Slope Deflection Equations for a Member:
In this section, we will develop a few general equations needed for further discussion in the moment distribution context.
Figure 5(a) shows a beam hinged at the end A and fixed at B. A clockwise moment M_{A} at A produces a clockwise rotation θ_{A} at end A, the moment at end B is assumed M_{B} while the slope at the fixed end B is zero. The vertical reactions at V_{A} and V_{B} are equal in magnitude but their directions will be opposite as shown in the diagram producing an anticlockwise couple to balance the couple due to the end moments M_{A} and M_{B}.
i.e. - V_{A} = V_{B} = M_{A}+M_{B}/L
To find out M_{A} and M_{B} we shall apply the moment area method. The bending moment (÷ EI) diagram is shown in Figure. At the left hand end clockwise M_{A} is positive (sagging) BM while at the right hand end clockwise MB is a negative (sagging) BM.