Page 38 - Failure Analysis Case Studies II

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top of wall

- bottom of wall
-.
0 a 2a X
Fig. 3. Elevation and reference axes.

mean rotation on its inclined and vertical sides, and it is a good approximation to treat it as clamped
on all three sides.
The next step is to relate the maximum moment stress resultant to the loading. Consider first a
series of geometrically-similar plates, each characterized by an area A and loaded by a uniform
pressure p, and made of the same material. It can then be shown by dimensional analysis that the
maximum value of m must be proportional to PA. The form of the relationship is therefore:
m = pA/k (2)
The value of k depends on the material properties, on the shape of the plate, and on how the
edges are fixed.
We can calibrate this relationship by using analytic solutions for simple shapes. Values derived
in this way are listed in Table 2. Each solution takes the plate edges as clamped. The table could be
based on elastic solutions, for which the stress in the plate does not anywhere, reach the yield point,
or it could be based on plastic solutions, which correspond to a condition in which the plate yields
and a collapse mechanism develops. Since we wish to focus on the conditions that are present when
the plate fails, the second plastic option is chosen. The values of k are derived from solutions to the
problem of plastic collapse of a thin plate, within the well-established theoretical framework of
plastic analysis of plates.
The analytic solution for a circular plate is exact. The other solutions are based on lower and
upper bounds on collapse pressure, which can be derived from the lower and upper bound theorems
of plasticity theory.
The table shows that the value of k does not depend strongly on the shape of the plate. This
suggests that we can adopt a single value of k, and can use it to derive an approximate general
relationship between pressure, area and maximum value of the moment stress resultant. The
relationship ought to be applicable under the following conditions:
I. the plate is only supported at its edges, and not by internal supports;
,
2. the breadth and width of the plate are comparable, so that the plate is not long in one direction
and narrow in the transverse direction: Table 2 suggests a maximum length/breadth ratio of 2;
3. the shape is convex.

Table 2
shape k=pA/m source yield condition notes
square 42.8 t6. 7 Johansen refined upper bound
square 32.0 [7,81 Johansen lower bound
circle 35.4 r91 Tresca exact
21 rectangle 56.6 t8, 101 Johansen upper bound
hexagon 40.0 [Ill Johansen lower bound
eauilateral trianale 41.6 161 Johansen uuwr bound