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1.12 Three-Dimensional Stress Transformation

The physical elements studied are always three dimensional, and hence it is desirable to consider three planes and their associated stresses, as illustrated in Fig. 1.2. We note that equations governing the transformation of stress in the three-dimensional case may be obtained by the use of a similar approach to that used for the two-dimensional state of stress.

Consider a small tetrahedron isolated from a continuous medium (Fig. 1.19a), subject to a general state of stress. The body forces are taken to be negligible. In the figure, px , py , and pz are the Cartesian components of stress resultant p 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 on ABC.

Figure 1.19

Figure 1.19 Stress components on a tetrahedron.

The orientation of plane ABC may be defined in terms of the angles between a unit normal n to the plane and the x, y, and z directions (Fig. 1.19b). The direction cosines associated with these angles are

Equation 1.24

01equ24.jpg


The three direction cosines for the n direction are related by

Equation 1.25

01equ25.jpg


The area of the perpendicular plane QAB, QAC, QBC may now be expressed in terms of A, the area of ABC, and the direction cosines:

AQAB = Ax = A · i = A(li + mj + nk ) · i = Al

The other two areas are similarly obtained. In so doing, we have altogether

Equation a

034equ01.jpg


Here i , j , and k are unit vectors in the x, y, and z directions, respectively.

Next, from the equilibrium of x, y, z-directed forces together with Eq. (a), we obtain, after canceling A,

Equation 1.26

01equ26.jpg


The stress resultant on A is thus determined on the basis of known stresses s x , s y , s z , t xy , t xz and t yz and a knowledge of the orientation of A. In the limit as the sides of the tetrahedron approach zero, plane A contains point Q. It is thus demonstrated that the stress resultant at a point is specified. This in turn gives the stress components acting on any three mutually perpendicular planes passing through Q as shown next. Although perpendicular planes have been used there for convenience, these planes need not be perpendicular to define the stress at a point.

Consider now a Cartesian coordinate system x', y', z', wherein x' coincides with n and y', z' lie on an oblique plane. The x' y' z' and xyz systems are related by the direction cosines: l 1 = cos (x', x), m 1 = cos(x', y), and so on. The notation corresponding to a complete set of direction cosines is shown in Table 1.2. The normal stress s x' is found by projecting px , py , and pz in the x' direction and adding

Equation 1.27

01equ27.jpg


Table 1.2. Notation for Direction Cosines

x

y

z

x'

l 1

m 1

n 1

y'

l 2

m 2

n 2

z'

l 3

m 3

n 3

Equations (1.26) and (1.27) are combined to yield

Equation 1.28a

01equ28a.jpg


Similarly, by projecting px , py , and pz in the y' and z' directions, we obtain, respectively,

Equation 1.28b

01equ28b.jpg


Equation 1.28c

01equ28c.jpg


Recalling that the stresses on three mutually perpendicular planes are required to specify the stress at a point (one of these planes being the oblique plane in question), the remaining components are found by considering those planes perpendicular to the oblique plane. For one such plane, n would now coincide with the y' direction, and expressions for the stresses s y' , t y'x' , and t y'z' would be derived. In a similar manner, the stresses s z' , t z'x' , and t z'y' are determined when n coincides with the z' direction. Owing to the symmetry of the stress tensor, only six of the nine stress components thus developed are unique. The remaining stress components are as follows:

Equation 1.28d

01equ28d.jpg


Equation 1.28e

01equ28e.jpg


Equation 1.28f

01equ28f.jpg


Equations (1.28) represent expressions transforming the quantities s x , s y , s z , t xy , t xz , and t yz which, as we have noted, completely define the state of stress. Quantities such as stress (and moment of inertia, Appendix C), which are subject to such transformations, are tensors of second rank (see Sec. 1.9).

The equations of transformation of the components of a stress tensor, in indicial notation, are represented by

Equation 1.29a

01equ29a.jpg


Alternatively,

Equation 1.29b

01equ29b.jpg


The repeated subscripts i and j imply the double summation in Eq. (1.29a), which, upon expansion, yields

Equation 1.29c

01equ29c.jpg


By assigning r, s = x, y, z and noting that t rs = t sr , the foregoing leads to the six expressions of Eq. (1.28).

It is interesting to note that, because x', y', and z' are orthogonal, the nine direction cosines must satisfy trigonometric relations of the following form:

Equation 1.30a

01equ30a.jpg


and

Equation 1.30b

01equ30b.jpg


From Table 1.2, observe that Eqs. (1.30a) are the sums of the squares of the cosines in each row, and Eqs. (1.30b) are the sums of the products of the adjacent cosines in any two rows.

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