= Raised Bed Design =

|| 16 to 24 inches high <<BR>> Water saturated silt loam <<BR>> 1.83 g/cm^3^, 114 lb/ft^3^, 1830kg/m^3^ <<BR>> 18 kN/m^2^ per meter depth <<BR>> 2.6 psi per meter depth, 0.794 psi per foot depth <<BR>><<BR>>source:[[attachment:raisedbed-8hoop.pov]]||                   {{attachment:raisedbed480.png | | width=360}} ||
------
Summary: 2 stacked 8x12x2 (96 x 11.5 x 1.5 inch) cedar board sides, less than 1/8 inch differential bulge
------
 * 8 foot long, 23 inches high, two boards, 1.522 psi at bottom
  . w =  4.38 lbf (pounds force) per lineal inch top board
  . w = 13.13 lbf per lineal inch bottom board
 * cedar modulus: E = 1M psi [[http://www.wrcea.org/technical-specifications/engineering_data.htm | Table 9]]
 * 3 point, end and center post support - see [[http://civilengineer.webinfolist.com/str/prob72.htm]]
  . L = 96 inch, w = 4.4 lbf/in top, 13 lbf/in bottom,  EI tbd
   . wL = 420 lbf top, 1260 lbf bottom
  . deflection of two point supported board, distributed load  d = 5 w L^4^/384EI
  . deflection of two point supported board, point load R in middle d = R L^3^/48EI
  . therefore, R = -(5/8)(wL)
   . top . . . :   79 lbf both ends, 262 lbf middle
   . bottom  :  236 lbf both ends, 788 lbf middle
 * deflection for uniformly loaded beam: y = ( w x / 24 EI ) ( L^3^ - 2 Lx^2^ + x^3^ )
 * deflection for center point loaded beam: y = ( R x / 48 EI ) ( 3 L^2^ - 4 x^2^ )  ... 0 < x < L/2
  . see [[http://en.wikipedia.org/wiki/Deflection_(engineering)]]
 * compound force deflection: y = ( w / 384 EI ) ( L^3^ x - 12 L x^3^ + 16 x^4^ )
  . test, x=L/2 : ( L^4^/2 - 3/2 L^3^ + L^3^ ) = 0 deflection at center
  . max deflection, first derivative = 0 = L^3^ -36Lx^2^ + 64x^3^, x=0.210767582704314 L
  . max deflection = 3.3851E-4 ( L^3^ (wL) / E I )
 * I = w t^3^ / 12 for beam width w, thickness t 
 * I = 3.23 in^4^ for '''11.5 x 1.5''' inch side board
  . (3.3851E-4*96^3^/(3.23*1E6)) ( wL ) = 9.2E-5 in/lb * (W/L)
  * 8 foot bed deflection '''0.042 inch top, 0.127 inch bottom'''
 * I = 0.404 in^4^ for 11.5 x 0.75 inch side board
  . (3.3851E-4*96^3^/(0.404*1E6)) ( wL ) = 7.4E-4 in/lb * (W/L)
  * 8 foot bed deflection 0.311 inch top, 0.934 inch bottom
  . (3.3851E-4*72^3^/(0.404*1E6)) ( wL ) = 3.13E-4 in/lb * (W/L)
  * 6 foot bed deflection 0.144 inch top, 0.432 inch bottom