directions, the resulting nodal forces cancel. In

of the rock is fairly well known. This is especially

order to keep the nodal forces due to uplift pres-

true if the permeability is thought to be anisotropic.

sures from canceling, a thin pervious elastic region

is placed between the two impervious regions (mon-

olith and foundation). The pervious elastic region

simulates the pipe in the pipe flow analogy shown

(1) The monolith, the backfill above the stair-

in Figure A-3. This region corresponds to the ele-

step pattern on the back side of the monolith, and a

ments where pore pressures are prescribed as

portion of the foundation near the monolith are

described in paragraph 3*c*. The result of this

idealized by the finite element mesh shown in Fig-

approach is that the nodal forces due to the uplift

ure A-4. This mesh consists of 1020 elements and

pressures are transferred to the monolith and the

1236 nodes. A rectangular region of the rock foun-

foundation one row of elements away from the

dation, 192 ft wide and 92 ft in depth, is modeled.

interface. There is also a small horizontal force due

The vertical plane on the left side of this region is

to the pore pressure gradient in the horizontal direc-

aligned with the center line of the main lock cham-

tion transmitted to the monolith and the foundation

ber. The left-most vertical face of the monolith is

which results in some shear stresses. This has only

52 ft to the right of the main lock chamber center

a minor influence on the computation of final crack

line. The entire mesh of the model consists of

length.

4-node quadrilateral elements which include a

selective-reduced integration scheme to improve

(4) The uplift model described by Case 3 is

performance for bending modes (Reich 1992). The

only practical for the traditional method of analysis.

plane strain idealization is used to define the stress-

An infinitely rigid foundation would be modeled in

strain relations for the entire model. Strain and

a finite element analysis using either pinned sup-

stress values are computed at the nodes using a

ports or a combination of pinned and roller supports

variational recovery technique (Zienkiewicz et al.

located along the base of the monolith. Since the

1985).

nodes on the uncracked portion of the base of the

monolith are fixed in the vertical direction, it is

(2) The material properties for the concrete,

impossible to prescribe pressures on the base of the

rock, soil, and interface are given in Table A-1,

where *E *is the modulus of elasticity, υ the

monolith. The addition of a thin pervious region at

Poisson's ratio, γ the dry unit weight, γs the satu-

the base, described above for Cases 1 and 2, could

rated unit weight, γw the unit weight of water, *K*H

also be applied for Case 3.

the horizontal earth pressure coefficient, and *K*V the

vertical earth pressure coefficient. Elastic properties

(5) Regardless of how the analysis is per-

for the concrete and rock are assumed to be identi-

formed, be it either the traditional method or finite

cal. The loads applied to the mesh correspond to

element analysis, the most fundamental issue to be

those shown in Figure A-2. A lock water elevation

addressed in the modeling of uplift pressures is the

of 340.0 ft and a backfill saturation elevation of

flow regime. Once the flow regime has been estab-

396 ft are assumed. For the prediction of cracking,

lished, the determination of the uplift pressures is

the crack induced by this force system is confined

rather straightforward. In the absence of field data

to the interface between the foundation and the

from piezometers, the assumption of a linear

monolith.

variation in pressure, which corresponds to a true

steady-state condition, is the most reasonable.

(a) Bi-material interface. The modeled region

When piezometric data are available, the assump-

includes bi-material interfaces at the base of the

tion of a linear variation in pressure between adja-

monolith (concrete/rock interface) and at the back

cent piezometers may be reasonable, provided the

of the monolith (concrete/soil interface). Special

resulting uplift pressure profile is relatively smooth.

considerations are required to enable the computa-

If drains are present and functioning properly, a

tion of unique nodal strain and stress values for

seepage flow analysis may be required as the

each of the two materials on a bi-material interface.

assumption of a linear variation in pressure would

To enforce continuity, yet maintain the ability to

be unconservative for this situation. In general,

compute unique strain and stress values, master and

determining uplift pressures from a seepage flow

slave nodes have been utilized. Master and slave

analysis should be avoided unless the permeability

A-4

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