A certain material has a yield stress limit in simple tension of 387 N/mm^2. The yield limit in compression can be taken to be equal to that in tension. The material is subjected to three stresses in mutually perpendicular directions, the stresses being in the ratio 3 : 2 : -1.8. Determine the

Question

A certain material has a yield stress limit in simple tension of [latex]387 N / mm ^{2}[/latex]. The yield limit in compression can be taken to be equal to that in tension. The material is subjected to three stresses in mutually perpendicular directions, the stresses being in the ratio 3 : 2 : -1.8.
Determine the stresses that will cause failure according to the von Mises and Tresca theories of elastic failure.
Ans. Tresca: [latex]\sigma_{ I }=241.8 N / mm ^{2}, \sigma_{ II }=161.2 N / mm ^{2}, \sigma_{ III }=-145.1 N / mm ^{2}[/latex].
von Mises: [latex]\sigma_{ I }=264.0 N / mm ^{2}, \sigma_{ II }=176.0 N / mm ^{2}, \sigma_{ III }=-158.4 N / mm ^{2}[/latex].

Accepted Answer

Let the unit stress be σ then the actual stresses are

[latex]\sigma_{ I }=3 \sigma, \quad \sigma_{ II }=2 \sigma, \quad \sigma_{ III }=-1.8 \sigma[/latex]

From the Tresca theory, since [latex]\sigma_{ III }[/latex] is negative

[latex]\sigma_{ I }-\sigma_{ III }=\sigma_{ Y }[/latex]

i.e.
[latex]3 \sigma+1.8 \sigma=387[/latex]

Then

[latex]\sigma=80.6 N / mm ^{2}[/latex]

and

[latex]\sigma_{ I }=241.8 N / mm ^{2}, \quad \sigma_{ II }=161.2 N / mm ^{2}, \quad \sigma_{ III }=-145.1 N / mm ^{2}[/latex]

From the von Mises theory, Eq. (14.54)

[latex]\left(\sigma_{ I }-\sigma_{ II } ight)^{2}+\left(\sigma_{ II }-\sigma_{ III } ight)^{2}+\left(\sigma_{ III }-\sigma_{ I } ight)^{2}=2 \sigma_{ Y }^{2}[/latex] (14.54)

[latex](3 \sigma-2 \sigma)^{2}+(2 \sigma+1.8 \sigma)^{2}+(-1.8 \sigma-3 \sigma)^{2}=2 imes 387^{2}[/latex]

from which

[latex]\sigma=88.0 N / mm ^{2}[/latex]

Then

[latex]\sigma_{ I }=264.0 N / mm ^{2}, \quad \sigma_{ II }=176.0 N / mm ^{2}, \quad \sigma_{ III }=-158.4 N / mm ^{2}[/latex] .

Question 14.P.19: A certain material has a yield stress limit in simple tensio...

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