Members of a truss are pin connected at joint O


The members of a truss are pin connected at joint O. Determine the magnitudes of F_1 and F_2 for equilibrium. Set ϴ = 60°.

Members of a truss are pin connected at joint O

Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.

Solution:

Let us first draw a free body diagram and label our forces like so:

Members of a truss are pin connected at joint O

As stated in the question, we set ϴ = 60° in our diagram.

We will also pick which sides are positive. In this case, we will assume \rightarrow^+ is positive and \uparrow+ is positive.

 

Now, we will write each force in unit vector form as follows:

F_1=(F_1\text{cos}60^0)i\,+\,(-F_1\text{sin}60^0)j

F_2=(F_2\text{sin}60^0)i\,+\,(F_2\text{cos}60^0)j

F_3=(-5\text{cos}30^0)i\,+\,(-5\text{sin}30^0)j

F_3=(-7\frac{4}{5})i\,+\,(-7\frac{3}{5})j

(Note the negative signs. When the component of a force is towards the negative quadrant, we set it as negative, since we assumed that \rightarrow^+ is positive and \uparrow+ is positive)

 

For the forces to be in equilibrium, the x and y (i and j in unit vector form) forces must add up to zero. Mathematically, we can write this as:

 

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

\sum \text{F}_\text{y}=0

 

Let us now only look at the x forces (unit vector i):

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

0\,=\,0.5F_1\,+\,0.866F_2\,-\,4.33\,-\,5.6

(Here, we simplified all the trigonometric values) Simplify further:

0\,=\,0.5F_1\,+\,0.866F_2\,-\,9.93 ——- Eq.(1)

 

Now, we will look at the y forces (unit vector j):

 

\sum \text{F}_\text{y}=0

0\,=\,-0.866F_1\,+\,0.5F_2\,+\,2.5\,-\,4.2

(Again, we simplified all the trigonometric values) Simplify further:

0\,=\,-0.866F_1\,+\,0.5F_2\,-\,1.7 ——- Eq.(2)

 

We can now solve both equations simultaneously which gives us:

F_1\,=\,3.49 kN

F_2\,=\,9.45 kN

This question can be found in Engineering Mechanics: Statics (SI edition), 13th edition, chapter 3, question 3-1

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