Advanced Methods of Structural Analysis

82 4 Three-Hinged Arches

Internal Forces

In any sectionkof the arch, the following internal forces arise: the bending moment

Mk, shearQk, and axial forceNk. The positive directions of internal forces are

shown in Fig.4.4b. Internal forces acting over a cross sectionkmay be obtained

considering the equilibrium of free-bodydiagram of the left or right part of the

arch. It is convenient to use the left part of the arch. By definition

MkDRAxk

X

left

Pi.xkxi/Hyk;

QkD

RA

X

left

P

!

cos'kHsin'k;

NkD

RA

X

left

P

!

sin'kHcos'k; (4.9)

wherePiare forces which are located atthe left side of the sectionk;xiare corre-

sponding abscises of the points of application;xk,ykare coordinates of the point

k;and'kis angle between the tangent to the center line of the arch at pointkand a

horizontal.

These equations may be represented in the following convenient form

MkDMk^0 Hyk;

QkDQ^0 kcos'kHsin'k;

NkDQk^0 sin'kHcos'k; (4.10)

where expressionsMk^0 andQ^0 krepresent the bending moment and shear force at the

sectionkfor the reference beam (beam’s bending moment and beam’s shear).

Analysis of Formulae [(4.8), (4.10)]

1.Thrust of the arch is inversely proportional to the rise of the arch.
2.In order to calculate the bending moment in any cross section of the three-hinged
arch, the bending moment at the same section of the reference beam should be
decreased by valueHyk. Therefore, the bending moment in the arch is less than
in the reference beam. This is the reasonwhy the three-hinged arch is more eco-
nomical than simply supported beam, especially for large-span structures.
In order to calculate shear force in any cross section of the three-hinged arch, the
shear force at the same section of the reference beam should be multiplied by
cos'kand this value should be decreased byHsin'k.
3.Unlike beams loaded by vertical loads, there are axial forces, which arise in
arches loaded by vertical loads only. These axial forces are always compressed.