Theory of Structures II (ENCE 252): Moment Distribution Method

An Introduction to the Moment Distribution Method

The Moment Distribution Method is a powerful iterative technique for analyzing statically indeterminate beams and frames. Developed by Professor Hardy Cross in 1930, it revolutionized structural analysis by providing a practical, intuitive, and less computationally intensive alternative to classical methods. The method involves successively “locking” and “unlocking” joints to distribute moments until the system reaches equilibrium.

The core of the method relies on a few fundamental concepts:

• Fixed-End Moments (FEMs): Initial moments at the ends of members, assuming all joints are fixed against rotation.
• Stiffness (k): The moment required to produce a unit rotation at one end of a member. For a member with far-end fixed, $k = \frac{4EI}{L}$. For a member with far-end pinned, $k = \frac{3EI}{L}$.
• Distribution Factor (DF): The proportion of an unbalanced moment at a joint that is distributed to a specific member connected to that joint. It’s calculated as the ratio of the member’s stiffness to the total stiffness of all members meeting at the joint: $DF = \frac{k}{\sum k}$.
• Carry-Over Moment: When a moment is applied to one end of a prismatic member, a moment is induced at the other (fixed) end. The carry-over factor is typically 0.5, meaning half the moment is “carried over”: $M_{carry-over} = 0.5 \times M_{distributed}$.

By systematically applying these principles, engineers can accurately determine the final moments in complex structures, paving the way for detailed shear, moment, and thrust diagram construction.