Moment Distribution Method

Moment Distribution Method

An iterative method for analyzing continuous beams and rigid frames.

Introduction

An iterative displacement-based method, developed by Hardy Cross, used to analyze continuous beams and rigid frames. It does not require solving simultaneous equations, making it historically significant for hand calculations.

Key Concepts

Fundamental Terms

The Hardy Cross Method relies on defining the rotational characteristics of the members meeting at a joint.

Fixed-End Moments (FEM): The initial moments produced at the ends of a member by external loads, assuming all joints are artificially locked against rotation and translation.
Stiffness Factor ( $K$ ): The amount of moment required to rotate the near end of a member by one radian while the far end remains fixed. For a prismatic member: $K = \frac{4EI}{L}$ .
Distribution Factor (DF): When a joint is unlocked, any unbalanced moment is distributed among the connecting members in proportion to their relative stiffnesses. $DF = \frac{K}{\sum K}$ . The sum of DFs at any rigid joint must equal 1.0. A fixed support has $DF = 0$ , and a pinned support has $DF = 1$ .
Carry-Over Factor (COF): The fraction of the distributed moment applied at the near end that is "carried over" to the fixed far end. For prismatic members, $COF = 0.5$ .

Procedure

Procedure

STEP-BY-STEP

Determine the Fixed-End Moments (FEM) for each loaded member, assuming all joints are locked (fixed).
Calculate the Stiffness Factors ( $k$ ) for all members.
Calculate the Distribution Factors (DF) for each member at every joint. A fixed support has $DF = 0$ , and a pinned support has $DF = 1$ .
Release the joints one at a time. Calculate the unbalanced moment at the released joint (the algebraic sum of the FEMs).
Distribute this unbalanced moment to the connected members using their respective Distribution Factors. This "balancing" moment will be opposite in sign to the unbalanced moment.
Carry over half of the distributed moment ( $COF = 0.5$ ) to the far end of each member.
Re-lock the joint.
Repeat steps 4-7 for all joints until the unbalanced moments are negligibly small.
Sum the initial FEMs, all distributed moments, and all carried-over moments at each member end to find the final internal end moments.

Frames with Sidesway

Analyzing frames that can translate laterally requires an additional correction step because the standard moment distribution process only accounts for joint rotations, not joint translations.

Procedure

STEP-BY-STEP

No-Sway Analysis: Apply artificial lateral supports to prevent the frame from swaying. Perform a standard moment distribution analysis for the applied loads. Calculate the artificial holding force ( $R$ ) required at the support using statics.
Sway Analysis: Remove the applied loads. Assume an arbitrary lateral displacement ( $\Delta'$ ), which induces arbitrary fixed-end moments in the columns (e.g., $M' = -\frac{6EI\Delta'}{L^2}$ ). Perform a second moment distribution on these arbitrary moments. Calculate the arbitrary lateral force ( $R'$ ) corresponding to this sway.
Correction Factor: Since the structure is linear-elastic, the actual sway forces are proportional to the arbitrary sway forces. Determine the correction factor $c = \frac{R}{R'}$ , where $R$ is the unbalanced lateral force from the no-sway analysis that must be negated.
Final Moments: The true final moments are the sum of the moments from the no-sway analysis and the moments from the sway analysis multiplied by the correction factor: $M_{final} = M_{no-sway} + c(M_{sway})$ .

Interactive Tool: Moment Distribution Simulator

Visualize the iterative distribution and carry-over of moments in a continuous beam.

Moment Distribution Interactive Lab

Adjust the initial Fixed End Moments (FEM) and the Distribution Factor at Joint B for a two-span continuous beam (fixed at A and C). Watch how the unbalanced moment at B is iteratively distributed and carried over.

Distribution Factor DF_BA: 0.50(DF_BC = 0.50)

FEM_AB

FEM_BA

FEM_BC

FEM_CB

Joint	A	B		C
Member	AB	BA	BC	CB
DF	0	0.50	0.50	0

* Check Equilibrium at B: + = NaN

Distribution Factor and Carry-Over Factor

Definitions

Distribution Factor (DF): The ratio of the stiffness of a specific member to the sum of the stiffnesses of all members meeting at a joint. It dictates how an unbalanced moment is distributed among connecting members. $DF_i = K_i / \sum K$ .
Carry-Over Factor (COF): The fraction of the distributed moment at one end of a member that is transferred (carried over) to the far end. For a prismatic member with a fixed far end, the COF is exactly $+0.5$ .

Modified Stiffness

Simplification for Pinned Far Ends

To speed up convergence and eliminate the need to balance joints at pinned or roller supports, the stiffness

K

of a member can be modified.

If the far end is fixed: $K = 4EI / L$
If the far end is pinned/roller: $K_{mod} = 3EI / L$ (and the carry-over factor to the pin becomes zero).

Key Takeaways

Moment Distribution is an iterative displacement method for analyzing frames and continuous beams.
It relies on calculating Fixed-End Moments, Stiffness Factors, and Distribution Factors.
The process involves successively locking and releasing joints, distributing unbalanced moments, and carrying over moments to adjacent joints until the unbalanced moments are negligible.
The Moment Distribution Method (Hardy Cross Method) is an iterative technique that avoids solving large systems of simultaneous equations.
It relies on successive locking and unlocking of joints, distributing unbalanced moments according to member stiffnesses.
Key parameters include Fixed-End Moments (FEM), Stiffness Factors ( $K$ ), Distribution Factors ( $DF$ ), and Carry-Over Factors ( $COF$ ).
Frames with sidesway require a two-stage analysis: a "no-sway" analysis followed by an artificial "sway" correction scaled to satisfy story shear equilibrium.

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