Slope & Moment Distribution

Cheatsheet Content

### Moment Distribution: Step-by-Step (Table Method) **Goal:** Find moments at member ends using an iterative table. 1. **List FEMs:** * Draw the beam/frame. * For *each member*, calculate `Fixed-End Moments (FEMs)` assuming all ends are fixed. * *Example (UDL)*: $FEM_{AB} = -wL^2/12$, $FEM_{BA} = +wL^2/12$. * *Example (P at mid)*: $FEM_{AB} = -PL/8$, $FEM_{BA} = +PL/8$. 2. **Calculate Stiffness (K):** * For *each member*, calculate `K = I/L`. (Can use $4EI/L$ or $3EI/L$ if $E$ is not constant, but $I/L$ is common for relative stiffness). * *Special Case (Far end hinged/roller)*: Use `K = (3/4)I/L` or `3EI/L` effectively. 3. **Calculate Distribution Factors (DF):** * For *each joint*, sum the K values of all members meeting at that joint ($\sum K_{joint}$). * For *each member end at that joint*: $DF = K_{member} / \sum K_{joint}$. * *Special Cases:* * `Fixed end`: DF = 0. * `Simply supported/hinged end (at the very end of the structure)`: DF = 1. 4. **Set Up Table:** * Draw a table. Each *column* represents a member end (e.g., AB, BA, BC, CB). * **Row 1:** Joint Name (e.g., A, B, C, D). *Write above the member ends it applies to.* * **Row 2:** Member End (e.g., AB, BA, BC, CB). * **Row 3:** DF (from Step 3). *Place under the corresponding member end.* * **Row 4:** FEM (from Step 1). *Place under the corresponding member end.* 5. **Iterative Process (Balance & Carry Over):** * **a. Balance Joints:** * For *each internal joint* (where DF is not 0 or 1): * Sum the moments currently in the "working" row for all member ends meeting at that joint. This is the `Unbalanced Moment`. * For each member end at that joint, calculate the `Distributed Moment = -Unbalanced Moment * DF`. * Write these `Distributed Moments` in a new row below the FEMs. * **b. Carry Over (CO):** * For *each Distributed Moment* you just wrote: * Calculate `Carry Over Moment = 0.5 * Distributed Moment`. * Write this `Carry Over Moment` in the *next* row, under the *opposite end* of the member. * *Important:* If a member's far end is a simply supported end (DF=1 at the structure's edge), you do NOT carry over *to* it. * **c. Repeat:** * Continue steps `a` and `b` in cycles. Each cycle will start by balancing the `Carry Over Moments` from the previous step. * Stop when the distributed and carry-over moments are negligibly small (e.g.,