Direct Variational and Weighted Residual Methods: Classical Trial Function Methods - Mechanics of Structures: Variational and Computational Methods - page 444

subject to the conditions (constraints) of Eq. (7.22b). This problem statement is readily cast
into a standard linear programming format.
The minimax method, as well as the least squares method, can be extended to apply to
more complicated problems taking advantage of the option to introduce constraints. For
example, structures with offset supports are readily analyzed. As with the other weighted-
residual methods, the minimax method can be applied to systems wherein separate partic-
ular trial solutions are established for each element forming the system [Park and Pilkey,
1982].

7.3.4 Subdomain Method
Suppose the domain or region ( V ) is subdivided into as many subdomains as there are
free (unknown) parameters. Then choose the parameters so that the average value of the
residual over each subdomain is zero. Thus, if there are mp subdomains V j , the integral of
the residual over each subdomain is set equal to zero, i.e.,

R dV = 0 j = 1 , 2 , ... ,mp (7.24a)

V j

With R = R V of Eq. (7.10) and the trial solution of Eq. (7.9),

LN u

V j (

u − f ) dV = 0 j = 1 , 2 , ... ,mp (7.24b)

This leads to mp simultaneous equations for the ,mp unknowns.

In terms of Eq. (7.14), this subdomain method is equivalent to setting

u i ,i = 1 , 2 , ...

1if x is in the j th subdomain
0if x is outside the j th subdomain

h ( R ) = R , W j = (7.25)

where x is the coordinate in V . This approach is sometimes referred to as the method of
Biezeno 5 and Koch 6 (1923) or the method of integral relations . The subdomains can be chosen
to be continuously adjacent, overlapping, separated, of equal size, or of different sizes.

EXAMPLE 7.3 Beam with Linearly Varying Load
In the case of a beam, the linear simultaneous equations corresponding to the subdomain
method are readily obtained. For a simple beam,

dx 2 EI d 2

d 2

w
dx 2 −

R = p z (1)

N u Use w = w , so that Eq. (7.24) leads to linear simultaneous equations

d 2
dx 2 EI N u dx

w − p z ( x ) dx = 0 j = 1 , 2 , ... ,m (2)

j j

k u w p u

or k u w = p u , where the beam is considered to be formed of m segments .

j

5 Cornelius Benjamin Biezeno (1888-1975) was a mechanical engineer who held for many years a chair in applied
mechanics at the University of Technology in Delft, Holland. He co-authored with R. Grammel the influential,
monumental treatise Technische Dynamik, in which his disdain for energy principles is evident. It would appear as
though he abhorred the notion of “virtual” work in so an exact a science as engineering. He preferred to rely on
geometric concepts.
6 J.J. Koch was a student and later colleague of Biezeno. Together they were a very effective research team in
applied mechanics. Koch had a reputation for having a fine-tuned physical intuition for mechanical phenomena.