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.

members of a truss are connected to the gusset plate

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:

Gusset plate is subjected to the forces of four members

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

This question can be found in Engineering Mechanics: Statics (SI edition), 13th edition, chapter 3, question 3-6.

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