Advanced Methods of Structural Analysis

10.1 Construction of Influence Lines by the Force Method 337

for moment of inertia of cross section may be assumed:Ixcos'x DIc.This
expression corresponds to increasing of bending stiffness of the arch from crown
to supports.
3.In arch with pinned supports, the zeros bending moments arise at supports. For
these cases, the following law for moment of inertia of cross section may be
taken as:Iccos'xDIx. This expression corresponds to decreasing of bending
stiffness of the arch from crown to supports. Both types of arches are presented
in Fig.10.7.

x

a

Ix=

IC

cosjx

IC

x

b

Ix = IC cosjx IC

Fig. 10.7 Types of nonuniform arches

Thus, it can be observed, that shape (10.7a) is not wise to use for pinned type of

supports, while the shape (10.7b) is dangerous to use in case of clamped supports.

It is obvious, that the laws for moment of inertia of cross section in real structures

are not limited to two considered cases above.

Example 10.2.Design diagram of symmetric nonuniform parabolic arch with

clamped ends is presented in Fig.10.8. The cross-sectional moments of inertia

varies by lawIxDIC=cos'xas considered above. The arch is subjected to con-

centrated loadPD 30 kN and uniformly distributed loadqD 2 kN=m, as shown on

the design diagram. Calculate the reactions of supportAand internal forces (shear

and bending moment) at the crownC. Use the influence lines obtained above.

0.1l

u=0.1 0.2 0.3 0.4 0.6 0.7 0.8 0.9

A

C

P= 30 kN

RA

0.25l= 6 m

H B H

f=6m

l/2=12m l/2=12m

MA MB

q= 2 kN/m

RB

0.5

Fig. 10.8 Design diagram of parabolic nonuniform arch

Solution.Influence lines for reactions and bending moment at the crownCare

showninFig.10.6c–f. For design diagram in Fig.10.8, the following factors should

be taken into account:l=fD 4 ,andlD 24 m.