Advanced Methods of Structural Analysis

386 11 Matrix Stiffness Method

11.4.2 Geometrical Equations and Deformation Matrix

These equations present the relationships between deformationseof the elements
and displacementsZof the joints. The required relationships is

EeDBZE: (11.2)

Vector of deformation isEe.n1/D
e 1 e 2 en

̆T

. The entries of this vector are
deformation of the elements at the sections with unknown internal forces. There-
fore, a dimension of this vector equals to the number of unknown internal forces

S. Vector of joint displacement isZE.m1/ D
Z 1 Z 2 Zm

̆T
; dimension of
this vector equals to the number of primary unknownsZof displacement method.
MatrixB.mn/presents the matrix of deformation. The entrybij(i-th row andj-
th column) means deformation in direction unknown internal forceSicaused by
displacementZj:In fact, the matrix equation (11.2) describes the conditions of the
deformationcontinuityof the elements.
Let us show the construction of matrix deformationBfor continuous beam
(Fig.11.20a). The primary system, the positive angular displacementsZand pos-
itive direction of the unknown bending momentsMi(iD1;:::;5) at the ends of
each member are shown in Fig.11.20b.

a

F

b

12

M 1

M 3

M 2

Z 1 Z 2

M 4

M 5

Fig. 11.20 Construction of the deformation matrix

If introduced constraint 1 has angular displacementZ 1 then deformation (angle
of rotation) at the section 1 will be zero, i.e.,e 1 D 0 Z 1. If introduced constraint
2 has angular displacementZ 2 then deformation at the section 1 will be zero, also.
The deformations in directionMiare

e 1 D 0 Z 1 C 0 Z 2

e 2 D 1 Z 1 C 0 Z 2

e 3 D 1 Z 1 C 0 Z 2

e 4 D 0 Z 1 C 1 Z 2

e 5 D 0 Z 1 C 1 Z 2

or in matrix form

2 6 6 6 6 6 6 6 6

e 1

e 2

e 3

e 4

e 5

3 7 7 7 7 7 7 7 7

D

2 6 6 6 6 6 4

00

10

10

01

01

3 7 7 7 7 7 5

„ ƒ‚ ...

B



Z 1

Z 2