Advanced Methods of Structural Analysis

7.3 Analysis of Statically Indeterminate Structures 235

Kinematical verification of computed internal forces may be done using the follow-
ing formula
X
NNN l
EA

D0: (7.9)

If a primary unknown is thereactionof support, then this equation means that dis-
placement in the direction of the primary unknown due to primary unknown and
given loads is zero. If the primary unknown is the internal force in any redundant
member of a web, then (7.9) means that amutualdisplacement in the direction of
the primary unknown due to this unknown and given loads is zero.
Let us consider the symmetrical truss that carries two equal forcesPD 120 kN
(Fig.7.13a). The axial stiffness for all members isEA.This structure is first degree
of statically indeterminacy. The reaction of the intermediate support is being as the
primary unknown. The primary system is shown in Fig.7.13b.

ab

3m

4m 4m 4m 4m

04

PP

123

5

6

7

X 1

PP

Primary system

P=12 P=120

Loaded state

X 1 =1

cdUnit state

e V6-2=0.0

U6-7= 63.71 U6-5= 63.71

D6-1=120.25 D6-5=120.25

a

R 6

U0-7=63.71

D0-1=79.75

a

R 0

H 0

Fig. 7.13 (a,b) Redundant truss and primary system. (c,d) Unit and loaded states. (e) The free-
body diagrams of joint 0 and 6

Canonical equation of the force method and primary unknown are

ı 11 X 1 C1PD 0 !X 1 D

1P

ı 11

:

For calculation ofı 11 ,1P, it is necessary to show the unit and loading states
(Fig.7.13c, d).
The analysis of this statically indeterminate truss is presented in the tabulated
form (Table7.5). Column 1 contains the flexibility for each member; the factor 1/EA
is omitted. Internal forces for all members in unit and loaded states are presented in