Advanced Methods of Structural Analysis

238 7 The Force Method

moments for the arch itself and an axial force for the tie, while the shear and axial

forces for arch may be neglected. Since the axis of the arch is curvilinear, then the

graph multiplication method leads to approximate results.

Unlike three-hinged arches, in redundant arches, as for any statically indetermi-

nate structure, the internal forces arise in case of displacements of supports, changes

of temperature, and errors of fabrication. For masonry or concrete arches, a concrete

shrinkage should be taken into account, since this property of material leads to the

appearance of additional stresses.

Procedure for analysis of statically indeterminate arches is as follows:

1.Choose the primary system of the force method.
2.Accept the simplified model of the arch, i.e., the arch is divided into several
portions and each curvilinear portion is changed by straight member. Calculate
the geometrical parameters of the arch at specified points.
3.Calculate the unit and loaded displacements, neglecting the shear and axial forces
in arch:
ıikD

Z

.s/

MNiMNk

EI

dsI iPD

Z

.s/

MNiMP^0

EI

ds:

Computation of these displacements may be performed using the graph multiplica-

tion method.

4.Find the primary unknown using canonical equation (7.4) of the force method.

5.Construct the internal force diagrams; the following formulas may be applied:

MDMN 1 X 1 CMN 2 X 2 CCMP^0

QDQN 1 X 1 CQN 2 X 2 CCQP^0

NDNN 1 X 1 CNN 2 X 2 CCNP^0 ;

whereXiare primary unknowns;MNi,QNi,NNiare bending moment, shear, and

axial force caused by unitith primary unknownXi D 1 ;andMP^0 ;Q^0 P;NP^0

are bending moment, shear, and axial force caused by given load in primary

system.

6.Calculate the reactions of supports and provide their verifications.

Let us show this procedure for analysis of the parabolic two-hinged uniform arch

shown in Fig.7.15a. The flexural stiffness of the cross section of the arch isEI.

The equation of the neutral line of the arch isyD.4f = l^2 /x.lx/. The arch is

subjected to uniformly distributed loadqwithin the all span. It is necessary to find

the distribution of internal forces.

The arch under investigation is statically indeterminate of the first degree. The

primary system is shown in Fig.7.15b; the primary unknownX 1 is the hori-

zontal reaction of the right support. Canonical equation of the force method is

ı 11 X 1 C1PD 0. The primary unknownX 1 D1P=ı 11.

Specified points of the archThespanof the arch is divided into eight equal parts;

the specified points are labeled 0-8. Parameters of the arch for these sections are