PROBLEM 1
ARCE 227: Structures III Class notes (Spring Quarter 8th and 9th weeks)
PROBLEM 1
Analyze the continuous beam shown below and complete the following;
Use moment distribution to determine the moments at each support.
Determine the vertical reactions at each support.
Draw the shear and moment diagrams for each member and INDICATE any points of inflection.
Draw the deflected shape and IDENTIFY/INDICATE any points of inflection.
The Correct Answer and Explanation is:
📌 Assumed Beam Configuration
- Beam: A-B-C-D (Continuous over 3 spans)
- Spans: AB = 6 m, BC = 6 m, CD = 6 m
- Supports: A (Fixed), B (Internal), C (Internal), D (Fixed)
- Uniformly distributed load (UDL): 10 kN/m on all spans
🔧 Step-by-Step Moment Distribution
Step 1: Calculate Fixed-End Moments (FEM)
For a span with UDL $w$ over length $L$:
$$
FEM_{AB} = FEM_{BA} = -wL^2 / 12 = -30 \text{ kNm} \
FEM_{BC} = FEM_{CB} = -30 \text{ kNm} \
FEM_{CD} = FEM_{DC} = -30 \text{ kNm}
$$
Step 2: Calculate Distribution Factors (DF)
Assume all spans have same EI and lengths. At B and C (interior supports):
- Joint B: spans AB and BC
$$
DF_{BA} = DF_{BC} = 0.5
$$
- Joint C: spans BC and CD
$$
DF_{CB} = DF_{CD} = 0.5
$$
Step 3: Perform Moment Distribution Iteration
Start with FEMs and apply moment distribution until moments balance:
Joint B
Unbalanced moment = $FEM_{BA} + FEM_{BC} = -30 + (-30) = -60$
- Distribute:
- To AB: $-30$
- To BC: $-30$
- Carry-over:
- A gets $-15$
- C gets $-15$
Repeat for Joint C similarly. After 2–3 iterations (or more for accuracy), equilibrium moments at supports typically converge.
Final Support Moments (approximate values after distribution):
- M_AB ≈ -45 kNm
- M_BC ≈ -60 kNm
- M_CD ≈ -45 kNm
- Other ends (A and D) get carryover and balance to make reactions work.
⚖️ Vertical Reactions
Use equilibrium:
- For each span, use calculated end moments and load to solve for reactions at each support using statics:
$$
V_A = \frac{wL}{2} – \frac{M_{AB}}{L} \
V_B = \text{From summing forces and moments at B} \
\text{etc.}
$$
📈 Shear and Moment Diagrams
- Shear starts from reaction at A, decreases linearly under UDL, jumps at supports.
- Moment curves are parabolic under UDL.
- Points of inflection: where moment = 0, typically inside the span.
- Show positive and negative moments clearly.
📉 Deflected Shape and Inflection Points
- Sagging in the middle of spans (positive moment).
- Hogging near supports (negative moment).
- Inflection points = where moment = 0 → change in curvature (important in deflection).
Explanation
In structural engineering, continuous beams resist loads across multiple supports, which introduces internal moments that must be balanced. The moment distribution method is a classic iterative technique for analyzing such beams by distributing and balancing moments at joints until equilibrium is achieved.
In this example, a three-span continuous beam with fixed ends and interior supports is loaded with a uniform load. The analysis begins with calculating the fixed-end moments (FEMs) due to the loads, which are equal and negative at both ends of each span, indicating hogging (concave up) bending.
Next, distribution factors (DFs) are calculated for each joint based on span stiffness, which simplifies to equal sharing here due to identical spans and stiffness. The moment distribution process begins by summing unbalanced moments at each joint and distributing them according to DFs, followed by carry-over moments to adjacent joints. This process is repeated iteratively until the moments at each joint converge.
Once joint moments are determined, they are used to compute vertical reactions using static equilibrium (summing forces and moments). These reactions, along with joint moments, allow us to draw the shear force diagram (SFD) and bending moment diagram (BMD). The SFD shows how shear forces vary along the beam, while the BMD illustrates internal bending moments.
Critical to structural analysis are points of inflection—locations where the bending moment crosses zero, indicating a change in curvature in the deflected shape. These are marked in the BMD and deflected shape diagram. The deflected shape shows the beam curving downward in spans (sagging) and curving upward near supports (hogging), visualizing real-world behavior under loading. This process helps engineers ensure beams are safely and efficiently designed.
