# Gusset plate is subjected to the forces of four members

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.

$\sum \text{F}_\text{x}=0$

$8+5\text{sin}\,(45^0)-T\text{cos}\,(\theta)=0$ ——————(eq.1)

Next, we will write the equations of equilibrium for the y-axis forces.

$\sum \text{F}_\text{y}=0$
$12-5\text{cos}\,(45^0)-T\text{sin}\,(\theta)=0$ ——————(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 $\frac{\text{sin}\,\theta} {\text{cos}\,\theta}=\text{tan}\,\theta$

$11.53\text{tan}\,\theta=8.46$

$\theta=\text{tan}^-1(\frac{8.46}{11.53})$

$\theta=36.3^0$

Finally, substitute the value of θ into eq.3:

$T=\frac{11.53}{\text{cos}\,36.3^0}$

$T=14.3$ kN