L-02-006, Part 1 of 2, Diablo Canyon Independent Spent Fuel Storage Installation - Submittal of Diablo Independent Spent Fuel Storage Installation Safety Analysis Report Reference Documents
| ML021630191 | |
| Person / Time | |
|---|---|
| Site: | Diablo Canyon  |
| Issue date: | 05/23/2002 |
| From: | Womack L Pacific Gas & Electric Co |
| To: | Document Control Desk, Office of Nuclear Material Safety and Safeguards |
| References | |
| +sispmjr200505, -nr, -RFPFR, DIL-02-006 BC-TOP-9-A, Rev 2, UCB/EERC-94/05 | |
| Download: ML021630191 (193) | |
Text
{{#Wiki_filter:W PacificGas and Electric Company Lawrence F. Womack D~iablo Canyon Power Plant Vice President PO. Box 56 Nuclear Services Avila Beach, CA 93424 805.545.4600 May 23, 2002 Fax: 805.545.4234 PG&E Letter DIL-02-006 U.S. Nuclear Regulatory Commission ATTN: Document Control Desk Washington, DC 20555-0001 Docket No. 72-26 Diablo Canyon Independent Spent Fuel Storage Installation Submittal of Diablo Canyon Independent Spent Fuel Storage Installation Safety Analysis Report Reference Documents
Dear Commissioners and Staff:
On December 21, 2001, Pacific Gas and Electric Company (PG&E) submitted an application to the Nuclear Regulatory Commission, in PG&E Letter DIL-01 -002, requesting a site-specific license for an Independent Spent Fuel Storage Installation (ISFSI) at the Diablo Canyon Power Plant. The application included a Safety Analysis Report (SAR), Environmental Report, and other required documents in accordance with 10 CFR 72. As requested by Mr. S. Baggett, four documents referenced in the Diablo Canyon ISFSi SAR are enclosed for use by the NRC staff in their review of the application. If you have any questions regarding this matter, please contact Mr. Terence Grebel at (805) 595-6382. Sincerely, Lawrence F. Womack Enclosure GWH2 cc: Diablo Distribution Ellis W. Merschoff David L. Proulx Girija S. Shukla cc/enc: James R. Hall David A. Repka John Stamatakos A member of the STARS (Strategic Teaming and Resource Sharing) Attiance Callaway
- Comanche Peak 9 Diablo Canyon
- Palo Verde
- South Texas Project e Wolf Creek
Enclosure PG&E Letter DIL-02-006 Sheet 1 of 1 LIST OF ATTACHED DIABLO CANYON ISFSI SAR REFERENCE DOCUMENTS
- 1. Harding, Miller, Lawson & Associates, Soil Investigation Landslide, Diablo Canyon Site, San Luis Obispo County, California, July 29, 1970.
- 2. Rocscience, Inc., SWEDGE Probabilistic Analysis of the Geometry and Stability of Surface Wedges, Version 3.06, 1999 (Users Guide).
- 3. S. A. Ashford and N. Sitar, Seismic Response of Steep Natural Slopes, Report No. UCB/EERC-94/05, College of Engineering, University of California, Berkeley, May 1994.
- 4. Design of Structures for Missile Impact, BC-TOP-9A, Bechtel Power Corporation Topical Report, Revision 2, September 1974.
HARDING, MILLER, LAWSON & ASSOCIATES SOIL INVESTIGATION LANDSLIDE, DIABLO CANYON SITE SAN LUIS OBISPO COUiNTY, CALIFORNIA Project Number 569,010.04 Prepared for Pacific Gas & Electric Company 245 Market Street San Francisco, California by Stephen R. Korbay, j Geologist - 853 Heg~y T. Taylo, Civil Engineer - 8787 Harding, Miller, Lawson & Associates 155 Montgomery Street San Francisco, California 94104 July 29, 1970
HARDING, MILLER, LAWSON & ASSOCIATES TABLE OF CONTENTS LIST OF ILLUSTRATIONS........ ............. I INTRODUCTION......................1 II FIELD EXPLORATION AND LABORATORY TESTS ....... ........ 2 III SITE CONDITIONS ......................... 3 IV SOIL AND ROCK CONDITIONS. ............... 4 V GEOLOGY . ........... ............ 5 VI DISCUSSION . . . . . . . . ....... . . . .. . 6 VII CONCLUSIONS 7......................7 VIII RECOMMENDATIONS ............ ........... 8 A. Scheme No. 1 . . . . .................. 8 B. Scheme No. 2 . . . . . . . . . . ............ 9 C. Scheme No. 3 .................. ................... 9 IX ILLUSTRATIONS . . . . . . . . . . . . . . . . . . . . . 11 Appendix FIELD AND LABORATORY DATA .. ............. 17 DISTRIBUTION . . . . . . . . . . . . . . . . .......... 23 ii
HARDING, MILLER, LAWSON & ASSOCIATES LIST OF ILLUSTRATIONS Plate 1 Site Plan . . . . . . .... S........... 12 Plate 2 Cross Sections ....... .............. 13 Plate 3 Slope Stabilization Scheme 1 ..
. . ........ 14 Plate 4 Slope Stabilization Scheme 2 *.. *..........15 Plate 5 Slope Stabilization Scheme 3 . ............ 16 Plate 6 Log of Boring 1 .......... . . . . Appendix S....
Plate 7 Log of Boring 2 ... ........ Appendix .. Plate 8 Log of Boring 3 . . . .... *.*....... Appendix Plate 9 Log of Boring 4 . . . .... . .
. Appendix Plate 10 Soil Classification Chart and Key to Test Data . . .... ...... . Appendix .
iii
HARDING, MILLER, LAWSON & ASSOCIATES I INTRODUCTION This report presents the results of our soil investigation of the recent landslide in the coastal bluff at the Diablo Canyon site, San Luis Obispo County, California. We understand that the landslide occurred during the early part of this year in the cove located adjacent to the Pacific Ocean and west of the existing plant access road near the warehouse and batch plant. We also understand that this area is not presently part of the nuclear plant construction; however, it may be the location for future cooling water discharge conduits. The purpose of our work was to investigate the probable cause, extent, and condition of the recent slide in order to provide you with conclusions and recommendations for stabilization of the land slide area. 1
HARDING, MILLER, LAWSON & ASSOCIATES II FIELD EXPLORATION AND LABORATORY TESTS We performed a field investigation of the landslide and surround ing area by conducting a geologic reconnaissance and by drilling four test borings. The general site conditions and boring locations are shown on the accompanying Site Plan, Plate 1, Section IX. The borings were drilled with a 24-inch diameter bucket auger drill rig to depths ranging from 13 to 46 feet. Each boring was logged by our geologist who also obtained representative samples of the soil and rock pene trated. These samples were tested in our laboratory to determine moisture content, dry density, and shear strength. The results of these tests are shown on the boring logs, Plates 6 through 9 in the Appendix. 2
HARDING, MILLER, LAWSON & ASSOCIATES III SITE CONDITIONS The landslide is approximately 200 feet wide and occupies the full height of the bluff face above a cove that is contiguous to the Ocean. The existing slope in places is as steep as 1-1/3 horizontal to 1 vertical. The top of slope is approximately 105 feet above Sea Level. The top of the slide is approximately 15 feet laterally from the toe of the existing fill for the plant access road. The slide surface contains numerous tension cracks and scarps especially along the upper limits. Some of the slide debris has moved to the base of the slope along the beach where wave action continuously removes loose material. A 15-foot deep and 25-foot wide erosion gully is present in the center of the slide. The gully is a result of outlet flow from an existing 4-1/2 foot diameter culvert pipe located at the top of slope. The culvert inlet is located across the plant access road near the warehouse. The flow has since been diverted around the slide area and is presently contained in a temporary drainage ditch leading to the cliff edge to the south. Numerous shallow mud flows are present on the slope outside the present slide limits. Evidence of soil creep, the gradual movement of soil on slopes due to shrinkage and expansion resulting from mois ture changes, is also present on the slope. Seepage and springs exist in the area, especially along the toe of slope. Existing erosion gullies expose the deeply weathered bedrock. 3
HARDING, MILLER, LAWSON & ASSOCIATES IV SOIL AND ROCK CONDITIONS According to boring data and surface exposures, the landslide and adjacent slope is underlain by soils consisting of silt, clay, and sand; the soils overlie volcanic bedrock. The rock consists of altered tuff, generally sheared and deeply weathered, with occasional tuffaceous siltstone and shale interbeds. The east and west cliff faces of the cove contain hard vitric tuff and shale bedrock, producing steep rock slopes which are rela tively resistant to erosion. Seepage was observed in Borings 1 and 2; however, only two feet of water accumulated in Boring 1 after two days. Both Borings 1 and 2 have been converted to observation wells by installing 12-inch diameter PMP casing and backfilling the sides with drain rock. Since borings were not drilled within the slide mass due to inaccessibility, the location of the slip plane can only be inferred. Surface condi tions indicate slide debris extending to the beach at the base of the slope, suggesting the presence of a rotational-type slide. Sub surface conditions are illustrated on the accompanying Cross Sections, Plate 2. 4
HARDING, MILLER, LAWSON & ASSOCIATES V GEOLOGY The bedrock in the slide area is assigned to the Obispo tuff member of the Monterey Formation of Miocene Age. The Obispo tuff stratigraphically underlies the sedimentary rocks of the Monterey formation found to the north. The contact between these two units is located north of the slide. The altered tuff underlying the north slope of the cove and beach is apparently in fault or intrusive contact with the stronger vitric tuff forming the adjacent steep cliffs. The altered tuff is highly sheared adjacent to the vitric tuff, indicating intense frac turing due to past movement along the contact. Evidence of slicken sides, fault gouge, and breccia is present along the contact at the base of the west cliff face. The sand and gravel overlying the bedrock represents marine terrace deposits on an ancient wave cut bench formed during the Pleistocene. The thickness of overburden at this location suggests the presence of an ancient ravine area, probably extending farther upslope. The ravine probably formed as a result of the deep erosion of the relatively weak tuff. The ancient ravine area later became filled and covered with younger alluvium and is presently not. distinguishable on the surface. The upper portions of the soil overburden are part of the large alluvial fan forming most of the gentle slopes in the area to the north. 5
HARDING, MILLER, LAWSON & ASSOCIATES VI DISCUSSION The landslide above the cove represents an accelerated form of the natural process of bluff regression; it occurred at this location for the following reasons:
- 1. The thick alluvial soil is unstable on the steep existing slope, especially when it overlies weak material such as the tuff rock-type present.
- 2. The terrace deposit of sand and gravel overlying the bedrock probably carries water acquired from surface infiltration of the upper alluvial fan, especially during heavy rainfall.
- 3. Erosion and removal of the slope toe by wave action continuously produces the existing steep slope, preventing natural stabilization by normal slope flattening due to continued slide activity.
- 4. The concentrated flow of water from the existing culvert hastened the sliding process.
6
HARDING, MILLER, LAWSON & ASSOCIATES VII CONCLUSIONS On the basis of our investigation, we conclude that the land slide can be stabilized. Corrective measures should be taken since it is probable that the slide area will enlarge with time unless the stability is improved. The completeness of the corrective measures (and therefore the cost) can vary through wide limits depending upon the importance of minimizing future movements. We do not believe the slide represents a threat to the safety of persons or existing or proposed structures. The access road could become undermined with time but it appears relatively simple to relocate it farther uphill should this occur. Three alternate schemes of correction are presented in subse quent parts of this report. Scheme 1 is the minimum we believe should be done if the enlargement of the slide area is to be retarded. Future movement could occur but the chance of it happening and the magnitude if it happens should be greatly reduced. Scheme 3 should prevent future sliding. While no slide correction is fool proof, experience has been excellent with slide areas retained by the extensive method shown. Scheme 2 is an intermediate method in terms of cost and expected performance. 7
HARDING, MILLER, LAWSON & ASSOCIATES VIII RECOMMENDATIONS A. Scheme No. 1 Objective: To perform a limited amount of slope reconstruction and drainage installation necessary to reduce but not completely eliminate continued slide movement. Corrective measures should include
- 1. Resurface the existing slope of the landslide and surrounding area by removing steep scarps, loose surface material, and filling erosion gullies and tension cracks with compacted soil.
- 2. Remove excess water in the existing observation wells by pumping during periods of rainfall or when required.
- 3. In lieu of No. 2, provide gravity drainage of the
-wells by installing perforated tar-coated pipe in hydrauger borings drilled from the base of the slope up to the bottom of each well.
- 4. Provide slope protection at the landslide toe in the form of a 30-foot wide and 15-foot high berm of riprap to reduce continued erosion by wave action. Riprap should consist of 1/2-ton class material placed by clam-type equipment.
- 5. Concrete line the existing drainage ditch located along the south top of slope to prevent any further infiltration of surface water from the culvert.
Illustration of the above scheme is presented on Plate 3 in Section IX. 8
HARDING, MILLER, LAWSON & ASSOCIATES B. Scheme No. 2 Objective: Stabilization of the landslide by buttressing and surfacing the slope with riprap. Corrective measures should include
- 1. If sufficient riprap is available, provide a slope buttress and surface cover as shown on Plate 4, Section IX. Approximately 16,000 to 18,000 cubic yards of light class material will be required and should be placed by Method B as specified in the State of California Standard Specifications. The buttress portion (behind the 1:1 slope) should consist of 1/2-ton class material placed by clam type equipment.
- 2. Prior to the placement of riprap, dress and smooth the existing slope by grading loose slide debris and soil.
- 3. The drainage ditch located on the top of slope should be concrete-lined as previously recommended under Scheme No. 1.
C. Scheme No. 3 Objective: To perform the most thorough slope reconstruction necessary for stabilization of the landslide area. Corrective measures should include
- 1. Excavate the unstable slide debris (approximately 30,000 cubic yards) and stockpile in an area south of the existing leach field to allow for drying.
- 2. Excavate keyways into firm soil or bedrock below the slide plane, as shown on Plate 5, Section IX.
- 3. Install subdrainage in the base of the excavation and along keyways where required. Approximately 600 feet of six-inch diameter perforated metal pipe will be required with 180 feet of six-inch diameter CMP.
Approximately 500 cubic yards of crushed drain rock also will be required. Provide subdrain cleanout risers. 9
HARDING, MILLER, LAWSON & ASSOCIATES
- 4. Reconstruct the slope with compacted fill using the stockpiled material after moisture conditioning as required. Compact the fill to at least 90 percent of the maximum dry density determined by the ASTM D1557-66T(C) compaction test method. Fill placement should be in lifts no greater than eight inches loose thickness and compacted with sheepsfoot rollers.
- 5. Provide riprap slope protection at the toe to reduce the amount of erosion due to wave action. Approxi mately 1500 cubic yards will be required. Riprap should consist of material as specified in Scheme No. 2, paragraph 1.
- 6. Provide an 8- to 10-foot wide bench approximately halfway up the slope to allow for collection and diversion of surface water.
- 7. Upon completion of the compacted fill, the slope surface should be planted with deep-rooted vegetation to retard erosion and sloughing.
- 8. The drainage ditch located on the top of slope should be concrete-lined as previously recommended under Scheme No. 1.
10
HARDING, MILLER, LAWSON & ASSOCIATES IX ILLUSTRATIONS 11
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. JobNo:5 6 9 ,.0 1 0 . 0 4 Appr: '/io. Date 5/15/70
0- -6LOG OF BORING i Shear Strength (lbs/sq ft) "
,* 4- " . Equipment 24" Diameter Buc!.et Auger U Elevation 111.8 Date 2/17/70
" DARK BROWN SANDY CLAY (CH) soft, damp, with occasional angular gravel LIGHT BROWN SANDY SILT (ML) 5 stiff, dry, with occasional angular gravel change to hard at 5', with occasional caliche cementation increased gravel up to 8" 10- from 9 to 10.5' change to soft, wel at 10.5' 26.0 86 15- LIGHT BROWN SANDflY SILTY CLAY (CL) - soft, wet, with occasional 25.8 94 angular gravel 20-f change to stiff at 22.5' 25 * /-CRAY BROWN SILTY SAND (SM) dense, wet, with abundant angular gravel
"/-YELLOW BROWN SAND, CLAY (CL) 30- /stiff, wet, with occasional gravel and shell fragments MOTTLED YELLOW GRAY SILTY CLAY (CL) - stiff, wet, with abundant weathered roc, 35" fragments MOTTLED YELLOW GRAY TUFF friable, low hardness, sheared, altered and deeply weathered
- change to gray, weak at 44'
"-- 1 I'I--,- water level 2/18/70 HARDING, MILLER, LAWSON & ASSOCIATES (ConsultihgEngineers LOG OF BORING 1 PLATE Landslide Job No: 569, 210. 04 Appr: __'_'iw Date 3/18/70 PG&E - Diablo Canyon 18
U LOG OF BORING - Shear Strength (lbs/sq ft) V CL4 In E) Equipment 24" Diameter Bucket Auger 00 0 C a C 0 0 0 (I Elevation 1"10.8 Date 2/17/7c 0 U C
- 0. wwUwmI DARK BROWN SANDY CLAY (OH) soft, damp, with occasional angular gravel LIGHT BROWN SANDY SILT (ML) stiff, damp, with abundant 5 angular grave I with occasional gravel up to 6" at 5' U LIGHT BROWN SANDY CLAY (CL) 25.4 c3 U soft, wet, with occasicnal gravel z
10 change to firm ot 9.5' change to soft a1 1 ' r LIGHT BROWN SANDY CLAYEY SILT (ML) - soft to medium stiff, 15 wet, with occasional angular 27.9 87 I grave I LIGHT BROWN SILTY CLAY (CL) stiff, moist, with slight porosity 201 / I /~
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I LIGHT BROWN SILTY SAND (SM) 25- dense, wet BPOWN GRAVELLY SAND (SP) 2
* . medium dense, moist, with silt binder 30-j-,MOTTLED GRAY BROWN CLAYEY "SILT (ML) - soft, moist, with occasional angular gravel MOTTLED ORANGE GRAY GRAVELLY 35- CLAY (CL) - firm, moist, with z iabundant rock fragments
-z'-MOTTLED ORANGE GRAY TUFF soft to friable, low hardness,
- 1 ___________ 1 .5. 77 sheared, altered and moderately Ui4 3 2 01 Ul l 33.0 87 41 *! weathered HARDING, MILLER, LAWSON & ASSOCIATES Consiting Engineers LOG OF BORING 2 PLATE ,,_____Landslide Job No:69,01- 14 Appr: f-4-/'w Date 3/18/70 PG&E - Diablo Canyon "7
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- 5. firm, damp, with occasional angular grave I up to 4" gravel at 7' grading clayey at 8', with I caliche cementation 10 slightly porous and moist at 14' 15-
.1i BROWN SILTY SAND (SM) medium dense, moist, with 20-20.0 83 'III occasional gravel I
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I stiff, damp, with abundant 2 rock fragments 25- 46 OPRANGE BPOWN TUFFACEOUS SILTSTONE - weaL-, moderately hard, closely fractured, I moderately weathered 301 (no free water observed) 35- ____________4U-* HARDING, MILLER, LAWSON & ASSOCIATES
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.4 Shear Strength (lbs/sq ft) C 0 4 0. Equipment 24" Diameter Bucket Auger
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BLACK SANDY CLAY tCH) e1/ soft, moist, with occasional 24.8 77 / angulai grOvel 5. 0 X/O MOTTLED YELLOW BROWN SILTY. I CLAY tCHt - soft, wet, with I 2 occasional rock fragments change to stof at 8'
- 10. BROWN GRAVELLY SANDY CLAY iCL) - firm, moist, with abundant rock fragments II BLACK. SHALE moderately strong, hard, closely 15 fractured, slightly weathered change to strong, very hard a,1 15' auger refusal at 16' 20 (no free vwater observed) 25 30 35-HARDING, MILLER, LAWSON & ASSOCIATES Consulting Engineers , LOG OF BORING 4 PLATE Job No: 569,010.04 Appr: 51'-jiw Date 3/18/70 PG&E LandslIide Diablo Canyon 9
MAJOR DIVISIONS TYPICAL NAMES CLEAN GAVELS # WELLGRUADED GRAVELS, GRAVEL - SAND MIXTURES WITH LITTLEOl (4 GRAVELS NO FINES 8 POORLY GRADED GRLAVELS, GRAVEL -S$AND _~ MIX TUNES
- MORE THAN CAS FRAT HALF SILTY GRAVELS, POORLY GRADED GRAVEL - SAND CASFRCINI GM SILTMIXTURES Z IS LARGER THAN GRAVEL WITH SILTMXU___
SNO. 4 SIEVE SIZE ova 12% PINES w z; GC CLAEY CLAY GRAVELS, POORLY MIXTURES GRADED GRAVEL - SAND CLEAN SANDS SW .I WELL GRADED SANDS, GRAVELLY SANDS WITH LITTLE OR SANDS z N FN iPS . POORLY GRADED &ANDS, GRAVELLY SANDS v ,CO MORE COAJSETHAN HALF FRACTION Im
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- IS SMALLERTHAN SANDS WITH MIXTURES NO. 4 SIEVESIZE OVER 12% FINES SC CLAYEY $ANDS, POORLY GRtADED SAND - CLAY I i MIXTURES I INORGANIC SILTSAND VERY FINE SANDS, ROCK (OI ML FLOUR, SILTY OR CLAYEY FINE SANDS, OR
=j I CLAYEY SILTS WITH SLIGHT PLASTICITY SILTS AND CLAYS INORGANIC CLAYS OF LOW TO MEDIUM PLASTICITY, Mz LIQUID LIMIT LESSTHANCL GRAVELLY CLAYS, SANDY CLAYS, SILTY CLAYS, LD CL LEAN CLAYS 0L I IIII ORGANIC CLAYS AND ORGANIC SILTY CLAYS OF l-II LOW PLASTICITY CC R ~INORGANIC FINE SANDY SILTS, MICACEOUS ORtDIATOMACIOUS OR SILTY SOILS, ELASTIC SILTS K< SILTS AND CLAYS 000, Sz z
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INORGANIC CLAYS OF HIGH PLASTICITY, FAT CLAYS _ __/, ORGANIC CLAYS OF MEDIUM TO HIGH PlASTICITY, __ ORGANIC SILTS HIGHLY ORGANIC SOILS PEATAND OTHER HIGHLY ORGANIC SOILS UNIFIED SOIL CLASSIFICATION SYSTEM SAMPLE DESIGNATION U'Undisurbed" S-pie N] III, Cor ification Semple STRENGTH TESTS SVANE SHEAR TEST UNCONFINED COMPRESSION TEST F Field L Laboratory 1000 (30.0) DIRECT SHEAR TEST 1000 (30.0) X TRIAXIAL COMPRESSION TEST CD - Con..Iidol.d - Draine.d A AUU : Uncn,'ions'oed - UndroI, .d 1 Content ofter Test I%) CD . Con. oildoted - Dra in.d SMoisttre S tres Nirsm l to Sheer Piano (pi12Dsfoar tes pf DMltu .st. Conteit ft., Test( Confining Streis - 03 (p.f) KEY TO TEST DATA HARDING, MILLER, LAWSON & ASSOCIATES SOIL CLASSIFICATION CHART PLATE Uosllin, Epnrr -LC AS I CA TO CH R PL T KEY TO TEST DATA Job No: Appr: Date .1 7 I
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HARDING, MILLER, LAWSON & ASSOCIATES DISTRIBUTION 3 copies: Pacific Gas and Electric Company 245 Market Street San Francisco, California 94106 Attention: Mr. Richard Bettinger, Supervising Civil Engineer 23
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Geomechanics Software & Research - I E. II Swedge Probabilisticanalysis of the geometry and stability of surface wedges
Swedge Probabilistic analysis of the geometry and stability of surface wedges 44I e 0 User's Guide
© 1991-99 Rocscience, Inc.
Table of Contents i Table of Contents Getting Started I Hints about this Manual ......................................................................... 2 rp
- r. -3 Introduction 3 ro Swedge Input....................................4 Joint Sets ............................................................................................ 6 Upper Slope and Face Slope .............................................................. 7 Overhanging Slope ............................................................................ 7 Tension Crack ................................................................................... 7 Slope Height ....................................................................................... 7 External Force .................................................................................... 8
@ W ater Pressure ................................................................................. 8 Varying the W ater Pressure ........................................................ 9 Seismic Force .................................................................................... 10 Probabilistic Input ............................................................................ 11 Statistical Distributions .............................................................. 11 Normal ............................................................................ 12 Truncated Normal Distribution..................... 13 Uniform ............................................................................ 13 Triangular ......................................................................... 14 Beta ................................................................................. 15 Exponential ...................................................................... 16 Probability - Further Reading ........................................................... 17 Sw edge Analysis ................................................................................. 18 Geometry Validation ......................................................................... 18 Sliding Planes .................................................................................... 19 References ......................................................................................... . .. 20
ii Table of Contents Table of Contents iii N Probability of Failure .......................................................................... 45 Quick Tour of Swedge 21 Wedge Display ................................................................................. 46 e Histograms ....................................................................................... 46 Job Control ........................................................................................... 22 Mean Safety Factor .................................................................. 47 Input Data ............................................................................................. 23 Manipulating the Histogram View........................ 47 Manipulating the View .......................................................................... 24 Viewing Other Wedges ........................................................... 47 Rotating the Model ............................................................................. 24 e Histograms of Other Data ........................................................ 49 Moving the Wedge Out of the Slope............................ 25 Re-running the Analysis ................................................................... 50 Resetting the W edge ............................................................... 25 Cumulative Distributions (S-curves)............................ 52 Rotating and Moving .......................................................................... 25 Info Viewer ........................................................................................ 54 Resizing the Views................................... 26 Current Wedge Data .............................................................. 54 Zooming .................................................................................. 27 Sampling Method ............................................................................ 56 Display Options ................................................................................ 27 Changing the Input Data & Re-calculating the Safety Factor ................ 28 4 Removing the Tension Crack ............................................................ 29 Adding Support 57 Entering a New Wedge .................................................................... 30 re a Sliding Plane ............................................................................ 31 How Bolts are Implemented in SWEDGE.........................60 Water Pressure ............................................................................... 32 Capacity and Orientation ................................................................. 60 External Force ................................................................................... 33 Length and Location ......................................................................... 60 Seismic Force .................................................................................. 34 Bolts vs. External Force................................. 61 More About the Input Data Dialog..............................35 Multiple Bolts ..................................................................................... 61 Stereo Projection of Input Data Planes........................... 36 Deleting Bolts ....................................................................................... 62 Importing Data from a DIPS File ......................................................... 36 Editing Bolts .......................................................................................... 63 Info Viewer.............................................................................................. 37 Listing of Bolt Properties .................................................................... 63 Bolts in a Probabilistic Analysis ......................................................... 64 Probabilistic Analysis 39 Job Control ........................................................................................... 40 Probabilistic Input Data ....................................................................... 41 Defining Random Variables .............................................................. 41 Joint Set 1....................................................................................... 42 Joint Set2 ....................................................................................... 43 Tension Crack .................................................................................. 43 Slope ................................................................................................ 44 Forces ................................................................................................ 44 Sampling .......................................................................................... 44 Probabilistic Analysis ......................................................................... 45 Results ................................................................................................... 45
Getting Started Getting Started SWEDGE is designed to work on Windows 95, 98 and Windows NT 4.0 operating systems. To install SWEDGE on your computer:
- 1. Insert the CD-ROM.
- 2. Setup should begin automatically displaying the main Rocscience Installation window.
- 3. If not, select Add / Remove Programs from the Control Panel and click on the Install button. Follow the directions until the main Rocscience Installation window is displayed.
- 4. Click on the SWEDGE button.
C_ .3 5. Click on the INSTALL FULL VERSION button.
- 6. Follow the installation instructions. During C_ *installation you will be asked to enter your seventeen character alphanumeric serial number. Enter the serial number located on the outside of the CD case to install the program. Proceed until the installation is complete and you are back to the Rocscience Installation window.
- 7. Click on the RETURN button.
- 8. If you have NOT previously installed the hardlock driver software for any other Rocscience program proceed with step 9. Otherwise go to step 13.
- 9. Click on the HARDLOCK button.
- 10. Click on the INSTALL DRIVER FOR 95,98,NT button.
2 SWEDGE User's Guide Introduction 3 9
- 11. Proceed until the hardlock driver installation is 9 complete and you are back to the Rocscience Installation window. I.
Introduction
- 12. Click on the RETURN button. 4 SWEDGE is a quick, interactive and simple to use analysis tool for evaluating the stability of surface wedges
- 13. Click on the EXIT button. in rock slopes, defined by two intersecting discontinuity planes, the slope surface and an optional tension crack.
- 14. To run SWEDGE, you will also need the hardlock 9 Wedge stability can be assessed using either:
supplied with the program. The hardlock must be attached to the parallel port on your computer during BI "* DETERMINISTIC (safety factor), or execution of the program. Attach the SWEDGE hardlock to the parallel port of your computer. 9 "* PROBABILISTIC (probability of failure)
- 15. The installation process creates a ROCSCIENCE analysis methods. For a DETERMINISTIC analysis menu in your START...PROGRAMS menu. In the SWEDGE computes the factor of safety for a wedge of ROCSCIENCE menu there will be an SWEDGE menu known orientation. For a PROBABILISTIC analysis, containing the SWEDGE application. Run the statistical input data can be entered to account for SWEDGE application. uncertainty in joint orientation and strength values. This results in a safety factor distribution, from which a
- 16. If you are a first time user, read the Introduction and probability of failure is calculated.
Tutorial chapters of this manual, to get acquainted with the features of SWEDGE. Other modeling features include: Hints about this Manual "* water pressure,
"* external / seismic forces, This manual is intended as a hands-on, getting started user's guide. For more information on any SWEDGE "* rock bolt reinforcement.
options which are not discussed in these pages, consult the SWEDGE Help system. In the tutorial chapters, In all cases, the assumed failure mode of the wedge is instructions such as: translational slip - rotational slip and toppling are not taken into account. The stability method used in Select: Analysis --+ Input Data SWEDGE can be found in "Rock Slope Engineering" (Rev. 3rd edition, E. Hoek & J.W. Bray, pp 341-351). are used to navigate the menu selections. When a toolbar button is displayed in the margin, as shown above, this indicates that the option is available in the SWEDGE toolbar. This is always the recommended and quickest way to use the option.
Intro~duction 5 4 SWEDGE User's Guide (F When a pair of discontinuities are selected at random Swedge Input from a set of field data, it is not known whether: SWEDGE computes the factor of safety for translational " the planes could form a wedge (the line of intersection slip of a tetrahedralwedge formed in a rock slope by: may plunge too steeply to daylight in the slope face or it may be too flat to intersect the upper ground
"* two intersecting discontinuities (joint sets), surface).
"* the slope face, "* one of the planes overlies the other (this affects the calculation of the normal reactions on the plane).
"* the upper ground surface, and
"* one of the planes lies to the right or the left of the 0 a tension crack (optional). other plane when viewed from the bottom of the slope.
Typical problem geometry is illustrated below (Ref. 1). In order to resolve these uncertainties, the solution has been derived in such a way that:
"* Either of the planes may be labeled 1 (or 2).
"* Allowance has been made for one of the planes LEGEND overlying the other (this is illustrated in Figure 1-2) 1,2 = Failure planes (2 "* The crest can overhang the base of the slope.
intersecting joint sets) 3 = Upper ground surface "* Contact may be lost on either plane (this is dependent on wedge geometry, and also on the magnitude of the 4 = Slope face water pressures acting on the planes). 5 = Tension crack A check on whether the two planes do form a wedge is HI = Slope height referred to included in the solution at an early stage. In addition, plane I SWEDGE also examines how the tension crack intersects L = Distance of tension the other planes, accepting only those cases where the crack from crest, tension crack truncates the wedge in a kinematically measured along the admissible manner. trace of plane 1. The SWEDGE stability analysis has been derived from a solution presented in Ref. 1. For a complete and detailed description of this analysis, consult this reference. Figure 1-1: Typical wedge geometry for SWEDGE analysis (Ref.1) @ 1-
.4..
SWEDGE User's Guide Introduction Upper Slope and Face Slope
.4 Note that there is no restriction on the inclination of the (D e crest of the slope (the line of intersection of the upper and face slope planes), therefore the Dip Directions of the Upper Slope and Face Slope do not necessarily have to be the same.
9! The Upper Slope and Face Slope correspond to planes 3 and 4 in Figure 1-1. Overhanging Slope
- 0) If the crest overhangs the base of the slope, select the Overhanging checkbox in the Input Data dialog, and enter appropriate Dip and Dip Directions of the Upper and Face Slope planes.
Tension Crack The Trace Length of the Tension Crack is the distance of the tension crack from the crest, measured along the trace Figure 1-2: Situation where wedge is formed, and one plane of plane 1. See Figure 1-1. Length L is the trace length. overlies the other. SWEDGE examines how the tension crack intersects the other planes, and only accepts those cases where the Joint Sets tension crack truncates the wedge in the manner shown in Figure 1-1. If the tension crack plane does not form an Either joint set can be defined as Joint Set I or Joint Set 2 acceptable wedge with the other planes, a warning in the Input Data dialog. message will be displayed when you select the Apply button to compute. However, remember that the Slope Height and the Trace Length of the Tension Crack are measured with respect to A Tension Crack is optional in SWEDGE, and can be Joint Set 1- see Figgure 1-1. excluded from a model by de-selecting the Tension Crack checkbox in the Input Data dialog. Slope Height The Slope Height is the vertical distance Hi in Figure 1-1, referred to plane 1.
.4ý-
Introduction 9 SWEDGE User's Guide where: Ui, U2 and U5 are the average values of water External Force pressure on the failure planes 1 and 2, and the tension crack, respectively A single external force (eg. a blast acceleration acting in a known direction) can be applied to the wedge, by selecting the External Force checkbox, and entering a direction and yw = unit weight of water magnitude. H5w = depth of bottom vertex of the tension crack below External force can also be applied through the use of rock the upper ground surface bolts. See the Adding Support tutorial at the end of this manual for details. The above formulae are simple estimations which are useful in the absence of more precise information. Water Pressure Varying the Water Pressure In the SWEDGE factor of safety calculation, it is assumed that extreme conditions of very heavy rainfall occur, and "* To simulate a "dry" slope, the user can de-select the Water Pressure checkbox in the Input Data dialog, or that in consequence the fissures (failureplanes) are completely full of water. Further, it is assumed that the To vary water pressure in enter a Unit Weight of zero. SIWEDGE, alter the Unit pressure varies from zero at the free faces to a maximum Weight of water in the Input value at some point on the line of intersectionof the two Data dialog.
" To simulate intermediate water pressures, the user failure planes. can effectively vary the water pressure by varying the unit weight of water between zero and the actual unit SWEDGE calculates average values of water pressure weight. This allows the user to perform a sensitivity on each failure plane as follows (Ref. 1): analysis on the effect of water pressure on the safety factor of the wedge.
With NO Tension Crack UI= U2= 7w.H /6 Eqn 1.1a where: ui and U2 are the average values of water pressure on failure planes 1 and 2 yw = unit weight of water
= total height of the wedge With Tension Crack 7w. Hw/3 Eqn. I.lb UI = U2 = Us5=
10 SWEDGE User's Guide Introduction 11 I.!09 Seismic Force Probabilistic Input Seismic Force can be applied to the wedge, by selecting If the Analysis Type is Probabilistic, the user can define the Seismic checkbox in the Input Data dialog, and the following random variables in the Probabilistic Input entering the following data: Data dialog: Seismic Coefficient "* Dip and Dip Direction of all planes (ie. Joint Sets 1 and 2, Upper and Face Slope, and Tension Crack). A dimensionless number defining the seismic acceleration as a fraction of the acceleration due to gravity. Typically
"* The strength (Cohesion and Friction Angle) of Joint the Seismic Coefficient might be around 0.1 to 0.2. If a = Sets I and 2.
Seismic Coefficient, g = acceleration due to gravity = 9.81 m/s2 , and m = mass of the wedge, then the Seismic Force For each random variable, enter an appropriate: applied to the wedge, F = m a g.
"* Mean Direction "* Standard deviation (if applicable)
"* "Line of Intersection" will apply the Seismic Force in
"* Relative minimum and maximum values the direction (PLUNGE and TREND) of the Line of NOTE that the minimum / maximum values are specified Intersection of Joint Sets 1 and 2. as RELATIVE numbers (ie. distance from the mean),
rather than as absolute values. All references below to
"* "Horiz. & Inters. Trend" will apply the Seismic Force "minimum" and "maximum" values refer to the actual horizontally, but with the same TREND as the Line of values (ie. mean - rel. min and mean + rel. max), and Intersection of Joint Sets 1 and 2. not to the relative values entered in the Input Data dialog.
"* User Defined allows the user to define any direction for the Seismic Force.
Statistical Distributions To define a random variable, first choose a Statistical
- 4) Distribution (also known as "Probability Density Function" or "pdf'). The five available distributions are:
d4 "* Normal
- 4) "* Uniform
"* Triangular
"* Beta
"* Exponential (only available for cohesion and friction angle)
Introduction 13 12 SWEDGE Users Guide Truncated Normal Distribution Normal A truncated NORMAL distribution can be defined by The NORMAL (or Gaussian) distribution is the most setting the desired minimum and/or maximum values for common type of probability distribution function, and is the variable. For practical purposes, if the minimum and generally used for probabilistic studies in geotechnical maximum values are at least 3 standard deviations away engineering. Unless there is a good reason to use one of the other four PDFs available in SWEDGE, it is e 4 from the mean, you will obtain a complete normal distribution. If the minimum / maximum values are less recommended that the user choose the NORMAL "pdf'. than 3 standard deviations away from the mean, the distribution will be visibly truncated. For a NORMAL distribution, about 68% of observations should fall within one standard deviation of the mean, and about 95% of observations should fall within two Uniform standard deviations of the mean. A UNIFORM distribution can be used to simulate a random variation between two values, where all values in the range are equally probable. mean = p A UNIFORM distribution is entirely specified by the minimum and maximum values. The mean value of a UNIFORM distribution is simply the average of the minimum and maximum values, and cannot be f(x) independently specified. C- 4 p f(x) um x Figure 1-3: Normal probability density function, showing standard deviation ranges. C- 4 x Figure 1-4: Uniform probability density function. re
14 SWEDGE User's Guide Introduction 15 Triangular 1. If the distribution is symmetric, then the mean is equal to the mode. You may wish to use a TRIANGULAR distribution in 2. For a left triangular distribution, the mode some cases, as a rough approximation to a random minimum, and the mean = (2*minimum + maximum) / variable with an unknown distribution. 3.
- 3. For a right triangular distribution, the mode =
A TRIANGULAR distribution is specified by its minimum, maximum, and the mean = (2*maximum + minimum) maximum and mean values. It does not have to be /3. symmetric, it can be skewed to the left or right by entering a mean value less than or greater than the Beta average of the minimum and maximum values. @ . The BETA distribution is a very versatile function which can be used to model several different shapes of probability density curves, as shown in the figure below. c
* - .
- S*
- f(X)
-* o*+. ° + ,, 2. . . ;
(22) (b) 5 b X
@4 Figure 1-5: Triangular probability density function. Minimum = a, maximum = b, mode = c. For a symmetric distribution, mean @4 W}(d) mode.
@cm4 Figure 1-6: Beta (al, a*2) density functions (Ref. 3)
Note: for a non-symmetric TRIANGULAR distribution, the mean value is not equal to the mode. The mode is the The form of the BETA distribution is determined by the value of the variable at the peak of the TRIANGULAR shape parameters a I and o_2. Both a I and a2 are always distribution. In general for a TRIANGULAR distribution, > 0. The relationship between the BETA distribution the mean is given by: shape parameters and the SWEDGE input data is as minimum + maximum + mode Eqn. 1.2 follows: mean =n 3 et aj
Introduction 17 16 SWEDGE User's Guide
- 1. Since the range of values must always be positive for al Eqn. 1.3
-3 an EXPONENTIAL distribution, it has only been mean = made available for the strength parameters (cohesion al+a2 and friction angle), and not for the orientation ala2 parameters, since these may have negative ranges variance = Eqn. 1.4 which would be invalid for an EXPONENTIAL (al + a2) 2 (al + a2 + 1) distribution.
The standard deviation is the positive square root of the variance. 2. The mean is always equal to the standard deviation for an EXPONENTIAL distribution. This is a property Note that Equations 1.3 and 1.4 apply to a beta random of the EXPONENTIAL distribution, and cannot be variable on 10,1]. To rescale and relocate to obtain a beta altered by the user. random variable on [a,b] of the same shape, use the transformation a + (b-a)X. 3. Like the NORMAL distribution, the EXPONENTIAL distribution can be truncated by entering the .desired minimum and maximum values (the basic Exponential EXPONENTIAL distribution varies from zero to The EXPONENTIAL probability density function has also infinity). been made available in SWEDGE. The EXPONENTIAL distribution is sometimes used to define events, such as the occurrence of earthquakes or rockbursts, or quantities such as the length of joints in a 4 rockmass. Of the currently defined statistical variables in SWEDGE, you may occasionally find it useful for 4 modeling joint cohesion, for example. f(x) Probability - Further Reading 4 An excellent introduction to probability theory in a geotechnical engineering context, can be found in Chapter 4 2 of Ref. 2. 4 More comprehensive and detailed information can be found in statistics textbooks. For example, Chapter 6 of Ref. 3 is an excellent guide to the selection of input X probability distributions. Ref. 4 provides a summary of over 30 different probability density functions, in a quick Figure 1-7: Exponential probability density function. reference format. Note the following:
.4.-
Introduction 19 8 SWEDGE User's Guide Sliding Planes 3owedge Analysis 4) After a typical SWEDGE analysis, the analysis summary To run the SWEDGE analysis, simply select the Apply will indicate, for a given wedge: button in the Input Data dialog, after entering all your input data. Sliding along line of intersection (trend I plunge)
" If the Analysis Type = Deterministic, the Safety This indicates that the factor of safety accounts for sliding Factor will be immediately calculated and displayed Depending on wedge on both of the failure planes (joint sets). The line of in the lower right corner of the dialog, as well as in geometry and waterpressure, intersection refers to the line of intersection of the two sliding may take place along: failure planes (Joint Set 1 and Joint Set 2).
the toolbar.
. Both failure planes
"* If the Analysis Type = Probabilistic, the Probability of . One failure plane In some cases, depending on the geometry of the wedge Failure will be calculated and displayed in the toolbar. and the magnitude of the water pressure, contact may be
. None (loss of contact) lost on either failure plane. In such cases, the analysis Note that a Probabilistic Analysis can be re-run at any summary will show:
A ProbabilisticAnalysis can be re-run at any time by time, by selecting the Compute button in the toolbar. The selecting the Compute button Probability of Failure will not necessarily be the same, Sliding on Joint I or in the toolbar. each time a Probabilistic Analysis is re-run. Sliding on Joint 2 Geometry Validation If the water pressure is too high, the wedge will 'float', and the analysis summary will indicate: SWEDGE always checks if the model geometry is valid, before proceeding to calculate a Safety Factor for a given Contact Lost on Both Planes wedge. Finally, if tension in the rock bolts is too high, the
" If the Analysis Type = Deterministic, you will receive 20 analysis summary may indicate:
a warning message if there is a problem with your input data. Sliding UP Line of Intersection (trend I plunge)
" If the Analysis Type = Probabilistic, validation is first indicating that the total rock bolt tension is high enough performed on the mean Input Data. If the mean to potentially push the wedge 'up' the slope.
orientation data does not form a valid wedge, then the entire Probabilistic Analysis will be aborted, and you I, For a Deterministic Analysis, the sliding plane(s) will be will receive a warning message. If the mean wedge is indicated in the Input Data dialog, along with the Safety valid, but invalid wedges are generated during the I, Factor. For a Probabilistic Analysis, this information is statistical sampling, then these results are discarded, listed in the Info Viewer. but the analysis is allowed to proceed. The Number of U, Valid Wedges for a Probabilistic Analysis can be found listed in the Analysis -+ Info Viewer option.
Quick Start Tutorial 21 0 SWEDGE User's Guide teferences Quick Tour of Swedge
- 1. Hoek, E. and Bray, J.W. Rock Slope Engineering, Revised 3rd edition, The Institution of Mining and Da aa T 0 0.
Metallurgy, London, 1981, pp 341 - 351. e 4 I " '- . .. .. " : . .
- 2. Hoek, E., Kaiser, P.K. and Bawden, W.F. Support of Underground Excavations in Hard Rock, A.A.Balkema, Rotterdam, Brookfield, 1995. 4
- 3. Law, A.M. and Kelton, D.W. Simulation Modeling and 4 Analysis, 2nd edition, McGraw-Hill, Inc., New York, 1991.
4 Evans, M., Hastings, N. and Peacock, B. Statistical Distributions,2nd edition, John Wiley & Sons, Inc., New York, 1993. N. 4 This "quick tour" will familiarize the user with some of the basic features of SWEDGE. 9 If you have not already done so, run SWEDGE by double clicking on the SWEDGE icon in your installation folder. Or from the Start menu, select Programs --* Rocscience -* 4 Swedge -* Swedge. 4 If the SWEDGE application window is not already maximized, maximize it now, so that the full screen is 4 available for viewing the model. To begin creating a new model: Select: File -> New A wedge model will immediately appear on your screen, as shown in the above figure. Whenever a new file is opened, the default input data will form a valid wedge. 4',
22 SWEDGE User's Guide Quick Start Tutorial 23 The first thing you will notice is the four-view, split screen Input Data format of the display, which shows:
"* TOP Now let's see what input data was used to create this
"* FRONT model.
"* RIGHT and
"- PERSPECTIVE Select: Analysis -- Input Data views of the model. The Top, Front and Right views are uelýiftistic input Date -- x IFormeI A
orthogonal with respect to each other (ie. viewing angles Geometry differ by 90 degrees). Oip(deg) DimODrecton(deg) C esionrVm2) FnrictonAngle (deg) Joint Sel, Fs IT - F-- 25" JolrnrSeFl 70- s- Fs-i 30 Job Control UpperFeFe 2- lgs Slope Fece 116f
- -5 SlopePoo e .e .
Job Control allows the user to enter a Job Title, and select Slope Helrg(m)nt ' a Unit System and Analysis Type. P Tension CrOak Ur We,~ow3) Dip (deog) F75 I- ONtltnqri Select: Analysis --, Job Control DipDirnonri(deg) 65 Tace Len~gi(F) Fi SaoilyFortor 1 04026 Wedge Weigh- 159077lone Ope ne, m1O00k ( (1000 kg) Slrfngon Lneo oIttersecob, Job Tde: rSWPOGE Ouk SwTuoSOnl For'ceronToonno Ttond°-157232 Pluge. 31.965 Us . ... AndysisType
' Me.c 0COetem-aoeec App5' Oone r Impenal C'nPobab'distc Figure 2-1: Input Data dialog (Deterministic).
The Geometry input data which you see in this dialog is Distace unft inmeteers end Farcewits m tnnes (1000 Sg) the default input data, which forms a valid default wedge, each time a new file is started. Enter "SWEDGE Quick Start Tutorial" as the Job Title. Leave Units = Metric and Analysis Type = Deterministic. Quickly examine the input data in this dialog. See the Select OK. introductory section of this manual for definitions of the SWEDGE input data. Do not change any values just yet,
" The Job Title will appear in the Info Viewer listing, we will be coming back to this shortly.
discussed later in this tutorial.
"* Units determines the length and force units used in Before you close the dialog, notice the Safety Factor, the Input Data dialog (see the next section). Wedge Weight etc information displayed in the lower right corner. The Safety Factor (FS = ...) is also displayed
"* Probabilistic SWEDGE analysis is covered in the next in the SWEDGE toolbar, at the top of the screen.
tutorial.
24 SWEDGE User's Guide Quick Start Tutorial 25 Now close the dialog by selecting the X in the upper right Moving the Wedge Out of the Slope corner.
- 1. Press and HOLD the RIGHT mouse button anywhere Manipulating the View in ANY of the four views. Notice that the cursor changes to an "up-down arrow" symbol.
The LEFT and RIGHT mouse buttons can be used to t 2. Now, keep the RIGHT mouse button pressed, and interactively manipulate the view as follows: move the cursor UP or DOWN. The wedge will slide
"* The Perspective view of the model allows the model to UP or DOWN out of the slope. Note:
be rotated for viewing at any angle with the LEFT 4 "* If your model does NOT have a Tension Crack, mouse button. 4 then the wedge will slide UP or DOWN along the
"* The wedge can be moved out of the slope with the Line of Intersection of Joint 1 and Joint 2.
RIGHT mouse button in any of the four views.
" If your model DOES have a Tension Crack, then the wedge will slide DOWN along the Line of Rotating the Model Intersection of Joint 1 and Joint 2, and UP along the plane of the Tension Crack.
- 1. Press and HOLD the LEFT mouse button anywhere in the Perspective view. Notice that the cursor changes 3. To exit this mode, release the RIGHT mouse button.
to a "circular arrow" symbol to indicate that you may Notice that the cursor reverts to the normal arrow rotate the model. cursor.
- 2. Now keep the LEFT mouse button pressed, and move Resetting the Wedge the cursor around. The model is rotated according to the direction of movement of the cursor. To reset the wedge in its normal position, click and RELEASE the RIGHT mouse button in any of the four
- 3. To exit the rotation mode, release the LEFT mouse views. The wedge will snap back to its normal position.
button. Notice that the cursor reverts to the normal arrow cursor. 4 Rotating and Moving
- 4. Repeat the above steps to rotate the model for viewing 4 The rotate and move options, described above, can be used at any angle.
in any order. That is, the model can be rotated after 4 moving the wedge, and the wedge can be moved after rotating. This allows complete flexibility of viewing the 4 slope and wedge from all possible angles. 4 Note that rotating the model only affects the Perspective view, while moving the wedge out of the slope affects all 4 views (Top, Front, Right and Perspective). 4
Quick Start Tutorial 27 26 SWEDGE User's Guide Zooming Resizing the Views I. Zooming (from 50% to 800%) is available in the View --+ You can change the relative size of the Top / Front / Right Zoom menu, to increase or decrease the displayed size of
/ Perspective views in a number of ways: the model in all four views.
- 1. Double-clicking in any view will maximize that view. Individual views can be zoomed in or out using the Page e
Double-clicking again in the maximized view, will Up I Page Down keys, or the + or - numeric keypad keys. restore the four view display. You must first click in the view with the LEFT mouse button, to make it the active view.
- 4D v F-r5-Double-clicking in any view will maximize that view. Display Options Double-clicking again in the maximized view, will restore the four view display. You may change the colours of the Slope, Wedge,.
Background and Bolts, and the Drawing Mode (Shaded or Wireframe) in the Display Options dialog. Select: View -> Display Options 4J II awo. - :,. W.d"g. G ýA Sw,- c-Figure 2-2: Maximized Perspective view. 4
- 2. Alternatively, hover the cursor over the vertical or horizontal dividers between the views, or over the .4 intersection point of the four views. The cursor will change to a "parallel line" or "four arrow" symbol. .4 Select new slope, wedge and background colours, and hit Press and HOLD the LEFT mouse button, and drag to the Apply button. Now change the Drawing Mode from re-size the views. 4 Shaded to Wireframe, and hit Apply. Select the Defaults Maximizing views can also be accomplished with the V.4 button to restore the defaults, and hit OK or Cancel to exit the dialog.
3. View -+ Layout options. To reset the four views to equal size, select View-* Layout --* All Views. V.4 The "Selection" colour refers to the colour of selected bolts K while using the Delete Bolt and Edit Bolt options. See the last tutorial in this manual for more information. V4 S S
Quick Start Tutorial 29 -8 SWEDGE User's Guide Let's first remove the Tension Crack, and observe the Note that the Cancel button in the Display Options dialog effect on Safety Factor and the wedge geometry. does NOT cancel any changes once they have been applied with the Apply button. Removing the Tension Crack M,... e 1. To remove the Tension Crack from the model, simply de-select the Tension Crack checkbox in the Input 4 Data dialog.
- 2. Select the Apply button, and a new Safety Factor is 4 immediately calculated. Removing the Tension Crack 4 increased the Safety Factor from 1.04 to 1.13.
4 3. Select the Done button, to close the Input Data dialog, so that you can view the new wedge. It should appear 4 as below. ceeme.=led *- I-I. Figure 2-3: Wireframe Drawing Mode. e 4 Wedge Weight - 21483.3 lonnes SSelety Factor-=1.13191 Changing the Input Data & Re-calculating the Safety Factor Sliding on Line of Intersection: Trend= 157.732 Plunge - 31 .1965 Now let's experiment with changing the Input Data and re-calculating a new Safety Factor. 4 N.7. This is simply a matter of:
- 1. Entering the desired Input Data.
-I
- 2. Selecting the Apply button.
4 Figure 2-4: Wedge with tension crack removed. For a Deterministic analysis, the Safety Factor is immediately calculated and displayed in the lower right corner of the dialog. 4 Note that the Input Data dialog can also be minimized without closing it, by selecting the - arrow in the upper Fir1 Select: Analysis -+ Input Data 4 4 right corner of the dialog. 4
30 SWEDGE User's Guide Quick Start Tutorial 31 Entering a New Wedge 4 Let's enter data for a completely different wedge. 4 Select: Analysis --> Input Data 4
- 1. Enter the following data and select Apply.
4 Deterndnisfic InputDat. ... 4 G.o*ry I Fo.. I 4 Dip (d.9) DipOcioa(d4g Coh.son ) Fickoon-ll (deq) 4 st So a 70-- Pe- 1-. Uppe, Fam F 0- F 4 Slop. Face, Fs - 4 -5Slope Properties Slope H.eOltn) r' Tensn-Crac Uni0Weit(Trn,3) F7 -op r-Overhangig Figure 2-5: A new wedge.
.............. ..... .... S ty~vFe S....
ar - 0.$29299 C Sliding Plane WedgeWekilqt- 5531.5t5or. OWacein met.6; S dkg on Join1 Fow m i Tonnes (I000kq)
- 4) Notice that in this case, the analysis summary in the Input Data dialog indicates the failure mechanism as SDone "Sliding on Joint 1", rather than "Sliding along line of intersection". This is consistent with the model geometry,
- 2. Select the Done button to close the dialog, or minimize since Joint Set 2 dips at 70 degrees and has a cohesion of it by clicking on the - arrow, and you should see the zero, and therefore has little influence on the wedge following wedge, with a Safety Factor of 0.53. ,4 stability.
oI See the Introduction for more information on the sliding plane failure modes in SWEDGE. oI 4j Ef o II F' C. C
32 SWEDGE User's Guide Quick Start Tutorial 33 External Force Water Pressure
- 1. Toggle on the External Force checkbox.
When Water Pressure is toggled on and the Unit Weight 1 in the Input Data dialog, the factor of safety is 2. Enter Plunge = 20, Trend = 45 and Magnitude = 500. calculated assuming extreme conditions of heavy rainfall. This means that maximum (average) values of water 3. Select Apply. pressure are applied on the failure planes (and tension crack, if present). Parametric analysis of the effect of 4. The Safety Factor (with Water Pressure toggled off) varying water pressure, can be achieved by varying the drops to 0.97. Unit Weight of water between 0 and 1. Oeternninisuc TnpO t.- . .- - For example: Ge.erby ForcesI
- 1. Enter Unit Weight of water = 0.5. Select Apply. r- Walr Pres*sure.. .... . ... r- Sereriuc . . . . . . ... .
- 2. The Safety Factor increases from 0.53 to 0.81.
r- bw '"S.. I* ..
- 3. Now toggle Water Pressure off (or enter Unit Weight
= 0, this has the same effect). Select Apply.
@4 W Es~leredFor- .*. fl
- 4. The Safety Factor with no water pressure (ie. a completely dry slope) is 1.11. This is the maximum Trend (dsge I _
Safety Factor for this wedge, without installing rock bolt reinforcement.
@4@
@4 Magnihde 0) Is00 I SIW FeJ.'r- 0111111 WedgeWeight.- 53512 onnes Geereby Forces Oisttrcn ormeters SSidmgenaJo.J 1 Fo-ce iTonnes (1000k )
9 P Wate.,Piesresr . Seeu. UMWeigh,r(V3)F) .- " r F- Ep aJ F
'= .... i --.
... . - - @4 At this point we will note that rock bolts in SWEDGE are implemented in the analysis in exactly the same way as
@4 the External Force.
F-F That is, rock bolts can be simulated by an equivalent External Force, or an External Force can be simulated by Sael yFeclor" 1.10801 rock bolts. See the last tutorial in this manual, Adding WedgeWeigh.* 5365.12 *stas Dist .e . meiers I
/Sfidig onmJinti Support, for more information.
FoceinToen$ssfOOikg)9 .3
@ 4
Quick Start Tutorial 35 w4 SWEDGE User's Guide More About the Input Data Dialog Seismic Force You may not have noticed, but the Input Data dialog in
- 1. Toggle off the External Force checkbox, and toggle on SWEDGE works a little differently than a regular dialog:
the Seismic Force checkbox.
- 1. It is known as a "roll-up" dialog, since it can be
- 2. Enter a Seismic Coefficient of 0.2, and select the 4 "rolled-up" (minimized) or "rolled-down" again, by Direction as "User Defined", and enter Plunge 0 and selecting the - or arrow in the upper right corner Trend = 52. of the dialog.
t0 I ep*DI. e-M--- -- * :-P ..... :.-
.. DOterminisicInpuwOData- . . - . .
Geo-ey IFoe, I
-I- W Presure
.......... . - Setsrnic. . . . .... . . .
- 2. It can be left up on the screen while performing other tasks. When not needed, it can be "rolled-up" and
, c o,*wnt
, . .siti 02 dragged out of the way (for example, the top of the Diecton User Dond -_ screen) with the LEFT mouse button.
- 3. If multiple files are open, the Input Data dialog will always display the data in the active file.
4 You may find these properties of the Input Data dialog ISefiy F (- 3i24 09i useful, for example, when performing parametric WedgeWeight - 536.i i o1?W 4 analysis, or when working with multiple files. istat inm M momSltting on JotI I Fo,,Ton-,. I[tfl kg) 4
- 3. This will apply a force on the wedge F = 0.2
- g
- 9 where g = acceleration due to gravity and m = mass of the wedge. Note that the Trend is equal to the Dip 4 Direction of Joint Set 1, which is the worst possible direction in this case, since the failure mode for this wedge already indicates Sliding on Joint 1.
4
- 4. Select Apply, and the Safety Factor now drops to 0.84.
'4 K9 K9 I. -.
36 SWEDGE User's Guide Quick Start Tutorial 37 0 4 Stereo Projection of Input Data Planes Info Viewer 4 To view a stereographic projection of your SWEDGE Before we conclude this "quick tour", let's examine the F@-ý Input Data planes, select the Stereonet button in the toolbar. The great circles on the stereonet are identified Info Viewer option. by labels - 1, 2, TC, US, and FS - to indicate failure planes 1 and 2, the tension crack plane, and the upper and face slope planes. 1WJ Select: Analysis -4 Info Viewer A convenient summary of model and analysis parameters is displayed in its own view. Scroll down to view all of the OS; M-6'* Van .. information. This can be printed if desired. NJ _'.. .... .. - S.... LJ , r.,85 Dag &a] vam. Io gCed .WAnalysi Stýaffm*31. di
'be *TMlNISn¢C 1t*v Cagfl ~t( 4teS e n12 2)zwgN S?'7 I $0 edgea oreg(,aneS*eSI568 35tn'S Figure 2-6: Stereonet projection of SWEDGE input data planes.
Figure 2-7: Info Viewer listing. Importing Data from a DIPS File 4 That concludes this "quick tour" of SWEDGE. To exit the The planes forming the wedge geometry can also be read program: into SWEDGE from a DIPS planes file (".dwp" filename 4 Select: File -- 'Exit extension), with the Import option in the File menu. 4 DIPS is a program for the graphical and statistical analysis of structural geology data using spherical 4 projection techniques. Visit the Rocscience website at www.rocscience.com for more information. 4 4
38 SWEDGE User's Guide Probabilistic Tutorial 39 Probabilistic Analysis i.,IgF 4 4 4
° 5-',- . . ...
4 This tutorial will familiarize the user with the Probabilistic analysis features of SWEDGE. 4 If you have not already done so, run SWEDGE by double clicking on the SWEDGE icon in your installation folder. Or from the Start menu, select Programs --) Rocscience - 4 Swedge -+ Swedge. 4 If the SWEDGE application window is not already maximized, maximize it now, so that the full screen is 4 available for viewing the model. To begin creating a new model: 4 Select: File -4 New 4 A default wedge model will immediately appear on your screen. Whenever a new file is opened, the default input 4 data will form a valid wedge. 4 4
- 9 41
Probabilistic Tutorial 41 40 SWEDGE User's Guide Job Control Probabilistic Input Data Now let's look at the input data. Job Control allows the user to enter a Job Title, and select a Unit System and Analysis Type. Let's switch the Analysis Type to Probabilistic. Select: Analysis -> Input Data 4 E!1 Select: Analysis --. Job Control You should see the Probabilistic Input Data dialog shown below. 4 Frobabilistic Input Da t.. JobTde JSWEDGEOb Pobib~lsbcTutonal JoOIS.1 IorASetl j ISlope I upperFae I rTension .c, Il Fo=.ýI Sn PhigI DOn . . . . .. . .. O...Oimcon
.Dp .. . . -.... . .
.-CUn*s -.... nmeysit Type
- r* Mehic e' Deteminis Mewovaoue. [q dog i.n V.
M : r10 deg C Pobabulstc Stalitin]J OSibidO5 .WeSltiend~isikbsdn:Foo: Zj t-Imperial "r-F deg ".. .. F. dog F deg .dog
. ' deg " . . . dog Dismceu ýns mimeterssndForcetiesuintnnes (1000kg)
Cohesion . FdchonAngle. . meo Vaoue.. "2 I/.2 MeanssVoe ji deg Enter "SWEDGE Probabilistic Tutorial"as the Job Title. 4 SWISIhce! Otib on - Stetidca J ibuto: rNo Leave Units = Metric and change the Analysis Type to . "/ m.2 - 'deg " Probabilistic.Select OK. 4
........ ...... F i. .d .g d Note:
4nalysis Type can be selected' "* Analysis Type can also be changed at any time, using from the drop-down list box in the drop-down list box in the middle of the SWEDGE the toolbar toolbar. This is a convenient shortcut. Defining Random Variables
"* The Job Title will appear in the Info Viewer listing, 4) To define a random variable in SWEDGE:
discussed later in this tutorial. 4* 1. First select a Statistical Distribution for the variable.
"* Units determines the length and force units used in (In most cases a Normal distribution will be the Input Data dialog and the analysis. adequate.)
4}
- 2. Enter Standard Deviation, Minimum and Maximum values. NOTE that the Minimum / Maximum values Minimum/I Maximum are specified values 1 as RELATIVE are specified as RELATIVE distances from the mean, distances from the mean rather than absolute values.
4-
0 42 SWEDGE Users Guide Probabilistic Tutorial 43
- 3. Any variable for which the Statistical Distribution= C- Joint Set 2 "None tt will be assumed to be "exactly" known, and will not be involved in the statistical sampling. Select the Joint Set 2 tab and enter the following data:
See the Introduction for information about the properties 4 of the statistical distributions available in SWEDGE. IUggotdonnlTno,ooCoodFotolSmvPttgjh e JAWtStO Sins
,Joo,tSetl 4 Oqp MooVai. .:
_0._ 70- dog igDrea M-onolo 235 dog For this example, we will use the default Mean Input Data, and define Normal Statistical Distributions for the S Storolod Dertoun jr3-dog StodoodD. . . F3V dog following variables: R.1.6" A.-nmm j-h dog Roe]nýtjtm- j15-dog 4 R.1onroeknt.-m [FO- 4.9 Slo old-acnn FS- dog
"* Joint Set 1 Dip and Dip Direction
"* Joint Set 1 Cohesion and Friction Angle S Colasnot Moon F-Vo.od2 F-ton Agin MoonMok. F30-dog
"* Joint Set 2 Dip and Dip Direction
"* Joint Set 2 Cohesion and Friction Angle e S Sfontdoo0--ma-n o-s 5-2 Stndowd~ffietba.F2- dog
"* Tension Crack Dip and Dip Direction S S AWtnnoteo F2-n m? SZoo. Med.-
tatm dog Joint Set I e 4 Make sure the Joint Set 1 tab is selected in the Input S 4 Data dialog, and enter the following data: Tension Crack Poobodi.odotpidDe1.ft S 4 tiIo is~eI Slo. jI Ppotooj T-.On tfloItjF.-ooI St-nt*ng Select the Tension Crack tab and enter the following data: oip JOSS Ooooo S. Moot~Voo.
.. dog i MotVo" de Ptobohhoiclo.. pOlDoe - -ý. . ... -
St~hdaiotý0%boO- StvAooma0Sttnnt ntt S. JonnSotsIjJintcSWtzjSlope I jppetForeTwsonsnCnmorkelp SopnphgI1 Stottdnatdfl~ontoiFnn - dog Siodotdao~meoob rj3 - dog W TeontsOoWUmt, Soodo
] a ntut- O -- dog Relý t~otM att 15r dog @4 RoloonoMaeatFlo- dog PR~e-so F15-o~dog
- TtonLongdtm) ji ColtnonFmnoodntgto- SI, MoonVolun. 70 dog M-oonvbj dog
- M-ooD iojo n. r dn2 Moo Dobjo, fi dogel
@4 Stodomdawaofl F3n deg Stodomd 00.5cm t dog donsDeina. jSr 11.2 Swodood~otoas F2 dog Stol"W. o-o tnm jTW dog PoloonoOdammuto S dog SWOibuM.-tt t f2- ldm2 Roiawbdom~n- Fi-sog 5ii9 Roloing MV--oot VetO jW 5.50.,Maoo. Or-deg SolnunooMotnum F -0 dog Roaýml--ctt, i dog
@4 SI',
@9 5119
Probabilistic Tutorial 45 44 SWEDGE User's Guide lo ! Slope Probabilistic Analysis We will assume that the orientation of the slope planes is To carry out the SWEDGE Probabilistic Analysis: exactly known, so we will not enter statistical data for the I. upper slope or face slope orientations (ie. Statistical 0 Select the Apply button in the Input Datadialog. Distribution = None for these variables). The analysis will be run using the parameters you have Forces just entered. Calculation should only take a few seconds. The progress of the calculation is indicated in the status We will not be using the Forces options in this tutorial. bar. See the Quick Start Tutorial for a discussion of Forces in SWEDGE. Close the dialog by selecting the Done button. Results Sampling We will use the default Sampling Method and Number of Samples (ie. Monte Carlo method, 1000 samples). Probability of Failure The primary result of interest from a Probabilistic 4 Analysis is the Probability of Failure. This is displayed in the toolbar at the top of the screen. I. 4 TkEAei s au Pr wiSasbsm W'. do elp -U For this example, if you entered the Input Date correctly, 4 you should obtain a Probability of Failure of around 6%. (eg. PF = 0.061 means 6.1% Probability of Failure). 4 However, remember that the sampling of the Input Data 4 is based on the generation of random numbers by the Monte Carlo analysis. Therefore the Probability of Failure 4 will not necessarily be the same each time you compute with the same data. 49 See the section on Re-running the Analysis later in this C '4 tutorial, for a demonstration. K. K. K. 4.-
46 SWEDGE User's Guide Probabilistic Tutorial 47 Wedge Display Mean Safety Factor The wedge initially displayed after a Probabilistic Notice the mean, standard deviation, min and max values Analysis, is based on the mean input values. Therefore, displayed below the histogram. the wedge will appear exactly the same as one based on The mean Safety Factoris Deterministic Input Data with the same orientation as not necessarily the same as Keep in mind that the mean Safety Factor from a the mean Probabilistic Input Data. the Deterministic Safety Probabilistic Analysis is not necessarily the same as the Factorbased on the mean Input Data values. Deterministic Safety Factor based on the mean Input However, other wedges generated from the Probabilistic Data values. In general, these two values will not be equal Analysis can be displayed as described below. to each other. Histog rams Manipulating the Histogram View To plot histograms of results after a Probabilistic 1. If you right-click on a histogram and select 3D Histogram, you can apply a 3D effect. Analysis:
- 2. If you click and HOLD the LEFT mouse button on the Select: Statistics --* Plot Histogram histogram and move the mouse, you can change the "viewing angle" of the 3-D histogram.
OotmType: ISafely Feawor 3. To restore the default viewing angle of a 3D Histogram, right-click and select Reset View. Numbeoflntmrs. 130 4 r o Viewing Other Wedges 7 oon~noT~ 4 Let's now tile the Histogram and Wedge views, so that both are visible. Select OK to plot a histogram of Safety Factor. It 4 Fl-i1 Select: Window --> Tile Vertically A useful property of the Histogram view is the following: The histogram represents the distribution of Safety 4
- If you double-click the LEFT mouse button anywhere Factor, for all valid wedges generated by the Monte Carlo on the histogram, the nearest corresponding wedge sampling of the Input Data. The red bars at the left of the 4 will be displayed in the Wedge view.
distribution represent wedges with Safety Factor less than 1.0. 4 For example: eL 4 1. Double-click on the histogram at approximately Safety Factor = 1. 4 4 jI
.4.,
48 SWEDGE User's Guide Probabilistic Tutorial 49
- 2. Notice that a different wedge is now displayed. Histograms of Other Data
- 3. The actual Safety Factor ofthis wedge is displayed in .j In addition to Safety Factor, you may also plot histograms the title bar of the Wedge view. It will probably not be of:
exactly = 1, since it depends on exactly where you clicked, and the actual safety factor of the nearest "* Wedge Weight wedge.
"* Plunge or Trend of Line of Intersection of Joint Sets 1 and 2 D$0am T i ,d..m 43 4
"* Any random variable (ie. any Input Data variable which was assigned a Statistical Distribution) di For example:
r' e- .- : .,.,. Select: Statistics Plot Histogram In the dialog, set the Data Type = Wedge Weight, and select OK. A." A histogram of the Wedge Weight distribution will be generated. Note that all of the features described above for the Safety Factor histogram, apply to any other Data Type. For example, if you double-click on the Wedge Weight Figure 3-1: Safety Factor histogram and wedge view. histogram, the nearest corresponding wedge will be displayed in the Wedge View. In any case, this feature is meant to give you a general -4 idea of the shape and orientation of wedges corresponding Let's generate one more histogram. to locations along the histogram. For example, you will probably want to double-click in the "red" Safety Factor Select: Statistics -- 4 Plot Histogram region, to see the wedges with a Safety Factor < 1. This time we will plot one of our Input Data random To reset the Wedge view so that the mean wedge is variables. Set the Data Type = Dip of Joint 1. Check the displayed: Plot Sampled Distribution checkbox. Select OK.
.r4 Select: View -+ Reset Wedge e'4 This will display the wedge corresponding to the mean Probabilistic Input Data. E'4
.4..
Probabilistic Tutorial 51 ro SWEDGE User's Guide
'JI . The Wedge View and 1 - . '- . -
1 I
- The Safety Factor, Wedge Weight, and Joint 1 Dip Sri. Angle Histograms.
a", I If you closed any of the histograms, re-generate them as described above. Now tile the four views.
@~ta Select: Window -4 Tile Vertically a""
a'aazSL.)A .b',,
, M i, k Figure 3-2: Joint 1 Dip Angle - Monte Carlo sampling of normal distribution.
The histogram shows how the Dip of Joint 1 Input Data variable was sampled by the Monte Carlo analysis. The curve superimposed over the histogram is the Normal distribution you defined when you entered the mean, standard deviation, min and max values for Dip of Joint 1 in the Input Data dialog. Figure 3-3: Tiled histogram and wedge views. Re-running the Analysis Now select the Compute button in the SWEDGE toolbar. The Probabilistic Analysis can be re-run at any time, by selecting the Compute button in the toolbar. In general, the Probability of Failure will be different each E-1 Select: Analysis -+ Compute Notice that the Histograms and Probability of Failure are time the analysis is re-run. updated with the new analysis results. Let's demonstrate this, but first let's tile the views again. Now continue to select Compute several times, and If you have not closed any views, you should still have on observe the variation in the Histograms and the your screen: Probability of Failure. This graphically demonstrates the SWEDGE Monte Carlo analysis.
.461.
52 SWEDGE User's Guide Probabilistic Tutorial 53 Note that the Wedge view does not change when you re Alternatively, press and HOLD the LEFT mouse compute, since the default wedge displayed is based on button on the plot, and you will see the double-arrow the mean Input Data, which is not affected by re-running I. icon. Move the mouse left or right, and the sampler the analysis. will continuously display the values of points along 9 the curve. For this example, if you re-run the analysis several times, you will find that the Probability of Failure will vary 9 Y - - .. XL QUa5S. Zl-eu a. S-r--- 0
- 0d O.
aIL FF.- between about 4 and 8%. Cumulative Distributions (S-curves) :.,I In addition to the histograms, cumulative distributions (S curves) of the statistical results can also be plotted. i0, 0I FLOSI, Select: Statistics -> Plot Cumulative 4 DmtType: lSaetsyFacoss op I iii Ia ma Nanber of Intervos. 30 9 4 I.-, S f techMeekets t ODislibuion 4 Figure 3-4: Cumulative safety factor distribution. The display of the Sampler can be turned on or off in the right-click menu or the Statistics menu. I, Select OK. Now tile the views one more time, and re-compute the 4 analysis. The cumulative Safety Factor distribution will be generated, as shown in Figure 3-4. 4 Select: Window -- Tile Vertically 4 Fo-1 Notice the vertical dotted line visible on the plot. This is Select: Analysis -- Compute the Sampler, and allows you to obtain the coordinates of any point on the cumulative distribution curve. 4 Notice that the cumulative distribution gets updated along with the histograms, each time the analysis is re
- To use the sampler, just SINGLE click the LEFT U, run.
mouse button anywhere on the plot, and the sampler will jump to that location, and display the results. 4 4 4 lb-
Probabilistic Tutorial 55 54 SWEDGE User's Guide
- 1. Close (or minimize) all views you may have on the Info Viewer I screen, EXCEPT the Info Viewer and the Safety Factor Histogram.
Let's examine the Info Viewer listing for a Probabilistic Analysis. 2. Select the Tile Vertically toolbar button. Select: Analysis --> Info Viewer 3. If necessary, scroll down in the Info Viewer view, so that the Current Wedge Data is visible. A convenient summary of model and analysis parameters is displayed in its own view. Scroll down to view all of the 4. Double-click at different points on the Safety Factor information. This can be printed if desired. histogram, and notice that the Current Wedge Data is updated to show the data for the "picked" wedge. Notice the summary of Valid, Failed and Safe Wedges. Depending on your geometry input, it is possible for the Probabilistic Sampling of the Input Data to generate
$edge ,e~ysislnformaLvn invalid wedge geometries. In general:
Number of Failed Wedges + 4., W.WI Number of Safe Wedges = Number of Valid Wedges Number of Samples 2 WzM2331.11 Number of Valid Wedges = 2DN .. 2' 80 Number of Invalid Wedges W 8236.C'W4 4 obt40j0 As with the Histograms and S-curves, if you re-compute w DeoD-J-10 )=5105ft9 the analysis, the Info Viewer listing is automatically 4e J I 0o. 2921 1 I F,A-g-22 621t IIn 0r o updated to reflect the latest results. Current Wedge Data Notice the Current Wedge Data listing in the Info Viewer. Figure 3-5: Current Wedge Data for Picked Wedge. By default, the mean wedge data is displayed after a Probabilistic analysis. 5. To reset the Current Wedge Data to the mean data: Remember we pointed out earlier that if you double-click Select: View - Reset Wedge on a Histogram, the nearest wedge will be displayed in the Wedge View. The Current Wedge Data will also be updated, to reflect the data for the "picked" wedge. Let's demonstrate this.
.4-
Support Tutorial 57 56 SWEDGE User's Guide Sampling Method Adding Support As a final exercise, set the Sampling Method to Latin Hypercube, and re-run the analysis. Rock bolts are added to an SWEDGE model with the Add Bolt option. This allows the user to evaluate the number, Select: Analysis --3 Input Data location, length and capacity of bolts necessary to stabilize a wedge (ie. increase the Safety Factor to a required In the Input Data dialog, select the Sampling tab, and set amount). the Sampling Method to Latin Hypercube. Select the Apply button. Let's start with a new (Deterministic) file for the purposes
@ . of the following demonstration.
Examine the Probability of Failure, and the Safety Factor Histogram. The results should be very similar to the Monte Carlo analysis. The difference is in the sampling of the Input Data lk M LDi1 Select: File -), New To add a rock bolt: random variables. For example, generate a Histogram of Joint Set 1 Dip Angle. Select: Support -+ Add Bolt
- 1. Move the cursor into the Top or Front orthogonal Select: Statistics -* Plot Histogram views.
Set the Data Type = Dip of Joint 1. Select the Plot
- 2. Notice that the cursor changes to an "arrow / rockbolt" Sampled Distributioncheckbox. Select OK. icon.
Now compare the histogram with Figure 3-2. The Latin
- 3. As you move the cursor over the wedge, notice that Hypercube sampling method results in a much smoother the "rockbolt" and "arrow" now line up - this indicates sampling of the Input Data distribution, compared to the that you may add the bolt to the wedge.
Monte Carlo method.
-xll~~l,. 4. Click the LEFT mouse button at a point on the wedge I, Lemp h(m): ~ where you want the bolt installed.
CTpen (tonnes:) F120-
,4 Trend (deg): J 5. The bolt will be installed NORMAL to the face of the wedge on which you clicked (ie. normal to the Upper Plunge(deg): 125
- 4) or Face slope), however you can modify the Facbo oItaled :1.0428 orientation using the Bolt Properties dialog which you 6 4) F-071AFip-ý1r~ will see in the middle of the screen.
4) 4) 4P
Support Tutorial 59 58 SWEDGE User's Guide
- 6. The Bolt Properties dialog works as follows:
"* If you modify the Capacity, Trend or Plunge with the "arrow" buttons at the right of the dialog, the Safety Factor is immediately recalculated and displayed in the dialog as the values are being changed. This allows the user to interactively modify the bolt properties, and immediately see the effect on the Safety Factor.
" Alternatively, values can be typed in to the dialog.
In this case, the Safety Factor is NOT automatically re-calculated, the user must select the Apply button to apply typed in values.
" As the bolt Trend and Plunge are changed, you will see the orientation of the bolt updated on the screen. Figure 4-1: Adding a bolt.
"* Changing the Length of the bolt will be visible on Note:
the model, but has NO effect on the Safety Factor
- see the next section for details. 4 The Right orthogonal view can also be used for adding bolts, however this is not recommended, as correct placement may be difficult or impossible. (If the Dip
- 7. When the bolt orientation, length and capacity are Directions of the Upper and Face Slope are the same satisfactory, select OK, and the bolt will be added to it will NOT be possible to add a bolt in the Right the model. orthogonal view.)
- 8. If you are not happy with the location of the bolt,
- Bolts can NOT be added in the Perspective view.
select Cancel, and the bolt will be deleted. E6d C.,
60 SWEDGE User's Guide Support Tutorial 61 How Bolts are Implemented in SWEDGE Bolts vs. External Force
- 1. A bolt is therefore exactly equivalent to adding an Bolts are implemented in the SWEDGE stability analysis External Force with the same magnitude and as follows: orientation. (See the Quick Start Tutorial for an example of adding an External Force).
Capacity and Orientation
- 2. It is left as an exercise for the user to verify that a
- 1. Bolts affect the Safety Factor through their Capacity bolt, and an equivalent External Force, result in the and Orientation (Trend / Plunge) only. same Safety Factor.
- 2. Bolt capacities and orientations are added vectorially, Multiple Bolts and are included in the Safety Factor calculation as a single, equivalent force passing through the centroid Any number of bolts can be added to a model, by of the wedge. repeating the steps outlined above.
- 3. Multiple bolts with the same orientation can therefore However, remember that bolts in SWEDGE simply be simulated by a single bolt having the same total behave as force vectors passing through the centroid of capacity. the wedge. The applied force is equal to the bolt capacity.
Length and Location Therefore, in terms of the effect on the Safety Factor, multiple bolts can be simulated by:
- 1. Bolt Length and Location (on the face of the wedge) have NO effect on the Safety Factor. E 9 "* a fewer number of bolts, or even a single bolt, with equivalent capacity and direction,
- 2. The Length and Location of bolts allows the user to C, visualize the practical problems of installing the bolts. "* or an equivalent External Force.
"'9
- 3. Even bolts which do not pass through the wedge, will Installation of multiple bolts is useful for visualizing the affect the Safety Factor (ie. SWEDGE does NOT check 54 practical problems of bolt installation, and the necessary for valid bolt lengths). So do NOT assume that "short" bolt lengths and spacing. Or for back-calculating the bolts will be filtered out - they will have exactly the 54 Safety Factor of an existing wedge support system.
same effect as longer bolts with the same capacity and orientation. I ill.
.-4..
I 62 SWEDGE User's Guide Support Tutorial 63 Deleting Bolts Editing Bolts To delete bolts: To edit the properties of a bolt: Select: Support -. Delete Bolt SSelect: Support -4 Edit Bolt Bolts can be deleted in the Top, Front or Right views as Bolts are selected for editing in the same manner as for follows (bolts cannot be deleted in the Perspective view): deleting - see the previous page for instructions. fig ps$
- 1. Move the cursor in the Top, Front or Right views. Once a bolt has been selected for editing:
- 2. The cursor will change to a small "box". 1. You will see the Bolt Properties dialog in the middle of the screen, displaying the properties of the bolt.
- 3. Hover the cursor over a bolt that you wish to delete.
- 2. You can modify the Capacity, Trend, Plunge or
- 4. The bolt will change colour, to indicate that it is Length of the bolt, in the same manner as when you "selected". originally added the bolt. See the previous pages for details.
- 5. When the correct bolt is selected, click the LEFT mouse button, and the bolt will be deleted. e- I 3. When you are finished editing the properties, select OK to save your changes.
- 6. A new Safety Factor will immediately be calculated.
- 4. If you select Cancel, all changes will be cancelled,
- 7. Repeat steps 3 to 5 to continue deleting bolts. even if you used the Apply button to apply the changes.
- 8. Press Escape to exit the Delete Bolts option. Sm4 Bolts can only be edited one at a time in this manner. It is To delete ALL bolts at once: not possible to edit the properties of multiple bolts simultaneously.
E1 Select: Support -+ Delete Bolt
- 1. Enter the asterisk ( * ) character on the keyboard.
Listing of Bolt Properties A listing of all bolts and their properties (Capacity,
- 2. ALL bolts will be deleted from the model.
Length, Trend and Plunge) can be found in the Info Viewer listing. Note that the bolt colour and the "selected" bolt colour, can be modified in the View -+ Display Options dialog, if necessary, for easier viewing. Fd-1 The Info Viewer option is available in the Analysis menu, and in the toolbar.
e. 64 SWEDGE User's Guide Bolts in a Probabilistic Analysis The above discussion of bolts in SWEDGE assumes a Deterministic Analysis of a single wedge. If the Analysis Type is PROBABILISTIC: ' 9
"* the Probabilistic Analysis will be run EACH time a bolt is added or edited (ie. when OK is selected on the Bolt Properties dialog).
"* Selecting Apply in the Bolt Properties dialog will * "9 calculate a new Safety Factor for the MEAN wedge, but will NOT run the Probabilistic Analysis.
"*Ifyou are deleting bolts, the Safety Factor for the *'
MEAN wedge will be re-calculated as each bolt is deleted, but the Probabilistic Analysis will only be run when you exit the Delete Bolts option. NOTE: Bolts should be used with some caution in a Probabilistic Analysis if your random variables include the orientation of the planes forming the wedge.
- Since the bolts are added while viewing the mean wedge, the orientations of bolts added on the mean wedge may no
- longer be appropriate in terms of support to wedges of other orientations generated by the Probabilistic Analysis. f 4 If the only random variables in the Probabilistic Analysis 4 are the strength parameters (cohesion and friction angle) of the failure planes, then this will not be an issue, since
- the wedge geometry will remain constant.
C.
*6.'
80.1 73 DISPLAY COPY 994 .4 PLEASE DO NOT REMOVE National Information Service for Earthquake Engineering REPORT NO. UCB/EERC-94/05 EARTHQUAKE ENGINEERING RESEARCH CENTER MAY 1994 N. SEISMIC RESPONSE OF STEEP NATURAL SLOPES UNWV6t1i Y u0- UiLIFOHNIA by .~r..-tquak'e Engnering SCOTT A. ASHFORD JUNj 2 71994 NICHOLAS SITAR Lt[m-V ,+ Final report on research sponsored by the United States Geological Survey under USGS Award Number 14-08-0001 -G2127 1 I I P I "r COLLEGE OF ENGINEERING UNIVERSITY OF CALIFORNIA AT BERKELEY
For sale by the National Technical Information Service, U.S. Department of Commerce, Spring field, Virginia 22161 See back of report for up to date listing of EERC reports. DISCLAIMER Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the Sponsor or the Earthquake Engineering Research Center, University of California at Berkeley.
EARTHQUAKE ENGINEERING RESEARCH CENTER SEISMIC RESPONSE OF STEEP NATURAL SLOPES by Scott A. Ashford Nicholas Sitar Report No. UCB/EERC-94/05 Earthquake Engineering Research Center College of Engineering University of California at Berkeley May 1994 Final report on research supported by the U.S. Geological Survey under USGS award number 14-08-0001-G2127. The views and conclusions contained in this document are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of the U.S. Government. EARTHQUAKE ENG. RES. CTR. LIBRARY Univ. of Calif. - 453 R.F.S. 1301 So. 46th St. Richmond, CA 94804-4698 USA (510) 231-9403
ABSTRACT The results of a research program to evaluate the seismic response of steep slopes are presented. The impetus for this work was the October 17, 1989, Loma Prieta Earthquake which caused extensive landsliding along the coastal bluffs from San Francisco to Santa Cruz. While this research is specific to the bluffs in the San Francisco Bay region, the methods developed are generally applicable to stability analyses of steep, natural slopes. A frequency domain parametric study on topographic effects, using the generalized hyperelement method, shows that the peak amplification of motion at the crest occurs at a normalized frequency H/X = 0.2, where H is the slope height and X is the wavelength of the motion. Amplification was found to increase with inclined waves traveling into the slope crest, and to decrease with inclined waves traveling away from the crest. More importantly, the natural frequency of the site behind the crest dominates the response, relative to the topographic effect. The importance of the natural frequency is illustrated by the time domain response of 3 prototype sites using actual seismograms. The results show that the topographic amplification at the crest of a steep slope can be reasonably estimated by increasing the peak acceleration obtained from a one-dimensional site response analysis in the free field behind the crest by 50 percent. Then, for use in limit equilibrium slope stability analyses, the seismically induced force on a potential sliding mass can be estimated using profiles of the average seismic coefficient developed from the analytical results. i
One of the objectives of the research was to develop practical analysis guidelines for evaluation of seismic response of steep slopes in weakly cemented natural deposits. A review of the laboratory behavior of weakly cemented sands shows that these materials exhibit brittle behavior under low confining stress, typical of a near-slope environment. This assessment of the behavior is supported by numerous field observations of seismically induced failures. Therefore, a limit equilibrium approach, rather than a deformation based analysis, is recommended. ii
ACKNOWLEDGMENTS Professor John Lysmer made a significant contribution to this research effort in the area of two-dimensional seismic site response; his assistance is greatly appreciated. This research was sponsored by the United States Geological Survey under USGS award number 14-08-0001-G2127. iii
I-9 AT
TABLE OF CONTENTS Page
. . i Abstract.
..i Acknowledgments .............................
V Table of Contents ............................. S.. . . . . . . . . . . . . . ix List of Tables ................................ S.. . . . . . . . . . . . . . xi List of Figures ................................ 1
- 1. Introduction ............................
*. ... . . . . . . . . . . . 7
- 2. Behavior of Weakly Cemented Sand ...........
2.1 Review of Static Properties ............
...... . . . ..... 10 2.2 Review of Dynamic Properties ..........
S.ands ........... 13 2.3 Observed Slope Failures in Weakly Cemented
.............. 20 2.4 Conclusions .......................
. ... .. .. ... ... 23
- 3. Static Finite Element Analysis of Steep Slopes ...
24 3.1 Background .......................
. .. .. .. .. ...... 29 3.2 Physical Model .....................
.. ... .. .. ...... 32 3.3 Finite Element Model ................
35 3.4 Results ........................... 45 3.5 Conclusions ....................... S.. . . . . . . . . . . . .. 47
- 4. The Generalized Hyperelement Method .........
S........ ....... 47 4.1 Computational Model ................ 4.1.1 The Complex Response Method ......... 49 v
Page 4.1.2 Equations of Motion . ................................. 51 4.1.3 Eigenvalue Problem of a Layered Halfspace .................. 53 4.1.4 Generalized Transmitting Elements ......................... 56 4.1.5 The Generalized Hyperelement ........................... 58 4.2 Free-Field Motions in the Layered Regions .................. 61 4.2.1 Inclined In-Plane Waves (SV- and P-Waves) ........... 62 4.2.2 Inclined Out-of-Plane Waves (SH-Waves) .............. 65 4.3 Simulation of Semi-Infinite Halfspace at Base ................ 65 4.4 Input Ground Motion .................................. 68 4.5 Generation of Ground Motions ............................ 69
- 5. Topographic Effects ........................................ 71 5.1 Background . ....................................... 71 5.2 Analysis of a Stepped Halfspace .......................... 76 5.2.1 Effect of SH- (Out-of-Plane) Waves on a Vertical Slope . . . 78 5.2.2 Effect of SV- (In-Plane) Waves on a Vertical Slope ...... 84 5.2.3 Effect of the Slope Angle . ........................ 94 5.2.4 Effect of the Incident Angle ........................ 94 5.3 Analysis of a Stepped Layer over a Halfspace ............... 103 5.4 Conclusions ...................................... 107
- 6. The "Average Seismic Coefficient" for Steep Slopes ................ 109 6.1 Studies of Slope Response .............................. 109 6.2 Studies of Embankment Dam Response ..................... 111 vi
Page 6.3 The Seed and Martin (1966) Approach .... . . . . . . . . . . 114 6.4 The Makdisi and Seed (1978) Method ..... . . . . . . . . . . 121 Application of ka, to Steep Slopes ........ 123 6.5 Sliding Mass 123 6.5.1 Distribution of Weight in the Potential 126 6.5.2 Selection of an Acceleration Profile . 6.5.3 Evaluation of Crest Acceleration .... 127 135 6.6 Conclusions ....................... * . . . . . . . ... 137
- 7. The Seismic Response of Steep Slopes ..........
137 7.1 Prototype Site Characterization .......... 139 7.1.1 Seacliff State Beach Site ......... Daly City Site ................ 140 7.1.2 144 7.2 Site Specific Analyses ................ 149 7.2.1 Seismograms Used in Analysis ..... 150 7.2.2 Slope Models ................. 155 7.3 Results ........................... ts. . . . .. . . . 7.3.1 Slope Crest Amplification 155 7.3.2 Maximum Average Seismic Coefficien 161 169 7.4 Influence of Inclined Waves .... *s . . . ... .
............... 172 7.5 Conclusions 7.6 Implications for Stability Analyses of Steep S*lopes ...... 174
.. .. . . . . . . . . . . . .. ... ..... 177 References 185 Appendix A: Soil Borings ............... . . . . . . . . . .
191 Appendix B: Shear Wave Velocity Testing ... vii
viii LIST OF TABLES Page Table 25 3.1 Summary of finite element analyses of slopes ................ 35 3.2 Summary of meshes used in study ........................ 106 5.1 Transfer functions for stepped layer over halfspace ........... 6.1 Comparison of amax for GHE method and Makdisi and 134 Seed procedure ...................................... 156 7.1 Summary of results for 2-D site response analysis ............. 7.2 Summary of horizontal results for inclined wave analysis 170 of Seacliff model with Z/H = 1.0......................... 7.3 Summary of vertical results for inclined wave analysis 171 of Seacliff model with Z/H = 1.0 ......................... ix
LIST OF FIGURES Figure Page 1.1 Distribution of landslides in the San Francisco area caused by the 1989 Loma Prieta Earthquake (from Seed et al., 1990) ...... 2 1.2 Slope failures along the Ocean Shore Railroad north of Mussel Rock caused by the 1906 San Francisco Earthquake (courtesy of the Bancroft Library, U.C. Berkeley) ......................... 4 1.3 Landslides along bluffs in Daly City caused by the 1957 San Francisco Earthquake (courtesy of the California Department of Transportation) ...................................... 4 1.4 Localized minor failures of marine terrace deposits in Pacifica caused by the 1989 Loma Prieta Earthquake ................... 5 1.5 Failure of bluffs in Daly City caused by the 1989 Loma Prieta Earthquake ...................................... 5 2.1 Typical stress-strain curves for artificially cemented sand (from Clough et al., 1981) ................................. 9 2.2 Summary plot of shear modulus versus shear strain for cemented sands (from Wang, 1986) ................................. 11 2.3 Summary plot of damping ratio versus shear strain for cemented sands (from Wang, 1986) ................................. 11 2.4 Typical static and cyclic stress-strain curves from simple shear tests on cemented sands (from Wang, 1986) ............... 12 2.5 Dynamic strength envelopes for cemented sands under cyclic simple shear loading (from Wang, 1986) ...................... 12 2.6 Landslides at Centerville Beach caused by the 1992 Petrolia Earthquake. ......................................... 16 2.7 Intact blocks in landslide debris at Centerville Beach ............. 16 2.8 Landslide at Pacific Palisades caused by the 1994 Northridge Earthquake .......................................... 17 xi
Figure Page 2.9 Intact blocks in slide debris at Pacific Palisades ................. 17 2.10 Failure mode for moderately steep slopes in cemented sands (after Sitar, 1990) ....................................... 21 2.11 Failure mode for very steep slopes in cemented sands (after Sitar, 1990) ....................................... 21 3.1 Gelatine model (from La Rouchelle, 1960) .................... 30 3.2 Typical FEM mesh and parameter definition .................. 33 3.3 Curved FEM mesh ...................................... 34 3.4 Variable sized element mesh.............................. 34 3.5 Comparison of al-03 between gelatine model and FEM using element n4al ......................................... 36 3.6 Comparison of ol-a 3 between gelatine model and FEM using element nl6al ........................................ 36 3.7 Comparison of a 3 between gelatine model and FEM using element n4al .......................................... 38 3.8 Comparison of a3 between gelatine model and FEM using elem ent nl6al ......................................... 38 3.9 Comparison of a1 between gelatine model and FEM using elem ent n4al .......................................... 39 3.10 Comparison of a1 between gelatine model and FEM using elem ent nl6al ......................................... 39 3.11 Comparison of al-a3 between gelatine model and FEM using curved mesh ......................................... 41 3.12 Comparison of a3 between gelatine model and FEM model using curved mesh ......................................... 41 3.13 Comparison of a1 between gelatine model and FEM using curved mesh ......................................... 42 xii
Figure Page 3.14 Comparison of al-a3 between gelatine model and FEM using variable sized element mesh ............................... 42 3.15 Comparison of a3 between gelatine model and FEM using variable sized element mesh ............................... 43 3.16 Comparison of a1 between gelatine model and FEM using variable sized element mesh ............................... 43 3.17 Variation of maximum tensile stress along top of slope with varying aspect ratio ...................................... 44 3.18 Variation of maximum tensile stress along top of slope with varying element size ..................................... 44 4.1 Site for two-dimensional seismic site response analysis (after D eng, 1991) . ..................................... 48 4.2 Typical model of site using GROUND2D (after Deng, 1991) ....... 48 4.3 Semi-infinite layered region (after Deng, 1991) ................. 54 4.4 Layered blocky region bounded by irregular boundaries (from D eng, 1991) ........................................... 54 4.5 Wave-scattering in a layered system (after Chen, 1981) ........... 63 4.6 Variation of surface amplitude with incident angle ............... 63 5.1 Definition of wave types used in study ........................ 77 5.2 Stepped halfspace model for a vertical slope ................... 77 5.3 Horizontal transfer functions (a) and amplifications (b) for vertically incident SH-wave on a stepped halfspace for various distances behind crest, o = 1% ............................ 80 5.4 Horizontal transfer functions (a) and amplifications (b) for vertically incident SH-wave on a stepped halfspace for various distances behind crest, 6 = 5% ............................ 81 xiii
Figure Page 5.5 Horizontal transfer functions (a) and amplifications (b) for vertically incident SH-wave on a stepped halfspace for various distances behind crest, a = 10% ........................... 82 5.6 Horizontal transfer functions (a) and amplifications (b) for vertically incident SH-wave on a stepped halfspace for various distances behind crest, p = 20% ........................... 83 5.7 Horizontal transfer functions (a) and amplifications (b) for vertically incident SV-wave on a stepped halfspace for various distances behind crest, 0 = 1% ........................... 86 5.8 Horizontal transfer functions (a) and amplifications (b) for vertically incident SV-wave on a stepped halfspace for various distances behind crest, p = 5% ............................. 87 5.9 Horizontal transfer functions (a) and amplifications (b) for vertically incident SV-wave on a stepped halfspace for various distances behind crest, p = 10% ............................ 88 5.10 Horizontal transfer functions (a) and amplifications (b) for vertically incident SV-wave on a stepped halfspace for various distances behind crest, p = 20% ............................ 89 5.11 Vertical transfer functions (a) and amplifications (b) for vertically incident SV-wave on a stepped halfspace for various distances behind crest, p = 1% ........................... 90 5.12 Vertical transfer functions (a) and amplifications (b) for vertically incident SV-wave on a stepped halfspace for various distances behind crest, P = 5% ........................... 91 5.13 Vertical transfer functions (a) and amplifications (b) for vertically incident SV-wave on a stepped halfspace for various distances behind crest, P = 10% ............................ 92 5.14 Vertical transfer functions (a) and amplifications (b) for vertically incident SV-wave on a stepped halfspace for various distances behind crest, / = 20% ............................ 93 5.15 Stepped halfspace model for an inclined slope ................ 95 xiv
Figure Page 5.16 Horizontal amplification at the crest for a vertically incident SH-wave on an inclined slope, a = 1% ...................... 95 5.17 Horizontal amplification at the crest for a vertically incident SV-wave on an inclined slope, / = 1% ....................... 96 5.18 Vertical amplification at the crest for a vertically incident SV-wave on an inclined slope, / = 1% ....................... 96 5.19 Stepped halfspace model for inclined wave incident on a vertical slope ........................................... 98 5.20 Amplifications at the crest for inclined SH-wave incident on a vertical slope, F = 00, p = 1% ............................. 99 5.21 Amplifications at the crest for inclined SH-wave incident on a vertical slope, F = -10° and + 10, /3 = 1% .................... 99 5.22 Amplifications at the crest for inclined SH-wave incident on a vertical slope, F = -20' and +20, /3 = 1%................... 100 5.23 Amplifications at the crest for inclined SH-wave incident on a vertical slope, F = -30° and +30°, 6 = 1% ................... 100 5.24 Amplifications at the crest for inclined SV-wave incident on a vertical slope, F = 0, p = 1% ............................ 101 5.25 Amplifications at the crest for inclined SV-wave incident on a vertical slope, F = -10' and + 100, /3 = 1% ................... 101 5.26 Amplifications at the crest for inclined SV-wave incident on a vertical slope, F = -20° and +20, /3 = 1% ................... 102 5.27 Amplifications at the crest for inclined SV-wave incident on a vertical slope, F = -30° and +300, p = 1%................... 102 5.28 Model for vertically stepped layer over a halfspace ............. 104 5.29 Comparison of transfer function ratio Tnc/Tnf as a function of frequency ratio, ('Yon . ................................. 104 6.1 The one-dimensional shear slice method ..................... 116 xv
Figure Page 6.2 The forces acting on a potential sliding mass (after Seed and Martin, 1966) .......................................... 116 6.3 Assumed shape of potential sliding mass in Seed and Martin (1966). 119 6.4 Relationship between kmax/iumax and depth of sliding mass (from Makdisi and Seed, 1978) ................................. 119 6.5 Wedged-shaped failure surface for a steep slope ............... 125 6.6 Shear wave velocity profile used for comparison of kmax profiles... 128 6.7 Comparison of kmax profiles at slope crest and H/4 behind slope crest............................................ 129 6.8 Shear modulus reduction (a) and damping (b) curves for weakly cemented sand (after Wang, 1986) .......................... 131 6.9 Acceleration response spectra for El Centro N/S seismogram (a) and UCSCO seismogram (b) ........................... 132 7.1 Location map for prototype sites ........................... 138 7.2 Downhole travel times of compression and shear waves signals, together with corresponding velocities, for Seacliff State Beach site ............................................ 141 7.3 Average and interval shear wave velocities for Seacliff State Beach site ........................................ 142 7.4 Downhole travel times of compression and shear waves signals, together with corresponding velocities, for Daly City site ......... 145 7.5 Average and interval shear wave velocities for Daly City site ...... 146 7.6 Seismograms used in analysis .............................. 147 7.7 Acceleration response spectra for JOS90 seismogram ........... 148 7.8 Shear wave velocity profile used in analysis of Seacliff model ...... 151 7.9 Shear wave velocity profile used in analysis of Daly City model .... 153 xvi
Figure Page 7.10 Shear wave velocity profile used in analysis of Pacific Palisades model ....................................... 154 7.11 Acceleration locations calculated in study .................... 158 7.12 Normalized maximum seismic coefficient profile for Pacific Palisades model, Z/H = 1.5 ............................... 162 7.13 Normalized maximum seismic coefficient profile for Daly City model, Z/H = 1.16 ..................................... 163 7.14 Normalized maximum seismic coefficient profile for Seacliff model, Z/H = 2.44 ..................................... 164 7.15 Normalized maximum seismic coefficient profile for Seacliff model, Z/H = 1.5 ...................................... 165 7.16 Normalized maximum seismic coefficient profile for Seacliff model, Z/H = 1.00 ...................................... 166 7.17 Summary plot of all results compared to Makdisi and Seed (1978). 167 xvii
xviii
- 1. INTRODUCTION The results of a research program to evaluate the seismic response of steep slopes in weakly cemented granular soils are presented in this report. The main objective of the research program was to develop practical analysis guidelines for evaluation of seismic response of steep slopes in weakly cemented natural deposits.
The impetus for this work has been the October 17, 1989, Loma Prieta Earthquake which caused extensive landsliding in the epicentral region and along the coastal bluffs from Seaside, south of Santa Cruz, to Daly City (Figure 1.1). While this research is specific to the coastal bluffs in the San Francisco Bay region, the methods developed are applicable to analysis of seismic response of similar marine terrace bluffs along the coast of Southern California, Oregon, and Washington, and should be generally applicable to stability analyses of steep, natural slopes. The California coastline from Moss Landing northward to San Francisco is characterized by extensive stretches of steep coastal bluffs in marine terrace deposits, ranging from 20 to 200 meters in height. The appearance of the bluffs along this entire stretch of the coast shows evidence of active erosion, and there is abundant historical evidence of slope failures caused by earthquakes, wave erosion, and intense rainfall. Records indicate that seismically-induced slope failures along different portions of this coastline occurred during earthquakes in 1865 (Plant and Griggs, 1990), in 1906 (Lawson, 1908), in 1957 (Bonilla, 1957) and, most recently, during the Loma Prieta earthquake of October 17, 1989 (Plant and Griggs, 1990; Sitar, 1990). No loss of property was recorded in 1865, probably due to very sparse 1
L~arqe Coastal Landslide Dosrupinlq Hiqhwuy I
/
8th AvenUe Landslide-- V 0 o Fon1: GO
*U 0
\M Figure1.1"Ds*
tribution of landslides in the San Francisco area caused by the 1989 Lonza PrietaEarthquake (from Seed et al., 1990). 2
population of the area. In 1906, extensive landsliding was observed, particularly along the bluffs in Daly City, where the railbed of the Ocean Shore Railroad was extensively damaged (Figure 1.2). In 1957, extensive landslides along the bluffs in Daly City blocked State Route 1 for about two weeks (Figure 1.3) and led to the eventual abandonment of the highway by the California Department of Transportation. Most importantly, while cracking was detected along the crest of the bluffs, there was no direct damage to dwellings because the bluff crests were still largely undeveloped. Since then the bluff crests along the coast have been extensively developed, particularly in Daly City, Pacifica, Half Moon Bay, Santa Cruz, Capitola, and Seaside. The risk posed by seismically-induced slope failures to these new developments was amply demonstrated by the October 17, 1989, Loma Prieta earthquake. Fortunately, the extent of damage was surprisingly minor considering the severity of other damage in the epicentral region and in San Francisco. Figure 1.4 shows a shallow failure of the bluffs at Pacifica, on the San Francisco peninsula. Similar slides at Rio Del Mar, south of Santa Cruz, were responsible for relatively minor damage to structures at toe of the slope; however, tensile cracking and loss of crest left many structures more vulnerable to future events. By far the largest failure occurred along the Daly City bluffs, some 55 miles from the epicenter (Figure 1.5). Cracking along the crest of the bluffs was also observed. Thus, given the apparent potential for damaging landslides during earthquakes, there is a need to develop adequate understanding of the bluff response under seismic loading. Only then can rational procedures for stability evaluation of these slopes be developed. 3
Figure 1.2: Slope failures along the Ocean Shore Railroad north of Mussel Rock caused by the 1906 San FranciscoEarthquake (courtesy of the Bancroft Libraiy, U.C. Berkeley). Figure 1.3: Landslides along bluffs in Daly City caused by the 1957 San Francisco Earthquake (courtesy of the CaliforniaDepartment of Transportation). 4
Figure 1.4: Localized minorfailures of marine terrace deposits in Pacificacaused by the 1989 Lonia Prieta Earthquake. Figure 1.5: Failureof bluffs in Daly City caused by the 1989 Loma PrietaEarthquake. 5
This report begins with a review of the behavior of weakly cemented sands, both in the laboratory and in the field. The results of the review show that these materials exhibit brittle behavior under low confining stress, typical of a near-slope environment, and this behavior is confirmed by field observations of seismically induced failures. Since the stress conditions in the vicinity of the slope are an important aspect of a slope stability analysis, an evaluation of the accuracy of computed stress distributions using the finite element method is performed. This evaluation shows some of the limitations of a finite element method analysis in this case. The Generalized Hyperelement Method (Deng, 1991) is then presented for the analysis of the seismic response of these steep slopes. The analysis of steep slopes to seismic loading begins with a frequency domain parametric study of a stepped halfspace and a stepped layer over a halfspace, in order to develop fundamental relationships between response, slope geometry, and material properties. Following this parametric study, the use and applicability of the "average seismic coefficient", originally developed for the seismic response analysis of embankments and dams, is evaluated. Seismic response characteristics of steep slopes in the time domain are then realistically examined using actual slope profiles, material properties, and seismograms. These results are used to develop a simplified methodology, similar to that presented by Makdisi and Seed (1978) to determine the equivalent seismic force induced by an earthquake on a steep slope. Finally, recommendations on the use of the results of this study in appropriate pseudo-static limit equilibrium stability analyses are presented. 6
- 2. BEHAVIOR OF WEAKLY CEMENTED SANDS Weakly cemented granular deposits composed of various proportions of sand, gravel, and silt can be classified as either soft rock or hard soil, depending on the degree of compaction and the degree of cementation. Typical cementing agents include silica, calcium carbonate, clay, and iron. In addition, apparent cementation is achieved by mechanical interlocking of the soil grains or by capillary tension of pore water. Examples of such materials include marine terrace deposits along the Pacific coast of the United States, loess deposits in the mid-western United States and China, and volcanic ash deposits in Japan and Guatemala (Sitar 1990). Though examples of these materials are found around the world, the emphasis in this study is on the marine terrace deposits in the San Francisco Bay area, which are mainly composed of weakly cemented sands.
Nearly vertical natural slopes in weakly cemented sands have been observed in excess of 30 m in height, and slopes steeper than 30 degrees have been observed in excess of 150 m. In addition, the ability of these materials to stand in steep slopes has often been exploited to cut nearly vertical slopes for highways or roadways. Under the low confining pressures encountered near slope faces, cemented sands exhibit brittle behavior and low tensile strength. As a result, tension cracks are typically observed behind the crests of the slopes, and the brittle behavior makes for spectacular and potentially devastating slope failures during dynamic, earthquake loading. As a preface to the study of the slope response, the following sections of 7
this chapter contain a review of the static and dynamic behavior of the material to the extent necessary for slope stability evaluation. 2.1 REVIEW OF STATIC PROPERTIES The static behavior of weakly cemented soils has been the subject of numerous studies in the recent past (Clough et al. 1981, Haruyama 1973, Murata and Yamanouchi 1978, O'Rourke and Crespo 1988, Saxena and Lastrico 1978, and Wang 1986). One of the earliest studies devoted to cemented sands was performed by Saxena and Lastrico (1978) who tested the static stress-strain behavior of lightly naturally cemented sand with calcite as a cementing agent. They found that the cohesion caused by cementation was the predominant strength component at low strain levels (below 1 percent), and at high strain levels the frictional component of strength became predominant. They also found that very high confining stress could destroy the cementation. Clough et al. (1981) reported on the results of over 100 tests on naturally and artificially cemented sand. They noted that cemented sand tends to behave in a brittle fashion, with brittleness increasing with cement content and decreasing with increasing confining pressure. The relationship between brittleness and confining pressure is apparent from Figure 2.1 which shows a set of typical stress-strain curves for an artificially cemented sand. At low confining pressures, the cementation tends to control behavior, making the material more brittle. As confining pressure increases, the ductility of the material also increases, as intergranular friction 8
LO N rz (jU
<0 QLA Q- wL a_ 0 O f I I I 0 5 10 15 2(
5.0 4.0 ccI-z 35 3.0 )D S2.C 103 tw 1.0
-JU) 207 414 0- I--0.0' 0 5 10 15 2(
%/
AXIAL STRAIN Figure 2.1: Typical stress-straincurves for artificially cemented sand (from Clough et al., 1981). 9
becomes more important. Also, the material exhibits nearly linear behavior until failure. Typical tensile strength, determined using the Brazilian tensile test, is on the order of 10 percent of the unconfined compressive strength. Thus, the failure envelope curves in the tensile region and gives a lower tensile strength than would be estimated using a straight-line extrapolation of the compression test results. 2.2 REVIEW OF DYNAMIC PROPERTIES Fewer studies addressing the dynamic properties of cemented sands are available. Acar and El-Tahir (1986) studied the low strain dynamic properties of artificially cemented sands, while Frydman et al. (1980) and Clough et al. (1989) studied the effects of cementation on liquefaction. Studies that are most relevant to dynamic slope response in cemented sands were reported by Sitar and Clough (1983), Sitar (1990), and Wang (1986). Wang (1986) conducted a comprehensive laboratory study on the dynamic behavior of cemented sand, consisting of over 80 dynamic tests on sands which were naturally and artificially cemented. He found that the shear modulus decreases and damping increases with increasing strain, as is the case with most soil during cyclic loading. Summary plots of shear modulus and damping ratio varying with shear strain are presented in Figures 2.2 and 2.3, respectively. A typical result of a cyclic simple shear test on cemented sand is shown in Figure 2.4. These results, together with results of cyclic triaxial tests, show that the stress-strain curve from static tests tend to provide an envelope for the hysteresis loops from the cyclic stress-strain test. 10
RESONANT COLUMN (ACAR e t e I . 1986) OrMEAN NORMAL STRESS - 103 KPA MONTEREY NO. 0 SAND 2s CEMENT, Dr- = 203 TRIAXIAL cN a MEAN NORMAL STRESS - 103 KPA SIMPLE SHEAR c CONFINING STRESS = 103 KPA O NORMAL STRESS w 207 KPA 0~ 0C,, w a A r Q) F-P V 1 1 i'e a1 6-4 "ib- ." b ... ...... i SHEAR STRAIN for cemented sands Figure 2.2: Summary plot of shear modulus versus shear strain (from Wang, 1986). 0 1--4 I z 0_ 0-0 SHEAR STRAIN x sands Figure 2.3: Summary plot of damping ratio versus shear strainfor cemented (from Wang, 1986). 11
250 0 200 CL Itf
"-Y 150 1oo Cf) 100
(') s50 LLJ 0 Ct) -50 L.J I -150 C/) -200
-250 L
-5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 SHEAR STRAIN (%)
Figure2.4: Typical static and cyclic stress-strain curves from simple shear tests on iI[ 1.16 cemented sands (from Wang, 1986). 1.05 a C-WIN[NG STRESS - 207 EPA
& CO*FINING STRESS - 163 KPA
- COFINING STRESS - 69 ZPA o RMAL STRESS - 2,7 KPA N ORAL STRESS - 163 EPA SORMAL STRESS - 69 CPA 1i, 10.95 1'
iiI 0.60 0.75 0.706 0 10 20 36 40 50 66 70 Be NI113ER OF CYCLES TO FAILURE Figure 2.5: Dynamic strength envelopes for cemented sands under cyclic simple shear 1 loading (from Wang, 1986). 12
Thus, Sitar (1990) has suggested that the large strain cyclic stress-strain behavior can be estimated from the results of static testing. Figure 2.5 presents a plot of the ratio of static to cyclic simple shear strength with respect to the number of cycles to failure as a function of confining pressure (Sitar 1990). It is apparent that there is a trend for reduction in dynamic strength with increasing number of cycles. The effect is most pronounced at low confining pressures where the reduction can be as much as 15 percent. At higher confining pressures, the effect seems to be less than 10 percent. 2.3 OBSERVED SLOPE FAILURES IN WEAKLY CEMENTED SANDS Failures of steep slopes in weakly cemented granular soils during seismic events have been recorded in many parts of the world. In California, in the San Francisco Bay region, landslides in coastal bluffs due to a seismic event were first recorded following an earthquake in 1865 (Plant and Griggs, 1990), though no property damage was reported. The first noted failures causing property damage occurred during the San Francisco Earthquake of 1906. "Rockfalls and dry sand flows were particularly disruptive" to highways and railroad grades (Youd and Hoose, 1978). Five kilometers of the Oceanshore Railway between Lake Merced and Mussel Rock were closed due to failure of coastal bluffs. In this area, large cracks were observed extending several hundred feet behind the slope crest (Lawson, 1908). These slopes are mostly in the Merced Formation which is primarily composed of uncemented and weakly cemented sand with interbedded clay layers. Further south, 13
near Capitola, slope failures were also observed in coastal bluffs in marine terrace deposits of weakly cemented sand (Youd and Hoose, 1978). However, due to the sparse population along the coast at the time, the available information is quite sketchy. Numerous failures, which closed the coast highway, also occurred in the coastal bluffs between Lake Merced and Mussel Rock in the 1957 San Francisco Earthquake (Bonilla, 1959). The largest slide was several hundred feet wide and 700 feet from top to bottom along a 40 degree slope. The slide material appeared to be dry, and dust was observed rising from the slopes during failure. Cracks were also observed along the coast highway and behind the crests of the failed slopes. The Loma Prieta Earthquake caused hundreds of failures in marine terrace deposits and coastal bluffs between Marin County and Big Sur (Sitar, 1990). This included a large slide in the bluffs in Daly City, near the site of earlier failures recorded in 1906 and 1957. Closer to the epicenter, slides were mapped all along the coast of Santa Cruz County by Plant and Griggs (1990). At Seacliff State Beach, many slides were observed occurring in the upper 12 m of the 30 m high cliffs. These slides were observed to be up to 60 m wide with tension cracks extending 1 to 6 m behind the crest. The slopes in this area are composed of up to 5 m of Quaternary marine terrace deposits, underlain by moderately indurated, weakly jointed sandstone member of the Purisima Formation. It is interesting to note that the types of failures, based on aerial photographs, appear similar to those which occurred during heavy rains in the winter of 1982. 14
More recently, failures in steep coastal bluffs occurred during the Petrolia Earthquakes of April 24 and 25, 1992. Failures in coastal bluffs composed of weakly cemented sand were observed at Centerville Beach, located approximately 6 km west of hard-hit Ferndale, California (Figure 2.6). These bluffs, consisting of Pliocene marine terrace deposits, are 10 to 50 m in height and slope angles range from 450 to nearly vertical. Strong motion instrumentation at the Oceanographic Naval Station, just behind the crest of the slope, indicated horizontal peak ground acceleration of 0.5g for the main shock (Shakal et al. 1992). The failures appeared to be relatively shallow and occurred in the upper portion of the slopes. Most of the material in the failure mass seemed to have lost its cementation, though several small Three intact blocks up to 1 m in diameter and larger were found (Figure 2.7). slope tension cracks were observed at 2 to 3 m intervals behind the crest of the 50 m adjacent to the west of the Naval Station. No other failures were observed in directly materials. Most recently, slope failures occurred in the Pacific Palisades due to the January 17, 1994, Northridge Earthquake near Los Angeles California. These coastal
= 6.7 bluffs are located approximately 30 km south of the epicenter of the Mw indicate earthquake. Strong motion records at the nearby Santa Monica Fire Station 0.25g.
a peak horizontal acceleration of 0.93g and a peak vertical acceleration of (State The bluff failures closed the northbound lanes of the Pacific Coast Highway were Route 1) for at least 4 days following the earthquake. Four large landslides observed in this area, along with several smaller slides. One of the large slides carried a portion of a house down the slope, as shown in Figure 2.8. On properties 15
Figure 2.6: Landslides at Centerville Beach caused by the 1992 PetroliaEarthquake. Figure 2.7. Intact blocks in landslide debris at Centerville Beach. 16
Earthquake. Figure 2.8: Landslide at PacificPalisadescaused by the 1994 Northridge Figure 2.9: Intact blocks in slide debris at Pacific Palisades. 17
adjacent to this house, shallow concrete piers and H-piles were observed hanging in mid-air at the crest of the slope, implying that they provided little benefit. The failures occurred in Quaternary age deposits of weakly cemented sand (Jennings and Strand, 1969). The slopes on which the failures occurred were 40 to 60 m in height and moderately steep (between 45 and 60 degrees). The failure masses appeared to be only a few yards thick, subparallel to the slope, and had widths on the order of 100 m. The slide debris was predominately loose sand with a few intact blocks, as shown in Figure 2.9. Failures of this type are not limited to California. Brittle, tensile failures of steep slopes in cemented volcanic ash deposits in Japan following the 1968 Ebino Earthquake were documented by Yamanouchi and Murata (1973) and Yamanouchi (1977). This material, called Shirasu, is a Pleistocene volcaniclastic deposit apparently cemented by welding, interlocking, or electro-static bonding. Again, similar types of failures were observed following heavy rains in 1949 and 1969. Harp et al. (1978) documented slope failures in the February 4, 1976, Guatemala Earthquake. Landslides in Pleistocene pumice deposits blocked major highways and a railway, stalling relief efforts. This pumice has a very low tensile strength, but derives apparent cohesion from the mechanical interlocking of the angular particles. Nearly all failures occurred in steep-sided canyons. On slopes steeper than 50', the pumice appeared to fail in tension by spalling off into nearly vertical slabs less than 6 m thick. Tension cracks were observed to extend 15 to 30 m back behind the up to 100 m high, nearly vertical slopes. On 300 to 50' slopes, debris slides less than 1 m thick were observed in sandy soil overlying the pumice. 18
Almost all individual slides were limited in size to less than 15,000 n3. Harp noted that both types of failures were heavily concentrated on narrow ridges and spurs, and suggested that topography may have amplified the ground motions. O'Rourke and Crespo (1988) described similar type of landslides in the Cangahua formation in Ecuador and southern Colombia. This volcaniclastic formation is characterized as a loess-like tephra with silica as a cementation agent. The material has the ability to stand in nearly vertical slopes up to 50 m high. An earthquake in 1987 closed the Pan-American Highway due to landslides in this material. The "Conglomerate of Lima" has failed in several earthquakes (Carrillo and Garcia, 1985). This material is a coarse-grained granular soil, including gravel, cemented with "fine soils mixed with calcium carbonate". It is of Quaternary age, well jointed, and forms steep coastal bluffs outside of Lima, Peru. Following the failures, tension cracks were typically observed 2 to 4 m back from the slope crests, with some cracks as much as 10 to 20 m behind the slope crests. The slopes were also observed to fail in heavy rainstorms and due to sewer leaks. Based on a review of documented failures, Sitar (1990) classified slope failures into two general categories. Moderately steep slopes, with slopes angles between 30' and 600, tend to experience shallow planar failures, subparallel to the slope face. The second category, the very steep slopes with slope angles greater than 60'. tend to develop tension cracks behind the slope crest, and then fail in block toppling or by in shear at the base of the tension cracks. The failure modes for the moderately 19
steep and very steep slopes are schematically depicted in Figures 2.10 and 2.11, respectively. In both cases, the failure planes tend to be only a few meters deep.
2.4 CONCLUSION
S A review of laboratory studies of weakly cemented sands shows that the material exhibits brittle behavior, particularly at low confining stresses, as would be anticipated near the face of a steep slope. Dynamic studies show that there is a reduction in strength due to cyclic loading, typically on the order of 85 to 90 percent of the static simple shear strength. In addition, results of dynamic tests have led to the development of shear modulus reduction and damping curves which are suitable for seismic site response analyses. Numerous observations of seismically induced failures confirm the inference about the brittle behavior of the material based on the results of laboratory tests. Typically, the failure mass at the base of the slide shows an almost complete loss of cementation, with occasional intact blocks. There is little evidence of incremental permanent deformations eventually leading up to failure, though tension cracks are frequently observed at the crest of the slopes. These observations indicate that a failure based stability analysis rather than a deformation based analysis would be more appropriate for these type of slopes. 20
Shear Plane"- f
>200 mn sands (after Sitar, Figure 2.10: Failuremode for moderately steep slopes in cemented 1990).
Tension Cracks Shear Plane >30 rn sands (after Sitar, 1990). Figure 2.11: Failuremode for very steep slopes in cemented 21
22
- 3. STATIC FINITE ELEMENT ANALYSIS OF STEEP SLOPES Limit equilibrium analyses are often used in the assessment of slope stability.
However, this method of analysis does not provide any opportunity to assess the actual stress distribution within the slope. Since the deformation characteristics of cemented sands depend on the actual stress conditions, other methods have to be employed to look at the stress distributions. The finite element method has been used extensively to analyze stresses in a variety of man-made and natural slopes. However, most of the work to date has concentrated on embankment slopes, where the desired stresses are often located along some curved failure surface through the interior of the embankment. For steep slopes, for example in weakly cemented soil, the failure surface tends to be shallow and planar, initiating in a zone of tension near the slope face, as already discussed (Sitar and Clough, 1983). Consequently, the critical stresses in a steep slope are located near a free boundary and are of relatively small magnitude; whereas in the flatter embankment slopes, the stresses are larger, since they are averaged along a deep curved surface within the interior of the embankment. Because of these differences, and since most of the finite element work to date has been directed toward embankment analysis, a study of the suitability of the finite element method for analysis of steep slopes was performed. Specifically, the accuracy of the computed stress distribution near the free surface of the slope was evaluated as a function of the element size and shape. 23
3.1 BACKGROUND
The accuracy of a linear elastic finite element analysis depends on the type of element, fineness of mesh, mesh layout, and the geometry of the problem. The effect of these variables can be estimated empirically, and a review of published FEM analyses of slopes has been carried out. Table 3.1 lists the papers reviewed and also gives the pertinent information about the fineness of the mesh and type of element used. Though the list is not exhaustive, it gives an overview of the history of the development of the finite element method over the last 25 years. Only two dimensional studies are included, since three-dimensional studies were found not to be applicable to this research effort at this stage (e.g. Lefebrve 1973). In this review, the aspect ration is defined as the ratio of element height to element width. The first application of the FEM to the analysis of stresses in a slope was performed by Clough and Woodward (1967) in the context of an analysis of stresses in an embankment dam. In their study, 3-node constant stress triangular elements were used with an aspect ratio of 1/2. The analyzed embankments were divided into 7 to 14 layers, so that the element height ranged from H/7 to H/14. In addition to studying the discretization effects, Clough and Woodward compared the effects of incremental construction to single step, or "gravity turn-on," loading and evaluated the effects of soil nonlinearity on an idealized dam. The study was validated by comparison of the computed deformations to deformations observed in an actual dam. 24
Table 3.1 Summary of Finite Element Analyses of Slopes Source Layers Nodes Aspect Notes Ratio Clough and Woodward (1967) 7,10,14 3 1/2 EM Idriss and Seed (1967) 4 3 1/2 EM,D Duncan and Goodman (1968) 10 4 1 EX Zienkiewicz et at. (1968) 11 3 1 EX Seed et al. (1969) 5,7 3 1 EM,D Boughton (1970) 6 3 1/2 EM Kovacs et al. (1971) 4 3 1/3 EM,D Kulhawy and Duncan (1972) 9,12 3 NS EM Lefebvre et al. (1973) 8 4 1 EM Smith and Hobbs (1974) 10 4 1/2,1 EM Vrymoed (1981) 25 4 1/3 EM,D Sitar and Clough (1983) 4 4 3 N,D Naylor et al. (1986) 5,9 8,4 1 EM Acar et al. (1988) 6,8 9 1 EM Kuwano and Ishihara (1988) 10 4 1/2,1 EM,D Griffiths and Prevost (1988) 4,6 4 1/2,1 EM,D Naylor and Mattar (1988) 4,6 8 1 EM 3 4 5 1 Embankment, 2Included dynamic analysis, Excavation, Not shown, Natural slope Based on the results of this study, Clough and Woodward concluded that a staged analysis, in which layers of elements were added to the model to simulate incremental construction, was crucial in predicting deformations during construction. Stresses were affected to a lesser extent, though a staged analysis led to a more to accurate prediction of stresses. As for the number of stages, or layers, necessary 25
model deformations, virtually no difference was found between 7 and 14 layers for two horizontal sections through the idealized dam. Furthermore, they found that nonlinearity could easily be accounted for by changing the soil properties between stages in the analysis. Duncan and Goodman (1968) used the FEM to analyze stresses and deformations in excavated rock slopes. For analysis of homogenous rock, they used a 4-node quadrilateral linear stress element with a height of H/10 and an aspect ratio of one. In their study, two sequences of analysis were considered: the "gravity turn on" method, in which gravity was applied to the finite element mesh with the excavation at its final geometry; and the staged analysis, in which stresses were reduced according to the sequence of excavation. They found that for the purposes of estimating the stress distribution around an excavation, the simpler gravity turn-on method was adequate. However, for estimating displacements, the staged procedure was necessary to obtain reasonable accuracy. They concluded that the coefficient of lateral earth pressure, K, was critical for determining the stresses in the rock mass, and that the aspect ratio of elements should be between 1/5 and 5. In addition, they investigated the effect of joints on stresses and deformations. For a joint set occurring in only one direction, an equivalent anisotropy was used to model the joints. Joints were also modeled using one- and two-dimensional elements (1-D and 2-D, respectively). They concluded that the 1-D joint element was more versatile than the 2-D element for modelling joints, and that joints had little effect on initial stresses in the rock mass. 26
Excavation in rock has been analyzed by Zienkiewicz (1968) assuming the rock was a "no-tension" material. In his study, a 4-node quadrilateral element was used to with a height of H/11 and an aspect ratio of one. The rock mass was considered an be unable to carry tensile stress due to the formation of joints and fractures, and in the iterative process was used transfer stress to other regions of the rock such that end, no tensile stresses existed. Zienkiewicz concluded that this type of analogy rock mass. represented a "lower bound" solution to the stress distribution in the Sitar and Clough (1983) used the FEM to specifically model naturally with occurring steep slopes in weakly cemented soils. A 4-node quadrilateral element that a an aspect ratio of 3 was used in their analyses. Sitar and Clough concluded on the zone of tension occurs behind the crest of the slope, as well as in a small zone these types face of a vertical slope. This appeared to concur with observed failures in aspect of materials. However, it should be noted that they used elements with an ratio greater than one. Smith and Hobbs (1974) used the FEM to analyze the observed behavior of compared. model slopes in a centrifuge. In particular, aspect ratios of 1/2 and 1 were used. In Discrepancies were noted when the coarser mesh (aspect ratio of 1/2) was They addition, the effect of overall width of the finite element mesh was studied. crest, the showed that, even with boundaries as close as 1H from the toe and of proximity of the boundary had little effect on the stress distribution in the vicinity relatively the slope. Of particular interest to the current research, is the fact that centrifuge poor agreement was observed between the finite element model and the models of steep slopes. 27
Naylor and Mattar (1988) studied the effects of element height when using FEM to model embankment dams. They concluded that only 4 to 6 layers are necessary in most cases to properly analyze stresses in an embankment, not the 10 layers as is the common practice. However, in their analyses, 8-node serendipity elements were used, not the 4-node quadrilateral elements used in most of the previous studies. The different discretization schemes used in FEM analyses of slopes during the 25 years since the publication of Clough and Woodward's pioneering work are summarized on Table 5-1. It can be seen that there is little consistency in the size and shape of the elements. The initial studies used element heights of approximately H/10. Since then heights have ranged from H/4 to H/25, though H/10 appears to be close to the most commonly used value. The most commonly used element in the studies reviewed herein is the 4-node element, though the 3-node element was common in earlier studies, and 8- and 9-node elements have become more common over the past 5 years. The aspect ratios (height/width) have typically been close to one, though they vary from approximately 1/3 to 3. Most of the above studies were analyses of embankments, where stresses in the interior of the slope were of greatest interest. In addition, many of the studies performed were subject to a limitation of the total number of elements due to computer costs. Most importantly, it is evident from this review that there is no commonly agreed upon approach to the discretization of the modelled domain. Therefore, a study was deemed necessary to determine the optimum size and aspect ratio of elements for use in studying steep slopes, where shallow zones of tension may occur. 28
3.2 PHYSICAL MODEL Before the advent of computers, methods of studying stress distributions were were often used relatively limited, and physical models using photoelastic materials materials, such as to obtain stresses and stress distributions directly. Photoelastic polarized light is gelatine, bakelite, and glass show contours of equal stress when gelatine is the only shone through them (Timoshenko and Goodier, 1970). Of these, stresses caused by material sensitive enough to be of practical use when looking at the weight of the material itself. is used for Therefore, the gelatine slope model built by La Rochelle (1960) of our study. La comparison with the numerical analyses performed as a part gelatine models Rochelle studied the stability of excavations in London clay and used to determine stress distributions. He considered four different slopes: a vertical slope, and a "Bradwell" slope, a 1 horizontal to 2 vertical (1H:2V) slope, a 2H:IV (benched) slope. Only the vertical slope is of interest herein. gelatine La Rochelle's model was constructed by pouring a heated liquid mounted on a wooden mixture into a mold comprised of 13-mm thick perspex plates perspex plates. Prior frame. All wood within the mold was lined with 2-mm thick the mold were coated to pouring the mixture into the mold, all interior surfaces of plates were removed with silicon grease. Once the mixture had dried, the perspex removed any friction and coated with heavy gear oil. The grease and oil effectively presented in Figure 3.1. between the mold and model. A schematic of the mold is 29
74 I ýt" H Ofes $Ir p6upit'l SOCLtime- . V4" SCPCWA for SeCU)-ittj jwe,,,,&bee bear-h.
'mto- . ( ý'Ll, Mch)
Remova6te vwoojeL Leock fell 9,t"%tLe&t;ml excavariok. kotes fol. Iftur', Hotes thromIk Peates anct ft&Mt for IN" Stett 1-041 1/4" witk Pmls 61% 6oth emcis.
-Wood fr&vtie.
('S "s 4" s ect lam) INKeg- t&ees to ve P-eA Rkkbber 5CLS ket v/itk pet-sper Pt&tes. ( I/st 11 ( IVII&', th ic, k ) V4 " 8 o Ct S joiftr wLth Vit" ru6ber YaLsket. Figure 3.1: Gelatine model (from La Rouchelle, 1960).
The gelatine mix contained 12% leaf gelatine, 28% glycerine, and 60% water. 3 , and Poisson's ratio of the mix was approximately 0.5, the unit weight was 10.7 kN/m to be 62 kPa for the analysis presented herein, Young's modulus will be assumed (Farquharson and Hennes, 1940). none were Prior to testing, the model was checked for initial stresses and then removed to found. The wooden block supporting the molded slope face was mix to determine the simulate excavation and polarized light was shone through the stress, and the "isoclinics", contours of constant direction of major principal was then able to "isochromatics", contours of constant shear stress. La Rochelle Laplace equation for separate the principal stresses using a numerical solution of the the sum of the stresses. in the vicinity La Rochelle concluded that boundary effects were eliminated 2H; however, he noted of the slope by locating model boundaries at a distance of to severe deformations. some difficulties with modelling of the vertical slopes due the slope with a 6-mm This problem was somewhat eliminated by cutting the toe of 5 percent deformation radius instead of a right angle. Nevertheless, approximately H/5 between the mold still remained, and a tension crack opened up to a depth of Rochelle concluded that and the top of the model behind the crest of the slope. La up to approximately 10 as a result, stresses in the vicinity of the slope included included approximately 2 percent error, while stresses at the base of the model percent error. 31
3.3 FINITE ELEMENT MODEL The computer program FEAP (Taylor 1977) was used to model the gelatine slopes. An enhanced, plane strain 4-node quadrilateral element capable of modeling a Poisson's ratio very close to 0.5 was used, and the actual value of Poisson's ratio used in the analyses was 0.499. The left and right boundaries of the mesh were restrained from lateral movement, and the bottom of the mesh was restrained from vertical movement to simulate the gelatine model. The gravity turn-on method was used, in which full gravity was applied to the entire mesh at once. Nine different meshes are discussed herein, though many others were used in the course of the study. Most of the analyses were performed using meshes containing elements of uniform size, as shown in Figure 3.2. A mesh curved at the base of the slope and an extremely fine mesh with variably-sized elements were also used to complete the study (Figures 3.3 and 3.4, respectively). A description of each mesh is given in Table 3.2. The number of uniformly-sized elements required to model the slope height is given by the variable n. As can be seen from Table 3.2, values of n ranged from 4 to 32. The aspect ratio, a, is defined as the vertical dimension, h, of a given element divided by its width, w. Aspect ratios (height/width) from 0.25 to 4 were used in the analyses. 32
I mD 2H1 2H Aspect Ratio = h/w = a 2 Elements per Slope Height = h/H = n H 2H w h C DID 4 Restrained Boundary Figure 3.2: Typical FEM mesh and parameterdefinition.
Figure 3.3: Curved FEM mesh. Figure 3.4: Variable sized element mesh. 34
Table 3.2 Summary of Meshes Used in Study Mesh Description Element
~Height I_ Aspect Ratio n4al H1/4 1 n4a4 H/4 4 n8al H/8 1 nl2al H/12 1 nl6aO.25 H/16 0.25 nl6al H/16 1 n32al H/32 1 n64a0.25 H/64 0.25 n128a0.25 H/128 0.25 curved NA NA variably-sized element NA NA 3.4 RESULTS The most direct method to compare the gelatine and FEM model is to compare the shear stress, a1 a 3 , because this is the only stress which is obtained directly from the gelatine model, while the values of the principal stresses must be calculated numerically from the shear stress measurements. Shear stresses computed using the meshes n4al and nl6al are compared to the shear stresses from the gelatine model in Figures 3.5 and 3.6, respectively. These meshes are composed of square elements with element heights of H/4 and H/16, respectively. Figure 3.5 shows only a general agreement with the gelatine model for the larger element size, 35
Geiatine Model F.E.M. 0.4-I" 0.4"W 0.6W I \ Figure 3.5: Comparison of 01-a3 between gelatine model and FEM using element n4al. Gelatine Model F.E.M. 0.47 0.4yH 0.67 Figure 3.6: Comparison of aJ-a 3 btween gelatine model and FEM using element n,6a1. 36
and this general agreement tends to decrease with increasing shear stress. The difference near the base of the slope is in excess of 25 percent, with the FEM model always underestimating the stress. The agreement is somewhat improved by quartering the element size (nl6al). The difference at the base of the slope is less than 20 percent. The principal stresses were obtained from the gelatine model by applying the finite difference method to the results of the photoelastic test. The comparisons between the principal stresses computed from the gelatine model and those obtained from the FEM analyses are presented in Figures 3.7 through 3.10. Figures 3.7 and 3.8 show the comparison between the minimum principal stress, a 3 , contours for the large and small elements, respectively, and those obtained from the gelatine model. Similarly, Figures 3.9 and 3.10 compare the maximum principal stress, ap, contours between the finite element and gelatine models. All of these results indicate that there is a better agreement between the gelatine model and the finite element model for principal stresses than for shear stresses. There is excellent agreement between the minimum principal stresses in Figures 3.7 and 3.8, except within about 0.2H of the slope face. Good agreement is also found in Figures 3.9 and 3.10 for the maximum principal stresses. The figures also show that there is better agreement in both cases for the finer mesh (n=16). However, even with n=4, the overall agreement is still good. The major differences between the gelatine and finite element model occur along the slope face for a3 , and near the base of the slope for a1 and the shear stresses. Though it is not apparent from the stress contours, the finite element model 37
II Gelatine Model F.E.M. Figure 3.7: Comparison of T3 between gelatine model and FEM using element n4al. Gelatine Model F.E.M. Figure 3.8: Comparison of T3 between gelatine model and FEM using element n16a1. 38
Gelatin.e Model F.E.M. Figure 3.9: Comparison of a2 between gelatine model and FEM using element n4al. Geatin*Model F.LM. Figure 3.10: Comparison of a 1 between gelatine model and FEM using element n16a1. 39
breaks down near the base of the slope. A review of stresses at the Gauss integration points reveals an oscillation between compression and tension within elements near the face of the slope. This oscillation dies out rather quickly, within about 0.2H of the face. The oscillation is apparently caused by a singularity in the solution matrix originating in the element at the base of the slope. It is important to note that stresses in this zone may not be critical in a typical slope stability analysis because the interior stresses are well defined. However, for shallow shear or tensile failures, these stresses are critical and must be quantified. In an attempt to reduce this error, two additional meshes were used to compare the FEM model to the gelatine model: a mesh curved at the slope base and a very fine mesh with elements increasing in size away from the slope base. The results of these comparisons are shown for all stresses in Figures 3.11 through 3.13, and 3.14 through 3.16, respectively. It is apparent from these figures that not much improvement is achieved from using these more complicated meshes. In fact, Figures 3.15 and 3.16 match very closely the results for nl6al shown in Figures 3.9 and 3.10. Tensile stresses along the top of the slope from the model edge to the crest of the slope are compared in Figure 3.17. The stresses from four uniform meshes with different sized elements are shown. Meshes indicated by n4al and nl6al used square elements. Element n4a4 is a tall element with the same height as n4al and the same width as nl6al. Conversely, nl6aO.25 is a short element, with the same height as nl6a1 and the same width as n4al. The figure shows that elements with the same heights give similar results, and the square elements bound the results. Since the results obtained using the meshes n4al and n4a4 give similar results, as 40
Gelatine Model F.E.M. and FEM using curved Figure 3.11: Comparison of c1)-3 between gelatine model mesh. Gelatine Model F..M. Figure 3.12: Comparison of 03 between gelatine model and FEM model using curved mesh. 41
Gelatine Model F.E.M. Figtre 3.13: Comparison of a1 between gelatine model and FEM using curved mesh. Gelatine Model F.EM. Figure3.14: Comparisonof O1-a3 between gelatine model and FEM using variable sized element mesh. 42
Gelatine Model F.E.M. O.OyH Figure 3.15: Comparison of (Y3 between gelatine model and FEM using variable sized element mesh. Gelatine Model F.E.M. Lo.y* 1.O.51*H
- ~~~~2.OOf -- --
m _...
-- 2.5)f Figure 3.16: Comparison of a, between gelatine model and FEM using variable sized element mesh.
43
2.50E-02 2.OOE-02
--- n4al S1.50E-02 0---- n16aO.25 (0 S n4a4 *i1OE0 C, - nl6al 5.00E-03 O.OOE+00 0 0.5 1 1.5 2 Distance from Model Edge, D/H Figure 3.17: Variation of maximum tensile stress along top of slope with varying aspect ratio.
3.OOE-02 2.50E-02 Z. 2.00E-02 S1.50E-02 "1.OOE-02 5.OOE-03 O.OOE+00 0 0.5 1 1.5 2 Distance from Model Edge, D/H Figure 3.18: Variation of maximum tensile stress along top of slope with varying element size. 44
do nl6al and nl6aO.25, it appears that the element height has more effect than aspect ratio when computing tensile stresses behind the crest of the slope. The effect of element size on the magnitude of tensile stress is shown in Figure 3.18. All meshes are composed of uniformly sized square elements, except n64aO.25 and n128a0.25 which contained short rectangular elements in the upper portion of the mesh. Figure 3.18 shows that progressively finer meshes yield higher tensile stresses at a decreasing rate, with the finest mesh approaching a practical upper-bound. Based on these results, it appears that the element height of H/10 typically used in many studies predicts a much lower tensile stress behind the crest than may be actually present, and an element height of H1/32 may be required to get within 10 percent of the upper-bound.
3.5 CONCLUSION
S The results of the analyses indicate a reasonable agreement between the physical (gelatine) model and the numerical (FEM) model for shear stresses, and an overall good agreement between the two models for the principal stresses, even for coarse meshes. When looking at stresses along the top of the slope, the height of the element tends to be more important than the aspect ratio, at least for aspect ratios up to 4. In all cases, the greatest difference between the two models occurs in the vicinity of the slope. Therefore, due to limitations in the FEM analyses used herein, accurate stresses could not be determined within 0.2H of the slope face. Finally, an element height of H/10 commonly used in FEM analyses of slopes does not appear 45
to accurately define tensile stresses behind the crest of the slope, and an element as small as H/32, or higher order elements, may be necessary to determine stresses within 10 percent of the upper-bound. The difficulty of modeling stresses behind the free face of steep slopes, and the fineness of mess required to accurately model stresses behind the crest of steep slopes are of concern. In light of these results, an approach other than the finite element method, and in fact, other than detailed stress analysis, may be advantageous to the study of the response of steep slopes in weakly cemented soil, as is explored next. 46
- 4. THE GENERALIZED HYPERELEMENT METHOD SThe generalized hyperelement (GHE) method was developed by Deng (1991) that this method works for two-dimensional seismic response analysis. Deng showed slopes (less than about well for steep slopes, and in fact, is not applicable to shallow program 20 degrees). The GHE method, coded by Deng into the computer number of degrees of freedom GROUND2D (Deng et al., 1994), greatly reduces the as compared to the finite required for the analysis of two-dimensional site response, offers greater capability for element method. In addition, the current program finite element programs.
analysis of a variety of seismic waves than readily available GROUND2D, is used for the For these reasons, the GHE method, as coded in herein. analysis of the seismic response of steep slopes presented 4.1 COMPUTATIONAL MODEL model, consider In order to illustrate the concepts used in the computational in a site shown in Figure 4.1. A model of this site using the methods coded is divided into two large blocky GROUND2D is presented in Figure 4.2. The site right sides, respectively. Within regions and two semi-infinite regions on the left and a group of perfectly horizontal each region, the soil and rock strata are divided into layer to layer. The boundaries layers, with material properties perhaps varying from whole model rests on a visco between the regions can be of arbitrary shape. The elastic halfspace. 47
Soft Soil Hard Rock Figure 4.1: Site for two-dimensionalseismic site response analysis (after Deng, 1991). 1 0 I 'I in L H1 H2 R 0k _ 0 Half Space Figure 4.2: Typical model of site using GROUND2D (after Deng, 1991). 48
In general, each blocky region is simulated by a generalized hyperelement, H 1 , H 2 5 ... , Hn. The semi-infinite regions, L and R, are simulated by generalized transmitting elements. The prefix "generalized" refers to the ability of these elements to model arbitrarily shaped boundaries. Nodal points exist only at the boundaries between any two regions, and only the motions at the nodal points need to be solved in the global equations of motion. Once the nodal point motions are obtained, the motions within each region can be recovered through a nodal expansion process. 4.1.1 The Complex Response Method The techniques used in formulation of the generalized transmitting element (GTE) and the generalized hyperelement (GHE) utilize the complex response method to simulate viscous damping within the elements. For simplicity, the complex response method is described below for a simple damped oscillator. The general form of the equation of motion for the simple damped oscillator is: Mfi + C4* + Ku = q(t) (4.1) where M, C, and K are mass, viscous damping and stiffness, respectively; q(t) is the driving force; and ii, a, and u are the acceleration, velocity, and displacement of the system. The solution of the equation of motion for harmonic motion at circular frequency o is given by: (K + ioC - W2M)Ul = QI (4.2) 49
where UP and Q' are the complex amplitudes of the displacement and force, respectively. Employing the concept of complex stiffness, the above equation can be reduced to the following: (K' - ci2M)Ul = Q1 (4.3) where K' = K + iwC is the complex stiffness which can be formulated using the complex modulus for linear visco-elastic materials. In the context of the methodology used herein, viscous damping is employed through the use of the complex shear modulus G',
' 0,(1_~2I32 + i2f3-/2) (4.4) and complex constrained modulus M, = Mc(1 -2p 2 + i2 i)p2f3 (4.5) where p is the ratio of critical damping for S- and P-waves. Though some laboratory studies indicate that p may be different for S- and P-waves, there are treated as equal in this study. It is possible, however, to define separate values in GROUND2D. It should be noted that MC' = V. + 2G', where the complex value of Lame's constant is
- 1. = X(1 -2p32 + i2f3Vi-p ) (4.6)
Using definitions of complex modulii leads to: a real value of Poisson's ratio, 50
;/1(4.7)
V = 2(;L'÷ýL (X an exact value for amplitude, because IK'I = K (4.8) and, for small values of p, a small error in the phase of the solution: 02 - 0 1-- 2P3
+a (4.9) where 0 1 and 02 are the phase lags between the displacement and the driving forces using the complex response method and modal analysis, respectively. The phase lag is greatest at low frequencies (for the static case a = 0); is approximately 0 (in radians) near the natural frequency; and disappears with high frequencies. For the sake of simplicity, the prime symbol (') indicating complex values will be dropped from here on, though it should be understood that the complex values of these variables are used throughout. For a detailed discussion of the complex response method, see Deng (1991).
4.1.2 Equations of Motion The equations of motion for the model shown in Figure 4.2 can be written generally as 51
n ([LIL+[R]R+E [I]I) { U} = (PI (4.10) where [L]L is the stiffness matrix for the left semi-infinite region, [RIR is the stiffness matrix for the right semi-infinite region, and [H]i is the stiffness matrix for the i-th blocky region Hi. {U} is vector containing the motions at all nodal points, and {P} is the force vector, which is of different form for different incident wave cases. For surface wave incidence, assuming the wave propagates from left to right, the vector {P} can be expressed as {P} =([L]L +[R] 1){ U}R (4.11) where {U}R is the incident surface wave vector, the subscript L refers to the left boundary, and the superscript R refers to the direction of wave propagation. Similarly, if the incident surface wave propagates from right to the left, the {P} vector becomes {P} =([L]R+[RJR){UI} (4.12) For body wave incidence, the {P} vector is n {IP =[L]L, Uf) LfR, RI UR+E [ttJ{ Ufjj i-1 (4.13)
+[LolL{ UfIL-[ IR]R{ U>R+. [Hoji U~f, i=1 where {Uf}L, {Uf}R, {Uf}i are the "free field" motion vectors for the left, right, and Hi regions, respectively (i.e. the response vectors of the regions corresponding to the incident wave, if the motions in the regions are computed using one-dimensional layer 52
models with no lateral boundaries). The matrices [Lo], [Ro] and [Ho]i are defined as follows: [LOIL=i1Ca[AIL +[DIL (4.14) [RO]R=iKotA]R+[D]R (4.15)
-(ijK[AIL,+[D]L [ 0
[H°] 0 iWa[AjiR,+[D] R, (4.16) where the subscripts Li and Ri are for the left and right boundaries of the Hi region, respectively. The matrices [A] and [D] are related to the eigenvalue problem of a layered system and will be defined in the following section. The parameter ica is the apparent wavenumber of the incident body wave in the underlying halfspace and is defined as 1C. = I,0 sin 0 0 (4.17) where ico = ris for SH- and SV-waves, and xo = xp for P-waves. 00 is the incident angle, measured from the z-axis, with the positive value corresponding to a wave propagating toward the positive x-direction. 4.1.3 Eigenvalue Problem of the Layered Halfspace Consider a semi-infinite layered region bounded at the left end by an irregular boundary S, shown in Figure 4.3. According to Deng (1991), the surface wave motions in the region along a curve S*, which is parallel to the boundary S at a 53
fx Free Surface IT z 1I+ Figure 4.3: Semi-infinite layered region (after Deng, 1991). SL H SR Figure 4.4: Layered blocky region bounded by irregular boundaries (from Deng, 1991). 54
surface distance x, can be generally expressed as a superposition of the generalized wave modes: N (4.18) (U(w,x)I=F tv), e-iKIxas s=1 where {U(wx)} is the frequency-dependent displacement vector, {v}, is the s-th wave mode mode of the layered region, xs is the s-th wavenumber, and as is the in participation factor for the mode. Equation (4.18) is the general form for both plane (e.g. SV-waves) and out-of-plane motions (e.g. SH-waves). Assuming, for example, that the bottom of the layered system is fixed, then N = 4n for in-plane motions, where n is the number of discretized nodal points along the boundary S, and N = 2n for out-of-plane motions. The eigenvalues and eigenvectors must be solved for each frequency from the following generalized eigenvalue problem
-w2 [M])IvI (4.19)
([A]Kx +i[B]Y÷+[G-* =101 where ca is the circular frequency, [A], [B], [G], and [Af] are frequency independent geometry matrices related to the material properties of the layered region and the of the irregular boundary S. Depending on the different type of motions concerned, (e.g. 1st- or (e.g. in-plane or out-of-plane) and different order of the discretization 2nd-order), the matrices may have different forms and dimensions. This Equation (4.19) can be solved by a method proposed by Deng (1991). motions solution yields 4n pairs of the wave modes and wavenumbers for the in-plane Half (2n pairs for the wave modes and wavenumbers for the out-of-plane motions). 55
of the wave modes represents the surface waves propagating in the positive x direction and the other half represent the waves in the negative x-direction. The wave modes and wavenumbers are used in generating the transmitting boundary matrices and in computing the motions in any part of the layered regions. 4.1.4 Generalized Transmitting Elements The generalized transmitting elements (GTE) are formulated by using the exact analytical solution in the horizontal direction and a discretized displacement shape function along the irregular boundary S. The boundaries of these elements transmit energy accurately in the horizontal direction and represent the perfect "infinite" boundary condition. The GTE stiffness matrix represents the response of the semi-infinite region to the boundary nodal forces. For each irregular boundary S, two stiffness matrices exist. One is for the waves propagating toward the positive x-direction, i.e., the semi-infinite region is at the RIGHT of the boundary, and is denoted [R]. The other is for the waves propagating toward the negative x-direction, or the semi-infinite region is at the LEFT of the boundary, and is denoted [L]. We have generally the following force-displacement relationship for the right region (PJ=[R]IU) (4.20) and, for the left region 56
(P)=[L]{ U) (4.21) where {P} is the nodal force vector along the boundary, {U} is the nodal displacement vector along the boundary respectively, and [L] and [R] are the GTE stiffness matrices. According to Deng (1991), the GTE stiffness matrices for both the in-plane and the out-of-plane motions can generally be written in the following form: [R] =i[A][V[ic/][ -]- +[D] (4.22) (4.23) [L] =i[A]J[W]*]cJ[W1-j-[D] In the two equations, the dimensions of all matrices are 2n x 2n for the in-plane motions (n x n for the out-of-plane motions). [V] is the matrix containing all right eigenvectors and [W] is the matrix containing all left eigenvectors, respectively; and [_K] is a diagonal matrix containing all the wavenumbers (i.e., the eigenvalues) for the corresponding modes. These eigenvalues and eigenvectors serve as the basis for the mode superposition of the motions in the layered region, as defined in Eq.(4.19). The matrix [D] is related to the geometry of the lateral boundary and the material properties of the layered region, and is assembled from the submatrices of individual layers. 57
4.1.5 The Generalized Hyperelement Each of the blocky regions in Figure 4.2 is simulated by a generalized hyperelement (GHE) which is an extension of the generalized transmitting element. The following gives a brief outline of the formulations of GHE. The detailed derivation can be found in Deng (1991). Consider a layered blocky region bounded laterally by two irregular boundaries, SL and SR, respectively, as shown in Figure 4.4. The layers inside the region are assumed to be perfectly horizontal, but material properties may differ from layer to layer. Nodal points exist only along the boundaries. The displacement shape functions in the region, i.e., the correlations between the displacement vectors of the two boundaries, are obtained by the superposition of all right-propagating modes from SL and all left-propagating modes from SR. After considering the effect of the boundary geometry, the general form of the displacement vector within the region can be written as {U(*) = {U(t)LR+{U(4)}RL (4.24)
= [Q(4)]tR{ U(L)IL+[Q(&)]RL{U(R)}L where the vector {U(L)}LR is the right-propagating component of the displacement vector defined at the left boundary, the vector {U(R)}RL is the left-propagating component of the displacement vector defined at the right boundary. {I} is the vector which defines the distance from the left boundary to the line of interest, and
{*} = {d}-{*} is the vector which defines the distance from the right boundary to 58
the left and the right the line of interest, where {d} is the distance vector between boundaries. of the The matrix operator o is defined as a term-by-term multiplication x bij. In this manner, [Q]LR matrix entries. i.e., [C] = [A] o [B] implies that cij = aij and [QIRL are matrices of the following form: [QIt*=([VJt[4t)[P]L 1 (4.25) along SL, [WMR is the left where [11L is the right eigenvector matrix defined the matrices [W]L and [FIR are eigenvector matrix defined along SR. The entries in defined as (W,*)=e -'*,* (R=e (4.26) s,t=l, ..., n for the out-of-plane motions, and (*,-I_1)L=e -itA*, (*ýS)L =e -'EA (4.27) (*,-l_ t)R=e -it ' ,, (*,.ts)R=e -it 'k s=1, ..., n; t=1, ..., 2n for in-plane motions, where {K} and {K} are defined from the left and right n x n for the out-of-plane boundaries, respectively. All the matrices are of dimension motions. motions, and of dimension 2n x 2n for the in-plane 59
For the left boundary, {f} ={O} and {f}={d}. For the right boundary, {f}={d} and {I}={0}. Thus the nodal displacement vector of the blocky region can be written as I=U IL)}+{U(LI}RI {U1= I U()I itlU(R)) J(=/RIu).+(U(R)l RLJ (4.28) J U<R)I L [1 [11 where [I] is the identity matrix. Similarly, applying the boundary conditions of the GTE on both lateral boundaries, SL and SR, and considering all propagating components of the displacement field, the nodal force vector of the blocky region can be written as IF[IRL -[L]L[QJRLI{U(L)}r" (4.29) where [R] and [L] are the left and the right GTE stiffness matrices, respectively. The subscripts L and R denote that the matrices are defined at SL and SR, respectively. Combination of the Equations (4.28) and (4.29) leads to the following relationship: IF) =[H]{U) (4.30) where the matrix [H] is the hyperelement matrix: [H] [R] 2 [R12221 (4.31)
=[[H]1 [ Hi!
60
The components of the matrix [HI] are [H]11 =[RIL[*] +[L]L[Q]j[I] 2[QIR [R12=-[RIL[4*l][Q]n-[L]L[Q][ *] 2 (4.32) ("121=-[R)1= [ILQ]JI -[L]R[12[(QILR [H]2 =[RIR[IQ]lI[[QIQ*+[L]R[I)2 The matrices [I]1 [1T12 are defined as the following [] 2 =([]- [Q Q][ )(4.33) [P'12=([/1Ql-(OQt.Rd]* The component matrices [1111, [H112 , WN]21, [H)22 are of dimension n x n for out-of plane motions, and 2n x 2n for in-plane motions. 4.2 FREE FIELD MOTIONS IN THE LAYERED REGIONS The term "free field" herein denotes the responses of a layered system to a incident wave field, either an inclined body wave, or a surface wave, computed by one-dimensional model, i.e., without the lateral boundaries. The free field motions will be different for the different types of incident waves. The procedures used in GROUND2D in determining the free field motions are described below. 61
4.2.1 Inclined In-Plane Waves (SV- and P-Waves) Consider a one-dimensional layered system, shown in Figure 4.5. Using a method similar to Chen et al (1981), the response of the layered system to incident in-plane (SV- or P-waves) can be obtained by solving the following equation: 2+]2[M])U o, (4.34) where the matrices [A], [B], [G] and [Al] are the same matrices as defined in the eigenvalue problem, as shown in Equation (4.19), except that the dimension of the matrices are (2n +2) x (2n+2), since the motions at the interface between the layered system and the underlying halfspace are now taken into account. ca is the apparent wavenumber for the incident wave, as defined in Equation (4.27). Uf is the to-be solved free field displacement vector in the layered system;" Ub is the interface displacement vector; and Pb is the interface force vector. The vectors Ub and Pb each have two components, one for horizontal motion and one for vertical motion, and they define the influence of the incident waves upon the layered system. The vectors are dependent on the properties of the underlying halfspace, the incident angle, and the type of the impinging wave, and must be determined from case to case. The procedure used in determining Ub and Pb in GROUND2D is a variation of the technique developed by Chen et al (1981). The detailed formulation can be found there. Solution of Equation (4.34) yields the free-field displacement vector { U}. which is defined along S. Assuming the x-coordinate of the reference curve at top 62
so Uf,1 (1) z z __Uf,2 Layer 1 (2) 2 Uf,2j-1 (J) IrU f,2j (j+l) j+l (n) HalfspaCesv* * *Ub,1 (n+1) ir S P n+1 Figure 4.5: Wave-scattering in a layered system (after Chen, 1981). 2 , 1 LU L*1.5 CL LU
*- 0.5 U/)
0 0 30 60 90 INCIDENT ANGLE Figure 4.6: Variation of suiface amplitude with incident angle. 63
of the layers is x0, the free-field displacements along any curve which is parallel to the reference curve and at a distance x-xo can then be obtained using the relation {U(x) ].= IU(x o)I., e - o -. (4.35 ) It is worthy to note that Equation (4.35) not only defines the free field motions along a curve which is parallel to the reference curve, it can also be used to determine the free field motions along any curves in the layered system, since the relationship defined in Equation (4.35) as a vector can also be applied for each individual term as well. Suppose it is necessary to determine the free field motions along S* (see Figure 4.3) which is not parallel to the reference curve S. The vector defining the distances of the nodal points of the two curves is
-XO,2 10}='t*3S1 '= *Xo,3-A (4 .36 )
The free-field displacement vector along S*, {U(X1 )}f can then be determined as
.e e
U(X1) jI e - U(Xd) if (4.37) e 64
4.2.2 Inclined Out-of-Plane Waves (SH-Waves) The free-field displacement vector {U}f due to SH-wave (out-of-plane) incidence can also be solved in the same way as the SV- and P-wave case. That is, to solve the equation 2IJ~~BIC+G C2[MD{I {Pb}1 (4.38) where the matrices [A], [B], [G] and [M] are defined for the out-of-plane motion, and rca is the apparent wavenumber of the inclined SH-wave in the underlying halfspace. The determination of the terms of Ub and Pb is much simpler, however, because each term contains only one component. In fact, the interaction of the SH-wave at the interface of the layered system and the underlying halfspace can be represented by a simple viscous force. 4.3 SIMULATION OF SEMI-INFINITE HALFSPACE AT BASE The approach described above was first developed for layered systems resting on a rigid base. A rigid base will reflect the scattered energy back into the system and will cause the site to have erroneous natural frequencies which will affect the overall response. This becomes especially critical for two-dimensional site response analysis. Since the model often covers a large distance, any small deviation from the true solutions is likely to be amplified. However, at some sites, soil layers may extend 65
to such a great depth that an artificial boundary must be introduced at a certain depth. The following two techniques are used in GROUND2D to remedy these problems in simulating the semi-infinite halfspace at the base of the layered system. The first technique used in the program is to add some additional layers to the original model of the site. The total thickness, h, of the additional layers varies with frequency and is set to h=1.5_.VV (4.39) f where f is the frequency of analysis in Hz, and Vs is the shear wave velocity of the underlying halfspace. The choice of this thickness is based on the observation that fundamental mode Rayleigh waves in a halfspace decay exponentially with depth and essentially vanish at a depth corresponding to one and a half wave length. Since the scattering motions generated due to the geometrical and geological irregularities in a site can be expressed as generalized surface wave motions, i.e., the generalized Rayleigh wave motions in the in-plane motion case, and higher modes usually decay faster than the fundamental mode, only minimal error is introduced by placing a rigid base at this depth. In GROUND2D, the total thickness of the layers is automatically adjusted according to the frequency under consideration, and all layers in the extended region are of uniform thickness. This type of discretization provides sufficient depth for the scattered motions to decay, and also allows the pathway of the incident wave be accurately modeled. The second technique used in the program is to attach viscous dashpots at the base, thus the base becomes a viscous boundary instead of a rigid boundary. The 66
dashpots are formulated into the eigenvalue problem for each region, and Equation (4.19) becomes ([A]r +i[B]iC+[G] -W2 [MJ +[CH]) (v} =101 (4.40) where, for out-of-plane motion, [Cn]=diag{O, ..., 0, icapVs (4.41) with dimensions (n+1) x (n+1); and for in-plane motions [CHl]=diag{O, ..., 0, iWpVV, iapVp} (4.42) with dimensions (2n +2) x (2n +2). Vs and Vp are the shear wave velocity and P-wave velocity of the halfspace, respectively; p is the mass density of the halfspace; and ( is the circular frequency. The mode vectors and wavenumbers are obtained by solving Equation (4.40), are usually different from the solutions of Eq.(4.19), and are used as the base in the computation of all the boundary matrices, the hyperelement matrices, and the mode superposition for expansion of the motions within a layered block region. Thus, the effect of the radiation damping of a perfect halfspace is built in. Since the dashpot representation of the halfspace is only exact for the vertically propagating P- and S-waves, and the directions of the scattered motions are usually unknown, this technique is approximate in the sense that some of the scattered energy may still be able to bounce back into the system. However, the use of both techniques gives very satisfactory results in most practical problems. 67
4.4 INPUT GROUND MOTION Ever since one-dimensional site response analyses became commonplace with the advent of readily available computer programs like SHAKE (Schabel et al, 1972), the ground motions used as input into the analyses are derived from a reference "outcrop" motion. That is, the data input into the computer program is the motion recorded on a rock outcrop. The actual motion used by these earlier programs at the base of the soil model is equal to 0.5 of the input "outcrop" motion, because for the vertically propagating waves considered by these programs, the amplitude of the incident wave is equal to 0.5 of the amplitude of the outcrop motion. The outcrop motion is assumed to occur on the surface of a homogeneous, isotropic halfspace. The use of the outcrop motion as the reference motion for site response analyses is reasonable for analyses that do not consider inclined incident waves. The approach is also reasonable for inclined SH-waves. However, it has been shown (Lysmer et al, 1994) that for the same amplitude of the outcrop motion, the amplitude of incident P- and SV-wave is extremely dependent on the angle of incidence. An example of this relationship between incident angle and incident wave amplitude for a given outcrop motion is shown in Figure 4.6, after work performed by Knopoff (1957). This figure indicates that the amplitude of the incident wave back-calculated from a given outcrop motion can vary several-fold, depending on the incident angle. The most extreme case is for an incident angle of 45 degrees, where no horizontal surface motions are generated, thus leading to an incident wave amplitude of infinity. The effect of inclined incident waves is most noticeable at soil 68
sites, because as the seismic waves travel upward through less and less stiff materials, they tend to become more vertical due to Snells Law. Thus, by the time the seismic waves reach the ground surface, they may be near vertical and there is no reduction in surface amplitude due to hitting the free surface at an inclination. Therefore, Deng et al. (1994) recommended that the amplitude of the incident wave, and not the outcrop motion, be used as the basis for analysis using inclined waves. In particular, the recommended amplitude of incident wave is that which would be backcalculated for a vertically propagating wave incident on a rock outcrop, which is the same amplitude as would be used in SHAKE. This procedure then allows the user to directly consider the effect of inclined incident waves without the added variable of incident wave amplitude. 4.5 GENERATION OF GROUND MOTIONS After all boundary matrices, hyperelement matrices and load vectors are computed for a particular frequency, the global equations of motion are assembled according to Equation (4.20) and the final equations are solved. Since the use of the generalized boundary element and the generalized hyperelement greatly reduces the total degrees-of-freedom in the global equations, only an in-core active column solver is coded in GROUND2D. The solution process is repeated for each of the frequencies specified. All of the solution vectors, which consist of the displacements at all nodal points, and free field motions for all blocky regions in case of a body 69
wave incidence are stored in external files for post-processing to obtain required ground motions. The post-processing of the nodal point solutions is an integral part of the solution process to generate necessary results due to the uniqueness of the analytical model adopted in the program. Three types of post-processing techniques are required, namely: (1) the spatial interpolation through modal superposition to extract the motions within blocky regions from nodal point solutions at each specified frequency; (2) the frequency domain interpolation through, discrete Fourier wavenumber transform to obtain a continuous form of transfer functions at a specified point from solutions at discrete frequencies; and (3) the Fourier transform to convert frequency domain (steady state) solutions to time domain (transient state) solutions, or from transfer functions to acceleration time histories and acceleration response spectra, given a reference earthquake motion. Details of the three post processing techniques can be found in Deng et al. (1994). 70
- 5. TOPOGRAPHIC EFFECTS
"...The effect of the vibration on the hard primary slate, which composes the foundation of the island, was still more curious: the superficialparts of some narrow ridges were as completely shivered as if they had been blasted by gunpowder. This effect, which was rendered conspicuous by the fresh fractures and displaced soil, must be confined to the near surface, for otherwise there would not exist a block of solid rock throughout Chile; nor is this improbable, as it is known that surface of a vibratingbody is affected differently form the centralpart. It is, perhaps,owing to this same reason that earthquakes do not cause such temific havoc within deep mines as would be expected..."
(Charles Darwin, 1835).
5.1 BACKGROUND
The above quote by Darwin (Barlow, 1933) describing the effects of the February 20, 1835 Chilean earthquake suggests that topographic amplification of seismic motions is a phenomenon that has been well recognized for some time. Certainly, in the recent past, there have been numerous cases of observed earthquake damage pointing to topographic amplification as an important effect. As a result, a considerable amount of work has been done in an attempt to model, quantify, and predict these effects. Some of the earliest experiments aimed at evaluating topographic amplification were performed by F. J. Rogers (Lawson, 1908) following the 1906 San Francisco earthquake. These early experiments were conducted with buckets of sand on a shaking table. Rogers observed vibrations on the top of the sand pile to be greater than those at the base. These results were later discussed by Reid (1910) who suggested that the observed effects could be the result of irregular reflections 71
and refractions in the immediate neighborhood of the slope. Similar experiments were repeated later by Goodman and Seed (1966) with comparable results. The apparent effects of topographic amplification were observed by Celebi (1987) following the Ms=7.8 1985 Chile Earthquake. While four- and five-story buildings on ridgetops were extensively damaged, similar buildings in adjacent canyons suffered no damage. In addition, one- and two-story buildings along ridgetops suffered only minor damage. The concentration of damage along the ridgetops prompted the deployment of an array to measure possible topographic amplification. The ridges in question were approximately 20 m in height with side slopes of 10 to 15 degrees. The seismograph stations were set on alluvial deposits or weathered granite, with care taken so that a station in the base of a canyon was set upon similar material as the station on the corresponding ridgetop. Aftershock data was then collected for a period of 5 months following the main shock. Results were presented in the form of spectral ratios between the ridge crests and canyon bases for a frequency range of 0 to 10 Hz. These results indicated considerable frequency-dependent amplification, particularly in the range of 2 to 4 Hz, and 8 Hz. Spectral amplifications up to 10 and above were noted. Similar amplification was noted between the canyon station and a nearby reference station sited on bedrock. No consideration was given in the study for differences in soil amplification between the canyon and ridge sites. Celebi (1991) presented additional evidence of topographic amplification observed in aftershock data following the 1983 Coalinga and 1987 Superstition Hills 72
earthquakes. A 3 station array (1 crest and 2 opposing gully stations) set across a ridge was able to record motions from a MS=5.3 aftershock of the Coalinga event. The ridge was approximately 30 m high, with average side slopes of 10 to 15 degrees. The geology was described as weathered sandstone near the surface, with harder Pliocene sandstone at depth. Amplification was calculated between the crest and gully stations. Spectral amplifications of up to 10 were noted, with most amplification occurring between 1 and 6 Hz, and at 7.5 Hz. Eleven aftershocks of the 1987 Superstition Hills earthquake were recorded at the base and crest of Superstition Mountain. The mountain has side slopes of less than 5 degrees and the observed amplifications were as high as 20 in the 2 to 12 Hz range. One of the first numerical studies of the effect of topography on seismic response was carried out by Boore (1972). This study, prompted by observations of high accelerations near Pacoima Dam during the 1971 San Fernando earthquake, considered the effect of simple topography on vertically propagating SH-waves. Boore noted that numerical models were necessary when considering steep slopes, or when the wavelength and size of topographic feature are similar (i.e. analytical solutions are not possible). Boore used the finite difference method to model 20 m high ridges with side slopes of 23 and 35 degrees. The medium was assumed to be homogeneous, isotropic, and linearly elastic. The shear wave velocity of the material was 500 m/s, and the frequency range under consideration was approximately 1 to 10 Hz. Damping values ranged from 2 to 20 percent, depending on the frequency. Since no layering was included in the model, the level of damping had little effect on the spectral ratios. 73
Boore concluded that the motion within the ridge consisted of 3 phases: a direct wave, a reflected wave, and a diffracted wave. The results showed that there was amplification at the ridge crest, and that both amplification and attenuation could occur along the side slopes, depending on the slope geometry and the frequency of motion. The effect of topography was found to vary with frequency, and amplification up to 100 percent was noted over the free-field. The amplification was found to decrease with slope angle and as the wavelength became large compared to the characteristic length. Rogers et al. (1974) performed experiments using a physical model to study ridge effects on P-waves. Amplifications on the order of 50 percent were found on broadband input motion, and on the order of 200 percent on band limited input. When this data was compared to actual field measurements of ground motion (Davis and West, 1973), qualitative, but not quantitative, agreement was found. The field studies showed amplifications of 400 percent for peak velocities, and amplifications as high as 20 times in relative spectral velocity from base to crest. May (1980) studied the effectiveness of vertical scarps on reducing the seismic energy transmitted to a site above or below the scarp. May used the finite element method to analyses horizontally propagating SH- and Love waves passing through 60 to 150-m high vertical scarps in a halfspace and a layer over a halfspace. The frequencies of motion considered ranged from 1.5 to 6 Hz. May found that refection off the scarp face played a large role in the response, and that the effect of the scarp could be related to the ratio of slope height, H, and the wavelength of the motion 74
under consideration. May performed tests using an instrumented granite block to validate his numerical model, and found a good comparison between the two models. Geli et al. (1988) reviewed previous analytical studies of topographic amplification, which used the following methods: finite difference (Boore, 1972; Zahradnik and Urban, 1984), finite elements (Smith, 1975), integral equation method (Sills, 1978), boundary methods (Sanchez-Sesma et al., 1982), and discrete wavenumber methods (Bouchon, 1973, Bard, 1982). All of these studies considered the analysis of an isolated two-dimensional ridge on the surface of a homogeneous halfspace and all yielded consistent results: the amplification of acceleration of no more than 2 at the crest, peaking when the wavelength is about equal to the ridge width; and varying amplification and attenuation along the surface of the slope from the crest to the base. However, these results considerably under-estimate amplifications observed in the field, which mostly range from 2 to 10, and up to as much as 30. Geli et al. then analyzed a more detailed model configuration using a layered profile and introduced nearby ridge effects, but arrived at conclusions similar to those of the previous researchers. In addition, they found that neighboring ridges may have greater effect on site response than layering, and concluded that future models should be able to analyze SV- and surface waves and three dimensional geologic configurations. Sitar and Clough (1983) used a two-dimensional finite element model to analyze the seismic response of steep slopes in weakly cemented sands. They found that accelerations tended to be amplified in the vicinity of the slope face. However, in contrast to Geli et al. (1991) they noted that these topographic effects tended to 75
be small relative to the amplification that occurs in the free field due to the site period. Most recently, various methods of analyzing topographic effects were reviewed in an NSF/EPRI workshop (EPRI 1991). Recommendations resulting from the workshop include the need for instrumented sites to verify numerical analyses of topographic effects. It was also suggested that there is a need for simple, easy to measure parameters and empirical correction factors for determining topographic effects, since 2- and 3-dimensional modeling can become quite cumbersome. 5.2 ANALYSIS OF A STEPPED HALFSPACE To determine the effect of a steep cliff on the dynamic response of a uniform visco-elastic material subject to out-of-plane (SH) waves and in-plane (SV) waves, a parametric study using the computer program GROUND2D has been performed as part of this study. For clarity, the definition of the wave types as used herein is illustrated in Figure 5.1. A SV-wave is the in-plane shear wave with displacement in the plane of the slope cross-section, i.e. within the plane shown in Figure 5.1. The SH-wave is the out-of-plane shear wave with displacement normal to the slope cross section, i.e. out of the plane shown in Figure 5.1. These definitions are consistent with those commonly used for the case of a wave traveling normal to the slope face, i.e. in the plane of Figure 5.1. The problem of a steep slope in a uniform visco-elastic material can be simplified to that of a stepped uniform halfspace. The analysis of this problem is 76
Slope SV-WAVE SH-WAVE (In-Plane) (Out-of-Plane) Figure5.1: Definition of wave types used in study.
-.HH I
k I. Left GTE Right GTE ___________________________________________________ t S 4' S ___________________________________________________ h Control Point for Input Motion
-Simulated Halfspace Figure 5.2: Stepped halfspace model for a vertical slope.
77
very useful for the development of an understanding of the fundamental parameters necessary to quantify the effect of topography on seismic response, because the only variables are the slope height and the wavelength. This allows the analysis to focus on the relationship between these two parameters without having to incorporate the natural frequency of the site. Once this relationship is examined, then other variables such as slope angle and wave inclination can be incorporated. The development of the generalized hyperelement and the generalized transmitting element and their incorporation into GROUND2D make two dimensional modelling relatively simple, particularly for steep slopes. To consider the effect of a slope on a uniform soil deposit, only a left and a right GTE are required, as shown in Figure 5.2. The results of the analyses are presented as a function of H/I), i.e. the ratio of the slope height and the wavelength of the motion under consideration. This definition of the normalized wavelength is in contrast to earlier studies of ridge effects (e.g. Boore, 1972; Geli et al., 1988) and dams (e.g. Gazetas and Dakoulas, 1993), in which the correlation was made between the wavelength and the width of the topographic feature, but is similar to the "dimensionless frequency" proposed by Dakoulas (1993) for the study of SH-waves in earth dams. 5.2.1 Effect of SH- (Out-of-Plane) Waves on a Vertical Slope The effect of vertically propagating SH-waves on the seismic response of a vertically stepped halfspace is evaluated in the frequency domain over the range of 78
velocity of 300 m/s, a 0.5 to 10 Hz. The uniform halfspace has a shear wave ranging from 1 to 20 Poisson's Ratio of 0.3, and the fraction of critical damping percent. The height of the slope is 30 m. for the normalized The results are presented in the form of transfer functions of the free field motion frequencies of motion, and in terms of the amplification to transfer the input behind the crest. The transfer function is the multiple required Figure 5.2, to the output motion, at a given frequency, from the control point in is a complex number motion at the point of interest. The actual transfer function however, only the which accounts for the phase difference between the motions; the amplification of magnitudes of the transfer functions are needed to compare range of motion. The frequency range of 0.5 to 10 Hz includes the typical frequencies most often engineering interest and spans the range of dominant observed in large earthquakes. 5.3 through 5.6, for The results of the analyses are presented in Figures varying from the slope damping values of 1, 5, 10, and 20 percent, and for distances plotted in Figures 5.3a crest to 4H behind the slope crest. The transfer functions reduces the response of the through 5.6a show that increased damping significantly The transfer functions free-field and of the slope, particularly at higher frequencies. at low levels of damping. also show that the effect of the slope is more pronounced at various distances A comparison of the amplification of the free field motion damping does not greatly behind the slope (Figures 5.3b through 5.6b) shows that slightly with increased affect the amplification, though the amplification decreases at higher frequencies. damping. Again, the effect of damping is more pronounced 79
I 2 Z 1.5 0 F 0 z U E1 LL LU C0 IL C') z 0.5 0 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH 2 w UJ U-U 1.5 - L" LU LL 0 1 "z 0 f]0.5 CL 0-r 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH Figure 5.3: Horizontal transfer fiinctions (a) and amplifications (b) for vertically incident SH-wave on a stepped halfspace for various distances behind crest, P = 1%. 80
2 Crest 0-1H (a)
.. a-.
2H Z 1.5 0 4H Free Field U-LL LU C/) z -U-i- I I
- 0. 5 0
0.10l 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH 2 o 0 a: [L 1.5 IU L.M ILL 0 1 Z 0 LL
-05 -J 0_
0
- 0. 01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH Figure 5.4: Horizontal transfer functions (a) and amplifications (b) for vertically incident SH-wave on a stepped halfspace for various distances behind crest, P = 5%.
81
2 Crest 1H (a) 2H Z 1.5 0 4H l
-a----Field Free z
LL z 0:. I--0.5 A3 0 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH 2 Crest S~(b)
-J 1H _
w "2H LL 1.5 w 4H w cc LL L,, - -6 0 _- z 0 L
- 0.5 C-1A 0
0.01 0.03 0.05 0.1 0.2 0.3 0.5 1 0.02 SLOPE HEIGHT/WAVELENGTH Figure 5.5: Horizontal transfer functions (a) and amplifications (b) for vertically incident SH-wave on a stepped halfspace for various distances behind crest, P = 10%. 82
2 Z 1.5 0 0 Z 1 U 0.5_ L, I,-0.5 w E1.5_ w 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 LL SLOPE HEIGHT/WAVELENGTH LL1.5 Z LU 0 0L U LU 01 Fi 0 H IL_
-i 0.5 0
0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH Figure S.6: Horizontal transfer functions (a) and amplifications (b) for vertically incident SH-wave on a stepped halfspace for various distances behind crest, P = 20%. 83
Looking only at the amplification at the crest, a peak amplification of nearly 30 percent occurs at approximately H/I. = 0.2, a secondary peak of 15 percent at about HA). = 0.7, and a null at HIA. = 0.45. The peaks correspond somewhat to the natural frequency of the soil column behind the crest for the height of the slope, which would occur at H/A. = 0.25 and H/). = 0.75 for the first and second modes. This implies that the relationship between the slope height and the shear wave velocity of the soil behind the slope is very important in quantifying the effect of topography. The two peaks are seen at all levels of damping, though the peaks appear to shift to slightly lower values of H/I. with increased damping. The magnitude of the amplification decreases away from the slope crest, primarily as a function of damping. The peak amplification seems to decrease and occur at a lower frequency with increasing distance from the crest, though in any case, the amplification is on the order of 15 to 20 percent of the free field motion. In addition, attenuation occurs at certain frequencies with increasing distance from the crest. At low values of H/!., where the topographic step is small compared to the wavelength, the slope has little effect on the response. 5.2.2 Effect of SV- (In-Plane) Waves on a Vertical Slope For the analysis of the response to SV-waves, both a horizontal and vertical component need to be considered. Since the input motion only consists of horizontal motion, the transfer functions for the vertical response are given relative to the 84
horizontal input motion, and the vertical amplification is relative to the free field horizontal response. The results of the horizontal response due to SV-waves are shown in Figures 5.7 through 5.10 for frequencies ranging from 0.1 to 10 Hz (H/I) = 0.01 to 1.0). Considering first the horizontal response, the results are similar to those obtained for the SH-waves. The first peak amplification occurs at H/I = 0.2, and the second peak occurs at HA = 1.0. For all levels of damping, the magnitude of both amplification peaks are on the order of 50 percent, which is higher than those observed for SH waves, the second peak significantly so. However, as with SH-waves, increased damping significantly reduces the response at higher frequencies, so the second peak has a lesser importance in the overall response at higher damping levels. The pattern of attenuation and amplification with increasing distance away from the slope is also similar to the SH-wave case, though the magnitudes are greater for the SV-case. The results showing the vertical response are presented in Figures 5.11 through 5.14. The vertical response is most pronounced at the crest of the slope, and at H/I.> 0.2, it is greater than the free field horizontal response. The amplitude at the crest does not seem to be effected by damping. The amplification of the vertical response away from the crest is never greater than about 50 percent the free field motion, and decreases with increased damping. Finally, it appears that the amplitude of the vertical response at the crest tends to increase with increasing frequency, and seems to be independent of the horizontal response at frequencies above H/I. > 0.2. 85
2 z 0 S1.5 D LL IL 0 zi F Z 0.5 0 0 0 0.01 0.02- 0.03 0.05 0.1 0.2 0.3 0.5 SLOPE HEIGHT/WAVELENGTH 2 z 0 H1.5 a F 0 F0.5 0I 0 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH Figure 5.7: Horizontal transfer functions (a) and amplifications (b) for vertically incident SV-wave on a stepped halfspace for various distances behind crest, P = 1%. 86
2 z 0 C-) 1.5 U w U C/) zi F 0.51 0 0 0 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 SLOPE HEIGHT/WAVELENGTH 2 z H 1.5 0 LL
<1 -J z
0 N 0 0-r 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH Figure 5.8: Horizontal transfer functions (a) and amplifications (b) for vertically incident SV-wave on a stepped halfspace for various distances behind crest, P = 5%. 87
2 z 0 Z 1.5 U cc I F Uj U C/)
"Z0.5 N
0 0 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH 2 ( Z o
. - 1.5 "
L 0
"1
_No. 0 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH Figure 5.9: Horizontal transfer functions (a) and amplifications (b) for vertically incident SV-wave on a stepped halfspace for various distances behind crest, P = 10%. 88
2 z "Crest 0 "0- (a) F: 1H 0 2H Z 1.5 4H LL CC Free Field w LU L.L U/) Z F
-J I
Z 0.5 0* N Wfa 0 M 0
- 0. 01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH 2
z 0 0.5 IL 0i a_ 0 z 0 N 0 . 0 I 0 0.I 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH Figure 5.10: Horizontal transfer fiinctions (a) and amplifications (b) for vertically incident SV-wave on a stepped halfspace for various distances behind crest, P3= 20%. 89
2 Crest z - (a) 0 1H 2H 1.5 z 4H U Free Field w LL C/) 1 z I 0 0.5 w 0 U3 A.U 3 aU 05. 1
- 0. 01 U. SL P.U H I.UGHT/WAVELEN v.v SLOPE HEIGHT/WAVELENGTH 2
z Q1.5 ILL E 0.5 w 0 0 0..01 0.02 0.03 005 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH Figure 5.11: Vertical transferfunctions (a) and amplifications(b)for vertically incident SV-wave on a stepped halfspacefor various distances behind crest, P=1%. 90
z 0 o 1.5 z LL F wU LL. z I-So.0.5 w 0 0.01 SLOPE HEIGHT/WAVELENGTH 2 C z _J H U FC 00.5 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH Figure 5.12: Vertical transferfunctions (a) and amplifications (b)for vertically incident SV-wave on a stepped halfspacefor various distances behind crest, P=5%. 91
2 z 0 z. ILJ Fr Ur:" w U) C/) 1 z F CC: w 0 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 SLOPE HEIGHT/WAVELENGTH 2 1.o5 C z 0 U 0j l-J i0 w 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 SLOPE HEIGHT/WAVELENGTH Figure 5.13: Vertical transferfunctions (a)and amplifications (b)for vertically incident SV-wave on a stepped halfspace for various distances behind crest, f3=10%. 92
2 Crest
"'.w-0 z 1H (a)
F (O 1.5 2H zZ) 4H ILL Free Field LU ILL cO I z a: c oL 0. UJ a: A 00.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 I SLOPE HEIGHT/WAVELENGTH 2 z 01.5 ILL F a: 0.5 w 0 A 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH Figure 5.14: Vertical transferfunctions (a) and amplifications (b)for vertically incident SV-wave on a stepped halfspace for various distances behind crest, P3=20%. 93
5.2.3 Effect of Slope Angle The effect of slope angle on topographic amplification was considered by varying the slope angle, S, as shown in Figure 5.15. Since steep slopes are the subject of this study, only slopes between 45 and 90 degrees were considered (30 and 90 degrees for SV-waves). The slope-crest amplification of the SH-wave free field motion is shown in Figure 5.16. With decreasing slope angle, the magnitude of the amplification at the first peak decreases from about 25 percent to about 15 percent, while the response at higher frequencies tends to increase to about 50 percent, with no apparent second peak. The horizontal response due to SV-waves is shown in Figure 5.17. In general, the magnitude of the amplification decreases with decreasing slope angle, from about 55 percent to about 15 percent, for H/I. < 0.4. Results at higher frequencies, above H/I. = 0.4, indicate no clear trend. The vertical response due to SV-waves is presented in Figure 5.18. Again, the vertical response decreases with decreasing slope angle. 5.2.4 Effect of the Incident Angle The effect of varying the incident angle of SV-waves has until recently been a subject of little understanding. With the development of GROUND2D, the analysis of inclined SV-waves is made relatively easy, and provides us with an opportunity to consider their effect on the seismic response of steep slopes. Though our ability to determine the angle of incidence for the purposes of a site-specific stability analysis 94
H S 9
*6 0
4 9 16 I... - - Left GTE *
- E'Innt nmum IlL SGTE 6
0 9 9 4. d, ________________________________________________________ I.
*Control Point for Input Motion
-- Simulated Halfspace -
Figure 5.15: Stepped halfspace model for an inclined slope. 2 1.5 Z 0 I a_ 1 a 4: 0.5 0 " 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH Figure 5.16: Horizontal amplification at the crestfor a vertically incident SH-wave on an inclined slope, P = 1%. 95
2 1.5 z 0 LL 0.5 0 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH Figure 5.17: Horizontal amplification at the crestfor a vertically incident SV-wave on an inclined slope, P = 1%. 2 1.5 z 0 LL 1 i CL 0.5 0 bw 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH Figure 5.18: Vertical amplification at the crestfor a vertically incident SV-wave on an inclined slope, P = 1%. 96
of an actual slope is questionable, the purpose of the analyses presented herein is to determine if the incident angle is, in fact, important to the response. The angle of incidence, F, is measured clockwise from the z-axis (Figure 5.19). Waves with positive incident angles will be referred to as travelling away from the slope, and those with negative incident angles are referred to as travelling into the slope. The model characteristics are the same as previously analyzed, with damping equal to 1 percent, and the angle of incidence ranging from +30 to -30 degrees. The response to SH-waves is presented in Figures 5.20 through 5.23. In each case, the response due to the wave traveling into the slope is greater than for the wave angle traveling away from the slope. For all angles considered, waves traveling into the slope result in greater amplification than for vertically propagating waves, and this effect increases with increasing frequency. The opposite is true for waves traveling away from the slope. The motion is attenuated with increasing incident angle, and the attenuation increases with frequency. Similar results are obtained for the horizontal component of the SV-wave response, presented in Figures 5.24 through 5.27. However, in contrast, the direction of wave propagation appears to make little difference in the vertical response to SV waves. Although, there is a notable increase in the vertical response due to SV-waves at low frequencies, which increases with incident angle independent of the direction of propagation due to wave splitting on the free surface. An SV-wave of amplitude 0.5 incident on a free surface will result in both horizontal and vertical motions, depending on Poisson's ratio, as shown in Figure 4.6. For material with a Poisson's 97
H t IL Left GTE Right GTE 41 6 H
'*-IControl Point for Input Motion
--Simuulated Halfspace - - ----
Figure 5.19: Stepped halfspace model for inclined wave incident on a vertical slope. 98
3 2.5 Z 2 0 9.. 0151.5 LL a_ 0.5 0 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH Figure5.20: Amplifications at the crest for inclined SH-wave incident on a vertical slope, F = 00, P = 1%. 3 2.5 Z 2 0 U a
<1 0.5 0-L 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH Figure 5.21. Amplifications at the crest for inclined SH-wave incident on a vertical slope, F = -10' and +100, ý = 1%.
99
3 2.5 Z 2 0 U-0.5 0 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH Figure 5.22: Amplifications at the crest for inclined SH-wave incident on a vertical slope, F = -20" and +200, 0 = 1%. 3 2.5 Z 2 0
< 1 0.5 0 "
0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH Figure 5.23: Amplifications at the crest for inclined SH-wave incident on a vertical slope, F = -300 and +30',[ = 1%. 100
3 2.5 Z 2 0 LL Fiue .4 0.5 21.5 o0 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH 0 1.5 I Figure 5.24: Amplifications at the crest for inclined SV-wave incident on a vertical
,lm slope, F = 00, P = 1%.
3-7 2.5 Z 2 0 L 0 .5 0~ 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH Figure 5.25: Amplifications at the crest for inclined SV-wave incident on a vertical slope, F = -100 and +10°, P = 1%. 101
3 2.5 Z 2 0 F S1.5 I LL 15_ 0.5 0 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH Figure 5.26: Amplifications at the crest for inclined SH-wave incident on a vertical slope, F = -20° and +200, P = 1%. 3 2.5 Z 2 0 o1.5 LL 0.5 0 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 1 SLOPE HEIGHT/WAVELENGTH Figure 5.27: Amplifications at the crest for inclined SV-wave incident on a vertical slope, F = -300 and +300, P = 1%. 102
Ratio = 0.3, the effect is relatively minor on horizontal motion, but is pronounced on the vertical response for the angles of incidence considered in this study. Overall, the amplification of inclined SV-waves traveling into the slope may partially explain field observations of failures on slopes facing in a particular direction, while slopes in the same material, but of different orientation, showed no distress. Consequently, these analytical results suggest the need to account for wave orientation in relation to the slope in performing stability analyses. 5.3 ANALYSIS OF A STEPPED LAYER OVER A HALFSPACE The preceding parametric study of the stepped halfspace provides a fundamental understanding of the topographic effect. The next step is to evaluate the relationship between the natural frequency of the site and the topographic amplification effect. To this end, the analysis of a stepped layer over a halfspace is presented. A vertically stepped layer over a halfspace was used, as shown in Figure 5.28. The layer has the same material properties as used in the halfspace study discussed previously, while the underlying halfspace has properties as shown in the figure, resulting in an impedance between the layer and the halfspace of 3. The natural frequency of the model behind the crest of the step is varied by changing the thickness of the layer, Z, from H to 5H. The thickness Z is varied because the topographic effect is normalized as a function of HIA which is dependent on Vs. 103
....---- Left GTE ---
-::i --- '-i : !.:* ::- - :.- -
- : * " ""- ::" ::' "::: ": """ -"'" *-': . :::: -.-
- i LQ9i~olPoint for Input moton
-Simulated. Halfspase -
Figure 5.28: Model for vertically stepped layer over a halfspace. 2 1.75
-- 1.5 1.25 1
0 0.25 0.5 0.75 1 1.25 1.5 O)tIO)n Figure 5.29: Comparison of transferfunction ratio, TfrTf, as a function of frequency ratio, co3,,1 104
Changing Vs, of the layer, therefore, would not have allowed for the separation of the topographic effect and the resonance at the natural frequency. The results are presented in Table 5.1 in the form of the horizontal transfer functions for the free field behind the crest and at the crest of the slope. The transfer functions shown are those at the natural frequency of the free field behind the crest, (n, defined as cn
, (5.1) 4Z and at the topographic frequency, wt, defined as V
V (5.2) 5H since the peak effect of topography occurs at about HIA = 0.2. A review of the transfer functions for the response at the crest shows that the transfer function at the topographic frequency, Ttc is never greater than the transfer function at the natural frequency of the site, Tnc. The results also show that, in the free field, Tnf remains relatively constant for all values of wn" However, at the crest, Tnc increases as (n approaches wt (i.e. as Z/H approaches 1.00). This trend is clearly shown in Figure 5.29, in which the ratio of the transfer functions, TnclTnf, is plotted versus the ratio of the frequencies, wn/(O. At low values of wn/ot, where slope height is small compared to the wavelength at the natural frequency, the transfer function at the crest, Tnc, is approximately equal to the free field transfer function, Tnf. However, when the natural frequency of the site occurs near the topographic frequency, the free field motion is amplified by over 50 percent. This amount of amplification is 105
similar to the amount observed at the topography frequency of the stepped halfspace. It would seem, therefore, that the effects of the natural frequency and those of topography may work independently. Table 5.1 Transfer Functionsfor Stepped Layer over Halfspace CREST FREE FIELD Z/H Z] ~ntTnc (an ~~ Ttc Tnf Ttf 1.00 2.0 2.5 5.2 4.3 2.9 2.2 1.30 2.0 1.92 5.0 4.8 2.9 2.8 1.60 2.0 1.56 4.9 3.0 3.0 1.9 2.00 2.0 1.25 4.3 2.6 3.0 1.2 3.00 2.0 0.83 3.3 1.6 2.9 1.2 4.00 2.0 0.62 3.0 2.1 2.8 2.1 5.00 2.0 0.5 3.0 2.3 2.8 1.0 In general, the results of the frequency domain analysis of a stepped layer over a halfspace indicate two important points. First, the natural frequency of the site has a greater effect on surface amplification than does the effect of topography. Second, it appears that the topographic amplification can be added onto the amplification caused by the natural frequency, as is indicated in Figure 5.29. This concept of separating the amplification caused by topography from that caused by the natural frequency is advantageous to the development of a simplified method to estimate topographic effects. 106
5A CONCLUSIONS The parametric study of the seismic response of a stepped halfspace and a stepped layer over a halfspace shows that the topographic effect of a steep slope on the seismic response of that slope can be normalized as a function of the ratio of the slope height (H) and the wavelength of the motion (1). Such relationship between slope height and wavelength was also noted by May (1980) for horizontally propagating SH-waves incident on a vertical scarp, and similar relationships were observed between structure dimension and wavelength by others (e.g. Boore, 1972; Geli et al., 1988; Dakoulas, 1993). For both out-of-plane (SH) waves and in-plane (SV) waves, the magnitude of the response at the crest of the slope is significantly reduced by increased damping, particularly at higher frequencies. However, the amplification of the motion at the crest over that in the free field behind the crest is relatively unaffected by damping. The fact that amplification is relatively unaffected by damping in a homogeneous system was also observed by Boore (1972). The peak topographic effect occurs at a H/ X = 0.2. This amplification is on the order of 25% for SH-waves, and 50% for SV-waves. The peak at H/ X z 0.2 approximately corresponds to the first mode of vibration of a soil column of thickness H (H/ X = 0.25), which is the frequency at which Boore (1972) and Geli et al. (1988) observed the peak response in their studies of ridges. Secondary peaks occur near H/,X = 0.7 for SH-waves and H/.I z 1.0 for SV-waves. The vertical component of the topographic effect occurs independently of the natural frequency of the site. 107
The topographic effect is most apparent for slopes steeper than 60 degrees, and is greater for inclined waves travelling into the slope than away from the slope. For a stepped layer over a halfspace, the natural frequency of the site behind the crest dominates the response, which agrees with observations by Sitar and Clough (1983). If the natural frequency of the site is approximately equal to the topographic frequency, i.e. cantwr then that response is amplified. In no case is the topographic effect greater than the response at the natural frequency. 108}}