  New resonance $\Lambda_c(2860)^+$, reported by LHCb, is a key to establish
\emph{D}-wave charmed and charmed-strange baryon families. In this work, we
first carry out an analysis of mass spectra of the $\lambda$-mode excited
charmed and charmed-strange baryon states, which may reveal the relation of
$\Lambda_c(2860)^+$, $\Lambda_c(2880)^+$, $\Xi_c(3055)^+$ and $\Xi_c(3080)^+$,
$i.e.$, $\Lambda_c(2860)^+$ and $\Lambda_c(2880)^+$ can form a \emph{D}-wave
doublet $[3/2^+,5/2^+]$ while $\Xi_c(3055)^+$ and $\Xi_c(3080)^+$ are the
strange partners of $\Lambda_c(2860)^+$ and $\Lambda_c(2880)^+$, respectively.
Further study of their two-body Okubo-Zweig-Iizuka-allowed decays supports
$\Lambda_c(2860)^+$ and $\Xi_c(3055)^+$ as the \emph{D}-wave charmed and
charmed-strange baryons, respectively, with $J^P=3/2^+$. For
$\Lambda_c(2880)^+$ and $\Xi_c(3080)^+$, however, there exist some difficulties
under the \emph{D}-wave assignment with $J^P=5/2^+$ since the experimental
widths and some ratios of partial width of $\Lambda_c(2880)^+$ and
$\Xi_c(3080)^+$ cannot be reproduced in our calculation.
