22)

where M is the magnitude of the earthquake.

Known that γz is equal to σv and dividing both sides of the eq. (2.21) by the effective vertical stress (σ’v), it becomes:

𝜏_𝑚𝑎𝑥

𝜎′_𝑣 = 𝑟_𝑑 ∙𝑎_𝑚𝑎𝑥 𝑔 ∙ 𝜎_𝑣

𝜎′_𝑣 (2.23)

As a matter of the fact that in situ the seismic solicitation is made of irregular cyclic loading and that the τmax is reached only in few moments of the earthquake, CSR can be conventionally assumed as the 65% of τmax/ σ’v:

𝐶𝑆𝑅 =𝜏_𝑎𝑣𝑒

𝜎′_𝑣 = 0.65 · 𝑟_𝑑∙𝑎_𝑚𝑎𝑥 𝑔 ∙ 𝜎_𝑣

𝜎′_𝑣 (2.24) The capacity CRR may be evaluated from laboratory or in situ tests.

Dealing with laboratory, as already mentioned, cyclic tests allow to determinate a cyclic resistance curve for a sandy soil in a fixed state condition (Dr and σ’0). CSR should be compared to the value of CRR at a fixed number of cycles at liquefaction (Nliq). The latter

44

is chosen from the relationship (Fig. 2.36; Idriss, 1999) between the magnitude of the expected earthquake in situ and the number of equivalent cycles (Neq).

Figure 2.36. Relationship between the earthquake magnitude (M) with number of equivalent cycles (Neq) (Idriss, 1999).

Once the Magnitude of the expected seismic event is known, Neq can be determined and thus the corresponding CRR from the cyclic resistance curve. Therefore, the factor of safety can be evaluated.

The described procedure is easy to apply, but as mentioned in paragraph 2.2.2.1, undisturbed specimens should be tested in laboratory to evaluate precisely the cyclic behaviour of sandy soils and to estimate as correctly as possible the cyclic resistance to liquefaction. The problem is that not always it is possible to recover undisturbed samples, especially because of high costs of these operations.

To overcome this drawback, the capacity of the deposit can be evaluated from in situ tests, such as SPT or CPT. The factor of safety (FS) can be expressed as:

𝐹𝑆 =𝐶𝑅𝑅

𝐶𝑆𝑅 = (𝐶𝑅𝑅𝑀=7.5,𝜎′𝑣=1

𝐶𝑆𝑅 ) ∙ 𝑀𝑆𝐹 ∙ 𝐾_𝜎 ∙ 𝐾_𝛼 (2.25)

Where the CSR is evaluated as described above (eq. (2.24)), while CRRM=7.5, σ’v=1 is the liquefaction resistance referred to a magnitude M=7.5 and to ’v=103 kPa, MSF is the magnitude scaling factor, introduced to account for the effect of the duration of the seismic event, K and K are correcting factors to account respectively for the effective overburden stress and for an initial static shear stress on the horizontal plane. The expressions of all the factors of equation (2.25) are not reported here for the sake of brevity, and can be easily found in literature (e.g., National Academies, 2016).

The capacity (CRRM=7.5, σ’v=1) of soils can be evaluated by means of in situ tests, such as SPT or CPT. Among other, the following relationships can be used (Idriss and Boulanger, 2004):

𝐶𝑅𝑅𝑀=7.5,𝜎′𝑣=1 = 𝑒𝑥𝑝 (𝑁_1,60𝑐𝑠

14.1 + (𝑁_1,60𝑐𝑠 126 )

2

− (𝑁_1,60𝑐𝑠 23.6 )

3

+ (𝑁_1,60𝑐𝑠 25.4 )

4

− 2.8) 𝑓𝑜𝑟 𝑆𝑃𝑇 (2.26)

45 𝐶𝑅𝑅𝑀=7.5,𝜎′𝑣=1 = 𝑒𝑥𝑝 (𝑞_{𝑐1𝑁𝑐𝑠}

113 + (𝑞_{𝑐1𝑁𝑐𝑠} 1000)

2

− (𝑞_{𝑐1𝑁𝑐𝑠} 140 )

3

+ (𝑞_{𝑐1𝑁𝑐𝑠} 137 )

4

− 2.8) 𝑓𝑜𝑟 𝐶𝑃𝑇 (2.27)

where N1,60cs and qc1Ncs have been introduced to take into account the effect of fines content (FC) and they can be calculated from SPT or CPT results: number of blows (N60) or tip resistance (qc), respectively. Firstly, N60 and qc, have to be normalized as N1,60 and qc1N, by using respectively, the factors CN and Cq, as reported below:

𝑁_1,60 = 𝐶_𝑁∙ 𝑁₆₀ 𝐶_𝑁= ( 𝑝_𝑎 𝜎′_𝑣0)

0.5

𝑓𝑜𝑟 𝑆𝑃𝑇 (2.28)

𝑞_𝑐1𝑁 = 𝐶_𝑞∙ 𝑞_𝑐 𝐶_𝑞= ( 𝑝_𝑎 𝜎′_𝑣0)

𝑛

𝑓𝑜𝑟 𝐶𝑃𝑇 (2.29)

where 0.5 ≤ n ≤1.0.

N1,60 and qc1N can be used for clean sand, but for soils with a fines content (FC), N1,60 and qc1N have to be corrected as N1,60cs and qc1Ncs, according the following equations (Idriss and Boulanger, 2004):

𝑁_1,60𝑐𝑠= 𝑁_1,60+ 𝑒𝑥𝑝 (1.63 +9.7

𝐹𝐶− (15.7 𝐹𝐶 )

2

) 𝑓𝑜𝑟 𝑆𝑃𝑇 (2.30)

𝑞_{𝑐1𝑁𝑐𝑠}= 𝑞_𝑐1𝑁+ (11.9 +𝑞_𝑐1𝑁

14.6) ∙ 𝑒𝑥𝑝 (1.63 − 9.7

𝐹𝐶 + 2− ( 15.7 𝐹𝐶 + 2)

2

) 𝑓𝑜𝑟 𝐶𝑃𝑇 (2.31)

Eqs. (2.26 and 2.27) have been obtained starting from experimental observations carried out in sites where liquefaction occurs and does not, known the results of SPT or CPT. The curves of Figure 2.37 split the plane into two parts, above the curves, liquefaction was recorded, while below no cases of liquefaction were observed. These charts are used to carry out a fast and a simplified liquefaction analysis of the tested site.

(a) (b)

Figure 2.37. Charts to evaluate the liquefaction resistance of sandy soil deposits for SPT (a) and CPT (b).

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

0 10 20 30 40

CRR

N_1,60 SPT

eq. (2.26) LIQ

NO LIQ

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

0 50 100 150 200

CRR

qc1N

CPT

eq. (2.27) LIQ

NO LIQ

46 2.3.2 STRAIN-BASED METHODS

In strain-based methods, earthquake induced loading is expressed in terms of cyclic strains and the factor safety (FS) is defined as:

𝐹𝑆 = 𝛾_𝑡ℎ

𝛾_𝑐𝑦𝑐 (2.32) where γth is the capacity and γcyc is the demand.

The capacity γth is defined as the shear strain amplitude required to cause sliding across grain-to-grain contact surfaces. Dobry et al. (1982), carried out several strain-controlled undrained cyclic tests and they noted that a threshold strain exists. Regardless of the specimen preparation technique, when shear strain is 0.01% excess pore pressure starts to increase. Therefore, this value is assumed as the capacity of the soil in these kinds of models.

The amplitude of the uniform cyclic strain (demand) can be evaluated from the following equation (Dobry et al., 1982):

𝛾_𝑐𝑦𝑐 = 0.65 · 𝑟_𝑑∙𝑎_𝑚𝑎𝑥 𝑔 ∙ 𝜎_𝑣

𝐺(𝛾_𝑐𝑦𝑐) (2.33)

where G(γcyc) is the shear modulus of the soil at γ=γcyc. It means that both sides of eq.

(2.33) depends on γcyc, so G(γcyc) must be obtained iteratively from a measured Gmax

profile and appropriate modulus reduction curves.

2.3.3 ENERGY-BASED METHODS

Recently, new models are developing increasingly. They are based on the concept of energy, in particular the concept of dissipated energy, already described in paragraph 2.2.2.2. These models are based on the observation that the parameters used to quantify an earthquake are almost always linked to energetic concepts, first of all the magnitude, defined as the energy released during a seismic event.

Unlike stress and strain methods, the energy-based models do not need to convert the earthquake motion to an equivalent number of uniform cycles (Neq), as a matter of the fact that the specific deviatoric energy does not depend on the shape of the cyclic loading.

This is why several researchers have suggested the energy-based approach as an innovative and promising method for evaluating liquefaction potential in situ (Law et al., 1990; Desai, 2000; Kokusho, 2013; Kokusho, 2017).

In such models, the demand can be estimated from a large number of correlations based on the concept of magnitude or Arias intensity. For sake of brevity, the most important relationships will be described, while other correlations can be found in Green (2001).

One of the first and most important formulas was provided by Gutenberg and Richter (1956). According to them:

47

𝐸₀ = 10^1.5𝑀+1.8 (2.34)

where E0 is the total radiated energy from the source (expressed in kJ), while M is the magnitude of the earthquake (Richter scale).

Later, Davis and Berrill (1982) modified the original formula of Gutenberg and Richter (1956) (eq. (2.34)), taking into account the distance from site to the center of energy release (r) and the initial effective overburden stress (σ’v0):

𝐸 = (𝑟²∙ 𝜎′^1.5_𝑣0 10^1.5𝑀 )

−1

(2.35)

On the other hand, Kayen and Mitchell (1997) quantified the demand by using the Arias Intensity, while according to Kokusho (2013; 2017), the demand can be computed from the multiple reflection theory of SH waves, and finally Baziar and Jafarian (2007) used the recorded accelerations at ground surface and downhole arrays of Wildlife and Kobe earthquakes to assess average shear stress and strain history at several earthquake sites.

The demand in this case was estimated from the area of stress-strain loops.

Regarding the capacity, it has been evaluated from different correlations based on the results of SPT, that for sake of brevity have not been reported. However, one of the first proposed expressions of the capacity, which was introduced by Gutenberg and Richter (1956), has been shown as an example:

𝐶𝑎𝑝𝑎𝑐𝑖𝑡𝑦 = (450 𝑁₁²)

−1

(2.36)

where N1 is the measured SPT N-value adjusted to 1 tsf (ton per square foot).In the previous equation no hammer-energy correction factors or fines correction were applied.

After that, several modifications of this equation followed in literature.

Finally, the factor of safety by using energy-based methods can be assessed as the ratio of the capacity and the demand.

In this thesis, further considerations on these promising methods will be done (Chapters 8 and 9), where starting from the results of laboratory tests, a new and easy methodology will be proposed to assess the liquefaction potential in situ.

Nel documento Experimental and theoretical investigation on cyclic liquefaction mechanisms and on the effects of some mitigation measures (pagine 54-59)