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A Dynamic Coupled Model for Shanghai Soils
© 2001 Elsevier Science Ltd. All rights reserved. Computational Mechanics - New Frontiers for New Millennium S. Valliappan and N. Khalili, editors.
357
A DYNAMIC COUPLED MODEL FOR SHANGHAI SOILS
Y. Huang, W. M. Ye, Y. Q. Tang and X. Shu Department of Geotechnical Engineering, Tongji University, Shanghai, 200092, People's Republic of China
ABSTRACT The behaviour of geomaterials, and in particular of soils, is governed by the interaction of their solid skeleton with the pore fluid. The behaviour of soils under dynamic loadings can only be achieved through coupled analysis for the interaction of the soil skeleton and the pore fluid. A coupled model for Shanghai saturated soils is presented according to large amount of experimental data. The characters of stress, strain, pore water pressure and deformation can be well described by several specific empirical equations. The proposed model can be used in the dynamic soil-pore water coupled analysis. The dynamic coupled model is used in a free field earthquake response of Shanghai horizontal soil layer of depth 280m subjected to a strong base excitation.
KEYWORDS Soil dynamics, Constitutive model, Coupled problem, Saturated soil, Laboratory test, Shanghai
INTRODUCTION Shanghai is located on the East Coast of China at the mouth of the Yangtze River at the Donghai Sea. The alluvial soil deposit of Shanghai is 150-400m deep with an about 100m thick saturated soft soil, which has a large water content and a high compressibility. It has great importance to explore a suitable soil dynamic coupled model in earthquake engineering, traffic cyclic loading and ocean engineering analysis. The interaction of the pore water with the solid soil skeleton belongs to the Class II coupled problems (Zienkiewicz and Taylor, 2000). The coupling occurs through the governing differential equations of both solid mechanics and transient seepage. A hierarchy of constitutive models is available for the dynamic response of soils to cyclic loading. The models range from
358
the relatively simple hysteretic nonlinear models to complex elastic-kinematic hardening plastic models. Equivalent linear procedures appear to work quite well provided the behaviour of the structure is not strongly nonlinear and significant pore pressures do not develop. To simplify the research relatively, in the paper, a viscoelastic model was developed in this paper, which can model nonlinear behaviour in terms of effective stresses and to provide the generation and dissipation of pore water pressures and to predict the soil behaviour under cyclic loading. The dynamic calculation model includs a set of relationships of stress, strain, pore water pressure buildup and permanent deformation. Compared with elasto-plastic models, the model simulates the soil dynamic behavior in a cycle of loading as a whole, not in detail. The following discussions about soil properties were based on triaxial and resonant column test data for Shanghai soils.
MAXIMUM SHEAR MODULUS Maximum shear modulus, Gmax, is also called initial shear modulus or low-amplitude shear modulus. It represents the stress-strain backbone curve's slope at the origin. Laboratory tests have shown that soil stiffness is influenced by cyclic strain amplitude, void ratio, mean principal effective stress, plasticity index, overconsolidation ratio, and number of loading cycles. A vast majority of experimental data has been accumulated to evaluate the shear modulus of various soils at very small levels of strains. Gmax can be estimated in several different ways. Generally, the measured shear wave velocities Vs obtained by most seismic geophysical tests can be used to compute Gmax as (1)
Gm^=pv]
Another way is the empirical relationships between Gmax and parameters of various in situ test (SPT, CPT, DMT, PMT et al.). In this paper, the following relationship between Gmax, the void ratio e, the over consolidation ratio OCR, and the confining pressure σ0 has been used:
«. OCR* 0.3 + 0.76
/ · \°· 5 (σ0) {pj
where Pa = atmospheric pressure, and k = a coefficient which relates to plasticity index of soil. Based on the above relationship and experimental data, an acceptable fit for Shanghai saturated soft soil is obtained, as follows: D = 353 for clay, D = 451 for silt, and D = 485 for sand.
STRAIN-DEPENDENT MODULUS AND DAMPING It is well known that the deformation characteristics of soil are highly nonlinear and this is manifested in the shear modulus and damping ratio, which vary significantly with the amplitude of shear strain under cyclic loading. The proposed expression of the secant shear modulus G at a strain amplitude / i s
359
\-H(y)
(3)
AlTrf
(4)
In the model, the function H (γ) is
H(y) =
^ΑΦ,Τ
where yr = a reference or yield strain; and A and B = two dimensionless parameters. It is suggested that values of yr for Shanghai saturated soft soil could be determined by the empirical relationship Yr =C-\¡a0
(5)
where σ0 = the effective mean principal stress in kPa, and C = an empirical parameter. Table 1 shows a summary of the experimental numerical values for the three parameters A, B, and C obtained for Shanghai saturated soft soil. TABLE 1 REFERENCE VALUE OF PARAMETERS: A, B, AND C Soil type Clay Silt Sand
1.62 1.12 1.10
0.42 0.44 0.48
0.00013 0.00017 0.00022
The curve of shear modulus ratio G/Gm!LX of Shanghai clay with γ is compared with the experimental data in Figure 1. 1.0
0.30"
0.8
0.25" 0.20"
a0·6"
Q0.15"
§0.4
0.10' 0.05"
0.2" on" 1E-6
1i 1E-5
i
i
1E-4 Y
1E-3
0.00"
0.01
Figure 1: The relationship between shear modulus ratio and shear strain of Shanghai clay
1E-6
1E-5
1E-4 Y
1E-3
0.01
Figure 2: The relationship between damping ratio and shear strain of Shanghai clay
For Shanghai saturated soft soil, the variation of the damping ratio, Z), with strain level is
360 D Dm
(6)
where Z)max = 0.30 for clay, Z)max = 0.25 for silt and sand; and ß= 1.0. A comparison between the proposed model and experimental data is shown in Figure 2.
PORE WATER PRESSURE BUILDUP Laboratory tests can reveal the manner in which excess pore pressure is generated. On the basis of the results from undrained cyclic triaxial test data, the pore water pressure buildup of Shanghai clay and silt may be expressed as
4 = aiV>
(7)
where p = the pore water pressure; N = the number of uniform stress cycles; and a and b = two experimental parameters which are determined by the ratio of the dynamic shear stress to the effective confining pressure. Table 2 shows the reference value of a and b for Shanghai clay and silt. The curve of pore water pressure ratio ρ/σ0
and N of Shanghai mucky clay is compared
with the experimental data in Figure 3. TABLE 2 REFERENCE VALUE OF a AND b FOR SHANGHAI CLAY Soil type Clay Silt
a 0.274 rTu 0.273 rT°
b 0.375 r? 0.348 rT°
rx= rd / σ0; Td = dynamic shear stress.
i—·—i—·—r 400 600 800 Cyclic Number N
1000
1200
Figure 3: Variation of pore water pressure ratio and N of Shanghai mucky clay
361 For Shanghai sand, the development of pore water pressure in laboratory cyclic loading tests is of the form 'ΝΛ -*-r = (l - msl)— arcsinl N π ση fJ
(8)
where s¡ = static stress level; m and 0= experimental parameters, for Shanghai sand m = 1.1 and 0=0.7; and Nf = the accumulative number of cycles at the same stress level required to produce a peak cyclic pore water pressure ratio of 100% under undrained conditions.
PERMANENT DEFORMATION The deformation under cyclic loading can be divided into the volumetric and deviatoric components, εp and γρ. The volume change, ερ, comes from dissipation of excess pore water pressure and can be calculated through consolidation equation. For Shanghai saturated soft soil (Zhou and Hu, 1998), γρ is formulated according to the undrained cyclic triaxial tests as follow
r
"~d-{d-2oy
(9)
where d = an experimental parameter, d = 8 for Shanghai sand, d = 3 for Shanghai clay, r* is r =
r r
~s
(10)
where rs = initial dynamic stress ratio; yy= failure dynamic stress ratio.
APPLICATION The couple model presented allows any problem of earthquake response to be studied quantitatively via the finite element method (Zienkiewicz and Taylor, 2000). In the example the Shanghai horizontal soil layer of depth 280m, which is subjected to a base motion of the first 10s N-S component of the El Centro Earthquake (May, 1940) with the maximum acceleration scaled to O.lg, is studied in terms of the dynamic calculation model. The base motion input for analysis is shown in Figure 4. The ground acceleration history and the distribution of maximum accelerations are shown in Figs. 5 and 6 respectively. It is interesting that the surface maximum acceleration is only 0.92 m/s2. The main reason is probably that the buildup of pore water pressures causes a decrease in stiffness of Shanghai soil. The ground and bedrock response spectra with 5 percent damping ratio are shown in Figure 7. The spectral values of the ground acceleration are less than those of the bedrock accelerations between the period of 0.1 and 1.2 seconds. The soil deposit acts as a filter when the bedrock earthquake acceleration is transmitted through it. The soil deposit filters out a significant portion of the high frequency content of the bedrock acceleration. During the earthquake, the pore water pressure of deep soil is very small. But the pore water pressures from the depth less than 20m to the ground is
362 gradually increasing. The maximum pore water pressure ratio equals 0.35 in the depth of 9m. The calculated maximum earthquake subsidence is about 15mm until the dissipation of pore water pressure completes.
Figure 5: Predicted acceleration time history of ground
Figure 4: Input earthquake wave
Bedrock Ground
0.5
1.0
1.5
2.0
2.5
Maximum Acceleration (m/ s2)
Figure 6: Predicted variation of the maximum acceleration with depth
Period (s)
Figure 7: Predicted response spectra of ground and bedrock
CONCLUSION In this paper, the saturated soil is modeled as a two-phase porous media system consisting of solid and fluid phases. On the basis of resonant column test and dynamic triaxial test data of Shanghai saturated soft soil, the dynamic calculation model including a set of relationships of stress, strain, pore water pressure and earthquake subsidence is developed to compute the seismic response of soil. The procedure to identify soil constants for the dynamic calculation model is also reported in detail. The developed dynamic calculation model together with the dynamic coupled analysis is utilized to predict the seismic response of Shanghai soil strata through the finite element method.
REFERENCES Zienkiewicz, O.C., and Taylor, R.L.(2000), The Finite Element Method. Vol. 1: The basis, Fifth Edition, Butterworth-Heinemann, Oxford, UK Zhou, J. and Hu XY. (1998). Analysis of earthquake resistance of underground construction surrounded by soft soil in Shanghai. Journal ofTongji University 26: 5, 492-497.
357
A DYNAMIC COUPLED MODEL FOR SHANGHAI SOILS
Y. Huang, W. M. Ye, Y. Q. Tang and X. Shu Department of Geotechnical Engineering, Tongji University, Shanghai, 200092, People's Republic of China
ABSTRACT The behaviour of geomaterials, and in particular of soils, is governed by the interaction of their solid skeleton with the pore fluid. The behaviour of soils under dynamic loadings can only be achieved through coupled analysis for the interaction of the soil skeleton and the pore fluid. A coupled model for Shanghai saturated soils is presented according to large amount of experimental data. The characters of stress, strain, pore water pressure and deformation can be well described by several specific empirical equations. The proposed model can be used in the dynamic soil-pore water coupled analysis. The dynamic coupled model is used in a free field earthquake response of Shanghai horizontal soil layer of depth 280m subjected to a strong base excitation.
KEYWORDS Soil dynamics, Constitutive model, Coupled problem, Saturated soil, Laboratory test, Shanghai
INTRODUCTION Shanghai is located on the East Coast of China at the mouth of the Yangtze River at the Donghai Sea. The alluvial soil deposit of Shanghai is 150-400m deep with an about 100m thick saturated soft soil, which has a large water content and a high compressibility. It has great importance to explore a suitable soil dynamic coupled model in earthquake engineering, traffic cyclic loading and ocean engineering analysis. The interaction of the pore water with the solid soil skeleton belongs to the Class II coupled problems (Zienkiewicz and Taylor, 2000). The coupling occurs through the governing differential equations of both solid mechanics and transient seepage. A hierarchy of constitutive models is available for the dynamic response of soils to cyclic loading. The models range from
358
the relatively simple hysteretic nonlinear models to complex elastic-kinematic hardening plastic models. Equivalent linear procedures appear to work quite well provided the behaviour of the structure is not strongly nonlinear and significant pore pressures do not develop. To simplify the research relatively, in the paper, a viscoelastic model was developed in this paper, which can model nonlinear behaviour in terms of effective stresses and to provide the generation and dissipation of pore water pressures and to predict the soil behaviour under cyclic loading. The dynamic calculation model includs a set of relationships of stress, strain, pore water pressure buildup and permanent deformation. Compared with elasto-plastic models, the model simulates the soil dynamic behavior in a cycle of loading as a whole, not in detail. The following discussions about soil properties were based on triaxial and resonant column test data for Shanghai soils.
MAXIMUM SHEAR MODULUS Maximum shear modulus, Gmax, is also called initial shear modulus or low-amplitude shear modulus. It represents the stress-strain backbone curve's slope at the origin. Laboratory tests have shown that soil stiffness is influenced by cyclic strain amplitude, void ratio, mean principal effective stress, plasticity index, overconsolidation ratio, and number of loading cycles. A vast majority of experimental data has been accumulated to evaluate the shear modulus of various soils at very small levels of strains. Gmax can be estimated in several different ways. Generally, the measured shear wave velocities Vs obtained by most seismic geophysical tests can be used to compute Gmax as (1)
Gm^=pv]
Another way is the empirical relationships between Gmax and parameters of various in situ test (SPT, CPT, DMT, PMT et al.). In this paper, the following relationship between Gmax, the void ratio e, the over consolidation ratio OCR, and the confining pressure σ0 has been used:
«. OCR* 0.3 + 0.76
/ · \°· 5 (σ0) {pj
where Pa = atmospheric pressure, and k = a coefficient which relates to plasticity index of soil. Based on the above relationship and experimental data, an acceptable fit for Shanghai saturated soft soil is obtained, as follows: D = 353 for clay, D = 451 for silt, and D = 485 for sand.
STRAIN-DEPENDENT MODULUS AND DAMPING It is well known that the deformation characteristics of soil are highly nonlinear and this is manifested in the shear modulus and damping ratio, which vary significantly with the amplitude of shear strain under cyclic loading. The proposed expression of the secant shear modulus G at a strain amplitude / i s
359
\-H(y)
(3)
AlTrf
(4)
In the model, the function H (γ) is
H(y) =
^ΑΦ,Τ
where yr = a reference or yield strain; and A and B = two dimensionless parameters. It is suggested that values of yr for Shanghai saturated soft soil could be determined by the empirical relationship Yr =C-\¡a0
(5)
where σ0 = the effective mean principal stress in kPa, and C = an empirical parameter. Table 1 shows a summary of the experimental numerical values for the three parameters A, B, and C obtained for Shanghai saturated soft soil. TABLE 1 REFERENCE VALUE OF PARAMETERS: A, B, AND C Soil type Clay Silt Sand
1.62 1.12 1.10
0.42 0.44 0.48
0.00013 0.00017 0.00022
The curve of shear modulus ratio G/Gm!LX of Shanghai clay with γ is compared with the experimental data in Figure 1. 1.0
0.30"
0.8
0.25" 0.20"
a0·6"
Q0.15"
§0.4
0.10' 0.05"
0.2" on" 1E-6
1i 1E-5
i
i
1E-4 Y
1E-3
0.00"
0.01
Figure 1: The relationship between shear modulus ratio and shear strain of Shanghai clay
1E-6
1E-5
1E-4 Y
1E-3
0.01
Figure 2: The relationship between damping ratio and shear strain of Shanghai clay
For Shanghai saturated soft soil, the variation of the damping ratio, Z), with strain level is
360 D Dm
(6)
where Z)max = 0.30 for clay, Z)max = 0.25 for silt and sand; and ß= 1.0. A comparison between the proposed model and experimental data is shown in Figure 2.
PORE WATER PRESSURE BUILDUP Laboratory tests can reveal the manner in which excess pore pressure is generated. On the basis of the results from undrained cyclic triaxial test data, the pore water pressure buildup of Shanghai clay and silt may be expressed as
4 = aiV>
(7)
where p = the pore water pressure; N = the number of uniform stress cycles; and a and b = two experimental parameters which are determined by the ratio of the dynamic shear stress to the effective confining pressure. Table 2 shows the reference value of a and b for Shanghai clay and silt. The curve of pore water pressure ratio ρ/σ0
and N of Shanghai mucky clay is compared
with the experimental data in Figure 3. TABLE 2 REFERENCE VALUE OF a AND b FOR SHANGHAI CLAY Soil type Clay Silt
a 0.274 rTu 0.273 rT°
b 0.375 r? 0.348 rT°
rx= rd / σ0; Td = dynamic shear stress.
i—·—i—·—r 400 600 800 Cyclic Number N
1000
1200
Figure 3: Variation of pore water pressure ratio and N of Shanghai mucky clay
361 For Shanghai sand, the development of pore water pressure in laboratory cyclic loading tests is of the form 'ΝΛ -*-r = (l - msl)— arcsinl N π ση fJ
(8)
where s¡ = static stress level; m and 0= experimental parameters, for Shanghai sand m = 1.1 and 0=0.7; and Nf = the accumulative number of cycles at the same stress level required to produce a peak cyclic pore water pressure ratio of 100% under undrained conditions.
PERMANENT DEFORMATION The deformation under cyclic loading can be divided into the volumetric and deviatoric components, εp and γρ. The volume change, ερ, comes from dissipation of excess pore water pressure and can be calculated through consolidation equation. For Shanghai saturated soft soil (Zhou and Hu, 1998), γρ is formulated according to the undrained cyclic triaxial tests as follow
r
"~d-{d-2oy
(9)
where d = an experimental parameter, d = 8 for Shanghai sand, d = 3 for Shanghai clay, r* is r =
r r
~s
(10)
where rs = initial dynamic stress ratio; yy= failure dynamic stress ratio.
APPLICATION The couple model presented allows any problem of earthquake response to be studied quantitatively via the finite element method (Zienkiewicz and Taylor, 2000). In the example the Shanghai horizontal soil layer of depth 280m, which is subjected to a base motion of the first 10s N-S component of the El Centro Earthquake (May, 1940) with the maximum acceleration scaled to O.lg, is studied in terms of the dynamic calculation model. The base motion input for analysis is shown in Figure 4. The ground acceleration history and the distribution of maximum accelerations are shown in Figs. 5 and 6 respectively. It is interesting that the surface maximum acceleration is only 0.92 m/s2. The main reason is probably that the buildup of pore water pressures causes a decrease in stiffness of Shanghai soil. The ground and bedrock response spectra with 5 percent damping ratio are shown in Figure 7. The spectral values of the ground acceleration are less than those of the bedrock accelerations between the period of 0.1 and 1.2 seconds. The soil deposit acts as a filter when the bedrock earthquake acceleration is transmitted through it. The soil deposit filters out a significant portion of the high frequency content of the bedrock acceleration. During the earthquake, the pore water pressure of deep soil is very small. But the pore water pressures from the depth less than 20m to the ground is
362 gradually increasing. The maximum pore water pressure ratio equals 0.35 in the depth of 9m. The calculated maximum earthquake subsidence is about 15mm until the dissipation of pore water pressure completes.
Figure 5: Predicted acceleration time history of ground
Figure 4: Input earthquake wave
Bedrock Ground
0.5
1.0
1.5
2.0
2.5
Maximum Acceleration (m/ s2)
Figure 6: Predicted variation of the maximum acceleration with depth
Period (s)
Figure 7: Predicted response spectra of ground and bedrock
CONCLUSION In this paper, the saturated soil is modeled as a two-phase porous media system consisting of solid and fluid phases. On the basis of resonant column test and dynamic triaxial test data of Shanghai saturated soft soil, the dynamic calculation model including a set of relationships of stress, strain, pore water pressure and earthquake subsidence is developed to compute the seismic response of soil. The procedure to identify soil constants for the dynamic calculation model is also reported in detail. The developed dynamic calculation model together with the dynamic coupled analysis is utilized to predict the seismic response of Shanghai soil strata through the finite element method.
REFERENCES Zienkiewicz, O.C., and Taylor, R.L.(2000), The Finite Element Method. Vol. 1: The basis, Fifth Edition, Butterworth-Heinemann, Oxford, UK Zhou, J. and Hu XY. (1998). Analysis of earthquake resistance of underground construction surrounded by soft soil in Shanghai. Journal ofTongji University 26: 5, 492-497.