The 2-mm-thick bar shown in Fig. 3–30 is loaded axially with a constant force of 10 kN.The bar material has been heat treated and quenched to raise its strength, but as a consequence it has lost most of its ductility. It is desired to drill a hole through the centerof the 40-mm face of the plate to

Question

The 2-mm-thick bar shown in Fig. 3–30 is loaded axially with a constant force of 10 kN.
The bar material has been heat treated and quenched to raise its strength, but as a consequence it has lost most of its ductility. It is desired to drill a hole through the center of the 40-mm face of the plate to allow a cable to pass through it. A 4-mm hole is sufficient for the cable to fit, but an 8-mm drill is readily available. Will a crack be more likely to initiate at the larger hole, the smaller hole, or at the fillet?

Accepted Answer

Since the material is brittle, the effect of stress concentrations near the discontinuities must be considered.

Dealing with the hole first, for a 4-mm hole, the nominal stress is

[latex]\sigma _0=\frac{F}{A}=\frac{F}{\left(\omega -d\right)t } =\frac{10000}{\left(40-4\right)2 } =139 \ MPa[/latex]

The theoretical stress concentration factor, from Fig. A–15–1, with [latex] {d}/{\omega ={4}/{40}}=0.1 , \ is \ K_t=2.7[/latex].

The maximum stress is

[latex] \sigma_{max} =K_t\sigma _0=2.7\left(139\right) =380 \ MPa [/latex]

Similarly, for an 8-mm hole,

[latex]\sigma _0=\frac{F}{A}=\frac{F}{\left(\omega -d\right)t } =\frac{10000}{\left(40-8\right)2 } =156\ MPa[/latex]

with [latex] {d}/{\omega ={8}/{40}}=0.2, \ is \ K_t=2.5 [/latex].

The maximum stress is

[latex] \sigma_{max} =K_t\sigma _0=2.5\left(156\right) =390\ MPa [/latex]

Though the stress concentration is higher with the 4-mm hole, in this case the increased nominal stress with the 8-mm hole has more effect on the maximum stress. For the fillet,

[latex]\sigma _0=\frac{F}{A} =\frac{10000}{\left(34\right)2 } =147 \ MPa[/latex]

From Table A–15–5, [latex][{D}/{d}={40}/{34}=1.18 , \ and \ {r}/{d}={1}/{34}=0.026.[/latex].

Then [latex]K_t=[/latex]

[latex]\sigma _{max} =K_t\sigma _0=2.5\left(147\right) =368 \ MPa [/latex]

The crack will most likely occur with the 8-mm hole, next likely would be the 4-mm
hole, and least likely at the fillet.

Question 1.3.13: The 2-mm-thick bar shown in Fig. 3–30 is loaded axially with...

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