Soil Mechanics and FoundationEngineering, Vol. 33, No. 6, 1996

SOIL MECHANICS DISPLACEMENTS OF AN ENCLOSING WALL DURING TRENCH EXCAVATION

B. I. Didukh and V. A. loselevich

UDC 624.131:624.134.4

The problem of the development of a stress-strain state of a bounded soil mass undergoing trench excavation, one of the edges of which supports a rigid enclosing wall, is addressed. An effective iteration algorithm is developed for the numerical, solution of the problem of the wall's displacement and stress variation in the soil with deepening of the trench. It is demonstrated that there is a critical depth of excavation, which when exceeded, equilibrium of the wall is impossible, and the edge of the trench supported by the wall collapses together with the latter.

The problem of the interaction of soil with an enclosing structure is classed among problems most critical to soil mechanics owing to its major practical significance [1-3]. The problem of the lateral displacements of a rigid enclosing wall during the excavation of a trench, one of whose edges it supports, is discussed below. A schematic section of a soil mass with a retaining wall is shown in Fig. 1. It is assumed that a vertical wall with a height h is constructed prior to the start of trench excavation. The soil mass in which the wall is located is bounded to the left (region 1) and right (region 2) of the wall by stationary smooth vertical planes situated at comparatively small distances l 1 and/2 from the wall. These distances are commensurate with the height of the wall, i.e., l 1 - h and l2 ~ h. This enables us to disregard small stress and strain variations of the soil in the direction of the x-axis at these distances. The displacements of the wall are examined under plane deformation (there are no soil displacements along the y-axis, and the stresses and strains are independent of the y coordinate. The wall is considered smooth, and there is no friction along either of its surfaces. The horizontal displacements of the wall u(z), which develop during excavation of the trench, are assumed small u(z) << h. In connection with these assumptions, it is possible to consider that the stress-strain state of the soil at any fixed depth is uniform (ultimately, it differs in regions 1 and 2), and is determined by the depth z, the specific gravity 3, of the soil, and surcharges Pl and P2, which are uniformly distributed along the bottom of the trench and along the surface of the mass to the right of the wall, as well as by the lateral displacements u(z) of the wall. In that case, the principal axes of the stresses and strains are oriented along the coordinate lines x, y, and z. The compressive stresses and compressive strains are considered positive. The normal stresses azl and az2 in regions 1 and 2, respectively, are related to depth z by the expressions cr~ =(z-h~)y+p~

(z > h~);

(1)

crz2 =zy+p2

(z >_0),

(2)

where h w is the depth of the trench. It is assumed that the sublimiting deformation of the soil is described by relationships of a linearly deformable medium, i.e., is characterized by a compression modulus E and transverse-expansion ratio v. In the region of the limiting

Translated from Osnovaniya, Fundamenty i Mekhanika Gruntov, No. 6, pp. 2-6, November-December, 1996. 0038-0741/96/3306-0187515.00

9

Plenum Publishing Corporation

187

X---

-7 .,~e 171

Fig. 1

m

Fig. 2

Fig. 1. Computational diagram of enclosing wall in soil mass. Fig. 2. Plots showing dependence of horizontal normal stresses to left axl and right ax2 of wall on its displacement.

state, the stresses satisfy the Mohr-Coulomb condition in which the angle of internal friction r is the only characteristic of the soil's shear strength that is other than zero. The assumption concerning the absence of cohesion c is not principal in nature, and can be easily deleted (in truth, a certain complication to the equations). In the sublimiting region, the lateral normal stresses axl and ax2, which define the soil pressure against the left and right faces of the wall, are calculated from conditions of plane strain from the equations

= ~1

E V2 ~xl

+ ~Crzl;

E cry2 = -1 -- v2 e n

+ ~O'z2,

O'xl

- -

(3)

(4)

where ~ = v / O - v ) is the lateral-pressure coefficient, and ~xl and ex2 are the horizontal (lateral) strains of the soil in regions 1 and 2 at the corresponding depth. With the absence of lateral strains, the familiar relation axl = ~azl and ax2 = ~az2, which indicate that compression of the soil occurs in the mass to both the left and right of the wall, follows from (3) and (4). The lateral pressure against the wall increases with increasing strain ex (compression of the soil in the horizontal direction), and decreases with decreasing ex (tension). The ratio ff = ax/az varies in both cases; this may lead to attainment of a limiting state. The Mohr-Coulomb condition can be represented as

or3= rnrr~; m=(1-sin(o) / ( l +sinq~),

(5)

where a 1 and a 3 are the maximum and minimum principal stresses. A limiting state is attained when ff = m (a 1 = % and a 3 = ax) as ff decreases, and when ak = 1/m (a 1 = a x and o3 = az) as ff increases.

188

The compressive strain of the soil 8xl = 11/ll corresponds to a positive displacement u of the wall, i.e., in the direction of the trench in region 1, and the tensile strain ex2 = - u / l 2 in region 2. In the limiting state, the quantity r remains constant, irrespective of ex. Therefore (Fig. 2),

mCrzt, u
(6)

cr:~t =~A~u+~cG~, U~L
(7)

[ m O'z2 , Zl> U2R,

where A 1 = E/[(1- ~)li], A2 = E/[(1.-~)12], ulL = ( m - O a z l / A 1, UlR = ( 1 / m - O a z l / A I, U2L = ( ~ - 1/m)azJA 2, U2R = (~ -- m)azE/A2. On the assumption of the wall's rigidity, its position for each h w value is defined by thedisplacement of its upper point B = u(0) (see Fig. 1) and by the angular coefficient t~ = u'(0), i.e.,

(8)

u(z)=za+/~. The equations h

h

- ~ o-,:~ (z)dz + [~.2 ( z ) d z &, o h

=

O;

(9)

h

,~xl ( z ) z d z . f ,rx2 (z)edz = O. h~ o

(10)

follow from equations of the wall's equilibrium Partitioning the wall over its height into n segments of similar length Ah = h/n, the integrals in (9) and (10) can be replaced by the sums of a finite number of terms n

-ql/o-

n

~ qli + ~ q2i = 0; i=j+l

(11)

i=1 ~

n

q l j z j ( - O - ~. q l i z i - - ~ q2izi = O. i=j+l i=l

(12)

In these equations, j is the number of segments within whose bounds the level of the bottom of the trench is located to the left of the wall, (j - - 1)Ah < law _< jAh, and

co=]-h~ l a b ; z i =(/Ah+h~ ) / 2 ; z i - ( 2 i - 1 ) A h / 2 ; q~] =cr*l(z/); qli='~l(zi ); q2i =crn(zi). Generally speaking, a nonlinear system of two equations for determination of the parameters a and 15 of the wall's displacements, which correspond to a given trench depth law, emerges after substitution of (6) and (7) for axl and %2 in (11) and (12) in lieu of (8). The nonlinearity of this system is dictated by the existence of three different segments in the % 1 - u and ax2-U curves ((6) and (7)). The proposed method of solving this system consists in the following. As a first approximation, the segments where a x is independent of u in (6) and (7) are ignored for all i values, and only the median lines of these equations with no constraints on wall displacement u(zi) are used, i.e., initially, a sublimidng soil state is assumed everywhere in the mass. The.system of constitutive equations obtains the form 189

alla + a12fl= bl +Abl; a21a + a n t i = b2 + Ab2,]

(13)

where

all = _[Al(h 2 -hw)+A2h 2 2] / 2 A h ;

cq2 =-[A 1(h - h ~ ) + A2h]/2Ah; I,l

n

a2, =A,[z}co+ Z z~]+A2 Zz2i; a22 =-ct,,; i=]+1

i=1

n

n

bl =.~qu 09+ ~.. qli-~.q21]; i=j+l

i=1 /7

?l

b2 = ~-qljzj(.o - ff~ qlizi i=j+l

+

~"~q212i], iM

and the additional terms Ab 1 and Ab2 are equal to zero in the initial approximation. Not only the values of a and B of the first approximation, but also the pressures qu and q2i against the wall for all i (including for i = j) are found from solution (13). It is natural that for any i values, the values of u(zi) are outside the limits of the corresponding regions (UlL , UlR, and u2L, U2R). In this connection, the correctives Ab 1 and Ab2, , which are related to the differences in the pressures ql and q2 obtained and the stress values axl and ax2, which develop in the zones of the soil's limiting state, i.e.,

[AI(ulL - u ) ; u
[-&(u-ulR), u>ulR;

f-A2(U2L-U); '/,/< U2L; Aq2 =J0, U2L
[-&(u'u2R), u>u2R

(14)

(15)

are introduced to the right sides of Eqs. (13) to obtain the next approximation. Using (14) and (15), we then have

Ab,

q,i +2 Aq2,; i=]+l

(16)

i=|

n

Ab2 = -Aqqzyco + ~ Aqllzi - Y'. Aq2i2 i. i~j+l

i=1

(17)

Corrected in this manner, system (13) makes it possible to determine the values of a and B in second approximation. The iteration process described is continued until the value ofV'(Ao02 + (AB/h) 2, which earl be computed from the c~ and B values in two successive iterations, is no longer less than the assigned accuracy of the calculation. It should be pointed out that in each iteration, only the right sides of Eqs. (13) are corrected, and the coefficients before et and B remain unchanged. In the sample calculation cited below for h = 20 m, l 1 and l2 are equal to 10 m, and there are no surcharges Pl and P2" The following characteristics were adopted for the soil: 3' = 20 kN/m 3, E = 104 kN/m 2, v = 0.45, and ~ = 30 ~ The number of computed segments n = 20, i.e., Ah = 1 m. 190

TABLE 1 hw, m


1T00 2,00 3,00 4~00 5,00 6~00

7,00 8,00 9,00

--

.~oo ;

-1268,0 -3642,0

200 .-

"100 .

0,0145 0,0243 0r0371 0,0547 0r0830 0,1351 0,2559 0,6743

N 10 13 19 24 35 62 97 220 781

0~ r

eo0 kN/in 2 ~"

,8, II'). 0v0068

t,0o

\

9,

3,

4

~, m .

f~

1t~0 I

0

400

r

q,,

to

?o0

30okN/m2

,JI-s.,,,,.~'S-,'/,'/It,'~/H,'H~, "~

"

2..

in

Fig. 3. Diagram of pressures against teft q~ and right q2 faces of wall.

The dependence of the parameters ot and B, which characterize the lateral displacements of the wall on the depth law of the trench is apparent from Table 1, where the number N of iterations required to achieve a relative computational error of --10 ---6 is also given. The increase in the displacement of the top of the wall in the direction of the trench progresses, and the slope of the wall increases with increasing h w. It is interesting that where all of 19 approximations were required to obtain the given accuracy when h w = 3 m, as many as 781 iterations were required for this purpose when h w = 9 m. The zones of the soil's limiting state to the left and right of the wall, which develop downward from the bottom of the trench in region 1, and from the free surface of the mass in region 2 will increase gradually with deepening trench. When the depth of the trench reaches approximately half the height of the wall, these zones are already so significant that equilibrium of the wall becomes impossible. This indicates, among other things, vigorous growth in the number of iterations as the depth of the trench approaches = 10 m. The distributions of the pressures ql and qz on the left and right faces of the wall when h w = 4 m (a) and 8 m (b) are shown in Fig. 3. When law = 4 m, the zone of the soil's limiting state to the left of the wall extends to z = 5 m, and its thickness beneath the bottom of the trench amounts to all of 1 m. When h w = 8 m, the left zone of limiting equilibrium 191

-200

-~00

J

I

7illllll/ll)l/llllJ

tOO

200 ~'~/m2

~ ' / / / / / / / A / / / / / / / / / X / ~ '~

1

~oo

200

f J

l

~

~00 i

0

':

J00

e00

300 tN/nr~

2 3

/7 /

Pig. 4

5

Fig. 5

Fig. 4. Diagrams of resultant pressure on wall q = cb - - ql when h w = 8 m. Fig. 5. Variation of pressure curves to left ql and right q2 with increasing l]w. 1-6) h w = 0, 2, 4, 6, 8, and 9 m, respectively.

extends to as much as z = 11 m, while its thickness is 3 m. A similar pattern is also observed to the right of the wall. When h w = 8 m, the zone from the surface of the mass to a depth of 13.5 m is embraced by the limiting state. The distribution of the resultant pressure q = q2 m q], which is applied to the enclosing wall, is shown in Fig. 4 for h w = 8 m. The diagram obtained, which does not differ all that much from the limiting diagram, agrees well with familiar heuristic representations of the pressure against a retaining wall [4]. The computational method proposed here, however, also makes it possible, in addition to the pressure distributions, to calculate the displacements of the enclosing wall, which correspond to these pressures, and in this lies its principal advantage. As is apparent from Fig. 5, a gradual decrease in axl and Ox2 in both the sublimiting zones and zones of limiting state occurs in the soil mass on both sides of the wall with deepening trench and decreasing azl. When complicated models of a soil's elastoplastic deformation are used to design enclosing structures, this critical situation, i.e., unloading of the mass, must be taken into account.

REFERENCES

,

2. 3. 4.

192

P. L. Ivanov, Soils and Beds of Hydraulic Structures [in Russian], Vysshaya Shkola, Moscow (1985). S. B. Ukhov, Soil Mechanics, Beds, and Foundations [in Russian], Izdatel'stvo, ASB, Moscow (1994). V. N. Rengach, Sheet Pilings [in Russian], Stroiizdat, Leningrad (1970). N. I. Bezukhov, Retaining Walls [in Russian], Gosudarstvennoe Izdatel'stvo, Leningrad (1930).