The gusset plate is subjected to the forces of three members. Determine the tension force in member C and its angle θ for equilibrium. The forces are concurrent at point O. Take F=8 kN.
Solution:
Free-body diagram:
Equations of Equilibrium:
Taking the sum of forces in the x-direction:
\begin{aligned}
T \cos \beta-\frac{4}{5}(8) & = 0 & & \\
T \cos \beta& = \frac{4}{5}(8) & &\\
T \cos \beta & = 6.4 \qquad \qquad & & (1)\\
\end{aligned}Taking the sum of forces in the y-direction:
\begin{aligned}
9-\frac{3}{5}(8)-T \sin \beta & = 0 & &\\
T \sin \beta & =9-\frac{3}{5}(8) & & \\
T \sin \beta & =4.2 & &\qquad \qquad (2)
\end{aligned}Equation (2) divided by equation (1) to solve for angle β.
\begin{aligned}
\dfrac{T \sin \beta}{T \cos \beta} & = \frac{4.2}{6.4} \\
\tan \beta & = 0.65625 \\
\beta & =\tan ^{-1}(0.65625) \\
\beta & =33.27 \degree
\end{aligned}Substitute the solved value of the angle β to equation (1) to solve for T.
\begin{aligned}
T \cos \beta & =6.4 \\
T & = \dfrac{6.4}{\cos \beta}\\
T & = \dfrac{6.4}{\cos 33.27 \degree}\\
T & =7.65 \ \text{kN}
\end{aligned}Solve for the value of the unknown angle θ:
\begin{aligned}
\theta & = \beta +\tan ^{-1} \left( \frac{3}{4}\right) \\
\theta & = 33.27 \degree + 36.87 \degree \\
\theta & =70.14\degree
\end{aligned}Therefore, the tension force in member C is 7.65 kN and its angle θ is 70.14°.