Dear MoverZ, Shabeer,

I should include the end condition when deriving natural frequency in my posting before. Thanks MoverZ. It is quite clear in that book. Below is the condition.

Although this seem too low, in practice the natural frequency will be higher because (1) end moments, neglected here, will raise frequency by more than 15%; (2)the critical span is usually limited by stress and is rarely reached; and (3) the piping weight is assume is often larger than the actual load.