10th Asian Regional Conference of IAEG (2015)

Permanent displacement of rock slope considering degradation of slide surface during
earthquake
Shengwen QI(1),Chunling LIU(2)(1) Key Laboratory of Shale Gas and Geoengineering, Institute of Geolgoy and Geophysics, Chinese Academyof Sciences, China
E-mail: qishengwen@mail.iggcas.ac.cn(2) Aero-Geophysical Survey and Remote Sensing Center, the Ministry of Land and Resource, China
Abstract
A new approach for evaluating the permanent displacement including planar blockslide and wedge block slide of rock mass slope during earthquake is presented, which takes
degradation of sliding surface (friction weakening with sliding velocity and displacement)
during earthquake into account. The flow for putting the method into effect is put forward,
and three cases are studied. The results indicate that the approach is available andreasonable to assess the value of the permanent displacement of rock mass slope. The result
reached by traditional Newmark method(1965) is too small in general compare to the lab
test result. Traditional Newmark method(1965) is only available to evaluating the rock massslope with small permanent displacement during earthquake.
Keywords: permanent displacement, rock mass slope, degradation of slide surface, Newmark
method1.

Introduction

moderate-dense saturated sandy soil bring about
serious liquefaction and the yield acceleration
decreased obviously accompanied with theaccumulation of pore pressure during earthquake. For
this reason, many attempts were made to improve thismodel, resulting in models for obtaining theearthquake induced displacement (Franklin andChang, 1977; Makdisi and Seed, 1978; Sarma, 1975;
Sarma, 1981; Seed et al., 1969; Seed, 1979; Richard
and Elms, 1979; Lin and Whitman, 1986; Nadim andWhitman, 1983; Kramer and Smith, 1997; Rathje andBrady, 1999). Among them, Seed-Lee-Idriss method
proposed by Seed in the 19th Rankine Lecture is themost representative (Seed, 1979).
Rock slope dynamic response is controlled bydiscontinuities. Hendron (1971) introduced Newmark
method (Newmark, 1965) to predict permanent
displacement of rock block during earthquake.
Crawford (1981, 1982) did dynamic test of the rockmass and found for a flat structural surface that theearthquake yield acceleration changes with the
cumulative displacement and rate of the rockmovement, rather than the constant. On this basis, thecalculating method of earthquake permanent
displacement of planar rock block was put forward,
which took the cumulative displacement and rate of
the rock movement of the discontinuities seismic
yield acceleration into consideration. For a granite

Earthquake can trigger a large number of earth
and or rock slides, some of which are capable ofcausing deaths and major damages to major structures,
important roadways, dam reservoir etc. For example,
more than 15000 landslides induced by 2008
Wenchuan 7.9 Mw earthquake, claimed deaths over
20,000, damaged road 220,000 km and 2,900 bridges(Yin et al., 2009; Qi et al, 2010).
Newmark (1965) has suggested a famous
approach to obtain the earthquake induced
displacement suitable for rigid plastic materials in theFifth Rankine Lecture. He realized that whether theartificial earth fill dam was stable or not depended on
the deformation during earthquake, rather than
traditional slope safety factor less than 1.0 and what
related to the seismic deformation directly was thechanges of the stress history, rather than the
maximum stress. This approach has been usedsuccessfully to predict the surface displacements ofbanks of dry, cohensionless soils subjected to known
series of base motions (Goodman and Seed, 1966;
Bustamente, 1965). However, the method assumes
that yield acceleration is constant and the material isrigid, which does not meet the real behavior of rockmass.
The dams filled with loose sand and1

10th Asian Regional Conference of IAEG (2015)

mAgcos(θ - β) + mgsinβ - Ntanφ = ma

block placed on a flat plane in lab test, Wang andZhang (1982) found that the dynamic friction
coefficient is lessening gradually with increasing of
the displacement and rate of the block movement,
Based on this, they put forward a method to evaluate
the permanent displacement of the planar rock block.
Discontinuities are not smooth but fluctuant in therock mass (Patton, 1966). Dynamic tests of the rockmass showed that the discontinuities deteriorated
gradually under the cycle load (Hutson, 1987; Plesha,
1987; Hutson and Dowding, 1990; Jing et al., 1993;
Kana et al., 1996; Fox et al., 1998; Lee et al., 2001;
Homand et al., 2001; Yang et al., 2001), the asperitiesof the discontinuities plane reduced and undulation
angles deteriorated gradually. The asperities of
discontinuities are closely related to the shear strengthof surface (Barton, 1971, 1973, 1976), thus thedegradation of the discontinuities would affect thepermanent displacement of rock mass slope directly.
This paper presents a new approach to evaluate
the permanent displacement of rock slope including
planar block slide and wedge block slide, which
considers the degradation of asperities of sliding
surface during earthquake. The procedure of themethod is put forward and some cases are illustrated
to proof its effectiveness.
2.

(2)

where φ is the angle of internal friction, x is thedisplacement of the slide block, a is the accelerationof the slide block. Then we can get
a = Agcos(θ - β) + gsinβ - [gcosβ + Agsin(θ (3)β)]tanφ
When a = 0, A becomes the yield acceleration
coefficient, Ac:
Ac = (cosβtanφ - sinβ) / [cos(θ - β) - sin(θ - β)(4)tanφ] = sin(β -φ) / cos(θ - β +φ)a = [Agcos(θ - β +φ) + gsin(β -φ)] / cosφ =(5)gcos(θ - β +φ)·(A - Ac) / cosφ
where,

φ = φb + αk(6)

φb is the basic friction angle of the slide surface, αkis the first order asperity angle of the slide surface(Patton, 1966), as shown in Fig. 2.

Rock slope model

2.1 Rock block slide modelFig. 1 is the rock block slope model whichassumes that the mass of slide block is m, the dip
angle of the slide surface is β and the seismic
acceleration is Ag where A is seismic coefficient. The
acute angle formed by seismic force and horizontal
line is θ. The coordinate system is shown in Fig. 1.

Fig. 2 Conception model of one and two grades
asperity angle (Patton, 1966)
Substituting Eq. (6) into Eq. (5) gives
Fig. 1 Rock block slope modela = Agcos(θ - β) + gsinβ - [gcosβ + Agsin(θ β)]·tan(φb + αk) = Agcos(θ - β) + gsinβ - [gcosβ +Agsin(θ - β)]·[tanφb + tanαk] / [1 - tanφb·tanαk] (7)Considering the equilibrium conditions of force inY direction, we can get(1)

For flat slide surface αk = 0, thenφ=φb. Assume
that a = 0, and we can get

Where N is supportive force on which the slide planegives the slide.
In the X direction, we can havetanφb = [cos(θ - β)Ac + sinβcosβ] / [cosβ + sin(θ(8)- β)Ac]

N = mgcosβ + mAgsin(θ - β)

2.2 Wedge block slide model2

10th Asian Regional Conference of IAEG (2015)

Wedge block slide failure is a common failuremode for rock slope. Static problem for wedge blockslide was discussed by Hoek and Brown (1977).
Fig. 3 is a wedge block formed by two penetrative
discontinuities JA and JB of rock slope. Assuming that
under a seismic acceleration A, the wedge block only
vibrates in the vertical plane crossing the intersection
line of the two discontinuities. The angle ofinclination of intersection line is β, the mass of wedge
is m, the normal force on JA and JB are RA and RB,
both the internal friction angle of two discontinuities
are φ. The coordinate system is shown in Fig. 3.
In the X direction, we can have

Assuming that a = 0 and αk = 0, the Eq. (13) canbe expressed
tanφb = [sinβsinγ + Accos(θ - β)sinγ] / [sinδcosβ +(15)Acsin(θ - β)sinδ]3.

3.1 Law of degradation of the asperity angle (αk) of
slide surface
Under the role of the cyclic shear, asperities of theslide surface are deteriorated, as αk decreased
gradually. According to Plesha (1987), we can havemgsinβ + mAgcos(θ - β) - (RA + RB)tanφ = ma (9)Consider equilibrium conditions of the plane
normal to the intersection line of two discontinuities,
and assume N is component of mass force on thisplane, we can get
N = mgcosβ + mAgsin(θ - β)αk = (αk)0exp(-cWp)(16)

Where (αk)0 is the initial asperity angle, c is the joint
damage coefficient (m2/J) that is a test constant and
Wp is the plastic work.
According to Hutson (1987) and Hutson andDowding (1990),(10)

According to the equilibrium conditions, we canget

RA = [Nsin(δ + γ)] / sin2γ
RB = [Nsin(δ - γ)] / sin2γ

Approach to evaluate permanent
displacement
considering
degradation of slide surface
during earthquake(11)

c = 0.114JRC(σn / σc)(17)where σc is the uniaxial compressive strength of thematerial, σn is the normal force of the sliding surface
during the shearing process.

Thus,

RA + RB = Nsinδ / sinγ = mgsinδ[cosβ + Asin(θ β)] / sinγ
(12)3.2 Law of degradation of the basic friction angle(φb) of slide surface
For flat discontinuities, the asperity angle (αk)
equals to 0, and the yield acceleration Ac is dependent
on the basic friction angle ofφb.
Through extensive researches, Crawford andCurran (1982) found that for the flat discontinuities
the yield acceleration Ac was the function of the
cumulative displacement x and displacement rate v.
Thus it can be expressed as Ac(x, v). For the effect of
cumulative displacement, Crawford and Curran(1982) adopted the model of post-peak constantdisplacement which is rigid and strain-softening todescribe as

Substituting Eq. (12) into Eq. (9) gives
a = gsinβ + Agcos(θ - β) - gsinδ[cosβ + Asin(θ (13)β)]tanφ / sinγ
Substituting Eq. (6) into Eq. (13) gives

Ac(x) = Am[1 - (1 - p)(x / x0)], x ≤ x0
Ac(x) = Amp, x > x0(18)where p is the reduction coefficient which can be
determined by tests, Am is the maximum accelerationof input seismic waves. According to extensive
researches, Crawford and Curran (1982) believed thatx0 takes 50mm appropriately for most of the rock.
According to research of Crawford and Curran(1981, 1982), the rate effect of yield acceleration Ac(v)Fig. 3 Wedge slope modela = gsinβ + Agcos(θ - β) - gsinδ[cosβ + Asin(θ β)]·[tanφb + tanαk] / sinγ / [1 - tanφb·tanαk] (14)3

10th Asian Regional Conference of IAEG (2015)

can be expressed as

Ac(v) = Am[1.0 ± 0.20log(v / v0)], v ≥ v0
Ac(v) = Am, v < v0

(1) Substitute initial asperity angle of sliding
surface (αk)0 and initial basic friction angle of sliding
surface (φb)0 into Eq. (7) or Eq. (14) assume a = 0
we can get Ac(0, 0) of the block in Fig. 1 or Fig. 3;(2) Discretize acceleration time-travel history inton∆t as A(0), A(∆t), A(2∆t), ···, A(m1∆t), ···,
A(m2∆t), ···, A(n∆t);(3) Substitute A(0), initial asperity angle ofsliding surface (αk)0, initial basic friction angle ofsliding surface (φb)0 and Ac(0, 0) into Eq. (7) or Eq.
(14), we can get (a)0, and then figure out (v)1. If(v)1 > 0, continue next step (4); if not, forward
timestep, update corresponded time history of A(t),and continue step (3) till (v)m1 ≥ 0;(4) Integrate and obtain m1 steps’ displacementsof (x)m1;(5) According to displacement (x)m1, we can get
the work done by slide force and obtain thedeteriorated asperity angle of sliding surface (αk)m1;(6) Substituting (x)m1 and (v)m1 into Eq. (21) andEq. (8) or Eq. (15) gives tan(φb)m1;(7) Substituting (αk)m1 and (φb)m1 into Eq. (7) or
Eq. (14) gives m1 steps’ acceleration (a)m1, and thenfigure out (v)m1+1. If (v)m1+1 ≥ 0, continue step (4); if
not, forward timestep, update corresponded time
history of A(t), and continue step (3) till (v)m1+m2 ≥ 0;
(8) Repeating the above steps (3)-(7) till the nstep, we can get the slope seismic permanent
displacements (x)n = (x)n-1 + (v)n∆t.

(19)

Where “+” stands for different material. For weakmaterials (such as dolomite), take “+” while for hardmaterials (such as black granite, sandstone) take “-“.
Crawford and Curran (1982) found that v0 is usually
10mm/s through abundant test results.
If not consider cumulative displacement effectand rate effect, we can get Ac(x, v) = Am according to
Eq. (18) and Eq. (19). That means that yieldacceleration is equal to maximum seismic
acceleration, and this is obviously unreasonable. Eq.
(18) and Eq. (19) proposed by Crawford and Curran(1982) have conceptual errors, and Am should bechanged into initial acceleration Ac(0, 0). Meanwhile,
Crawford and Curran (1982) did not consider about
cumulative displacement effect and rate effect for Eq.
(18) and Eq. (19) at the same time. Therefore, thepaper considers initially the rate effect and revised Eq.
(18) asAc(v, x) = Ac(0, x)[1.0 ± 0.20log(v / v0)], v ≥ v0
(20)
Ac(v, x) = A(0, x), v < v0
Then considering the effects of cumulative
displacement again, Eq. (20) can be expressed as

Ac(v, x) = pAc(0, 0), x > 50mm, v < 10mm/s
Ac(v, x) = Ac(0, 0)[1 - (1 - p)(x / 50)], x ≤ 50mm,v < 10mm/s
Ac(v, x) = pAc(0, 0)[1 ± 0.20log(v / 10)], x >50mm, v ≥ 10mm/s
Ac(v, x) = Ac(0, 0)[1 - (1 - p)(x / 50)][1 ±(21)
0.20log(v / 10)], x ≤ 50mm, v ≥ 10mm/s

5.

(1) Case 1
The case is shown as Fig. 1 and is from Wang andZhang (1982), where  = 20°, vibration direction ishorizontal (θ = 0), acceleration time travel history of
bedrock is shown in Fig. 4 with maximum
acceleration of 10.20m/s2, sliding surface is flat andwithout asperity (αk = 0), tanφb =0.75 and time stepΔt = 0.001s. The lithology of the bed rock is granite.

Thus substituting Eq. (21) into Eq. (8) and Eq.
(15), we can get the law of degradation of the basic
friction angle (φb) of slide surface for models shownin Fig. 1 and Fig. 3 respectively.
Substituting Eq. (16), Eq. (17), Eq. (21) and Eq.
(8) into Eq. (7) gives acceleration time travel historyof the slide. By double integration, we can getseismic permanent displacement of planar sliding
block as shown in Fig. 1 considering the degradation
process of slide surface during earthquake.
Substituting Eq. (16), Eq. (17) and Eq. (21) andEq. (15) into Eq. (14), we can obtain motion
acceleration of wedge under dynamic load. By double
integration, we can obtain seismic permanent
displacement of rock wedge sliding block as shown in
Fig. 3 considering the degradation process of the slide
surface during earthquake.
4.

Cases

1500

1000

s)/
/s500
m
c(
n
o
it
a
re 0
elc
c
A
-500

-1000
0.00

0.50

1.00

Time (s)

1.50

2.00

2.50

Fig. 4 Acceleration time travel history of Bedrock forcase 1 and case 2

Procedures for calculating seismic permanent
displacement of rock slope

Take p = 1 (not considering the deteriorated effectof the sliding surface ), p = 0.65 and p = 0.70 in Eq.

410th Asian Regional Conference of IAEG (2015)

tan(φb), and the rock density is 2.2×103 kg/m3.
Acceleration time travel history of bedrock is shownin Fig. 7, with the peak acceleration of 0.1g,
horizontal vibration direction (θ = 0) and time step Δtof 0.005s. The calculated results can refer to Fig.8.
As it can be seen in the figure, the results oftraditional Newmark method(1965) is still near 2mm
less than the results (p = 1) of the paper. What not toconsider degradation of discontinuities has significant
impact on the seismic permanent displacement of theslope.

(21) respectively, and use procedures as detailed in
section 4, we can evaluate the permanent
displacement of the block as shown in Fig. 5. As it
can be seen in the figure, the result reached based on
p = 0.65 (805mm) or p = 0.70 (638mm) is muchcloser the lab tested result (730 mm) given by Wang
and Zhang (1982). While the result calculated byNewmark (1965) or the result reached based on p = 1
is much less that the tested result (see Fig. 5). The
traditional Newmark method(1965) which didn’t
consider the deteriorated effect of the sliding surface
will give much less estimated seismic permanent
displacement of the slope.

Fig. 7 Acceleration time travel history of Bedrock forcase 316

Fig. 5 Comparison of permanent displacement of case1 reached by different measures(2) Case 2
The case is identical to case 1 except for  = 30°,
and the calculated result is shown in Fig. 6.

14

Permanent displacement (mm)

12
10
8
6
4
2
0
-2
0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

Time (s)Newmark (1965)

This approach (p=1)

This approach (p=0.45)

Fig. 8 Comparison of permanent displacement of case
3 reached by Newmark (1965) and the approachpresented in the paper
6.

Discussions and conclusions

The paper puts forward the algorithm forevaluating seismic permanent displacement of theplane block sliding and wedge block sliding of therock slope which is taken degradation of the sliding
surface into consideration. Through three cases
studies, the paper shows that:(1) Three cases indicate that the degradation of
the sliding surface has significant impact on seismic
permanent displacement of the slope. The
degradation of sliding surface must be taken into
account when estimating the slope seismic permanent
displacement. The method presented in the paper isable to estimate the value of slope seismic permanent
displacement well, and the minimum value can begiven when p = 1.0, while the maximum can be taken

Fig. 6 Comparison of permanent displacement of case2 reached by different measures
As it can be seen from Fig. 6, the result reachedbased on p = 0.1 is much closer to the tested result
given by Wang and Zhang (1982). The result ofNewmark method(1965) is much smaller than the test
result, and even result given based on p = 1, see Fig.
6.
(3) Case 3
The case is a wedge shaped block as shown in Fig.
3, where  = 43°,  = 70°,  = 35°, the discontinuity
JA and JB have the same friction parameters with c = 0,

5

4.00

10th Asian Regional Conference of IAEG (2015)

when p = 0.1.
(2) The traditional Newmark method(1965) does
not consider degradation of sliding surface. The
seismic permanent displacement of the slope given by
this method is too small. For stable slope or one withsmall value of seismic slope permanent displacement,
result given by the traditional Newmark method(1965)
is a little closer to the result given by the method
presented in the paper which considered thedegradation of sliding surface. Case 3 clearly
indicated this point.
Acknowledgementsof Engineers, Nuclear Creating Group.
Hoek, E. and Bray, J. (1977): Rock SlopeEngineering, Institution of Mining and Metallurgy.
Homand, F., Belem, T. and Souley, M. (2001):
Friction and degradation of rock joints surfaces
under shear loads, International Journal for
Numerical
andAnalytical
MethodsinGeomechanics, Vol. 25, pp. 973-999.
Hutson, R.W. (1987): Preparation of duplicate rockjoints and their changing dilatancy under cyclic
shear, Ph. D Thesis, Northwestern University,
Evanston, Illinois.
Hutson, R.W. and Dowding, C.H. (1990): Joint
asperity degradation during cyclic shear, Int. J.
Rock Mech.. Min. Sci. & Geomech. Abstr., Vol. 27,
pp. 109-119.
Jing, L., Stephansson, O. and Nordlund, E. (1993):
Study of rock joints under cyclic loading conditions,Rock Mechanics and Rock Engineering, Vol. 26, pp.
215-232.
Kana, D.D., Fox, D.J. and Hsiung, S.M. (1996):
Interlock/friction model for dynamic shear response
in natural jointed rock, International Journal ofRock Mechanics and Mining Sciences andGeomechanics Abstracts, Vol. 33, pp. 371-386.
Kramer, S.L. and Smith, M.W. (1997): Modified
Newmark model for seismic displacements ofcompliant slopes, J. Geotech. Engrg. ASCE, Vol.
123, pp. 635-644.
Kramer, S.L. and Lindwall, N.W. (2004):
Dimensionality and directionality effects in
Newmark sliding block analyses, J. Geotech. &Geoenvir. Engrn., Vol. 130, pp. 303-315.
Lee, H.S., Park, Y.J., Cho, T.F. and You, K.H.
(2001): Influence of asperity degradation on themechanical behavior of rough rock joints undercyclic shear loading, Int. J. Rock Mech.. Min. Sci.,
Vol. 38, pp. 967-980.
Lin, J.S. and Whitman, R.V. (1986): Earthquake
induced displacements of sliding blocks, J. Geotech.
Engng. Div., Vol. 112, pp. 44-59.
Makdisi, F.I. and Seed, H.B. (1978): Simplified
procedure for estimating dam and embankment
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Engng. Div., Vol. 104, pp. 569-587.
Nadim, F. and Whitman, R.V. (1983): Seismicallyinduced movement of retaining walls, J. Geotech.
Engng. Div., Vol. 109, pp. 915-931.
Newmark, N.M. (1965): Effects of earthquakes ondams and embankments, Geotechnique, Vol. 15, pp.
139-160.
Patton, F.D. (1966): Multiple modes of shear failurein rock and related materials, Ph.D Thesis Univ. of
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Thanks for kind assistance from academician
Wang S. Foundations under the grant of No.
KZCX2-EW-QN108 from Chinese Academy of
Sciences and Nos. 41322020 and 41172272 from
National Nature Science Foundation of China are also
appreciated.
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