18
2 Analysis of Stress
In Fig. 2.9, T x , T y , T z are the Cartesian components of stress resultant T, acting
on oblique plane ABC. It is required to relate the stresses on the perpendicular planes
intersecting at the origin to the normal and shear stresses acting on ABC.
The orientation of the plane ABC may be defined in terms of the angle between
a unit normal n to the plane and the x-, y- and z-directions. The direction cosines
associated with these angles are
cos(n, x) = l
cos(n, y) = m and
cos(n, z) = n
(2.21)
But the three direction cosines are related by
l
2
+ m
2
+ n
2
= 1
(2.22)
The area of the perpendicular plane PAB, PAC, PBC may now be expressed in
terms of A, the area of ABC, and the direction cosines.
Therefore,
Area of PAB = A P AB = A x = A. i
= A(li + m j + nk) . i
Hence, A PAB = Al
The other two areas are similarly obtained. In doing so, we have altogether
A PAB = Al, A PAC = Am and A PBC = An
(2.23)
Here, i, j and k are unit vectors in x-, y- and z-directions, respectively.
Now, for equilibrium of the tetrahedron, the sum of forces in x-, y- and z-directions
must be zero.
Therefore,
T x A = σ x Al + τ xy Am + τ xz An
(2.24)
Dividing throughout by A, we get
T x = σ x l + τ xy m + τ xz n
(2.24a)
Similarly, for equilibrium in y- and z-directions,
T y = τ xy l + σ y m + τ yz n
(2.24b)
and