Advanced Methods of Structural Analysis

13.5 Stability of Arches 483

Nontrivial solution of canonical equations of the displacements method leads to
the following stability equation
ˇ
ˇ
ˇ
ˇ

r 11 r 12

r 21 r 22

ˇ

ˇ

ˇ

ˇD^0

If we assume thatlDh, and take into account the expressions for functions' 1 . /
and
1 . /, then stability equation after rearrangements becomes

^3 .3ktan/D0:

The root of this equation isD 0 and corresponds to initial condition of the frame.
Condition
3ktanD 0

allows to calculate the critical parameterfor any value ofk. Some results are
presented in Table13.2.

Table 13.2 Critical load in terms of parameterk

Parameterk Root of equation Critical loadPcr(factorEI=h^2 )

1 1.193 1.423

10 1.521 2.313

1 1.57 2.465

Second approach. In this case only constraint 1 is introduced (Fig.13.18e). How-
ever, this constrainallowsthe linear displacement. This case is presented in
Table A.22, row 3. Elastic curve caused by unit angular displacements (if linear
displacementoccurs) and corresponding bending moment diagram are shown in
Fig.13.18f, g.
Unit reaction
r 11 D

EI

h

tanC

3kEI

l

so the stability equation becomes

tanC3k

h

l

D0:

IflDh, then this stability equation is the same as was obtained above. The second
approach is more effective than the first one.

13.5 Stability of Arches.....................................................

Stability analysis of the different types of arches is based on a solution of a dif-
ferential equation. Precise analytical solution may be obtained only for specific
arches and their loading. In this section, we will consider a plane uniform arch with