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Re: Diaphragm Calculations

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Mike, I was reviewing your calculation for the conversion of the diaphragm 
deflection (simple span part) and came up with a different solution than you. 
Can you please review my notes below and let me know where I may have gone 
wrong?

In a message dated 7/8/99 6:54:19 PM Pacific Daylight Time, mtv(--nospam--at)skilling.com 
writes:

<< Dennis:
 
 The deflection term you are asking about represents the flexural 
 deformation of the diaphragm.  Here is how the equation is derived.
 
 Assume:
 Simple span beam
 Uniformly distributed load
 Base diaphragm moment of inertia on chords only
 
 Maximum deflection = ( 5 w L^4 ) / ( 384 E I )
 Maximum shear force, V = w L / 2
 Maximum unit shear, v = V /  b
 
 I = sum (A d^2), where d = b / 2
 
 So, I = (A b^2 ) / 2
 
 Substituting (and simplifying),
 
 Maximum deflection = (5 v L^3 ) / ( 96 E A b ) 
 Given the units noted in the code, the deflection is in feet.
*******************************************************
<<Dennis>> Mike, I may be confused by your Moment of Inertia term I which you 
define as: 

I = (A b^2 ) / 2

Since the basic formula for I is (base * height^3)/12  we can assume the base 
to be the span between shear elements or gridlines (L) and the height to be 
the depth of the diaphragm (b). Therefore the formula changes to I = (L 
b^3)/12. 
If the area is L*b the formula reduces to:  I = (A*b^2)/12

combining all of the terms:

Deflection = 5*2 v b L^4 12  / (384 E A b^2) which then reduces to:
 
Deflection = 5 v L^3 / (16 E A b)

This still does not balance to match the deflection formula 5 v L^3 / (8 E A 
b)

What am I missing???????

Dennis

 *************************************************************
 Max defl (inch) = (5 v L^3 ) / ( 8 E A b )
 
 or,  = (5/8) (v L^3 ) / ( E A b )
 
 As a point of interest, this also highlights the limitations 
 (assumptions) of the formula.  If we say that for all load conditions 
 and all boundary conditions,
 
 Max defl (inch) = X (v L^3 ) / ( E A b )
 
 X is 5/8 for single, simple span with uniformly dist load, but can 
 range from 1/8 (fixed-fixed, unif load) to 1 (pin-roller, centered 
 point load).
 
 The second term of the deflection is based on a similar derivation 
 for the deflection due to shearing of the "web".
 
 The equation for shear wall deflection (flexural and shear) is 
 derived based on a cantilever column with a concentrated load at the 
 top.
 
 -Mike
  >>