Advanced Methods of Structural Analysis

7.3 Analysis of Statically Indeterminate Structures 227

 1 D

MPMN 1

EI

D

a

6 EI



0  0 C 4 

1

2

a

7

64

qa^2 a

1

32

qa^2



„ ƒ‚ ...

horizontal portion;Simpson rule



1

EI

aa

1

32

qa^2

„ ƒ‚ ...

vertical element

D

qa^4

32 EI



qa^4

32 EI

D0:

Horizontal displacement of the cross barUnit state is shown in Fig.7.9f. Required
displacement is

horD

MPMN

EI

D

1

EI

1

2

a 1 a

qa^2

32

D

qa^4

64 EI

:

The positive result means that crossbar shifted from right to left. Corresponding
elastic curve for frame in whole is shown in Fig.7.9e by dashed line. Note that
inflection point for vertical member is absent.
Another version of the primary system and corresponding bending moment dia-
gramsMN 1 andMP^0 are shown in Fig.7.9g.
In this case
ı 11 D

4

3

a

EI

;1PD

qa^3

24 EI

:

A primary unknownX 1 Dqa^2 =32; this is bending moment at the rigid joint, as
presented in Fig.7.9e.

Property of Statically Indeterminate Frames of the First Degree
of Redundancy

Let us consider important property of any statically indeterminate frames of the first
degree of redundancy. Design diagram of the frame is presented in Fig.7.10.
Two primary systems and corresponding bending moment diagrams for the unit
conditions are presented in Fig.7.10and are denoted as versions 1–2. For version 1,
the primary system is obtained by eliminating of the support constraint; and for

q

A

C B

l

Version 2

X 1 =1

1/l

1/h

1/l

1/h

1

X 1 =1 1

Version 1

1

h

h h/l

h/l

Fig. 7.10 Property of a primary system for structure of the first degree of redundancy