Advanced Methods of Structural Analysis

7.3 Analysis of Statically Indeterminate Structures 239

l=24m

y

x

f=6m

EI = const

q=2kN/m q=2kN/m

j

Primary system

(^0) X 1
1
(^23456)
7
l/ 8 l/ (^88)
a b
y
N 1
Q 1
M 1
H=1
j
j
x
C
N(kN)
33.9330.0026.6424.7324.0024.7326.7330.0033.93
−
l=24m
f=6m
EA
q=2kN/m
N 0
A
RA
RB
H B H
N 8
x
y
d
Fig. 7.15 (a,b) Two-hinged parabolic arch. Design diagram and primary system, (c) Positive
directions of internal forces, (d) Final axial force diagram and reaction of supports
presented in Table7.6; the following formulas for calculation of trigonometric func-
tions of the angle'between the tangent to the arch andx-axis have been used:
tan'Dy^0 D
4f .l2x/
l^2
; cos'D
1
p
1 Ctan^2 '
; sin'Dcos'tan':
Ta b l e 7. 6 Geometrical
parameters of parabolic arch Points
Coordinates (m)
x y tan' cos' sin'
0 0 0:0 1:00 0.7070 0.7070
1 3 2:625 0:75 0.800 0.6000
2 6 4:500 0:50 0.8944 0.4472
3 9 5:625 0:25 0.9701 0.2425
4 12 6:000 0:0 1.0 0.0
5 15 5:625 0:25 0.9701 0.2425
6 18 4:500 0:5 0.8944 0.4472
7 21 2:625 0:75 0.800 0.6000
8 24 0:0 1:00 0.7070 0.7070
The length of the chord between pointsnandn-1 equals
ln;n 1 D
q
.xnxn 1 /^2 C.ynyn 1 /^2 : (7.10)
The chord lengths of each portion of the arch are presented in Table7.7.