Basic Equations: Differential Form - Mechanics of Structures: Variational and Computational Methods - page 26

These are the mechanical or static (force, stress) boundary conditions . Continuity requires that
on the remaini ng portion of the surface S u the displacement u be equal to the prescribed
displacements u :

u = u on S u (1.61)

These are the (kinematic) displacement boundary conditions .
We are now able to define more precisely the concepts of admissible stresses and dis-
placements. Statically admissible (or consistent) stresses (forces) are the stresses that satisfy
the conditions of equilibrium [Eq. (1.54)] and the static boundary conditions [Eq. (1.60)].
Kinematically admissible (or consistent) displacements are displacements that satisfy the kine-
matic (strain-displacement) conditions [Eq. (1.21)] and the kinematic boundary conditions
[Eq. (1.61)].

1.6 Other Forms of the Governing Differential Equations

In the previous sections, the three types of fundamental equations used in the mechanics of
solids have been derived. These are the static equat io ns, the kinematic equations, and the
constitutive equations. The static equations D T σ 0 [Eq. (1.54)] are written in terms

of the six static (stress or force) variables, and the kinematic equations

+ p V =

Du [Eq. (1.21)]
are expressed in terms of the kinematic (three displacement and six strain) variables. The
constitutive equations σ

=

E [Eq. (1.34b)] provide unique relations between the static and
kinematic variables. The general problem of elasticity theory is to calculate the stresses and
strains, as well as the displacements, throughout a body. Although in theory solutions for
the fifteen unknowns which satisfy these fifteen equations in their present form and the
boundary conditions [Eqs. (1.60) and (1.61)] can be found, in practice it is convenient to
combine some of the equations to obtain alternative forms of governing equations.

=

1.6.1 Displacement Formulation

The first formulation involves the development of a system of differential equations that
are referred to as the displacement, stiffness, or equilibrium formulations. These equations,
which are expressed in terms of displacements, are obtained by forming stress displacement
equations and then substituting these into the differential equations of equilibrium. That
is, substitute the strain displacement relations = Du [Eq. (1.21)] into σ = E [Eq. (1.34b)]

to obtain the stress displacement relations

σ = E = EDu (1.62)

Substitute this into the differential equations of equilibrium of Eq. (1.54) to obtain

D T σ D T EDu + p V = + p V = 0 (1.63)

These are the governing differential equations of the displacement formulation. Note that
Eq. (1.63) applies as well to solids made of anisotropic material if E is taken from Eq. (1.48).
It is of interest that Eq. (1.63), which gives three equations for the three unknown displace-
ments, involves various combinations of first and second derivatives of the displacements
and the material characteristics.