Design and Stress Analysis of Extraterrestrial ... - The Black Vault

dr dr r
d• _._._.• .d~t 1 •t--%)"(3.61)
Thus, in order to determine the radial and circular stresses
in a disk we derived a system of two differential equations (3.55)
and (3.61)
System of equations (3.55) and (3.61) can be replaced by one
differential equation of the second order relative to displacement u.
For this we solve a system of equations (3.60) relative to or and
a with equalities (3.56) and (3.57) taken into account:
E (du u u dE u
"-+1 ; ' -- 'iht'rJ"_ + 1 (3.62)
12 dr r I'~ dr
If we substitute equalities (3.62) into (3.55) we obtain a
differential equation of the second order relative to u:
d1 2 t --- I dhh ldu .drdh I \ u Isu2
'
dr dr r ]dr h dr .r -- r = --F
Cr, (3.63)
where
C--•w2. (3.64)
If we can obtain a solution to equation (3.63) relative to the
radial displacement of u for a disk of any shape (defined by function
h = h(r)), then stresses or and 1f are determined from expressions
(3.62).
Thus, stresses in a rotating disk can be obtained either
directly from equations (3.55) and (3.61) or from expressions (3.62)
after determining u from equation (3.63).
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