This is method of structural analysis which is given by the Prof. Maney in early of the 20th century (b/w 1910 to 1920). This is a displacement method of structural analysis, in which the end moment at the joints of the beams are expressed in the terms of the displacements and the fixed end moments. These expressions are known as the slope deflection equations.

Then the compatibility or the equilibrium equations are found for the structures, and these equilibrium equations are solved to get the unknown displacement at the joints. When the unknown displacements are found we can put these displacements in the slope deflection equations to find the final end moments. Once the final end moments are found, we can find the vertical reactions and the horizontal reactions at the various supports, to draw the shear force and the Bending moment diagrams.

A simple form of the slope deflection equation for a span AB of a continuous beam is written in the form as shown in the picture below.

In the beam shown in the figure, the kinematic indeterminacy is only two, i.e. angle at B and angle at C. So you need two equilibrium equations to get these values. Once you get the values, you can find the end moments by using the equations above.

The 2 equilibrium equations for the structure shown above can be created,on the basis of the fact that the sum of the left end moment and the right end moments at the joints B and C will be zero, because both of the joints are hinges, so they will not take any moment. So by equating the sum of the end moments at the two joints, and solving the two linear equations, simultaneously, I can get the unknown values of the slopes.

So now putting these values in the equations above, I can get the final end moments.

Then the compatibility or the equilibrium equations are found for the structures, and these equilibrium equations are solved to get the unknown displacement at the joints. When the unknown displacements are found we can put these displacements in the slope deflection equations to find the final end moments. Once the final end moments are found, we can find the vertical reactions and the horizontal reactions at the various supports, to draw the shear force and the Bending moment diagrams.

A simple form of the slope deflection equation for a span AB of a continuous beam is written in the form as shown in the picture below.

In the beam shown in the figure, the kinematic indeterminacy is only two, i.e. angle at B and angle at C. So you need two equilibrium equations to get these values. Once you get the values, you can find the end moments by using the equations above.

The 2 equilibrium equations for the structure shown above can be created,on the basis of the fact that the sum of the left end moment and the right end moments at the joints B and C will be zero, because both of the joints are hinges, so they will not take any moment. So by equating the sum of the end moments at the two joints, and solving the two linear equations, simultaneously, I can get the unknown values of the slopes.

So now putting these values in the equations above, I can get the final end moments.