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Technical Note

The inﬂuence of the degree of saturation on dynamic response of a cylindrical lined cavity in a nearly saturated medium Y. Wang a,b, G.Y. Gao a,b,n, J. Yang c, J. Song a,b a

Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai 200092, China c Department of Civil Engineering, The University of Hong Kong, Hong Kong b

art ic l e i nf o

a b s t r a c t

Article history: Received 2 December 2014 Received in revised form 6 January 2015 Accepted 8 January 2015

In most previous studies on the dynamic response of a long cylindrical cavity subjected to internal transient dynamic loads, the porous medium was usually assumed to be completely saturated by ground water. In practice, however, the full saturation condition does not always exist. In this paper the surrounding soil and the lining of the cavity are respectively treated as a nearly saturated porous medium and an elastic material, and the governing equations for the dynamic problem are derived. A set of exact solutions are obtained in the Laplace transform domain for three types of transient loads, i.e. suddenly applied constant load, gradually applied step load and triangular pulse load. By utilizing a reliable numerical method of inverse Laplace transforms, the time-domain solutions are then presented. The inﬂuence of the degree of saturation of the surrounding soil on the dynamic response of the lined cavity is examined for numerical examples. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Nearly saturated soil Lined cavity Dynamic response Degree of saturation Laplace transform Inverse Laplace transform

1. Introduction Evaluation of the response of a cylindrical cavity subjected to internal transient loads is a classical dynamic problem in geotechnical engineering. A large number of studies have dealt with the analytical solution on the dynamic problem. These previous studies can be classiﬁed into two types. One is to treat the soil surrounding the cavity as an elastic solid so as to simplify the problem (e.g. Refs. [1,2]). The other type focuses on dynamic response of saturated soil surrounding a cavity under internal transient loading. In these studies soil was assumed to be a poroelastic material composed of a solid matrix with ﬂuid and Biot's theory was used (e.g. Refs. [3–6]). In reality soil is a three-phased material, comprising solid particles, water and air. Even for soil under water table the condition of full saturation may not exist (Ref. [7]). In this paper, therefore, the effect of the degree of saturation on the response of a lined cylindrical cavity under various types of transient loading is investigated on the basis of Biot's theory and the assumption of homogenous pore ﬂuid. The lining is treated as an elastic medium and the one-dimensional wave equation is established for the lining. A set of exact solutions on the response of the lining and the surrounding poroelastic soil is derived in the

n Corresponding author at: College of Civil Engineering, Tongji University, Shanghai, 200092, China. Tel./fax: 86 21 55058685. E-mail address: [email protected] (G.Y. Gao).

http://dx.doi.org/10.1016/j.soildyn.2015.01.002 0267-7261/& 2015 Elsevier Ltd. All rights reserved.

Laplace transform domain. The method of Durbin [8] is then used to transform the Laplace space solutions into the time-domain solutions. Numerical examples are given to illustrate the inﬂuence of the degree of saturation.

2. Theoretical solutions A cylindrical lined cavity subjected to internal transient loads is presented herein, as shown in Fig. 1. Based on the second Newton's Law, the equilibrium equation for the surrounding soil can be written as ∂σ rr σ rr σ θθ ∂2 ur ∂2 wr þ ¼ ρ 2 þ ρf S r 2 ∂r r ∂t ∂t

ð1Þ

Where σ rr and σ θθ are the radial and hoop total stresses, respectively; ur and wr are respectively the radial displacements of solid skeleton and pore ﬂuid with respect to solid skeleton. ρf and ρ are respectively the mass density of pore ﬂuid and soil; Sr is the degree of saturation. For the pore ﬂuid, the equilibrium equation can be written as ρf ∂ 2 w r ∂σ f ∂2 ur η ∂wr ¼ ρf 2 ∂r n ∂t ∂t ks ∂t

ð2Þ

where σf is the absolute pore ﬂuid pressure at the interface between the lining and the soil; η is the coefﬁcient of viscosity of the ﬂuid; n and ks are the permeability and porosity of soil, respectively.

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Y. Wang et al. / Soil Dynamics and Earthquake Engineering 71 (2015) 27–30

0.05

Sr=95% Sr=98% Sr=99% Sr=99.9% Sr=100%

Gs*u(a)/(f0*a)

0.04 0.03 0.02 0.01 0

Fig. 1. Diagram of forces acting on an element in soil around a cylindrical lined cavity.

-0.01 0

∂ ξ ρf ∂ 2 w r ∂2 ur η ∂wr ¼ þ ρf 2 þ 2 ∂r n ∂t ks ∂t ∂t

ð4Þ

where λs and Gs are Lamé constants; α and M are Biot's coefﬁcients; and the variables e and ξ are deﬁned as follows e ¼ ð∂ur =∂rÞ þ ur =r, ξ ¼ ð∂wr =∂r þwr =rÞ. The governing equations of lining are shown in Ref. [4]. The solutions in Laplace transform domain can then be obtained based on procedures from Gao et al. [4]. Note that if introducing a Sr ¼100% into the solutions for radial displacement, hoop stress and pore pressure, the proposed solutions exactly degenerate to the ones given by Ref. [3].

0.05

0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0

0

5

10

15

t* 0.04

Sr=95% Sr=98% Sr=99% Sr=99.9% Sr=100%

0.035

3. Results and discussion

0.03 Gs*u(a)/(f0*a)

The effect of degree of saturation (Sr) of the surrounding soil on dynamic response of the lined cavity is now investigated based on the solutions derived. The parameters used in the calculations are the same as Ref. [3]. Fig. 2 depicts the time history of radial displacement at the cavity surface with respect to the degree of saturation for the three types of transient loads, i.e. suddenly applied constant load, gradually applied step load and triangular pulse load. In Fig. 2 u(a) is the radial displacement at cavity surface; a is the radius of the cavity; f0 is the maximum dynamic loading amplitude at the inner surface of the lined cavity; tn is the normalized time [4]. It is seen that there is much less impact on the radial displacement when Sr is less than 99%, but when Sr is larger than 99%, it has a signiﬁcant effect on the peak value of the radial displacement. The solutions for the case of a suddenly applied constant load are found to be very close to those for the case of a gradually applied step load. The peak value of the radial displacement goes up with an increase of the degree of saturation, while the radial displacement at higher degree of saturation needs more time to reach the peak value, meaning that the degree of saturation signiﬁcantly affects the velocity of propagation of the dilatational waves. This result is consistent with that in Ref. [9]. Fig. 3 shows the time history of hoop stress at the cavity surface at various degrees of saturation under the three types of transient loads, in which σθθ(a) is the hoop stress at cavity surface. It is noted that the peak values of the hoop stress corresponding to the three types of loads increases with the degree of saturation. However, the effect of saturation is more signiﬁcant in the latter two load cases. It is also found that the impact of degree of saturation on the hoop

15

Sr=95% Sr=98% Sr=99% Sr=99.9 Sr=100%

0.045

Gs*u(a)/(f0*a)

∂e ∂r

10 t*

On the basis of the constitutive equations and the straindisplacement relations, the governing equations for the soil and ﬂuid can be written as 1 ∂e ∂ξ ∂ 2 ur ∂2 wr ð3Þ Gs ∇2 ur þðλs þ α2 M þ Gs Þ αM ¼ ρ 2 þ ρf Sr 2 r ∂r ∂r ∂t ∂t

αM M

5

0.025 0.02 0.015 0.01 0.005 0 -0.005 0

5

10

15

t*

Fig. 2. The time history of radial displacement at the cavity surface for various degrees of saturation for three types of transient loads. (a) Sudden constant load (b) Gradually applied step load and (c) Triangular pulse load.

stress is less for Sr varying from 95% to 99% in comparison with Sr varying from 99% to 100%. Fig. 4 shows the time history of pore pressure at the interface between the soil and the lining at various degrees of saturation, in which σf (a þh) is the absolute pore ﬂuid pressure at the interface between the lining and the soil; h is the thickness of the lining. It can be seen from Fig. 4(a) that the pore pressure drops sharply with time. Fig. 4(b) shows that two peak values exist in a very short time for the case of gradually applied step load. In Fig. 4(c) three peak values are observed for the case of triangular pulse load. It is noted that an increase of degree of saturation leads to a higher peak value and to a delayed arrival of the peak value.

Y. Wang et al. / Soil Dynamics and Earthquake Engineering 71 (2015) 27–30

0.3

0.7

Sr=95% Sr=98% Sr=99% Sr=99.9% Sr=100%

0.2 0.1

0.5 0.4

-0.1

σf(a+h)/f0

σθθ(a)/f0

Sr=95% Sr=98% Sr=99% Sr=99.9% Sr=100%

0.6

0

-0.2 -0.3

0.3 0.2 0.1

-0.4

0

-0.5

-0.1

-0.6 -0.7

-0.2

0

5

10

15

0

5

0.1

0.25

Sr=95% Sr=98% Sr=99% Sr=99.9% Sr=100%

0.2

0.06 0.04

0.1 0.05

0.02 0 -0.02

0

-0.04 -0.06

-0.05

-0.08

0

5

10

15

0

5

10

0.15

0.3

Sr=95% Sr=98% Sr=99% Sr=99.9% Sr=100%

0.25

Sr=95% Sr=98% Sr=99% Sr=99.9% Sr=100%

0.1 0.05 σf(a+h)/f0

0.2 0.15 0.1 0.05

15

t*

t*

σθθ(a)/f0

15

Sr=95% Sr=98% Sr=99% Sr=99.9% Sr=100%

0.08

σf(a+h)/f0

σθθ(a)/f0

0.15

0 -0.05 -0.1

0

-0.15

-0.05 -0.1

10 t*

t*

-0.1

29

-0.2

0

5

10

0

15

t*

Fig. 3. The time history of hoop stress at the cavity surface for various degrees of saturation for three types of transient loads. (a) Sudden constant load (b) Gradually applied step load (c) Triangular pulse load.

4. Conclusions The problem of a lined cylindrical cavity subjected to internal transient loads is investigated in this study by assuming the surrounding soil to be a nearly-saturated porous material and the lining to be an elastic solid. Exact solutions in the Laplace space have been derived for three types of transient loads and the solutions in the time domain have been obtained by using an efﬁcient numerical inversion method. The effects of the degree of saturation on dynamic response of the lined cylindrical cavity have been examined numerically. It is found that a small change in the degree of saturation (from 99% to 100%) may signiﬁcantly affect peak values of the stress and

5

10

15

t*

Fig. 4. The time history of pore pressure at the interface between the soil and the lining for various degrees of saturation for three types of transient loads. (a) Sudden constant load (b) Gradually applied step load (c) Triangular pulse load.

displacement as well as the arrival times of these peaks. However, the inﬂuence is found to be little when the degree of saturation varies from 95% to 99%.

Acknowledgments The research was supported by the Natural Science Foundation of China (No. 51178342 and 51428901), the Specialized Research Fund for the Doctoral Program of Higher Education, China (20130072110016) and the Key Laboratory of Geotechnical and Underground Engineering of the Ministry of Education (Tongji University) under the International Cooperation and Exchange Program (KLE-TJGE-C1301).

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