and one coordinate defines the motion relative to the floors of the wail slab between ficors for
each story of both the front and rear faces, The positive direction of motion is as shown in
Fig. 2.53.
The details of construction of Building 2 make it possible to reduce the analysis of this
structure to a consideration of a three-ccgree-of-freedom system by neglecting the effect of
the front and rear faces. The basis for tus approachis presented in Appendix E, The analysis
of Building 3 1s complicated by the continuous construction of the front and rear faces. This
detail makes it necessary to include the fro.it- and rear-wall motion because these elements
significantly aid the frame in resisting the horizontal motion of the floors,
The approach e:nployed in this report :s based on the principle of conservation of energy.
There are only three basic quantities to be considered in this method. These are (1) the potential energy of the external loads, (2) the strain energy stored in the various reslating elements, and (3) the kinetic energy associated with the motion of the various masses.
The evaluation of the three basic energy terma is accomplished by standard relations as
shown in Appendix C. The fundamental equation which relates kinetic energy, strain energy,
and the potential energy of the external load for any coordinate Ym 18 given in Eq. 2.1,
a a(KE) _ a{KE) . a(PE) ae au,
at ay,
Im
avn,
29m
where KE =
PE =
U, =
Ym
(2.1)
.
total kinetic energy of the system
total potential energy of the system
potential energy of all external loads
first derivative of y,, with respect tot
Equation 2.1 is written for each coordinate of the system so that an equation of motion ia ubtained for each independent coordinz .e. This relation ia known ag the Lagranyian form of the
differential equation of motion.'?
The detailed comoutationa of deflection: for narticular loads are accomplished by a numerical method of analysis called the “Accelera‘ton Impulse Extrapolation Method.” This
method is rigorously presented in reference 4. ‘he basic recurrence formuta relating three
successive displacemeats separated by a time interval At is
a
Yarn = “Yn Yann * Yn At
3
The degree of accuracy obtained with thia procedure is a function of the leagt: of the time
interval.
For time intervals short fn relation to the natural frequency of the element, the error ia
small.
A detailed derivation of the equationa governing tne behavior of Ruildings 2 and 3 is
presented in Appendix C. The cor.putations of response of thege a! sucturcs to the various
loading conditions are presented in Appendix E,
20
Fy
“y
AD
Shera,
SECRET ~ RESTRICTED DATA
ieledete Sars te Line b
Evaluation of Rebound
The maximum displacement of any eleme* {s obtained from ‘he numerical analysis of the
dynamic response. The only deflection records available for Cuildings 4 iad 3 for shot Mike
of Operation Ivy are the permanent displacements. In order to obtain the permanent displacement of an element {rom the dynamic analysis, it is necessary to reduce the m2xi- um deflection by the amount of recovery to be expected, The computed recovery (or rebcund) {s obtained
by dividing the resistance at the time of maximum deflection vy the elastic spcing constant, k.
The Greenhouse data and the numerical analysis {ur Greenheuse presertca in Appendix 2 are
examined to determine the validity of this rebound coinputation.
The maximum relative deflection and final relative defiection of exch of the three stories
of Building 2 as recorded for Operation Greenhouse are prezcrted in Table 2.3. The computed
stich ai ittasinkrakidthaitatail
2.3.5
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