A column with both ends built in $($both ends fixed$)$ is modified during construction. Instead of building the column with both ends fixed, the top end of the column is left free $($rotation free and translation free$).$ The resulting change in design compressive strength is most nearly:

A$.$ reduced by $\frac{1}{2}.$

B$.$ reduced by $\frac{1}{4}.$

C$.$ reduced by $\frac{1}{8}.$

D$.$ reduced by $\frac{1}{16}.$