Determine the required diameter of the bolt at B. Determine the required outer diameter of the washers at B.

Steps to Solve

1. Determine the applied forces First, we need to establish the tensile force acting on the bolt due to the applied vertical loads. From the problem, the total vertical force $F$ on the connection at $B$ can be found by adding the vertical loads.

2. Calculate the required diameter of the bolt The tensile stress ($\sigma_t$) in the bolt is defined as: $$ \sigma_t = \frac{F}{A} $$ where $A$ is the cross-sectional area of the bolt, given by $A = \frac{\pi}{4}d_B^2$. Rearranging for $d_B$ gives: $$ d_B = \sqrt{\frac{4F}{\pi \sigma_{t\text{allow}}}} $$

3. Input the allowable tensile stress The allowable tensile stress for the bolt is given as $\sigma_{t\text{allow}} = 180 \text{ MPa}$.

4. Determine the required outer diameter of the washers For the outer diameter of the washers ($d_w$), the bearing stress ($\sigma_b$) is defined similarly: $$ \sigma_b = \frac{F}{A_w} $$ where $A_w$ is the contact area of the washer (considered to be the outer surface of the washer), which can be simplified as: $$ A_w = d_w \cdot t $$ where $t$ is the thickness of the washer. We arrange for that to get: $$ d_w = \frac{F}{\sigma_{b\text{allow}} \cdot t} $$

5. Input the allowable bearing stress The allowable bearing stress for the wood is given as $\sigma_{b\text{allow}} = 36 \text{ MPa}$.

6. Perform calculations Now substitute values to find the required diameters.