Where: ya yb are the End rotations
r = the relative deflection between the ends
mab1 Mb2 = induced end moments
Sign Convention:
M. 1. moment acting on the end of a member (not
joint) is positive when clockwise
y 2. Rotation: @ the end of a member
is positive when the tangent to the deformed
curve @ the end Rotates clockwise from its Original position:
r
3. the relative deflection between ends of a member is Positive,
when it corresponds to a clockwise rotation of the member
In Fig.
The end Moments mab and Mba may be considered as the Algebraic sum of Four separate effect
1. the moment due to end rotation ya. while the other end b is fixed
2. the moment due to end rotation yb, while end a is Fixed
3. the moment due to relative deflection r between the ends of the member without
altering the existing slopes of tangents @ the ends.
4. the moment caused by placing the actual load
on the span without altering the end distortions
Ma =
0] : ½ [m’ab/EI] [L] [L/3] – ½ [m’ba/EI] [L] [2/3L] = 0 à 1
Mb =
0]: ya(L) – ½ [m’ab/EI] [L] [2/3L] + ½ [m’ba/ei] [L] [L/3] = 0 à
2
Eq.1 m’ab
= 2M’ba or M’ba = ½ M’ab --- subst to eq 2
* M’ab = 4EI y a/L
*m’ba = 2Ei y a/L
Ma = 0: ½ [M”ab/EI] [L] [2/3L] = 0
and:
b= 0]: ½ [M”ab/EI]
[L] [2/3L] = i/2 [m”ba/EI] [L] [L/3]
2M”ab = M”ba
M’ab = ½ M”ba
subst:
½ [M”ab/EI]
[L] [L/3] - y b (L) – [M”ab/EI]
[L] [2/3] = 0
M”ab/EI
[L/6 – 2L/3] = y b
½ [M/EI]
[L] [L/3] + r - ½ [M/EI] [L] [2/3L] =
0
ML2/6EI
+ r - ML2/6EI (2) = 0
ML2/6EI
= r
: M = 6EIr / L2
*
M”’ab = M”’ba = - M = - 6EIr /L2