Introduction
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The method of consistent deformations, or sometimes referred
to as the force or flexibility method, is one of the several techniques available to
analyze indeterminate structures. The following is the procedure that describes the
concept of this method for analyzing externally indeterminate structures
with single or double degrees of indeterminacy.
- determine the degree of indeterminacy
Determine the degree of indeterminacy of a given structure.
This can be accomplished by calculating the number of unknown reactions, r, minus the
number of static equilibrium equations, e. For example, considering the frame shown
below (Fig.1), the number of unknown external reactions, r, equals 5, (XA,
YA, MA, XB, and YB). The number of static
equilibrium equations, e, equals 3,
( Fx = 0,
Fy = 0 and
M = 0).
Therefore, the number of degree of indeterminacy, n, is calculated as:
n = r - e
= 5 - 3
= 2

Figure 1 - Indeterminate frame structure
In the frame shown in Fig. 2, there is another equation of statics that can be
written at the hinge h. In other words, the fact that the moment at h = zero for
either part of the structure on the right or the left side of the hinge h can be used.
This equation is referred to as equation of condition, ec.
In this case, the number of unknown reactions is r = 6, the number of equations of
statics is e = 3, and the number of equations of condition ec = 1. The
degree of indeterminacy, n, is calculated as:
n = r - (e + ec)
= 6 - (3 + 1)
= 2

Figure 2 - Indeterminate frame structure with hinge
Select a number of the support reactions equal to the degree of
indeterminacy
as redundants. The choice of the redundants will vary since any of the
unknown reactions can be utilized as a redundant. In the example shown in Fig. 1,
the X and Y reactions at Support B can be selected as redundants. Another
alternative is to select the X reaction at B and the moment at A as redundants.
The later choice was selected and utilized through the remainder of this procedure.
- remove restraints at the redundants
Remove the support reactions (restraints) corresponding to the selected redundants
from the indeterminate structure to obtain a primary determinate structure, or
sometimes referred to as a released structure.
This determinate system must represent a stable and admissible system.
Sketch the deflected shape of the primary determinate structure under the applied
loads, and label the deformations at the removed restraints, (see Fig. 3).
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Figure 3(a) - Primary structure
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Figure 3(b) - Primary structure deflected shape
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- calculate deformations at redundants
Calculate the deformations corresponding to the redundants, i.e., the rotation at
Support A, A0, and the translation,
B0, at Support B.
This can be accomplished as follows; using the virtual work method,
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(a) |
Draw the moment diagram, M0, for the primary structure under
the applied loads, (see Fig. 4(a)(i)). The method of superposition can also be utilized when
drawing the M0 diagram, (see Fig. 4(a)(ii). This will simplify the integration needed to calculate
the deformation A0 and
B0.
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Figure 4(a)(i) - Moment diagram of primary structure
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Figure 4(a)(ii) - Moment diagram by superposition
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(b) |
Apply a unit load at the location of the first redundant. In this case, apply a
unit moment, MA = 1 ft-k at Support A. Sketch the deflected shape, label the
deformation at the removed restraints and draw the moment diagram
of the primary structure when subjected to this load, see Fig. 4(b).
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Figure 4(b)(i) - Moment diagram with MA = 1 ft-k
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Figure 4(b)(ii) - Deflected shape with MA = 1 ft-k
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(c) |
Calculate the rotation, A0,
at Support A using the following equation:
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(d) |
Apply a unit load at the location of the next redundant. i.e., apply a
unit force, XB = 1 k at Support B. Sketch the deflected shape, label the
deformation at the removed restraints and draw the moment diagram
of the primary structure when subjected to this load, see Fig. 4(c).
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Figure 4(c)(i) - Moment diagram with XB = 1 k
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Figure 4(c)(ii) - Deflected shape with XB = 1 k
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(e) |
Calculate the translation, B0,
at Support B using the following equation:
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(f) |
Calculate the deformations of the primary structure when subjected to the
redundant MA, see Fig. 4(b), or the redundant XB, see Fig. 4(c).
This is accomplished by using the following relationships:


The above relationships yield the flexibility coefficients faa, fab,
fba, and fbb. The flexibility coefficient fij is
defined as the deformation corresponding to the redundant i, due to a unit value of
the redundant j.
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- write consistent deformation equations
Write consistent deformation equations that correspond to each redundant. In this
case:
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(a) |
Rotation at Support A = 0 since Support A is a fixed
support that prevents rotations.
(1)
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(b) |
Translation at Support B = 0 since Support B does not
allow horizontal translation.
(2)
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- solve deformation equations
Solve Equations (1) and (2) in the previous step to obtain the unknown redundants
M
A and X
B. Notice that if the answers of M
A and X
B
are positive, this means that the assumed directions of the applied force in Figures 4(b)
and 4(c) are correct.
- determine support reactions
Determine the remaining support reactions, i.e., X
A, Y
A, and
Y
B of the indeterminate structure by imposing the calculated values of
M
A and X
B in the correct directions and utilizing the
three equilibrium equations,
(

F
x = 0,

F
y = 0 and

M = 0).
- draw resultant structural diagrams
Once all reactions have been evaluated, the axial, shear, and moments diagrams can be
drawn. With this information, an approximate deflected shape can also be sketched.
The following examples illustrate the application of the consistent deformation
method to analyze statically indeterminate structures.
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