BASIC CONCEPTS OF DISPLACEMENT OR STIFFNESS METHOD:

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CHAPTER 2
BASIC CONCEPTS OF DISPLACEMENT OR STIFFNESS
METHOD 2.1 INTRODUCTION Displacement or
stiffness method allows one to use the same
method to analyse both statically determinate and
indeterminate structures, whereas the force or
the flexibility method requires a different
procedure for each of these two cases.
Furthermore, it is generally easier to formulate
the necessary matrices for the computer
operations using the displacement method. Once
these matrices are formulated, the computer
calculations can be performed efficiently. As
discussed previously in this method nodal
displacements are the basic unknown. However like
slope deflection and moment distribution methods,
the stiffness method does not involve the ideas
of redundancy and indeterminacy. Equilibrium
equations in terms of unknown nodal displacements
and known stiffness coefficients (force due a
unit displacement) are written. These equations
are solved for nodal displacements and when the
nodal displacements are known the forces in the
members of the structure can be calculated from
force displacement relationship
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2.2 STIFFNESS, STIFFNESS COEFFICIENT AND
STIFFNESS MATRIX The stiffness of a member is
defined as the force which is to be applied at
some point to produce a unit displacement when
all other displacement are restrained to be zero.
If a member which behaves elastically is
subjected to varying axial tensile load (W) as
shown
2.2 STIFFNESS, STIFFNESS COEFFICIENT AND
STIFFNESS MATRIX The stiffness of a member is
defined as the force which is to be applied at
some point to produce a unit displacement when
all other displacement are restrained to be zero.
If a member which behaves elastically is
subjected to varying axial tensile load (W) as
shown
2.2 STIFFNESS, STIFFNESS COEFFICIENT AND
STIFFNESS MATRIX The stiffness of a member is
defined as the force which is to be applied at
some point to produce a unit displacement when
all other displacement are restrained to be zero.
If a member which
behaves elastically is subjected to varying axial
tensile load (W) as shown in fig. 2.1 and a
graph is drawn of load (W) versus displacement
(?) the result will be a straight line as shown
in fig. 2.2, the slope of this line is called
stiffness.

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Mathematically it can be expressed
as KW/? -------- 2.1 In other words
Stiffness K is the force required at a certain
point to cause a unit displacement at that
point. Equation 2.1 can be written in the
following form W K ? ---------
2.2

FIG.2.1 FIG.2.2 (Members
subjected to varying axial load ( (Graph of
load verses

displacement)
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• Mathematically it can be expressed as
• KW/? ------ 2.1
• In other words Stiffness K is the force
  required at a certain point to cause a unit
  displacement at that point.
• Equation 2.1 can be written in the following
  form
• W K ? ----- 2.2
• Where,
• W Force at a particular point
• K Stiffness
• ? Unit displacement of the particular point.
• The above equation relates the force and
  displacement at a single point. This can be
  extended for the development of a relationship
  between load and displacement for more than one
  point on a structure.

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Let us consider a beam of fig. 2.3 and two points
(nodes) 1, and 2. If a unit displacement is
induced at point 1 while point 2 is
restrained from deflecting up or down (see the
definition of stiffness). then the forces W1and
W2 can be expressed in terms of ?1 in
equation 2.2 as W K ? ----------
(2.2) when ?1 1 W1 K11. ?1 K11 See
fig. 2.3(b) W2 K21. ?1 Kwhere, K11
force at 1 due to unit displacement at 1 K21
force at 2 due to unit displacement at 1 These
are known as stiffness co-efficients. If a unit
displacement is induced at a point 2 while
point 1 is restrained from deflecting up or
down,then the forces W1 and W2 can be expressed
in terms of D2 in equation 2.2 as
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• when ?2 1
• W1 K12. ?2 K12
• See fig. 2.3(c)
• W2 K22. ?2 K22
• where,
• K12 force at 1 due to unit displacement at 2
• K22 force at 2 due to unit displacement at 2
• First subscript indicates the point of force and
  second the point of deformation. As forces W1, W2
  are proportional to the deformations ?1 and ?2,
  the following equation for the beam of fig.2.3
  can be written as
• W1 K11 ?1 K12 ?2 ---------- (2.3)
• W2 K21 ?1 K22 ?2 ---------- (2.4)
• Rewriting this in matrix form

--------- (2.5)
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where is called stiffness matrix
Elements of the stiffness matrix are known
as stiffness coefficients. So stiffness
coefficients can be defined as the forces at
points (nodes) caused by introducing various
unit deformations one at a time. is
called force vector and is
called displacement or deformation vector.
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• The expression (2.5) expresses the equilibrium at
  each of the node points in terms of stiffness
  co-efficients and the unknown nodal deformation
  and can be written as
• W K ? ---------- (2.6)
• The matrix K contains the stiffness co-efficients
  and it relates the forces W to the deformations ?
  and is called stiffness matrix. W and ? are
  called force and deformation vectors. The term
  force and the symbol W refers to the moments
  as well as forces and the term deformation and
  symbol ? refer to the both rotations and
  deflection.
• 2.3 STIFFNESS OR DISPLACEMENT METHOD FOR TRUSSES
• 2.3.1 Element and structure stiffness matrix.
• Application of stiffness method requires
  subdividing the structure into series of
  elements. The load-deformation characteristics of
  a structure are obtained from load-deformation
  characteristics of elements. It means that
  stiffness matrix of a structure K is formed
  from the stiffness matrices of the individual
  elements which make up the structure. Therefore
  it is important first to develop element
  stiffness matrix. The stiffness matrix for a
  truss element is developed in subsequent section.

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• 2.3.2 Stiffness Matrix of an Axially loaded
  Element (An Individual Truss Member)
• For the development of an element stiffness
  matrix for a truss member, let us consider an
  axially load member of length L, area A and
  modulus of elasticity E. The ends (nodes) of
  the member are denoted by 1 and 2 as shown in
  fig. 2.4(a).
• fig2,.4(a)
• (a) Element forces, w1, w2 and deformations, ?1,
  ?2
• (b) Deformation introduced at node 1 with node
  2 restrained.

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• (b) Deformation introduced at node 1 with node
  2 restrained
• (c)Deformation introduced at node 2 with node
  1 restrained.

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)d)Member forces and deformations of the actual
members. The vectors in fig. 2.4(a) define the
forces w1, w2 and the corresponding deformation
?1 and ?2 at the ends of the member. These also
define their positive directions. As shown in
fig. 2.4(b) a positive deformation ?1, at node
1 is introduced. while node 2 is assumed to
be restrained by a temporary pin support.
Expressing the end forces in terms of ? As
(from stress-strain
relationship) --------(2.7)
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when d 1

when ?1 1

-------(2.9) where ,
k11 is the force at 1 due to unit displacement
at 1 k21 is the force at 2 due to unit
displacement at 1 The first subscript denotes
the location of the node at which the force acts
and second subscript indicates the location of
displacement. As forces and deformations are
positive when they act to the right, so k11 is
positive while k21 is negative.
--------(2.8)
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Similarly if end 1 is restrained while end 2
is deformed in the positive direction a distance
d2 1 from fig. 2.4(c).
----- (2.10) where, k12 is
the force at 1 due to unit displacement at 2
k22 is the force at 2 due to unit
displacement at 2 To evaluate the resultant
forces w1 and w2 in terms of displacement ?1 and
?2 w1 k11 ?1 k12 ?2 ----------
(2.11) w2 k21 ?1 k22 ?2 ---------- (2.12)
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Expressing in matrix form
---------- (2.13)
---------- (2.14) It can be written as w
k ? ---------- (2.15)
---------- (2.16)
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This k is called element stiffness matrix. It
can be observed that sum of the elements in each
column of element stiffness matrix k equals
zero. It is due to the reason that co-efficients
in each column represent the forces produced by a
unit displacement of one end while the other end
is restrained (see fig. 2.4(b)). Since the bar is
in equilibrium in the x-direction the forces must
be equal to zero. Similarly all co-efficients
along the main diagonal must be positive because
these terms are associated with the forces acting
at the node at which a positive displacement is
introduced into the structure and correspondingly
the force is the same (positive) as the
displacement.
2.3.3 Composite stiffness matrix Equation 2.16
gives the stiffness matrix for an element of a
truss. A great advantage of subdividing a
structure into a series of elements is that the
same element stiffness matrix can be used for all
the elements of a structure. Stiffness matrix
comprising of all the element stiffness matrices
is called composite stiffness matrix. Composite
stiffness matrix is a square matrix and its size
depends upon number of members. Order of the the
composite stiffness matrix is 2m ? 2m, where m is
the number of members. Let us consider a truss
shown in fig. 2.5.
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This truss is subdivided into three elements.
Forces and deformations are shown in fig.
2.5(b). Stiffness matrix of element no.1
Stiffness matrix of element no.2
Stiffness matrix of element no.3
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Composite stiffness matrix of all elements is
given by
Relationship between forces and displacement from
equation w kc ? ----------- (2.17)
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It can be seen that the some of the elements in
each column of matrix kc is zero. It is due to
the reason that the stiffness co-efficient in
each column represents the force produced by unit
deformation of one end while other is restrained.
Since the member is in equilibrium the sum of the
forces must be zero. However all co-efficients
along the main diagonal must be positive because
these terms are associated with the force acting
at the end at which positive deformation is
introduced. As deformation is positive so force
produced is also positive. 2.3.4 Structure
stiffness matrix Stiffness matrix of a
structure can be generated from stiffness
matrices of the elements into which a structure
has been subdivided. The composite stiffness
matrix kc describes the force deformation
relationship of the individual elements taken one
at a time, whereas structure stiffness matrix K
describes the load deformation characteristics of
the entire structure. In order to obtain
structure stiffness matrix K from composite
stiffness matrix kc a deformation
transformation matrix is used which is described
in the subsequent section.
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• 2.3.5 Deformation transformation matrix
• Deformation transformation matrix relates
  internal element or member deformation to the
  external nodal structure deformation. It is
  simply a geometric transformation of co-ordinates
  representing the compatibility of the
  deformations of the system.
• Following is the relationship between element
  and structure deformation.
• ? T ? ------------ (2.18)
• where
• ? element deformation
• ? structure deformation
• T deformation transformation matrix
• As work done by structure forces work done by
  element forces

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------------ (2.19) as
------------ (2.6)
------------ (2.17) substituting
values of W and w from equation (2.6) and (2.17)
into equation (2.19)
------------ (2.20)

-------------(2.18)
-------------(2.21)
so
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• Therefore structure stiffness matrix K can be
  obtained from composite element stiffness matrix
  kc if latter is pre-multiplied by TT and
  post-multiplied by T.
• 2.3.6 Formation of deformation transformation
  matrix
• As we know
• ? T ?
• Let Tij represent the value of
  element deformation ?i caused by a unit
  structure displacement ?j. The total value of
  each element deformation caused by all structure
  deformations may be written as

where ?1, ?2, ?3 --- ?n represent set of element
deformations and ?1, ?2 --- ?n the set of
structure deformations. In matrix form
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so
------------(2.22) is called
deformation transformation matrix. Matrix T is
usually a rectangular matrix. Its columns are
obtained by applying a unit values of structure
deformation ?1 through ?n one at a time and
determining the corresponding element
deformations ?1 through ?m. Formation of the
transformation matrix of a truss, beam and a
frame element is explained in the subsequent
chapters.