Advanced Methods of Structural Analysis

84 4 Three-Hinged Arches

The bending momentM^0 and shearQ^0 diagrams for reference beam are pre-
sented in Fig.4.5. At pointC.xD 16 m/the bending moment isMC^0 D 152 kN m.

Three-hinged archThe vertical reactions and thrust of the arch are

RADRA^0 D14:5kN;RBDR^0 BD19:5kN;HD

MC^0

f

D

152

8

D 19 kN:

For construction of internal forces diagrams of the arch, a set of sections has to
be considered and for each section the internal forces should be calculated. All
computations concerning geometrical parameters and internal forces of the arch are
presented in Table4.1.

Ta b l e 4. 1 Internal forces in three-hinged circular arch (Fig.4.5); (RAD14:5kNIRBD19:5kNI
HD 19 kN)

Section x.m/y.m/ sin' cos'

Mx^0

.kNm/

Hy

.kNm/

Mx

.kNm/

Q^0 x

.kN/

Qx

.kN /

Nx
.kN /
0 1 234 550 678 9
A 0.0 0:0 0:8 0:6 0 0:0 0 14:5 6:5  23
1 4 4 0:6 0:8 58 76  18 14:5 0:2 23:9

2 8 6:330 0:4 0:9165 116 120:27 4:27
14:5
4:5

5:6892

3:4757

23:213
19:213
k^10 7:0788 0:3 0:9539^125 134:497 9:497 4:5 1:407419:474
3 12 7:596 0:2 0:9798 134 144:32410:324 4:5 0:609119:516
4(C) 16 8:0 0:0 1:0 152 152 0:0 4:5 4:5 19:00
5 20 7:596 0:20 0:9798 154 144:324 9:676 3:5 0:370719:316
6 24 6:330 0:40 0:9165 124 120:27 3:73 11:5 2:939722:013
n 26 5:3205 0:50 0:8660 101 101:089 0:089 11:5 0:459 22:204

7 28 4 0:6 0:8 78 76 2 11:5
19:5

2:2

4:2

22:1
26:9
19:5 4:2 26:9
B 32 0:0 0:8 0:6 0 0:0 0 19:5 3:5  27
Notes: Values in nominator and denominator (columns 8 and 9) mean value of the force to the left
and to the right of corresponding section. Values of discontinuity due to concentrated load equal
Pcos'andPsin'in shear and normal force diagrams, respectively

The column 0 contains the numbers of sections. For specified sectionsA,
1–7, andBthe abscissaxand corresponding ordinatey(in meters) are presented
in columns 1 and 2, respectively.Radius of curvature of the arch is

RD

f

2

C

l^2

8f

D

8

2

C

322

8  8

D 20 m:

Coordinatesyare calculated using the following expression

yDy.x/D

s

R^2 



l

2

x

 2

RCfD

q

400 .16x/^2 12: