Left and right focal point ratios

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): $k_1 = 0$, $k_2 = 0.25$, $k_3 = 0.2667$, $k_4 = 0.2679$.