Module 2: Analysis of Stress
2.2.1 PRINCIPAL STRESS IN THREE DIMENSIONS
For the three-dimensional case, for principal stresses it is required that three planes of zero
shear stress exist, that these planes are mutually perpendicular, and that on these planes the
normal stresses have maximum or minimum values. As discussed earlier, these normal
stresses are referred to as principal stresses, usually denoted by s1, s2 and s3. The largest
stress is represented by s1 and the smallest by s3.
Again considering an oblique plane
x ¢ , the normal stress acting on this plane is given by the
s x¢ = sx l2 + sy m2 + sz n2 + 2 (txy lm + tyz mn + txz ln)
The problem here is to determine the extreme or stationary values of s x¢ . To accomplish
this, we examine the variation of s x¢ relative to the direction cosines. As l, m and n are not
independent, but connected by l2 + m2 + n2 = 1, only l and m may be regarded as
¶s x '
¶s x '
Differentiating Equation (2.27), in terms of the quantities in Equations (2.22a), (2.22b),
(2.22c), we obtain
Ty + Tz
T x+ T z
From n2 = 1 - l2 - m2, we have
Introducing the above into Equation (2.27b), the following relationship between the
components of T and n is determined
Applied Elasticity for Engineers
T.G.Sitharam & L.GovindaRaju