Advanced Methods of Structural Analysis

6.3 Maxwell–Mohr Method 167

with neutral axis). If the cross section is nonsymmetrical about its neutral axis, then

the term.t 1 Ct 2 /=2must be replaced byt 2 C..t 1 t 2 /=2/y,whereyis the distance

of the lower fiber tothe neutral axis.

The term.t 1 Ct 2 /=2means that a bar is subjected to uniform thermal effect; in

this case all fibers are expanded by the same values. The termjt 1 t 2 j=h 0 means

that a bar is subjected to nonuniform thermal effect; in this case a bar is subjected to

bending in such way that the fibers on the neutral line have no thermal elongation.

So, the first and second terms in (6.15) present displacements inkth direction due

to uniform and nonuniform change of temperature, respectively. Integrals

R N

Mkds

and

R

NNkdspresent the areas of bending moment and axial force diagram in unit

condition, which corresponds to required displacement.

The presentation of Maxwell–Mohr integral in formula (6.15) allows us to cal-

culate any displacement (linear, angular, mutual linear, mutual angular) caused by

uniform or nonuniform change of temperature. This formula does not take into ac-

count the influence of shear. The procedure of summation in formula (6.15)must

be carried over all members of the system. The signs at all terms in this formula

will be obtained as follows: if the displacements of the element induced by both the

change of temperature and by the unit load occur at the same direction, then the

corresponding term of the equation will be positive.

Procedure for analysis is as follows:

1.Construct the unit state. For this we should apply unit generalized forceX,which
corresponds to the required displacement
2.Construct the bending moment and axial force diagrams in the unit state
3.For each member of a structure to compute the term

R N
Nkdx, which is the area
of axial force diagram in the unit state
4.For each member of a structure to compute the term

R N
Mkdx, which is the area
of bending moment diagram in the unit state
5.Apply formula (6.15).

Example 6.9.Determine the vertical displacement of pointCat the free end of

thekneeframeshowninFig.6.13, when the indoor temperature rises by 20 ıCand

outdoor temperature remains constant. The height of the elementABandBCareb

andd, respectively.

Unit state for ΔC

1

X=1 X=1

MX=1 NX=1

l

Design diagram

l

h

A

B C

DC

+0° d

b

+0°+20°

+20°

Fig. 6.13 Design diagram of the frame and unit state forC