--- title: "Neutral point in unloaded span of continuous beam" subject: "Theory of Structures II" chapter: 4 key: "semester_5/theory_of_structures_ii:neutral-point-continuous-beam" source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/neutral-point-continuous-beam site: BCT Engineering --- # Neutral point in unloaded span of continuous beam Theory of Structures II · Chapter 4: Influence line (IL) for continuous beams - Asked in: 2069 Chaitra, 5 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/neutral-point-continuous-beam - Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-4 ## Question Define and explain what is neutral point in an unloaded span of a continuous beam. Derive the recurrent formula for its determination. ## Short answer Neutral point is the zero-moment point on an unloaded span of a continuous beam; its position depends only on spans, stiffnesses and end condition, not on the load. From the three-moment equation, k_{n+1} = [L_{n+1}/I_{n+1}] / [(2-k_n)L_n/I_n + 2L_{n+1}/I_{n+1}], with k_n = -M_{n-1}/M_n; equal spans give k_{n+1}=1/(4-k_n). ## Answer #### Neutral point When one span of a continuous beam is loaded, the bending moment diagram of every unloaded span is a **straight line** with moments of **opposite sign** at its two ends (the beam is hogging at one support and sagging at the other end of the span). The point on the unloaded span where this line crosses the axis, that is, where the bending moment is zero, is the **neutral point** (also called the focal point). It is a point of contraflexure in that span. Properties: - Its position depends only on the spans, the stiffnesses ($EI$) and the end condition, not on the load: it is the same for any loading on the other spans lying on the same side of it. - Once it is known, the whole BMD of the unloaded span follows from one support moment, since the line passes through the neutral point. - There is one neutral point (left) for loading on the right, and another (right) for loading on the left. #### Recurrent formula Consider a continuous beam with spans numbered from the left; span $n$ lies between supports $n-1$ and $n$, with length $L_n$ and moment of inertia $I_n$. Let the load be on a span to the right of span $n+1$ (all spans from the left end up to $n+1$ unloaded), and let moments be taken with one sign convention (hogging or sagging) at all supports. **Three-moment equation** at support $n$ for the unloaded spans $n$ and $n+1$ (the load terms are zero): $$ M_{n-1}\frac{L_n}{I_n} + 2M_n\left(\frac{L_n}{I_n}+\frac{L_{n+1}}{I_{n+1}}\right) + M_{n+1}\frac{L_{n+1}}{I_{n+1}} = 0 $$ Because the moment in an unloaded span is linear, the moments at its two ends have opposite signs and a fixed ratio. Define the **left focal ratio** of span $n$: $$ k_n = -\frac{M_{n-1}}{M_n} $$ and similarly for span $n+1$: $M_n = -k_{n+1}M_{n+1}$. Substitute $M_{n-1} = -k_nM_n$ in the equation: $$ M_n\left[(2-k_n)\frac{L_n}{I_n} + 2\frac{L_{n+1}}{I_{n+1}}\right] = -M_{n+1}\frac{L_{n+1}}{I_{n+1}} $$ so that $M_n/M_{n+1} = -k_{n+1}$ gives the **recurrence formula** $$ \boxed{\,k_{n+1} = \frac{L_{n+1}/I_{n+1}}{(2-k_n)\,L_n/I_n + 2\,L_{n+1}/I_{n+1}}\,} $$ For equal spans with constant $EI$ this becomes $k_{n+1} = \dfrac{1}{4-k_n}$. **Starting value:** if the left end of the beam is hinged, $M_0 = 0$ so $k_1 = 0$; if it is fixed, the slope there is zero, $2M_0 + M_1 = 0$, so $k_1 = \tfrac12$. **Focal point:** the straight BMD of unloaded span $n$ is $M(x) = M_n\left[1 - (1+k_n)\dfrac{x}{L_n}\right]$, with $x$ measured from support $n$, so it crosses the beam axis at $x = \dfrac{L_n}{1+k_n}$ from support $n$. This point is the left focal point, and it is the same for any loading on the spans to the right of span $n$. #### Example (equal spans, constant $EI$, hinged end) $k_1 = 0$, $k_2 = \dfrac{1}{4} = 0.25$, $k_3 = \dfrac{1}{4-0.25} = 0.2667$. In the second span the neutral point lies $L/(1+k_2) = 0.8L$ from its right support (0.2L from the left support). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The neutral point of an unloaded span is A. Where its straight BMD crosses zero B. Where deflection is zero C. Where shear is maximum D. The span mid-point Answer: A. It is a point of contraflexure. (Key point: Definition) 2. Its position depends on A. Magnitude of load B. Spans, EI and end condition, not the load C. Position of the load D. Only the load span Answer: B. The same for any loading on that side. (Key point: Independence) 3. Equal spans, constant EI, hinged end: k_3 is A. 0.25 B. 0.2679 C. 0.2667 D. 0.5 Answer: C. 1/(4-0.25) = 0.2667. 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