This method is a powerful tool for analyzing indeterminate structures. One of its advantages over the flexibility method is that it is conducive to computer programming.
Stiffness method the unknowns are the joint displacements in the structure, which are automatically specified.
2. STIFFNESS METHOD
β’ This method is a powerful tool for analyzing indeterminate structures. One of its
advantages over the flexibility method is that it is conducive to computer
programming.
β’ Once the analytical model of structure has been defined, no further decisions are
required in the stiffness method in order to carry out the analysis.
β’ Stiffness method the unknowns are the joint displacements in the structure, which
are automatically specified.
β’ In the stiffness method the number of unknowns to be calculated is the same as
the degree of kinematic indeterminacy of the structure.
β’ Stiffness =
Action
Displacement
3. The essential features of stiffness method :-
β’ This method is also known as displacement method or equilibrium method.
β’ This method is a matrix version of classical generalized slope-deflection method.
β’ Kinematically indeterminate structures are solved using this method.
β’ Joint displacements are treated as primary unknowns in this method.
β’ Numbers of unknowns id equal to the degree of kinematic indeterminacy of the
structure.
β’ The unknown joint displacements for a particular structure are uniquely defined.
β’ Conditions of joint equilibrium are used to form equations in unknown
displacement.
4. ACTIONS AND DISPLACEMENT
β’ The term action and displacement are the fundamental concepts in structural
analysis. An action(force) is most commonly a single force or couple. However,
an, action may be also a combination of force and couple, a distributed loading, or
a combination of these actions.
β’ In addition to actions that are external to a structure, it is necessary to deal also
with internal actions. These actions are the bending moment, shear force, axial
force and twisting moment.
5. β’ The cantilever beam is subjected at end B to
loads in the form of action π1 and π1. At the
fixed end A the reactive force and reactive
couple are denoted π π΄ and ππ΄, respectively.
In calculating the axial force N, bending
moment M, and shear force V at any section
of the beam such as midpoint, it is necessary
to consider the static equilibrium of the
beam. One possibility to construct a free
body diagram of the right-hand half of the
beam, as show in fig-b. in so doing, it is
evident that each of the internal actions
appears in the diagram as a single force or
couple.
6. β’ There situations, however, in which the
internal actions appear as two forces or
couples. This case occurs most commonly
in structure analysis when a βreleaseβ is
made at some point in a structure as shown
in a fig for a continuous beam. If the
bending moment is released at joint B of the
beam, the result is the same as if a hinge
were placed in the beam at the joint. In the
order to take account of B.M. in the beam,
it must be considered as consisting of two
equal and opposite couples π π΅ that act on
the left and right hand positions of the beam
with the hinge at B.
7. β’ A displacement, which is most commonly a deflection or a rotation at some
point in a structure. A deflection refer to the distance moved by a point in the
structure, and a rotation means the angle of rotation of the tangent to the
elastic curve at a point.
8. β’ Action is noted by A and displacement is noted by D.
β’ Portrays a cantilever beam subjected to action π΄1, π΄2 and π΄3. The displacement
corresponding to π΄1 and due to all loads acting simultaneously is denoted by π·1
in fig-a, similarly, the displacements corresponding to π΄2and π΄3 are denoted by
π·2 and π·3.
9. β’ Now consider the cantilever beam subjected to action π΄1 only the displacement
corresponding to π΄1 in this beam is denoted by π·11. The significance of the two
subscripts is as follows.
β’ the first subscript indicates that the displacement correspond to action π΄1 and
the second indicates that the cause of the displacement is action π΄1. In a similar
manner, the displacement corresponding to π΄2 in this beam is demoted by
π·21, where the first subscript shows that the displacement correspond to π΄2 and
the second shows that it is caused by π΄1.also show in fig-b is the displacement
π·31 corresponding to the couple π΄3 and caused by π΄1.
β’ π·11 =
π΄1 πΏ3
24πΈπΌ
π·21 =
5π΄1 πΏ3
48πΈπΌ
π·31 =
π΄1 πΏ2
8πΈπΌ
10. SUPERPOSITION
β’ In using the principle of
superposition it is assumed that
certain action and displacements
cause other action and
displacements to be developed
in the structure.
β’ In general terms principle states
that the effect produced by
several causes can be obtained
by combining the effects due to
the individual causes.
11. β’ The beam is subjected to load π΄1 and π΄2 which produce various action and
displacement through out the structure.
β’ for reaction π π΄, π π΅ and π π΅ are developed at the support, and displacement D
is produced at the midpoint of the beam. The effect of the action π΄1 and π΄2
acting separately are shows in fig-b and fig-c.
12. β’ The beam has constant flexural rigidity EI and is subjected to the loads π1 , M, π2,
and π3 . since rotation can occur at joints B and C ,the structure is kinematically
indeterminate to the second degree when axial deformation are neglected. Let the
unknown rotation at these joints be π·1 πππ π·2, respectively, and assume that
counterclockwise rotations are positive . These unknown displacement may be
determined by solving equations of superposition for the action at joint B and C,
described in the following discussion.
β’ The restrained structure which is obtained by this means is shown in fig-b and
consist of two fixed end beams. The restrained structure is assumed to be acted
upon by all of the loads except those that correspond to the unknown displacement ,
thus, only the loads π1 , π2, and π3 are shows in fig-b. all loads that correspond to
the unknown joints displacement, such as the couple Min this example, are taken
into account later. The moments π΄ π·πΏ1 and π΄ π·πΏ2 are the action of the restrained
corresponding to π·1 and π·2, respectively, and caused by loads acting on the
structure .
13. β’ For example, the
restrained action π΄ π·πΏ1 Is
the sum of reactive
moments at B due to the
load π1 acting on member
AB and the reactive
moment at B due to the
π2 Acting on member BC.
14.
15. EXAMPLE
β’ K.I. = 2
Let, ΞΈ π΅ = π·1
ΞΈ π = π·2
AD = actions in actual structure corresponding
to redundant
AD1 = 0
AD2 = 0
ADL = actions in restrained structure due to
loads corresponding to redundant.
ADL1 =
π€π
8
-
π€π
8
=
24 β10
8
-
12 β10
8
= 15KN.m
ADL1 =
π€π
8
=
12 β10
8
= 15KN.m