Principles of Virtual Work: Integral Form of the Basic Equations - Mechanics of Structures: Variational and Computational Methods - page 107

Thus, for a beam the principle of stationary complementary energy leads to the kinematical
relations and the geometrical boundary conditions, i.e.,

u 0 , γ = w + θ κ = θ = , along x

0 x

(10)

u 0 = u 0 , w = w , θ = θ at x = 0 ,L

2.3 Generalized Variational Principles

In Section 2.2, the most important classical variational principles were developed. Now, it
will be shown how the integral expressions of the classical principles can be modified to
achieve more comprehensive forms which are referred to as generalized variational princi-
ples. Whereas the classical variational principles can be considered as single field principles
involving either displacements [Eq. (2.58a)] or forces [Eq. (2.81)] as unknowns, the gener-
alized principles may involve two fields such as displacements and forces simultaneously
as unknowns. These derived principles, like the classical principles, are useful for formu-
lations and numerical solutions of structural mechanics problems and, in some cases, offer
advantages over the classical principles (Wunderlich, 1970 and 1973).
In establishing generalized variational principles, it is instructive to start with the set of
fundamental equations of the theory of elasticity and to write them as shown in Table 2.1,
where a distinction is made between equations expressed in terms of stress and displace-
ment variables. As indicated, the two sets of equations are related through the material
law. Typically, with a generalized principle, the equations of equilibrium, the kinemati-
cal relations, and the collective boundary conditions will all be fulfilled simultaneously.
Displacements and forces will be varied independently of each other.

TABLE 2.1
Local Form of the Fundamental Equations for an Elastic Continuum

Force or Stress Variables Displacement Variables

Equilibrium equations (Chapter 1, Kinematical (strain-displacement)

Eq. 1.54): equations (Chapter 1, Eq. 1.21):

D T σ + p V = 0 in V = Du in V

Force (static or mechanical) boundary Kinematic (displacement) boundary

conditions (Chapter 1, Eqs. 1.57 and 1.60): conditions (Chapter 1, Eq. 1.61):

A T σ =

= =

p p on S p u u on S u

Material Law (linear) (Chapter 1, Eq. 1.34):
σ =

E
or

E − 1 σ =