Title 24 Three Moment Equation.pdf ✅

If EI is not constant for both spans, 1 6E1I1 ( 6A1a1 L1 + M1L1 + 2M2L1) + 1 6E2I2 ( 6A2b2 L2 + M3L2 + 2M2L2) = h1 L1 + h3 L2 If the beam is prismatic, EI is constant, 1 6EI ( 6A1a1 L1 + M1L1 + 2M2L1) + 1 6EI ( 6A2b2 L2 + M3L2 + 2M2L2) = h1 L1 + h3 L2 6A1a1 L1 + M1L1 + 2M2L1 + 6A2b2 L2 + M3L2 + 2M2L2 = 6EI ( h1 L1 + h3 L2 ) M1L1 + 2M2 (L1 +L2 )+ M3L2 + 6A1a1 L1 + 6A2b2 L2 = 6EI ( h1 L1 + h3 L2 ) If the points 1, 2, and 3 are taken at supports and therefore unyielding, h1 = h3 = 0, M1L1 + 2M2 (L1 +L2 )+ M3L2 + 6A1a1 L1 + 6A2b2 L2 = 0 2. Three Moment Factors For the value of the area moment factors 6A1a1 L1 and 6A2b2 L2 , consider a simply supported span with a concentrated load. 6Aa L = ( 6 L ) ( 1 2 ) ( Pab L ) (L) ( a +L 3 ) = Pa(L −a) L (L +a) 6Aa L = Pa L (L 2 − a 2 ) 6Ab L = ( 6 L ) ( 1 2 ) ( Pab L ) (L) ( b + L 3 ) = Pb(L − b) L (L + b) 6Ab L = Pb L (L 2 − b 2 ) For a uniformly distributed load, using integration,