Advanced Methods of Structural Analysis

258 7 The Force Method

X 1

2 ′

1

234

5

EA

3 ′ 4 ′

d=4m

h=3m

2 EA

2 EA

EA

Design diagram Primary system

Fig. 7.21 Externally statically indeterminate truss. Design diagram and primary system

Solution.The primary system is obtained by eliminating the middle support. The
primary unknownX 1 is the reaction of the middle support (Fig.7.21).

Canonical equation of the force method isı 11 X 1 C1t D 0. The primary un-
known isX 1 D1t=ı 11. Final internal forces may be calculated by formula
NtDNN 1 X 1
Results of the analysis are presented in Table7.15. The column 1 contains lengths
of all members of the truss. Internal forces in the primary system caused by unit
primary unknownX 1 D 1 are presented in column 2. The column 3 serves for
calculation of the free term of the canonical equation for the case whenall mem-
bers of the trussare subjected to temperature gradientt. In this case,1t D
18:99419:002Š 0 (relative error equals 0.04%), and for primary unknown we
getX 1 D 0. It means that if the change of temperature is related toallmembers of
the truss, then internal forces in the members of the truss do not arise.

Ta b l e 7. 1 5 Statically indeterminate truss subjected to temperature change – calculation of internal
forces in all members of the truss

Members

L.m/NN 1

̨NN 1 tl

trelated toall

members

̨NN 1 tl

trelated to

lower chord onlyNN 12 

l

EA

NtD

NN 1 X 1

NN 1 
Nt
l
EA
1 2 3 4 5 6 7
Upper chord 20 - 30 4 1.333 5.332 0 3.554 0.580 1.546
30 - 40 4 1.333 5.332 0 3.554 0.580 1.546
Lower chord 1-2 4 0:6672:668 2:668 0.890 0:290 0.387
2-3 4 0:6672:668 2:668 0.890 0:290 0.387
3-4 4 0:6672:668 2:668 0.890 0:290 0.387
4-5 4 0:6672:668 2:668 0.890 0:290 0.387
Vertical
members

2- 20 3 0 0 0 0 0 0
3- 30 3 0 0 0 0 0 0
4- 40 3 0 0 0 0 0 0
Diagonals 1- 20 5 0.833 4.165 0 3.470 0.362 1.508
3- 20 5 0.833 4.165 0 3.470 0.362 1.508
3- 40 5 0.833 4.165 0 3.470 0.362 1.508
5- 40 5 0.833 4.165 0 3.470 0.362 1.508
1t 0 1t ı 11
D10:672 D24; 548 10.672
Factor – ̨t ̨t 1=EA ̨tEA ̨t

The column 4 serves for calculation of free term of the canonical equation ifonly
members of the lower chordof the truss are subjected to temperature changet.