 On tape one of this program, Dr. Idris covered observed behavior of embankment dams in previous earthquakes. He also presented a review of geologic, seismologic, and earthquake ground motion issues. In this tape, Dr. Idris will describe methods for analyzing and evaluating embankment dam performance and the potential for deformations, including soil strength. He will also cover instrumentation and make concluding remarks. It gives me great pleasure to once again introduce Dr. Ed Idris. The case history summarized earlier indicated that the primary cause of damage or failure of dams with saturated cohesionless soils has been the buildup of excess pore water pressure due to the strong shaking and the possible loss of strength. This buildup of excess pore water pressure can lead to the phenomenon called liquefaction. In many instances, liquefaction is manifested by the formation of sand boils and mud spouts at the ground surface as shown in this photograph, which was taken during an earthquake in northern Japan in 1978. The consequences of liquefaction have been settlement, uniform in a few cases, but mostly abrupt and non-uniform. Lateral spread, limited lateral movement, for example, upper San Fernando Dam. Lateral flow, extensive lateral movement, for example, lower San Fernando Dam. It may be useful to briefly review this phenomenon at this time. When a saturated sand is subjected to ground vibrations, it tends to compact and decrease in volume. If drainage is unable to occur, the tendency to decrease in volume results in an increase in pore water pressure. Thus, the basic cause of liquefaction in saturated cohesionless soils during earthquakes is the buildup of excess pore water pressure due to the application of cyclic shear stresses induced by the ground motions. As a consequence, the structure of the cohesionless soil tends to become more compact with a resulting transfer of stress to the pore water and a reduction in the stress on the soil grains. The soil grain structure must rebound to the extent required to keep the volume constant as illustrated in this figure. As the pore water pressure approaches a value equal to the initial effective confining pressure, the sand begins to undergo deformations. If the sand is loose, the pore pressure will increase suddenly to a value equal to the initial effective confining pressure and the sand would rapidly begin to undergo large deformations with shear strains that may exceed 20%. If the sand will undergo virtually unlimited deformations without mobilizing significant resistance to deformation, the sand can be said to have liquefied. If on the other hand the sand is dense, it may develop a residual pore water pressure on completion of a full stress cycle. This pore water pressure would be equal to the initial effective confining pressure but when the cyclic stress is reapplied on the next stress cycle or if the sand is subjected to monotonic loading, the soil will tend to dilate, the pore pressure will drop if the sand is undrained and the soil will ultimately develop enough resistance to withstand the applied stress. However, the sand will have to undergo some degree of deformation to develop resistance and as the cyclic loading continues, the amount of deformation required to produce a stable condition may increase. Liquefaction of sand in this way may develop in any zone of a deposit where the necessary combination of in situ conditions and vibratory deformations may occur. Such a zone may be at the surface or at some depth below the ground surface depending only on the state of the sand and the induced motions. Liquefaction of the upper layers of a deposit may also occur not as a direct result of the ground motions to which they are subjected but because of the development of liquefaction in the underlying zone of the deposit. Once liquefaction develops at some depth in a mass of sand the excess hydrostatic pressures in the liquefied zone will dissipate by flow of water in an upward direction. If the hydraulic gradient becomes sufficiently large the upward flow of water will induce a quick or liquefied condition in the surface layers of the deposit. Similarly, liquefaction and embankment may be initiated at some location then spread to other parts as the water flows away from the initially liquefied zone. Liquefaction of this type will depend on the extent to which the necessary hydraulic gradient can be developed and maintained. The behavior of sand specimen was examined in the laboratory by conducting cyclic loading tests on reconstituted as well as some relatively undisturbed samples to represent various initial as well as loading conditions. These tests have been conducted in traxial, in simple shear and in torsional shear testing machines as well as in a shaking table which may be thought of as a large simple shear test. Most tests have been conducted under stress controlled conditions. A typical test result for a relatively loose sand sample is shown in the figure. The lower part of the figure shows the applied cyclic shear stress, the middle part shows the amount of shear strain induced in the sample and the upper part shows the amount of excess pore water pressure generated during the test. It's interesting to note that until the excess pore water pressure reaches a value equal to the initial confining pressure which is defined as 100% pore pressure ratio, the amount of induced strain is negligible. When the pore pressure becomes equal to the initial confining pressure, the induced strain in the sample increases very rapidly. For this test, reaching a condition of 100% pore pressure ratio coincides with the onset of inducing very large strains in the sample. This behavior is typical of a sample of loose sands. Medium dense and dense samples of sand behave significantly differently as shown in this figure. A dense sand can continue to sustain an applied load, although 100% pore pressure ratio had been reached. This ratio typically is reached at about 2-3% strain for many cohesionless soils. Reaching 100% pore pressure ratio has been defined as a condition of initial liquefaction. Tests completed in the past some 35 years indicate that the tendency for a sand to liquefy is affected by 1. Density, relative density or relative compaction 2. Effective confining pressure 3. Level of shaking, applied shear stress, often expressed in terms of the ratio of the cyclic shear stress divided by the initial effective normal stress on the plane of failure 4. Duration of shaking, or number of cycles 5. Soil type and soil structure or fabric 6. Stress conditions before onset of shaking 7. Other factors such as seismic history, geologic history, which may reflect aging, cementation and other factors For example, this figure shows the variation of cyclic stress ratio with a number of cycles required to reach initial liquefaction depending on the relative density of the soil The greater the relative density, the larger is the stress required to cause liquefaction Field performance data for cases of liquefaction or no liquefaction at sites in Japan, California and other parts of the world have also been used to establish a relationship between the cyclic stress ratio required to cause liquefaction and some index property that reflects the various parameters we just discussed Initially, relative density was used as the index, but was quickly abandoned because it was insufficient to reflect many of the key parameters such as fabric and aging SPT blow count and CPT tip resistance can reflect most of the important factors that affect susceptibility to liquefaction This figure summarizes the field cases of liquefaction and no liquefaction for essentially clean sands That is, fines content less than 5% in terms of the cyclic stress ratio CSR versus modified SPT blow count N160 Similar field cases facility sands with various fines content are also available and were used to develop the relationships shown in this figure The lower curve in this figure shows the relationship developed for relating the cyclic stress ratio to the corrected SPT blow count N160 for essentially clean sands The middle curve is for fines content of 15% and the upper curve is for cohesion less soils with fines content greater than or equal to 35% Similar curves relating cyclic stress ratio to CPT tip resistance and curves relating cyclic stress ratio to shear wear velocity have also been developed and can be used in many cases It has always been my belief that if the cyclic resistance of a soil is to be based on such curves the use of the SPT based curves is probably sufficient by itself because the sample is usually obtained when the SPT blow count is collected and the soil type can then be ascertained and the fines content can be measured if needed Such information is not explicitly available for CPT or shear wear velocity measurements Therefore I believe it is crucial to augment a CPT based or shear wear velocity based evaluation by sampling When using laboratory test data such as those shown in this figure it is necessary to select a number of cycles for evaluating the cyclic resistance to liquefaction The number of cycles has been related to earthquake magnitude as follows Magnitude 6.5, number of cycles equals to 7 Magnitude 7.5, number of cycles equals 15 Magnitude 8.5, number of cycles equals 33 The curves based on observed field cases apply to an earthquake magnitude 7.5 an effective confining pressure of 1 ton per square foot, approximately 100 kPa and essentially conditions with no static shear stress on the plane of failure that is alpha equals to zero For other magnitude earthquakes the curves can be adjusted by applying a magnitude scaling factor MSF which is given by this equation To obtain the cyclic stress ratio for confining pressure other than 1 ton per square foot the cyclic stress ratio is multiplied by a factor K sigma which depends on the value of the effective confining pressure as shown in this figure The symbols in the figure represent the results from a number of cyclic tests The curves represent values recommended in various studies or projects The most recent work on this subject however suggests that the variation of K sigma with the effective confining pressure also depends on the relative density of the soil with the effect being leased for looser soils This subject is still under evaluation and I suggest that the latest work we consulted prior to selecting a K sigma relationship for use in a specific application The initial static shear stress on the potential failure plane also influences the resistance to liquefaction The ratio of this static shear stress divided by initial effective vertical stress on the plane of failure is defined as alpha as shown in the figure Adjustments for the effects of the initial shear stress on the potential failure plane that is alpha not equal to zero are accommodated by multiplying the cyclic stress ratio for alpha equal to zero by a factor K alpha which appears to depend on alpha as well as the denseness of the soil as shown in this figure It should be noted that the relationships shown in this figure are for an effective confining pressure less than 300 kPa The values for K alpha shown in this figure will be different for other values of confining pressure For confining pressure greater than 300 kPa the curves in the figure would have to be rotated clockwise that is giving a lower value of K alpha The ratio of K alpha with N 160 or some other measure of denseness and with confining pressure is also another subject that is being evaluated further Again, it is recommended that the latest work be consulted prior to selecting a K alpha relationship for use in a specific application Hence the cyclic stress ratio for a part of the dam or its foundation at a given effective vertical stress and an alpha and with a corrected SPT blow count N 160 is given by this equation This value of the cyclic stress ratio will then be multiplied by the effective vertical stress to obtain the cyclic resistance at that location if the duration of shaking were to be representative of a magnitude seven and a half earthquake Usually this value of cyclic resistance is to be compared to the stresses induced by the earthquake at that location of the dam or its foundation The values of stresses induced by the earthquake in the dam or its foundation are usually obtained from a dynamic analysis which will be discussed later In some applications it may be appropriate to utilize the seed address simplified procedure to obtain a first order approximation of the value of the induced stress at a specific depth Caution should be exercised when using the latter procedure because it is quite possible to overestimate the inertial force acting at that depth since the simplified procedure would be right for a level of ground If the size of the earthquake being considered is not seven and a half then I prefer to divide the induced stress by the corresponding magnitude scaling factor This will lead to a value of induced stress equivalent to that induced by an earthquake whose magnitude is seven and a half Thus the equivalent induced stress is given by this equation This equivalent induced stress can then be compared to the cyclic resistance obtained for magnitude seven and a half to assess the potential for liquefaction at that location within the dam or its foundation The ratio of the available cyclic resistance divided by the induced stress is typically defined as a factor of safety against liquefaction When this factor of safety is equal to or less than one liquefaction is considered to be triggered within that part of the dam or its foundation The actual performance of the dam and its foundation, however, is dependent on the location and extent of the zone where liquefaction is triggered and on the strength of the soil that liquefied The sheer resistance of the liquefied soils has been estimated based on back analysis of available case histories The results expressed in terms of residual strength versus equivalent clean sand N160 are shown in the figure Note that points A and B are for the upper and lower San Fernando dams The available data from actual slides are for equivalent clean sand N160 up to 15 The curve presented in this figure is extended to a residual strength of 1,000 PSF at an equivalent clean sand N160 value of 20 Additional studies are needed to ascertain how this curve might be extended beyond N160 of 20 The equivalent clean sand N160 is obtained from the measured N160 by adding a delta N1 that depends on the fines content of the soils as follows For a fines content equal to 10%, delta N1 equals 1 For a fines content equals to 25%, delta N1 equals 2 For a fines content equals to 50%, delta N1 equals 4 And for a fines content equals 75%, delta N1 equals 5 These values of delta N1 were initially proposed by the late professor H. Bolton Seed who proposed basing the residual soil strength on evaluation of actual slides A limit to the residual strength should always be the drained strength of the soil For cohesionless soils having fines content greater than or equal to 25%, it is useful to re-plot the data without including delta N1 and using the resultant relationship for a specific application that involves cohesionless soils having such high fines content The use of the steady state strength has also been proposed to estimate the residual strength of a liquefied soil The measurements would be made on very high quality, relatively undisturbed samples of the soils under consideration The presence of less permeable soil layers overlying the liquefied soils that could impede drainage and result in void ratio redistribution cannot properly be accounted for in a typical steady state test This possibility is illustrated in the figure and shows how, due to shaking the upper portion of the sand layer would tend to become looser while the bottom portion would tend to become denser The steady state strength of the upper portion is significantly lower than that for a soil having the initial void ratio of the entire cohesionless soil layer Accordingly at this time, it is preferable to use the residual strength based on case histories rather than relying on laboratory test data This figure illustrates the behavior of a cohesionless soil depending on its denseness The bottom curve in this figure is for a specimen of loose sand which is compactive and hence it would tend to stabilize the residual strength far lower than its peak strength The top curve is for a specimen of very dense sand, which is dilative and its shear strength would tend to increase as it is sheared further The middle curve is for a specimen of sand of intermediate denseness which could initially reduce the strength to a quasi-steady state beyond which the soil would dilate thus increasing its shear strength as shown in the figure Therefore, when using residual strength in calculating post-earthquake performance it is necessary to keep in mind that relatively loose sand must experience large strains to achieve its residual strength Hence, it is more important to estimate the formation following the occurrence of an earthquake than to just calculate the factor of safety against sliding based on limited equilibrium This aspect will be discussed later when we turn our attention to analysis and evaluation procedures The above discussion pertains to cohesionless soils Occasionally we are concerned also with the behavior of cohesive soils Most cohesive soils do not suffer appreciable reduction in strength due to cyclic loading Typically a reduction in strength of the order of 10-20% may be reasonable to assume depending on the intensity of shaking We must keep in mind, however, that some cohesive soils with very high water content have exhibited significant loss of strength due to shaking These soils are usually evaluated using the so-called Chinese criteria In most cases, however, results of cyclic tests on properly prepared samples of such soils may prove to be more appropriate to use than applying the Chinese criteria The Chinese criteria are most useful for screening purposes and I strongly suggest that the criteria be applied with a great deal of caution Since the Chinese criteria utilize water content, liquid limit, and clay fraction some uncertainty should be applied to each of these parameters when applying these criteria And now we need to discuss methods of analysis The case histories examined earlier indicate that it is necessary to analyze the dam and its foundation not only during shaking, but also after the shaking stops The methods that are used to evaluate the performance of embankment dams during shaking are one, pseudo-static analyses, two, dynamic analyses and those used to evaluate the performance after the shaking stops are static analyses The pseudo-static method of analysis has been used for the past several decades to evaluate the stability of earth dams against sliding during an earthquake The effects of the earthquake on a potential sliding mass are represented by a static horizontal force applied at the center of gravity, 0.01, in the figure of the potential sliding mass This horizontal force is the product of the weight of the potential sliding mass and its seismic coefficient K The value of K was intended to reflect the severity of the earthquake ground motion that may be felt by the dam during future earthquakes In the U.S., seismic coefficients have varied from 0.05 to 0.15 even in highly seismic areas such as California and Alaska Incorporating these forces, a factor of safety is calculated using available slope stability programs When this factor of safety equals or exceeds a designated minimum the dam is considered safe against failure during an earthquake On the other hand, if the calculated factor of safety is less than the designated minimum this section is modified, typically by flattening the slopes Analysis of the landslides that occurred in Alaska during the 1964 earthquake as well as the failure of the Sheffield Dam in the 1925 Santa Barbara earthquake led to the conclusion that the pseudostatic approach would not have predicted these failures The evaluations of the slides of the upper and lower San Fernando dams in the 1971 earthquake also led to the same conclusion Therefore, this method of analysis should not be used to judge the stability of a slope under earthquake loading conditions Nevertheless, as we will discuss later, this method has proven to be useful when used in conjunction with the Newmark analysis to estimate movement of a potential sliding mass As for dynamic analyses The material characteristics of soils are non-linear Therefore, any analysis procedure must be able to accommodate these characteristics Initially, dynamic analyses for embankments were conducted using an equivalent linear procedure This procedure utilizes modulus and damping values for the soils which are dependent on the strains induced in each part of the embankment These procedures have been found to provide reasonably accurate means for calculating shear stresses Fully non-linear analyses, however, are needed to obtain reasonably accurate estimates of movement Equivalent linear approaches have relied on frequency domain solutions For example, programs Flush and Sassy are on time domain solutions For example, programs Quad 4 and Quad 4M Non-linear analyses can either include or not include the generation and dissipation of excess pore water pressure The simplest of these analyses is the equivalent linear approach which has been used extensively in the past 35 years with the seismic evaluations of numerous dams both in the U.S. and in other countries Its first application was in the evaluation of the failure of the Sheffield Dam which resulted in the development of what is known as a seedly address analysis procedure This procedure was refined based on the evaluations of the slides of the San Fernando dam The steps involved in this analysis procedure are the following 1. Develop the cross-section of the dam to be used in the analysis 2. Select an accelerogram for use as input at the base of the dam foundation system This selection is to be completed in cooperation with geologists and seismologists 3. Determine as accurately as possible the stresses existing in the embankment before the earthquake Currently available non-linear finite element procedures are usually used to complete this step 4. Determine the strain-compatible dynamic properties of the soils comprising the dam 5. Compute using an appropriate dynamic finite element analysis that stresses induced in the embankment by the selected base excitation 6. Assertain the cyclic resistance of the soils comprising the embankment This may be done by testing appropriate samples of the soils or by relying on correlations with in-situ properties such as SPT or CPT 7. Evaluate the potential for triggering liquefaction in the saturated cohesion in soils and the potential reduction in strength of the cohesive soils of the dam or foundation 8. Check the stability of the cross-section after the shaking has stopped and incorporating the results of step 7 The application of the seed-lea-address analysis procedure is illustrated in the next few figures as it was applied to the lower San Fernando dam which suffered a massive upstream slide as a result of the shaking it's received during the 1971 San Fernando earthquake The upper part of the first figure shows the maximum cross-section of the dam and its foundation The lower part of the figure shows the calculated initial that is prior to the earthquake vertical normal stresses sigma v' along the horizontal profile depicted in the upper part of the figure The next figure shows the calculated initial shear stresses tau along the same profile The ratio of sigma v' over tau that is alpha is depicted in the lower part of this figure The static stresses shown in these figures were calculated using a non-linear finite element program developed by Professor Duncan and his colleagues The accelerogram used as input to the base of the dam foundation is shown in this figure This accelerogram was constructed by Professor R. F. Scott using the seismoscope record attained at the abutment of the dam The program quad 4 was used to calculate the shear stresses induced by the shaking The equivalent uniform shear stresses induced along the same horizontal profile shown previously after the first 10.5 seconds of shaking are presented in the lower part of this figure The number of equivalent uniform cycles estimated for this duration using 65% of the maximum induced shear stress is 2 cycles Also shown in the lower part of the figure are the values of the cyclic stress required to cause liquefaction in 2 cycles The latter values had been determined from the results of cyclic triaxial test conducted on samples of the shell material The portions along the horizontal profile where the induced stresses exceed the stresses required to cause liquefaction are identified as having liquefied and are depicted in red in the upper part of the figure Similar evaluations were made along additional horizontal profiles within this section of the dam to identify the portion of the shells that would liquefy during this earthquake These are shown in red in this figure As the shaking continues, the extent of liquefaction is increased within the shell as shown in this figure The full extent of liquefaction within the shells is depicted in this figure for the entire duration of shaking The results of the dynamic analyses coupled with the laboratory cyclic test results provided the means to estimate the extent of liquefaction in the shells of the lower San Fernando Dam The shear strength within the liquefied portion is reduced and the possibility of a slide is checked by conducting a limit equilibrium static slope stability analysis as shown in this figure This analysis indicates that the slope is at best on the verge of slipping More recent work on the post-earthquake stability of this cross-section provided an estimate of the residual strength of the liquefied soils within the upstream shell of this dam As I noted earlier, equivalent linear procedures are adequate for calculating shear stresses induced by the earthquake ground motions The equivalent linear procedure is useful in assessing the extent of concern with an existing embankment dam If results indicate that no significant portions of the dam are likely to have significant reduction in strength due to shaking then the dam will most likely perform quite well during future earthquakes If, on the other hand, this type of analysis indicates that large portions of the dam can have major reduction in strength due to liquefaction being triggered then further evaluations are needed to judge how poorly the dam would perform if no remediations are implemented Limit equilibrium slope stability analyses would be useful to conduct in the following sequence 1. Assign the appropriate residual strength of the soils in the liquefied zones If the calculated factor of safety is less than or slightly above unity, then this dam is likely to have a flow slide and remediation is necessary If, on the other hand, the factor of safety is well above unity, conduct the analysis suggested in step 2 Assign zero or near zero strength of the soils in the liquefied zones If the calculated factor of safety is still well above unity, then the dam is likely to experience only minor movements and no extensive remediation is needed If, on the other hand, the calculated factor of safety is now less than or slightly above unity, then this dam is likely to have major movement and remediation may be necessary To judge the need and extent of remediation, deformation analyses are required To evaluate deformations of a dam, it is necessary to utilize a fully non-linear procedure to estimate movements or deformations In many cases, it is sufficient to compute these deformations for the post-earthquake conditions Static non-linear analyses can be used for this purpose This would be the case if the duration of shaking is relatively short so that liquefaction is most likely to occur near the end of shaking In some instances, it is necessary to estimate the deformed shape of the embankment during shaking and after shaking stops Dynamic non-linear analyses are needed for such calculations These analyses are usually needed when the duration of shaking is relatively long and liquefaction is likely to occur well before shaking stops At this time, there are several procedures that have been used for calculating the deformed shape for an earth dam to the earthquake When using non-linear procedures, it is important that appropriate calibration of the procedure and the associated computer program have been done and verified The end of this video includes references to non-linear procedures that have been used most widely in the past few years To examine the usefulness of using non-linear procedures, I will summarize two cases One case involves the Sardis Dam This dam is located in northwestern Mississippi It was built in the 1930s using hydraulic filling techniques As shown in the cross-section, the dam has a silt core which is surrounded by the hydraulically placed sand shells The upper part of the foundation consists of what is known as the top stratum clay Which is underlain by a relatively competent alluvial deposit consisting of sands underlain by tertiary sills and clays Sardis Dam is a U.S. Army Corps of Engineers Dam and was evaluated by the Vicksburg District for a magnitude 8 earthquake Occurring in the Neumatrid Science Week zone and resulting in a peak acceleration of 0.2 G in the free field at the dam side The Corps of Engineers was assisted in this evaluation by Woodward Client Consultants, the Waterways Experiment Station, Professor James K. Mitchell, and Professor Liam Finn An equivalent linear analysis indicated that liquefaction is likely to occur in the hydraulically placed silt core and in the lower portions of the hydraulically placed upstream shell The upper part of the top stratum clay was judged to have the potential to suffer major loss of strength and was assigned a residual strength corresponding to an effective angle of friction of about 4.5 degrees The post-earthquake factor of safety of the weakened cross-section was well below 1 indicating that the dam is likely to form significantly due to the earthquake ground motions under consideration The deformations of the dam due to shaking and post-shaking conditions were calculated using the computer program TARA The results of this analysis are shown in the figure and indicate that the upstream shell would slide sufficiently to result in a precariously low freeboard which could impair the stability of the dam Accordingly, the Corps of Engineers decided to remediate the dam After evaluating many of the potential remediation options, concrete piles were selected to be driven through the lower part of the upstream shell and well below the top stratum clay as shown in the figure thus providing the additional resistance to constrain movement of the upstream shell The layout of the piles is shown in this figure The dam is approximately 4,600 meters long but only two segments of the dam required remediation One segment is about 213 meters long where 676 piles were driven The other segment is 610 meters long where 1,918 piles were driven for a total of 2,594 piles It is noteworthy that the non-linear deformation analyses were extremely helpful in optimizing the location and the spacing of the piles The second example is the wiki-up dam in northern Oregon A cross-section of the dam is shown in the figure The performance of the dam during a great earthquake occurring on the Cascadia subduction zone producing a peak acceleration of only 0.1 g but with a duration of over two minutes was evaluated by the Bureau of Reclamation The Bureau is assisted in this evaluation by URS Woodward Clyde The computer program FLAC Incorporating a Non-Linear Soil Model was used to assess the deformation pattern of the dam in its existing condition and as remediated Again, the non-linear analyses were very helpful in optimizing the solution which consists of using jet ground to create two blocks within the foundation and the placement of a downstream buttress as shown in the figure Let me now take a few minutes to talk about the Numark analysis Professor Numark proposed a simple method for computing the displacement of an embankment This method is based on the concept that movement would be initiated when the inertial force acting on a potential sliding mass exceeds the yield resistance of that mass This movement would persist so long as the inertial force is larger than the yield resistance The movement would stop when the inertial force becomes less than yield resistance This approach is illustrated schematically in the figure The weight of the block is w and the inertial force is given by the weight multiplied by seismic coefficient k which represents the acceleration of the block The seismic coefficient is expressed as a function of time kT as shown in the figure The block would move down the slope when the inertial force exceeds the yield resistance shown as the weight of the block times the yield coefficient ky The yield resistance for the schematic shown is dependent on the inclination of the slope and on the frictional resistance between the mass and the slope For an actual slope the value of ky is obtained by conducting a series of pseudo-static analyses each with an increasing value of the applied seismic coefficient k This figure is schematic of the forces involved in the pseudo-static analysis The calculated factor of safety is plotted as a function of k to obtain a graph as shown The yield coefficient ky is the value of the applied seismic coefficient that corresponds to a calculated factor of safety of 1 as shown in the figure Numark considered the sliding mass to be rigid and hence assumed that the inertial force is caused by the acceleration applied at the base of the block that is, k as a function of time is equal to the acceleration of the base as a function of time Many analyses have been conducted using this assumption The modeling work by Goodman and Seed in 1966, later refinements by Seed and his colleagues by Embrazi's and Sarma, suggested that the inertial forces need to incorporate the dynamic response characteristics of the potential sliding mass Therefore, they suggested using essentially the average acceleration within this mass to represent the seismic coefficient k as a function of time Many of the analyses using the Numark method have incorporated the LiDAR modification in the past 30 plus years The values of k as a function of time for a potential sliding mass in an embankment can be obtained using one of the dynamic analysis programs discussed earlier One-dimensional programs, such as Shake, have been used for this purpose My personal preference is for the use of a two-dimensional program to account for the correct geometry of the potential sliding mass Once k as a function of time is obtained for a potential sliding mass the Numark analysis is carried out by integrating the portions of k from t1 to t2 that exceed ky as shown in the figure This integration provides values of velocity over a time increment, t1 to t3 An integration of the velocity trace provides the value of permanent displacement the potential sliding mass would experience up to time t3 As shaking continues and when k again exceeds ky, additional permanent displacement occurs as shown This process is continued until the end of shaking As noted in the figure, the value of ky need not be kept constant for the entire duration of shaking Reduction in ky, to account for possible loss of strength, can be incorporated in such an analysis However, care should be exercised if this approach is attempted and not only will ky decrease if the strength is reduced but there would also be changes in the value of the seismic coefficient Approximate values of potential movements can be obtained this way but to obtain a more accurate estimate and to examine the sense of potential movements in nonlinear dynamic analysis is required The Numark analysis has been simplified by McDixie and Seed and by Franklin and Chang who produced charts such as that shown in this figure This chart provides estimates of displacement of a potential sliding mass if the value of ky is known for this mass and if the maximum value of the seismic coefficient k max is also known k max can be obtained directly from the time history of k as a function of time or using approximate procedures such as that suggested by McDixie and Seed Note that the value of k max to be used with such charts is the peak and not an equivalent uniform value Charts such as that shown in this figure have proven to be useful for screening purposes and for initial estimates of potential deformations of an embankment dam They're also useful for making a quick check of whether the potential movements would be shallow or deep seated by examining various potential sliding masses Note that reduction in ky due to the loss of strength caused by shaking can also be accommodated using this chart by carrying out the analysis per cycle instead of for the full duration of shaking For example, the range of displacement for a magnitude 7.5 earthquake can be divided by the number of cycles for this magnitude which is estimated to be 15 cycles as we discussed earlier in this video A new plot can then be created to exhibit the displacement per cycle for magnitude 7.5 as a function of the ratio ky divided by k max The engineer can then estimate the value of ky and k max for each cycle starting with cycle number one which would have the largest value of ky that is no strength reduction as yet The value of ky can be adjusted for subsequent cycles depending on the extent of the strength reduction estimated up to that cycle Adjustment to ky max can also be incorporated Displacement of the potential sliding mass can then be calculated for each cycle and the total displacement would be the sum for all cycles This approach was used to back calculate the displacement in the 4th avenue slide in Anchorage during the 1964 earthquake where a strength loss of 2.3 was applied when a movement of the slope exceeded 1.5 foot Further details about this analysis are given in the reference listed at the end of this video This approach is not recommended for cases where the strength loss in a potential sliding mass is extensive and can occur early in the shaking For example, when the soil has very low penetration resistance Instrumentation plays a vital role in embankment engineering My reference here is specific to providing instrumentation for taking measurements during and or after an earthquake Pizometers, settlement monuments, and strong motion recorders are key instruments that need to be installed at strategic locations Practically all dams have pizometers and settlement monuments Many of these instruments, however, may not have been installed at locations that provide optimum information following the occurrence of an earthquake It would be useful to re-evaluate such instrumentation and adjust, modify, or augment the existing number and layout for many existing dams Strong motion recorders have been placed at many dam sites but numerous dams remain uninstrumented These instruments provide extremely valuable data when an earthquake occurs within certain distance of the dam Downhole arrays with the crest of the dam and at other locations in the dam or downstream of the dam are also very valuable to have At this time, 2002, only a few dams have such downhole arrays installed The San Husto Dam in California was equipped with such an array prior to the 1989 Loma Prieta earthquake The location of this dam with respect to the rupture zone in the Loma Prieta earthquake is shown in this figure The next figure shows the dam footprint in the locations of strong motion instruments along the crest, along the slope, at the toe, and in the free field Another strong motion instrument was placed into the core of the dam some 50 feet below the crest The two horizontal and vertical accelerograms recorded at the crest of the dam and those recorded within the core are shown in the next two figures The recordings obtained from these instruments during the Loma Prieta earthquake have been very valuable understanding the variations in motions at various locations in the dam and for calibrating available analytical procedures A useful application of the strong motion data obtained at several dams in California is illustrated in this figure Values of the recorded peak transverse acceleration at the crest of the dam are plotted versus the corresponding values of the toe or some distance away from the toe of the dam The data were obtained at dams ranging in height from a little less than 100 feet to several hundred feet and during earthquakes ranging in magnitude from a little less than 6 to magnitude 7 This figure shows the nonlinear trend of the data and the possible variations in peak crest acceleration as a function of the acceleration at or near the toe of the dam The information in this figure provides an excellent means for checking the reasonableness of any analysis completed for a dam if the toe acceleration is used as an approximate proxy for the input acceleration to the dam under consideration For example, if the peak input acceleration is 0.2G the crest acceleration may vary from about 0.25 to 0.45G depending on the height of the dam and the frequency content of the input motion If the input motion is rich in high frequencies which would be expected from a small magnitude earthquake occurring close to the dam then the low dam would experience a peak crest acceleration of about 0.4 plus or minus G while a high dam would experience a peak crest acceleration closer to 0.2G If on the other hand the input motion is richer in the intermediate and low frequencies which would be expected from a high magnitude earthquake the opposite trend in peak crest acceleration would be the case At this time I would like to offer the following concluding remarks The case histories reviewed earlier and the lessons learned from these case histories emphasize the need for our concern with the effects of earthquakes on embankment dams The work that has been going on over the past 35 or so years in terms of testing, analytical studies and physical modeling have provided us with the insight and the means to make reasonable assessments of the performance of such dams during and following the occurrence of an earthquake When the performance has been judged inadequate and the consequences unacceptable the following options are available Remediation of existing dams are designed and constructed Examples of such remediation solutions at two existing dams were presented earlier and Professor Mitchell and his video presentation covered available remediation techniques and several examples For a new dam modifications to design and construction are made Such modifications could be in terms of changing compaction requirements changing material types placing the least vulnerable material where it does the most good or other defensive measures For a new dam the technology available today including testing, analysis and construction equipment makes it possible to design and construct an embankment dam that will perform very well in the event of an earthquake Two excellent examples of such new designs are the recently completed Seven Oaks Dam and the East Side Reservoir in Southern California The Seven Oaks Dam was designed and constructed by the Corps of Engineers It is almost 600 feet high and is about 3 kilometers from the San Andreas Fault The East Side Reservoir, which is owned by the Metropolitan Water District includes three dams with heights up to about 300 feet and is 5 to 8 kilometers from the San Jacinto Fault and only a little more distant from the San Andreas Fault It is important to emphasize that in no case should any changes made to accommodate seismic performance be allowed to impair the behavior of the dam under normal operating or other conditions Above all, a great deal of judgment is required in every step when we are evaluating a dam under static or earthquake loading conditions Professor Beck concluded his video presentation by reminding us that I'm quoting him The risk of failure of any dam cannot be reduced to zero End of quote He also indicated that quote Fortunately, the likelihood of a failure can be minimized by intelligent and conscientious surveillance and timely maintenance, rehabilitation, or repair of the structure End of quote These are timeless statements and apply for all loading conditions They should be heeded by all persons involved in dams