Advanced Methods of Structural Analysis

11.5 Set of Formulas and Procedure for Analysis 391

This equation may be rewritten in the form

PEDKZE: (11.13)

The matrixKpresents the stiffness matrix of a structure in global coordinates (ex-
ternal stiffness matrix)

KDAkAQ

T

: (11.14)

This matrix is a symmetrical one and it has the strictly positive entries on the main
diagonal. Dimension of this matrix is.nn/,wherenis a number of unknown end
displacements. Thus, for calculation of stiffness matrixKin global coordinates, we
need to know a static matrixAand stiffness matrixkQ(11.4) of a structure in local
coordinates.

11.5.2 Unknown Displacements and Internal Forces

Equation (11.13) allows to calculate the vector of ends displacements

ZEDK^1 PE; (11.15)

whereK^1 is inverse stiffness matrix in global coordinates.
For truss the vector of possible joint external loadsPis formed on the basis of
the entire design diagram andZ-Pdiagram. For beams and frames, the vectorPis
formed on the basis of the joint-load andZ-Pdiagrams. If a structure is subjected
tomdifferent groups of loading, then matrixPcontainsmcolumns. Each column
corresponds to specified group loading.
Knowing vectorZwe can calculate, according to (11.11), the unknown internal
forces of the second state
SE 2 DkAQ TZE: (11.16)

The formula (11.8) may be used for verification of obtained internal forces, i.e.,
ASE 2 DPE.
Final internal forces may be calculated by the formula

SEfinDSE 1 CSE 2 ; (11.17)

where the vectorSE 1 presents the internal forces at specified sections in the first state.
It is obvious that for trussesSE 1 is a zero-vector. For bending elements, the entries of
ES 1 are bending moments at the end sections. This vector forms on the basis of the

MP^0 diagram (1 state) andS-ediagram. If final ordinateMat any section is positive,
then this ordinate should be plotted at the same side of the member as on theS-e
diagram.