The gusset plate is subjected to the forces of four members. Determine the force in member B and its proper orientation ϴ for equilibrium. The forces are concurrent at point O. Take F = 12 kN.
Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.
Solution:
We will first draw the free body diagram like so:
We will assume forces going \rightarrow^+ to be positive and \uparrow+ to be positive.
Now, we can write the equations of equilibrium for the x-axis forces.
8+5\text{sin}\,(45^0)-T\text{cos}\,(\theta)=0 ——————(eq.1)
Next, we will write the equations of equilibrium for the y-axis forces.
12-5\text{cos}\,(45^0)-T\text{sin}\,(\theta)=0 ——————(eq.2)
(Remember F=12 kN as stated in the question)
We can now solve for T and θ.
From eq.1, we will isolate for T.
T=\frac{11.53}{\text{cos}\,\theta} ——————(eq.3)
We will now substitute this value into eq.2.
8.46-\frac{(11.53)(\text{sin}\,\theta)}{\text{cos}\,\theta}=0
Now, we can solve for θ. Remember that \frac{\text{sin}\,\theta} {\text{cos}\,\theta}=\text{tan}\,\theta
11.53\text{tan}\,\theta=8.46
\theta=\text{tan}^-1(\frac{8.46}{11.53})
Finally, substitute the value of θ into eq.3:
T=14.3 kN