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23 ments. Whereas design charts or simplified expressions are which can be carried out with most two-dimensional slope available to provide design guidance, improvements were stability programs, a wedge failure under seismic excitation needed to better reflect the above variables and to provide is not widely analyzed. Analyses for the toppling failure, a basis for nationwide application and to use as a screening which generally involves moment equilibrium, rarely are tool to establish "no seismic analysis" criteria based on used in practice due to the complexity of the problem and appropriate serviceability criteria. Caltrans guidelines, for lack of adequate rock properties for carrying out meaning- example, use a "no analysis" screening criteria based on ful solutions. pseudo-static factors of safety greater than 1.1 when a seis- Often the seismic performance of the rock slope is expressed mic coefficient of 1/3 of the maximum ground acceleration in terms of a pseudo-static factor of safety. The challenge was used. faced by the practicing engineer involves assigning appro- For slopes and embankments of limited height, say less than priate shear strength parameters on the weakness plane about 30 to 40 feet, the assumption of a rigid sliding block where sliding is anticipated. Some engineers may be reluc- and the use of ground acceleration parameters to define tant to assign cohesion to the joint surface due to lack of inertial lateral forces was thought to be a reasonable approx- `stickiness' as found in a clayey soil. In fact, this assumed imation. For higher slopes and embankments, however, cohesive strength is defined by the intercept on the shear where the dynamic response of the sliding mass may influ- strength axis, of a tangent of a curvilinear Mohr envelope. ence displacement magnitudes, modifications to computed This curvature is the result of the interlocking of aspirates Newmark displacements were required, depending on the on the matching surface of the joints. Furthermore, labo- comparative natural period characteristics of the earth- ratory direct shear tests are generally conducted on small quake ground motion and the slope. Such modifications are rock specimens, and thus dilation due to waviness (undu- included, for example, in the design methods documented in latory nature) of the joint that has a wave length longer the SCEC (2002) recommended procedures. An approach than the specimen size is not captured in the test. These for analytical development is described in Chapter 4 to conditions would increase the gross shear strength proper- address this issue. ties of the joint planes when a large failure surface is consid- ered. When a large block failure is considered, the potential failure plane is likely to go through the existing discontinu- 3.2.2 Seismic Considerations for Rock Slopes ities and to shear the intact rock that bridges the joint planes. Rock slopes are encountered in many situations--both In this case, the shear strength parameters assigned to the urban and mountainous terrain. Some considerations related potential failure plane in a limit equilibrium analysis to these types of slopes are summarized below. should include some portion of the intact rock strength. These increases in shear strength play a crucial role in the In regularly bedded or foliated rock, cut by joints, there are stability of rock slope. many possibilities for block movement along weak planes. The seismic design of the rock slope can be further improved Where there are multiple sets of discontinuous planes by a deformation analysis involving a Newmark sliding intersecting at oblique angles, three failure modes must be block analysis on the failure plane. The Newmark sliding examined: plane sliding, wedge sliding, and toppling. A analysis for a plane failure is relatively simple to perform; plane slide can form where a block of rock rests on an however, for the wedge failure, it requires modification to inclined plane that dips downward and intersects the face deal with sliding on two planes under three-directional of the slope. A wedge slide can occur where two planes of loading. The resultant vector of the inertial body forces act- weakness intersect to define a tetrahedral block. Toppling ing onto each joint plane due to the three-component failure can develop from overturning of certain types of acceleration is checked against the yield acceleration of the rock, such as slates and schists, that have bedding planes joint. Sliding can take place on either plane or along the inclined steeply into the hillside. interception of the two planes depending on the direction In practical solutions, the plane failure is examined using a of the loads at any given instance of time. This type of two-dimensional limit equilibrium approach treating the analysis provides a rational basis for deformation analysis seismic inertial load as a constant horizontal acceleration of the wedge failure. acting on the potential failure block. For the wedge failure, three-dimensional limit equilibrium wedge analyses using Although these seismic performance considerations can be stereographic projection of joints and open free surface identified, it was also apparent that a transparent approach orientations are used for gravity loading. While the con- for evaluating the seismic response of rock slopes could not sideration of seismic loads in terms of pseudo-static accel- be developed into a guideline consistent with the simplified eration can readily be implemented for the plane failure approaches needed for these AASHTO LRFD Bridge Design