Problem 920 | Additional Centroidal Load to Eliminate Tensile Stress Anywhere Over the Cross Section | Strength of Materials Review at MATHalino

Problem 920
A compressive load P = 100 kN is applied, as shown in Fig. 9-8a, at a point 70 mm to the left and 30 mm above the centroid of a rectangular section for which h = 300 mm and b = 250 mm. What additional load, acting normal to the cross section at its centroid, will eliminate tensile stress anywhere over the cross section?

Solution 920

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Stresses due to the given load, see figure below for reference

$\sigma_1 = \dfrac{P}{\text{Area}} = \dfrac{100,000}{150(300)}$

$\sigma_1 = 20/9 ~ \text{MPa}$

$f_{bx} = \dfrac{6M_x}{hb^2} = \dfrac{6(100,000 \times 30)}{300(150^2)}$

$f_{bx} = 8/3 ~ \text{MPa}$

$f_{by} = \dfrac{6M_y}{bh^2} = \dfrac{6(100,000 \times 70)}{150(300^2)}$

$f_{by} = 28/9 ~ \text{MPa}$

At corner C, both f_bx and f_by are tensile stresses, hence, corner C is carrying the largest tension

$\sigma_C = -\sigma_1 + f_{bx} + f_{by} = -20/9 + 8/3 + 28/9$

$\sigma_C = 32/9 ~ \text{MPa}$

Additional axial compression load P₂ to eliminate tensile stress anywhere over the cross section.

$\sigma_C - \sigma_2 = 0$

$32/9 - \dfrac{P_2 (1000)}{300(150)} = 0$

$P_2 = 160 ~ \text{kN}$ ← answer

Problem 920 | Additional Centroidal Load to Eliminate Tensile Stress Anywhere Over the Cross Section

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