Page 39 - Failure Analysis Case Studies II

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An approximate relationship between the area of a section of firewall and the maximum pressure it
can sustain can be derived by bringing these results together. Taking the smallest value of 829N
from Table 1, and taking k as 50 from Table 2, the relationship is
p = 4 1450/A, (3)
wherep is in N/m2 and A is in m2. Most of the triangular segments have an area of aboout 14.55m2.
the precise value depending on the detailed layout. The corresponding breakup pressure is therefore
approximately 2.8 kN/m2 (0.028 bars), much smaller than the calculated maximum pressure at point
P1. This indicates that the firewall cannot withstand the pressure of the explosion in module C. A
small number of triangular segments are slightly larger at 16.2 m2, and have a correspondingly
smaller breakup pressure.

7. DYNAMIC RESPONSE: ELASTIC MODEL

Section 6 gives us an estimate of breakup pressure under slow loading, in which the loading time
is long by comparison with the lowest natural period of flexural oscillations. The next step is to
consider the dynamic response of the firewall to the actual pressure pulse, which is quite short
(between 100 and 200 ms), so that the dynamic response may be quite different from the response
to the same maximum pressure applied slowly.
Two idealisations were used. The first idealisation treats the deflection of the firewall as elastic,
but treats the critical deflection at which breakup begins as having both elastic and plastic
components, since the bolts have some capacity to extend plastically before they break. The second
more complete idealisation treats the wall as elastic-plastic, and is examined in Section 8.
The first step is to determine the natural frequency for a firewall segment, so that the loading time
can be compared with the period corresponding to the lowest natural frequency. Appendix A is a
summary of this calculation, which was carried out using the Rayleigh method.
The calculation idealises each firewall segment as a uniform plate with clamped edges. The mass
is taken as uniformly distributed and equal to the average mass per unit area. A comparison “exact”
calculation based on the actual distribution of mass in a typical segment confirms that this is an
excellent approximation: the difference between the “exact” and “averaged” natural frequencies is
0.8%. The equivalent stiffness is more difficult to estimate, because the absence of structural
continuity between adjacent frames leads to a significant contribution to the firewall flexibility from
torsion in the angle sections between the frame corners and the nearest frame bolts. The equivalent
plate flexural rigidity D was estimated as IO 000 N m. This was taken as the base case, but the study
examined the sensitivity of the conclusions to the assumed value of D: this point is returned to later.
The estimated lowest frequency is 73rad/s, which corresponds to a natural period of 86ms.
Looking back to Fig. 2, we can see that the loading time is of the same order as the natural period,
neither much longer (so that the response would be quasi-static) nor much shorter (so that the
response would correspond to impulsive loading).
The next step is to calculate the dynamic response. Pressure loading which is nearly uniform over
a firewall segment primarily excites the lowest mode (corresponding to the lowest frequency). The
lowest-mode response for central deflection can be written down as a formula which is a multiple
of two terms. The first term is the deflection that would occur if the loading were applied slowly.
The second term multiplies the first, and accounts for dynamic effects: it is a function of the natural
frequency, thc time that has elapsed since the pressure pulse began, and the duration and shape of
the pulse. The multiplying second term is identical to the corresponding formula for a simple one-
degree-of-freedom mass-on-spring system.
The results are shown in Fig. 4, which plots deflection at the centroid of a triangular segment
against time; time is measured from the start of the triangular pulse in Fig. 2.
The deflection when the wall begins to break up can be estimated as the sum of two components:
I. the elastic deflection of a segment under the estimated collapse pressure under quasi-static
loading, represented by xy in Fig. 5;
2. the additional deflection associated with plastic elongation of the frame bolts until they reach
their specified minimum elongation, represented as xF-xy in Fig. 5.