Moment Area Method -Cont.

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EQUATION:
Xm = Z1A1 - Z2A2
(8)
If an impulse, ib, is applied at time ti in addition to the load shown
in Figure 3, then Equation (8) becomes:
EQUATION:
Xm = (tm - ti)(ib/m) + Z1A1 - Z2A2
(9)
where Z1 and Z2 are the distances to the centroids of the areas A1 and
A2, respectively;
(c) The moment-area method assumes usually a linear variation
in resistance with time.  This approximation introduces very little error
unless tm is equal to tE.  For this approximation, the time tE can
_
be calculated from the following relationship.  Let p be the effective
pressure during the time interval 0 < t < tE, then
_
EQUATION:
XE = [(p/m)tE]tE/2
(10a)
multiplying by KE:
_2
EQUATION:
XEKE = ru = (pt KE)/2m
(10b)
E
but:
EQUATION:
KE = (4[pi]2m)/(TN)2
(10c)
therefore:
_2
EQUATION:
ru = 2pt [pi]2/(TN)2
(10d)
E
rearranging terms:
_
EQUATION:
tE = 0.226TN(ru/p)1/2
(10e)
_
Let pE be the pressure at time t = tE.
Then p is expressed as:
_
EQUATION:
p = (2B + pE)/3
(11)
where,
pE = B(1 - tE/T)
2.08-9