Response:
3.03 INCHES
Clarification:
Following ASTM D198;
The formula for the modulus of rupture is: ( M / I ) * y ----- ( 1 )
Where M (bending moment) is calculated as R * span length / 2
= (120 * 10^3 ) * 48 / 2 = 288 * 10^4 Ib-in
I (moment of inertia) is derived from the formula bd^3 / 12
= ( 2 )*( d )^3 / 12 = 2d^3 / 12
Given b = 2 in, and d is unknown
The span length = 4 * 12 = 48 inches
R = P / 2 = 240 * 10^3 / 2 = 120 * 10^3 Ib
y (distance to centroid) equals d / 2 inches
Refer back to equation ( 1 )
( M / I ) * y
940.3 ksi = ( 288 * 10^4 / 2d^3 / 12 ) * d / 2
= ( 288 * 10^4 * 12 ) / 2d^3 ) * d / 2
940300 = 34560000* d / 4d^3
4d^3 ( 940300 ) = 34560000 d (by dividing both sides by d )
4d^2 = 34560000 / 940300
d^2 = 9.188 ∴ Thus, d is approximately 3.03 in