Buckling loads and natural frequencies and the corresponding modal sha
pes and: forms for thin-walled tapered beams of open sections are exam
ined using the finite-element method. A tapered thin-walled bar finite
element with seven degrees of freedom at each node is adopted. In the
virtual work formulation, the updated Lagrangian approach is adopted
in which the effect Of geometric nonlinearity is considered. A rigorou
s expression for strains based on membrane theory of shells is conside
red. The flexural stiffness matrix, geometric stiffness matrix, and co
nsistent mass matrices are derived in a companion paper. The convergen
ce and accuracy of the method is tested based on other numerical resul
ts. Using the present theory, one is able to investigate various torsi
onal and flexural static and dynamic instability problems. Examples ar
e presented and comparisons are made with the existing solutions.