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SLOPE DEFLECTION METHOD

The slope deflection equations express the member end moments in terms of rotation angles and deflections.The slope deflection method is used for analysis of statically indeterminate structures such as beams and frames. This method was  introduced  by Prof.George A. Maney,in 1915.
Sign Convention:
1. The clock wise moments,and clock wise rotations/slopes are taken as positive ones.
2. The down ward displacements of the right end with respect to the left end of horizontal  member is considered as positive.
3. The right ward displacement of upper end with respect to lower end of a vertical member is taken as positive. 
Assumptions in the slope- deflection method:
1. The material of the structure is linearly elastic.
2. The structure is loaded with in elastic limit.
3. Axial displacements,Shear displacements are neglected.
4. Only flexural deformations are considered
The slope deflection equations of member AB of flexural rigidityEabIab and length Lab and the four parts represent by:

  1. The moment caused by the rotation at the near end A.
  2. The moment caused by the rotation at the far end B.
  3. The moment created by translation.
  4. The fixed end moment created by the loading.
M_{ab} = \frac{E_{ab} I_{ab}}{L_{ab}} \left( 4 \theta_a + 2 \theta_b - 6 \frac{\Delta}{L_{ab}} \right)
M_{ba} = \frac{E_{ab} I_{ab}}{L_{ab}} \left( 2 \theta_a + 4 \theta_b - 6 \frac{\Delta}{L_{ab}} \right)
where θa, θb are the slope angles of ends a and b respectively, Δ is the relative lateral displacement of ends a and b. The absence of cross-sectional area of the member in these equations implies that the slope deflection method neglects the effect of shear and axial deformations.
Total end moment                     M'ab = Mfab + Mab
                                M'ba = Mfba + Mba

Joint equilibrium

Joint equilibrium conditions imply that each joint with a degree of freedom should have no unbalanced moments i.e. be in equilibrium. Therefore,
\Sigma \left( M^{f} + M_{member} \right) = \Sigma M_{joint}
Here, Mmember are the member end moments,
 Mf are the fixed end moments, and Mjoint are the external moments directly applied at the joint.
The general procedure using slope- deflection method:
  1. Draw the separate the parts of the beam in  between supports.
  2. Compute the  fixed end moments 
  3. Write the moment equations for each beam segment.
    Mab = 2 EI/ L[ 2 qn + qf - 3 Dnf/l] + Mfab

  4. Write the compatibility condition/boundary conditions. The joints are combined to get zero moment, end conditions (moment = 0 or slope, q= 0). Shear condition if dealing with a sway in the column.
  5. Write the equations in matrix form and solve for the components.
  6. Solve the moment equations.
  7. Solve for the reactions for the beam/frame. By using the forces and the moments to find the individual components.
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