--- title: "Focal point ratio: define and derive left ratio" subject: "Theory of Structures II" chapter: 4 key: "semester_5/theory_of_structures_ii:focal-point-ratio-left-derive" source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/focal-point-ratio-left-derive site: BCT Engineering --- # Focal point ratio: define and derive left ratio Theory of Structures II · Chapter 4: Influence line (IL) for continuous beams - Asked in: 2070 Asar, 7 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/focal-point-ratio-left-derive - Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-4 ## Question Define focal point ratio and derive the expression to determine the left focal point ratio. ## Short answer Left focal ratio k_n = -M_{n-1}/M_n for an unloaded span left of the loaded span. 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}]; for equal spans k_{n+1}=1/(4-k_n), with k_1=0 for a hinged left end or 1/2 for a fixed end. ## Answer #### Definition In a continuous beam loaded on one span only, the bending moment diagram in every unloaded span is a straight line, and it cuts the beam axis at a fixed point called the **focal point** of that span. The **focal point ratio** is the ratio of the moments at the two ends of such an unloaded span (far-end moment to near-end moment, with opposite signs): - **Left focal ratio** $k_n$: for an unloaded span to the **left** of the loaded span, $k_n = -\dfrac{M_{n-1}}{M_n}$ (moment at the left end divided by the moment at the right end). - **Right focal ratio** $k'_n$: for an unloaded span to the **right** of the loaded span, $k'_n = -\dfrac{M_n}{M_{n-1}}$. #### Derivation of the left focal ratio 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 $n$. #### Numerical example Hinged left end A, span 1: $L_1 = 6$ m, $I_1 = I$; span 2: $L_2 = 4$ m, $I_2 = 2I$. - $k_1 = 0$ (hinged end). - $k_2 = \dfrac{4/2}{(2-0)\times6/1 + 2\times4/2} = \dfrac{2}{12+4} = 0.125$. - The focal point of span 2 lies $L_2/(1+k_2) = 4/1.125 = 3.56$ m from its right end (support 2), that is 0.44 m from support 1. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Focal point ratio is the ratio of A. Span length to inertia B. Load to reaction C. Left span to right span D. Far-end to near-end moment of an unloaded span (opposite signs) Answer: D. The unloaded span BMD is a straight line with fixed end-moment ratio. (Key point: Definition) 2. Equation used in the derivation A. Three-moment equation with load terms zero B. Slope-deflection with sway C. Castigliano on one span D. Moment area for a single span Answer: A. For unloaded spans the load terms vanish. (Key point: Start) 3. Left focal point lies from support n at A. L_n/2 B. L_n/(1+k_n) C. L_n k_n D. L_n(1+k_n) Answer: B. Set M(x) = M_n[1-(1+k_n)x/L_n] = 0. (Key point: Focal point) 4. With k_1 = 0, L1 = 6, I; L2 = 4, 2I, k_2 = A. 0.25 B. 0.5 C. 0.125 D. 0.0625 Answer: C. 2/(12+4) = 0.125. (Key point: Example) ## Related in Theory of Structures II - [Left and right focal point ratios](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Afocal-point-ratios-left-right) — Theory of Structures II, chapter 4 - [ILD as system-specific; focal point recurrence](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aild-system-specific-focal-point-recurrence) — Theory of Structures II, chapter 4 - [ILD by focal point method: moment at support 2](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aild-moment-support-2-focal-point) — Theory of Structures II, chapter 4 ## Related in other subjects - [Derive tractive force ratio (shear reduction factor)](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Atractive-force-ratio-derivation) — Hydraulics, chapter 10, asked 5× - [Derive expression for tile drain spacing](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Atile-drain-spacing-derivation) — Irrigation and Drainage Engineering, chapter 9, asked 8× - [Derive development length expression](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Aderive-development-length-expression) — Design of RCC Structure, chapter 6, asked 6× - [Derive siphon inlet to outlet leg ratio](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Asiphon-inlet-outlet-leg-ratio) — Hydraulics, chapter 2