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World Academy of Science, Engineering and Technology
International Journal of Environmental, Chemical, Ecological, Geological and Geophysical Engineering Vol:9, No:12, 2015
IInfluennce off Interm
mediaate Prinncipall Stresss on S
Solutioon
of Plaanar S
Stabilitty Prooblemss
M. Jahananddish, M. B. Zeydabadinej
Z
ad

International Science Index, Geological and Environmental Engineering Vol:9, No:12, 2015 waset.org/Publication/10003155
Abstract—In this paper, voon Mises and Drucker-Prageer yield
criiteria, as typiccal ones that cconsider the effect
e
of interm
mediate
priincipal stress σ2, have been
b
selected and employeed for
invvestigating the influence of σ22 on the solution of a typical stability
s
prooblem. The beearing capacity factors have bbeen calculatedd under
plaane strain condition (strip foooting) and axxisymmetric coondition
(ciircular footing)) using the methhod of stress characteristics toogether
wiith the criteria mentioned.
m
Diffferent levels of σ
σ2 relative to thhe other
tw
wo principal streesses have beenn considered. While
W
a higher σ2 entry
in yield criterion gives a higgher bearing capacity;
c
its enntry in
eqquilibrium equattions (axisymmetric) causes suubstantial reducttion.
Keywords—IIntermediate
axxisymmetric, yieeld criteria.
M
principal
strress,
plane
obtaained by the envelope of ssuch circles iin - diagram
m as
show
wn in Fig. 2. Any failure criterion sugggested for soiil, as
applied to planee strain or axxisymmetric conditions shhould
appear as a funcctional relationnship
or
in
this diagram. Herre,  and n arre the shear annd normal streesses
t failure plaane at failure. R and  are the radius of such
on the
circcle and the distance of its center from the orrigin,
resppectively.
strain,
I. INTR
RODUCTION
ANY stabbility problem
ms in geotechnnical engineeriing are
in plane sstrain or axisyymmetric condditions. An exxample
woould be the sstrip and circcular footingss on the surfa
face of
grround. At the failure
f
state inn these problem
ms; the soil floows in
planes in which the displaccements devellop. In plane strain
coondition; thesee are planes iin which the strains devellop. In
axxisymmetric prroblems; they are redial plannes (Fig. 1).
g
Mohr circle
c
Figg. 2 Failure enveelope obtained bby pushing the greatest
A
As seen; the inntermediate principal stresss 2, can takee any
valuue between 3 and 1 whenn point B movves from C to A in
Fig.. 2. The effectt of 2 and itss variation onn the solution have
not been studiedd before. In triaxial
t
test for
f example;
2 is
eithher assumed to be equall to 3 (com
mpression) orr 1
(exttension). The ambiguity aabout the valuue of 2 andd its
effeects have madde the researchhers to develoop the true triaxial
device [1]. The ratio b is usuaally used to reppresent its efffects.
Thiss ratio is definned as:
(1)
Fig. 1 Planes of flow in a) Plane strain b) axxisymmetric casses
The failure sstate is represented by the Mohr circle hhaving
the largest diaameter, i.e., 1-3. The failure functiion is
Inncrease in 2 from 3 to 1 is equivalennt to increase in b
from
m 0 to 1. It iis noteworthy that there is a unique relaation
betw
ween the Lodee angle , and the stress ratioo b, as [2]:
√
R
Related values of
M. Jahanandishh is with the D
Department of C
Civil and Environmental
Enngineering, Shirazz University, Shhiraz, Iran (correesponding authorr phone:
+998-917-716-7268; fax: +98-713-647-3161; e-mail: [email protected] shirazuu.ac.ir).
M. B. Zeydabaddinejad is with thee Department of Civil and Environmental
Enngineering,
S
Shiraz
Univeersity,
Shirazz,
Iran
(e-mail:
[email protected]).
International Scholarly and Scientific Research & Innovation 9(12) 2015
1368
√
and b aree typically givven in Table I.
scholar.waset.org/1999.6/10003155
(2)
World Academy of Science, Engineering and Technology
International Journal of Environmental, Chemical, Ecological, Geological and Geophysical Engineering Vol:9, No:12, 2015
TA
ABLE I
APPROPRIATE VALUES OF B AND
D

-30
-60
-90
b
1
0.5
0
We begin wiith criteria thaat do not deppend on stresss level
firrst. This is heelpful in realizzing the elabooration that is added
duue to stress levvel dependenccy later. The simplest
s
form is that
suuggested by Tresca [3] iin 1864; whhich is relevaant to
unndrained shearring of saturated clay soils. It can be conssidered
ass a special case of M-C critterion when = 0. This is usually
u
wrritten as:
Fig. 3 Compariison between Trresca and von Mises
M
at uniaxiaal
comprression
1,2
(3)
RvonMises/RTresca
International Science Index, Geological and Environmental Engineering Vol:9, No:12, 2015 waset.org/Publication/10003155
II. STRESSS LEVEL INDEEPENDENT YIEELD CRITERIA
whhere qu is tthe soil streength in uniiaxial comprression
(uunconfined com
mpressive streength).
Writing this ccriterion in thee form of
will resuult in:
(4)
whhere cu is the undrained sheear strength. A
As seen; theree is no
deependency on the intermeddiate principal stress 2, as is the
caase with its parrent M-C criteerion.
The other criterion is the oone suggested by von Misess [4] in
19913. For satuurated clay with strengthh qu in unddrained
coondition it wouuld be in the fo
following form
m:
√
(5)
whhere k is the von Mises coonstant that takkes different values
deepending on thhe condition of the problem..
In contrast to that of T
Tresca; the vvon Mises criterion
deepends on 2. Writing it inn the form off
using b
deefined above, we
w get:
.
1,15
2
√3
1.15
55
1,1
1,05
1
b
0
0,25
0,5
0,755
1
Fig. 4 The ratioo of strengths byy von Mises to tthat of Tresca as
a
function oof 2 and b.
A
As mentioned;; it is customaary in the axissymmetric casse to
assuume 2=3 whhen the flow of
o soil is awayy from the axxis of
sym
mmetry and to assume 2=1 when it is tooward it. These are
knoown as von Kaarman regime. It may be ratiional to assum
me 2
to be
b the averagge of 3 andd 1 in planee strain conddition
becaause, in this case, the floow neither can be considdered
outw
ward nor inwaard. But let us assume it iss a factor K of the
averrage of the othher two princippal stresses soo that:
(8)
(6)
A
Accordingly w
we can write:
The von Misees constant k, is usually obttained from unniaxial
coompression tesst. In such a ttest;
and
are taken eqqual to
zeero. The valuee of b is zero aand the constaant k would bee equal
to the unconfiined compresssive strengthh, qu. If these two
crriteria are set equal
e
at uniaxiial compressioon as it is usuaal (Fig.
3)); envelops off both appear aas straight linees parallel to -axis
onn the - diagrram.
While the raddius of Mohr ccircle at failure given by Treesca is
independent off 2 and b; thhe radius giveen by von Miises is
T ratio of thee radii of thesee two failure ccriteria
fuunction of b. The
in Mohr diagram
m would be a function of b so
s that:
(7)
This functionn is drawn vs. b in Fig. 4.
International Scholarly and Scientific Research & Innovation 9(12) 2015
.
(9)
Iff the material is linear elasstic; K=2; annd if the Poissson’s
ratioo ; is 0.5; K would be 1. The value off b relevant too this
condition would be
b 0.5.
Inn general; we can say K0 iss a reasonablee assumption ffor K
in pplane strain coondition. This is consistent w
with the conddition
=00 relevant to undrained behhavior of satuurated clay which
w
givees K=1 and bb=0.5; i.e.; thhe intermediatte principal stress
s
wouuld be equal to the averaage of the othher two princcipal
stresses.
Iff we assume associated fflow rule andd take the pllastic
poteential functionn g the same aas von Mises yield functionn, we
can investigate thhe plane strainn condition 2=y=0, by seetting
1369
scholar.waset.org/1999.6/10003155
World Academy of Science, Engineering and Technology
International Journal of Environmental, Chemical, Ecological, Geological and Geophysical Engineering Vol:9, No:12, 2015
⁄
equal too zero [5], [6]]. The followiing will result from
this operation onn (5):
International Science Index, Geological and Environmental Engineering Vol:9, No:12, 2015 waset.org/Publication/10003155
0
(10)
This gives bb=0.5 and = -60o; where there is maxximum
deeviation of vonn Mises from Tresca (Fig. 3). Therefore;; if the
sooil behavior obbeys associated flow rule annd the yield criiterion
is von Mises; b should be 0.55 (Fig. 5). Suubstituting thiss value
/√3. If we compaare this with tthat of
off b in (6) givees
Trresca, we cann conclude thaat if these tw
wo criteria are to be
m
matched in planne strain conddition; k shoulld be equal too √3 .
W
We should rem
member that this result has been obbtained
asssuming assocciativity. But the soil behaavior is usuallly not
asssociated and the value of b in this casse may be diffferent
froom 0.5. It is innteresting to ssee the effect of this deviation on
the solution off typical planne strain probblems. The bbearing
wn in Fig. 1 onn a saturated cclay in
caapacity of stripp footing show
unndrained conddition can be considered aas an examplee. The
am
mount the valuue of b is diffferent from 0.55 can be conssidered
ass the degree off nonassociativvity of soil behhavior. The soolution
is obtained usinng the rigorouus method off characteristiccs [2],
[77], [8]. For thee associated caase; the value of bearing caapacity
factor Nc is 5.14. The soluution for nonaassociative caases is
obbtained when different values of b aree put in (6) using
√3 . The result is shown in Fig. 6 and this figure
indicates reducttion in bearingg capacity withh increase in degree
d
off nonassociativvity of soil.
The axisymm
metric case is not as straighht forward as plane
strrain. The princcipal strain 2= is not zeroo in this case tto help
finnding the proper path. Thee b value can vary with diistance
froom the axis off symmetry. A
Another imporrtant point is thhat the
intermediate principal stress enters the eqquilibrium equuations
in this case. Theerefore; in conntrast to the pplane strain casse; the
efffect of 2 andd b in this caase is debatabble even if thee yield
crriterion dos noot include 2. Fig. 7 showss the variationn of Nc
foor a circular fo
footing with b as predicted by Tresca annd von
M
Mises criteria. The startting point oof all curvves is
appproximately Nc=5.7. As (3) indicates; Trresca yield criiterion
dooes not dependd on 2. Therefore; the curvve related to Tresca
T
in Fig. 7 indiicates that enntry of higher values off b in
eqquilibrium equuations causes reduction inn bearing cappacity.
Thhis is also cleaar when we ccompare the cuurves related tto von
M
Mises. The dashhed line curvee indicates thatt the entry of higher
vaalues of b in yyield criterionn has an increeasing effect oon Nc;
buut when their eeffect in equillibrium equatiions is also alllowed;
their increasing effect is substtantially ceaseed (full line cuurve).
Fig. 5 Comparisoon between Tresca and von Miises at plane straain
conddition
Nc
Smoooth Strip Footinng on Saturated clay
5,14
5,1
5,06
5,02
4,98
b
4,94
0,25 0,33 0,35 0,4 0,45 00,5 0,55 0,6 0,65 0,7 0,75
Fig. 6 Effect off variation of b on Nc factor off a saturated clayy
Nc
Smoooth Circular Foooting on Saturatted Clay
b=0 inn
equil.eeq.
(V)
6,6
6,4
6,2
5,8
b>0 inn
equil.
V)
Eq. (V
5,6
Trescaa
6
5,4
5,2
5
0
0,1
0,2
0,3
0,4
0,55
b
Fig. 7 Effecct of b on Nc viaa yield and equiilibrium eqs.
III. STRESS LEVEL DEPPENDENT YIELD CRITERIA
W
We have seleected Mohr-C
Coulomb (M--C) and DrucckerPragger (D-P) yield criteria for oour investigatiion in this partt [9];
but the discussionn can be extennded to Lade-D
Duncan, MasuuokaNakkai and others as well. Heree, the M-C cann be considereed as
an eextension of T
Tresca becausee it does not ddepend on 2. D-P
is sttress level deppendent form of von Mises and considerrs 2.
For the sake of clarity; we lim
mit our discuussion to frictiional
soil. The M-C in this case can bbe written as:
(11)
whiich can be easiily written in
International Scholarly and Scientific Research & Innovation 9(12) 2015
1370
scholar.waset.org/1999.6/10003155
form
m as:
World Academy of Science, Engineering and Technology
International Journal of Environmental, Chemical, Ecological, Geological and Geophysical Engineering Vol:9, No:12, 2015
The D-P criteerion in the abbsence of coheesion can be w
written
ass [9], [10]:
(12)
whhere I1 is the first invariant of stress tenssor and
paarameter. It caan be easily shhown that thee
D--P in this casee, is:
is thhe D-P
foorm of
(13)
International Science Index, Geological and Environmental Engineering Vol:9, No:12, 2015 waset.org/Publication/10003155
√
Comparison w
with similar foorm of M-C, inndicates that:
√
√
the D-P parameteer , on the basis of (15). W
We then substtitute
this value of  inn (14) and finnd an equivallent value for  in
term
ms of b. In thiis way; the vaariation of b aaffects the matterial
equivalent strenggth parameter . Values off b different from
whaat is given by (16) will resuult in equivaleent friction anngles
lesss than the origginal. This is cclear from Figg. 8 as we seee that
the D-P circle liees inside of M
M-C hexagon. Fig. 9 showss the
variiation of bearinng capacity faactors Nq and N for smooth strip
footting with b when =30. The value of b relevannt to
assoociative behaavior in this case is 0.755 and the fiigure
indiicates lower bbearing capacitty factors wheen the behavioor of
soil is nonassociated; i.e.; w
when b is diffferent from 0.75.
0
Theerefore; it is nnot conservativve to go with the usual beaaring
capaacity factors iff the soil behaavior is really nnonassociatedd.
(14)
Nq
Strip Footing D-P (=30)
18,9
whhich gives thee D-P parameeter , for maatching with M
M-C in
geeneral.
Assuming thhe behavior tto be associaated; we agaain set
⁄
equal to zero [3], [[4] to get thee matching in plane
strrain conditionn. This operatioon on (12) willl result in:
18,4
17,9
18,4
4069
17,4
16,9
16,4
b
15,9
(15)
15,4
0,5 0,55 0,6
0
0,65 0,7 0,75 0,8 0,85 0,9 0,95
1
Applying thee requirementt for general m
matching exppressed
byy (14) to this equation
e
resultts in:
((a)
(16)
Nγ
whhich gives thee value of b inn terms of  ffor matching oof D-P
wiith M-C in pplane strain condition.
c
In other words; both
crriteria predict the
t same strenngth for the soil if the value of b is
asssumed to be thhat given by (16).
(
Fig. 8 shoows the matchhing of
D--P and M-C inn plane strain condition.
c
Strip Footingg D-P ( =30)
7,9
7,65
7,4
7,,657
7,15
6,9
6,65
6,4
b
6,15
0,5 0,55 0,6 0,65 0,7 00,75 0,8 0,85 00,9 0,95
1
(b)
Fig. 9 Effect of varriation of b on ((a) Nq and (b) N for strip footinng (
=330o)
Fig. 8 Matchinng of Drucker-P
Prager and Mohhr-Coulomb critteria
The actual vaalue of b can bbe different froom what is givven by
(16) due to nonnassociativity in soil behavvior. To investigate
the effect of vvariation of b on the soluttion of plane strain
prroblems like bbearing capacitty; we can asssume we have found
International Scholarly and Scientific Research & Innovation 9(12) 2015
A
As mentionedd before; thee intermediatee principal stress
s
partticipates in thhe equilibrium
m equations in case of axial
sym
mmetry. In ordder to see the eeffect of its enntry in equilibrium
equations on beaaring capacityy calculations;; we have chhosen
the M-C criterionn because it dooes not includde 2. It is usuual to
assuume 2=3 in this conditionn i.e.; b=0. Buut this is merelly an
assuumption and it
i is interestinng to see if 2 is really greater
thann 3; how aree the values oof Nq and N influenced
i
byy this
mattter. The resultt of investigattion for  = 300o is demonstrrated
by tthe curve draw
wn for M-C iin Fig. 10. Thhe curve indiccates
entrry of higher values of b in equilibrium eqquations resullts in
1371
scholar.waset.org/1999.6/10003155
International Science Index, Geological and Environmental Engineering Vol:9, No:12, 2015 waset.org/Publication/10003155
World Academy of Science, Engineering and Technology
International Journal of Environmental, Chemical, Ecological, Geological and Geophysical Engineering Vol:9, No:12, 2015
lower values for bearing capacity factors, Nq and N. Matching
of other yield criteria like D-P with M-C under axial
compression relevant to bearing capacity problem of circular
footings is made at the top point of Fig. 8, where b is zero. For
other points; the D-P circles lies outside of the M-C hexagon.
This indicates higher strength with increase in b. This effect is
demonstrated by the dashed line curve in Fig. 10. When the
effect of 2 being greater than 3 is considered both in
equilibrium and yield equations; the full line curve of Fig. 10
is obtained. Comparison of these two curves indicates again,
the reduction in Nq and N due to participation of 2 in
equilibrium equations. The full line curve indicates the net
effect has been increase in bearing capacity factors in case of
 = 30o.
Nq
Circular Footing (=30)
120
b=0 in equil.eq. (D-P)
105
b>0 in equil. Eq. (D-P)
90
plane strain and axisymmetric conditions. It was found that in
plane strain condition; the value of 2 may be different from
what is usually assumed and this may be due to nonassociativity in soil behavior. It was shown that under such a
condition, a lower bearing capacity is obtained. Therefore; the
usual solution to bearing capacity of strip footings would not
be conservative.
In the axisymmetric conditions however; 2 enters the
equilibrium equations as well; and if the Mohr-Coulomb
criterion is employed for bearing capacity calculation of
circular footings; increase in 2 relative to other principal
stresses results in decrease in bearing capacity. Therefore; if in
reality, 2 is greater than 3; this effect should be considered
in calculations. If other criteria that include 2 are used instead
of Mohr-Coulomb; the calculated bearing capacity may be less
or more than what is usually calculated, depending on the
criterion used, and the amount 2 has been taken more than 3.
In case of Drucker-Prager for example; the net effect has been
shown here to be a little increase in the calculated bearing
capacity.
M-C
75
REFERENCES
60
[1]
45
30
b
15
0
0,1
0,2
0,3
0,4
(a)
Circular Footing (=30)
N
25
23
21
19
17
15
13
11
9
7
5
3
b=0 in equil.eq. (D-P)
b>0 in equil. Eq. (D-P)
M-C
b
0
0,1
0,2
0,3
P. V. Lade “Assessment of test data for selection of 3-D failure criterion
for sand.” Int. J. Numer. Anal. Meth. Geomech, 2006; 30:307–333.
[2] R. O. Davis and A. P. S. Selvadurai, Plasticity and Geomechanics.
Cambridge University Press, 2002, 287 p.
[3] H. Tresca, Sur l’ecoulement des corps solids soumis `a de fortes
pression, Comptes Rendus Acad. Sci. Paris, 1864, 59, 754.
[4] R. von Mises, Mechanik der festen Koerper im plastisch-deformablen
Zustand, Nachr. d. K. Ges. d. Wiss G¨ottingen, Math.-Phys. Kl., 1913,
582–592.
[5] V. G. Solberg “Development and Implementation of Effective Stress
Soil Models.” M.Sc. Thesis, Under Supervision of Dr. S. Nordal,
Department of Civil and Transport Engineering, Norwegian University
of Science and Technology. (2014).188 pp.
[6] G. Vikash, A. Prashant “Calibration of 3-D Failure Criteria for Soils
Using Plane Strain Shear Strength Data.” GeoShanghai 2010
International Conference, pp.86-91.
[7] M. E. Harr, Foundation of Theoretical Soil Mechanics. McGraw-Hill,
1966, New York, 381 p.
[8] C. M. Martin, “Exact bearing capacity calculations using the method of
characteristics.” Turin 4, 2005, pp. 441-450.
[9] H. Jiang, Y. Xie “A note on the Mohr-Coulomb and Drucker-Prager
strength criteria.” Mech. Research Comunications, Vol. 38, 2011, pp.
309-314.
[10] D. C. Drucker, W. Prager, ‘Soil mechanics and plastic analysis or limit
design’, Quarterly of applied mathematics 10(2), 157–165. (1952).
0,4
(b)
Fig. 10 Effect of variation of b on (a) Nq and (b) N for circular
footing
IV. CONCLUSION
Tresca and Mohr-Coulomb yield criteria have long been
criticized for not considering the effect of intermediate
principal stress. As typical stress level- dependent and
independent yield criteria that consider the effect of 2; von
Mises and Drucker-Prager were selected for investigating the
effect of 2 on solution of bearing capacity problem under
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