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Design of waterfront retaining wall for the passive case under earthquake and tsunami
Applied Ocean Research 29 (2007) 37–44 www.elsevier.com/locate/apor
Design of waterfront retaining wall for the passive case under earthquake and tsunami Deepankar Choudhury ∗ , Syed Mohd. Ahmad Department of Civil Engineering, Indian Institute of Technology Bombay, Mumbai, India Received 12 February 2007; received in revised form 5 August 2007; accepted 5 August 2007 Available online 7 September 2007
Abstract The paper pertains to a study of analysing a waterfront retaining wall under the combined action of tsunami and earthquake forces. The stability of the waterfront retaining wall is assessed in terms of its sliding and overturning modes of failure. Pseudo-static approach has been used for the calculation of the passive seismic earth pressure. Hydrodynamic pressure generated behind the backfill due to shaking of the wet backfill soil is considered in the analysis. Tsunami force is considered to be an additional force acting on the upstream face of the wall and is calculated using a simple formula. It is observed that the factor of safety in sliding mode of failure decreases by about 70% when the ratio of tsunami water height to initial water height is changed from 0.375 to 1.125. Variations of different parameters involved in the analysis suggest sensitiveness of the factor of safety against both the sliding and overturning modes of failure of the wall and provides a better guideline for design. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Seismic passive earth pressure; Wall inertia; Hydrodynamic pressure; Factor of safety; Sliding; Overturning
1. Introduction Tsunamis triggered by an earthquake (or otherwise too) cause severe damage to the waterfront retaining structures. From the point of view of a geotechnical engineer, one of the important structures situated at the waterfront, which gets affected due to the combined action of an earthquake and a tsunami is a retaining wall. Sheth et al. [1] had reported the recent events of tsunami and subsequent damages to the waterfront retaining structures at the Indian sub-continent in December 2004. The combination of earthquake and tsunami forces on a waterfront retaining wall severely challenges its stability, both in terms of sliding and overturning modes of failure. A typical waterfront retaining wall subjected to earthquake and tsunami is basically under: the seismic forces, the hydrostatic and hydrodynamic pressures, and the tsunami force. Due to the complications of the combination of these forces simultaneously, the design of the waterfront retaining wall becomes more complicated and challenging to the civil ∗ Corresponding author. Tel.: +91 22 2576 7335 (O), 8335 (R); fax: +91 22 2576 7302. E-mail address: [email protected] (D. Choudhury).
c 2007 Elsevier Ltd. All rights reserved. 0141-1187/$ - see front matter doi:10.1016/j.apor.2007.08.001
engineers. In spite of being so important and relevant to the civil engineering fraternity, the available literature suggests that the area of study is not thoroughly researched and whatever has previously been done in this area is generally confined to the consideration of the above mentioned forces individually i.e. one at a time acting on the retaining structures. For example, many researchers in the past considered the problem of retaining wall to compute the seismic lateral earth pressure and had given different solutions based on different approaches. The work of Okabe [2] and Mononobe and Matsuo [3] by adopting the pseudo-static seismic accelerations (commonly known as Mononobe–Okabe method; see Kramer [4]) is the pioneering work in this field and is generally used worldwide. Approximate elastic solutions were given by Matsuo and Ohara [5]; while limit equilibrium analyses were considered by Richards and Elms [6]; Choudhury and Nimbalkar [7,8]. In the recent past, Morrison and Ebeling [9], Choudhury and Subba Rao [10], Choudhury et al. [11], Subba Rao and Choudhury [12], Choudhury and Nimbalkar [13], Nimbalkar and Choudhury [14] proposed the seismic design of a retaining wall under passive earth pressure condition. However, these theories considered only the seismic earth pressures, acting on the retaining wall subjected
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Notations b, H width and height of the wall ht tsunami water height h wd , h wu height of the water on downstream and upstream sides of the wall Fd , Fr total driving and resisting forces FSo , FSs factor of safety against overturning and sliding modes of failure FSoEM , FSsEM factor of safety against overturning and sliding modes of failure by Ebeling and Morrison’s approach [16] kh , kv horizontal and vertical seismic acceleration coefficients K a constant = 0.5K pe γ¯ (1 − kv ) (1 − ru ) K0 a constant = 0.5K 0pe γ¯ (1 − kv ) (1 − ru ) K pe seismic passive earth pressure coefficient from Subba Rao and Choudhury’s approach [12] K 0pe seismic passive earth pressure coefficient calculated using Mononobe–Okabe’s approach (see Kramer [4]) Pdyn hydrodynamic pressure Ppe seismic passive earth resistance/force Pstd , Pstu hydrostatic pressure on downstream and upstream sides of the wall Pt force due to tsunami ru pore pressure ratio Ww weight of the wall y point of application of Ppe δ, φ wall and soil friction angle γc specific weight of concrete γd , γsat dry and saturated specific weight of the soil γw specific weight of water γwe , γ¯ equivalent specific weight of water and soil due to submergence µ coefficient of base friction to earthquake only with dry-soil condition. Chakrabarti et al. [15] were probably the first researchers who took into account both the seismic lateral earth pressure and additional water pressure generated due to seismic shaking of the water on the quay wall. But again, this theory dealt with only the hydrodynamic pressure on the waterfront retaining wall, in addition to the seismic lateral earth pressure and the tsunami force was not considered in the analysis. Other researchers like Ebeling and Morrison [16], Kim et al. [17], and Nozu et al. [18] also studied the effect of hydrodynamic pressure on the waterfront retaining wall along with the seismic earth pressure under earthquake condition. Solutions for the interaction of tsunami with coastal defence structures were presented by Silva et al. [19], while Bullock et al. [20] detailed the wave impact characteristics by conducting the experiments for impact pressures and forces breaking on the vertical and inclined walls and concluded that the largest impact pressure occur near the still water level. Considering the action of the tsunami force on the waterfront retaining wall, there have been studies in the
Fig. 1. Typical waterfront retaining wall subjected to different forces during earthquake and tsunami.
past, but unlike the case of hydrodynamic pressure, the seismic lateral earth pressure was not considered in the analysis. For example, Hinwood [21] discussed the effect of tsunami forces and suggested expression for the calculation of the tsunami force on the retaining structure and other coastal structures, but did not consider the effect of seismic earth pressures. Simplified expressions for the calculation of tsunami force on the vertical face of rigid walls were also given by CRATER [22] and Yeh [23]. Hence the combined effect of the seismic forces, hydrostatic and hydrodynamic forces and tsunami force acting on the waterfront retaining wall during earthquake and tsunami is not yet well investigated. The present study describes a simplified design approach in which the stability of a typical waterfront retaining wall subjected to the earthquake and tsunami forces is checked in terms of its sliding and overturning modes of failure under passive case of earth pressure, which is one of the most critical case for failure of waterfront retaining wall. 2. Method of analysis A typical waterfront retaining wall with vertical face having width ‘b’ and height ‘H ’ is shown in Fig. 1. It retains backfill to its full height on one side, referred as ‘downstream side’ and water to a height of ‘h wu ’ on the other side, called as ‘upstream side’ of the wall. The ground surface of the backfill (submerged to a level ‘h wd ’) is assumed to be horizontal. During the occurrence of a tsunami (which may be due to an earthquake or otherwise too), there may be a rise in the water level on the upstream face of the wall; this rise is denoted as ‘h t ’ in Fig. 1. The wall is subjected to the lateral seismic earth pressure, seismic inertia force on the wall, hydrodynamic pressure (on the downstream face), hydrostatic pressure (both on the upstream and downstream sides), and force due to tsunami on the upstream face. The respective points of application of these forces and pressures are shown in Fig. 1, while the calculation of these above mentioned forces is detailed in the following sections. 2.1. Seismic passive earth pressure During the tsunami, the wall moves towards the downstream side of the wall, i.e. towards the backfill, which is going
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to be the most critical case in terms of the stability of the waterfront retaining wall. Hence, for the similar type of the wall movement, passive case of earth pressure is going to be generated on the wall. To account for the effects of submergence of the backfill and excess pore pressure, the expression for the total seismic passive earth pressure on the wall is modified after Ebeling and Morrison [16] and Kramer [4] and is given as
as shown in Fig. 1. It acts at a height of 0.4h wd from the base of the wall [16].
1 K pe H 2 γ¯ (1 − kv )(1 − ru ), (1) 2 where, γ¯ is the equivalent unit weight of the backfill soil due to submergence and is given as ! h wd 2 h wd 2 γsat + 1 − γd . (2) γ¯ = H H
1 γw (h wu )2 . (4) 2 For calculating the hydrostatic pressure on the wall from the downstream side (Pstd ), γw in Eq. (4) is replaced by γwe [16], which can be calculated as,
The values of the seismic passive earth pressure coefficient (K pe ) are taken from Choudhury and Subba Rao [10] and Subba Rao and Choudhury [12] with the point of application of the seismic passive forces acting on the wall by considering the composite (logspiral + planar) failure surface instead of the planar failure surface considered by Mononobe–Okabe (see Kramer [4]). Because, as proposed by Terzaghi [24], for the passive case of earth pressure, mostly the curved rupture surface is generated instead of planar failure surface. It is to be noted that the pore pressure ratio (ru ), which is defined as the ratio of excess pore pressure to the initial vertical stress, incorporated in Eq. (1) above is a simplified way [16] of simulating the effect of the excess pore pressure generated due to cyclic shaking of the soil during an earthquake.
Thus,
Ppe =
2.2. Seismic inertia force on the wall Due to the earthquake, wall will also experience additional seismic forces due to the inertia of the wall, which need to be considered in the analysis. These seismic wall inertia forces are assumed to be pseudo-static in nature and are given by kv Ww and kh Ww for the vertical and horizontal directions respectively. Depending upon the direction of the vertical and horizontal seismic acceleration coefficients (kv and kh ), there could be four possible combinations of these pseudo-static wall inertia forces, however only the critical combination for the design is shown in Fig. 1. 2.3. Hydrodynamic pressure Due to the seismic shaking of the standing water, the total hydrodynamic pressure (Pdyn ) will be generated, which is calculated by using the following formula proposed by Westergaard [25], 7 kh γw (h wd )2 . (3) 12 This hydrodynamic pressure (see Kramer [4]) is assumed to act on the downstream side of the wall. For analysing the worst case with respect to the stability of the wall, this pressure is considered along with the disturbing forces coming on the wall Pdyn =
2.4. Hydrostatic pressure Due to the presence of water, both on the upstream and downstream faces of the wall, there will be hydrostatic pressures acting on the wall. On upstream face it is calculated as, Pstu =
γwe = γw + (γ¯ − γw ) ru .
Pstd =
(5)
1 γwe (h wd )2 . 2
Pstu and Pstd respectively acting at heights of from the base of the wall.
(6) h wu 3
and
h wd 3
2.5. Force due to tsunami Upstream face of the wall will be subjected to a tsunami. As per CRATER [22] this total tsunami force per unit length of the wall is given as, Pt = 4.5γw (h t )2 .
(7)
This force acts at mid-height of the tsunami water height (i.e., acting at h t /2). 3. Stability of the wall Adopting the limit equilibrium method, the stability of the wall is checked for both the sliding and overturning modes of failure. The expressions of factor of safety for each of these two modes of failure of the wall are derived as given in the following sections. 3.1. Factor of safety against sliding mode of failure Considering the equilibrium of all the forces acting in the horizontal direction (Fig. 1), one can write Total resisting force, Fr = µ Ww − kv Ww − Ppe sin δ + Pstd + Ppe cos δ. (8) Total driving force, Fd = Pstu + Pt + kh Ww + Pdyn .
(9)
The factor of safety (FSs ) of the wall against sliding mode of failure can now be written as, FSs = =
Fr Fd µ Ww − kv Ww − Ppe sin δ + Pstd + Ppe cos δ Pstu + Pt + kh Ww + Pdyn
(10)
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where, µ = coefficient of base friction, and Ww = weight of the retaining wall = bH γc . Substituting different values from Eqs. (1), (3), (4), (6) and (7) in Eq. (10), and writing in nondimensional form (for the generalized design purpose), the final simplified expression for the factor of safety of the retaining wall against sliding mode of failure (FSs ) is, 2 γc (1 − kv ) − K sin δ + 12 γwe hHwd + K cos δ FSs = , 2 2 2 h wd h wu ht 1 b 7 γ + 4.5γ γ + k γ + k h we w c h w H 2 h wu H 12 H µ
b H
(11) where, K is a constant = 0.5K pe γ¯ (1 − kv ) (1 − ru ).
Fig. 2. Factor of safety against sliding mode of failure for different h t / h wu values.
3.2. Factor of safety against overturning mode of failure Similarly, by considering that the total seismic passive earth pressure (Ppe ) acts at height ‘y’ above the base of the wall, the expression for the factor of safety against overturning mode of failure of the wall (FSo ) in simplified non-dimensional form can be written as, h wd 3 b 2 γ (1 − k ) + 1 γ + K y cos δ c v H H 6 we FSo = . 2 3 h wd 3 ht h wu 1 h t + 1 + 1 k b γ + 2.8 k γ 1 γ + 4.5γ w h wu H H 6 w 2 h wu 2 h H c 12 h w 1 2
(12) In Eqs. (11) and (12), the values of the seismic passive earth pressure coefficient (K pe ) and the point of application (y) of the total seismic passive earth pressure (Ppe ) are dependent on different combinations of soil friction angle (φ), wall friction angle (δ) and vertical and horizontal seismic acceleration coefficients (kv and kh ). Both of these values are taken from Choudhury and Subba Rao [10] and Subba Rao and Choudhury [12]. Eqs. (11) and (12) can be easily used for the design of the section of the waterfront retaining wall subjected to the combined seismic (both passive earth pressure and wall inertia), hydrostatic, hydrodynamic and tsunami forces. It can be mentioned that for computation of the hydrodynamic pressure and the tsunami force, already existing individual empirical formulae are used in the present study. For example, the calculation of the hydrodynamic pressure is carried out by the proposed method of Westergaard [25] and subsequent use of the same method by other researchers like Ebeling and Morrison [16], Kramer [4], and Choudhury and Ahmad [26]. Similarly, the chosen empirical formula for calculation of the tsunami force in the present study is taken from CRATER [22], which was also used by The US Army Corps of Engineers [27]. Though the above empirical relations are suffering from the inherent limitations like any other empirical relation, but still can be used for the estimation of the tsunami force and hydrodynamic pressure on the vertical wall. However, for the estimation of seismic earth pressure, conventional closed-form solution is used (see Choudhury and Subba Rao [10], Subba Rao and Choudhury [12]) and for the wall inertia component the procedure adopted by Ebeling and Morrison [16] is used individually in the present study. Considering the limitations of the above individual methods, in
Fig. 3. Factor of safety against overturning mode of failure for different h t / h wu values.
the present study, an attempt to find the stability of the wall under the combined action of all the above forces has been made. Results obtained from the proposed methodology shows the simple design technique for the typical waterfront retaining wall subjected to combined action of earthquake and tsunami forces. 4. Results and discussions The stability of the wall can be predicted by substituting the practically acceptable values of the different parameters involved in Eqs. (11) and (12). The values and ranges of these parameters for the present study are shown in Table 1. Effects of these parameters on both the sliding as well as the overturning stability are presented in Figs. 2–13. 4.1. Effect of the tsunami water height (h t ) Figs. 2 and 3 respectively present the effect of tsunami water height on the factor of safety against sliding and overturning modes of failure of the wall. It is observed from these Figs. 2 and 3 that as the ratio of tsunami water height to the water height on the upstream face (h t / h wu ) increases, there is a significant decrease in the value of the factor of safety. As an illustration from Fig. 2, for the typical values of b/H = 0.2, h wu /H = 0.4, h wd /H = 0.75, φ = 30◦ , δ = φ/2,
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Fig. 4. Factor of safety against sliding mode of failure for different h wd /H values.
Fig. 7. Factor of safety against overturning mode of failure for different φ values.
Fig. 5. Factor of safety against overturning mode of failure for different h wd /H values.
Fig. 8. Factor of safety against sliding mode of failure for different δ values.
Fig. 6. Factor of safety against sliding mode of failure for different φ values.
Fig. 9. Factor of safety against overturning mode of failure for different δ values.
Table 1 Values/Range of different parameters chosen for the present study Parameter
Value/Range
b/H h t / h wu h wd /H φ (◦ ) δ (◦ ) kh kv ru γc , γsat , γd , γw (kN/m3 )
0.2 0 (no tsunami), 0.375, 0.750, 1.125, 1.500 0 (dry), 0.25, 0.50, 0.75, 1.00 (fully wet) 25, 30, 35, 40 −φ/2, 0, φ/2 0, 0.1, 0.2, 0.3, 0.4 0, kh /2, kh 0, 0.2, 0.4 25, 19, 16 and 10 respectively
kv = kh /2, ru = 0.2, kh = 0.2, and h t / h wu = 0.375, the factor of safety against sliding mode of failure of the wall is 6.68, while for the same combination of all the parameters it gets reduced to around 2.00 when the value of h t / h wu is increased to 1.125. Hence, about 70% decrease in the factor of safety against the sliding mode of failure occurs for an increase in h t / h wu from 0.375 to 1.125. A similar trend is noted against the overturning mode of failure also (Fig. 3). This important observation prompts the judicious selection of the wall section of the retaining wall situated at the waterfront as the increase in tsunami water height seriously challenges the wall stability.
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Fig. 10. Factor of safety against sliding mode of failure for different kv values.
Fig. 13. Factor of safety against overturning mode of failure for different ru values.
4.3. Effect of submergence of the backfill (h wd )
Fig. 11. Factor of safety against overturning mode of failure for different kv values.
The backfill submergence gives rise to the additional hydrostatic and hydrodynamic pressures on the wall. For the present case, any additional water pressure coming from the backfill side would tend to act as a stabilizing force for the passive case. This is presented in Figs. 4 and 5 for the sliding and overturning modes of failure of the wall respectively. As can be observed from Fig. 4, the factor of safety against sliding mode of failure of the wall for typical values of b/H = 0.2, h wu /H = 0.4, h t / h ws = 1.125, φ = 30◦ , δ = φ/2, kh = 0.1, kv = kh /2, ru = 0.2 and h wd /H = 0.0 (i.e. dry backfill) is 2.18; while it increases to 2.94 when the ratio of h wd /H = 1.00 (fully wet backfill). The similar trend is observed for the overturning mode of failure (Fig. 5). 4.4. Effect of soil friction angle (φ)
Fig. 12. Factor of safety against sliding mode of failure for different ru values.
4.2. Effect of horizontal seismic acceleration coefficient (kh ) The effect of the horizontal seismic acceleration coefficient on the factor of safety against sliding and overturning modes of failure of the wall can be interpreted from Figs. 2 and 3. It is observed that for the typical values of b/H = 0.2, h wu /H = 0.4, h wd /H = 0.75, φ = 30◦ , δ = φ/2, kv = kh /2, ru = 0.2, h t / h wu = 0.75, and for kh = 0.1, the factor of safety against sliding mode of failure is 4.94 (Fig. 2), and it gets reduced to 1.64 when kh is increased to a value of 0.4. Similarly for the overturning mode of failure and for the same combination of the parameters, the factor of safety for kh = 0.1 is 3.69, reducing to a value of 1.07 for an increase in the kh value to 0.4.
Figs. 6 and 7 respectively show the variation of factor of safety against the sliding and overturning modes of failure for different φ values. With the increase in the value of φ from 25◦ to 40◦ , there is 83% increase in the factor of safety against the sliding mode of failure (Fig. 6) for kh = 0.1, b/H = 0.2, h wu /H = 0.4, h t / h wu = 1.125, h wd /H = 0.75, δ = φ/2, kv = kh /2, ru = 0.2. The rate of decrease in factor of safety value with decrease in the value of soil friction angle φ is nearly constant for all values of kh and is true for the overturning case also (Fig. 7). 4.5. Effect of wall friction angle (δ) Figs. 8 and 9 illustrate the effect of wall friction angle (δ) on the sliding and overturning stability of the wall. It is observed that as δ increases from −φ/2 to φ/2, the stability of the wall increases. For example, for kh = 0.1, b/H = 0.2, h ws /H = 0.4, h t / h wu = 1.125, h wd /H = 0.75, φ = 30◦ , kv = kh /2 and ru = 0.2 in Fig. 8, the factor of safety against the sliding mode is 1.83 when δ is −φ/2 and increases to 2.61 when δ is changed to φ/2, i.e., an increase of about 42.6% in the factor of safety value for a change in δ from −φ/2 to φ/2. For the overturning mode (Fig. 9), the similar trend is observed.
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4.6. Effect of the vertical seismic acceleration coefficient (kv ) The vertical seismic acceleration coefficient (kv ) has significant effect on the factor of safety against sliding and overturning modes of failure of the wall. In Fig. 10 for an increase in the vertical seismic acceleration coefficient (kv ) from 0 to kh , the factor of safety against sliding mode reduces (for kh = 0.3, b/H = 0.2, h wu /H = 0.4, h t / h wu = 1.125, h wd /H = 0.75, φ = 30◦ , δ = φ/2, ru = 0.2) by about 43%. Overturning mode of failure of the wall shows the similar behaviour as can be seen from Fig. 11. 4.7. Effect of the pore pressure ratio (ru ) Effect of the pore pressure ratio (ru ) on the stability against sliding and overturning modes of failure of the wall is shown in Figs. 12 and 13 respectively. For the sliding case, it is observed that for kh = 0.2, b/H = 0.2, h wu /H = 0.4, h t / h wu = 1.125, h wd /H = 0.75, φ = 30◦ , δ = φ/2, kv = kh /2, when ru is increased from 0 to 0.4, the factor of safety decreases from 2.4 to 1.7. A similar trend is observed for the overturning mode of failure also (Fig. 13). 5. Comparison of results Literature review showed that no previous work has been done in which the combination of both the tsunami and earthquake forces are considered simultaneously on the waterfront retaining wall during earthquake and tsunami. However, as mentioned earlier, the work done by Ebeling and Morrison [16] presents the design of a typical waterfront retaining wall subjected to the seismic passive earth pressure, seismic wall inertia force and the hydrodynamic and hydrostatic pressures. But no tsunami force was considered in that analysis. Rewriting the expressions in terms of non-dimensional parameters, the factors of safety in both the sliding and overturning modes of failures using the methodology proposed by Ebeling and Morrison [16] can be given by Eqs. (13) and (14). The expression for the factor of safety in sliding mode of failure (FSsEM ) is, µ FSsEM =
b H
2 γc (1 − kv ) − K sin δ + 21 γwe hHwd + K 0 cos δ . 2 2 h wu 1 7 kh γw hHwd + kh Hb γc + 12 2 γw H
(13) Similarly, the factor of safety for the overturning mode of failure (FSoEM ) is, 3 h wd 1 b 2 1 γ − k + γ + K 0 y cos δ (1 ) c v 2 H 6 we H FSoEM = 3 3 , h wd h wu 1 2.8 b 1 γ k γ + k γ + w h c h w 6 H 2 H 12 H (14) where, K 0 is a constant = 0.5K 0pe γ¯ (1 − kv ) (1 − ru ) with K 0pe is the seismic passive earth pressure coefficient calculated using the Mononobe–Okabe’s pseudo-static method (see Kramer [4]). However, the use of planar rupture surface for the computation of passive earth pressure is erroneous (see
43
Terzaghi [24], Choudhury and Subba Rao [10] and Subba Rao and Choudhury [12]). Correcting this error, in the present study, the curved rupture surface is used for computation of seismic passive earth pressure. Hence, it can be seen that, the equations of factor of safety under sliding mode of failure viz. Eqs. (11) and (13) by the present study and the same given by Ebeling and Morrison [16] are found to be identical in the absence of tsunami force (i.e. h t = 0). Similarly, the equations of factor of safety under overturning mode of failure viz. Eqs. (12) and (14) by the present study and the same given by Ebeling and Morrison [16] are identical in absence of tsunami force (i.e. h t = 0) as expected. The only variation in results can be attributed to the fact of using different failure surfaces in the two analyses. Hence, it validates the use of the proposed methodology for the design of waterfront retaining wall under earthquake condition, however, the consideration of the combined effect of the earthquake and tsunami force for the present design methodology seems to produce unique result as already mentioned. 6. Conclusions The present study gives a design methodology for analysing the waterfront retaining wall subjected to both the earthquake and tsunami forces simultaneously against the sliding and overturning modes of failure. Through a simple analytical approach, the present work shows that the stability of the waterfront retaining wall decreases significantly when subjected to the earthquake and tsunami forces. From the typical parametric study, it is observed that for an increase in h t / h wu from 0.375 to 1.125, there is a decrease of about 70% in the factor of safety value against the sliding mode of failure of the wall. The factor of safety against sliding mode of failure gets reduced from 4.94 to 1.64 when an increase in the horizontal seismic acceleration coefficient (kh ) is made from 0.1 to 0.4; while there is a 43% reduction in the factor of safety value against sliding mode when vertical seismic acceleration coefficient (kv ) is increased from 0 to kh . The soil friction angle (φ) of the backfill also has significant effect on the stability of the wall as can be observed from a change in the value of the factor of safety against sliding mode of failure by about 83% for a change in φ from 25◦ to 40◦ ; while on changing the soil friction angle (δ) from −φ/2 to φ/2, the factor of safety against sliding mode of failure increases by 42.6%. On increasing the pore pressure ratio (ru ) from 0 to 0.4, the factor of safety against mode of failure gets reduced from 2.4 to 1.7. For all the typical values, similar trends are observed for the overturning mode of failure of the wall. Present method matches well with the existing method under earthquake condition, however, due to non availability of the results under the combined action of earthquake and tsunami, the present study seems to develop a new simplified design technique for the waterfront retaining wall. References [1] Sheth A, Sanyal S, Jaiswal A, Gandhi P. Effects of the December 2004 Indian ocean tsunami on the Indian Mainland. Earthquake Spectra 2006; 22(S3):s435–73.
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seismic design criteria for category I cofferdams. Nuclear Engineering and Design 1978;45:277–83. Ebeling RM, Morrison EE Jr. The seismic design of waterfront retaining structures. US Army Technical Report ITL-92-11, Washington DC, 1992. Kim S, Jang I, Chung C, Kim M. Evaluation of seismic displacements of quay walls. In: Proceedings of the international conference on geotechnical engineering for disaster mitigation & rehabilitation. 2005. p. 84–93. Nozu A, Ichii K, Sugano T. Seismic design of port structures. Journal of Japan Association for Earthquake Engineering 2004;4(3-special issue): 195–208. Silva R, Losada IJ, Losada MA. Reflection and transmission of tsunami waves by coastal structures. Applied Ocean Research 2000;22:215–23. Bullock GN, Obhrai C, Peregrine DH, Bredmose H. Violent breaking wave impacts. Part 1: Results from large-scale regular wave tests on vertical and sloping walls. Coastal Engineering 2007; Available online since 9 February 2007. Hinwood J. Design for tsunamis — coastal engineering considerations. Structural Engineering International 2005;3:189–93. CRATER: Coastal risk analysis of tsunamis and environmental remediation. Italian Ministry for the Environment and the Territory (IMET) 2006. Yeh H. Maximum fluid forces in the tsunami runup zone. Journal of Waterway, Port, Coastal, and Ocean Engineering 2006;132(6):495–500. Terzaghi K. Theoretical soil mechanics. New York: Wiley; 1943. Westergaard HM. Water pressures on dams during earthquakes. Transactions, ASCE 1933;98:418–33. Choudhury D, Ahmad SM. Stability of waterfront retaining wall subjected to pseudo-static earthquake forces. Ocean Engineering 2007;34(14–15): 1947–54. US Army Corps of Engineers. Wave forces on a wall shoreward of the still-water level. Technical Note III-29, Coastal Engineering Research Centre 1990.
Design of waterfront retaining wall for the passive case under earthquake and tsunami Deepankar Choudhury ∗ , Syed Mohd. Ahmad Department of Civil Engineering, Indian Institute of Technology Bombay, Mumbai, India Received 12 February 2007; received in revised form 5 August 2007; accepted 5 August 2007 Available online 7 September 2007
Abstract The paper pertains to a study of analysing a waterfront retaining wall under the combined action of tsunami and earthquake forces. The stability of the waterfront retaining wall is assessed in terms of its sliding and overturning modes of failure. Pseudo-static approach has been used for the calculation of the passive seismic earth pressure. Hydrodynamic pressure generated behind the backfill due to shaking of the wet backfill soil is considered in the analysis. Tsunami force is considered to be an additional force acting on the upstream face of the wall and is calculated using a simple formula. It is observed that the factor of safety in sliding mode of failure decreases by about 70% when the ratio of tsunami water height to initial water height is changed from 0.375 to 1.125. Variations of different parameters involved in the analysis suggest sensitiveness of the factor of safety against both the sliding and overturning modes of failure of the wall and provides a better guideline for design. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Seismic passive earth pressure; Wall inertia; Hydrodynamic pressure; Factor of safety; Sliding; Overturning
1. Introduction Tsunamis triggered by an earthquake (or otherwise too) cause severe damage to the waterfront retaining structures. From the point of view of a geotechnical engineer, one of the important structures situated at the waterfront, which gets affected due to the combined action of an earthquake and a tsunami is a retaining wall. Sheth et al. [1] had reported the recent events of tsunami and subsequent damages to the waterfront retaining structures at the Indian sub-continent in December 2004. The combination of earthquake and tsunami forces on a waterfront retaining wall severely challenges its stability, both in terms of sliding and overturning modes of failure. A typical waterfront retaining wall subjected to earthquake and tsunami is basically under: the seismic forces, the hydrostatic and hydrodynamic pressures, and the tsunami force. Due to the complications of the combination of these forces simultaneously, the design of the waterfront retaining wall becomes more complicated and challenging to the civil ∗ Corresponding author. Tel.: +91 22 2576 7335 (O), 8335 (R); fax: +91 22 2576 7302. E-mail address: [email protected] (D. Choudhury).
c 2007 Elsevier Ltd. All rights reserved. 0141-1187/$ - see front matter doi:10.1016/j.apor.2007.08.001
engineers. In spite of being so important and relevant to the civil engineering fraternity, the available literature suggests that the area of study is not thoroughly researched and whatever has previously been done in this area is generally confined to the consideration of the above mentioned forces individually i.e. one at a time acting on the retaining structures. For example, many researchers in the past considered the problem of retaining wall to compute the seismic lateral earth pressure and had given different solutions based on different approaches. The work of Okabe [2] and Mononobe and Matsuo [3] by adopting the pseudo-static seismic accelerations (commonly known as Mononobe–Okabe method; see Kramer [4]) is the pioneering work in this field and is generally used worldwide. Approximate elastic solutions were given by Matsuo and Ohara [5]; while limit equilibrium analyses were considered by Richards and Elms [6]; Choudhury and Nimbalkar [7,8]. In the recent past, Morrison and Ebeling [9], Choudhury and Subba Rao [10], Choudhury et al. [11], Subba Rao and Choudhury [12], Choudhury and Nimbalkar [13], Nimbalkar and Choudhury [14] proposed the seismic design of a retaining wall under passive earth pressure condition. However, these theories considered only the seismic earth pressures, acting on the retaining wall subjected
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Notations b, H width and height of the wall ht tsunami water height h wd , h wu height of the water on downstream and upstream sides of the wall Fd , Fr total driving and resisting forces FSo , FSs factor of safety against overturning and sliding modes of failure FSoEM , FSsEM factor of safety against overturning and sliding modes of failure by Ebeling and Morrison’s approach [16] kh , kv horizontal and vertical seismic acceleration coefficients K a constant = 0.5K pe γ¯ (1 − kv ) (1 − ru ) K0 a constant = 0.5K 0pe γ¯ (1 − kv ) (1 − ru ) K pe seismic passive earth pressure coefficient from Subba Rao and Choudhury’s approach [12] K 0pe seismic passive earth pressure coefficient calculated using Mononobe–Okabe’s approach (see Kramer [4]) Pdyn hydrodynamic pressure Ppe seismic passive earth resistance/force Pstd , Pstu hydrostatic pressure on downstream and upstream sides of the wall Pt force due to tsunami ru pore pressure ratio Ww weight of the wall y point of application of Ppe δ, φ wall and soil friction angle γc specific weight of concrete γd , γsat dry and saturated specific weight of the soil γw specific weight of water γwe , γ¯ equivalent specific weight of water and soil due to submergence µ coefficient of base friction to earthquake only with dry-soil condition. Chakrabarti et al. [15] were probably the first researchers who took into account both the seismic lateral earth pressure and additional water pressure generated due to seismic shaking of the water on the quay wall. But again, this theory dealt with only the hydrodynamic pressure on the waterfront retaining wall, in addition to the seismic lateral earth pressure and the tsunami force was not considered in the analysis. Other researchers like Ebeling and Morrison [16], Kim et al. [17], and Nozu et al. [18] also studied the effect of hydrodynamic pressure on the waterfront retaining wall along with the seismic earth pressure under earthquake condition. Solutions for the interaction of tsunami with coastal defence structures were presented by Silva et al. [19], while Bullock et al. [20] detailed the wave impact characteristics by conducting the experiments for impact pressures and forces breaking on the vertical and inclined walls and concluded that the largest impact pressure occur near the still water level. Considering the action of the tsunami force on the waterfront retaining wall, there have been studies in the
Fig. 1. Typical waterfront retaining wall subjected to different forces during earthquake and tsunami.
past, but unlike the case of hydrodynamic pressure, the seismic lateral earth pressure was not considered in the analysis. For example, Hinwood [21] discussed the effect of tsunami forces and suggested expression for the calculation of the tsunami force on the retaining structure and other coastal structures, but did not consider the effect of seismic earth pressures. Simplified expressions for the calculation of tsunami force on the vertical face of rigid walls were also given by CRATER [22] and Yeh [23]. Hence the combined effect of the seismic forces, hydrostatic and hydrodynamic forces and tsunami force acting on the waterfront retaining wall during earthquake and tsunami is not yet well investigated. The present study describes a simplified design approach in which the stability of a typical waterfront retaining wall subjected to the earthquake and tsunami forces is checked in terms of its sliding and overturning modes of failure under passive case of earth pressure, which is one of the most critical case for failure of waterfront retaining wall. 2. Method of analysis A typical waterfront retaining wall with vertical face having width ‘b’ and height ‘H ’ is shown in Fig. 1. It retains backfill to its full height on one side, referred as ‘downstream side’ and water to a height of ‘h wu ’ on the other side, called as ‘upstream side’ of the wall. The ground surface of the backfill (submerged to a level ‘h wd ’) is assumed to be horizontal. During the occurrence of a tsunami (which may be due to an earthquake or otherwise too), there may be a rise in the water level on the upstream face of the wall; this rise is denoted as ‘h t ’ in Fig. 1. The wall is subjected to the lateral seismic earth pressure, seismic inertia force on the wall, hydrodynamic pressure (on the downstream face), hydrostatic pressure (both on the upstream and downstream sides), and force due to tsunami on the upstream face. The respective points of application of these forces and pressures are shown in Fig. 1, while the calculation of these above mentioned forces is detailed in the following sections. 2.1. Seismic passive earth pressure During the tsunami, the wall moves towards the downstream side of the wall, i.e. towards the backfill, which is going
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to be the most critical case in terms of the stability of the waterfront retaining wall. Hence, for the similar type of the wall movement, passive case of earth pressure is going to be generated on the wall. To account for the effects of submergence of the backfill and excess pore pressure, the expression for the total seismic passive earth pressure on the wall is modified after Ebeling and Morrison [16] and Kramer [4] and is given as
as shown in Fig. 1. It acts at a height of 0.4h wd from the base of the wall [16].
1 K pe H 2 γ¯ (1 − kv )(1 − ru ), (1) 2 where, γ¯ is the equivalent unit weight of the backfill soil due to submergence and is given as ! h wd 2 h wd 2 γsat + 1 − γd . (2) γ¯ = H H
1 γw (h wu )2 . (4) 2 For calculating the hydrostatic pressure on the wall from the downstream side (Pstd ), γw in Eq. (4) is replaced by γwe [16], which can be calculated as,
The values of the seismic passive earth pressure coefficient (K pe ) are taken from Choudhury and Subba Rao [10] and Subba Rao and Choudhury [12] with the point of application of the seismic passive forces acting on the wall by considering the composite (logspiral + planar) failure surface instead of the planar failure surface considered by Mononobe–Okabe (see Kramer [4]). Because, as proposed by Terzaghi [24], for the passive case of earth pressure, mostly the curved rupture surface is generated instead of planar failure surface. It is to be noted that the pore pressure ratio (ru ), which is defined as the ratio of excess pore pressure to the initial vertical stress, incorporated in Eq. (1) above is a simplified way [16] of simulating the effect of the excess pore pressure generated due to cyclic shaking of the soil during an earthquake.
Thus,
Ppe =
2.2. Seismic inertia force on the wall Due to the earthquake, wall will also experience additional seismic forces due to the inertia of the wall, which need to be considered in the analysis. These seismic wall inertia forces are assumed to be pseudo-static in nature and are given by kv Ww and kh Ww for the vertical and horizontal directions respectively. Depending upon the direction of the vertical and horizontal seismic acceleration coefficients (kv and kh ), there could be four possible combinations of these pseudo-static wall inertia forces, however only the critical combination for the design is shown in Fig. 1. 2.3. Hydrodynamic pressure Due to the seismic shaking of the standing water, the total hydrodynamic pressure (Pdyn ) will be generated, which is calculated by using the following formula proposed by Westergaard [25], 7 kh γw (h wd )2 . (3) 12 This hydrodynamic pressure (see Kramer [4]) is assumed to act on the downstream side of the wall. For analysing the worst case with respect to the stability of the wall, this pressure is considered along with the disturbing forces coming on the wall Pdyn =
2.4. Hydrostatic pressure Due to the presence of water, both on the upstream and downstream faces of the wall, there will be hydrostatic pressures acting on the wall. On upstream face it is calculated as, Pstu =
γwe = γw + (γ¯ − γw ) ru .
Pstd =
(5)
1 γwe (h wd )2 . 2
Pstu and Pstd respectively acting at heights of from the base of the wall.
(6) h wu 3
and
h wd 3
2.5. Force due to tsunami Upstream face of the wall will be subjected to a tsunami. As per CRATER [22] this total tsunami force per unit length of the wall is given as, Pt = 4.5γw (h t )2 .
(7)
This force acts at mid-height of the tsunami water height (i.e., acting at h t /2). 3. Stability of the wall Adopting the limit equilibrium method, the stability of the wall is checked for both the sliding and overturning modes of failure. The expressions of factor of safety for each of these two modes of failure of the wall are derived as given in the following sections. 3.1. Factor of safety against sliding mode of failure Considering the equilibrium of all the forces acting in the horizontal direction (Fig. 1), one can write Total resisting force, Fr = µ Ww − kv Ww − Ppe sin δ + Pstd + Ppe cos δ. (8) Total driving force, Fd = Pstu + Pt + kh Ww + Pdyn .
(9)
The factor of safety (FSs ) of the wall against sliding mode of failure can now be written as, FSs = =
Fr Fd µ Ww − kv Ww − Ppe sin δ + Pstd + Ppe cos δ Pstu + Pt + kh Ww + Pdyn
(10)
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where, µ = coefficient of base friction, and Ww = weight of the retaining wall = bH γc . Substituting different values from Eqs. (1), (3), (4), (6) and (7) in Eq. (10), and writing in nondimensional form (for the generalized design purpose), the final simplified expression for the factor of safety of the retaining wall against sliding mode of failure (FSs ) is, 2 γc (1 − kv ) − K sin δ + 12 γwe hHwd + K cos δ FSs = , 2 2 2 h wd h wu ht 1 b 7 γ + 4.5γ γ + k γ + k h we w c h w H 2 h wu H 12 H µ
b H
(11) where, K is a constant = 0.5K pe γ¯ (1 − kv ) (1 − ru ).
Fig. 2. Factor of safety against sliding mode of failure for different h t / h wu values.
3.2. Factor of safety against overturning mode of failure Similarly, by considering that the total seismic passive earth pressure (Ppe ) acts at height ‘y’ above the base of the wall, the expression for the factor of safety against overturning mode of failure of the wall (FSo ) in simplified non-dimensional form can be written as, h wd 3 b 2 γ (1 − k ) + 1 γ + K y cos δ c v H H 6 we FSo = . 2 3 h wd 3 ht h wu 1 h t + 1 + 1 k b γ + 2.8 k γ 1 γ + 4.5γ w h wu H H 6 w 2 h wu 2 h H c 12 h w 1 2
(12) In Eqs. (11) and (12), the values of the seismic passive earth pressure coefficient (K pe ) and the point of application (y) of the total seismic passive earth pressure (Ppe ) are dependent on different combinations of soil friction angle (φ), wall friction angle (δ) and vertical and horizontal seismic acceleration coefficients (kv and kh ). Both of these values are taken from Choudhury and Subba Rao [10] and Subba Rao and Choudhury [12]. Eqs. (11) and (12) can be easily used for the design of the section of the waterfront retaining wall subjected to the combined seismic (both passive earth pressure and wall inertia), hydrostatic, hydrodynamic and tsunami forces. It can be mentioned that for computation of the hydrodynamic pressure and the tsunami force, already existing individual empirical formulae are used in the present study. For example, the calculation of the hydrodynamic pressure is carried out by the proposed method of Westergaard [25] and subsequent use of the same method by other researchers like Ebeling and Morrison [16], Kramer [4], and Choudhury and Ahmad [26]. Similarly, the chosen empirical formula for calculation of the tsunami force in the present study is taken from CRATER [22], which was also used by The US Army Corps of Engineers [27]. Though the above empirical relations are suffering from the inherent limitations like any other empirical relation, but still can be used for the estimation of the tsunami force and hydrodynamic pressure on the vertical wall. However, for the estimation of seismic earth pressure, conventional closed-form solution is used (see Choudhury and Subba Rao [10], Subba Rao and Choudhury [12]) and for the wall inertia component the procedure adopted by Ebeling and Morrison [16] is used individually in the present study. Considering the limitations of the above individual methods, in
Fig. 3. Factor of safety against overturning mode of failure for different h t / h wu values.
the present study, an attempt to find the stability of the wall under the combined action of all the above forces has been made. Results obtained from the proposed methodology shows the simple design technique for the typical waterfront retaining wall subjected to combined action of earthquake and tsunami forces. 4. Results and discussions The stability of the wall can be predicted by substituting the practically acceptable values of the different parameters involved in Eqs. (11) and (12). The values and ranges of these parameters for the present study are shown in Table 1. Effects of these parameters on both the sliding as well as the overturning stability are presented in Figs. 2–13. 4.1. Effect of the tsunami water height (h t ) Figs. 2 and 3 respectively present the effect of tsunami water height on the factor of safety against sliding and overturning modes of failure of the wall. It is observed from these Figs. 2 and 3 that as the ratio of tsunami water height to the water height on the upstream face (h t / h wu ) increases, there is a significant decrease in the value of the factor of safety. As an illustration from Fig. 2, for the typical values of b/H = 0.2, h wu /H = 0.4, h wd /H = 0.75, φ = 30◦ , δ = φ/2,
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Fig. 4. Factor of safety against sliding mode of failure for different h wd /H values.
Fig. 7. Factor of safety against overturning mode of failure for different φ values.
Fig. 5. Factor of safety against overturning mode of failure for different h wd /H values.
Fig. 8. Factor of safety against sliding mode of failure for different δ values.
Fig. 6. Factor of safety against sliding mode of failure for different φ values.
Fig. 9. Factor of safety against overturning mode of failure for different δ values.
Table 1 Values/Range of different parameters chosen for the present study Parameter
Value/Range
b/H h t / h wu h wd /H φ (◦ ) δ (◦ ) kh kv ru γc , γsat , γd , γw (kN/m3 )
0.2 0 (no tsunami), 0.375, 0.750, 1.125, 1.500 0 (dry), 0.25, 0.50, 0.75, 1.00 (fully wet) 25, 30, 35, 40 −φ/2, 0, φ/2 0, 0.1, 0.2, 0.3, 0.4 0, kh /2, kh 0, 0.2, 0.4 25, 19, 16 and 10 respectively
kv = kh /2, ru = 0.2, kh = 0.2, and h t / h wu = 0.375, the factor of safety against sliding mode of failure of the wall is 6.68, while for the same combination of all the parameters it gets reduced to around 2.00 when the value of h t / h wu is increased to 1.125. Hence, about 70% decrease in the factor of safety against the sliding mode of failure occurs for an increase in h t / h wu from 0.375 to 1.125. A similar trend is noted against the overturning mode of failure also (Fig. 3). This important observation prompts the judicious selection of the wall section of the retaining wall situated at the waterfront as the increase in tsunami water height seriously challenges the wall stability.
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Fig. 10. Factor of safety against sliding mode of failure for different kv values.
Fig. 13. Factor of safety against overturning mode of failure for different ru values.
4.3. Effect of submergence of the backfill (h wd )
Fig. 11. Factor of safety against overturning mode of failure for different kv values.
The backfill submergence gives rise to the additional hydrostatic and hydrodynamic pressures on the wall. For the present case, any additional water pressure coming from the backfill side would tend to act as a stabilizing force for the passive case. This is presented in Figs. 4 and 5 for the sliding and overturning modes of failure of the wall respectively. As can be observed from Fig. 4, the factor of safety against sliding mode of failure of the wall for typical values of b/H = 0.2, h wu /H = 0.4, h t / h ws = 1.125, φ = 30◦ , δ = φ/2, kh = 0.1, kv = kh /2, ru = 0.2 and h wd /H = 0.0 (i.e. dry backfill) is 2.18; while it increases to 2.94 when the ratio of h wd /H = 1.00 (fully wet backfill). The similar trend is observed for the overturning mode of failure (Fig. 5). 4.4. Effect of soil friction angle (φ)
Fig. 12. Factor of safety against sliding mode of failure for different ru values.
4.2. Effect of horizontal seismic acceleration coefficient (kh ) The effect of the horizontal seismic acceleration coefficient on the factor of safety against sliding and overturning modes of failure of the wall can be interpreted from Figs. 2 and 3. It is observed that for the typical values of b/H = 0.2, h wu /H = 0.4, h wd /H = 0.75, φ = 30◦ , δ = φ/2, kv = kh /2, ru = 0.2, h t / h wu = 0.75, and for kh = 0.1, the factor of safety against sliding mode of failure is 4.94 (Fig. 2), and it gets reduced to 1.64 when kh is increased to a value of 0.4. Similarly for the overturning mode of failure and for the same combination of the parameters, the factor of safety for kh = 0.1 is 3.69, reducing to a value of 1.07 for an increase in the kh value to 0.4.
Figs. 6 and 7 respectively show the variation of factor of safety against the sliding and overturning modes of failure for different φ values. With the increase in the value of φ from 25◦ to 40◦ , there is 83% increase in the factor of safety against the sliding mode of failure (Fig. 6) for kh = 0.1, b/H = 0.2, h wu /H = 0.4, h t / h wu = 1.125, h wd /H = 0.75, δ = φ/2, kv = kh /2, ru = 0.2. The rate of decrease in factor of safety value with decrease in the value of soil friction angle φ is nearly constant for all values of kh and is true for the overturning case also (Fig. 7). 4.5. Effect of wall friction angle (δ) Figs. 8 and 9 illustrate the effect of wall friction angle (δ) on the sliding and overturning stability of the wall. It is observed that as δ increases from −φ/2 to φ/2, the stability of the wall increases. For example, for kh = 0.1, b/H = 0.2, h ws /H = 0.4, h t / h wu = 1.125, h wd /H = 0.75, φ = 30◦ , kv = kh /2 and ru = 0.2 in Fig. 8, the factor of safety against the sliding mode is 1.83 when δ is −φ/2 and increases to 2.61 when δ is changed to φ/2, i.e., an increase of about 42.6% in the factor of safety value for a change in δ from −φ/2 to φ/2. For the overturning mode (Fig. 9), the similar trend is observed.
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4.6. Effect of the vertical seismic acceleration coefficient (kv ) The vertical seismic acceleration coefficient (kv ) has significant effect on the factor of safety against sliding and overturning modes of failure of the wall. In Fig. 10 for an increase in the vertical seismic acceleration coefficient (kv ) from 0 to kh , the factor of safety against sliding mode reduces (for kh = 0.3, b/H = 0.2, h wu /H = 0.4, h t / h wu = 1.125, h wd /H = 0.75, φ = 30◦ , δ = φ/2, ru = 0.2) by about 43%. Overturning mode of failure of the wall shows the similar behaviour as can be seen from Fig. 11. 4.7. Effect of the pore pressure ratio (ru ) Effect of the pore pressure ratio (ru ) on the stability against sliding and overturning modes of failure of the wall is shown in Figs. 12 and 13 respectively. For the sliding case, it is observed that for kh = 0.2, b/H = 0.2, h wu /H = 0.4, h t / h wu = 1.125, h wd /H = 0.75, φ = 30◦ , δ = φ/2, kv = kh /2, when ru is increased from 0 to 0.4, the factor of safety decreases from 2.4 to 1.7. A similar trend is observed for the overturning mode of failure also (Fig. 13). 5. Comparison of results Literature review showed that no previous work has been done in which the combination of both the tsunami and earthquake forces are considered simultaneously on the waterfront retaining wall during earthquake and tsunami. However, as mentioned earlier, the work done by Ebeling and Morrison [16] presents the design of a typical waterfront retaining wall subjected to the seismic passive earth pressure, seismic wall inertia force and the hydrodynamic and hydrostatic pressures. But no tsunami force was considered in that analysis. Rewriting the expressions in terms of non-dimensional parameters, the factors of safety in both the sliding and overturning modes of failures using the methodology proposed by Ebeling and Morrison [16] can be given by Eqs. (13) and (14). The expression for the factor of safety in sliding mode of failure (FSsEM ) is, µ FSsEM =
b H
2 γc (1 − kv ) − K sin δ + 21 γwe hHwd + K 0 cos δ . 2 2 h wu 1 7 kh γw hHwd + kh Hb γc + 12 2 γw H
(13) Similarly, the factor of safety for the overturning mode of failure (FSoEM ) is, 3 h wd 1 b 2 1 γ − k + γ + K 0 y cos δ (1 ) c v 2 H 6 we H FSoEM = 3 3 , h wd h wu 1 2.8 b 1 γ k γ + k γ + w h c h w 6 H 2 H 12 H (14) where, K 0 is a constant = 0.5K 0pe γ¯ (1 − kv ) (1 − ru ) with K 0pe is the seismic passive earth pressure coefficient calculated using the Mononobe–Okabe’s pseudo-static method (see Kramer [4]). However, the use of planar rupture surface for the computation of passive earth pressure is erroneous (see
43
Terzaghi [24], Choudhury and Subba Rao [10] and Subba Rao and Choudhury [12]). Correcting this error, in the present study, the curved rupture surface is used for computation of seismic passive earth pressure. Hence, it can be seen that, the equations of factor of safety under sliding mode of failure viz. Eqs. (11) and (13) by the present study and the same given by Ebeling and Morrison [16] are found to be identical in the absence of tsunami force (i.e. h t = 0). Similarly, the equations of factor of safety under overturning mode of failure viz. Eqs. (12) and (14) by the present study and the same given by Ebeling and Morrison [16] are identical in absence of tsunami force (i.e. h t = 0) as expected. The only variation in results can be attributed to the fact of using different failure surfaces in the two analyses. Hence, it validates the use of the proposed methodology for the design of waterfront retaining wall under earthquake condition, however, the consideration of the combined effect of the earthquake and tsunami force for the present design methodology seems to produce unique result as already mentioned. 6. Conclusions The present study gives a design methodology for analysing the waterfront retaining wall subjected to both the earthquake and tsunami forces simultaneously against the sliding and overturning modes of failure. Through a simple analytical approach, the present work shows that the stability of the waterfront retaining wall decreases significantly when subjected to the earthquake and tsunami forces. From the typical parametric study, it is observed that for an increase in h t / h wu from 0.375 to 1.125, there is a decrease of about 70% in the factor of safety value against the sliding mode of failure of the wall. The factor of safety against sliding mode of failure gets reduced from 4.94 to 1.64 when an increase in the horizontal seismic acceleration coefficient (kh ) is made from 0.1 to 0.4; while there is a 43% reduction in the factor of safety value against sliding mode when vertical seismic acceleration coefficient (kv ) is increased from 0 to kh . The soil friction angle (φ) of the backfill also has significant effect on the stability of the wall as can be observed from a change in the value of the factor of safety against sliding mode of failure by about 83% for a change in φ from 25◦ to 40◦ ; while on changing the soil friction angle (δ) from −φ/2 to φ/2, the factor of safety against sliding mode of failure increases by 42.6%. On increasing the pore pressure ratio (ru ) from 0 to 0.4, the factor of safety against mode of failure gets reduced from 2.4 to 1.7. For all the typical values, similar trends are observed for the overturning mode of failure of the wall. Present method matches well with the existing method under earthquake condition, however, due to non availability of the results under the combined action of earthquake and tsunami, the present study seems to develop a new simplified design technique for the waterfront retaining wall. References [1] Sheth A, Sanyal S, Jaiswal A, Gandhi P. Effects of the December 2004 Indian ocean tsunami on the Indian Mainland. Earthquake Spectra 2006; 22(S3):s435–73.
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