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A bar is subjected to a combination of a steady load of $60 \: kN$ and a load fluctuating between $-10 \: kN$ and $90 \: kN$. The corrected endurance limit of the bar is $150 \: MPa$, the yield strength of the material is $480 \: MPa$ and the ultimate strength of the material is $600 \: MPa$. The bar cross-section is square with side $a$. If the factor of safety is $2$, the value of $a$ (in $mm$), according to the modified Goodman's criterion, is _____ (correct to two decimal places).
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