Advanced Methods of Structural Analysis

4.2 Internal Forces 85

Column 3 and 4 contain values of sin'and cos', which are calculated by formulae

sin'D

l2x

2R

D

32 2x

40

;cos'D

yCRf

R

D

yC 12

20

:

Values of bending moment and shear for reference beam, which are presented in
columns 5 and 7, are taken directly from corresponding diagrams in Fig.4.5.Val-
ues forHyare contained in column 50. Columns containing separate terms for
Q^0 cos'; Q^0 sin'; Hcos'; Hsin'are not presented. Values of bending moment,
shear and normal forces for three-hinged arch are tabulated in columns 6, 8, and 9.
They have been computed using (4.10). For example, for sectionAwe have

QADQ^0 Acos'AHsin'AD14:50:6 19 0:8D6:5kN;

NADQA^0 sin'AHcos'AD14:50:8^19 0:6D^23 kN:

The final internal force diagrams for arch are presented in Fig.4.6.

RA RB

q=2kN/m

P 1 =10kN P 2 =8kN

A

l=32m

C

f=8m

B

HH

1

3 4 5

2

k

7

6

n

4m 4m

y

x

18

0.089

10.32

4.27

M (kNm)

2.0

9.676

3.73

9.49

22.1

26.9

N (kN)

23 −

23.21^27

19.21

19.47

P 1 sinj 2

+ Q (kN)

−

6.5

5.69

3.47

4.5

2.94 4.2

2.2 3.5

0.2

1.4

P 1 cosj 2

Fig. 4.6 Design diagram of three-hinged circular arch. Internal forces diagrams

Bending moment diagram is shown on the side of the extended fibers, thus the
signs of bending moments are omitted. As for beam, the bending moment and shear