Advanced Methods of Structural Analysis

7.4 Computation of Deflections of Redundant Structures 243

Corresponding axial force diagram is presented in Fig.7.15d.

Knowing the internal forces at points 0 and 8 we can calculate the reactions at

supportsAandB. Axial forceN 0 D33:936kN is shown at supportA(Fig.7.15d).

Reactions of this support areRADN 0 sin' 0 D 33:9360:707D 24 kN and

H DN 0 cos' 0 D 24 kN. For primary unknownX 1 we have obtained the same

result.

Verification of results.

(a)For arch in whole

P
YDRACRBq 24 D 0.
(b)The bending moment at any point of the arch

M.x/DRAxHy

qx^2

2

D24x 24

4f

l^2

x.lx/

qx^2

2

for given parametersf,l,andqindeed equals to zero foranyx.

Discussion:

1.If two-hinged uniform parabolic arch is subjected to uniformly distributed load
within all span, then this arch is rational since the bending moments and shear
forces are equal to zero in all sections of the arch. In this case, only axial com-
pressed forces arise in all section of the arch.
2.Procedure for analysis of nonuniform arch remains same. However, in this case
the Table7.6must contain additional column with parameterEIforeach point
0–8, Table7.7must contain parameterEIiformiddle pointof each portion, and
column 2 of the Table7.10should be replaced by columnli=6EIi.

7.4 Computation of Deflections of Redundant Structures................

As known, for calculation of deflection of any bending structure it is necessary to

construct the bending moment diagram in the actual state, then construct unit state

and corresponding bending moment diagram, and finally, both diagrams must be

multiplied

kD

Z

s

MPMN

EI

dsD

MPMN

EI

: (7.12)

HereMPis a bending moment diagram of a given statically indeterminate structure

due to applied load, while a bending moment diagramMNis pertaining to unit state.

The construction of bending moment diagramMPin the entire state is discussed

above using the superposition principle. Other presentation of superposition prin-

ciple will be considered in the following sections of the book. Now the following

principal question arises:how to construct the unit state? It is obvious that unit load

must correspond to the required deflection. But which structure must carry this unit

load? It is obvious that unit load may be applied to the given statically indeterminate

structure. For construction of bending moment diagram in unit state for statically in-

determinate structure the additional analysis is required. Therefore computation of