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9oct vajont lecture hatzor

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9oct vajont lecture hatzor

  1. 1. 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
  2. 2. 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
  3. 3. Dynamic Sliding: Verifications and Validations Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 3
  4. 4. Single Plane Sliding Photo courtesy of R. E. Goodman Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 4
  5. 5. 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 Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 6 5
  6. 6. Double Plane Sliding Photo courtesy of G. H. Shi Photo courtesy of R. E. Goodman Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 6
  7. 7. 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) Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 7
  8. 8. 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 Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 8
  9. 9. 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 Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 9
  10. 10. 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 Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 10
  11. 11. 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). Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 11
  12. 12. THERMAL VS. SEISMIC TRIGGERING Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 12
  13. 13. Masada World Heritage Site as Field Station Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 13
  14. 14. Six month monitoring in the East face: 1998 Hatzor (2003), JGGE, ASCE Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 14
  15. 15. Joint meters and pressure transducers Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 15
  16. 16. Monitoring Installation Program: East Face Block 3 Block 2 Block 1 Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 16
  17. 17. 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 Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 17
  18. 18. 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 Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 18
  19. 19. Influence of Climatic Changes on Block Displacement Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 19
  20. 20. 24 months of monitoring in West face: 2009 - 2011 Bakun-Mazor, Hatzor, Glaser, Santamarina (2012) IJRMMS Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 20
  21. 21. 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 Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 21
  22. 22. 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 Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 22
  23. 23. 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 Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 23
  24. 24. 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 Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 24
  25. 25. 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 Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 25
  26. 26. 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
  27. 27. Shear strength of bedding planes in Masada Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 27
  28. 28. 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) Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 28
  29. 29. Input Motion: Consideration of Topographic Site Effect Empirical response function for the topographic site effect at Masada (Zaslavsky and Shapira, 2000). Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 29
  30. 30. 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. Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 30
  31. 31. 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 Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 1 M = 7.0 rock, with site response w 31
  32. 32. Response of Block 1 to regional earthquakes 1 Mw = 7.5 rock, with site response Mw = 7.0 rock, with site response Dynamic Sliding of Block 1 ayield = 0.404 g Mw = 6.0 rock, no site response Peak Acceleration, g Masada Mw = 6.5 rock, with site response Mw = 6.0 rock, with site response 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. Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 32
  33. 33. 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. Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 33
  34. 34. 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 Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 34
  35. 35. 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) Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 35
  36. 36. Thank you! Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 36
  37. 37. Appendix Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 37
  38. 38. 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 Y. Hatzor: Thermally vs. seismically induced disaplecements. Vajont 1963 – 2013 Intl. Conference. October 8 – 10, 2013 , Padua, Italy 38
  39. 39. 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
  40. 40. 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

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