which was chosen based on the element size (i.e.,

Figure 1-2) are computed using a contour integral

those adjacent to the monolith/foundation). No

method based on a reciprocal work theorem (Stern,

analyses were performed for crack lengths greater

than 13.5 ft because the computed value of *K*I for a

Becker, and Dunham 1976). The method is modi-

crack length of 13.5 ft was negative and *K*I was

fied to include the contributions of body forces,

initial stresses, and pressures on the crack surfaces.

positive for all prior analyses. A final crack length

Comparisons between the modified contour integral

of 12.99 ft was estimated by interpolation of results

of *K*I for analyses with crack lengths between 12 ft

method and the displacement correlation method

and 13.5 ft (based on a value of zero for *K*Ic.)

have shown that the computed stress intensity

factors are within engineering accuracy

(ETL 1110-8-16(FR)). When initial stresses due to

(c) Additional analyses were performed to

pore pressures are present, the effective stresses are

determine a more precise value for the final crack

used to compute the contour integral. Since the

length. In these analyses, re-meshing was per-

deformations and stresses in the foundation due to

formed by relocating a pair of nodes on the inter-

self weight occurred before the lock was con-

face between the monolith and the foundation to the

structed, the corresponding body forces in the foun-

estimated position of the crack tip. To accommo-

dation do not contribute to the computed stress

date uplift forces, node pairs on the boundaries

intensity factors. Assigning a value of zero to the

between elements with and without initial stresses

unit weight of the rock eliminates body forces in

that were originally directly above or below the

the foundation and no special logic is required to

crack tip nodes are repositioned in the horizontal

eliminate their contribution from the computed

direction along with the crack tip nodes to keep all

stress intensity factors.

elements rectangular. The modified mesh was

reanalyzed to determine the corresponding stress

intensity factors. This procedure was repeated until

the value of *K*I was less than 0.001 ksi√in. (0.0011

MPa ). The final crack length computed using

√m

(1) Estimation of crack length.

this approach was 13.02 ft, which is 43.7% less

(a) For the method of analysis used in this

than the 23.13 ft predicted by the traditional method

study, the estimation of crack length for a crack at

of analysis. The results of the first six analyses and

the base of the monolith requires an iterative

the analysis yielding the final crack length are sum-

marized in Table A-2. For each crack length *a*, the

approach. The basic approach involves running

stress intensity factors for modes I (*K*I) and II (*K*II),

several analyses each with a different crack length,

crack mouth opening displacement (CMOD), and

with the goal of obtaining a calculated value of

the horizontal crest displacement ∆*H*crest are given.

computed and compared to *K*Ic. If *K*I is larger than

Negative values for the horizontal crest displace-

ments indicate movement into the lock chamber.

under the given conditions and a subsequent analy-

sis with a longer specified crack length is per-

(2) Variation of stress intensity factors. The

formed. If *K*I is less than *K*Ic, the crack will not

variations in *K*I and *K*II as a function of crack

length are shown in Figure A-8. The value of *K*I

propagate and a subsequent analysis with a shorter

crack length is performed. For all analyses per-

decreases with increasing crack length and equals

formed in this study, it was assumed that *K*Ic =

zero at a crack length of 13.02 ft. Since the frac-

0.0 ksi√in. The structural analysis is therefore

ture toughness is assumed to be zero, the final

reduced to a finite element analysis with fracture

estimated crack length corresponds to this crack

mechanics used as the criterion for crack extension.

length. If a less conservative value had been

assumed for the fracture toughness, the final pre-

(b) A series of six analyses, each with a dif-

dicted crack length would have been shorter. For

example, a realistic value of *K*Ic for concrete is

ferent specified crack length, was performed using

approximately 1.0 ksiin. and the resulting pre-

√

the MERLIN computer program (Reich, Cervenka,

dicted crack length for *K*Ic = 1.0 ksiin. would be

√

and Saouma 1991) to obtain an initial estimate of

less than 6.0 ft. Because *K*I decreases as the crack

the crack length. The crack lengths for these analy-

ses ranged from 6.0 to 13.5 ft in 1.5-ft increments,

length is increased, this particular combination of

A-9

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