Advanced Methods of Structural Analysis

392 11 Matrix Stiffness Method

11.5.3 Matrix Procedures............................................

The following procedure for analysis of any structure by MSM may be proposed:

1.Define the degree of kinematical indeterminacy and type of displacement for
each joint.
2.Calculate the fixed end moments to construct theJ-Ldiagram.
3.Numerate the possible displacementsof the joints, then construct theZ-Pdia-
gram and form the vectorPEof external joint loads.
4.Numerate the unknowns internal forcesS (for truss it is a number of the
elements; for frame it is a nonzero bending momentsMat the ends of each ele-
ments) to construct theS-ediagram and to form the vectorSE 1 of internal forces.
For this use the first state andS-ediagram; for trussES 1 D^0.
5.Consider the equilibrium conditions foreach possible displacement of the joint
and construct the static matrixA; the number of the rows of this matrix equals
to degree of kinematical indeterminacy and the number of the columns equals to
the number of the unknown internal forces.
6.Construct the stiffness matrix for each member and for all structure in local
coordinates.
7.Perform the following matrix procedures:

Compute the intermediate matrix complexkAQ

T

(this complex will be used in

the next steps)

Compute the stiffness matrix in global coordinatesKDAkAQ

T

and its inverse

matrixK^1

Calculate the vector of joint displacementsZEDK^1 EP

Calculate the vector of unknown internal forces of the second stateS 2 D

kAQ TZ

Calculate the vector of final internal forcesSEfinDES 1 CES 2

All matrix procedures (11.14)–(11.17) may be performed by standard programs us-

ing computer.

For trusses the procedure (11.17) leads to the axial forces at the each member. For

frames this formula leads to nonzero bending moments at the ends of each element.

To plot the final bending moment diagram, the signs of obtained final moments

should be consistent withS-ediagram.

The shear force can be calculated on the basis of the bending moment diagram

considering each member subjected to given loads and the end bending moments;

the axial forces can be calculated on the basis of the shear diagram consideration

of equilibrium of joints of the frame. Finally, having all internal force diagrams, we

can show the reactions of supports and checkthem using equilibrium conditions for

an entire structure as a whole or for any separated part.