UPORABA TEOREMA KOR6SPOND6NCNIH STANJ PRI IZRAČUNU ZGORNJIH VREDNOSTI PASIVNIH ZCMCLJSKiH TLAKOV_
BORUT MACUH Ln STANISLAV ŠKRABL
o avtorjih
Borut Macuh
Univerza v Mariboru,
Fakulteta za gradbeništvo
Smetanova ulica 17, 2000 Maribor, Slovenija
E-pošta: borut.macuh@uni-mb.si
Stanislav Škrabl
Univerza v Mariboru,
Fakulteta za gradbeništvo
Smetanova ulica 17, 2000 Maribor, Slovenija
E-pošta: stanislav.skrabl@uni-mb.si
izvleček
Rešitve nekaterih mejnih stanj (nosilnost temeljnih tal, aktivnih in pasivnih zemeljskih pritiskov itd.) za čiste-trenjske zemljine so za večji del praktičnih primerov enostavnejše v primerjavi z rešitvami enakih primerov za kohezijsko-trenjske zemljine. Teorem of Corresponding States nam v nekaterih primerih omogoča doseganje rešitev pripadajočih mejnih stanj za kohezijsko trenjske materiale s transformacijo že poznanih rešitev robnih elasto-plastičnih rešitev mejnih stanj za trenjske nekohezijske zemljine. Veljavnost oz. uporabnost osnovnega transformacijskega teorema (Caquot 1934) je omejena in velja le za enostavnejše primere mejnih stanj, kjer so napetostni vektorji pravokotni na robne površine ter kadar se pri transformaciji ohranjajo smeri trajektorij glavnih napetosti (Michalowski 2001). Pri aplikaciji kinematičnega pristopa mejne analize na osnovi teorema zgornje vrednosti je za primer določanja pasivnih zemeljskih pritiskov dokazano, da je za izbrani kinematični model tudi v splošnih primerih robnih napetosti dopustna posredna uporaba teorema korespondenčnih stanj v spremenjeni obliki. Rezultati opravljenih analiz pasivnih zemeljskih pritiskov kažejo, da je v splošnejših primerih nekritična uporaba teorema korespondenčnih stanj v osnovni obliki nedopustna, ker so dobljeni rezultati lahko pravilni le naključno ter v odvisnosti od robnih pogojev lahko pomenijo precenjene ali podcenjene vrednosti pasivnih zemeljskih pritiskov v geoteh-nični praksi.
Ključne besede
korespondenčno stanje, zemeljski tlak, pasivni tlak, mejna analiza, zgornja vrednost
16. ACTA GeOTeCHNICA SLOVENICA, 2010/2
PASSIVE EARTH PRESSURE DETERMINATION: APPLICATION OF THE CORRESPONDING STATE THEOREM FOR CALCULATING UPPER-BOUND VALUES
BORUT MACUH and STANISLAV ŠKRABL
About the authors
Borut Macuh
Univerza v Mariboru,
Fakulteta za gradbeništvo
Smetanova ulica 17, 2000 Maribor, Slovenija
E-mail: borut.macuh@uni-mb.si
Stanislav Škrabl
Univerza v Mariboru,
Fakulteta za gradbeništvo
Smetanova ulica 17, 2000 Maribor, Slovenija
E-mail: stanislav.skrabl@uni-mb.si
Abstract
The validity of some limit state solutions, when strictly applied to the basic corresponding state theorem (Caquot, 1934), is limited and valid only for simpler limit states, where stress vectors are either perpendicular to the boundary surfaces or when the direction of stress eigenvalue trajectories in transformation are preserved (Michalowski, 2001). The theorem of corresponding states allows us, in some cases, to attain solutions belonging to the limit states for cohesive-friction materials with the transformation of the known boundary of elasto-plastic solutions of limit states for pure friction materials. We demonstrated that for the selected kinematically admissible model, in general cases of boundary stresses, the indirect application of the corresponding states theorem in modified form is permitted. To determine this, we applied the kinematic approach of limit state analysis and used the upper-bound theorem for determining passive earth pressures. The results of our analyses show that incautious application of the corresponding state theorem in its basic form and for general cases is inadmissible because the results obtained can be correct only coincidentally, depending on the boundary conditions.
Keywords
corresponding state, earth pressure, passive pressure, limit analysis, upper-bound
1 INTRODUCTION
The corresponding state theorem (Caquot, 1934) is based on the fact that for a considered boundary problem the stress state of cohesive-friction soils in a limit state is similar to the sum of the stress states of the same boundary problem for non-cohesive soils and hydrostatic pressure p = c/tan 0 . The solutions of equal elasto-plastic boundary problems at limit states of cohesive-frictional and pure frictional material are undeniably similar. However, for more general and more complex boundary problems it is necessary to apply more exacting transformation relations to obtain solutions of limit states for cohesive-frictional materials, such as an inclined back fill or boundary conditions that require the transformation of limit state solutions for non-cohesive soils. The solutions of these limit states (bearing capacity of foundation ground, active and passive pressures, etc.) for pure friction soils are simpler for most practical examples compared with solutions of the same examples for cohesive-friction soils.
With the advancement of mathematical knowledge and numerical methods the practical significance of the corresponding state theorem has been reduced. However, it can frequently be found useful in the field of limit states, in investigating active and passive earth pressures and ground bearing capacities. Many authors, including Caquot (1934), Michalowski (2001) and Silvestri (2006), have suggested that there are limitations to applying the theorem in its basic form. In the past the corresponding state theorem was typically applied uncritically or unacceptably: Caquot and Kerisel (1948), Soubra and Regenass (2000), Škrabl and Macuh (2005), Vrecl-Kojc and Škrabl (2007) and many other authors.
The most practical use of the corresponding state theorem in limit state analysis using the upper-bound theorem most frequently occurs in three-dimensional cases where the transformation of known solutions compensate extensive integrations along individual discontinuity surfaces of deformation velocities. In the analyses of two-dimensional cases of limit-state analysis,
16. ACTA GeOTeCHNICA SLOVENICA, 2010/2
B. MflCUH & S. SKRABL: PASSIVE EARTH PRESSURE DETERMINATION: APPLICATION OF THE CORRESPONDING STATE THEOREM FOR CALCULATING
it is most successfully applied to control the results of mathematical analyses.
This article describes the procedure of determining the limit values of passive earth pressures for two-dimensional cases using the kinematic model of limit states with the upper-bound theorem. A comparison of several results of passive earth pressure coefficients, determined using the procedure of Kérisel and Absi (1990), shows that applying the corresponding state theorem, in its original form, to more general situations is not admissible.
2 KINEMATIC FAILURE MECHANISM
Figure 1 describes a general two-dimensional example of a rigid inclined wall having inclination a, height h with inclined backfill fi. The kinematical failure mechanism comprises n triangular rigid blocks. As presented in Figure 1b, the kinematically admissible deformation velocities of individual blocks act in directions that enclose angle 0 with individual discontinuity lines dt (i=1,2,.. .n). The velocities of individual rigid blocks are uniformly defined by the condition that relative velocity directions between individual rigid blocks should enclose angle 0 with lateral contact surfaces (i=1,2,___n). The ho dograph of individual rigid blocks is shown in Figure 1c.
The velocities of the whole failure mechanism can be uniformly determined from the chosen value of the deformation velocity of the first rigid block:
y 1 = 1 Vi+1 = V
sin/*+1 + a)
'sin(3w+1 + a+1) '
Vi +1 = V-
sin(a+1 - a)
(1)
sin(b,i+1 + a+1)
The resultant value of passive earth pressures (Pp) is defined by equation two:
h2
Pp = Kpg— + KPcCh + Kpqqh (2)
where Kpy denotes the coefficient of passive earth pressures due to soil self-weight, y denotes the soil unit weight, Kpc denotes the coefficient of passive earth pressures due to cohesion (c) and Kpq is the coefficient of passive earth pressures due to the surcharge q. The passive pressure distribution along a wall height for a part that belongs to soil self-weight is triangular, while
the part that belongs to cohesion and surcharge is rectangular or constant along the wall height.
This paper assumes that the backfill soil fulfills the Mohr-Coulomb yield criterion with the associative plastic flow rule (normality principle). The change of energy dissipation per volume unit of backfill soil can be
evaluated by (Michalowski, 2001): • • • •
D = -evc cos f = —(ei +63)0 cotf (3)
where 61 and 63 denote major and minor eigenvalues of strain rate; 6v rate of volumetric strain deformation, and 0 and 0 represent the cohesion and angle of inner friction of backfill soil.
3 WORKING EQUATION
For soils that follow the associative flow rule, the change of inner energy dissipation is never lower than the change of work of outer forces for an arbitrary kinematically admissible failure mechanism (Fig. 1):
f b('ej )dV = — [sin(a-b-^—Vn -
•J	tan f	cos 3
cos/3
h • h2
cos(a+ a)-V1 ]> g— K [cosdcos(a + a) —
cos a	2
sindsin(a + a) ]V 1 + chKpc [cosdcos(a + a) — sindsin(a + a) ]V 1 + qhKpq [cosdcos(a + a) —
• v^	•	l	•
sindsin(a + a) ]V1—> sin a Vi — q-sinanVn
1	=1	cos3 n
(4)
where V denotes the total volume of the failure mechanism. Provided that deformation velocity V1 = 1 and the generalized wall height h* = 1 equation 4 leads to:
—c— [sin( an — 3)—1— Vn — cos(a+a)—1— ]> tanf	cos/	cos a
Kp	*
cos(d+ a + a) + c Kpc cos(d+ a + a) +
* ^ * • * f q K cos(d + a + a) — > G* sin aVi — q -sin an Vn
q	i=\	cos/
(5)
* c	* q
where c = — and q =— denote normalized cohesion gh gh * G * * * l and normalized surcharge; G* = —= g V* and l = —
gh h
normalized weight of the individual triangular block of
48. ACTA GEOTECHNICA SLOVENICA, 2010/2
b. mflcuh & s. skrabl: passive earth pressure determination: application of the corresponding state theorem for calculating
backfill soil and the normalized length of the failure line	unit weight and unit height of the wall and the appurte-
(surface); y* = 1, h* = 1 and V* denote the generalized	nant volume of individual soil blocks.
Figure 1. Translational failure mechanism; (a) geometry, (b) absolute and relative velocities of individual rigid blocks and (c) hodograph.
16. ACTA GeOTeCHNICA SLOVENICA, 2010/2
B. MflCUH & S. SKRABL: PASSIVE EARTH PRESSURE DETERMINATION: APPLICATION OF THE CORRESPONDING STATE THEOREM FOR CALCULATING
4 NUMERICAL ANALYSES AND RESULTS
The original failure mechanism is completely defined by n coordinates that define the individual blocks (Fig. 1). They have to be selected in a way that ensures that the original failure mechanism is kinematically admissible. In numerical analyses using the process of mathematical optimization, the critical kinematical admissible failure mechanism is obtained by minimizing equation 6:
mation theorem (Caquot, 1934) that is used for passive pressure states given in equation 11.
Kpc ---ff
tanf
Kpq--d
cos 0
(11)
A comparison of equations 10 and 11 shows that the original transformation of equation 11 is applicable only for the simplest cases of passive earth pressure on vertical walls. Such cases do not consider the friction between the wall, backfill soil and horizontal backfill.
f - KPg + c * Kp + q * Kp -Y G*sina V, 2 pc ^ pq ^cosd+a + a)
l
cos bcos( d + a + a)
-sin a Vn +
[sin( a -b
l
tanf cos(a + a)
cosb cos( d+ a + a) 1
-Vn -
cos acos( d+ a + a) (6)
Where f represents the objective function of the optimization problem. The coefficients of passive earth pressures are defined by equations 7, 8 and 9.
Kp --Yg* sin a V. (7)
pl cos(d+ a + a) tf '
KPq -
l*
cos b cos(d + a + a) 1
-sin a Vn
(8)
K --
tanf cos(d+ a + a)
l*
[sin( a -b)-- Vn -
cosb
cos(a+a)- ]
cos a
(9)
Using equation 8, the coefficient of passive earth pressures, due to cohesion, can be given in the following form:	. „
^ =—{ Kpq(cosfi-—^)-
pc
tanf
tana
1
cosa[cosd- tan(a + a)sind] (10)
}
Equation 10 represents the transformation rule for determining the coefficient of passive earth pressures for
cohesive-frictional soils Kpc from the known and, as a
pc
rule, easier solutions for pure friction soils Kpq. We can establish that equation 10 is valid only for the selected failure mechanism and differs from the original transfor-
We numerically analyzed the kinematical admissible failure mechanisms using n = 30 triangular soil blocks (Fig. 1).
Equation 6 shows that for different ratios of generalized unit weights of soil blocks y*, the surcharge intensities q* and soil cohesions c*, we were able to obtain different geometries of the critical failure mechanism through the process of mathematical optimization. This enabled us to determine the lowest total value of passive pressures Pp.
Table 1 represents a comparison of passive earth pressure with coefficients Kpc calculated using equations 9 and 10 with the original transformation in equation 11. This was done in accordance with the procedure of Kéri-sel and Absi (1990). In the procedure of mathematical optimization, we first analyzed the equal conditions Y* = c* = 0 and q* > 0 that were considered in the method used by Kérisel and Absi (1990). For backfill soil analysis, we considered the inner friction angle 0 = 35°, 8 = 0/2, a = 0, and fi = 30° to 35° in increments of 5°.
Furthermore, we used the same set of coefficients of passive earth pressures and applied them to the cohesion for three different combinations of influences on soil unit weight, cohesion and surcharge (Table 1).
The calculations of coefficient Kpc using equations 9 and 10 give exactly the same results for all kinematical admissible failure mechanisms.
5 CONCLUSIONS
The results of our numerical analyses show that it is not admissible to determine the coefficient of passive earth pressures Kpc to cases of friction between wall and soil and inclined backfills when applying the original transformations according to the corresponding state theorem (Caquot, 1934). The results of the original transformations usage can indicate overestimated or underestimated values of passive earth pressures in geotechnical practice.
48. ACTA GEOTECHNICA SLOVENICA, 2010/2
B. MflCUH & S. SKRABL: PASSIVE EARTH PRESSURE DETERMINATION: APPLICATION OF THE CORRESPONDING STATE THEOREM FOR CALCULATING
Table 1. Comparison of passive pressure coefficients Kpq and Kpc obtained with the results of calculations using the method of Kérisel and Absi (1990) for f = 35°, S = f/2 and a = 0.
Rackfill	Kinematical model n-		30	Kérisel and Absi		Kinematical model n-		30
inclination		Y* =	c*=0 and q* > 0			y*=q*=0 and c*>0	c*=q*=0 and y*>0	c*=q*=0.2 and y*=1
P (°)	Kpq (8)	Kpc (11)	Kpc (9)	Kpq	Kpc (11)	Kpc (9)	Kpc (9)	Kpc (9)
-30	1.610	0.803	3.532	1.505	0.652	2.301	3.532	2.604
-25	2.153	1.577	3.376	2.092	1.490	2.764	3.371	2.975
-20	2.757	2.439	3.690	2.739	2.415	3.332	3.687	3.479
-15	3.474	3.464	4.255	3.448	3.427	4.038	4.222	4.125
-10	4.282	4.617	4.979	4.264	4.541	4.841	4.915	4.877
-5	5.179	5.898	5.824	5.155	5.864	5.747	5.786	5.760
0	6.167	7.310	6.824	6.116	7.237	6.769	6.839	6.791
5	7.246	8.851	7.985	7.220	8.814	7.920	8.203	7.986
10	8.412	10.516	9.338	8.368	10.454	9.221	9.630	9.375
15	9.657	12.294	10.923	9.615	12.235	10.688	11.413	10.986
20	10.962	14.158	12.801	10.929	14.112	12.346	13.605	12.853
25	12.298	16.067	15.068	12.270	16.025	14.215	16.277	15.053
30	13.620	17.956	17.955	13.605	17.933	16.331	19.736	17.594
35	14.823	19.673	21.829	14.706	19.505	18.718	23.489	20.556
We estimate that similar deviations and miscalculations will also appear in analyzing the limit states of ground bearing capacities for horizontally loaded shallow foundations, foundations near slopes and foundations with an inclined foundation base.
The largest deviations appeared in the limit states of passive earth pressures for inclined backfills, where the negative inclination approached the value of the soil's inner angle of friction. In such cases the coefficients of passive earth pressures, obtained from equation 11, are essentially lower from the actual deviations, which can reach up to 300% of the lowest values. The overestimated values of coefficient Kpc using the transformation expression (equation 11) also appear for horizontal backfills and backfills with a moderate inclination. The values come to 12% of the lowest value, determined according to the limit state method using the upper-bound theorem.
It is therefore false and unacceptable to calculate passive pressures for cohesive-friction material from solutions of pure friction material using the known procedure for calculating passive pressures described in Kerisel and Absi (1990). Slopes with decreasing inclinations are very
frequent in geotechnical practice. They are characteristic of embedded regions of embedded retaining structures on slopes (pile walls, sheet pile walls etc.). Such situations require detailed and systematic approaches of passive earth pressure. The results of our analyses also show that different geometries of the failure mechanism are critical for determining the different influences of soil self-weight, cohesion and surcharge (Table 1). We obtained the lowest expected values of coefficients Kpc when analyzing y* = q* = 0 and c* > 0. In our opinion, these values are generally applicable because the passive pressure coefficients, in practice, are a bit higher due to the cohesion that occurs in practice.
The transformation expression (equation 11), first proposed by Caquot (1934) and uncritically used in the procedure of Kérisel and Absi (1990), is not generally applicable. It should be replaced with the expression defined by equation 10 for determining passive pressures with the limit state method using the upper-bound theorem. This procedure is also applicable to three-dimensional limit state analyses where similar failure mechanisms are in accordance with the upper-bound theorem.
48. ACTA GEOTECHNICA SLOVENICA, 2010/2
B. MflCUH & S. SKRABL: PASSIVE EARTH PRESSURE DETERMINATION: APPLICATION OF THE CORRESPONDING STATE THEOREM FOR CALCULATING
REFERENCES
[1]	Caquot A. (1934). Equilibre des massifs au frottement interne. Stabilite des terres pulverulents et coherents. Gauthier-Villars, Paris.
[2]	Caquot A. and Kérisel J. (1948). Tables de poussée et de butée. Gauthier-Villars, Paris.
[3]	Kérisel, J., and Absi, E. (1990). Tables for the calculation of passive pressure, active pressure and bearing capacity of foundations. Gauthier-Villars, Paris, France.
[4]	Michalowski, R. L. (2001). Upper-bound load estimates on square and rectangular footings. Géotechnique, The Institution of Civil Engineering, London, England, 51(9), 787-798.
[5]	Silvestri, V. (2006). Limitations of the theorem of corresponding states in active pressure problems. Canadian Geotechnical Journal, 43, 704-713.
[6]	Škrabl, S., and Macuh, B. (2005). Upper-bound solutions of three-dimensional passive earth pressures. Canadian Geotechnical Journal, Ottawa, 42, 1449-1460.
[7]	Soubra, A. H., and Regenass, P. (2000). Three-dimensional passive earth pressure by kinematical approach. Journal of Geotechnical and Geoenvi-ronmental Engineering Division, ASCE, 126(11), 969-978.
[8]	Vrecl-Kojc, H., and Škrabl, S. (2007). Determination of passive earth pressure using three-dimensional failure mechanism. Acta Geotechnica Slovenica, 4(1), 10-23.
48. ACTA GEOTECHNICA SLOVENICA, 2010/2