Advanced Methods of Structural Analysis

11.4 Resolving Equations 381

c

M 2

M 3 M^5

M 4

M 1 M 6 M 10

M 7 M^9

M 8

b

1

2

3

4

5

6

7 8

10

9

a

H

1

5

Fig. 11.16 Formulation of the internal force vector

The signs of the moments are established on the basis of theMP^0 andS-e

diagrams. For example, since a momentM 1  5 D^6 kNm on the bending moment

diagramMP^0 is plotted left (Fig.11.5b), and the positive momentM 3 at the same

section on theS-ediagram is plotted right (Fig.11.16c), then third entry of theS

vector have a negative sign.

The vectorof required internal moments at the specified sections 1, ..., 10 is

SED

M 1 M 2 M 9 M 10

̆T

This vector will present a result of analysis of the frame by computer. If that

occurs, for example, the first entry (bending momentM 1 ) will be positive, then

according toS-ediagram (Fig.11.16c), the extended fibers at the section 1(bottom

of the left column) will be located at the right. Therefore, the internal momentM 1

should acts as shown in Fig.11.16b, i.e., clockwise. Computation of the vector of

internal forces in the second state will be discussed later.

Summary The following combination of two diagrams leads to the initial vectors:

1.Joint-load (J-L)diagramC(Z-P)diagram!Vector of external joint loadsPE.
2.MP^0 diagramC(S-e)diagram!Vector of internal forcesES 1 in the first state.

11.4 Resolving Equations...................................................

MSM considers three sides of problem: they are static, geometrical, and physical.

On their basis the second group of initial matrices may be constructed. They are

static matrix, deformation matrix, and stiffness matrix in local coordinates. These

matrices describe the different features of structure: configuration of structure, its

geometry and supports, stiffness of each element, and the order of theirs connection.

11.4.1 Static Equations and Static Matrix...........................

Assume that structure ism-times kinematically indeterminate. It means that struc-

ture hasmexternal joint loads. These loads, according toZ-Pdiagram, may be