Stability Analysis - Mechanics of Structures: Variational and Computational Methods - page 664

FIGURE 11.12
Sign conventions for beams.

can be rearranged leading either to the stiffness matrix k or to the transfer matrix U of the
bar including second order effects. For the transformation, the solutions contained in N u
and N s have to be evaluated at the ends of the bar. These discrete values are combined into
the matrices N u and N s (Sign Convention 1 for the transfer matrix) or into N v and N p (Sign
Convention 2 for the stiffness matrix) (Fig. 11.12). For the stiffness matrix

w( 0 )

)
···
w()
θ()

θ( 0 N u

v a
v b

( 0 )

= N v c

c

v = = = ···

N u ()

(11.36)

− V ( 0 )

− M ( 0 )

− N s

p a
p b

( 0 )

= N p c

c

p = = ···

V

= ···

N s ()

(11.37)

()
M ()

The stiffness matrix is defined as

=

p kv (11.38)

Substitute the value of the analytical solution at both edges of the bar, Eqs. (11.36) and
(11.37), into Eq. (11.38)

N p c

k N

= c (11.39)

v

where the conversion from forces in Sign Convention 1 to Sign Convention 2 has been taken
into account. It follows that

= N p N − 1 k (11.40)

v