COMPUTING THE DIMENSIONS OF SINK-HOLE COLLAPSES IN DESIGNING FOUNDATIONS IN AREAS PRONE TO SINK HOLES V. V. Tolmachev and G. M. Troitskii

UDC 699.8:551.448

The planform of sink holes is one of the basic initial parameters in designing antisink measures. For the majority of buildings and structures, it determines the volume and cost of these measures. At the present time, however, there is no method of determining the effective dimensions of a sink hole in plan. They are usually assigned without sufficient grounds, intuitively drawing attention to the mean-probable diameter d of the sink hole and its standard deviation ~, the rate of occurrence X of the sink holes, and the extent to which the buildings and structures are used for critical purposes [i]. A method of computing the planform dimensions of a sink hole in which an attempt is made to account for the above-indicated factors is outlined below. The essence of the method consists in the fact that sink holes and the structure under design are continuous within the framework of a single system. It is impossible to indicate the computed size of a sink hole for any site without reference to a specific structure, whereas in analyzing any_structure, we cannot disregard specific parameters of possible subsidence deformations (~, d, ~) for a certain combination of these parameters. The proposed method for determining the effective dimensions of a sink hole can be used for both the case of deterministic prediction (a sufficiently narrow class of values for the sink hole diameters on the basis of geomechanical models) and also probabilistic prediction (a function of the distribution of d) [2]. Let us examine the significance of the method as applies to probabilistic prediction. The basic initial data are: the rate of occurrence ~ of sink holes, a histogram or distribution curve of the diameters of sink holes of stable shape (Fig. I), the design service life of the structure T, the planform of the foundation, and its dimensions.

a

ZP~6'

3

6

9

/f

/5

/~'

2!

2~' 27

7ifT f l [7171 i'f

ItI~

÷lilt

II ~ I i / Hi N

l'r

iI

i'

l

rll I ill!

If ~I

Ii \ hl Y IL:I ~I ~Lk--~J+C! _ _JfLi__J~I ""

*

~ 6

II

II

)9 }

I

I

1

I

1

I

12

15

15

Zl

24

27

Fig. I

Fig. 2

Fig. i. Histogram and straightened distribution curve of sinkhole diameters (a), and integral distribution curve of sink holes Fig. 2.

(b)

Computed zone of building damaged by sink holes.

Industrial and Scientific-Research Institute for Engineering Surveys in Construction. Scientific-Research Institute of Bases and Underground Structures. Translated from Osnovaniya, Fundamenty i Mekhanika Gruntov, No. 2, pp. 22-24, March-April, 1983.

0038-0741/83/2002- 0069507.50

© 1983 Plenum Publishing Corporation

69

o,6

o,z 0

Fig. 3

3

6

g

12 15 lg

21

(.,Ill

Fig. 4

Fig. 3. Working diagram for determining deformation ~ beneath lower surface of foundation for sink hole with diameter d. Fig. 4.

Distribution function of parameter ~.

It is proven [2] that for sites with approximately the same conditions, which statistically determine the diameters of sink holes, their distribution is close to normal. On this basis, it is possible to estimate the maximum sink hole diameter dma x = d + 30. We can, with a certain amount of faith, assume that the rate of occurrence of sink holes and their diameters are independent quantities. When taking this situation into account, the working scheme of the zone of influence of sink holes beneath a structure may assume a form similar to that shown as an example in Fig. 2. When sink holes form in this zone, the probability of the appearance of sink holes beneath the foundations of the structure will be low, and if if does occur, the probability will not be manditory on the basis of hole diameter. It is precisely the term "computed diameter" of the sink hole, which is used in designing foundations, that is therefore invalid. It is expedient to indicate the "design span" I of a foundation above a sink hole, which is a parameter of the "cave--structure" system. In connection with the probilistic nature of this quantity, it is necessary to know the probability distribution of 7 for a specific foundation. Let us first determine the distribution of the parameter Z for the 0-0' axis of a single strip of a foundation of infinite length (Fig. 3). The quantity ~ is related to the distance x from the 0-0' axis to the center of the sink hole by the relationship l=2

-x~)

(1)

,

and the average value is

2d,i'2~(-- xe)l/2dx l=

0 d

(2)

__ ~ 4

d=O,785d.

2

It is obvious that the appearance of sink holes is uniformly probable within the limits of the strip

0,

. For this condition,

the distribution function is expressed as [3] (d~ _ f~)i/2

Pz = 1

d

'

(3)

and the distribution density P t = (d~ - - Z D - ~ / 2

.

(4)

The quantity ~ and the empirical distribution curves are also derived by the method of statistical testing [3]. The good agreement between these results and the values computed from Eqs° (2)-(4) make it possible to use with assurance the method of statistical testing for as complex a foundation planform as one pleases. The theoretical derivation of equations of the distribution functions and densities of I for foundations of any configuaration (for example, by the composition of laws governing the distribution of random values) will hardly

70

w

Ma,tons

1,2"

sooi

\ a

4

L=I2 m

~

+o,f 8

12

16 20 g,o

L=t2 m 2Og tOnS

,!--7--4

0

6

Fig. 5

;2

18

2~ ~,

rn

Fig. 6

Fig. 5. Dependence of effectiveness on computed deformation.

indicator of antisink foundation

Fig. 6. Dependence of bending moment M~ in continuous foundation on size of sink hole ~ beneath foundation. be justified from the practical standpoint. The curves showing the distribution function P{, which was obtained for one of the specific projects (see Fig. 2), is shown in Fig. 4. To determine the probability of the appearance of a sink hole beneath the foundation, let us assume that a hole did form. In this case, the conditional probability of an occurrence beneath the foundation will be P j . = -d/ d~.~ ,

(5)

When a sink hole with a diameter d forms in the zone of one of the sections where the dimensions of the foundation in plan are a: and bl (a: < b:), the probability of the occurrence of a hole beneath the foundation will be:

P~I= 1--(ai--di) (bl--di) al bl

di (ai~-bl--di) a:b~

=

'

(6)

and when the distribution curve of hole diameters is taken into account,

P/:= d a:+ albl b:-

dai

Pa: @1-- Pa:,

where da: is the average value of the hole diameters in the [0-a:] interval, probabiity of the formation of holes with diameters to a: (see Fig. !b).

(7) and Pa: is the

When a sink hole forms in the adjacent j-th section with the dimensions aj and bj, the probability of the occurrence of a hole beneath the foundation is determined in a similar manner; this is done, however, without consideration of the overall with the first part of the foundation:

PSi :-drnax 2 al ~' aibi b/-- 2 dal pai -.}-1 For the foundation sho:cn in Fig. 2, for example,

-

-

Pal"

this probability is:

Sks,

PJ: ~ ° So T +PJ:

(8)

(9)

S

where S O is the area of the section around the building to a distance dmax/2 , Skj is the area within the limits of the contour of the j-th sections, n is the number of sections, and n

S So ~-~ Ski. =

71

According to Tolmachev [2], the probability that the area S located in an area with a sink hole intensity % over the service life T of the structure will not be affected by sink holes is determined by the equation

Po----- exp (--~ST). The probability (unconditional) then be found as [4, 5]

(lO)

of the occurrence of holes beneath the foundation can

p e = (I --Po) PI.

(ii)

The reliability of the foundation as computed from the loading of the bed beneath its lower surface with a dimension ~ relative to the sink holes can be determined from the equation p = i--pF(i--Pl) ,

(12)

where Pl is the probability that when a sink hole forms, its plan dimension beneath the foundation will not be greater than ~. If a certain minimum allowable value of the reliability to determine the design value P~d" Pip --

[P] is given,

it is not difficult

[Pl + Pp --1 PF

(13)

The design (desired) value ~d can now be found from the integral distribution curve for (see Fig. 4). Thus, on examining the of the sink process within required for the design of most successful in economic

characteristics of the structure and the quantitative parameters the framework of a single system, assign the basic parameter Id single antisink foundations. This results in design solutions respects.

The question concerning the allowable value of the reliability [P] deserves special attention. For projects with a clear economic reliability, it can be found for a minimum of mean-probable reduced costs [P] = [Pec] [5]. For buildings and structures for which any damage produces a socioeconomic result, the approach to assigning an allowable reliability value should, in our opinion, be similar to the approach used in earthquake resistant construction [6, 7]. In this case antisink measures should not ensure the preservation of the building or structure, but guarantee it from rapid (catastrophic) collapse. In many cases, this required condition is also ensured for the quantity 7 as determined from economic considerations (a sufficient condition), and all the more so if the possibility of ensuring the reliability is taken into account during foundation construction in the plastic stage [8]. Where it is necessary to standardize the value of the allowable reliability [P], a volitional solution [9] could hardly be avoided up to the end. Using the effectiveness cator of antisink measures = A P / A K,

indi-

(14)

where AP and AK are differences, respectively, in the reliabilities and costs of antisink foundations computed for different spans, however, we can assign rather objectively an allowable value for [P]. In the example shown in Fig. 5, it is quite clear, therefore, that the effectiveness of the design of antisink foundations computed for spans 7 > 12 m is extremely negligible. As applies to continuous antisink foundations, values of the reliability P are determined from Eq. (12). The cost of the foundations, however, can be considered approximately proportional to the bending moments M~ that develop in the foundations around the center of a sink hole beneath the foundation. The M a = t(~) relationship is not difficult to derive as applies to specific buildings and structures (as, for example, the one shown in Fig. 6). Using the method that we have outlined,

72

it is possible to solve successfully a number of

engineering-economic problems that arise during construction in areas prone to sinking, as, for example, evaluation of the effectiveness of different antisink structural measures and selection of the most rational solution on this basis; development of conditions of uniform stability of buildings and structures in areas with different parameters of sink defomations; the selection of rational shapes and dimensions of buildings from the stand point of their sustaining the least damage from sink holes, etc. LITERATURE CITED io

2~ 3. 4o 5~ 6o

7.

8. 9.

Recommendations for Geologic Engineering Explorations and Evaluation of Areas for Industrial and Civil Construction in Regions Prone to Subsidence in the USSR [in Russian], Industrial and Scientific-Research Institute of Bases and Underground Structures, Moscow (1967). V. V. Tolmachev, "The probabilistic approach in evaluating the stability of areas prone to subsidence and the design of antisink measures," Inzh. Geol., No. 3, (1980). L. N. Bol'shev and N. V. Smirnov, Tables of Mathamatical Statistics [in Russian], Nauka, Moscow (1965). E. S. Venttsel', Probabilty Theory [in Russian], Nauka, Moscow (1969). V. V. Tolmachev, "Construction conditions on industrial projects in areas prone to subsidence," Prom. Stroit., No. 8 (1974). Ya. M. Aizenberg and A. I. Neiman, "Analysis of economic optimality criteria," in: Design and Construction of Earthquake-Resistant Buildings and Structures (Theses of papers presented s~ the All-Union Conference, Frunze, 1971) [in Russian], Moscow (1971). B. I. Snarskis, "Statistical-economic basis of the bearing-capacity reserve in structures (2. The case of noneconomical accountability)," Tr. Akad. Nauk LitSSR, Series B.2 (29) (1963). N. S. Metelyuk, Improving the Design of Structures Built under Complex Soil Conditions [in Russian], Budivel'nik, Kiev (1980). A. R. Rzhanitsyn, Theory of the Reliability Computation of Structural Designs [in Russian], Stroiizdat, Moscow (1978).

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