displacement of rock slope during seismic loading. ... The velocity and displacement histories (shown in Fig. 1(b) ~ Fig. 1(c)) could then ... calculate displacement.
Fifth International Young Geotechnical Engineering Conference - 5iYGEC’13
STUDY OF NEWMARK SLIDING BLOCK METHOD FOR RIGID SLIDING MASS DURING ASYMMETRIC DYNAMIC LOADING É TUDE DE LA MÉ THODE DE NEWMARK POUR UNE MASSE RIGIDE GLISSANTE SOUS CHARGE DYNAMIQUE ASSYMÉ TRIQUE 1
Chih-Chieh Lu 1 Geotechnical Engineering Department, Sinotech Engineering Consultants, Ltd., Taipei, ROC
ABSTRACT - Newmark sliding block method is a popular engineering method to evaluate permanent displacement of rock slope during seismic loading. In practice, a fixed threshold of horizontal seismic coefficient defined by the critical surface subject to factor of safety equal to 1.0 is used to examine the horizontal component of earthquake. The magnitude of displacement is obtained by integrating twice the difference of the applied acceleration and the critical acceleration with respect to time. However, this procedure does not rationally consider the variation of strength of rock mass during seismic loading and effect of vertical component of earthquake. The direction and quantity of displacement based on this approach is unclear. This study proposed modifications to improve the traditional Newmark sliding block method for consideration of the earthquake in horizontal and vertical directions and the variation of strength of Mohr-Coulomb material during earthquake. The proposed framework, compiled in the EXCEL program, is illustrated with an ideal model of sliding block. The yielding results are compared and discussed in this article. The modified analysis procedure is then applied to a real case of dip slope around reservoir in south of Taiwan for safety of reservoir. Satisfactory results are obtained in the demonstration example analyzed with the proposed framework. The effect of vertical earthquake is also discussed by this case.
1. Introduction Limit equilibrium method (LEM) is a conventional approach to evaluate safety of slope subjected to gravity using factor of safety (FOS) defined as a ratio of available shear resistance (capacity) to that required for equilibrium. The result of the deterministic assessment is binary: the slope is rated as stable if FOS≧0, or unstable if FOS 0
0
2
4 Time (sec)
6
(3)
(4)
Where Vmotion is vector of the block’s motion. After gaining the relative acceleration (a) to slope, the relative velocity (v) and displacement (d) could be obtained by way of integration:
0 -10
(2)
Where Vdriving force is vector of driving force, m is mass of block, and a is acceleration along the surface of slope. If the block is in motion at former stage, the relative acceleration of block can be computed as follows.
(b) Velocity
0.2
-20
Where W is self-weight of block, β is slope angle, ay is the vertical acceleration acting on the block, and ax is the horizontal acceleration acting on the block.
F max( 0, D R Vdriving force ) ma
-0.3 0.4
(1)
The relative motion of block follows Newton’s second law of motion. When the block is static at former stage, the relative acceleration of block is calculated by following equation.
Barking force appears while velocity is larger than 0 and acc. is smaller than athreshold
0.3
D W (1 a y ) sin( β ) Wa x cos( β )
8
Figure 1. Double integrating acceleration to calculate displacement
v v 0 at
(5)
d d 0 vt
(6)
Where vo and do are relative velocity and displacement at last stage, respectively, and t is time.
Wax W (1+ay) β W = weight of block ax = horizontal acceleration ay = vertical acceleration β = slope angle
Figure 2. Profile of the ideal model 2.1. Driving force The equation of driving force (D) applying on the slope block can be written as:
3. Behavior of slope block under dynamic loading 3.1. Basic information of the ideal model The research used the proposed Newmark sliding block method to analysis the seismic behavior of slope block, shown in Fig. 2, using the coded program Excel. The slope angle, block weight and contact length of block are 25°, 12t and 3m per
Fifth International Young Geotechnical Engineering Conference - 5iYGEC’13
-0.4 -0.8 0
1
2 Time (sec)
3
4
Figure 3. The acceleration acting on block
Force (kN)
0 Driving force Resistance Unbalance force
4
β =25°
Acceleration (m/s2)
Force (kN)
(b) Relative acc.
2 0 -2 (c) Relative velocity
Velocity (m/s)
Velocity (m/s)
0.0
Displacement (cm)
Acceleration (g)
0.4
12
Case 1
-6
0.8
0.8
-1.2
(a) Force
6
0.4 0.0 -0.4 30
(d) Relative disp.
Displacement (cm)
1.2
12
Acceleration (m/s2)
meter, respectively. Friction angle and cohesion of interface are 33° and 0. In order to make the results easy to interpret, only the horizontal acceleration with PGA=0.4g, shown as Fig. 3, was just adopted. Note the acceleration is acting on block, which the direction is right opposite to the base motion.
20 10 The variation of NF due to EQ is considered
0 -10
0
1
2 Time (sec)
4
3
(a) Force
Case 2
6 0
Driving force Resistance Unbalance force
-6 4
β =25°
(b) Relative acc.
2 0 -2 0.8
(c) Relative velocity
0.4 0.0 -0.4 30
(d) Relative disp.
20 10 The variation of NF due to EQ is NOT considered
0 -10
0
1
2 Time (sec)
3
Figure 4. Analyses results of slope block 3.2. Discussion of analyses results 4. Case study The analyzed results of the driving force, resistance, and unbalance force of slope block by the proposed procedure are shown in Fig. 4(a). Based on the unbalance force, the corresponding relative acceleration, velocity, and displacement can be computed as shown in Fig. 4(b)~Fig. 4(d), respectively. The results of case 1 are based on the analysis that considered the variation of normal force. During the period from 0 to 0.1sec, since the driving force does not change and is not larger than the limit of resistance, the forces are balance and the block remains static. Meanwhile, the quantity of resistance is proportional to driving force and the directions of both are opposite. While the dynamic loading is applied and breaks the balance condition, the slope block begins to move and the performance is affected by the interaction of driving force and resistance. Since the block is located on relative steep slope, the sliding potential of the slope block is downward during whole analysis due to gravity. After 1.55sec, the block slides downward with an acceleration motion until 1.88sec. The resistance is then larger than driving force, and the block decelerates until relative velocity becomes zero (t=2.31sec). The permanent displacement at this moment is 25.8cm. The analysis with a fixed maximum resistance was also adopted in case 2. From the results, the driving force is larger than the resistance during the period from 1.55 to 1.85sec, but the unbalance force between two is smaller than the case with considering the variation of normal force, which causes the integration results of relative velocity and displacement to be underestimated. The block becomes static after 2.14sec and the permanent displacement relative to slope is 11.5cm. It can be seen that the stable time is earlier and the permanent displacement is quite smaller than the one following the proposed procedure. The consideration of the variation of normal force is necessary based on the above comparison.
4.1. Basic information A real case of a dip slope around a reservoir located on the south part of Taiwan is used to demonstrate the proposed approach. The profile of the slope is shown in Fig. 5. Referring to detailed planning report of the site, the bedding of the rock 3 is N40°E/30°S, and the unit weight is 24kN/m . The bedding strength parameters of friction angle and 2 cohesion are 15 ° and 100kN/m , respectively. Based on the characteristic of this site, the block is divided into zone A and B. To zone A, the sliding 3 length is 449.3m/m, the volume is 6869m /m, and the dip is 30 ° . To zone B, the sliding length is 3 37.7m/m, the volume is 666m /m, and the dip is 0°. While conducting analyses, the driving forces and resistances of the two zones were calculated separately and then summed up for the evaluation of sliding behavior. Zone A Weight of block=6869m3/m Sliding length=449.3m/m Dip of sliding surface=30o
Be d
din
g
Strength parameters c = 100kN/m2 φ = 15o Unit weight of rock=24kN/m3
Zone B Weight of block=666m3/m Sliding length=37.7m/m Dip of sliding surface=0o
Figure 5. The numerical model of the site 4.2. Input motion The real record by the station near the site was used in this study. In light of the procedure mentioned above, the loadings acting on the slope block are shown in Fig. 6. Besides, based on the safety report of the site, the design earthquake is 0.35g in horizontal direction and 0.23g in vertical
4
Fifth International Young Geotechnical Engineering Conference - 5iYGEC’13
40 Time (sec)
60
80
Figure 6. The acceleration acting on the block 4.3. Results Fig. 7 shows all analyses results of the demonstrated example. Because of the gravity and the dip of bedding, the motion of block is always in the direction of fall during whole analysis. As the earthquake time passes 20sec, the ground motion enters main shock and the downward displacement quickly accumulates especially at timing of PGA (t=25.2sec). After that, the sliding of block comes stable, and the relative velocity slows down until stop. The permanent displacement based on the case considering both components of earthquake is 21.2cm, meanwhile, the yield permanent displacement is 19.4cm for the case that just considers horizontal earthquake only. The difference between two is about 1.8cm, which is about 10% of the results. Therefore, vertical earthquake is better taken into account since the analysis results might not tilt toward conservatism. From the analyses results, the mass dumping 3 into reservoir after earthquake is about 3m /m. With the slide width of 300m, the distance of 320m from the sliding area to main dam, and the propagation direction of 90°, the potential impulse wave height is about 10cm based on the formula suggested by ICOLD. Since the height is about 7.5m from high water level to the top of the dam, the safety of the dam could be guaranteed. 5. Conclusion and suggestion 1. This paper reviewed the Newmark sliding block method in detail. The operations of driving force and resistance are described and explained by the equations. The yielding results of tested examples were described with graphical display. Also, from the results, the normal force should be carefully considered. 2. The procedure to conduct the proposed Newmark sliding block method was
200
Driving force Force (MN)
0
Resistance
Unbalance force
-100 6
Acceleration (m/s2)
20
(a) Force
100
(b) Relative acceleration
3 0 -3 0.8
(c) Relative velocity Velocity (m/s)
0
Acceleration (m/s2)
(b) Horizontal direction
Force (MN)
200
Velocity (m/s)
0.3 0.2 0.1 0.0 - 0.1 - 0.2 - 0.3
(a) Vertical direction
0.4 0.0 - 0.4 30 20 10 Analysis by horizontal and vertical parts of earthquake
0 -10
0
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40 Time (sec)
(a) Force
60
80
Driving force
100 0
Resistance
Unbalance force
-100 6
(b) Relative acceleration
3 0 -3 0.8
(c) Relative velocity
0.4 0.0 -0.4 30
(d) Relative displacement
Displacement (cm)
0.3 0.2 0.1 0.0 - 0.1 - 0.2
demonstrated by a real case. The yielding results could be further used to evaluate impulse wave height for the safety of reservoir. 3. Based on the case study, the effect of vertical earthquake is about 10% of the results. 4. FEM or FDM could also be conducted in the future for comparison. 5. The coded program by EXCEL could easily and quickly repeat LEM analysis, which is beneficial to conduct seismic hazard analysis.
Displacement (cm)
Acceleration (g)
Acceleration (g)
direction. To this, the analysis multiplied the input motion by the absolute PGA ratio of the design earthquake to real record. In order to clarify the effect of vertical earthquake, the slope block subject to horizontal component of earthquake only or both components of earthquake were conducted for comparison.
(d) Relative displacement
20 10 Analysis by horizontal part of earthquake only
0 -10
0
20
40 Time (sec)
60
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Figure 7. Analysis results of the site 6. Reference Franklin, A.G. and Chang, F.K., 1977, “Permanent displacements of earth embankments by Newmark sliding block analysis,” Report 5, Miscellaneous Paper S-71-17, U.S. Army Corps of Engineers Waterways Experiment Station, Vicksburg, Mississippi. Newmark, N.M., 1965, “Effect of earthquake on dams and embankments,” Geotechnique, Vol. 15, pp. 139~159. Rathje, E.M. and Bray, J.D., 2000, “Nonlinear coupled seismic sliding analysis of earth structures,” Journal of Geotechnical and Geoenvironmental Engineering, Vol. 126, No. 11, pp. 1102~1014. Seed, H.B., Lee, K.L., Idriss, I.M., and Makdisi, F.I., 1973, “Analysis of the slides in the Sam Fernando dams during the earthquake of Feb. 9, 1971,” Report No. EERC 73-2, Earthquake Engineering Research Center, University of California, Berkeley. Seed, H.B. and Martin, G.R., 1966, “The seismic coefficient in earth dam design,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 97, No. 5, pp. 1199~1218. Terzaghi, K., 1950, “Mechanisms of landslides,” The Geological Survey of America, Engineering Geology (Berkley) Volume. Yegian, M.K., Marciano, E.A., and Ghahraman, V.G., 1991, “Earthquake-induced permanent deformations: probabilistic approach,” Journal of Geotechnical Engineering, ASCE, Vol. 117, No. 1, pp. 1158~1167.
