---
title: "Left and right focal point ratios"
subject: "Theory of Structures II"
chapter: 4
key: "semester_5/theory_of_structures_ii:focal-point-ratios-left-right"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/focal-point-ratios-left-right
site: BCT Engineering
---

# Left and right focal point ratios

Theory of Structures II · Chapter 4: Influence line (IL) for continuous beams

- Asked in: 2074 Asoj, 4 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-ratios-left-right
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-4

## Question

Define the term left and right focal point ratios. Also write their expressions.

## Short answer

Left focal ratio k_n = -M_{n-1}/M_n is the far-to-near end moment ratio of an unloaded span when loading lies to the right; right focal ratio k'_n = -M_n/M_{n-1} applies for loading to the left. Recurrences are k_{n+1} = (L_{n+1}/I_{n+1})/[(2-k_n)L_n/I_n + 2L_{n+1}/I_{n+1}] and the analogous right ratio formula.

## Answer

#### Focal point
When a continuous beam is loaded in one span only, the bending moment diagram in every unloaded span is a straight line. For a given unloaded span this line always passes through the same point of the beam axis, whatever the load position, and this point is called the **focal point** of the span.

#### Left and right focal ratios
- **Left focal ratio** ($k_n$ or $k^L_n$) of span $n$: the ratio of the moment at the left (far) end to the moment at the right (near) end of an unloaded span, when the load lies on spans to its **right**. The two end moments have opposite signs, so the ratio is defined with a minus sign to make it positive: $k_n = -M_{n-1}/M_n$. The left focal point of the span lies $L_n/(1+k_n)$ from its right support, i.e. $k_nL_n/(1+k_n)$ from its left support.
- **Right focal ratio** ($k'_n$ or $k^R_n$) of span $n$: the ratio of the moment at the right (far) end to the moment at the left (near) end of an unloaded span, when the load lies on spans to its **left**; $k'_n = -M_n/M_{n-1}$. The right focal point lies $L_n/(1+k'_n)$ from its left support.

#### Expressions
For span $n$ between supports $n-1$ and $n$, with $L_n/I_n$ the flexibility ratio of the span:

$$
k_{n+1} = \frac{L_{n+1}/I_{n+1}}{(2-k_n)\,L_n/I_n + 2\,L_{n+1}/I_{n+1}}\qquad(\text{left ratios, from the left end})
$$
$$
k'_{n-1} = \frac{L_{n-1}/I_{n-1}}{(2-k'_n)\,L_n/I_n + 2\,L_{n-1}/I_{n-1}}\qquad(\text{right ratios, from the right end})
$$
For equal spans with constant $EI$: $k_{n+1} = \dfrac{1}{4-k_n}$, $k'_{n-1} = \dfrac{1}{4-k'_n}$.

Starting values: hinged end $k = 0$; fixed end $k = \tfrac12$. Example (equal spans, constant $EI$, hinged end): $k_1 = 0$, $k_2 = 0.25$, $k_3 = 0.2667$, $k_4 = 0.2679$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The focal point of an unloaded span is
   A. The mid-point of the span
   B. The fixed point where its straight BMD crosses the axis for any loading on one side
   C. The point of maximum moment
   D. The support
   Answer: B. The BMD of an unloaded span is linear and always passes through the same point. (Key point: Focal point)

2. The left focal ratio k_n of span n is defined as
   A. -M_n/M_(n-1)
   B. M_n x M_(n-1)
   C. -M_(n-1)/M_n
   D. L_n/L_(n-1)
   Answer: C. It is the ratio of left-end to right-end moment, with load on the right. (Key point: Left ratio)

3. For constant EI the recurrence is
   A. k_(n+1) = 1/(2 - k_n)
   B. k_(n+1) = 4 - k_n
   C. k_(n+1) = k_n/4
   D. k_(n+1) = 1/(4 - k_n)
   Answer: D. Putting equal L/I in the general formula gives 1/(4-k_n). (Key point: Constant EI)

## Related in Theory of Structures II

- [Focal point ratio: define and derive left ratio](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Afocal-point-ratio-left-derive) — 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

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