Basic Concepts and Theorems
In structural theory, the members of a structure are not, in general, treated as three-dimensional continua but rather as continua of one or two dimensions. Rods, beams, and arches are representatives of the first class, and disks, plates, and shells, of
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of the considered element of the arc or shell, and
the term specific will be used in the sense of per unit volume. To introduce some concepts that will be used throughout these notes, consider a horizontal elastic beam that is built in at the end x simply supported at the end x
= ~ ,
=0
and
and subjected to a distributed verti-
cal, downward load of the specific intensity P 1 (x) and a distributed counterclockwise couple of the specific intensity P 2 (x) . To simplify the terminology, we shall speak of P (x), ( a = 1, 2 ) , as the generalized a loads acting on the beam. For the sake of brevity, we shall not discuss concentrated loads and couples in this chapter. With the generalized loads P
a
, we shall associate generalized displace-
W. Prager, Introduction to Structural Optimization © Springer-Verlag Wien 1972
2
Basic Concepts and Theorems
ments pa , which are supposed to be small; they will be defined in such a manner that the specific external work w(e} of the loads P placements p w
a
is given by
a
on the dis-
(e)
(1.1}
Here, and throughout these notes, a repeated letter subscript in a monomial implies summation over the range of the subscript unless the contrary is explicitly indicated by the words "no summation". For the
be~
example, p 1 (x} and p 2 (x} obviously are the vertical, downward deflection of the centerline and the counterclockwise rotation of the cross section at the abscissa x. The generalized displacements of a structure are subject to certain continuity requirements. For our beam, for instance, the displacements p (x} a will be required to be continuous and piecewise continuously differentiable. We shall refer to these requirements as the kinematic continuity conditions. Their general form will be discussed in Section 1.2. The generalized displacements are moreover subject to kinematic constraints that may be external or internal. For our beam, the external constraints are 0 ,
P2 (0}
0
0
(1.2}
Bernoulli's hypothesis, according to which material cross sections remain normal to the material centerline, would impose the internal constraint 0 ,
(1.3}
where the prime denotes differentiation with respect to x. At present, however, we do not impose this constraint. Generalized displacements that satisfy the kinematic continuity conditions and kinematic constraints will be called kinematically admissible.
3
Basic Concepts and Theorems
To each kinematic constraint, there corresponds a reaction. For example, the reactions associated with the external constraints (1.2) are vertical forces at x
0 and x
=t
, and a clamping couple at x
=0
. Note that
the work of any one of these reactions on arbitrary kinematically admissible displacements vanishes. It will be assumed in the following that all kinematic constraints that are imposed on the structure are workless in this sense. Reactions to internal constraints will be discussed at the end of Section 1.2. 1.2. Generalized stresses and strains. The state of stress at a typical point of a structural member is specified by a
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