Problem

In Example 14.1, let the hole be circular rather than elliptical. For stresses σ1, σ2 as g...

In Example 14.1, let the hole be circular rather than elliptical. For stresses σ1, σ2 as given,

a. determine the maximum tensile stress in the plate
b. determine the maximum compressive stress in the plate

Example 14.1 Narrow Elliptical Hole in a Plate

Consider an elliptical hole in a plate with ratio a/b = 100 (Figure 14.13). For this large value of a/b, the hole appears as a very narrow slit (crack) in the plate. Let compressive stresses σ1 = −20 MPa and σ2 = −75 MPa be applied to the plate edges.

(a) Determine the orientation of the hole (value of θ) for which the tensile stress at the perimeter of the hole is a maximum.
(b) Calculate the value of this tensile stress.
(c) Calculate the associated value of β (location of the point) for which this tensile stress occurs.

Solution:

Since a/b = 100, Eq. 14.10 indicates that coth α0 = 1/tanh α0 = a/b = 100. Hence, α0 = 0.0100 rad, sinh 2α0 = 0.0200, cosh 2α0 = 1.000, and coth 2α0 = 50.0. For these values of σ1, σ 2, and α0, Eq. 14.36 is satisfied.

(a) The value of θ is given by Eq. 14.34. Hence,
or
θ = 0.6535 rad
(b) The maximum value of the tensile stress is given by Eq. 14.38. Thus,

(c) This tensile stress is located on the perimeter of the hole at a value of β given by Eq. 14.35:

or
θ = 0.0342 rad

This small value of β means that the maximum tensile stress occurs very near the end of the major axis of the elliptical hole (see Figures 14.13 and 14.5).

The preceding computation shows that a slender elliptical hole (long narrow crack) in a plate may result in a high tensile stress concentration even when the applied edge stresses are compressive.

FIGURE 14.13 Infinite plate with inclined elliptical hole and uniformly distributed stresses σ1 > σ2.