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Strength of Materials

Maximum Principal Strain Theory

In a two dime...

Question

In a two dimensional stress system, the two principal stresses are $σ_{1}$ of 180 $N / m m^{2}$ (tensile) and $σ_{2}$ (compressive). For the material, yield stress in simple tension and compression is 240 $N / m m^{2}$ and Poisson's ratio is 0.25. According to maximum normal strain theory, the value of $σ_{2}$ at which yielding will commence, is

240

N / m m^{2}

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180

N / m m^{2}

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195

N / m m^{2}

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200

N / m m^{2}

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Solution

The correct option is C 195 $N / m m^{2}$
For yielding
(i) Maximum normal tensile strain
$\frac{240}{E} = \frac{(180 + 0.25 \times σ_{2})}{E}$
(ii) Maximum normal compressive strain
$\frac{240}{E} = (\frac{0.25 \times 180 + σ_{2}}{E})$
$∴ F r o m c o n d i t i o n (i) σ_{2} = 240 N / m m^{2}$

From condition $(i i) σ_{2} = 195 N / m m^{2}$
So the valueof $σ_{2}$ at which yielding will start is 195 $N / m m^{2}$

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The correct option is C 195 N/mm2For yielding (i) Maximum normal tensile strain 240E=(180+0.25×σ2)E (ii) Maximum normal compressive strain 240E=(0.25×180+σ2E) ∴From condition (i)σ2=240N/mm2 From condition (ii)σ2=195N/mm2 So the valueof σ2 at which yielding will start is 195 N/mm2