Special Analysis
Procedures
{d c }  [Scc ]1 ({Pc  Pfc }  [Scr ] {d r })
Condensation
The term condensation refers to
the contraction in size of a system
of equations by elimination of
certain degrees of freedom, i.e.
 Srr
S
 cr
Src   d r 
 Pr   Pfr 
        (1)

Scc  d c 
Pc  Pfc 
where {dr} = degrees of freedom
to be retained; and {dc} = degrees
of freedom to be condensed.
Condensation of (1) proceeds as 1
Observe that stiffness equation
condensation means satisfaction
of the equilibrium equations
corresponding to the eliminated
equations. Furthermore, “eliminated” (condensed) degrees of
freedom are not discarded. They
are expressed as functions of the
corresponding forces, the retained degrees of freedom, and the
equation stiffness coefficients.
Nothing is “lost” or “approximated”
in the process of condensation.
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(2)
where (2) is obtained by expanding
the lower partition of (1) and
solving. Expanding the upper
partition of (1) and substituting (2)
leads to
([Srr ]  [Src ][Scc ]1 [Scr ]) {d r }
 {Pr  Pfr }  [Src ][Scc ]1 {Pc  Pfc }
or
[Sˆ rr ]{d r }  {Pˆr }
(3)
Once (3) is solved for {dr}, {dc} can
be obtained from (2).
2
Finally, the selection of degrees of
freedom to be eliminated is at the
analyst’s discretion. Usually,
however, the selection is not
arbitrary; there are generally
logical reasons for choosing
certain degrees of freedom for
elimination. Condensation can
be applied at the element level
(local or global coordinates) as
well. Simply change the notation.
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1
Constraints
Constraint designates suppression
of displacement components and
defining the dependency relationships between displacements. In
general, a constraint can be
defined as a set of relationships
between degrees of freedom that
are supplemental to the basic
stiffness relationships. Considering
c constraint relationships, the
constraint equations can be written
as
(4)
[G]cxn {d}nx1  {H}cx1
where [G] = matrix of constraint
coefficients, {d} = displacement
vector, {H} = vector of known
constants, and n = number of
displacement degrees of freedom.
Figure 1 shows two examples of
situations where constraint equations can be used to reduce the
number of unknown displacement
degrees of freedom.
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5
freedom from the stiffness
equations. Expressing the stiffness
equations as
 Srr
S
 er
Figure 1: Representative Constraint
Conditions
Each constraint equation can be
viewed as a means to eliminate
one of the degrees of freedom in
favor of the remainder. Hence,
with c constraint equations, it is
possible to eliminate c degrees of7
Sre   d r 
 Pr  Pfr 





See  d e 
Pe  Pfe 
(5)
where subscript r = retained
degrees of freedom and subscript e
= eliminated degrees of freedom.
With the partitioning of (5), the
constraint equations of (4) can be
expressed as
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2
d r 
 [I] 
   
 {d r }
d e 
[er ]
 [] {d r }
(6)
The result of (7) is an (n-c) system
of symmetric equations expressed
in terms of a set of conjugate
vectors.
Substituting (6) into (5) and making
use of the contragradience principle leads to
As shown in (6), the transformation
matrix [er] is obtained from
[Sˆ rr ]{d r }  {Pˆr }
(7)
where
[Sˆ rr ]  [Srr ]  [Sre ][er ]  [er ]T [Ser ]
T
 [er ] [See ][er ]
{Pˆr }  {Pr } {Pfr }  [er ] {Pe } {Pfe }
T
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{d e }  [er ]{d r }
(8)
For example, the constraints of Fig.
1(a) can be expressed as
v1 v 2
v3
T
 0
1
3
2  v
4
3
Furthermore, v4 = 3a if it is desired
to express the displacements in
terms of .
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3