For example, the friction in FEM is a macro friction and it can be regarded as the macro effect of the mesoscopic friction on the particle surface and the engagement force between the particles in contact.
Therefore, the slope analysis method in DEM should be further discussed. The following is the detailed analysis method. In the slope project, sliding surface is a kind of the macro-contact mentioned above. Therefore, the characterization method of macro-contacts can characterize the sliding surface of the granular slope.
As shown in Figure 5 , the edge sequence in the form of end-to-end which is similar to the corresponding sliding surface in terms of the route can be found to characterize the sliding surface. Due to the randomness of granular materials, the edge sequence is not just right on the presupposed sliding surface but basically near the sliding surface.
The edge sequence can approximately characterize the presupposed sliding surface and it will separate all particles into sliding mass particles and sliding bed particles.
When the soil slope with the characteristic of strain softening slides, the stress of the soil near the sliding surface does not reach the shear strength at the same time. When the stress of the soil located at a certain place reaches the shear strength, the shear failure occurs at this place and corresponding peak strength will become residual strength.
At the same time, failure will extend to other places. If the shear failure occurs through a route from slope toe to slope crest, the slope will slide integrally. That is the whole progress of progressive landslide. The aim of this article is to simulate this progress in graph theory. In the characterization method of the contact surface, the vertexes of a certain graph characterize the void cell and the edges characterize the particle contacts of the granular material.
The vertexes in the slope toe are defined as vertex group V a and the vertexes in the slope crest are defined as vertex group V b. There are many routes from vertex group V a to vertex group V b and each route has a corresponding safety factor. Among all the whole safety factors, the minimum safety factor is the global safety factor of the slope and its corresponding route is the most dangerous sliding surface of the slope.
The sliding surface is the route which corresponds to the minimum safety factor among all the possible routes from vertexes V a to vertexes V b. There are many routes that need to be analyzed. In order to simplify the problem and improve calculation speed, a certain number of sliding surfaces are supposed. Referring to the slip circle method, a certain number of circle centers are supposed and every circle center responds to multiple sliding surfaces with different radiuses.
Referring to the characterization method above, the edge sequence, which is similar to the route of the sliding face, can be calculated. Each edge of the edge sequence has three weights. Weight E 1 is the sliding moment of the contact and weight E 2 is the anti-sliding moment of the contact.
The safety factor of a certain edge sequence is calculated by reducing weight E 2 repeatedly and this safety factor is stored in the vertexes forming this edge sequence. One vertex can be the component of a certain number of edge sequences, so one vertex can have many safety factors and the entire safety factor of the vertex are the minimum of these safety factors. Weight E 3 is contact-bond broken flag and value 1 means that a contact bond was present at the corresponding contact in the past but has been judged to be broken.
The detailed calculation method is shown in Figure 6. In the graph G , weight E 2 should be classified into two types: the anti-sliding moment before the corresponding contact bond break before softening and the anti-sliding moment after the corresponding contact bond break after softening.
Most weights of edge E 2 are the anti-sliding moment before the corresponding contact bond breaks. If weight E 2 is reduced to a certain extent, weights E 1 of the corresponding contact may exceed the weight E 2 , then the bond of the corresponding contact is judged to break and weight E 3 is changed to value 1 from value 0. The corresponding weight E 2 is changed to the anti-sliding moment after the corresponding contact bond breaks.
At this moment, weight E 1 and weight E 2 cannot maintain the balance but the slope may not slide integrally. The part of weights E 1 which cannot be balanced by the corresponding weights E 2 will be balanced by weights E 2 of other adjacent edges. If the ratio of the edges of which weight E 3 is value 1 to the number of the whole edges forming the edge sequence exceeds a certain extent, the slope is determined to slide and the corresponding reduction factor is the safety factor of the corresponding sliding surface.
Among the safety factors of all the supposed sliding surfaces, the minimum safety factor is the entire safety factor of the slope and the corresponding surface is the most dangerous sliding surface. The safety factors of all the location can be shown by color mapped figure. The parameters adopted to calculate the safety factor of the granular slope are given in Table 1.
As shown in Figure 7 , the particle contact surface may not overlap the preset sliding surface. The direction S and N are the tangential direction and the normal direction of particle contact surface, respectively. F s is the projection of the contact force in the tangential direction of particle contact surface. F n is the projection of the contact force in the normal direction of particle contact surface.
In the characterization of the contact surface of the granular material, according to the constitutive model of the granular material, the weight E 1 sliding moment of the edge is the moment of the contact force located at the contact characterized by the corresponding edge about the supposed slip circle center.
As mentioned above, the calculation method of the weight E 2 should be classified into the anti-sliding moment before the corresponding contact bond break and the anti-sliding moment after the corresponding contact bond break.
There are two types of bond strength, contact bond normal strength and contact bond shear strength. The calculation method of the anti-sliding moment before the contact bond break is relative to its stress state, which can be expressed as follows: for particles of which the contact force type is compression, the failure type is likely to be shear dislocation when the shear force reaches the contact bond shear strength.
This is because particles are inseparable and cannot be crushed. Q s is defined as the quotient of the contact bond shear strength divided by the corresponding shear force and the anti-sliding moment is the product of the sliding moment multiplied by Q s. For particles of which the contact force type is tension, there may be two kinds of failure types, shear dislocation and tension fracture.
If the shear force reaches the contact bond shear strength or the normal force reaches the contact bond normal strength, the bond will break. Thus, Q s is defined as the quotient of the contact bond shear strength divided by the corresponding shear force and Q n is defined as the quotient of the contact bond normal strength divided by the corresponding normal force.
The Q min is defined as the smaller one between Q s and Q n. The corresponding anti-sliding moment is the product of the sliding moment multiplied by Q min.
For particles of which the contact force type is compression, the calculation method of the anti-sliding moment after the contact bond break is the moment of the macro friction about the sliding circle center, which can be calculated by the following equation. For particles of which the contact force type is tension, the adjacent particles are likely to separate from each other and there may be no friction between the adjacent particles after the corresponding bond breaks, so the anti-sliding moment is zero.
Therefore, the computational formula of E 2 can be expressed as follows:. A numerical model is built to help state the computing method of slope stability on the basis of characterization method of macro-contact.
The detailed geometry profile of the example and the corresponding model composed of granular material are shown in Figure 8 and Figure 9 , respectively. The model is constructed by the expansion of the random particles inside the slope boundaries. The particles in the slope top do not touch the boundary in the top in case there is additional loads on the slope top.
The macro parameters and the corresponding mesoscopic parameters of the example are shown in Table 2 and Table 3 , respectively. The mesoscopic parameters are determined by the calibration in the virtual biaxial compression test in DEM [ 17 ]. Firstly, the virtual biaxial compression tests with different mesoscopic parameters should be done. Then, the macro mechanical responses of the simulated results will be compared with the macro parameters of the material.
Finally, the mesoscopic parameter that can present the macro mechanical responses consistent with the material property could be selected to simulate the material. At first, a series of sliding surfaces can be preset. For the simplification of the calculation, the circular sliding surface is adopted. The sketch map is shown in Figure 10 in which the coordinate of circle center in Figure 10 is 3. As shown in Figure 11 , due to the randomness of granular materials, the particle contact is not necessarily on the sliding surface, but basically near the sliding surface.
The enlarged drawing of the sliding surface and the corresponding edges characterizing it. There are two adjustable parameters in the computing method of the slope stability mentioned above: the impact load caused by the bond break and the computing method of macro friction. The following is the discussion of the value of these adjustable parameters.
When the contact bond breaks, the sliding moment that is originally undertaken by the anti-sliding moment at this contact will be undertaken by the anti-sliding moment of its adjacent contacts suddenly. Therefore, the load undertaken by its adjacent contact is an impact load. The slope studied in this article is cohesive soil slope. Its failure mode is mostly brittle failure and the particle is arranged closely, so the velocity of the particle is comparatively low before the sliding mass start to slide.
As shown in Figure 12 , if a weight with no velocity is placed above a spring suddenly, the weight will be in situation of simple harmonic vibration and the contact force will reach maximum when the particle reaches the lowest point. In Figure 12 , F is the contact force between the weight and the spring, a is the accelerated speed and g is the acceleration of gravity.
The max contact force is the double of the particle weight. Therefore, when weight E 2 is reduced to the extent that the bond of the corresponding contact is judged to break, the part of weights E 1 which cannot be balanced by the corresponding weights E 2 will be doubled and then transferred to be undertaken by weight E 2 of its adjacent edges.
No matter the bond of these adjacent edges is judged to break under the impact load or not, the doubled load will convert back to the original load after this impact. The calculation method of the anti-sliding moment after the contact bond break is the moment of the macro friction about the sliding circle center.
The macro friction has two parts: the mesoscopic friction on the particle surface and the engagement force between the particles in contact.
Since the macro friction in this computing method is a kind of macro effect, it is not appropriate to select the mesoscopic friction on the particle surface for calculating the anti-sliding moment. The friction adopted in this article can be get in the following equation:. Therefore, the computing method of macro friction f is consistent with the limit equilibrium method. The particle displacement at the 10,th step after the reduction of the bond strength and the friction coefficient is shown in Figure The bond in the slope toe breaks first, which indicated that it is a retrogressive landslide.
The location of the maximal displacement is not precisely the slope toe but about 1 m higher than the slope toe. In order to analyze the slope, the solid line in Figure 13 is marked as the most dangerous slip surface calculated by Janbu method.
The circle center of sliding surface A is 4, 9. It is observed that the displacement of the slope is consistent with the result of Janbu method. The shape of the sliding mass in DEM is consistent of the result of Janbu method. The x coordinate of pull cracks in the slope crest is about 11, which is larger than the x coordinate calculated by Janbu method.
The y coordinate of the bottom of the sliding mass is 2. The particle displacement at the 10,th step after the reduction of mesoscopic parameter. The calculation result of the slope safety factor on the basis of the characterization method of the macro-contact is shown in Figure The result is presented by the color map of the vertex safety factors and the most dangerous sliding surface calculated by Janbu method is also marked in Figure The safety factor of a certain vertex is the minimum of the safety factors of the all the routes formed by this vertex.
The safety factors of the slope example calculated by the characterization method of the contact surface. In DEM, the configuration of the particles is random, which can present the heterogeneity of granular materials. Therefore, the results will present a degree of heterogeneity and cannot form a continuous contour map like FEM.
As is shown in Figure 14 , the safety factors of the vertexes are discrete but most of the value of the adjacent vertexes are close in general and can be classified into one band in the color map. For example, the yellow band represents the safety factor value ranging from 1.
However, due to the heterogeneity of the granular materials, some safety factors of the vertexes may differ from other safety factors signally in the same area and they cannot be classified into one band, which leads to the mixing phenomena in the boundary of two adjacent bands.
For example, there are some green vertexes 1. In general, the difference between the bands in mixing does not exceed one level. The result of the method raised in this article can still present the consistency with the theoretical method.
The location of the critical sliding surface in this method yellow band is adjacent to that of the theoretical method. The color of some vertexes in the slope surface is red and orange which represents the value between 1.
The yellow band 1. The minimum safety factor in the yellow band is 1. The location and shape of the yellow band and the result of Janbu method is similar except some small differences. Compared with limit equilibrium method, the location of the slope toe in yellow band is more adjacent to the particles with the maximal displacement and the location of the slope crest in yellow band is more adjacent to the particles with the tension fracture.
However, this does not mean that the result calculated by limit equilibrium method is inaccurate. The location of the critical sliding surface will change with the configuration of the particles. Therefore, the result presents a certain degree of randomness. The green band 1. The scope of the upper one is large and regular, which means that the safety factor in this band increases steadily and the increasing rate is slow. The downside of the green band is narrower than the upside part.
Most bands in the downside are cyan 2. That is because the particles under the slope toe can resist the sliding. The slope is simulated by FEM. According to the result of Li et al. Therefore, the result of this paper is compared with that of strength reduction method in FEM. As is shown in Figure 15 , the plastic strain magnitude in the reduced model is shown in figure.
The safety factor calculated by FEM is 1. The location of the critical sliding surface of the two models is also consistent. The calculated overall safety factor is a little lager compared with the result of Janbu method. The method in this paper does not consider the impact of dilatation softening. Although the slope is a cohesive slope and its failure mode is mainly brittle failure, the dilatation softening also can have a certain amount impact on its stability.
In the later period of the strength reduction, most bonds between the particles in the sliding surface have broken. The property of the particles without bond will be more like the property of coarse-grained soil. The particles without bond can rotate relative to other particles to some extent, which causes the decrease of the internal friction angle. Figure 16 is the safety factors of the slope example on the condition that the internal friction angle is transferred to half of its previous value.
The safety factors on the condition that the internal friction angle is transferred to half of its previous value. As can be seen in Figure 16 , there is an orange band 1. Chapter First Online: 01 April This process is experimental and the keywords may be updated as the learning algorithm improves. This is a preview of subscription content, log in to check access. Archard JF Elastic deformations and the laws of friction. Barton NR Review of a new shear strength criterion for rock joints.
Byerlee JD Friction of rocks. Identification, movement and causes. Wiley, Chichester Google Scholar. Duncan JM Soil slope stability analysis. Investigation and mitigation.
Special Report Keefer DK Investigating landslides caused by earthquakes — a historical review. Alternatively, if the object consists of a collection of materials like soil, clay, sand, etc.
Thus, down-slope movement is favored by steeper slope angles which increase the shear stress, and anything that reduces the shear strength, such as lowering the cohesion among the particles or lowering the frictional resistance. This is often expressed as the safety factor, F s , the ratio of shear strength to shear stress. Shear strength consists of the forces holding the material on the slope and could include friction, and the cohesional forces that hold the rock or soil together.
If the safety factor becomes less than 1. Although water is not always directly involved as the transporting medium in mass movement processes, it does play an important role. Water becomes important for several reasons Addition of water from rainfall or snow melt adds weight to the slope.
Water can seep into the soil or rock and replace the air in the pore space or fractures. Since water is heavier than air, this increases the weight of the soil. Weight is force, and force is stress divided by area, so the stress increases and this can lead to slope instability. Water has the ability to change the angle of repose the slope angle which is the stable angle for the slope. Think about building a sand castle on the beach. If the sand is totally dry, it is impossible to build a pile of sand with a steep face like a castle wall.
If the sand is somewhat wet, however, one can build a vertical wall. If the sand is too wet, then it flows like a fluid and cannot remain in position as a wall. Dry unconsolidated grains will form a pile with a slope angle determined by the angle of repose. The angle of repose is the steepest angle at which a pile of unconsolidated grains remains stable, and is controlled by the frictional contact between the grains. In general, for dry materials the angle of repose increases with increasing grain size, but usually lies between about 30 and 45 o.
Slightly wet unconsolidated materials exhibit a very high angle of repose because surface tension between the water and the solid grains tends to hold the grains in place. Water can be adsorbed or absorbed by minerals in the soil.
Adsorption, causes the electronically polar water molecule to attach itself to the surface of the minerals. Absorption causes the minerals to take the water molecules into their structure. By adding water in this fashion, the weight of the soil or rock is increased. Furthermore, if adsorption occurs then the surface frictional contact between mineral grains could be lost resulting in a loss of cohesion, thus reducing the strength of the soil.
In general, wet clays have lower strength than dry clays, and thus adsorption of water leads to reduced strength of clay-rich soils. Water can dissolve the mineral cements that hold grains together. If the cement is made of calcite, gypsum, or halite, all of which are very soluble in water, water entering the soil can dissolve this cement and thus reduce the cohesion between the mineral grains.
Liquefaction - As we have already discussed, liquefaction occurs when loose sediment becomes oversaturated with water and individual grains loose grain to grain contact with one another as water gets between them. This can occur as a result of ground shaking, as we discussed during our exploration of earthquakes, or can occur as water is added as a result of heavy rainfall or melting of ice or snow.
It can also occur gradually by slow infiltration of water into loose sediments and soils. The amount of water necessary to transform the sediment or soil from a solid mass into a liquid mass varies with the type of material. Clay bearing sediments in general require more water because water is first absorbed onto the clay minerals, making them even more solid-like, then further water is needed to lift the individual grains away from each other. Another material that shows similar swelling and compaction as a result of addition or removal of water is peat.
Peat is organic-rich material accumulated in the bottoms of swamps as decaying vegetable matter. As water infiltrates into the pore spaces, as discussed above, it can both be absorbed onto the clay minerals, and can dissolve away the salts holding the "house of cards" together.
Compaction of the soil or shaking of the soil can thus cause a rapid change in the structure of the material. The clay minerals will then line up with one another and the open space will be reduced. But this may cause a loss in shear strength of the soil and result in slippage down slope or liquefaction.
This is referred to as remolding. Clays that are subject to remolding are called quick clays. Weak Materials and Structures. Weak Layers - Some rocks are stronger than others.
In particular, clay minerals generally tend to have a low shear strength. If a weak rock or soil occurs between stronger rocks or soils, the weak layer will be the most likely place for failure to occur, especially if the layer dips in a down-slope direction as in the illustration above.
Similarly, loose unconsolidated sand has no cohesive strength. A layer of such sand then becomes a weak layer in the slope. If the joints are parallel to the slope they may become a sliding surface.
Combined with joints running perpendicular to the slope as seen in the sandstone body in the illustration above , the joint pattern results in fractures along which blocks can become loosened to slide down-slope. A mass movement event can occur any time a slope becomes unstable. Sometimes, as in the case of creep or solifluction, the slope is unstable all of the time and the process is continuous.
But other times, triggering events can occur that cause a sudden instability to occur. Here we discuss major triggering events, but it should be noted that it if a slope is very close to instability, only a minor event may be necessary to cause a failure and disaster. This may be something as simple as an ant removing the single grain of sand that holds the slope in place.
Turnagain Heights Alaska, During the Good Friday earthquake on March 27, , a suburb of Anchorage, Alaska, known as Turnagain Heights broke into a series of slump blocks that slid toward the ocean.
This area was built on sands and gravels overlying marine clay. The upper clay layers were relatively stiff, but the lower layers consisted of a sensitive clay, as discussed above. The slide moved about m toward the ocean, breaking up into a series of blocks. It began at the sea cliffs on the ocean after about 1. As the slide moved into the ocean, clays were extruded from the toe of the slide. The blocks rotating near the front of the slide, eventually sealed off the sensitive clay layer preventing further extrusion.
This led to pull-apart basins being formed near the rear of the slide and the oozing upward of the sensitive clays into the space created by the extension. The peak consists of granite with nearly vertical joints fractures covered by glacial ice.
On January 10, a huge slab of rock and glacial ice suddenly fell, with no apparent triggering mechanism. This initiated a debris flow that moved rapidly into the valley below and killed 4, people in the town of Ranrahirca, but stopped when it reached the hill called Cerro de Aira, and did not reach the larger population center of Yungay.
On May 31, a magnitude 7. The avalanche then hit a small hill composed of glacially deposited sediment and was launched into the air as an airborne debris avalanche. From this airborne debris, blocks the size of large houses fell on real houses for another 4 km. The mass then recombined in the vicinity of Cerro de Aira and continued flowing as a debris flow, burying the town of Yungay and its 18, residents.
The debris flow reached the valley of the Rio Santa and climbed up the valley walls killing another people on the opposite side of the river. Example: Elm Switzerland, In s there was a large demand for slate to make blackboards throughout Europe. To meet this demand, miners near Elm, Switzerland began digging a slate quarry at the base of a steep cliff. Slate is a metamorphic rock with an excellent planar foliation that breaks smoothly along the foliation planes. By a "v" shaped fissure formed above the cliff, about meters above the quarry.
By September , the quarry had been excavated to where it was m long and 60 m into the hill below the cliff, and the "v" shaped fissure had opened to 30 m wide. Falling rocks were frequent in the quarry and their were almost continuous loud noises heard coming from the overhang above the quarry. Realizing that the slope had become unstable, the miners stopped working, thinking that the rock mass above the quarry would probably fall down. On September 11, the 10 million m 3 mass of rock above the quarry suddenly fell.
But, it did not stop when it hit the quarry floor. Instead, it broke into pieces and rebounded into the air.
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