! 
SIMPLE STRAINS AND STRESSES 67 
7+ may be neglected. The quantity z will of course be determined 
from the change of temperature and the coefficient of expansion of 
the bar. 
If a compound bar be made up of two bars of different -materials, 
firmly united at their ends, so that the component bars must suffer the 
same alteration of length when the compound bar is placed in tension or 
compression by a load W, then if a, and a, be the areas of the cross 
sections of the component bars, /, and /, the stresses produced in them 
by the load W, E, and E,, their coefficients of elasticity, / the original 
of the compound bar, and @ the alteration in length produced 
by the load W, then the following equations will obviously apply :— 
l l E 
E, B=, dng and W=a,/, + ayfy. 
If the foregoing is understood, the case of a compound bar made 
up of more than two bars of different materials presents no difficulty. 
Considering further the compound bar made up of two bars of 
different materials ; suppose that the compound bar is heated or cooled, 
so that the component bars, if entirely free, would expand or contract by 
amounts x, and x, respectively. Assuming that the two materials have 
different coefficients of expansion, then x, and 2, would not be equal. 
Let x, be the greater. The first bar will tend to lengthen or shorten by 
an amount 2,, but will be prevented by the other bar, which tends to 
alter by the amount , by the change of temperature. The first bar will 
therefore drag the other in one direction, while the second will drag the 
first in the opposite direction. The result will be that the alteration in 
length of the compound bar will be an amount x, which will lie between 
2, and z,. Also the stress produced in one bar will be tensile, while 
in the other it will be compressive. Using the same notation as 
before— 
Strain on first bar =“1—" &£, fill + #)) 
b+2, L,— 2 
‘Strain on second bar =" —*2 By =72(/+%) 
b+ x, L~ Ie 
and since the pull on the one bar must balance the thrust on the other 
af, =Ay)y : > 
Since 2, and x, are very smal] compared with /, the error introduced 
by putting / instead of 7+, and 7+, in the above equations may be 
ected. 
The method indicated above may easily be extended to determine the 
relations: between the various quantities when the compound bar is made 
up of more than two bars of different materials. 
84. Bars of Varying Cross Section.—If at any point in the length 
of a bar which is in tension the cross section suddenly changes, then the 
stress at that section will not be uniformly distributed over the section ; 
and at sections for some distance on each side of that section the stress 
will not be uniformly distributed, and the rules already demonstrated in 
this chapter will not apply. But if the several parts of a bar between 
the points where sudden changes of section occur be long compared with 
their cross sections, the elongations of these several parts of the bar may 
7 
