Stiffness Method
Overview
The stiffness and flexibility methods are the most common techniques for analysing
statically indeterminate structures. In the flexibility method the requirement that the
deflected shape of the structure being analysed be compatible with the boundary
conditions of the structure is used to generate additional equations. In the flexibility
approach the number of compatibility equations required (i.e. the size of the flexibility
matrix) is equal to the degree of static indeterminacy of the structure. Remember, you
introduce releases until the structure is statically determinate
In the stiffness method the equations being solved are equilibrium equations.
However, the unknown variables in the equations are not the member forces (as
would be used in a traditional static analysis) but are the nodal displacements. Once
the displaced shape of the structure is known the forces in the individual members can
be calculated. For example, in a pin-jointed truss change in member length will
dictate the member force. Similarly, in a frame structure the relative displacements,
lateral and rotational, of the members ends will, in combination with the loads applied
to the member, define the bending moments in the member.
In the stiffness method the number of equations equals the number of displacements
required to describe the displaced shape of the structure. The number of
displacements required to describe the displaced shape of a structure is called the
Degree of Kinematic Indeterminacy, this is also called the number of degrees of
freedom.
For example the statically determinate structure shown above has a degree of
kinematic indeterminacy of 11. To describe the displacement of each internal requires
two coordinates (i.e. a X and a Y ) plus the roller support can displace. Thus
making a total of 11 degrees of freedom.
Note: the structure shown below, which is statically indeterminate, has the same
degree of kinematic indeterminacy, and its solution will involve solving the same
number of equilibrium equations as the statically determinate structure.
Stiffness and flexibility methods are related.
1
General Procedure
1. Identify the degrees of freedom required to describe the deflected shape of the
structure. Assume that all the n degrees of freedom are restrained in position.
2. Apply the loads to the structure and calculate the forces FR  that need to be
applied at the restraints to prevent movement. In addition calculate AR m1 ,
the m required structural actions, for the restrained structure.
3. Generate the stiffness matrix S  . The stiffness matrix is generated by,
displacing each degree of freedom by one unit while preventing movement at
all other degrees of freedom. This is done for each degree of freedom in turn.
Thus, for each degree of freedom i we calculate sii , the force applied at i
associated with a unit displacement at i (this is the force required to cause the
displacement) and zero displacement at all other nodes. And we calculate a
series of values s ji , the force applied at j associated with a unit displacement
at i and zero displacement at all other nodes (including j ).
In addition, calculate AU mn the m structural actions associated with unit
displacements at each of the n degrees of freedom.
4. Find the unknown displacements D by solving the equation,
S D   FR 
5. Calculate the member forces and moments from the nodal displacements using
the equation,
Am1  AR m1  AU mn  Dn1
2
Example – 1
Note:

i 1
m
s 21 

Ai Ei
cos i sin  i
Li
m
Ai Ei
sin  i cos i
Li
i 1
s12 

i 1
m
s 22 
Ai Ei
cos 2  i
Li

i 1
Thet
m
s11 
a
For pin-jointed members
Ai Ei
sin 2  i
Li
 A1 E1
  L cos 1
1

  A2 E 2 cos
2
Au    L2


 A E
 m m cos m
Lm

A1 E1

sin  1 
L1

AE
 2 2 sin  2 

L2



Am E m

sin  m 
Lm


3