rev 0 to 'asw bypass liquefaction analysis

Harding Lawson AssociatesDiablo Canyon Power Plant

ASW Bypass 10183.ASWCALC

CALCULATIONCOVER SHEET I UEFACTION ANALYSIS

FaainninaryFina! QC

PROJECT: Diablo Canyon ASW Bypass

File Number

Calculation Number: 10183-96-02Number ofpages: Z:1

Date l LS 9b+1 e~+ l c 0 Z.+

This calculation set documents HLA's liquefaction analysis for medium dense cohesionless

fillmaterials below the water table encountered in borings drilled on or near the ASWbypass alignment. The calculation evaluates the potential for liquefaction at various peakground accelerations (PGAs) using procedures developed by Seed et aL (1985). Thevolumetric strain and settlement within the soil layer were calculated using proceduresdeveloped by Tokimatsu and Seed (1987). Ground surface settlements were estimated byHLAon the basis of the computed settlements within the liquefied soil layer, the depth ofthe layer, and observations by HLAand others regarding compaction methods used at thetime the fillmaterials were placed. A residual strength was calculated for the liquefiedlayer using procedures developed by Seed and Harder (1990),

Preparer:

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HARDINGLAWSON ASSOCIATES DIABLOCANYONPOWER PLANT

ASW B>~ Liquefaction Calculations HLAJob No.:10183.ASWCALC

CALCULATIONSHEETPreparer: Wayne MagnusenChecker: Robert PykeDate: l 1$ Ab

Calculation Number:10183-9642Revision: 0

Sheet ~of 2l

COVER SHEETTABLEOF CONTENTS

TABLEOF CONTENTS

Sheet ~lSheet ~2-

1.02.03,0

4.05.06.0

PURPOSEREFERENCESASSUMPTIONS AND DESIGN INPU3.1 Subsurface Condition3.2 Ground MotionsMETHOD AND EQUATION SUMMARYBODY OF CALCULATIONRESULTS AND CONCLUSIONS6,1 Uquefaction Potential6.2 Seismically-induced Settlement

5- l5-bIP- l 0l t-t&l l <I

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ATTACHMENTS

1. Site Plan, Plate 1

2. Log of Boring B-l, Plate 2

3. Log of Boring B-t, Plate 3

Figure 3 (Tokimatsu and Seed, 198

S. Figure 13 (Seed et aL 1985

6. Figure 6 (Tokimatsu and Seed, 198

Sheet ~~<Sheet ~3Sheet ~2<Sheet ~25Sheet ~24Sheet ~2

HARDINGLAWSON ASSOCIATES DIABLOCANYONPOWER PLANTASW Bi~s Liquefaction Calculations HLAJob No.:10183.ASWCALC

CALCULATIONSHEETPreparer: %aJ+e MagnusenChecker: Ro rt PykeDate:

Calculation Number:10183-96-02Revision: 0

Sheet 3 of Z~

1.0 PURPOSE:

This calculation set documents HLA's liquefaction analysis for medium dense cohesionless fillmaterials

below the water table encountered in borings drilled on or near the ASW bypass alignment. The potential

for liquefaction is evaluated and the volumetric strain and total settlement within the liquefied layer is

calculated. The purpose of the calculation set is to estimate the settlement ofportions of the ASW bypass

piping during various seismic events.

HARDINGLAWSON ASSOCIATES DIABLOCANYONPOWER PLANTASW B>~s Liquefaction Calculations HLAJob No.:10183.ASWCALC

CALCULATIONSHEETPreparer: Wayne MagnusenChecker: Robert PykeDate: 5


Sheet~of Ll

2.0 REFERENCES:

Harding Lawson Associates, 1996. Geotechnical Field and Laboratory Investigation, ASK

System Bypass, Units 1 &2, Diablo Canyon Power Plant, San Luis Obispo County,

California, May 8.

Seed, H. B., K. Tokimatsu, L.F. Harder, and R.M. Chung, 1985. Influence ofSPT Procedures

and Liquefaction Resistance Evaluations, American Society of CivilEngineers, Journal of

Geotechnical Engineering, Vol. 111, No. 12, December.

Tokimatsu, K, and H. B. Seed, 1987. Evaluation ofSettlement ofSands Due to Earthquake

Shaking, American Society of Civil Engineers, Journal of Geotechnical Engineering, Volume

113, No. 8, August.

PG&E Referencest

Mr. Al Tafoya's (of PG&E) draft memorandum entitled Background Information ofSoil Near

the I.S. and for Liquefaction Issue, dated May 31, 1996.

PG&E plan entitled Rock Topography near A%VBypass Piping Routing (SK-C-ASWBROCK),

Revision A.


ASW B>~ Liquefaction Calculations HLAJob No.:10183.ASWCALC

ALCULATIONSHEETPreparer: Wayne MagnuscnChecker: Robert PykeDate: a Wb


Sheet 5 of

3.0 ASSUMPTIONS AND DESIGN INPUTS:

3.1 Subsurface Conditions

In December of 1995, HLAdrilled 4 borings to characterize thc backfill for purposes of the dynamic

analyses of the proposed ASW Bypass being conducted by others and the slope stability evaluation being

conducted by HLA. The boring locations are shown on the Site Plan, Plate 1 (see Attachment 1). The

borings encountered fillthat ranged in depth from 6 feet at Boring B-2 to 31-1/3 feet (the depth explored)

at Boring B4. The fillgenerally consisted ofstiffclays and dense to very dense sands and gravel which is

consistent with the 1978 borings; these materials are not believed to be susceptible to liquefaction.

Unexpectedly however, two of the borings encountered medium dense sands below Mean Sca Level

(Borings B-1 and B-4, see Attachments 2 and 3) as indicated by relatively low blow counts obtained

during soil sampling. The relatively low blow counts occurred at a depth of25 feet (Elevation -1 feet,

Mean Sea Level), as shown the boring logs, and are in what is believed to be the original backfill placed

during the CWI conduit construction. Thc blow count of 18 at 25 feet in Boring B-I represents a

Standard Penetration Test (SPT) N-Value (Scc note 1, below) and the blow count of 15 at 25 feet in

Boring B4 represents a pseudo-SPT N-value that was obtained by multiplying thc field blow count by 0.7

to account for thc larger sampler size (3-inch outside diameter and 2.43-inch inside diameter). For our

liquefaction analyses we used the same unit weights for the cohesive and granular fillsas we used in our

, slope stability calculations (sec HLACalcuhtion No. 10183-9641),

Additional details regarding the field investigation are presented in HLA's May 8, 1996 report.

1 The SPT N-value is defined as the number of blows of a 140-potmd hammer, fallingfreely through a height of 30 inches, required to drive a standard split-barrel sampler (2-inch outside diameter and a 1-3/8-inch inside diameter) the final 12 inches of an 18-inchdrive. For SPT procedures, see ASTM D1588.


ASW Bypass Liquefaction Calculations HLAJob No.:10183.ASWCALC

CALCULATIONSHEETPreparer: Wayne MagnusenChecker: Robert PykeDate: Alc

Calculation Number:10183-9&42Revision: 0

Sheet 4 of ~~

3.1 Subsurface Conditions (continued)

The groundwater was encountered only in Boring B-4 at a depth of 24 feet (Elevation 0 feet).

Because of the proximity of the site to the ocean, the groundwater level is likely influenced by

tidal effects. In our liquefaction evaluation, we used a groundwater elevation of +3 feet

(based on Mean Sea Level).

The zone of the medium dense sands that are located below the water table are believed to be

confined to an'area that is approximately 10 to 20 feet wide and 100 feet long, This zone is

shown on Plate 1 (Attachment 1) and is defined by the followingboundaries: the edge of the

original excavation for the CWI conduit construction on the north and east, the 1980 backfill

placed during repair of the electrical conduits on the west, and the clayey backfill placed

during the CWI conduit and IS construction that was encountered by the 1978 Borings 4 and 5

on the south. Based on the recent Boring B-4, we have assumed that this zone is 5 feet thick,

from Elevation 0 to -5 feet, throughout the zone described above. This is a conservative

assumption, given that the fillwas reportedly compacted to at least 95 percent relative

compaction. It is more likely that the sands range from being medium dense to dense within

the zone defined above.

HARDINGLAWSON ASSOCIATES DIABLOCANYONPOWER PLANTASW Bypass Liquefaction Calculations HLAJob No.:10183.ASWCALC

CALCULATIONSHEET Calculation Number:10183-9642

Preparer: Wayne Magnusen Revision: 0

Checker: Robert PykeDate: Sheet I of ~

3.2 Ground Motions

In our analyses, we evaluated the liquefaction potential for three levels ofground motion:

~ A Magnitude (M, Richter Magnitude) 7-1/2 event with a peak ground acceleration (PGA) of 0.83

gravity (g)

~ An M6-1/2 event with a PGA of0.40g

~ An M6 event with a PGA of„0.35g.

The larger magnitude event is believe to be representative of the maximum credible earthquake defined in

the Long Term Seismic Program for the plant, while the smaller events are representative ofearthquakes

with higher probabilities ofoccurance.


ASW B<~s Liquefaction Calculations HLAJob No.:10183.ASWCALC

CALCULATIONSHEETPreparer: Wayne MagnusenChecker: Robert PykeDate: 1 t crab


Sheet ~e< ~~

4.0 METHOD AND EQUATION SUMIKQ(Y:

The calcuhtion evaluates liquefaction potential for using procedures developed by Seed et al. (1985).

Liquefaction settlements were calculated for the design event and for lesser events using procedures

developed by Tokimatsu and Seed (1987). The above referenced procedures use a dimensionless quantity

called the cyclic stress ratio along with corrected blow counts from standard penetration tests to evaluate

the potential for liquefaction and to predict liquefaction-induced settlement.

The cyclic stress ratio is defined as follows:

Where,

""~,'cyclicstress ratio

= peak ground surface acceleration (PGA) (See Section 3.2)

acceleration due to gravity

C7u uotai stresses at the depth under consideration

I~c ~ective stres<as at the depth under consideration

1 J m stress reduction factor which decreases from a value of 1 at the ground surface to 0.9

at a depth of35 feet. (we used 0.93 corresponding to a depth of25 feet)


CALC ATIONSHEETPreparer: Wayne MagnusenChecker Robert PykeDate: l 5 6b


Sheet ~ of ~

Blow counts are corrected using the following relation:

where

(~ l)QO

CawCw

corrected blow count for use mth Fig. 13 (Sheet ~~~

~rrection for hammer energy; we used 1.2 for the auto-trip hammer

=correction coefficient from Fig. 3(Sheet ~LA;we used 0.86.

measured blow counts (N=18 at 25 feet in B-I using an SPT sampler.N=15 at 25 feet in B< is a pseudo-SPT N-value obtained by multiplying the field blowcount by 0.7 to account for the larger sampler size (3-inch outside diameter and 2.43-inch inside diameter)

For the M7-1/2 event, the potential for liquefaction evaluated using Fig 13 (Sheet ~l~ . For all three

levels of ground motion, estimated strain within the liquefied layer is obtained from Fig. 6 (Sheet ~i&

For the lesser M6-1/2 and M6 events, the cyclic stress ratio is scaled using scaling factors of 1.19 and

1.32, respectively, as recommended by Tokimatsu and Seed (1987). Estimated settlement is calculated by

multipl)ing the strains by the 5-foot layer thickness.


CALCULATIONSHEETPreparer: Wayne MagnusenChecker: Robert PykeDate: l i


Sheet (0 of%~

Ground surface settlements were estimated by HLAon the basis of the computed settlements within the

liquefied, soil layer, the depth of the layer, and observations by HLAand others regarding compaction

methods used at the time the fillmaterials were placed. In evaluating seismically-induced settlement, we

utilized procedures developed by Tokimatsu and Seed (1987).

HARDINGLAWSON AS SOCIATES DIABLOCANYONPOWER PLANTASW Bypass Liquefaction Calculations HLAJob No.:10183.ASWCALC

CALCULATIONSHEETPreparer: Wayne MagnusenChecker: Robert PykeDate


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HARDINGLAWSON ASSOCIATES DIABLOCANYONPOWERPLANI'SW

Bypass Liquefaction Calculations HLAJob No.:10183.ASWCALC

CALCULATIONSHEETPreparer: Wayne MagnusenCbecker:- Robert PykeDate: 1 K ib


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CALCULATIONSHEETPreparer: Wayne MagnusenChecker: Ro rt PykeDate: t it Ab

Calcuhuon Number:101&3-96-02Revision: 0

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HARDINGLAWSON ASSOCIATES DIABLOCANYONPOWER PLANTASW Bypass Liquefmction Calculations HLAJob No.:10183.ASWCALC

CALCULATIONSHEETPnq~r: Wayne MagausenChecker: Robert PykeDate: 1 )


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ALCULATIONSHEETPreparer: Wayne MagnusenCbecter: bert PykeDate: 1 6b


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HARDINGLAWSON ASSOCIATES DIABLOCANYONPOWER PLANTASW Bypass Liquefaction Calculations HLAJob No.:101&3.ASWCALC

CALCULATIONSHEETPreparer. Wayne MagnusenCbccker. Robert PykeDate: 1 x


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HARDINGLAWSON ASSOCIATES DIABLOCANYONPOWER PLANTASW Bi~s Liquefaction Calculations HLAJob No.:10183.ASWCALC

CALCULATIONSHEETPreparer: Wayne MagnusenChecker: Robert PykeDate: a ie he


Sheet ~ l of ~~

6.0 RESULTS AND CONCLUSIONS:

6.1 Liquefaction Potential

Our analyses indicates that there is a high probability of liquefaction for the medium dense sands located

below the water table during the M7-1/2 event. For the M6 event, the data points plot near the border

between "liquefaction" and "no liquefaction" indicating that there is a much lower chance of liquefaction

occurring during the smaller event.

6.2 Seismically-induced Settlement

For the M7-1/2 event (PGA of 0.83g), the computed maximum settlement of the 5-foot layer

that liquefies is approximately 1 inch. Because of the depth of this layer and the limited

extent of the medium dense sands, we judged that the maximum ground surface settlements

could be up to approximately 50 percent of the computed settlements. This would result in a

maximum ground surface settlement on the order of 1/2 inch during the M7-1/2 event. For the

M6 event (PGA of 0.35g), the computed maximum settlement of the 5-foot layer that liquefies

is approximately 1/2 inch. As with the larger event, we judge that the maximum ground

surface settlement willbe approximately 50 percent of the computed value; approximately 1/4

inch.

HARDINGLAWSON ASSOCIATES DIABLOCANYONPOWER PLANTASW Bypass Liquefaction Calculations HLAJob No.: IOI83.ASWCALC

CALCULATIONSHEETPreparer: Wayne MagnusenChecker: Ro rt PykeDate: I KS %l


Sheet l~ of >"

Differential settlement of the proposed ASW Bypass could occur during an earthquake

because the proposed pipelines willcross over the zone of medium dense sands. We judge

that the magnitude of the maximum differential settlement willbe approximately equal to the

maximum ground surface settlements mentioned above. Because of the depth of the sands,

the differential settlement willnot occur abruptly, but willoccur gradually along the pipeline.

For design purposes, we recommend assuming that the estimated maximum differential

settlement occurs over a distance of 25 feet.

The settlements mentioned above are the result of densification of the sands following

dissipation of pore water pressures developed as a result of seismic shaking. Therefore, the

settlements will take place following the earthquake as pore water pressures are allowed to

dissipate. As a result, these settlements should not be added to the transient displacements

that are predicted for the pipelines during the earthquake shaking.

These estimated settlements are maximum values that could occur during a single event. It is

possible for liquefaction to occur during an earthquake, but to have little or no observable

settlement; this does not mean that settlement willnot occur during future earthquakes.

While liquefaction has been observed to occur repeatedly at the same site during multiple

earthquake, the settlement estimates represent the upper bound of accumulative settlement

during repeated events at any given point at the site.


CALCULATIONSHEETPreparer: Wayne MagnusenChecker: Robert PykeDate: iQ Rb


Sheet 2 l pf C1

We judge the risk of lateral movements due to liquefaction to be very low because of the

discontinuous nature of the medium dense sands and the fact that they are confined on all

sides.

ATTACHMENTS:

1. Site Plan, Plate 1 (Sheet ~i~

2. Log ofBoring B-l, Plate 2 (Sheet ~~

3. Log ofBoring B4, Plate 3 (Sheet ~~i

4. Figure 3 (Tokimatsu and Seed, 1987) (Sheet ~~>

5. Figure 13 (Seed et al. 1985) (Sheet ~~~

6. Figure 6 (Tokimatsu and Seed, 1987) (Sheet ~~

Calculation Number: 10)83-9~2Revision: 0ASW Bypass Liquefaction Calculations HLAJob No.:10183.ASWCALC

ATTACHMENTlVO.I

Preparer: Wayne MagnusenChecker: Robert PykeDate:

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TOP OFEXCAVATIONFOR ELECTRICALCONDUITS

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Noornoy Lowoon AOOool0100

Toyreeooa OndTnvooroNrfol sorvgol

ONowfrWM

REFERENCE:

ROCK TOPOGRAPHY NEAR ASW BYPASS PIP INE ROUTNIE.'GSEDRAWINGNO. SKWASBROCK REVISION A. (NO DATE)

HivilloOAISOAI0

yl960 I 0OOOf0C I OOVOSii

10183.ASW CALO

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HARDINGLAWSONASSOCIATES DIABLOCANYONPOWER PLANT

HARDINGLAWSONASSOCIATES .DIABLOCANYONPOWER PLANT

ASW Bypass Liquefaction Calculations HLAJob No.:10183.ASWCALC

ATTACIIMENTN0.2

Preparer. Wayne MagnusenChecker. Robert PykeDate: I 4


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HARDINGLAWSON ASSOCIATES DIABLOCANYONPOWER PLANTASW Bypass Liquefaction CaIculations HLAJob No.:10183.ASWCALC

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HARDINGLAWSON ASSOCIA'IZS DIABLOCANYONPOWER PLANTAS% Bypass Liquefaction Calculations HLAJob No.:10183.ASWCALC

ATTACHMENTXO.S

Preparer. %ayne MagnusenChecker: Robert PykeDate; "I 6(,

Calculation Number.10183-9&42Revision: 0

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HARDINGLAWSON ASSOCIATES DIABLOCANYONPOWER PLANTASW By~ Liquefaction Calculations HLAJob No.:10183.ASWCALC

ATTACIIMENTN0.6

Preparer. Wayne MagnusenCbecker. rt PykeDate:


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Attachment 2.2

. Terzaghi, Karl, Theoretical Soil Mechanics, John Wiley &, Sons, Inc.,New York, 1943, pages 66 through 76

THEORETICAL

BOOKS BY TEBZAG))l K. AND PECK, B. B.

Soil Mechanics in Engineering Practice

BOOKS BY TEllZAOI)l> K.

Theoretical Soil Mechanics

From Theory to Practice in Soil Mechanics

SOIL MECHANICS

xr )t p r wriD v, A c~HT

JOIN WILEY&SONS

New York ~ Chichester ~ Brisbane ~ Toronto

SFCTION 3

CONDITIONS FOR SHEAR FAILURE IN IDEAL SOILS

isa. 19 STATE OF STRESS lN THE ZONE OF ARCff 67

I9. State of stress in the@one of arching. The local yield of thehorizontal support of a bcd of sand shown in Figure 17a can be producedby gradually lowering a strip-shaped section ab of the support, Beforethe strip starts to yield, the vertical pre@sure per unit of area on thehorizontal support is cveiywhere equal to the depth of the layer of sand

CDCD

tsCP

CD

CHUTER V

ARCHING IN IDEAL SOILS

18. Definitions. If one part of the support of a mass of soil yieldwhile the remainder stays in place the soil adjoining the yielding partmoves out of its original position between adjacent stationary massesof soil. The relative movement within the soil is opposed by e shearingresistance wittun the zone of contact between the yielding and the sta-tionary masses. Since the shearing resistance tends to keep the yieldingmass in its original position, it reduces the pressure on the yielding partof the support and increases the prcssure on the adjoining stationarypart. This transfer of pressure from a yielding mass of soil onto ad-joining stationary parts is commonly called the arching effect, and thesoil is said to arch over the yielding part of the support. Arching alsotakes place if ono part of a yielding support moves out more than theadjoining parts.

Arching is one of the most universs1 phenomena encountered in soilsboth in the Geld and in the laboratory. Since arching is maintainedsolely by shearing stresses in the soil, it is no less permanent than anyother state of stress in the soil which depends on the existence ofshearingstresses, such as the state of stress beneath the footing of a column.For instance, ifno permanent shearing stresses were possible in a sand,footings on sand would settle indefinitely. On the other hand, everyexternal influence which causes a supplementary settlement of a footingor an additional outward movement of a retaining wall under unchangedstatic forces must also be expected to reduce tho intensity of existingarching effects. Vibrations are the most important influence of thissort.

In the following article two typical eases will be investigated, viz.,arching in an ideal sand due to the local yield of a horizontal supportand arching in the sand adjoining a vertica support whose lower partyields in an outward direction.

n

Igtulare/

I I Shen'sniI

(Dj

PI/

Eudial Sherr

La/grplcl2cc/ts 411

d.H~d Aib 44'-cy.I

cz 90~

Sofia'asz cc bl sWBQ

(~j

Fio. 17. Failure in cobesionless esnd preceded by arching. (o) Fsilum caused bydownward movement of s long nsrmw section of the base of a layer of sand;(b) enlarged detail of diagram (a); (c) shear failure in sand due to yield of lateraleuppert by tiltingabout its upper edge.

tiines its unit weight. However, a lowering of the strip causes the sandlocated above the strip to follow. This movement is opposed by fric-tional resistance along the boundaries between the moving and thestationary mass of sand. As a consequence the total pressure on theyielding strip decreases by an amount equal to the vertical componentof the shearing resistance which acts on the boundaries, and the totalpressure on the adjoining stationary parts of the support increases bythe same amount. In every point located immediately above the yield-ing strip the vertical principal stress decreases to a small fraction ofwhat it was before the yield commenced. The total vertical pressureon the base of the layer of sand remains unchanged, because it is alwaysequal to the weight of the sand. Therefore the decrease of the verticalpressure on the yielding strip must be associated with an increase 'of thevertical pressure on the adjoining parts of the rigid base, involving anabrupt increase of the intensity of the vertical pressure along the edgesof the strip. This discontinuity requires the existence of a zone ofradial shear comparable to that shown in Figure 15a. The radial shearis associated wit,h a lateral expansion of the sand located within thehigh-pressure zone, on both sides of the yielding strip towards the low-

sA

ARCHINQ IN IDEAL SOILS Am. 19 bar. 20 THEORIES OP ARCHINO 69

pressure zone located above the strip. If the base of the layer of sandwere perfectly smooth, the corresponding shear pattern should besimilar to that indicated in Figure 17a and, on a larger scale, in Figure17b.

As soon ss the strip has yielded sufliciently in a downward direction,a shear failure occurs along two surfaces of sliding which rise from theouter boundaries of tho strip to the surface of the sand. In the vicinityof the surface all the sand grains move vertically downward. This hssbeen demonstrated repeatedly by timmxposurc photographs. Such amovement is conceivable only iftile surfaces of sliding intersect the hori-z ntal medsce nf the.ssnrl st right annlne, Whc.p the fails'uzJus atroughlike depression appears on thc surface of the sand as indicated inFigure 17a. The slope of each side of the depression is greatest whereit intersects the surface of sliding. The distance between these steepestparts of thc trough can bo measured. Ithas been found that it is alwaysgreater than the width of the yielding strip. Hence, the surfaces ofsliding must have a shape similar to that indicated in Figum 17a by thelines ac and hf. The problem of deriving the equations of the surfacesof eliding ac snd bd has not yet been solved. However, experiments(Vollmy 1937) suggest that the average slope angle of these surfacesdecreases from almost 90'or low values of D/2B to values approach-ing 45'+ 4/2 for very high values of D/2B.

The vertical pressure on the lower part of the mass of sand locatedbetween the two surfaces of sliding, ac and bd in Figure 17a, is equal totho weight of the upper part reduced by'he v'ertical component of thefrictional resistance which acts on the adjoining surfaces of sliding.This transfer of part of tbc weight of the sand located above the yieldingstrip onto the adjoining masses of sand constitutes tho arching effect.

The preceding reasoning can also be applied to the analysis of thearching effect produced in a maes of sand by the lateral yield of thelower part of a vertical support. In Figure 17c the lateral support isrepresented by ak The surface of the sand is horizontal and the sup-port yields by tiMng around its upper edge. After the supporf; hasyielded suf6ciently, a shear failure occurs in tho sand along a surfaceof sliding 58 which extends from the foot b of the support to the surfaceof the sand. The stationary position of the upper edge, a, of the lateralsupport prevents a lateral expansion of the upper part of the slidingwedge. Therefore the sand grains located in the upper part of thewedge can move only in a downward direction. EIence the surface ofsliding inteizccts tho horizontal surface of thc sand at d at right angles.The corresponding subsidence of the surface of the sliding wedge isindicated in the figure by a dashed line.

Tho lateral expansion of the lower part of thc sliding wedge is associ-ated with a shortening in a vertical direction. The correspondingsubsidcncc of the upper part of the wcdgo is opposed by tho frictionalresistance along the adjoining steep part. of the surface of sliding. Asa consequence the vertical pressure on tho lower part of the wedge issmaller tlran the weight of the sand located above it. This phenomenonconstitutes the arching effect in the sand behind yielding lateral sup-ports whose upper part is stationary.

20. Theories of archiag. Most of the existing theories of archingdeal with the pressure of dry sand on yielding horizontal strips. They.can be diyidcd jato three groups, The author of the. theories nf thofirst group merely considered thc conditions for the equilibrium of thosand which is located immediately above the loaded strip withoutat tempting to investigate whether or not the results of the computationswere compatible with the conditions for the equilibrium of tho sand ata greater distance from the strip. The theories of the second group arebased on the unjusti6ed assumption that the entire mass of sand locatedabove the yielding strip is in a state of plastic equilibrium.

In the theories of a third group it is assumed that the vertical sectionsae and bf (Fig. 17a) through the outcr edges of the yielding strip repre-sent surfaces of shding and that the pressure on the yielding strip isequal to the difference between the weight of the sand located above thestrip and the full frictional resistance along the vertical sections (Cain1916 and others). The real surfaces of shding, ac and bd (Fig. 17a), arecurved arid at the surface of the sand their spacing is considerablygreater than the width of the yielding strip. Hence the friction alongthe vertical sections as and bf cannot be fullyactive. The error due toignoring this fact is on the unsafe side.

Tha followiag oommeats are Jateadsd to fafona the resder ia a general wsy ontho fuadameatal assumptions of tho theories of the first two groups. Eagcsser(1882) replaced the sand located immediately above the yielding ship by an imsgi-nary amh and computei the pressum on the strip on the bssis of the coadiuoas forthe squilibrium of the arch. Bierbaumer (1913) camped the sand locatsd fmms-distdy above the stri to the ksysloas in an arch. He assumed that the base of thekcystono coiacidcs with the surface of the strip, sad thst the sides of tha kcystoaesm plsao sad rise from tha outer bouadarics ol the strip towsnb tbs csaicv. Thepressure on the strip is equal aad opposite to the force requued to msiataia thekeystoae ia its position. Caquat (1934) rsplaaed the ca@re msss of ssad locstedabove the ylddiag strip by a system of arches. He assumed that tbs horisoatalnormal stree ia the arches above ths center llas of the strip ls equal to the corre-spoadiag verticsl normal stress times the How'alue Ns, equation?(4), sad hscomputed the pressure on the strip on the basis of the conditions for the aqui]ibriumof the arches. Vsllmy (1937) rephiced the curved surfaces of slidiag ac aad hd(Fig. 17a) by iadiaed plsae surfaces sad sssumed thst the aonasl stresses on these

70 ARCHlNO JN IDEhE SOILS ARv. 20

surfaces are identical with the normal stresses on similarly oriented sections througha semi-infinite mass of sand in an active Ranhine state. The slope of the surfac»s

of sliding is chosen such that the corresponding pressure on thc yielding strip is a

maximum. According to the results of some of his investigations an increase ofthe angle of internal frictionof the sand should cause an increase of tho pressure onthe yielding strip. According to all tho other thcorics and to tho existing test re-sults an increase of the angle of internal friction has the opposite cflcct. Vollmy(1937) also invcstigatcd the prcssure of the earth on rigid nnd on Qcxible culvertsand compared tho results of his analysis with those obtained by earlier investigators.However, under Geld conditions ths pressure on yielding horir~ntal supports such

as the roofs of culvcrts or of tunnels depends on many conditions other than those

vrhich have been considered so far in theoretical'investigations.

Allthe theories cited above are in accordance wii,h experience in thatthe pressure on a yielding, horizontal strip with a given width increases

less rapidly than the weight of the mess of sand located above the stripand approaches asymptotically s finite value. However, the valuesfurnished by ddferent thcorics for the presto on the strip are quitedifferent. In order to Gnd which of tho theories deserves preference itwould be necessary to investigate experimentally the state of stress

above yielding strips and to compare the results with the basic assump-

tions of the theories. Up to this time no complete investigation of thistypo has been made, and the relative merit of the several theories isstill unknown. The simplest thcorics are those in the third categorywhich are based on the assumption that the surfaces of sliding arevertical. Fortunately the sources of error associated with this assump-

tion are clearly visible. In spite of the errors the final results are fairlycompatible with the existing experimental data. Therefore the follow-ing analysis willbe based exclusively on the fundamental assumptionsof the theories in this category. In connection with a scientific studyof the subject Volimy's publication should be consulted (Vollmy 1937).

Ifwe assume that the surfaces of sliding arc vertical as indicated bythe lines ae and bf (Fig. 17a) the problem of computing the verticalpressure on the yielding strip bccomcs identical with the problem ofcomputing the vertical pressure on the yielding bottom of prismaticbins.

For cohcsionlcss materials this problem has boon solved rigorously by Kcttcr(1899). It has also been solved with diflcrcnt dcgrccs of approximation by otherinvestigators. The simplest of the solutions is based on the assumption that thevertical pi»ssure on any horisontal section through the 6ll is uniformly distributed(Janssen 1896, Kocncn 1896). This assumption fs incompatible with tho state ofstress on vertical sections through the soil, but the error due to this assumption isnot so important that the assumption cannot be used as a basis fora rough estimate.

Figure 18a is a section through the space between two vertical sur-

THEORIES OP ARCffI1Vg „

4 cry B

x-riBc+pp r&

f'fY~2

gdz

w 2

xc 40

)e 6

Id

2" cvcrndb -0 2- 3~ a crndb

c40

86

io

Ib

2

a4

On 6

0

8

peg) fn(10

(b)

~a~(1 gEn&CC)Et'biz-tnhef

ii 40;Enl

Fio. 18. (ci) Diagram illustrating assumptions on which computation of prcssure insand between two vertical surfaces of sliding is based; (c and d) reprcscntations oftho results of tho computations.

and the vertical prcssure is assumed to be equal to an empirical constantKat every point of the fill. The vertical stress on a horizontal sectionat any depth z below the surface is cr„and the corresponding normal

faces of sliding. The shearing resistance of the earth is determined bythe equation

s = c+crtangThe unit weight of the soil is y and the surface of the soil carries a

uniform surcharge q per unit of area. The ratio between the horizontal

CICD

CP

CD

Crl

'Ti

A

AO

r

ARCHING IN IDEAL SOILS

e,~q for z~0

Solving these equations we obtain

B(y —C/B) c, -/r Caag 0/Bx I )riant s/B t2))+q0Ztanp

Hy substituting in this equation in succession the values c = 0 and

q ~ 0, we obtain

(-Xtaa I) */B) t3]c)0 q 0 4), 1 —e

c 0 0>0 c,= —(1 —en e'~)+te """e'ec [4)

Ztan p

By -K6an P 0/8c=0 0 0 c,=—(1 —e )K tan@t5]

Ifthe shearing resistance in a bed of sand is fullyactive on the vertical

sections ae and bf (Fig. 17a), the vertical pressure e, per unit of area of

the yielding strip ab is determined by equation 5. Substituting in this

equationz~nB

we obtain

wherein

Ic = (1 —e

" 'eck'~ 't —(1 —e' [6b)

Ktcn4 Ktcnk

stress on the vertical surface of sliding is

d/I ~ ZO'tt t1)

The weight of the slice with a thickness dz at a depth z below thc

surface is 2By dz per unit of length perpendicular to the plane of thc

drawing. Thc slice is acted upon by the forces indicated in the figure.

The condition that the sum of the vertical components which act on thc

slice must be equal to zero can be expressed by the equation

2By dz = 2B(~, + d(),) —2Bn, + 2c dz + 2'„dz tan P

ord(), c tan 4

y ———Zt),—dz B B

THEORIM OI ARCHINCr

It'or z ~ we obtain a = 1/K tan 4 and

wherein

yBaz + yBnibz = yB(az + nibs j

1(1 —e "'~) and b> ——<

x" ~'~ t8b]Ktan @

For nz a> thc value az'becomes equal to

1a

K tan<

and the value bz equal to zero. The corresponding value of (K, is

yBe yBa

K tang

which is equal to the value given by equation 7. In other words. thevalue cr is independent of the depth zq in Figure 17a.

The relation between nz and az is identical with the relation between

yB<It ~ trna ~

K tan p

In Figurc 18b the ordinatcs of the curve marked a represent tho valuesof n z/B and the abscissas the corresponding values of a for P

30'nd

K = 1, or for K tan P = 0.58. Figure 18c contains the same datafor p = 40'nd K = 1 or for K tan p = 0.84.

Experimental investigations regarding the state of stress in the sandlocated above a yielding strip (Tcrzaghi 1936e) havo shown that thcceo) ek Ke»k~ S~ sLee e6 ena 4ck ~ 4 4 l i ki Ak lecnc~ keeeke kk eket e 44JP4 1 k eeekk ekkeee \4 et IklkllJ kekekkkI eke ekl Ice+ enlenl Y %j telic eckk net 1meko

of thc yielding strip to a maximum of about 1.5 at an elevation of ap-proximately 2B above the center line. At elevations of moro thanabout 5B above thc center line the lowering of the strip seems to haveno cflcct at all on the state of stress in the sand. Hence we are obligedto assume that the shearing resistance of the sand is activo only on thclower part of the vertical boundaries ae and bf of the prism of sandlocated above the yielding strip ab in Figure 17a. On this assumptionthe upper part of tho prism acts like a surcharge q on the lower part sndthe pressure on tho yielding strip is determined by equation 4. If

nrB is the depth to which there aro no shearing stresses on thovertical boundaries of the prism abje in Figure 17a the vertical prcssure

pcr unit of area of a horizontal section eifi through the prism at a depthz> below tho surface is q = yzq ynrB. Introducing this value and thevaluo z zz = nzB into equation 4 wc obtain

74 ARCHIE IN IDEAl SOILS Am. 20 Aar. 20 THEORIES OP ARCHINO 75

C),CO

ri and a, represented by equation 6b and by the plain curves in Figures

18Ir and 18e. The relation between the values n and the corresponding

values ofEtl 440 f0 =C

is reprc.ented inFigures 18b and 18c by the dash-dotted curves marked b.

Ia order to illustrate by means of a numerical example the influencc

of thc absence of shearing stresses on the upper part of tho vertical

sections oe and bf in Figure 17a wo assume P 40', K ~ 1, and ni ——4.

I3ctween the surfsco snd a depth zi njB ~ 48 the vertical pressure

on horizontal sections increases like a hydrostatic prcssure in simplo

proportion to depth, as indicated In l'iguie 18d by the straiglit linc oe.

Below a depth zi the vertical pressuro is determined by equations 8.

It decreases with increasing depth, as shown by the curve ef and itapproaches asymptotically the value cr (eq. 7).

The dashed line og in Figure 18d hss been plotted on the assumption

rii ~ 0. The abscissas of this curve are determined by equations 6.

With increasing depth they also approach thc value ~ (eq. 7). The

figure shows that the influence of the absence of arching in the upper

layers of the bed of sand on the pressure e, on a yielding strip practicallyceases to exist at a depth of more than about 8B. Similar investi-

gations for different values of P and of ni Ied to thc conclusion that the

pressure on a yielding strip is alinost independent of tho state of stress

which exists in the sand at an elevatioa of more than about 4B to 6B

above the strip (two or three times the width of the strip).Ifthere is a gradual transition from fullmobilization of the shearing

resistance of the sand on the lower part of the vertical sections ae and

bf in Figurc 17a to a state of zero shearing stress on the upper part„ the

change of the vertical normal stress with depth should be such as

indicated in Figure 18d by the hae cdf. This line is similar to the

pressure curve obtained by measuring the stresses in the sand above the

center line of a yielding strip (Terzagbi 1936e).

Less simple is the investigation of the clfcct of arching oa the pressure

of sand on a vertical support such as that showa in Figure 17c. The

first atteinpt to investigate this clfcct was made on the simplifyingassumption that the surface of sliding is plane (Terzaghi 1936c).

According to the results of tho investigation tho arching in the sand

behind a lateral support with a height 8'liminates the hydrostatic

pressure distribution and it increases the vertical distanco JI, between

the point of application of the lateral pressure and the lower edge of the

support. The intensity of the arching elfect and its influence on the

value of the ratio H,/H depends on the type of yield of the support. If

tho support yields by tilting around its lower edge ao arching occurs.

The distribution of the earth pressure is hydrostatic and the ratioII,/FIis equal to onc third. A.yield by tiltingaround tho upper edge isassociated witha roughlyparabolic pressure distribution and thc point ofapplication of tho lateral pressure is located near midheight. Finally,if the support yields parallel to its original positioa, the point of appli-cation of the lateral prcssure may be expected to descend graduallyfrom an initial position 'close to midheight to a final position at thelower third point. The investigation gave a satisfactory general con-

ception of the influence of the diifcrent factors involved, but, owing tothe assumption that the surface of sliding is plane, failed.to give in-.

formation regarding the effect of arching on the intensity of the lateralpressure.

In order to obtain'tho missing information it was necessary to takethe real shape of the surface of sliding into consideration. Since the

upper edge of the lateral support does aot yield, the surface of slidingmust intcrscct the top surface of the backfillat right angles (scc Art. 19).

Ohdc investigated the influence of this condition on tho intensityof the earth pressure on tho assumption that the trace of the surface ofsliding on a vertical plane is an aro of a circle which intersects the sur-face of the backfill at right angles (Ohde 1938). The correspondinglateral pressure and the location of the point of application of thelateral prcssure have been computed for an ideal sand, with an angleof internal friction p ~ 31', by three different methods.

In one of these, the location of the centroid of tho pressure has beendetermined in such a manner that the stresses along the surface ofsliding satisfy Kotter's equation, 17(10). In a second one it has beenassumed that tho normal stresses oa both the wall and tho surface ofsliding are a function of thc second power of the distance from the sur-faco of tho backfill, measured along the back of the lateral support andthe surface of sliding respectively. The values of the constants con-tained in the functions have been determined in such a way that thcconditions for tho equilibrium of the sliding wedge are satisfied. In athird investigation another function hss been selected, approximatelyexpressing the distribution of the normal stresses over the boundariesof the sliding wedge. In spite of the dBercnces between the funds-mental assumptions, tho values obtained by these methods for the ratiobetween the elevation of the centroid of the earth pressure and theheight of the bank range between the narrow limits 0.48 and 0.56.

They correspond to an angle of wall friction 5 = 0. However, the wallfrictionwas found to have little influence on the location of the centroidof the pressure. Hence we are entitled to assume that the centroid is

'76 ARCHING IN IDEALSOILS hiCX'. 20

csCD

located approximately at midheight of the support snd the corresponding

pressure distribution is roughly parabolic, as shown on the right-hand

side of Figure 17c. The investigation has also shown that an increase

of the ratio H,/H duo to srcbing is associated with an increase of the

horizontal prcssure on thc lateral support. A simple method of com-

puting the intensity of thc lateral pressure is described in Article 6?.

It is based on the assumption that the curve of sliding is a logarithmic

spiral, vrhich intersects the surface st right angles.

A general mathematical discussion of the infiuence of the wall move-

ment on the earth pressure hasbeen published by Jfiky (1938).

Cm.roan VI

RETAINING WALL PROBLEMS

21. Definitions. Retaining walls are used to provide lateral supportfor.masses of soil. The supported material is called the backfill. Fi-ures 19 and 27 represent sections through the two principal types ofretaining walls. The wall shown cvn uesea 1A toasnJ ~ 4os 11e a ~aava, o ys latowy

because the wall depends on its ownweight for stability against thehorizontal thrust produced by thelateral earth prcssure. On theother hand, the canfileirer retaining IPurall, shown in Figure 2?, derivespart of its stability from the weight

S'urfcfecFsliai'ng

6'lidi'ry

wed'f

the soil located above the foot-ing st the baclr. of the wall. The h art Pressures scag

retaining wail at instant of failure.side of a retaining wall againstwhich the fillis placed is called the bah of the uralL The back may be

plane or broken, and s plane back may bo vertical or inclined (chaffered).

The failure of s retaining wall can occur by tilting (h75ng failure) or bysliding along its base parallel to its original position (sliding failure).Either typo of failure of tho wall is associated with the downward move-

mcnt of a wedgeshaped body of soil (abcin Fig. 19) located immediatelyback of the wall. This body is called the sliding uredge.

22. Assumptions and conditions. Most of the theories of earthpressure are based on the following assumptions: The backfil of thewall is isotropic snd homogeneous; the deformation of the backfilloccurs exclusively parsUel to a vertical plane at right angles to the back

of the wall, and the neutral stresses in the backfillmaterial are negligible.

Any departure from these fundamental assumptions willbe mentioned

specifically. In this chapter it will be further assumed that the wallmoves to a position which is located entirely beyond the boundary aitrof the shaded ares in Fiigure 14c. This is the deformation condition.

The width of tho shaded area acib in Hguie 14c represents the amount by wh/ch

the horizontal dimensions of tho body of sand abc increue while the saad passes

from its fuiQal state of stress into that of plastic cclullibrium. Ifa hL!eral support

Attachment 4.1

Harding Lawson Associates Report, dated 7/3/96, entitled"Geotechnical Slope Stability Evaluation - ASW System Bypass, Unit 1-

Diablo Canyon Power Plant - San Luis Obispo County, California"

rev 0 to 'asw bypass liquefaction analysis

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