Flexibility Method - Overview
The flexibility method uses knowledge of the deformations of the structure to provide
additional equations. The method is sometimes called the force method because the
unknown variables that are solved for using the flexibility method are either member
forces or reactions. Remember that moments are forces too.
Consider a propped cantilever.
Qualitative analysis indicates that the directions of the external reactions are as
shown.
The beam has four unknown reactions and hence is indeterminate to degree one. This
implies that one additional independent equation is all that is necessary to analyse the
beam.
The cantilever’s end is propped and therefore does not deflect. Consider how the
structure would behave if the prop was removed.
Without the prop, the structure is statically determinate and it is possible to calculate
the value of the three remaining reactions, the bending moments and the deflected
shape of the structure.
However, the beam is propped and qualitative analysis indicates that the prop applies
a vertical reaction to the cantilever’s end. Consider the effect of this vertical reaction
if it were to act in isolation. The result of applying a vertical force at the cantilever’s
end can be predicted using the equations of statics.
If the structure is made from a linear-elastic material and the deflections are small
then superposition applies. If superposition can be applied then once the magnitude of
the propping force is known then the overall reactions can be calculated by adding the
reactions for the case without the propping force to the reactions due to the propping
force. In the same way the bending moment at any point on the beam will equal the
sum of the moments at that point for the two cases. More importantly the deflected
shape will be equal to the deflected shape for the cantilever without the prop plus the
deflected shape due to the propping force. That presupposes that the magnitude of the
propping force is known.
The magnitude of the propping force cannot be determined directly. However the
deflection at the beam’s end is known, the net deflection is zero. Hence the magnitude
of the propping force can be determined. This is the essence of the flexibility method.
Flexibility Method – General Procedure
Step 1
Introduce releases and/or remove external reactions until the structure is statically
determinate. For each release that has been introduced a corresponding bi-action of
unknown magnitude must be introduced. Similarly for each reaction that has been
removed a corresponding unknown force must be introduced. These are the unknown
that must be solved for.
Step 2
Calculate the reactions, internal forces and displacements
structure under the action of the applied known loads.
D
of the released
Step 3
Taking each of the releases that were introduced or reactions that were removed
separately, calculate the reactions, internal forces and displacements associated with
their corresponding unit forces and unit bi-actions. Using these results formulate the
flexibility matrix  f  .
This information is sufficient to generate the flexibility matrix.
Step 4
Solve for the unknown forces F  by solving
 f F  D
Step 5
Calculate the net deflections forces and reactions by using superposition and adding
the moments, displacements and reactions in the released structure to the moments,
displacements and reactions due to the now known forces F .
Releases, Reactions and Bi-actions
The first step of the generalised flexibility method is to introduce sufficient artificial
releases to make the structure determinate. Releases can be introduced by two means:
removing support reactions or adding internal releases.
Consider the encastre support shown below. This support provides vertical, lateral and
rotational support.
Rotational Release
If a rotational release is introduced then the support will provide restraint to vertical
movement and lateral displacement only.
Of course the real support does provide rotational restraint. This rotational restraint is
now represented by an external force whose magnitude must be solved for
Lateral Release
If a lateral release is introduced then the support will not provide axial restraint to the
member and will provide restraint vertical restraint and rotational restraint only.
In this case the lateral restraint that the real support provides is represented by an
unknown lateral force.
Vertical release
In a similar manner if the vertical release is removed then the support reactions
comprise a moment of resistance and a lateral restraint.
The vertical reaction is represented by a vertical force whose magnitude is unknown.
Combinations of releases
The external releases mentioned where shown singly. These releases can also be
applied together. For example the support could be removed completely in which case
three unknown forces would need to be applied.
Internal Releases
Releases can also be applied internally. In this case a corresponding pair of internal
forces (a bi-action) appropriate to the type of release must be introduced
If a hinge is introduced then a pair of moments are introduced.
If an axial release is introduced then the appropriate bi-action comprises a pair of
equal and opposite axial forces.
Similarly if a shear release is introduced then the appropriate bi-action comprises a
pair of equal and opposite shear forces.
Propped Cantilever Example