Advanced Methods of Structural Analysis

242 7 The Force Method

Ta b l e 7. 1 0 Calculation of coefficient and free term of canonical equation
Unit state Loaded state
Portion
li
6 EI
a 1 c 1 b 1
MN 1 MN 1
EI
aP cP bP
MN 1 MP^0
EI
1 2 3456 789 10
0-1 0.6644 0.0 1:3125 2:625 9:1563 0:0 31:5 63 219:75
1-2 0.5896 2:625 3:5625 4:500 45:9335 63 85:5 108 1; 102:40
2-3 0.5340 4:500 5:0625 5:625 82:4529 108 120:5 135 1; 978:87
3-4 0.5039 5:625 5:8125 6:000 102:1815 135 139:5 144 2; 452:35
4-5 0.5039 6:000 5:8125 5:625 102:1815 144 139:5 135 2; 452:35
5-6 0.5340 5:625 5:0625 4:500 82:4529 135 121:5 108 1; 978:87
6-7 0.5896 4:500 3:5625 2:625 45:9335 108 85:5 63 1; 102:40
7-8 0.6644 2:625 1:3125 0.0 9:1563 63 31:5 0:0 219:40
Factor 1=EI 1=EI 1=EI
ı 11 D479:4484
EI
.m=kN/ 1PD11;506:74
EI
.m/

Canonical equation and primary unknownCanonical equation and primary un-
known (thrust) are

479:4484

EI

X 1 

11;506:74

EI

D 0 !X 1 D24:00

!

kN:

Construction of internal force diagrams Internal forces, which arise in the entire
structure, may be calculated by formulas

MDMN 1 X 1 CMP^0

QDQN 1 X 1 CQ^0 P

NDNN 1 X 1 CNP^0 : (e)

Calculation of internal forces in the arch due to given fixed load is presented in
Ta b l e7.11; internal forcesMN 1 ,Q 1 ,andNN 1 due to unit primary unknownX 1 D 1
are presented earlier in Table7.8.

Ta b l e 7. 1 1 Calculation of internal forces at specified points of the arch

Points

M 1 X 1 Q 1 X 1 N 1 X 1 MP^0 Q^0 P NP^0 MQN
12 3 45 6 1 C 42 C 53 C 6
0 0.0 16:968 16:968 0:0 16:968 16:968 0.0 0.0 33:936
1  63 14:400 19:2 63 14:400 10:80 0.0 0.0 30:0
2 ^108 10:733 21:466^108 10:733 5:178 0.0 0.0 26:644
3  135 5:820 23:282 135 5:820 1:455 0.0 0.0 24:737
4  144 0:0 24:00 144 0:0 0:0 0.0 0.0 24:0
5  135 5:820 23:282 135 5:820 1:455 0.0 0.0 24:737
6  108 10:733 21:466 108 10:733 5:178 0.0 0.0 26:674
7  63 14:40 19:20 63 14:40 10:800 0.0 0.0 30:0
8 0.0 16:968 16:968 0:016:968 16:968 0.0 0.0 33:936