Basic Equations: Differential Form - Mechanics of Structures: Variational and Computational Methods - page 1

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Basic Equations: Differential Form

The fundamental equations for describing the behavior of a solid can be classified into the
three categories:
•

The conditions of equilibrium

•

The material law

•

The conditions of geometric fit (strain-displacement relations).

As will be shown in this chapter, these relationships are often expressed as local differen-
tial equations. They can also be written in global or integral form, corresponding to work
or variational principles, which are treated in Chapter 2. Both the local and global forms of
these relationships are used throughout this text.
To demonstrate that the local basic equations can be grouped into three categories, it is
worthwhile to consider a simple structure.

EXAMPLE 1.1 Illustration of the Basic Equations by a Simple Example
Consider a rigid block of weight W attached to two wires of the same initial length (Fig. 1.1).
The two wires are equal distances from the center point 0. The conditions of equilibrium
( 2. The equations of

equilibrium applied to the undeformed geometry were sufficient to find the forces P 1 ,P 2 .
This is not always the case, as an increase in the number of constraints in the problem
means that the forces cannot be obtained using equilibrium alone. For a system with a third
wire (Fig. 1.2), the number of unknown forces increases, while the number of conditions for
equilibrium remains the same. Thus, equilibrium requirements give the two relationships

F vert = 0 , M 0 = 0 ) give P 1 + P 2 = W, P 1 = P 2 . Hence, P 1 = P 2 = W /

P 1 + P 2 + P 3 = W, P 1 = P 2 (1)

which cannot be solved uniquely for the three forces P 1 ,P 2 , and P 3 . (Witness that P 1 =
P 2 = 2 are possible solutions.) Clearly there

are infinitely many statically acceptable or admissible solutions. Considerations other than
equilibrium must be introduced to identify the correct solution.
For the material at hand, observed or documented information of the deformation caused
by loads is usually available. The force-deformation effect is described by the constitutive or
material response relation . Often this is referred to as the material law . Assume in the present
problem that the elongation of each wire is proportional to the force in the wire, i.e.,

P 3 = W / 3 , and P 1 = P 2 = W / 4 ,P 3 = W /

= = =

f 1 P 1 , f 2 P 2 , f 3 P 3 (2)

1 2 3

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