9oct vajont lecture hatzor

International Conference Vajont 1963 – 2013
Thoughts and analyses after 50 years since the catastrophic landslide
October 8 – 10, 2013 , Padua, Italy

Thermally vs. Seismically
Induced Block Displacements in Jointed
Rock Slopes
Yossef H. Hatzor
Lemkin Professor of Rock Mechanics
Dept. of Geological and Environmental Sciences
Ben-Gurion University of the Negev, Israel

Talk Outline
Seismic Triggering: Verifications and Validations
 Single Plane Sliding
 Double Plane Sliding
 Shaking Table Experiments
 Velocity Dependent Friction Degradation

Climatic Triggering: Field Monitoring and Theoretical Model
 Masada World Heritage Site as a Field Station
 Monitored Rock Mass Response to Thermal Fluctuations
 Thermally Induced Ratcheting Mechanism
 Seismic vs. Thermal Triggering
Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy

2

Dynamic Sliding:
Verifications and Validations

3

Single Plane Sliding

Photo courtesy of R. E. Goodman


4

Verification of Single Plane Sliding
=22 0
=30 0
=35 0

Input
motion (m/s2)

10

DDA

Analytic

DDA

Analytic

DDA

Analytic

Input Motion

5
0

relative error (%)

Dynamic sliding under
gravitational load only was
studied originally by Mary
McLaughlin in her PhD thesis
(1996) (Berkeley) and
consequent publications with
Sitar and Doolin 2004 - 2006.
Sinusoidal input first studied by
Hatzor and Feintuch (2001),
IJRMMS. Improved 2D solution
presented by Kamai and Hatzor
(2008), NAG. Ning and Zhao
(2012), NAG (From NTU)
recently published a very
detailed study of this problem.

Displacement of
upper block (m)

-5
12
8
4
0
100
10
1
0.1
0.01

0

1

2

3

Time (sec)

4

5


6

5

Double Plane Sliding

Photo courtesy of G. H. Shi

Photo courtesy of R. E. Goodman

6

Verification of Dynamic Wedge Sliding
2.5

y

Wedge parameters:

4

2

Displacement (m)

x

6

P1=52/063, P2=52/296

2
1.5
0
1
-2
0.5

=30o

Analytical solution proposed and 3D
DDA validation performed by BakunMazor, Hatzor, and Glaser (2012),
NAG.

Relative Error (%)

0

DDA validation originally investigated
by Yeung M. R., Jiang Q. H., Sun N.,
(2003) IJRMMS using physical tests.

-6

100

100

10

10

1

1

0.1

0.1
0

A

-4

Analytical
3D-DDA
Input Motion (y)

Horizontal Input motion (m/s2)

z

0.4

0.8

1.2

1.6

2

Time (sec)


7

Acceleration of
Shaking Table, g

Shaking Table Experiments
0.2
0

Accumulated Displacement, mm
Relative Error, %

A

-0.2

60

Shaking Table
3D DDA ; Loading mode
3D DDA ; Displacement mode

40
20

B
0

10000

Erel; Loading Mode
Erel; Displacement Mode

1000
100

C

10
0

ti
10

20

30

40

Time, sec


8

Rate Dependent Friction
Upper Shear Box

Normal Cylinder
Shear System
20 cm

Concrete Samples

4
n

3
n

2

n
n

0
0

A

= 4.03 MPa

= 3.00 MPa

= 1.97 MPa

1
n

v = 0.002 mm/sec

= 5.02 MPa

Shear Stress, MPa

Shear Stress, MPa

4

Lower Shear Box
Roller Bearing
Shear Cylinder

v = 0.020 mm/sec

3

v = 0.100 mm/sec

2
1

= 0.98 MPa

0
1
2
3
4
Shear Displacement, mm

5

0

B

2
4
Normal Stress, MPa

6


9

100
80
60
40
20
0

Acceleration
of Shaking
Table, g

Up. Block
Velocity,
mm/sec

Up. Block Accum.
Displacement, mm

Observed Block “Run-out”
Measured
Calculated

= 29.0
= 29.5

o

o

60
40
20
0
0.2
0

-0.2
0

A

= 27.0o

20

40

60

Time, sec


10

Friction Angle Degradation
0.7

Direct shear test results

0.6

0.4
0.3
0.0001 0.001 0.01 0.1
1
Velocity, mm/sec

= -0.0174 * ln(V) + 0.5668
R2 = 0.948

0.6

0.5

10

100

Friction Coefficient

Friction Coefficient

0.7

= -0.0079 * ln(V) + 0.6071
R2 = 0.909
0.5

Shaking table experiments
0.4
Shaking Table
Coulomb-Mohr
0.3
0.0001

0.001

0.01

0.1
Velocity, mm/sec

1

10

100

Conclusion: frictional resistance of geological sliding interfaces may exhibit both velocity dependence as well
as degradation as a function of velocity and/or displacement. This is particularly relevant for dynamic
analysis of landslides, where sliding is assumed to have taken place under high velocities. Therefore, a
modification of DDA to account for friction angle degradation is called for. This has already been suggested
by Sitar et al. (2005), JGGE –ASCE; a new approach has recently been proposed by LZ Wang et al. (in press),
COGE (from Zhejiang University).

11

THERMAL VS. SEISMIC TRIGGERING

12

Masada World Heritage Site as Field
Station


13

Six month monitoring in the East face: 1998

Hatzor (2003), JGGE, ASCE

14

Joint meters and pressure transducers


15

Monitoring Installation Program: East Face

Block 3

Block 2

Block 1


16

Typical Displacement Output – Block 3
0

250
JM 10

-0.1
JM 11

200

-0.4

150

-0.5
FJ 3

-0.6

-0.8

JM = Joint Meter (LVDT)
FJ = Flat Jack

)

-0.7

100

Flatjack Pressure

-0.3

(
mm

Relative Displacement

-0.2

50

-0.9
0

18

14

/1

/9
/1 8
23 / 9
/1 8
27 / 9
/1 8
31 / 9
/1 8
4 / 98
/2
8 / 98
/2
12 / 9
/2 8
16 / 9
/2 8
21 / 9
/2 8
25 / 9
/2 8
1 / 98
/3
5 / 98
/3
9 / 98
/3
13 / 9
/3 8
18 / 9
/3 8
22 / 9
/3 8
28 / 9
/3 8
02 / 98
/0
14 4/9
/4 8
27 / 9
/4 8
9 / 98
/5
21 / 9
/5 8
3 / 98
/6
15 / 9
/6 8
28 / 9
/6 8
/9
8

-1

Date

Displacement in mm, pressure in kPa


17

Relative Displacement

-1.6
)

(
mm

-1
/1

/9
/1 8
23 / 9
/1 8
27 / 9
/1 8
31 / 9
/1 8
/
4
/ 2 98
8 / 98
/2
12 / 98
/2
16 / 9
/2 8
21 / 9
/2 8
25 / 9
/2 8
/
1
/ 3 98
5 / 98
/3
9 / 98
/3
13 / 98
/3
18 / 9
8
/
22 3 /
/ 3 98
28 / 9
/3 8
02 / 9
/0 8
4
14 /98
/4
27 / 9
/4 8
9 /9
/5 8
21 / 98
/5
/
3
/ 6 98
15 / 98
/6
28 / 9
/6 8
12 / 9
/7 8
/9
8
18

14

Superposition of outputs from 3 Blocks
Date

0.2

0
Bedrock

-0.2

-0.4
Block 2

-0.6

-0.8
Block 3

-1.2
Block 1

-1.4

Displacement in mm

18

Influence of Climatic Changes on Block
Displacement


19

24 months of monitoring in West face:
2009 - 2011

Bakun-Mazor, Hatzor, Glaser, Santamarina (2012) IJRMMS

20

Motivation: A sudden block failure in 2009

scar

Before the collapse

Precipitation,
mm/hr

After the collapse

Wind Velocity,
m/sec
Temperature, C

The west slope of Masada before and
after the storm of February 10, 2009
.
24
20
16
12
8
10
8
6
4
2
0
6
4
2
0
4-Feb-09

12-Feb-09

20-Feb-09

28-Feb-09

After the collapse

Temperature, wind velocity and
precipitation, as recorded in the west slope
of Masada, during February 2009.
After the collapse

Before the collapse


21

Monitoring installation in west face
a.

1m

Data
Logger
WJM
1
Joint Meter

WJM
WJM
2
4
WJM
3

j3
N

Temperature &
Relative
Humidity sensors

Rock Mass

Cliff face
j2
WJM 1

Fault plane
WJM 2,3

85/270

Road on Roman aqueduct


22

Joint opening, mm

40
20

RH, %

Temp., Co

Temperature and displacement monitoring output

80
40
0
0.2
0
-0.2
0.2
0
-0.2
0.2
0
-0.2
0.2
0
-0.2

WJM 1

WJM 2

WJM 3
WJM 4 ; “Dummy” JM connected to bedrock
Aug-09

Jan-10

Jun-10

Nov-10

Apr-11


23

Temperature dependent cyclic opening/closure
of joint aperture
40
Temperature

36

0.1
32
0

WJM 1

28

WJM 3

24
-0.1
20
-0.2

WJM 2

Aug-09

Jan-10

Jun-10

Nov-10

Air Temperature, Co

Joint Displacement, mm

0.2

16
Apr-11


24

Suggested Wedging - Ratcheting Mechanism
Wedge Block
East
Rock
Mass

0 meter 1

Sliding
Block

Sliding Surface
( )

Tension Crack

initial
condition

cooling

heating

cycle 1

cycle 2

25

Theoretical model for thermally induced sliding
If the external temperature change ΔT exceeds the maximum
temperature for elastic deformation ΔTmax the plastic
displacement δjp [m] that the block will experience is:

δT free thermal expansion
δσ elastic contraction
*
j

δ

p
j

δT

δσ

Field situation
(a)

δ

*
j

limiting joint elastic displacement

Plastic Displacement [mm]
(One Season)

0.8

η = 22
LW

0.6

L
LB

Masada
0.4

Sd

η = 19

(b)

Wedge

Block

H

0.2

Conceptual
model

η = 16
ΔT = 20 C

0.0

0.0

0.2

0.4

0.6

Sd: Thermal skin depth

Base

η

A

LW / LB
One-cycle plastic displacement for several plane inclinations.
Pasten, Santamarina, and Hatzor (in prep.)
Dolomite block-wedge system subjected to a seasonal temperature
Y. Hatzor: Thermally
26
change ΔT= 20°C. vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy

Shear strength of bedding planes in Masada


27

Failure envelope of smooth and rough surfaces
Direct Shear of Natural Bedding Planes
Triaxial Shear of Filled Saw-cut

Shear Stress (MPa)

12

8

4
peak=41
residual=

o

23o

B

0
0

5

10

15

20

25

Normal Stress (MPa)

28

Input Motion: Consideration of Topographic Site Effect

Empirical response function for the
topographic site effect at Masada
(Zaslavsky and Shapira, 2000).

29

Dynamic response to cyclic loading with DDA
0.6
0.4
0.03
0.2
0.02

0
input function, f = 1.3 Hz
Analytical solution
DDA output; k = 10 GN/m

0.01
0

Fixed
rock
mass

0

Wedges
Fixed rock
mass

= 19˚

0.8
Time, sec

1.2

H= 15.0 0.006 b.
m
Block Displacement, m

Block
1

0.4

-0.2
-0.4
-0.6
1.6

0.6
0.4

0.004

0.2
0

0.002

input function, f = 3.8 Hz
Analytical solution
DDA output; k = 500 GN/m

0
0

Input Acceleration, g

Sd

a.

0.2

0.4
Time, sec

0.6

-0.2
-0.4

Input Acceleration, g

LB = 7.5
m

0.04
Block Displacement, m

The geometry of Block 1 in the
East face of Masada is used for
Lw
modeling

-0.6
0.8

DDA results are strongly affected by the penalty, or contact spring stiffness, value, especially in dynamic
simulations. We optimize the contact spring value using the analytical (Newmark) solution and the two
measured resonance frequency modes of the mountain: 1.3 Hz and 3.8 Hz.

30

Scaling the input motion

a.
0.4
0.2
0
-0.2
-0.4

PGArock = 0.275g

Mw = 6.0 ; R = 1 km

0

20
40
Time, sec

60

c.
0.4
0.2
0
-0.2
-0.4

PGAtopo = 0.465g

0

d.

20
40
Time, sec

60

Masada site response

Horizontal acceleration.
including site effect, g

Horizontal acceleration.
de-conv. for rock, g

a) The Nuweiba earthquake as recorded in Eilat on a soil layer de-convoluted for bedrock response
[Zaslavsky and Shapira, 2000] and scaled to PGA = 0.275g, corresponding to a Mw= 6.0 earthquake at a
distance of 1 km from Masada

3

b.
b) an empirical site
response function for
Masada [after
Zasalavsky et al.
2002]

2
1
0
0

2

4
6
8
Frequency, Hz

10

12

c) convoluted time series of the modified Nuweiba record
(a) to include the empirical site response function for
Masada (b)

Mw = 7.5 rock, with site response
1
M = 7.0 rock, with site response
w

31

Response of Block 1 to regional earthquakes
1


Dynamic Sliding of Block 1

ayield = 0.404 g
Mw = 6.0 rock, no site response

Peak Acceleration, g

Masada


Mw = 7.5 rock, no site response

Static Stability of Block 1

0.1
1

10
Distance from epicenter, km

100

Assumed attenuation curves for Dead Sea Rift earthquakes [after
Boore et al., 1997] (dashed lines) with amplification due to
topographic site effect at Masada (solid lines and symbols). Shaded
region delineates conditions at which seismically-induced sliding of
Block 1 at Masada is not possible.

32

Maximum displacement of Block 1 in a single
earthquake
DDA Block displacement, mm

2000

M=7.5
M=7.0
M=6.5
M=6.0

1600

1667 mm

1200

800

447 mm

400

mapped joint opening in the field = 200 mm
42 mm
0.23 mm

0
0

20

40

60

Time, sec

DDA results for dynamic displacement of Block 1 when subjected to amplified Nuweiba records
corresponding to earthquakes with moment magnitude between 6.0 to 7.5 and epicenter distance of 1 km
from Masada. Mapped joint opening in the field is plotted (dashed) for reference.

33

Comparison between thermal and seismic
displacement rates for Block 1 in East Masada
1200

Thermal displacement rate is
calculated assuming = 0.3 and
0.5. Seismic displacement rate is
obtained by summation of
earthquake magnitudes 6.0 to
7.0 with epicenter located 1 km
from Masada based on the
seismicity of the region. The
seismic rates in the zoom-in box
are for the long term seismicity
(5000 years).

Tension crack opening, mm

thermal ; analytical model
seismic ; numerical DDA model

Monthly Temperature, C

36
32

August
September
October
November
December
January
February

28
24

After Carlslaw and
Jaeger , 1959

20
16
0

1

2

3
4
5
6
7
Depth into the rock, m

8

9

1000

0

500

1000

= 0.5

1500
200
= 0.3

800

100

600

0

400
200
0
0

1000

2000
3000
time, year

4000

5000

10


34

Summary and Conclusions
•

•

•

•

•

The numerical, discrete element DDA method is shown to be suitable for
performing accurate computation of dynamic interaction between blocks, making
it an attractive tool for performing dynamic rock slope stability studies.
In the DDA version used here a constant friction angle is assumed. It is shown
here however that friction angle degradation should be considered depending on
the interface properties and the sliding velocities. Therefore incorporating rate
and state effects into DDA would be a significant enhancement.
It has been shown through careful field measurements that rock joints are
subjected to annual cyclic opening and closing motions due to thermal effects of
climatic origin.
Tension cracks filled with rock fragments subjected to seasonal temperature
fluctuations may be prone to the described thermally induced ratcheting
mechanism which could lead to irreversible annual plastic displacement of rock
blocks.
We show that when everything else is kept equal, thermally induced
displacements may exceed seismically induced displacements over time in regions
subjected to moderate seismicity and where the temperature amplitude is
sufficiently high to induce thermal expansion.

This research has been partially funded by the US – Israel Binational Science Foundation (BSF)

35

Thank you!


36

Appendix


37

Analytical Model: Equilibrium and Compatibility
The maximum force per unit length parallel to the base Fmax [N/m] that the block frictional
resistance can sustain is:

Fmax

H

r

( LB

LW )(

cos

sin )

where a fraction θ< 1 of the wedge weight is transferred to the block through the shear stress
along the block-wedge interface. The ensuing thermal expansion is constrained by friction at
the base. Compatibility of displacements requires that the joint elastic displacement δje [m]
equals the displacement caused by the constrained thermal expansion of the block-wedge
system (δT − δσ), i.e., the displacement caused by free thermal expansion δT [m] minus the
elastic contraction δσ [m]:
e

δT

δσ

δj

The elastic contraction due to the force per unit length Fmax [N/m],
assuming that the block toe does not slide (point A in model), is:
On the other hand, the limiting joint elastic displacement
Therefore:

δ

*
j

1 Fmax
k j LB

δ j*

Fm ax
LW
H E

Fmax
[m] satisfies:
LB

LB
2

k j δ*j


38

Thermal skin depth
The rock temperature T(x,t) at distance x and time t responds to changes in boundary
temperature, as prescribed by the heat diffusion equation (Carslaw and Jaeger, 1986*):
2
T ( x, t )
T ( x, t )
DT
t
x2
The rock thermal diffusivity DT= kT/(ρ∙cp) [m2/s] is proportional to its thermal conductivity kT
[W/m/K] and inversely proportional to its mass density ρ [kg/m3] and specific heat capacity cp
[J/kg/K].

We define the homogenization time t* [s] as the time required to change the temperature at
the center x= L/2 of a one-dimensional rock element length L from an initial temperature T0
[°C] to 99 % the new boundary temperature T1 [°C] at x= 0 and x= L. The homogenization
time of the block and the wedge can be estimated as tB*= 0.5∙LB2/DT and tW*=
0.5∙LW2/DT, respectively, and the thermal skin depth Sd [m] for a certain exposure time texp [s]
is (Carslaw and Jaeger, 1986):
0.5 DT t exp
t exp 0.5 L2 / DT
Sd
L/2
t exp 0.5 L2 / DT
Its maximum value is half the length of the rock element Sd= L/2 when the exposure time
equals the homogenization time texp= t*.
* Carslaw, H. S., and J. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8
Y. Hatzor: Thermally vs.C. Jaeger (1986), Conduction of Heat in Solids, Oxford University Press, New York, NY. – 10, 2013 , Padua, Italy

39

Thermal expansion as a function of exposure time
Consider a block larger than the wedge (LB > LW) and an air temperature change from T0 to T1 (T1> T0).
When the exposure time is short, texp< 0.5LW2/DT < 0.5LB2/DT, the wedge, the block, and the left wall
behind the wedge have a transient non-homogeneous temperature distribution. The system tends to
expand upon heating; unconstrained, the displacement parallel to the base due to the expansion of the
four skin depths involved could reach:

Short exposure time

T

T (4

Sd )

which is proportional to ΔT = T1 – T0 [°C] and the rock thermal expansion coefficient α [1/°C]. The
dimensionless coefficient β ≤ 1.0 accounts for the non-uniform diffusive temperature distribution within
the skin depth of the rock element.
For exposure times texp longer than the time required to homogenize the wedge but shorter than that
required to reach a homogeneous block temperature, 0.5LW2/DT < texp < 0.5LB2/DT, the free thermal
displacement of the system combines the full expansion of the wedge and the partial expansion of the
block and the left wall:

Intermediate exposure time

T

T ( LW

2

Sd )

Assumed here

Finally, when the exposure time texp exceeds the time required for temperature homogenization in
the block and the wedge, 0.5LW2/DT < 0.5LB2/DT < texp, the thermal displacement is:

Long exposure time

T

T ( LW

LB

Sd )

where the dimensionless coefficient ξ ≤ 1.0 is introduced to account for the free thermal expansion of the
Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, expansion.
40
right portion of the block that does not contribute to constraining the system thermal 2013 , Padua, Italy

9oct vajont lecture hatzor

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