Method of Joints

When analyzing a joint you will solve for the forces in all of the members connected to it.

Lets start with H, draw the two unknown forces F_{HE} and F_{HF} in tension (away from H).

You already know the downward force is 5k. F_{HE} is the only member with a vertical component so you can solve for it.

\sum{F_y} = \frac{10}{\sqrt{1000}}F_{HE} - 5k=0

 F_{HE} = 5k\frac{\sqrt{1000}}{10}\approx 15.8k (Tension)

Ok so you know F_{HE}. Use the horizontal component of F_{HE} to solve for F_{HF}:

\sum{F_x} = -\frac{30}{\sqrt{1000}}F_{HE} - F_{HF} = 0

 F_{HF} =-\frac{30}{\sqrt{1000}} 15.8k \approx -15k (Tension) = 15k (Compression)

Note: \frac{10}{\sqrt{1000}} is from the geometry of the structure. \sqrt{1000} = \sqrt{10^2 + 30^2} and \frac{10}{\sqrt{1000}} = \sin\theta (where \theta is the angle of the pointy end).


Jumping over to B

You do not know any of these forces. F_{BA} is a  doozy though. Summing the vertical forces reveals that..

 \sum{F_y} = F_{BA} = 0

F_{BA} is zero. You can calculate the horizontal reaction at B by taking moments of the entire truss at A.

\sum{M_A} = 10k(10ft) + 10k(20ft) + 10k(30ft) - 10ft(R_{Bx}) = 0

 R_{Bx} = \dfrac{(100 + 200 + 300)k*ft}{10 ft} =60k

Now sum the horizontal forces to find F_{BD}

 \sum{F_x} = 60k + F_{BD} = 0

 F_{BD} = -60k (Tension) = 60k (Compression)


Sliding on over to D

At this joint we know that F_{DB} = 60k( Compression) (so the arrow should actually point inward for compression), and that the downward force is 5k. We have three unknowns and only two equations (\sum{F_x} and \sum{F_y}). This joint is unsolvable so you must move on to another joint before coming back to it.

After practice you start to see the paths to take. At this point E is solvable (secret zero-force member), and then F will be solvable, and then D would be too.

Combining the Method of Joints with the Method of Sections makes things even faster because you can often jump straight to a joint or area that you need for the problem (it may only ask you to find one member for example).

About Conrad

I am a Civil Engineer. I work in San Diego and am preparing to take the PE Exam. I am interested in surfing, business, travelling, and spending time with my wife. Thanks!

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