In Example 14.1, let the ratio b/a = 5. Determine the orientation of the hole (value of θ) for which the tensile stress at the perimeter of the hole is a maximum, the magnitude of the maximum tensile stress in the plate, and its location.
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,
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(c) This tensile stress is located on the perimeter of the hole at a value of β given by Eq. 14.35:
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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.
FIGURE 14.5 Elliptical hole in an infinite plate.
(14.10)
(14.36)
(14.34)
(14.38)
(14.35)
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