--- title: "Interaction diagram and failure modes" subject: "Design of RCC Structure" chapter: 10 key: "semester_7/design_of_rcc_structure:interaction-diagram-and-failure-modes" source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/interaction-diagram-and-failure-modes site: BCT Engineering --- # Interaction diagram and failure modes Design of RCC Structure · Chapter 10: Design of compression members: Columns - Asked in: 2080 Bhadra, 4 marks - Video: [Intro to Interaction Diagrams for Concrete Columns | Reinforced Concrete Column Design](https://www.youtube.com/watch?v=lyQ4WqXPTv8) — Tyler Ley - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/interaction-diagram-and-failure-modes - Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-10 ## Question Define interaction diagram with its features and neat sketches. Discuss on the modes of failure for compression members in eccentric compression within the interaction diagram. ## Short answer Interaction diagram plots combinations of axial load Pu and moment Mu a column can carry at failure; inside is safe. Compression failure occurs above balanced point with deep neutral axis and brittle crushing; tension failure below balanced point with steel yielding first and ductile behavior; balanced point has simultaneous crushing and yielding. ## Answer An **interaction diagram** is a curve (or a set of curves) that shows the combinations of axial load $P_u$ and bending moment $M_u$ that a given column section can carry at failure. Any pair $(M_u, P_u)$ lying inside the curve is safe; a point on or outside the curve means failure. It is drawn by moving the neutral axis from very deep (pure compression) to very shallow (pure bending) and finding $P_u$ and $M_u$ for each position from strain compatibility. SP-16 gives ready charts as non-dimensional plots of $P_u/(f_{ck}bD)$ against $M_u/(f_{ck}bD^2)$. #### Features ```text Pu ^ Puz* | * compression failure | * (small e, xu > xu,max) | * Pb |..........* balanced point (Mb, Pb) | * | * tension failure | * (large e, xu < xu,max) 0 +---*---------------------> Mu M0 (pure bending) ``` - The top point $P_{uz}$ is pure axial compression ($M = 0$); the point on the $M_u$ axis is pure bending ($M_0$). - The balanced point $(M_b, P_b)$ is where the concrete crushes ($\varepsilon_c = 0.0035$) and the tension steel yields at the same time. It separates the two failure zones, and the moment capacity is greatest there. - Below $P_b$, the moment capacity increases as the axial load increases; above $P_b$, it decreases. - A radial line from the origin has slope $1/e$, so it represents a constant eccentricity. - Curves are plotted for different values of $p/f_{ck}$, for a given $d'/D$ and steel grade. - A design point is safe if it lies inside the curve. #### Modes of failure (eccentric compression) 1. **Compression failure** (small eccentricity, $P_u > P_b$, neutral axis deep, $x_u > x_{u,max}$): the concrete crushes first at the more compressed face, while the steel on the far side has not yielded. The failure is sudden (brittle) and lies on the upper part of the curve. 2. **Balanced failure** ($P_u = P_b$): the concrete crushes and the tension steel yields simultaneously. 3. **Tension failure** (large eccentricity, $P_u < P_b$, $x_u < x_{u,max}$): the tension steel yields first, followed by crushing of the concrete. The failure is gradual (ductile) and lies on the lower part of the curve. 4. Pure axial compression ($e = 0$) is a limiting case at the top of the curve. Since IS 456 requires every column to be designed for at least the minimum eccentricity $e_{min}$, the top of the design curve is cut off (for an axially loaded short column, $P_u = 0.4 f_{ck} A_c + 0.67 f_y A_{sc}$). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does a point inside the interaction curve represent? A. Failure B. A safe combination of Pu and Mu C. Pure bending only D. Balanced failure Answer: B. The curve shows the capacity envelope; points inside are safe. (Key point: Meaning of the diagram) 2. What happens at the balanced point? A. Concrete crushes at 0.0035 as steel yields simultaneously B. Steel yields only C. Concrete never crushes D. Neutral axis is at the centroid Answer: A. It separates compression and tension failure zones. (Key point: Balanced point) 3. Which statement describes compression failure? A. Large eccentricity, ductile, steel yields first B. Small eccentricity, Pu > Pb, concrete crushes first and failure is brittle C. Pu < Pb with gradual failure D. Occurs only in pure bending Answer: B. With deep neutral axis and small eccentricity the compression face crushes before tension steel yields. (Key point: Compression failure) 4. How does moment capacity vary with axial load? A. Increases with Pu up to Pb, then decreases B. Always decreases C. Always increases D. Unaffected Answer: A. Below Pb axial compression helps moment capacity, above Pb it reduces it. 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