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 5(b). 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.