Structural Analysis Methods II: Structural Systems - Mechanics of Structures: Variational and Computational Methods - page 277

Then, if the applied loads take the values P X = P Z = 1, the global equations (7) become

= 2 √ 2

.

U Xd
···
U Zd

1
···
1

1

0

··· · ··· (11)

.

2 √ 2

1

0 1 +

from which the displacements of node d are found to be

=

U Xd
···
U Zd

1 . 414

··· V d = (12)

.

0 586

Now that the nodal displacements are known, other displacements, forces, and stresses
can be computed. For example, the reactions at nodes a , b , and c are found by placing the
values of the displacements V d from (12) in the first three equations of (6). The displace-
ments at the ends of the bars are found by recognizing, i.e., using v aV , that the global

displacements of the nodes (upper case U ) are the same as the corresponding element end
displacements referred to the global coordinates (lower case u ). Thus, for example,

=

U Xa = u Xa | bar 1 = 0 and U Xd = u Xd | bar 1 = u Xd | bar 2 = u Xd | bar 3 = 1 . 414

Equation (5.81) provides the displacements referred to the local coordinates x . Then the
forces at the ends of each bar can be computed using Eq. (5.84). For a two-force member
such as the bar of a truss, the end forces in local coordinates are the same as the internal
forces. For the truss of Fig. 5.17, we obtain

N | bar 1 = 1 . 0 , N | bar 2 = 0 . 586 , N | bar 3 =− 0 . 4142 (13)

Alternatively, the bar forces or, equivalently, the stresses can be calculated using the
material law. If the displacements of the joints of the truss have been computed, then the
elongation of an element, for example element 2, would be ( u xb − u xd ) | bar 2 and the stress σ

in bar 2 becomes

E
(

σ = E = u xb − u xd ) | bar 2 (14)

The force in a bar is obtained by multiplying the stress by the cross-sectional area.

EXAMPLE 5.4 Stiffness Matrix for a Five-Bar Truss
In Chapter 3, Example 3.5, the global equilibrium equations were established for the five-
bar truss of Fig. 5.19 using the principle of virtual work. The same relationships are easily
established by assembling the element stiffness matrices.
The most fundamental step in implementing a displacement solution is identifying where
an element stiffness matrix fits in the global stiffness matrix. The goal is to assign appropriate
global nodal numbers as subscripts for the element stiffness matrices. As in Example 5.3,