If we consider a circular curve of radius R₂, tangent to a straight and mark the offsets O₁, O₂ and O₃, measured perpendicular from the circle to the straight and placed at regular C/2 interval we will get something like this:

The offsets O₁ and O₃ are the bases of a trapezoid and O₂+V₂ is the median.

We can hence express V₂ as:

But V₂ is also the versine of the circle of radius R₂

Now, let’s bend the straight into a curve of radius R_M .

The median segment is formed of two sets of segments D + V_M = V_2E + O₂

D is the median of the new trapezoid defined by O₁ and O₃.

The unknown radius R_2E can be found if we know the versine V_2E.

But we now know that

hence R_2E can be expressed as

where each versine can be replaced with its equivalent formula

and if we simplify C² we get this nice formula of the equivalent radius for similar flexure bending:

If the straight would be bent opposite to the flexure of the R₂ radius, the same calculation would give us this formula of the equivalent radius for contra-flexure bending:

References:

Cope, Geoffrey H. (1993) British railway track: design, construction and maintenance. Permanent Way Institution. (In the book there are no sketches. Do you see why?).