Statics/Method of Joints

The method of joints is a way to find unknown forces in a truss structure. The principle behind this method is that all forces acting on a joint must add to zero. If there were a net force, the joint would move.

Question


Find the force in member BC of the truss pictured to the right.

Answer
Using the method of joints, the force could be found by isolating the joint at either end of the member (joint B or C). Neither joint can be solved without further analysis; however, joint B can be solved if the force in member $$ AB $$ and $$ BH $$ is found.

To find force $$ AB $$ analyze joint A. This joint has an external vertical force of 300N which must be countered by the members attached to the joint. Member $$ AE $$ cannot possibly support any vertical load, otherwise it would not be loaded axially and the entire structure would no longer be a truss. If $$ AE $$ has no load then member $$ AB $$ is in 300N of tension.

When joint H is analyzed it is found that the force in members $$ BH $$ and $$ HC $$ must be zero. The reason why neither member can carry any load is that member $$ BH $$ can only take a vertical load and member $$ HC $$ can only take a horizontal load. In a real world application this structure might be useful if there was a load applied at joint $$ H $$. Now joint $$ B $$ can be analyzed.



The picture to the left shows the forces affecting joint B.

$$ \sum F_y = 0 = BH - BA + BC \cos(50) - BE \cos(50)$$

$$ \sum F_x = 0 = BC \sin(50) + BE \sin(50)$$

Substitution
From analysis of joint $$ A $$

$$ \ BA = 300N (Tension) $$

From analysis of joint $$ H $$

$$ \ BH = 0N $$

Put values for $$ BA $$ and $$ BH $$ into the equilibrium equations for joint B.

$$ \sum F_y = 0 = 0 - 300N + BC \cos(50) - BE \cos(50)$$

$$ \sum F_x = 0 = BC \sin(50) + BE \sin(50)$$

$$ \ BC \sin(50) = -BE \sin(50) $$

$$ \ BC = -BE $$

Now $$ BC $$ can be inserted in place of $$ -BE $$ in $$ \sum F_y $$, which gives:

$$ 2BC \cos(50) - 300N = 0 $$

Finally, $$ BC $$ can be solved for as follows:

$$ BC = 233.4N $$