ECR values are defined for curvatures of cross sections - we need to be able to transform them into a span deflection multiplier for convenient design use.

The deflection of a span is proportional to the elastic energy in the span. Considering only bending energy of the gross section:

Considering the ECR as “softener” of the gross section stiffness, this equation becomes:

From these two equations we can create a span deflection multiplier for convenient design use:

Deflection Multiplier = Δ_ecr / Δ_g

Deflection Multiplier $=\frac{k\int \left(M\right)\left(ECR\right)(M/EI)dl}{k\int \left(M\right)(M/EI)dl}$

Deflection Multiplier $=\frac{\int \left(M\right)\left(ECR\right)(M/EI)dl}{\int \left(M\right)(M/EI)dl}$

Note that this multiplier will likely be conservative for indeterminate structures as the bending moments in the structure will become rearranged (stiffer sections will attract more moment) in a manner that will reduce Δ_ecr .