Advanced Methods of Structural Analysis

7.3 Analysis of Statically Indeterminate Structures 237

The last result may be checked. For joint 0, the equilibrium equation

P

XD 0

(Fig.7.13e) leads to the following result:H 0 C63:7179:750:8D 0 !H 0 D

63:863:71. The relative error is 0.14%.

Kinematical verification Displacement in the direction of the primary unknown due

to primary unknown and given loads is zero. According to (7.9), for computation of

this displacement we need to multiply the entries of column 7, 2, and 1 (column 8).

The sum of entries of this column equals

X

NN 1 Nl

EI

D1;344:24C1;347:52D3:28:

The relative error is

3:28 100 %

1;344

D0:24%:

Discussion:

1.If statically indeterminate truss is subjected to any loads, then distribution of
internal forces depends onrelativeaxial stiffnessEAof the members and does
not depend on their absolute valueEA.
2.Fundamental property of statically indeterminate trusses: any change of rigidity
EAof specified element of the truss cannot change thesignof internal force in
this element.

7.3.4 Analysis of Statically Indeterminate Arches................

Different types of statically indeterminate arches are presented in Fig.7.14.Theyare

two-hinged arches with or without tie (Fig.7.14a, b), one-hinged arch (Fig.7.14c),

and arch with fixed ends (Fig.7.14d).

abcd

Fig. 7.14 Types of the redundant arches

The most effective method for analytical analysis of statically indeterminate

arches is the force method in canonical form (7.4); for two-hinged arches we have

one primary unknown, two unknowns for one-hinged arch, and three unknowns for

hingeless arch.

As for three-hinged arch, the distribution of internal forces depends on a shape

of the neutral line (parabola, circular, etc.). This should be taken into account

when calculating unit coefficients and free terms of canonical equations. In gen-

eral case, these coefficients and free terms depend on bending moments, shear, and

axial forces. In calculating displacements, we will take into account only bending