Advanced Methods of Structural Analysis

10 1 Kinematical Analysis of Structures

Equilibrium conditions for jointsA,1,andBand corresponding results are pre-
sented below.

JointA:

P

XD 0 !NA 1 D0;

P

YD 0 W NC 1 sin ̨PD 0 !NC 1 D

P

sin ̨

Joint1:

P

XD 0 W N 1 BNC 1 cos ̨D^0 !N 1 BDNC 1 cos ̨DPcot ̨

P

YD 0 W NBCsinˇCRBsinˇD 0 !NBCDRB

JointB:

P

XD 0 WN 1 BNBCcosˇRBcosˇD 0 !N 1 BD 0

Two different results for internal forceN 1 – Bhave been obtained, i.e.,Pcot ̨and
zero. This indicates that the system is defective. From mathematical point of view,
this happens because the set of equilibrium equations for different parts of the
structure is incompatible. From physicalpoint of view, this happens because three
support constraints are concurrent in one pointCfor any angle'. Any variation
of this angle'remains the system as instantaneously changeable. If constraint at
supportCwill be removed to pointA, or angle of inclination of supportBwill be
different, then system becomes geometrically unchangeable structure.
Let us show this criterion for system presented in Fig.1.12. Note that hingesD
andEare multiple similarly to hingesCandF.
Reaction

RA!

X

MBD 0 !RAD

Pb

aCb

:

A

RA

j N^2

N 1

C

j

N 1

N 4

N 4

N 3

N 2 N

D 5

D

C

A B

a b

P

RA RB

j

2

1 3

4

E

F

5

Fig. 1.12 Kinematical analysis of geometrically changeable system