Where: ya yb are the End rotations

r = the relative deflection between the ends

mab_{1} Mb_{2 } = 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

ML_{}^{2}/6EI
+ r - ML^{2}/6EI (2) = 0

ML^{2}/6EI
= r
: M = 6EIr / L^{2}

^{
}

^{ }*
M”’ab = M”’ba = - M = - 6EIr /L^{2}