Site Response Analysis with 2D-DDA Yossef H. Hatzor Sam and Edna Lemkin Professor of Rock Mechanics Dept. of Geological and Environmental Sciences Ben-Gurion University of the Negev, Beer-Sheva, Israel
Talk Outline Dynamic displacement of discrete elements: review of some published DDA verifications and validations Dynamic sliding on a single plane Dynamic sliding of a wedge on two planes Block response to dynamic shaking of foundations Dynamic block rocking Accuracy of wave propagation modeling with DDA: recent results P wave propagation S wave propagation Site response analysis with DDA Case Study: The Western Wall Tunnels in Jerusalem the significance of local site response 2
Dynamic Displacement of Discrete Elements with DDA: Verifications and Validations 3
Single Face Sliding Input motion (m/s 2 ) 1 5 =22 =3 =35 DDA DDA DDA Analytic Analytic Analytic Input Motion 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 24-26. Sinusoidal input first studied by Hatzor and Feintuch (21), IJRMMS. Improved 2D solution presented by Kamai and Hatzor (28), NAG. Ning and Zhao (212), NAG (From NTU) recently published a very detailed study of this problem. Displacement of upper block (m) relative error (%) -5 12 8 4 1 1 1.1.1 1 2 3 4 5 6 Time (sec) 4
Double Face Sliding z 2.5 6 Wedge parameters: P 1 =52/63, P 2 =52/296 1 = 2 =3 o x y Displacement (m) 2 1.5 1.5 Analytical 3D-DDA Input Motion (y) 4 2-2 -4-6 Horizontal Input motion (m/s2) DDA validation originally investigated by Yeung M. R., Jiang Q. H., Sun N., (23) IJRMMS using physical tests. Analytical solution proposed and 3D DDA validation performed by Bakun- Mazor, Hatzor, and Glaser (212), NAG. Relative Error (%) A 1 1 1.1.4.8 1.2 1.6 2 Time (sec) 1 1 1.1 5
Shaking Table Experiments Acceleration of Shaking Table, g.2 -.2 A Accumulated Displacement, mm 6 4 2 B Shaking Table 3D DDA ; Loading mode 3D DDA ; Displacement mode Relative Error, % 1 1 1 1 C t i Erel; Loading Mode Erel; Displacement Mode 1 2 3 4 Time, sec 6
Rate Dependent Friction Upper Shear Box Normal Cylinder Shear System 2 cm Concrete Samples Lower Shear Box Roller Bearing Shear Cylinder Shear Stress, MPa 4 3 2 1 = 5.2 MPa n = 4.3 MPa n = 3. MPa n = 1.97 MPa n Shear Stress, MPa 4 3 2 1 v =.2 mm/sec v =.2 mm/sec v =.1 mm/sec = = 2 n =.98 MPa = 22 A 1 2 3 4 5 Shear Displacement, mm B 2 4 6 Normal Stress, MPa 7
Observed Block Run-out Up. Block Accum. Displacement, mm 1 8 6 4 2 Measured Calculated o = 29. = 29.5 o = 27. o Up. Block Velocity, mm/sec 6 4 2 Acceleration of Shaking Table, g.2 -.2 A 2 4 6 Time, sec 8
Friction Angle Degradation.7.7 Direct shear test results = -.174 * ln(v) +.5668 Friction Coefficient.6.5.4.3.1.1.1.1 1 1 1 Velocity, mm/sec Friction Coefficient.6.5 = -.79 * ln(v) +.671 R 2 =.99 R 2 =.948 Shaking table experiments.4 Shaking Table Coulomb-Mohr.3.1.1.1.1 1 1 1 Velocity, mm/sec 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. (25), JGGE ASCE; a new approach has recently been proposed by LZ Wang et al. (in press), COGE (from Zhejiang University). 9
Dynamic interaction of discrete blocks The dynamic interaction between discrete blocks subjected to dynamic loads such as earthquake vibrations is of high importance in seismic risk studies both for preservation of historic monuments as well as geotechnical earthquake engineering design. Several DDA research groups have began to explore this issue. Notably, Professor Yuzo Ohnishi s DDA research group has recently made some important contributions to this field, e.g. Miki et al. (21), IJCM; Sasaki et al. (211), IJCM. Kamai and Hatzor (28), NAG, have suggested to use this approach to constrain the paleoseismic PGA in seismic regions by back analysis of stone displacements in historic masonry structures. 1
Direct acceleration input simultaneously to all blocks: we call it QUAKE mode 11
Direct displacement input to foundation block: we call it DISP mode 12
Response of overriding block to cyclic motion of foundation block: DISP mode 1.8 D=.3m (1.2g) analytical DDA 1.6 1.4 1.2 D=.5m (2g) analytical DDA D=1m (4g) analytical DDA Input motion: d t = D(1- cos (2pt)); f = 1Hz; =.6 D 2 (m) 1.8.6.4.2.2.4.6.8 1 1.2 1.4 1.6 1.8 2 For complete analytical solution see: Kamai and Hatzor (28), NAG. time (sec) 13
Dynamic Block Rocking: QUAKE mode 2b 2h c.m. R h b ü(t) Analytical solution proposed by: Makris and Roussos (2), Geotechnique. DDA validation with applications: Yagoda-Biran and Hatzor (21), EESD. 14
a peak slightly lower than PGA required for toppling 15
a peak slightly higher than PGA required for toppling 16
Accuracy of Wave Propagation Modeling with DDA: Benchmark Tests and Field Investigations 17
DDA accuracy in simulations of P wave propagation: 1D elastic bar Loading point Measurement point.5m 1m 2m 1m 5m 5m 5m Work in progress with Huirong Bao, Xin Huang, and Ravit Zelig 18
Input function for P wave and model properties F t = 1 sin 2πt KN Load (KN) 1 5-5 -1..2.4.6.8.1 Time (s) Block material Joint material Unit mass (kg/m 3 ) 265 Young s modulus (GPa) 5 Poisson ratio.25 Friction angle 35 Cohesion (MPa) 24 Tensile strength (MPa) 18 Time history of input load Input parameters for block and joint materials 19
Time interval effect on the accuracy of P wave stress e = A 1 A A 1% where A 1 is the measured wave amplitude or calculated wave velocity at a reference measurement point in the model, and A o is the incident wave amplitude or analytical wave velocity at a given point. Stress relative error (%) 5 4 3 2 1 5m 2m 1m.5m Note very significant effect of time interval on P wave stress accuracy, block size is much less important...1.2.3.4.5 Time interval (ms) 2
Time interval effect on the accuracy of P wave velocity For the special case of a one dimensional bar (Kolsky, 1964): V p = E ρ Error increases with decreasing block size P-wave velocity relative error (%) 5 4 3 2 1-1 Error decreases with decreasing block size 5m 2m 1m.5m..1.2.3.4.5 Time interval (ms) Note complicated block size effect on P wave velocity; time interval is much less important. There seems to be an optimal block size below which the error increases! 21
Time interval effect on waveform accuracy Stress (Mpa) 1..5. -.5-1. Wave form accuracy greatly improves with decreasing time step! Analytical.5 ms.1 ms.5 ms.1 ms.1.2.3.4.5 Block length = 1 m Stress measured at mid section Time (s) 22
Contact stiffness (k) effect on P wave stress accuracy Stress Relative Error (%) 11 1 9 8 7 6 5 4 3 Not much effect of k value on P wave stress Error of Min Stress Error of Max Stress Error of Max-Min Block length = 1 m Stress measured at mid section 2 1E 2E 4E 8E 16E 32E 64E Contact Stiffness 23
Contact stiffness (k) effect on P wave velocity accuracy P-wave velocity relative error (%) 3.5 3. 2.5 2. 1.5 1..5. P-wave velocity relative error There clearly exists an optimal k value beyond which the error begins to increase Block length = 1m 1E 2E 4E 8E 16E 32E 64E Contact stiffness 24
Contact stiffness (k) effect on waveform accuracy Stress (MPa) 1..5. -.5-1. Wave from accuracy greatly improves with increasing k value! Analytical 1E 5E 1E 4E 1E Block length = 1m.1.2.3.4.5 Time (s) 25
Relationship between wavelength and block size The relationship between element (block) side length ( x) and wave length ( ), has a strong influence on numerical accuracy. In FEM the optimal ratio η should be smaller than 1/12 (.83) where: = x In our simulations: T (s).1 v (m/s) 4343 λ (m) 43.43 We have performed a series of tests for various values of η and obtained the following results: Δt (ms) Block length (m).5 1 2 5 1 η (Δx/λ).12.23.46.115.23.1.1 Velocity (m/s) 4219.16 431.34 4359.2 4436.36 4432.43 Error 2.87%.77%.36% 2.13% 2.4% Velocity (m/s) 4884.24 4699.91 4553.73 4493.57 449.35 Error 12% 8% 5% 3% 3% 26
Influence of η on velocity accuracy We see again that the error decreases with decreasing block length, but when η is smaller than 1/22 the numeric error in fact increases! This corresponds to the result we obtained regarding the effect of block size. η = 1/22 Clearly accuracy improves with decreasing time interval! η < 1/12 27
Influence of η on stress accuracy η = 1/22 Great improvement in stress accuracy with decreasing block length down to a minimum at η = 1/22 below which the error increases for both time intervals studied. Very significant accuracy improvement with decreasing time interval. Δt (ms) block length (m).5 1 2 5 1 η (Δx/λ).12.23.46.115.23.1 Amplitude (KPa) 983.6 985.4 999.8 981.6 927.5 error 1.6% 1.5%.% 1.8% 7.2%.1 Amplitude (KPa) 884.6 887.3 888.2 87.3 835.3 error 11.5% 11.3% 11.2% 13.% 16.5% 28
The problem with adding joints artificially Analytical model: DDA model: In the analytical model there is no additional stiffness from contact springs as in the DDA model, and the total stiffness of a block system of length L is: Kb K L = n where n b is the number of blocks in specified length L. b In the DDA model, the total stiffness of the block system in length L is: æ 1 K L = + n æ c æ K L æ 29 K c -1 < K L where n c is the number of contact springs in specified length L. Artificially decreasing the size of blocks down to a certain value may increase stress accuracy, but below a value of η 1/22 errors both in stress and velocity will increase because of the inaccurate representation of the real stiffness of the system due the large number of contacts.
S wave propagation: DDA vs. SHAKE DDA Model SHAKE model 15m block 1 block 15 M2 M1 1mx15=15m layer 1 layer 2 layer 15 bedrock ug 1mx15=15m ug Bao, Yagoda-Biran, and Hatzor, Earthquake Engineering and Structural Dynamics (in press) 3
Input Ground Motions acceleration (m/s 2 ) 2 acceleration 1-1 -2 1 2 3 4 5 6 time (s) displacement (m).1 displacement.5. -.5 -.1 1 2 3 4 5 6 time (s) CHI-CHI 9/2/99, ALS, E (CWB): acceleration for SHAKE and displacement for DDA 31
Modeling Procedure Damping ratio transfer from DDA into SHAKE utilizing DDA algorithmic damping (for details on the algorithmic in DDA See Doolin and Sitar 24) Time step size Damping ratio SHAKE DDA Modeled material parameters in layered model Layer/Block Unit mass (kg/m 3 ) Young s Modulus (GPa) Shear Modulus (GPa) Poisson ratio 1-3 243 4.5 1.8.25 4-6 2162 4.1 1.64.25 7-9 2243 4.2 1.68.25 1-12 2483 4. 1.6.25 13-15 2643 4.8 1.92.25 32
Spectral Amplification Ratio DDA numerical control parameters 3 25 time step size (s):.1 Total steps: 6 Spring stiffness (N/m): 1.5E+12 Calculated damping ratio: 2.3% DDA SHAKE Amplification ratio 2 15 1 5 DDA SHAKE Natural frequency (Hz) 14.23 14.5 Max amplification 28.76 27.79 5 1 15 2 25 Frequency (Hz) 33
2D Site Response: DDA vs. Field Test Static Push and Release at top column Dynamic blow with sledgehammer at column base Bao, Yagoda-Biran, and Hatzor, Earthquake Engineering and Structural Dynamics (in press) 34
Results of Geophysical Field Measurements Typical vibrations of the Column top and base in the X and Y directions due to force excitation in the Y direction by horizontal stroke of sledgehammer at the base of the Column a The corresponding Fourier amplitude spectra for the top and base of the Column. 35
The DDA model of a multi-drum column Parameter Young's modulus Value 17 GPa Poisson's ratio.22 Interface Friction 3 o Density 225 kg/m 3 Time step Displacement ratio k (penalty value).1-.1 sec..1.1 2x1 7-2x1 7 N/m 36
DDA response to dynamic pulse of 1, N load F.2 t 1 - t t 1 time t 1 + t Dynamic response of uppermost block in time domain displacement (cm).15.1.5 -.5 -.1 -.15 -.2 5 1 15 2 25 3 time (sec) 37
Top column response to static push: DDA vs. Geophysical survey k = 4x1 8 N/m DDA results experimental results 15 load F displacement amplitude (cm).1 1 5 velocity amplitude (m/s) t 1 t 2 time t 2 + t 1 2 3 4 5 6 7 8 9 1 frequency (Hz) 38
Top column response to dynamic blow: DDA vs. Geophysical survey k = 4x1 8 N/m DDA results experimental results 3 load F displacement amplitude (cm).1 2 1 velocity amplitude (m/s) t 1 - t t 1 time t 1 + t 1 2 3 4 5 6 7 8 9 1 frequency (Hz) 39
DDA sensitivity of resonance frequency to penalty value k FFT of uppermost block displacement Amplitude (cm).35.3.25.2.15 k = 1x1 8 N/m k = 2x1 8 N/m k = 4x1 8 N/m k = 7x1 8 N/m k = 1x1 9 N/m Decreasing resonance frequency and motion amplitude with increasing penalty value k.1.5 1 2 3 4 5 6 7 8 9 1 Frequency (Hz) Best agreement with field test 4