Advanced Methods of Structural Analysis

252 7 The Force Method

The unit displacements should be calculated as usual. The free terms
it.iD1;2;:::;n/should be calculated using the following expression

itD

XZ

̨tavNNidsC

XZ

̨

t

h

MNids; (7.18)

where ̨is the coefficient of thermal expansion;his a height of a cross section of a
member;NNi;MNiare normal force and bending moment in a primary system due to
action of unit primary unknownXi.
In case of constant ̨; tav;t;andhwithin the each member

itD

X

̨tav

Z

NNidsC

X

̨

t

h

Z

MNids: (7.18a)

Thus for computation of1t, the unit primary unknownX 1 D 1 should be ap-
plied to the primary system and then procedure (7.18a) should be performed; the
procedure of summation is related to all members.
The temperature at axial line (average temperature) and temperature gradient are

tavD

t 1 Ct 2

2

;tDjt 1 t 2 j; (7.19)

wheret 1 andt 2 are changes of temperature on the top and bottom fibers of the
member; the average temperaturetavand temperature gradienttare related to
uniform and nonuniform temperature changes, respectively.
The integrals

R

NNids;

R
MNidspresent area of corresponding diagram in the pri-
mary system.
Solution of (7.17) is the primary unknownsXi. Bending moment diagram is
constructed using the formula

MtDMN 1 X 1 CMN 2 X 2 CCMNnXn: (7.20)

Kinematical control of the final bending moment diagram may be performed using
the following expression

XZ MtMN†

EI

dsC

X

itD0; (7.21)

whereMN†is the summary unit bending moment diagram andMtis the resultant
bending moment diagram caused by change of temperature.
Procedure for analysis of redundant structures subjected to the change of temper-
atureisasfollows:

1.Provide the kinematical analysis, determine the degree of redundancy, choose

the primary system of the force method, and formulate the canonical equations

(7.17)