M 620.6 L521F 't'O 4 ';,'. '\ - :J'. \,' ' - ---------------------------------...=m==I Soil Mechanics, and Theories of Plasticity ON TH· LIMI NAL YSIS OF TA ILITY OF SLOPES by W. F. Chen M. W. Giger H. Y. Fang Fritz Engineering Laboratory Report No. 355.4
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M620.6L521F't'O 3~1: 4';,'. '\ -
:J'. \,' "-"~-'.' '
- ---------------------------------...=m==I
Soil Mechanics, and Theories of Plasticity
ON TH· LIMI NALYSIS OFTA ILITY OF SLOPES
byW. F. ChenM. W. GigerH. Y. Fang
Fritz Engineering Laboratory Report No. 355.4
Soil Mechanics, and Theories of Plasticity
ON THE LIMIT ANALYSIS OF STABILITY OF SLOPES
by
w. F. ChenM. W. GigerH. Y. Fang
This work has been carried out as part of aninvestigation sponsored by the EnvirotronicsCorporation.
Fritz Engineering LaboratoryDepartment of Civil Engineering
Lehigh UniversityBethlehem, Pennsylvania
March 1969
Fritz Engineering Laboratory Report No. 355.4
On the Limit Analysis of Stability of Slopes
By
w. F. Chen*, M. W. Giger* and H. Y. Fang*
ABSTRACT
The upper bound theorem of the generalized theory
of perfect plasticity is applied to obtain complete numerical
solutions for the critical height of an embankment. A rota-
tiona! discontinuity mechanism (logarithmic spiral) is assumed
in the a.nalysis. The analysis· includes the existing solutions
as a special case, so that it may be considered a generaliza-
tion of previous solutions.
* Assistant Professor of Civil Engineering, Research Assistantand Director of Geotechnical Engineering Division, respectivelyFritz Engineering Laboratory, Lehigh University.
TABLE OF CONTENTS
ABSTRACT Page
1. INTRODUCTION. 1
2 • CRITICAL HEIGHT· OF AN EMBANKMENT................. 4
Fig. 3 STABILITY FACTORS N , AS A FUNCTION OF SLOPE ANGLE SS
8.50
8.00
403510 15 20 25 30SLOPE ANGLE a IN DEGREES
5
----
-...-.....-.¢ =400
-
..........~
-
------ ep =350
-~
ep =300- -------.........- ep :II: 250
...............- I I~¢a200
...............~ ~ {j ~oo¢ =150
-- I
- ¢ =100
_ ¢ =50
""-""'l
I
7.00
5.50
6.50
4.00o
6.00
5.00
4.50
a:oIo<[I.L
>I-....Jmi=!(J)
>"10 7.50o
:I:II
enZ
Fig. 4: STABILITY FACTORS N , AS A FUNCTION OF SLOPE ANGLE as
c
H
Hc
L
N s
r ,r (8)o
v (8)
7 • NOMENCLATURE
cohesion
function defined in Eq. (4)
function defined in Eq. (5)
function defined in Eq. (6)
function defined in Eq. (9 )
vertical height of an embankment
critical height of an embankment
length ABo in ·Fige 1
stability factor
length variables of a log spiral curve
discontinuous velocity across the failure plane
angular parameters of an embankment
unit weight of soil
angular velocity
friction angle of soil
angular variables of a log spiral curve
- 12 -
8. REFERENCES
Chen, W. F. and C. R. Scawthorn (1968). Limit analysisand limit equilibrium solutions in soil mechanics,Fritz Engineering Laboratory Report No. 355.3, Lehigh.University, Bethlehem, Pa.
Chen, W. F. (1968). Discussion of application of limitplasticity in" soil mechanics by W. D. Liarn Finn, AmericanSociety of Civil Engineers, Journal of the Soil Mechanicsand Foundations Division, Vol. 94, No. SM2, AmericanSociety of Civil Engineers.
Chen, W. F. (1969). Soil mechanics and the theorems oflimit analysis, Journal of the S~il Mechanics andF~undations Dlvision, American Society of Civil Engineers,Vol. 95, No. 8M2.
Cox, A. D., G. Eason and H. G. ijopkins (1961). Axiallysymmetric plastic deformation. in soils/transaction, RoyalSociety of London, Vol. 254 A, p. 1.
Drucker, D. C., W. Prager and H. J. Greenberg (1952). Extendedlimit design theorems for continuous media, Quarterly ofApplied Mathematics, Vol. 9, pp. 381-389.
Drucker, ·D. C~ and W. Prager (1952). Soil mechanics and plasticanalysis or limit design, Quarterly of Applied Mathematics,Vol. 10, pp. 157-165.
Drucker, D. C. (1953). Limit analysis of two and three dimensional soil mechanics problems, Journal of the Mechanics andPhysics of Solids (Oxfor~), Vol. 1, No.4, pp. 217-226.
Drucker, D. C., R. E. Gibson and D. J. Henkel (1957). Soilmechanics and work-hardening theories of plasticity, Proceedings, American Society of Civil Engineers, Vol. 81, Separate798, 1955; Transactions; Vol. 122, pp. 338-346.
Fellenius, w. (1927). Erdstatische Berechnungen. Berlin:W, Ernst und Sohn.
Finn, Liam W. D. (1967). Application of limit plasticity insoil mechanics, Journal of the Soil Mechanics and FoundationsDivision, American Society.of Civil Engineers, Vol. 93, No. SM5,September, 1967, pp. 101-120.
Jenike, A. W. -and R. T. Shield (1959). On the plastic flow ofcoulomb solids beyond original failure, Journal of AppliedMechanics, Vol. 26, pp. 599-602.
Shield, R. T. (1954). Mixed boundary value problems in soilmechanics, Quarterly of Applied Mathematics, Vol. 11, No.1,pp. 61-76
- 13 -
Shield, R. T. (1955). On Coulomb's law of failure in soils,Journal of the Mechanics and Physics of Solids, Vol. 4,No.1, pp. lO-16e
Spencer, A. T. M. (1964)0 A theory of the kinematics ofideal soils under plane strain conditions, Journal ofMechanics and Physics of Solids, Vol. 12, pp. 337-351.
Terzaghi, K. (1943). Theoretical Soil Mechanics. New York:John Wiley and Sons.
Taylor, D. W. (1948). Fundamentals of Soil Mechanics. NewYork: John Wiley and Sons
Weidler, T. B. and Paslay (1966). An analytical descriptionof the behavior of granular media, Engineering, BrownUniversity Report 36, Arpa SD-86.
1. Varrin, R. D. ,and Fang, H. Y.DESIGN AND CONSTRUCTION OF HORIZONTAL VISCOUSFLUID MODEL, Ground Water, Vol. 5, No.3, July1967, (Fritz Laboratory Reprint No. 67-10)
2. Fang, H. Y. and Schaub, J. H.ANALYSIS OF THE ELASTIC BEHAVIOR OF FLEXIBLE PAVEMENT,Proc. 2nd International Conference on the StructuralDesign of Asphalt Pavements, pp. 719-729, August 1967,(Fritz Laboratory Reprint 'No. 67-10)
3. Chen, W. F.DISCUSSION OF APPLICATION OF LIMIT PLASTICITY INSOIL MECHANICS by W. D. Liam Finn, ASeE Proe., Journalof Soil Mechanics and Foundation Division, Vol. 94,8M2, March 1968, (Fritz Laboratory Reprint No. 68-7)
*4. K'rugmann, P., Boschuk, J., Jr. and, Fang, H. Y.ANNOTATED BIBLIOGRAPHY ON STEEL SHEET PILE STRUCTURES,March 1968, (Fritz Laboratory Report No. 342.1)
5. Drucker, D. C. and Chen, W. F.ON THE USE OF SIMPLE DISCONTINUOUS FIELDS TO BOUNDLIMIT LOADS, "Engineering P1asticity" edi te,d byJ. Heyman and F. A. Leckie, Cambridge University Press,pp. 129-145, March' _1968 (Fri tz Laboratory F.eprint No.68-8)
*6. Chen, W. F. and Scawthorn, C.LIMIT ANALYSIS AND LIMIT EQUILIBRIUM SOLUTIONS IN SOILMECHANICS, June 1968, (Fritz Laboratory Report No. 355.3)
7. Fang, H. Y. and Brewer, C. E.FIELD STUDY OF AN ANCHORED SHEET PILE BULKHEAD, ASeEAnnual Meetinq, Pittsburg, ·Pa., October 1968, (ASCEPreprint No. 749)
*8. Chen, W. F.EXTENSIBILITY OF CONCRETE OR ROCK AND THE THEOREMS OFLIMIT ANALYSIS, November 1968, ~ritz Laboratory ReportNo. 356.5)
*9. Moore, C. A. and Hirst, T. J.A LOW FRICTION SEAL FOR THE TRIAXIAL TEST LOADINGPISTON, December 1968, (Fritz Laboratory Report No.348.3)
*10. Hirst, T. J. and Mitchell, J. KoTHE INFLUENCE OF SUSTAINED LOAD ON SOIL STRENGTH,December 1968, (Fritz Laboratory Report No. 348.4)
*Not for general distribution
11. Chen, W. F.ON THE RATE OF DISSIPATION OF ENERGY OF SOILS,January 1969, Soils and Foundations, The JapaneseSociety of Soil Mechanics and Foundation Engineering,Vol. VIII, No.4, pp. 48-51, December 1968, (FritzLaboratory Reprint No. 68-24)
*12. Dismuke, T. D.FIELD STUDY OF SURCHARGE EFFECTS ON A STEEL SHEETPILE BULKHEAD, December 1968, (Fritz LaboratoryReport No. 342.4)
*13. Fang, H. Y. and Varrin, R. D.MODEL ANALYSIS OF HYDRAULIC CONDUCTIVITY OF ANACQUIFER, December 1968, -(Fri tz Laboratory ReportNo. 341.2) ,
14. Fang, H. Y.INFLUENCE OF TEMPERATURE AND OTHER CLIMATIC FACTORSON PERFORMANCE OF SOIL-PAVEMENT SYSTEMS, InternationalSymposium on the Effects of Temperature and Heat onthe Engineering Behavior of Soils, January 1969,(Fritz Laboratory Reprint No. 69-2)
*15. Brewer, C. E. and Fang, H. Y.FIELD STUDY OF SHEAR TRANSFER IN STEEL SHEET PILING,February 1969, (Fritz Laboratory Report No. 342.3)
16 '. Chen, W. F.SOIL MECHANICS AND THE THEOREM OF LIMIT ANALYSIS,ASCE Prac., Journal of Soil Mechanics and FoundationDivision, Vol. 95, No. 8M2, March 1969, (FritzLaboratory Reprint No. 69-4)
*17. Chen, W. F., Giger, M. and Fang, H. Y.ON THE LIMIT ANALYSIS OF STABILITY OF SLOPES, March 1969,(Fritz Laboratory Report No. 355.4)
18. Corman, R. E., Jr. and Miller, F. H.MODIFIED RUT DEPTH GAUGE, Envirotronics Bulletin No.2,March 1969. (available from Envirotronics Corporation,P. O. Box 594, Bethlehem, Pa. 18015)
19. Chen, W. F. and Drucker, D. C.THE BEARING CAPACITY OF CONCRETE BLOCKS OR ROCK, ASCEProc., Journal of Engineering Mechanics Division, Vol. 95,No. EM2, April 1969, (Fritz Laboratory Reprint No. 69-1)
20. Chen, W. F.THE PLASTIC INDENTATION OF METAL BLOCKS BY A FLAT PUNCH,October 1968, to be presented at the Plasticity Sessionof the Louisville, Ky. ASeE National Structural Conference, April 14-18, 1969