seismic passive earth pressure with varying shear

Seismic Passive Earth Pressure with Varying Shear Modulus: Pseudo-Dynamic Approach

IGC 2009, Guntur, INDIA

SEISMIC PASSIVE EARTH PRESSURE WITH VARYING SHEAR MODULUS: PSEUDO-DYNAMIC APPROACH P. Ghosh Assistant Professor, Department of Civil Engineering, Indian Institute of Technology Kanpur, Kanpur–208016, India. E-mail: [email protected] S. Kolathayar PG Student, Department of Civil Engineering, Indian Institute of Technology Kanpur, Kanpur–208016, India. E-mail: [email protected] ABSTRACT: Knowledge of seismic passive earth pressure is very much important to design the retaining wall in the earthquake prone region. Using pseudo-dynamic approach, a limited number of investigations have been performed to obtain the seismic passive earth pressures considering a constant magnitude of shear modulus throughout the backfill. Truly speaking, the shear modulus and thus shear and primary wave velocities in the soil medium vary with the depth. However, no significant attention has been provided by the researchers towards the determination of seismic passive earth pressure behind a non vertical cantilever retaining wall with varying shear modulus throughout the backfill. Using pseudo-dynamic approach, this paper presents a study on the seismic passive earth pressure behind a nonvertical rigid cantilever retaining wall by considering time and phase difference of acceleration in the backfill; and the variation of shear modulus with depth. The results are provided in tabular and graphical form. 1. INTRODUCTION The determination of passive resistance on a retaining wall, under both static and seismic conditions, is very much essential as the damage of such earth retaining structures may lead to significant loss of life and wealth. Several investigations have been performed by different researchers to determine the passive earth pressure on a rigid retaining wall under seismic condition. The early work on earthquake induced lateral earth pressure acting on a retaining wall under both active and passive conditions was reported by Okabe (1926), and Mononobe & Matsuo (1929) using pseudo-static approach. This analysis was later recognized as well known Mononobe-Okabe method (Kramer 1996) to compute the seismic earth pressure. In the pseudo-static analysis, the dynamic load induced by an earthquake is considered as time-independent, which eventually assumes that the magnitude and phase of acceleration are uniform throughout the backfill. Apart from this, pseudo-static analysis does not consider the variation of shear modulus throughout the backfill. To overcome this constraint, Steedman & Zeng (1990) used a pseudo-dynamic approach to predict the seismic active earth pressure behind a vertical cantilever retaining wall. In pseudo-dynamic analysis, the time and phase difference due to finite shear wave velocity were considered. Steedman & Zeng (1990) considered the effect of non-uniform shear modulus distribution and amplification of acceleration on the magnitude of active earth pressure behind a vertical wall. Choudhury & Nimbalkar

(2005) and Ghosh (2007) also used pseudo-dynamic approach to predict the seismic passive earth pressure behind a cantilever retaining wall. In general, the shear modulus and thus the shear and primary wave velocities in the ground vary with depth which affects the phase of acceleration. However, using pseudo-dynamic approach, the seismic passive earth pressure behind a nonvertical cantilever retaining wall by considering the influence of varying shear modulus in the backfill, has not drawn much attention from the researchers. The present study explores the effects of soil friction angle φ, wall inclination θ, wall friction angle δ, horizontal earthquake acceleration coefficient αh, vertical earthquake acceleration coefficient αv, amplification factor fa, shear wave velocity Vs, primary wave velocity Vp and depth exponent causing shear modulus variation β, on the seismic passive earth pressure using pseudo-dynamic approach. The limit equilibrium method, with a planar failure surface behind the retaining wall, has been considered to compute the passive resistance of the wall. 2. DEFINITION OF THE PROBLEM A rigid nonvertical cantilever retaining wall of height H is placed with a dry, cohesionless, horizontal backfill as shown in Figure 1. The wall face (AB) on the backfill side is inclined at an angle θ with the vertical and has a wall friction angle δ. The objective is to determine the passive earth

522


pressure coefficient and distribution by knowing the passive resistance Ppe per unit length of the wall in the presence of a sinusoidal base shaking subjected to linearly varying horizontal and vertical accelerations with amplitudes of (H − z ) ⎤ (H − z ) ⎤ ⎡ ⎡ and ⎢1 + ( f a −1) H ⎥α h g ⎢1 + ( f a −1) H ⎥α v g , ⎦ ⎦ ⎣ ⎣ respectively, where z is any depth below the ground surface and g is the acceleration due to gravity. The parameters shown in Figure 1 are considered as positive and the unit weight of the soil is taken as γ. ah= faαhg A

C

where G0 is a constant and z is the depth below the ground surface. The shear wave velocity may then be deduced as a function of depth z,

⎛G V s = ⎜⎜ 0 ⎝ ρ

av= faαvg

Δt s (z ) = −

dz

δ θ

ah = αhg

(4)

z

1/ 2

(5)

The pseudo-dynamic analysis, which considers finite shear and primary wave velocities, can be developed by taking into account the variation of shear modulus and amplification of acceleration in the backfill and thus both phase and magnitude of the accelerations vary. The present analysis considers both shear (Vs) and primary wave velocity (Vp) acting within the backfill during earthquake in the direction as shown in Figure 1. The analysis includes a period of lateral shaking T, which can be expressed as, (1)

where, ω is the angular frequency. In practice, both shear and primary wave velocities vary with depth as a result of non-uniform shear modulus distribution in the backfill, generally in sands, and the variation of shear modulus with depth can be expressed as, (2)

⎛G ⎞ H = H β / 2 (1 − β / 2) ⎜ 0 ⎟ Δts ( z = 0 ) ⎝ ρ ⎠

(6) V = 1.87Vsavg , pavg The variation of acceleration through a soil layer with varying shear modulus also depends on different factors such as damping and the interaction of reflected, refracted and surface waves in the vicinity of structure. For a sinusoidal base shaking, the horizontal and vertical accelerations at any depth z below the ground surface and time t can be expressed as, Vsavg =

A planar failure surface (BC in Fig. 1) at an angle α, with the horizontal has been considered in this analysis. The passive thrust, Ppe makes an angle δ, with the normal to the wall face AB. Here, the failure mechanism has been solved using pseudo-dynamic analysis to compute the passive resistance.

0≤β≤1

)

1/ 2

av = αvg

3. ANALYSIS

G = G 0z

(

However, as the shear modulus varies with depth, it is necessary to define an average magnitude of shear and primary wave velocities as.

Fig. 1: Failure Mechanism and Associated Forces

β

∫

z

dz ( ρ / G0 ) = ( H 1−β / 2 − z1−β / 2 ) H V 1.87 (1 − β / 2 ) p (z)

α

2π T= ω

(3)

(ρ / G0 )1 / 2 1− β / 2 1− β / 2 dz = H −z H V s (z ) (1 − β / 2)

Δt p ( z ) = − ∫

φ R Vs,Vp

B

zβ /2

Similarly, the time increment for the passage of a primary wave from the base to a depth z will be,

z

W

Pp

H

1/ 2

where, ρ is the mass density of backfill. Primary wave velocity Vp can be calculated as 1.87 Vs which is valid for most of the geological materials (Das, 1993). The time increment for the passage of a shear wave from the base to a depth z will then be

Qv Qh

⎞ ⎟⎟ ⎠

1/ 2 ⎡ ( H − z ) ⎤ sin ω ⎧⎪⎨t − ( ρ / G0 ) H 1− β / 2 − z1− β / 2 ah ( z , t ) = α h g ⎢1 + ( f a − 1) ⎥ (1 − β / 2 ) H ⎦ ⎩⎪ ⎣

(

⎫ )⎪⎬ (7)

⎫ ⎡ ( H − z ) ⎤ sin ω ⎧⎪t − ( ρ / G0 ) av ( z , t ) = α v g ⎢1 + ( f a − 1) ( H 1− β / 2 − z1− β / 2 )⎪⎬ ⎨ ⎥ 1.87 (1 − β / 2 ) H ⎪ ⎪⎭ ⎩ ⎣ ⎦ 1/ 2

⎭⎪

(8)

The mass of the small shaded part of thickness dz (Fig. 1) is given by, m(z) =

γ ( H − z )(1 + tan α tan θ ) dz g tan α

(9)

The total weight of the failure wedge W is given by, W=

γ H 2 (1 + tan α tan θ ) 2 tan α

(10)

The horizontal inertia force exerted on the small element due to horizontal earthquake acceleration can be expressed as m(z)αh(z,t). Therefore, the total horizontal inertia force Qh(t) acting in the failure wedge is given by the integral H

Qh (t ) = ∫ m ( z ) ah ( z , t ) dz

523

0

(11)


Similarly, the total vertical inertia force Qv(t) acting in the failure wedge is given by, H

Qv (t ) = ∫ m ( z ) av ( z , t ) dz

(12)

under the influence of amplification of excitation and variation of shear modulus with depth. Here, the distribution of passive earth pressure is found to be nonlinear in nature which is not the case in pseudo-static analysis.

0

⎡ Q ( t ) cos (α + φ ) + Qv ( t ) sin(α + φ ) ⎤ W sin(α + φ ) −⎢ h ⎥ cos (α + φ + δ − θ ) cos (α + φ + δ − θ ) ⎢⎣ ⎦⎥

(13)

The first term on the right hand side gives the static passive resistance; whereas the last term predicts the dynamic passive resistance due to earthquake loading. The direction of both horizontal and vertical inertia forces as shown in Figure 1 causes the most critical effect on the wall under seismic condition. The seismic passive earth pressure coefficient can then be defined as,

K pe =

2 Ppe ( t )

(14)

γH2

The computations have been performed by writing a computer code in MATLAB. To find the minimum value of Kpe, the magnitudes of α and t/T have been varied independently at intervals of 0.1° and 0.01 respectively. 4.1 Seismic Passive Earth Pressure Coefficient The variations of seismic passive earth pressure coefficient Kpe with changes in αh for different values of θ and fa are presented in Figure 2 for φ = 30˚, δ = 0.5φ, αv = 0.5αh, β = 0.5, H/TVs = 0.3 and H/TVp = 0.16. It can be seen that the magnitude of passive earth pressure coefficient decreases continuously with an increase in the magnitude of αh. It can also be observed that the value of Kpe decreases with increase in wall inclination θ. The lowermost line in Figure 2 is for θ = 10°; whereas the uppermost line is for θ = –10°. It has been seen that higher the amplification factor fa, the lesser is the magnitude of Kpe.

It can be observed that Kpe is a function of α, t/T, H/TVs, H/TVp and β. H/TVs is the ratio of time taken by the shear wave velocity to travel the full height to the period of lateral shaking T and H/TVp is the ratio of time taken by the primary wave to travel the full height to T. The optimization has been done with respect to α and t/T to get the minimum value of Kpe. During optimization, the values of α and t/T have been varied in the ranges of 0–90˚ and 0–1 respectively. The passive earth pressure distribution behind the wall can be determined by taking partial derivative of Ppe with respect to z and expressed as, p pe ( z , t ) =

6.0

(a)

0.0

0.2

8.0

0.3

0.4

θ = –10° – 5° 0° 5° 10° (lower most)

6.0 K

pe

⎛ z ⎞ cos(α + φ ) sin ω ⎜ t − ⎟ ⎝ Vs ⎠ cos (α + φ + δ − θ )

α γz z⎤ ⎡ − v (1 + tan α tan θ ) ⎢1 + ( f a − 1) ⎥ H⎦ tan α ⎣ ⎞ sin(α + φ ) ⎟⎟ ⎠ cos (α + φ + δ − θ )

0.1

αh

α hγ z z (1 + tan α tan θ ) ⎡⎢1 + ( f a − 1) ⎤⎥ tan α H⎦ ⎣

⎛ z sin ω ⎜ t − ⎜ Vp ⎝

4.0 2.0

sin(α + φ ) ∂Ppe ( z , t ) = γ z 1 + tan α tan θ ( ) α α +φ + δ −θ ) tan cos ( ∂z

−

θ = –10° –5° 0° 5° 10° (lower most)

8.0

pe

Ppe ( t ) =

4. RESULTS

K

For values of β in the range of interest, Equations (11) and (12) can be solved analytically only for β = 0 and 1 and must be solved numerically for intermediate values of β. The total passive resistance Ppe can then be determined by taking the horizontal as well as the vertical force equilibrium of the failure wedge and is given by,

4.0

(b) 2.0 0.0

(15)

0.1

0.2

0.3

αh

The first term on the right hand side gives the passive earth pressure under static condition and the second and third terms represent the dynamic passive earth pressure due to horizontal and vertical earthquake accelerations respectively, 524

Fig. 2: Variation of Passive Pressure Coefficient Kpe with αh for φ = 30°, δ = 0.5φ, αv = 0.5 αh, β = 0.5, H/TVs = 0.3 and H/TVp = 0.16. (a) fa = 1.0 (b) fa = 1.4.


4.2 Seismic Passive Earth Pressure Distribution The normalized passive earth pressure distribution is shown in Figure 3 for different values of amplification factor fa with φ = 30˚, δ = 0.5φ, θ = 10°, αh = 0.2, αv = 0.5αh, β = 0.5, H/TVs = 0.3 and H/TVp = 0.16. It can be observed from the figure that the value of passive earth pressure decreases marginally with increase in the magnitude of fa and the difference in pressure becomes a maximum at the base of wall for different values of The variation of ppe/γH with changes in z/H for different values of φ and δ with θ = 10°, αh = 0.2, αv = 0.5αh, β = 0.5, H/TVs = 0.3, H/TVp = 0.16 and fa = 1.4 is presented in Figure 4. It can be seen from the figure that the magnitude of passive earth pressure increases with increase in the value of φ; and for a particular value of φ, the magnitude of passive earth pressure increases with an increase in the value of δ. 0.0

various researchers for β = 0, as the values of Kpe for β > 0 are scarce in the literature. For β = 0, the obtained values are exactly same as those obtained by Ghosh (2007) for different values of θ. However, the comparison of the present values of Kpe for different values of β is shown in Table 1 for a range of various parameters. It has been seen that there is no significant difference in the values of Kpe with change in β for low values of H/TVs. The same observation was also made by Steedman & Zeng (1990) for vertical wall under active case. Table 1: Values of Passive Earth Pressure Coefficient Kpe for φ = 30°, δ = 0.5φ, αh = 0.2, αv = 0.5αh, fa = 1.4 H/TVs H/TVp

fa = 1.2 fa = 1.6

z/H

fa = 1.8 0.6

0.5

1.0

1.5

2.0

2.5

3.0

Fig. 3: Normalized Passive Earth Pressure Distribution for Different Values of fa (φ = 30°, δ = 0.5φ, θ = 10°, αh = 0.2, αv = 0.5αh, β = 0.5, H/TVs = 0.3 and H/TVp = 0.16)

δ=0 δ = 0.5φ δ=φ

z/H 0.6 0.8

1

2

ppe/γ H

3

4

2.72 5.07 2.81 5.32 2.93

2.71 5.04 2.8 5.29 2.91

2.70 5.02 2.79 5.25 2.89

2.69 4.96 2.77 5.17 2.86

0.21

0.5

0.27

–10° 10°

0.6

0.32

–10° 10°

5.64 3.07

5.61 3.05

5.57 3.03

5.51 3.00

5.40 2.96

0.7

0.37

–10° 10°

5.96 3.20

5.92 3.18

5.86 3.16

5.77 3.11

5.65 3.06

0.8

0.43

–10° 10°

6.28 3.34

6.23 3.31

6.16 3.28

6.04 3.23

5.88 3.16

10° –10° 10˚

Using the pseudo-dynamic approach, the effects of the soil friction angle, wall inclination, wall friction angle, horizontal and vertical earthquake acceleration, amplification of vibration, shear and primary wave velocity with their variation along the depth of the backfill on the seismic passive earth pressure behind a nonvertical cantilever retaining wall have been explored. It has been found that the magnitude of seismic passive earth pressure decreases with an increase in the wall inclination θ, amplification factor fa and the sesmic acceleration coefficients αh and αv. The passive earth pressure shows little variation with the change in magnitude of the depth exponent β. The nonlinearity of seismic passive earth pressure distribution increases with an increase in seismicity, which causes the point of application of the total passive thrust to be shifted.

0.4

1.0 0

β= 1.0 4.79

6. CONCLUSIONS

0.0 0.2

β= 0.75 4.81

0.4

3.5

ppe/γ H

β= 0.50 4.83

2.72 5.08 2.82 5.34 2.94

0.8 1.0 0.0

β= 0.25 4.85

0.16

fa = 1.4 0.4

–10°

Kpe β= 0.0 4.85

0.3 fa = 1.0

0.2

θ

5

Fig. 4: Normalized Passive Earth Pressure Distribution for Different Values of φ and δ (θ = 10°, αh = 0.2, αv = 0.5αh, β = 0.5, H/TVs = 0.3, H/TVp = 0.16 and fa = 1.4) 5. COMPARISON For nonvertical retaining wall, the present values of Kpe can only be compared with the existing values proposed by

Due to the use of more rational pseudo-dynamic approach over the pseudo-static approach, the present values of Kpe and pattern of passive earth pressure distribution could be used by the design engineers to design and assess the behaviour of nonvertical cantilever retaining wall under seismic condition.

525


REFERENCES Choudhary D. and Nimbalkar S. (2005). “Seismic Passive Resistance by Pseudo-dynamic Method”, Geotechnique, 55(9): 699–702. Das B.M. (1993). “Principles of Soil Dynamics”, PWSKENT Publishing Company”, Massachusetts. Okabe S. (1926). “General Theory of Earth Pressure”, Journal of the Japanese society of Civil Engineers, 12(1): 311. Ghosh P. (2007). “Seismic Passive Earth Pressure Behind Nonvertical Retaing Wall Using Pseudo-Dynamic

Analysis”, Geotechnical and Geological Engineering, 25:693–703. Kramer S.L. (1996). “Geotechnical Earthquake Engineering”, Prentice Hall, Englewood Cliffs, N.J. Mononobe, N. and Matsuo H. (1929). “On the Determination of Earth Pressure during Earthquakes”, Proceedings of the World Engineering Conference, Vol. 9, pp. 176. Steedman R.S. and Zeng X. (1990). “The Influence of Phase on the Calculation of Pseudo-static Earth Pressure on a Retaining Wall”, Geotechnique, 40: 103–112.

526