Advanced Methods of Structural Analysis

310 8 The Displacement Method

1

2

h^ =

5m

l = 6m 6m

P

12 3

456

Fig. P8.13

by constructing the internal force diagrams. Trace the elastic curve of the frame.
Calculate the reactions of supports. Provide static and kinematical verification.
Ans.Q 1 DQ 3 DC0:1945P; Q 2 DC0:611P:

8.14.The structure shown in Fig.P8.14contains a nonuniform parabolic arch. The
moment of inertia at the arbitrary section of the arch isIxDIC=cos ̨,whereICis
the moment of inertia of the cross section at the crownCand ̨is the angle between
the horizontal line and the tangent at any section of the arch. Uniformly distributed
loadq D4kN=mis placed within the left half-span of the arch. Construct the
bending moment diagram. Solve this problem in two versions: (1) The axial forces
in the cross section of the arch are ignored and (2) the axial forces in the cross
section of the arch are taken into account. The moment of inertia and the area of the
cross section at the crownCareICD 894  106 mm^4 ,ACD33; 400mm^2. Hint:
The reader can find all the required data forparabolic arches as standard members
in TableA.19(uniform arches) and TableA.20(non-uniform arches). Compare the
results obtained by each version.

8m

9m 18m

1 C D 4m

Ix a

q IC = 2 EI

EI

3 EI

A

B

Fig. P8.14

Ans. (1)Z 1 D

8:1

EI

.rad/; M1AD8:1kN m;M1BD4:05kN m;

M1CD12:15kN mI (2)Z 1 D

8:4996

EI

.rad/:

8.15.A frame with an infinitely rigid crossbar is subjected to horizontal loadP.
The connections between the vertical members and the crossbar are hinged
(Fig.P8.15). Calculate the support moments and the reactions. Construct the in-
ternal force diagram. Show the elastic curve.