Structural Analysis Methods II: Structural Systems - Mechanics of Structures: Variational and Computational Methods - page 262

k ij . Before carrying out the summation, it is necessary
to fit each element stiffness matrix into the global nodal numbering system. That is, the
subscripts of k 1 are the node identifiers a and b , and the subscripts for k 2 are b and c . The
incidence table of Section 5.3.4 can be helpful in associating each element with the nodal
numbering of the global system. We observe that the incidence table contains the same
transformation information as a , except there are no zero coefficients.

k ij +

Thus, in Eq. (5.49), K ij =

5.3.2 Direct Derivation of the Displacement Equilibrium Equations
It is important to understand that the principle of virtual work leads essentially to equations
of equilibrium. As in the case of Chapter 2, the displacements must satisfy kinematic or
compatibility requirements. To emphasize that the displacement relations of Eq. (5.44) are
based on equilibrium, we also can derive these equations directly from the conditions
of equilibrium. Consider again the four-element, one-node system of Fig. 5.11. At this
node, the forces must satisfy equilibrium. This is accomplished by summing all forces
contributed by the elements joined at the node. Before this addition can take place, all forces
have to be referred to the same reference frame by applying the local to global coordinate
transformation of Eq. (5.27). For the node k of Fig. 5.11, the condition of equilibrium will be

p k + p k + p k + p k =

P k (5.51)

For the three-node, two-element system of Fig. 5.12, the nodal equilibrium relations can be
expressed as

p a
p b
p b
p c

p a = P a I P a

P b
P c

=

p b + p b = P b or II

I

(5.52)

p c =

P c

b ∗

p = P

where b ∗ is the global statics or equilibrium matrix t ha t defines the conditions of equilibrium
between p , the unassembled element forces, and P . It is possible to establish a reciprocal
relation of the form

bP (5.53)
However, some care must be exercised, since, as can be observed in Eq. (5.52), b ∗ is not
necessarily a square matrix and, hence, cannot be obtained as the inverse of b . It is the case,
however, that

p =

b ∗ b = I (5.54)

whereas bb ∗ =

I . In general,

p 1
p 2
p M

b 1
b 2
b M

=

P (5.55)

p = b P

By comparison of Eqs. (5.37) and (5.52), we observe that

b ∗ = a T

(5.56)