Structural Analysis Methods II: Structural Systems - Mechanics of Structures: Variational and Computational Methods - page 253

Thus, six displacements and six (generalized) forces occur at each node

U X ,U Y ,U Z 3 translations

X , Y , Z 3 rotations

(5.14)

P X ,P Y ,P Z 3 forces

M X ,M Y ,M Z 3 moments

As in the case of the coordinates X, Y, Z, these system forces and displacements are desig-
nated by capital letters.
In vector form, the forces and displacements at each node are written as

j

P X
P Y
P Z
M X
M Y
M Z

U X
U Y
U Z

P j = V j = (5.15)

X

Y

Z

j

where the subscript j indicates the j th node. The nodal forces P and nodal displacements
V for the whole structure are expressed as

P 1
P 2
P j
P N

V 1
V 2
V j
V N

P = V = (5.16)

where N is the number of nodes.

State Variables for an Element
A global reference frame X , Y , Z was established in the previous section in order to define
system forces and displacements. It is also necessary to represent the displacements and
forces on an element in the directions of the global X , Y , Z system coordinates. We shall con-
tinue the practice here of using generalized nomenclature of letting the term forces include
moments and displacements include rotations. Suppose an arbitrary element i , with ends a
and b , belongs to a structural model. Element forces p i

and corresponding displacements

v i

at the ends a and b are shown in Fig. 5.8, where

F i Xa
F Ya
F i Za
F i Xb
F Yb
F i Zb

u i Xa
u i Ya
u i Za
u i Xb
u i Yb
u i Zb

p i a
p i b

v i a
v i b

p i v i

= = = = (5.17)

As derived in Chapter 4, the fundamental relationships of solid mechanics—equilibrium
conditions, material law, and kinematical conditions (compatibility)—provide the func-
tional connection between the end cross-sectional forces and the end displacements for an
element. These relationships can be pictured as
p i σ i

→ Equilibrium Conditions → → Material Law →

i

v i

→ → →

Strain-Displacement (Kinematic) Relations (5.18)