Sinning Elders Publicly Rebuked
1Tim 5:20 Those who are sinning rebuke in the presence of all, that the rest also may fear.
Even in the case of elders. Public rebuke is a fearful thing, but one in which Paul does not shy away, as we note for example of his rebuke of the apostle Peter in Galatians 2, and of the sexually immoral Christian in 1Cor 5.
Public rebuke is meant to evoke a sense of fear due to public humiliation. Such shame motivates the God fearing to repent. But beware of misapplying this else oneself be found a slanderer.