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Moment Distribution Method

The Moment Distribution Method, developed by Hardy Cross, is a powerful technique for analyzing indeterminate beams and frames. It simplifies solving by balancing moments at joints through successive approximations, eliminating the need to solve large systems of equations.

Key Concepts

• Stiffness (K): Measures a member’s resistance to rotation. Given by:
  $$ K = \frac{4EI}{L} $$
  $$ \text{Relative Stiffness} = \frac{I}{L} $$
• Distribution Factor (DF): Ratio of a member’s stiffness to the total stiffness at a joint.
  $$ DF = \frac{K}{\Sigma K} $$

  For fixed supports, DF = 0 --> no rotation.
  For hinged or roller ends, DF = 1 --> full moment transfer.

• Carry-Over Moment (COM): Half of the moment at one end is transferred to the other end of the member.
  $$ M_A = -\frac{1}{2} M_B $$
• Modified Stiffness: For members with a hinged or roller end, modify the relative stiffness, $I/L$, using:
  $$ K_{\text{mod}} = \frac{3}{4} \cdot \frac{EI}{L} = \frac{3EI}{4L} $$

Fixed-End Moment Formulas and Support Settlement

Free Ends

If a member terminates at a free end, no moment is distributed from that joint. However, you can compute the end moment directly using:

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