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 MA at A produces a clockwise rotation θA at end A, the moment at end B is assumed MB while the slope at the fixed end B is zero. The vertical reactions at VA and VB 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 MA and MB.
i.e. - VA = VB = MA+MB/L
To find out MA and MB we shall apply the moment area method. The bending moment (÷ EI) diagram is shown in Figure. At the left hand end clockwise MA is positive (sagging) BM while at the right hand end clockwise MB is a negative (sagging) BM.