Frames
Frames and machines like pliers and wrenches use pin connections to create mechanical advantage—they allow a small applied force at one point to produce a much larger force at another. This is especially helpful when we need to exert forces that would be difficult or impossible with our bare hands, such as tightening bolts or gripping pipes.
The pin joints act as pivots, allowing different members of the frame to rotate and transfer forces efficiently.
As in the case of the pliers, the input force (from our hand) travels through rigid links and pivots, generating a greater output force at the jaws due to the geometry and relative lengths of the members—shorter output arms and longer input arms multiply the applied force.
This principle is the same behind levers and is fundamental to machines like wrenches, clamps, scissors, and even robotic arms.
We first analyze member BCD. In our free-body diagram, we draw the reactions of the internal hinge at B (pin). We still base our assumptions on our positive convention, so we write Bx going to the right and By going downward. In the solution below, Bx came out as negative. This means that the internal reaction, Bx, is leftward. Next, we compute the resultant reaction at B by using the Pythagorean Theorem. RB is simply the hypotenuse of the force triangle formed by the components of the internal reaction at B.
Now, in member BCD, we obtained a negative value for Bx, which means that it is actually leftward. However, to avoid redrawing arrows, since this cannot be done conveniently when using a pen, we retain the original direction based on our initial assumptions on member BCD. In member BCD, we wrote Bx as rightward — but it came out to be negative, so it should be leftward; and reversing the sign for member AB, Bx shall act rightward.