3. Today’s presentation will include the stiffness method of Matrix Analysis.
MATRIX METHOD OF STRUCTURAL ANALYSIS
In matrix methods for analysis the mathematical
approach of matrices is used to solve complex
problems related to structural analysis. The rapid
development of computers and the need for complex
structures lead to the development of matrix methods
of structural analysis. The analysis procedure can be
concisely written using matrix notations and are
suitable for computer programming.
4. MATRIX METHODS
THE STIFFNESS METHOD THE FLEXIBILITY METHOD
● Stiffness of a structure is defined as
the force required to produce unit
displacement.
● Stiffness = k = Force / Displacement
● In this method results are obtained
by equating forces to the
displacement where stiffness acts as
a constant.
● The force-displacement relations are
presented in a matrix form.
● Flexibility of a structure is defined as
the displacement caused by a unit
force.
● Flexibility = δ = Displacement / Force
● In this method results are obtained
by equating forces to the
displacement where flexibility acts as
a constant.
● The force-displacement relations are
presented in a matrix form.
5. THE STIFFNESS METHOD
In the stiffness method of analysis the structure is
divided into finite elements , firstly. Then the force-
displacement relation is formulated for each finite
element. These results are combined to yield the
general structure responses.
6. Fundamental terms:
Nodes: A point where two or mode members meet.
Member coordinates : The coordinate system used solely for the isolated member
of a system. ( x’ and y’ )
Global coordinates : The coordinate system which is used to represent the whole
structure. (x and y)
Kinematic indeterminacy: The Degree of kinematic Indeterminacy is the total
number of independent degrees of freedom available in the structure.
(unconstrained degrees of freedom)
Near and Far node : The identification of the near and far node is necessary. An
arrowhead is used for this whose apex points towards far node.
8. GENERAL PROCEDURE FOR STIFFNESS
METHOD:
1. Discretization
2. Formulating the member stiffness matrix
3. Assembly of member stiffness matrix
4. Assigning boundary conditions and load vectors
5. Solution using Q=KD
6. Determination of support reactions and member forces.
9. 1. DISCRETIZATION.
● General concept of every finite element analysis method.
● Structure is divided into sub parts and each part is analysed
separately.
10. 2. Member Stiffness Matrix.
The terms in a member stiffness matrix equation represent the load
displacement relation of a single member of a structure.
q=k’d
qn
qf
AE
L
1 -1
-1 1
dn
df
MEMBER STIFFNESS MATRIX
In terms of local coordinates*
SYSTEM OF EQUATIONS
REPRESENTING LOAD
DISPLACEMENTS
RELATIONSHIPS AT TWO
NODES OF A TRUSS MEMBER.
qn , dn = force and displacement at near node
qf , df = force and displacement at far node
11. 3. The Structure Stiffness Matrix
● The final truss stiffness matrix is formed after combining the member stiffness matrices
together.
● The order of this matrix is equal to the total degrees of freedom of the structure.
● When the k matrices are assembled , each element in k will then be placed
in its same row and column designation in the structure stiffness matrix K.
K=k1 + k2 + … + kn
K= Structure Stiffness Matrix
ki = member stiffness matrices
12. 4. Final Solution
After the formation of structure stiffness matrix, the global force vectors
can be related to the global displacement vectors as,
Q=KD
STRUCTURE STIFFNESS EQUATION
The matrices in the above equation can be partitioned into
respective sub matrices as
Qk , Dk= Known external loads and displacements
Qu , Du= Unknown reactions and displacements
The elements of submatrix Qu , Du is the final requirement, which
represent the reaction forces and the unknown displacements at the
nodes.
13. Applications:
This methods involves a lot of mathematical calculations and is often used for the
analysis of complex structures. Furthermore, it is generally much easier to formulate the
necessary matrices for the computer operations using the stiffness method; and once
this is done, the computer calculations can be performed efficiently.
The matrix method forms the base of many commercially available softwares like
STAAD.pro , SAP2000 , RISA etc. Structural engineers nowadays are also using
programming languages like python to analyse structure while considering the matrix
approach.
