Advanced Methods of Structural Analysis

380 11 Matrix Stiffness Method

Now we compile the vector of external forces for frame shown in Fig.11.15a.
For this frame, the joint-load diagram (Fig.11.5e) is repeated in Fig.11.15b. The
Z-Pdiagram is shown in Fig.11.15c.

c

3

4

12

Z-P

diagram

a

5m

P 2

P 1

3kN/m

4m 4m

4m

3m

3m

b

Mj 2 =10kNm Mj 3 =10kNm

J-L

diagram

Mj 1 =6kNm

Pj 4 =12.5kN

Fig. 11.15 Joint-load andZ-Pdiagrams. Formulation of the external load vector

Comparing theJ-Ldiagram withZ-Pdiagram we can form the following vector
of external loads

PED

610 10 12:5

̆T

11.3.2 Vector of Internal Unknown Forces..........................

The required internal forces may be presented in the ordered mathematics form.
For this purpose serves the matrix-vector of unknown internal forcesS. The entries
of this vector are the axial forces for end-hinged members and the bending moments
at the ends of the bending members. Generally, the vectorESmay be presented as a
sum of the vectors of internal forces at the first and second states, i.e.ESDSE 1 CES 2.
Let us consider the frame in Fig.11.5. This scheme without external load, nu-
meration of sections, and positive directions of the end moments for each element
of the frame are shown in Fig.11.16a–c. Pay attention that the sections where the
moments are zeros have been eliminated from consideration. They are point at the
upper rolled support (5) and at hingeHthat belongs to the second column.
The bending moment diagram in primary systemMP^0 (Fig.11.5b) andS-edi-
agram allow constructing the vector of the moment at the indicated sections in
primary system (state 1) due to given load. This vector becomes

SE 1 D

00  600  18 10 10 0 0

̆T