Advanced Methods of Structural Analysis

11.3 Initial Matrices 379

11.3 Initial Matrices.........................................................

The first group of initial matrices describes the loading of structure and unknown
internal forces.

11.3.1 Vector of External Joint Loads...............................

The arbitrary exposure on a structure (fixed or moving load, settlement of supports,
change of temperature) should be presented in mathematics form. For this purpose
it serves a vectorPof the external joint loads. This vector may be constructed on
the basis of the joint-load (J-L) and the displacement-load(Z-P)diagrams.
The truss is loaded as shown in Fig.11.14a; theZ-Pdiagram is shown in
Fig.11.14b.

a

P 1 =10

P 2 =8

P 4 =20

P 6 =15

A

B

C

b

1

2

4

A

3

5

6

Fig. 11.14 Design diagram andZ-Pdiagram for truss

Since the truss has 6 possible displacements, then vector of external load will
contains 6 entries, i.e.,

PED

2

6

(^66)
P 1

P 6
3
7
7
7
For convenience, this vector we will present in transposed form as
PED
P 1 P 2 P 6
̆T
I
the symbolTis reserved for the matrix transpose procedure. Considering Fig.
11.14a, b we can compile the vectorPEof joint loads:
PED
1080 20 0  15
̆T
The first entry 10 means that in the first direction (Fig.11.14b) there acts the force
P 1 D 10. The negative sign at the fourth entry indicates that external forceP 4 D
20 is directed opposite the positive fourth possible displacement. Zeros for third
and fifth entries show that in directions 3 and 5 ofZ-Pdiagram the external forces
are absent.