Page 198 - Introduction to Continuum Mechanics

Maximum Shearing Stress 183

Thus, if T s denotes the magnitude of the total shearing stress on the plane, we have (see
Fig. 4.6)

i.e.,

Fig. 4.6

For known values of 7\, TI, and 73, Eq. (4.6.4) states that 7J is a function of
W
e
«!, « 2- and 3> i- ->

or
w
We wish to find the triple (rti,«2> 3) ^ which / attains a maximum. However,

thus, we are looking for a maximum for the value of the function f(n ^2^3) subjected to the
constraint that «i 4-«2+«3 = 1.
Taking the total derivative of Eq. (4.6.5), we obtain