Problem 906 | Combined Axial and Bending | Strength of Materials Review at MATHalino

Problem 906
For the 2-in. by 6-in. wooden beam shown in Fig. P-906. Determine the normal stress at A and B. Are these the points of maximum normal stress? If not, where are they located and what are their values?

Solution 906

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$\Sigma F_H = 0$

$F_H = 2000 \cos 15^\circ ~ \text{lb}$

$\Sigma M_E = 0$

$6F_V = \frac{1}{4}(2000 \cos 15^\circ) + 2(2000 \sin 15^\circ)$

$F_V = \frac{250}{3}\cos 15^\circ + \frac{2000}{3}\sin 15^\circ ~ \text{lb}$

$F_V = \frac{250}{3}(\cos 15^\circ + 8 \sin 15^\circ) ~ \text{lb}$

At section through C

$M_C = \Sigma M_\text{to the right of C}$

$M_C = 3F_V$

$M_C = 250(\cos 15^\circ + 8 \sin 15^\circ) ~ \text{lb}\cdot\text{ft}$

$\sigma_a = \dfrac{P}{A} = \dfrac{2000 \cos 15^\circ}{2(6)}$

$\sigma_a = 160.99 ~ \text{psi}$

$\sigma_f = \dfrac{6M}{bd^2} = \dfrac{6[ \, 250(\cos 15^\circ + 8\sin 15^\circ) \, ] (12)}{2(6^2)}$

$\sigma_f = 759.11 ~ \text{psi}$

$\sigma_A = -\sigma_a - \sigma_f = -160.99 - 759.11$

$\sigma_A = -920.10 ~ \text{psi}$ answer

$\sigma_B = -\sigma_a + \sigma_f = -160.99 + 759.11$

$\sigma_B = 598.12 ~ \text{psi}$ answer

At section through D

By inspection, the maximum moment is at D, hence, the maximum normal stress will occur at the extreme fibers of section through D.

$M_D = \Sigma M_\text{to the right of D}$

$M_D = 4F_V$

$M_D = \frac{1000}{3}(\cos 15^\circ + 8 \sin 15^\circ) ~ \text{lb}\cdot\text{ft}$

$\sigma_a = \dfrac{P}{A} = \dfrac{2000 \cos 15^\circ}{2(6)}$

$\sigma_a = 160.99 ~ \text{psi}$

$\sigma_f = \dfrac{6M}{bd^2} = \dfrac{6[ \, \frac{1000}{3}(\cos 15^\circ + 8\sin 15^\circ) \, ] (12)}{2(6^2)}$

$\sigma_f = 1012.16 ~ \text{psi}$

$\sigma_\text{top} = -\sigma_a - \sigma_f = -160.99 - 1012.16$

$\sigma_\text{top} = -1173.15 ~ \text{psi}$ answer

$\sigma_\text{bottom} = -\sigma_a + \sigma_f = -160.99 + 1012.16$

$\sigma_\text{bottom} = 851.17 ~ \text{psi}$ answer

Problem 906 | Combined Axial and Bending

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