ETL 1110-2-344
31 Dec 93
KII for shearing mode II; see ETL 1110-8-16(FR)
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 KI for a
Becker, and Dunham 1976). The method is modi-
crack length of 13.5 ft was negative and KI 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 KI 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 KIc.)
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
d. Analysis and results.
intensity factors. This procedure was repeated until
the value of KI 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 (KI) and II (KII),
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 ∆Hcrest are given.
KI = KIc. For each analysis, the value of KI is
computed and compared to KIc. If KI is larger than
Negative values for the horizontal crest displace-
KIc, this indicates that the crack will propagate
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 KI is less than KIc, the crack will not
variations in KI and KII as a function of crack
length are shown in Figure A-8. The value of KI
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 KIc =
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 KIc 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 KIc = 1.0 ksiin. would be
√
and Saouma 1991) to obtain an initial estimate of
less than 6.0 ft. Because KI 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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