Presentation on theme: "1 DESIGN CONSIDERATIONS 1. Assessment of liquefaction risk and subsequent ground settlement and effects of preloading on them 2. Effects of preloading."— Presentation transcript:
1 DESIGN CONSIDERATIONS 1. Assessment of liquefaction risk and subsequent ground settlement and effects of preloading on them 2. Effects of preloading on seismic motion 3. Computer program implementing 1, 2 4. Conclusions
2 1. Assessment of liquefaction risk and subsequent ground settlement and effects of preloading on them The liquefaction risk is expressed by the liquefaction factor of safety coefficient FS lique, where: τ cyc,N SR N FS lique = = (1) τ cyc SR where τ cyc and SR are the cyclic stress exerted and the respective cyclic stress ratio that is the ratio of the cyclic stress to effective stress at a certain depth, and τ cyc,N and SR N are the respective values causing liquefaction in N cycles. Due to uncertainties mainly in the assessment of cyclic soil strength, the Eurocode recommends that the liquefaction safety coefficient should take the value of 1.25.
3 Usually, the liquefaction factor of safety is calculated versus depth. The following equation can be used to estimate the maximum shearing stress versus depth: τ max (z) = α max r d σ ν / g(2a) where α max is the maximum value of the horizontal seismic acceleration exerted at the surface, σ ν is the total vertical stress and r d is a stress reduction coefficient gradually decreasing with depth z. For the coefficient r d an empirical expression has been proposed by Ishihara (1993): r d = z(2b)
4 In an earthquake, the shearing oscillation exerted is not uniform. To use equation (2), the non uniform cycles of the oscillation exerted are converted into uniform cycles as: (a) the value of τ cyc and the respective value of SR are determined by the maximum value of the cyclic shearing stress exerted, τ cyc-max and SR max, multiplied times the coefficient 0.65 and (b) the number of equivalent uniform lοading cycles, Ν, is equal to the cycle number of the not uniform excitation with a magnitude 0.65 times the maximum.
5 Assessment of cyclic strength Fig. 1 gives the cyclic strength for an earthquake with a magnitude of Μ=7.5, corresponding to about 15 uniform cyclic loading cycles and for sand with or without fine grains, as a function of value (Ν 1 ) 60. Coefficient (Ν 1 ) 60 that is corrected for depth SPT index is taken from the number of strokes of the standard penetration test (SPT) in the examined depth, Ν, as follows: (Ν 1 ) 60 = Ν C N with C N = (Pa/σ΄ v ) 0.5 (3) where σ΄ v is the active stress due to the overlying soils, Pa is the atmospheric pressure (=1Kg/cm 2 ). Fig. 1 Cyclic strength ratio for an earthquake M=7.5 as a function of N1 for sand or silt. The presence of fine grains in sandy soils affects the number of resistance strokes in a standard penetration (SPT) and the cyclic soil strength and should be taken into account for assessing liquefaction strength. This is done in the diagram of Fig.1 by selecting the curve with the actual fines content. Between curves interpolation is applied.
6 For the assessment of cyclic strength in an earthquake with M=7.5R through the CPT test the corrected tip resistance q c1 =q c1N defined by: q c1 =q c C q with C q = (P a /σ΄ v ) 0.5 (4) is employed. Then the diagrammatic relation shown in Fig.2 is used. Fines content varies between 0 and 35% and between the limiting curves interpolation is applied. Fig. 2 Cyclic strength ratio for an earthquake M=7.5 as a function of q c1 for pure and silty sand.
7 The propagation velocity of shearing waves, Vs, measured through geophysical methods is an additional useful means for assessing cyclic soil strength. To start with, measured speeds Vs are corrected with depth on the basis of the active vertical stress of the overlying soil σ΄ v, as follows: V sl = V s (Pa/σ΄ v ) 1/n (5) where Pa is the atmospheric pressure and coefficient n takes, in accordance with the Εurocode, the value of 4. Fig. 3 Cyclic strength ratio for an earthquake M=7.5 as a function of corrected shear wave velocity V s1.
8 Finally, an earthquake with a magnitude Μ, smaller or greater than 7.5R, gives a cycle number Ν with an acceleration equal to 0.65 times the maximum, which may be greater or smaller than the 15 cycles of an earthquake with M=7.5, and consequently with a different cyclic strength ratio SR N. Corrective coefficients presented by Idriss and Boulanger (2004) for soil cyclic shear stress ratio are given in table 1. Table 1 Corrective coefficients of cyclic stress ratio for earthquake magnitude
9 Prediction of liquefaction- induced settlement A method for predicting volumetric deformation in saturated sand has been suggested by Ishihara (1993). Volumetric deformation is assessed from (a) the relevant density, in accordance with the value (Ν 1 ) 60 in the standard penetration test or the value q c1 in the cone penetration test and (b) the liquefaction factor of safety. Fig. 4 Chart for determination of the post-liquefaction volumetric strain as a function of factor of safety
10 Effects of preloading on liquefaction Results of the project showed improvement of soil mechanical properties expressed as increase of (N 1 ) 60 index value, q c1 and V s1. This explicitly means that: PRELOADING INCREASES THE CYCLIC STRENGTH OF THE SOIL. In particular, the following equation has been proposed predicting the increase in cyclic liquefaction strength induced by preloading: SR 15-a / SR 15-b = PR 0.04/SR 15-b (6) where SR 15-a, SR 15-b is the cyclic strength after and before preloading respectively and PR is the preloading ratio
11 The effects of preloading on the amplification of the seismic motion were studied by an extensive parametric analysis study. The seismic response in free field condition is usually computed by the 1- dimensional equivalent-linear elastic method of analysis. In the present study, the program EERA, which is functioning at environment Windows EXCEL software is used. The seismic response in the free field of soils is affected by: (a) the soil profile defined as the shear wave velocity (Vs) and unit weight (γ) versus depth, (b) the applied seismic motion, (c) the shear velocity of the underlying rock and (d) the change in the shear modulus and the damping coefficient with shear strain. Regarding the soil profile, (i) the soil type and density, (ii) the effective stress level, which at given depth is affected by the depth of the water table, (iii) the depth of the underlying rock and (iv) the overconsolidation ratio of the soil are important parameters. In the present study three types of sands and three types of clays are considered. Regarding the sand types, three relative densities are considered: Dr equal to 0.2, 0.5 and 0.8. Regarding the clay types, the three cases considered are normally-consolidated clays of small, of medium and of high plasticity. Two water table depths were considered: at 2 and 10m. 2. Effects of preloading on seismic motion
12 7 different depths of bedrock are considered: d b = 5, 10, 15, 20, 30, 45, 60m. In order to investigate the effect of preloading, preloading is applied, as usually in practice, as an embankment which is placed temporarily at the top of the soil. In addition to the case without preloading, two cases of preloading are assumed with embankment height (h) 5 and 11m. Fig. 5 gives schematically the cases considered. Fig. 5 Embankment – soil cross-section with the full range of values of design parameters.
13 For all these 12X7X3=252 cases, the following seismic motions were applied at the bedrock: Aigion 15/6/1995, Friuli-San Rocco 15/9/1976, Kozani 13/5/1995, Umbria and Itea 1997, typical for Europe. To further extend the range of the applied seismic motions, the accelerograms of Aigion 15/6/1995 and Friuli-San Rocco were applied after being multiplied by a factor of 3. Fig. 6 Response spectra of considered seismic motions.
14 The shear wave velocity of the bedrock (V s-b ), according to typical values of rock, was taken equal to 1500m/s and 900m/s. The change in the shear modulus and the damping coefficient with shear strain, was taken according to the largely used and validated curves in terms of the Plasticity Index (PI) by Vucetic and Dobry. Furthermore, considering recent findings where the relationships depend on stress level, the Darendeli equations were applied. Fig. 8 Shear modulus variation as a function of cyclic shear strain for a range of stresses. Fig. 7 Shear modulus variation as a function of cyclic shear strain for a range of Plasticity Index values.
15 The shear velocity variation with depth for the different soils, as well as the effect of preloading on this variation were estimated using well- known soil mechanics relationships. Performance of the analyses and the study of the results gave the following functional relation for the ratio of maximum acceleration after and before preloading with respect to the embankment height h: a max-t / a max-t-h=0 = - h Aμ d +1 (7) where h is the height of the preload embankment in [m] and A=A PGA is a factor that depends on soil type and depth of the soil layer as indicated in the table below:
16 Furthermore, based on the results of liquefaction analyses, it is proposed that the decrease of a max induced by preloading should vary linearly from the surface value at depth d=0 to zero at depth d equal to 3.5 m. This is expressed in equ. (7) by the depth coefficient μ d given by: μ d = (3.5 - d) / 3.5d3.5m (8) μ d = 0 d>3.5m It is observed that PRELOADING REDUCES THE SEISMIC MOTION
17 3. Computer program It is useful to implement the equations (a) assessing the pre-improvement liquefaction risk and subsequent ground settlement and (b) giving the effect of preloading on liquefaction risk and subsequent ground settlement in a computer program easily accessible by the practicing engineer. In order to achieve this, the rules given in graphical form in section 1 were fitted using empirical equations.
18 Fig. 9a gives the cyclic strength in terms of fines content and N1 in equation form, assuming an exponential fit. Fig. 9b gives the parameters of the exponential approximation in terms of the fines content (f) assuming linear variation. In particular, the following equations best fit the results SR 15 =a 1 exp(a 2 *N 1 )(9a) where a 1 = f (9b) a 2 =0.062 f (9c) It can be observed that the coefficient of correlation of all three equations is very close to unity. This indicates good precision of the proposed equations. Fig. 9c gives the response predicted by equations (9) It can be observed that the obtained curves in terms of N 1 are reasonable, as they (i) do not cross each other in terms of fines content, and (ii) are equivalent with those of Fig. 1 for the specified fines content. Fig. 9 Cyclic strength curves as function of N 1 with parametric variation of fines content defined on basis of experimental results.
19 Fig. (10a) gives the cyclic strength in terms of fines content and CPT in equation form, assuming an exponential fit. Fig 10b gives the parameters of the exponential approximation in terms of the fines content (f) assuming linear variation. In particular, the following equations best fit the results: SR 15 =c 1 exp(c 2 *q c1 ) (10a) where c 1 = f+0.039(10b) c 2 =0.094 f+0.014(10c) It can be observed that the coefficient of correlation of all three equations is very close to unity. This indicates good precision of the proposed equations. Fig. 10c gives the response predicted by equations (10). It can be observed that the obtained curves in terms of f are reasonable, as they (i) do not cross each other in terms of fines content, and (ii) are equivalent with those of Fig. 2 for the specified fines content. Fig. 10 Cyclic strength curves as function of q c1 with parametric variation of fines content defined on basis of experimental results.
20 Fig. 11 gives the cyclic strength in terms of V s1 in equation form, assuming an exponential fit. It should be noted that this expression does not depend on fines content. In particular, the following equations best fits the results: SR 15 = 0.02 exp(0.013 * V s1 )(11) It can be observed that the coefficient of correlation is very close to unity. This indicates good precision of the proposed equation. Fig. 11 Cyclic strength curves as function of V s1 on basis of experimental results.
21 Regarding assessing the ground volumetric change in terms of FS and N 1 (or the equivalent q c1 value), according to Fig. 4, the following response is observed: (a) when the factor FS is smaller than a particular value, Fo, ε vol /ε vol-max equals to unity and (b) when FS is larger than Fo, as FS increases further, the ratio (ε vol / ε vol-max ) gradually decreases from unity towards zero, forming an S-shape curve (see Fig.12). The shape of this relation can be simulated with the equation: ε vol = ε vol-max * 0.5 ( 1 + tanh [ (ln(A/FS))B ] )(12) where ε vol-max, Α, Β are best-fit parameters. The parameters ε vol-max, A, B depend on soil density, or, equivalently on N 1 or q 1.
22 Fig. 12 gives the volumetric strain in terms of safety factor FS and N 1 in equation form, for each N 1 value separately, assuming the fit described above. Fig. 12 Parametric with N 1 variation of ratio ε vol / ε vol-max as a function of FS liq.
23 Fig. 13 gives ε vol-max and A, B that best fit the results in terms of N 1 assuming a logarithmic relationship for ε vol-max and linear relationships for A and B. In particular, the following equations are proposed for ε vol-max, A and B: ε vol-max = ln (N 1 ) (13a) A= N (13b) B= N (13c) It can be observed that the coefficient of correlation of all three equations is very close to unity. This indicates good precision of the proposed equations. Fig. 13 Best fit parameters ε vol-max, A,B evaluation as functions of N 1, and q c1 variation with N 1.
24 Finally, Fig. 14, presents the results of various investigations on the effect of earthquake magnitude on the cyclic shear strength. The ratio SR M / SR M=7.5 is defined as magnitude scaling factor (MSF). A mathematical approximation of its variation is that based on Idriss results and is given by the equation: MSF = 6.9 exp (-M/4) – (14) Values presented in table 1 are only slightly different as an average of results of Seed and Idriss (1982), Idriss (1999) and Seed et al (2003). Fig. 14 Magnitude scaling factor, MSF, values proposed by various investigators and mathematical approximation of MSF based on Idriss results.
25 Description of computer program All the equations above have been implemented in an excel worksheet predicting the factor of safety against liquefaction risk with depth and the subsequent ground settlement, as well as the effect of preloading on these quantities. The program divides the soil profile in 1m sublayers down to 20m depth. The reason is that liquefaction typically does not occur at depth below 10m and, if it occurs, its effect on the ground surface is minimal. The input needed in the program is: a -max, M, Water table level, Preload height, Preload unit weight, Depth of bedrock. In addition the variation with depth of the following are needed: (a) N1, (b) total unit weight, (c) % of fines, (d) Plasticity Index. The program estimates the pre-improvement and post- improvement liquefaction factor of safety with depth and ground subsidence in terms of N SPT, q CPT and Vs.
26 For the pre-improvement case, the steps performed for these estimates in the case of N SPT are: estimate (1) σ v, (2) σ' v, (3) N 1 60, (4) f, (5) SR N (M=7.5), (6), SR N (M), (7) r d, (8) SR, (9) FS, (10) ε vol, (11) ρ tot. The steps performed for these estimates in the case of q c are: estimate (1) σ v, (2) σ' v, (3) q c1, N 1 60 (4) f, (5) SR N (M=7.5), (6), SR N (M), (7) r d, (8) SR, (9) FS, (10) ε vol, (11) ρ tot. The steps performed for these estimates in the case of V s are: estimate (1) σ v, (2) σ' v, (3) V 1, (4) SR N (M=7.5), (5), SR N (M), (6) r d, (7) SR, (8) FS, (9) N 1 60, (10) ε vol, (11) ρ tot. For the post-improvement case, the steps performed for these estimates in the case of N SPT, q c1, V s are: estimate (1) σ' v-prel, (2) PR, (3) N 1 60new, q c1new, V s1new, (4) SR N - new, (5) SR- new, (6), Fs-new, (7) ε- vol-new, (8) ρ tot-new
27 In summing up the program estimates the factor of safety and the ground subsidence using relationships that have been proposed in the bibliography in graphical form developed here in equation form. The program estimates the post-improvement cyclic strength and seismic acceleration based on results of the project. As an example of application of the program, the field study of the present project is used for both the pre-improvement and post-improvement cases and for different earthquake magnitudes. A validation of the computer program was made by comparison of computed values with the values measured/estimated from the field test (WP2). Good agreement was found to exist.
28 4. Conclusions The basic conclusions established concerning how preloading affects liquefaction seismic risk and therefore design are: (a) It increases the soils cyclic strength (b) it dampens the seismic motion A computer program was developed during the project that can be used for geotechnical design on liquefaction- susceptible soils in terms of both the factor of safety and the ground subsidence. Results were validated with application on the field test.