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:
$$ M = F \cdot d $$
