Advanced Methods of Structural Analysis

404 11 Matrix Stiffness Method

It is possible to expand more the number of internal forces at the ends considering
also the axial forces (and corresponding axial displacements). It is obvious that di-
mension of all initial and intermediate matrices become very large. So in this chapter
the adopted stiffness matrix should be considered as a truncated matrix. Such form
leads to the short and readily available for visual analysis of matrix procedures.

11.7 Analysis of Redundant Frames........................................

Now we consider application of the matrix procedures for analysis of the simple
frame (Fig.11.26a);l Dh,EIis constant. The frame has two unknowns of the
displacement method. They are the angulardisplacement at joint 1 and linear dis-
placement of cross bar. The primary system is shown in Fig.11.26b.
The ancillaryJ-L, Z-P,andS-e diagrams are presented in Fig.11.26c–e,
respectively.
The vector of external equivalent joint loads on the basis of theJ-LandZ-Pdia-

grams becomesPED
0P

̆T

Since the loadPis applied at joint then the fixed-end

a

l

h=l

P

A

C

b

P

A

C

1

2

d

1

2

Z-P diagram

e

1

2

3

S-e diagram

c

M=0

J-L diagram

P

f

1

P 1

S 2

S 3

g

P 2

S 1

(S 1 +S 2 )/h S 2

(S 1 +S 2 )/h

S 1 /h+S 2 /h

h

5

MP

factor Ph/8

3

3

P

Fig. 11.26 (a,b) Design diagram of the frame, primary system and finite elements; (c–e) Ancillary
diagrams; (f) Construction of static matrix; (h) Final bending moment diagram