Statics

Frames and Machines: Take Them Apart at the Pins

Multi-force members, dismembering at pins, the pin action-reaction pair, a beam-brace frame solve, a machine's mechanical advantage

Unlike a truss, a frame or machine has members pushed at three or more points, so their forces no longer run along the bar. The move is to take the structure apart at its pins and balance each piece on its own.

Toggle the third force on. With forces at three points, a member can no longer push only along its line - it bends and shears. That is a multi-force member.

Take the frame apart at the pin. Each member gets its own free-body diagram, with the pin force where they used to join.

At a shared pin, the two members push on each other with equal and opposite forces. Switch members - the pin force just flips by Newton's third law.

Drag the load. With the brace as a two-force member, one moment equation on the beam gives the brace force, scaling with P.

A machine trades distance for force. Drag the input; the lever multiplies it by the ratio of the arms - that is mechanical advantage.

In PracticeFrames and machines contain multi-force members - loaded at three or more points - so unlike a truss their forces do not lie along the member. Solve them by dismembering: draw a separate FBD for each member and apply ΣFx=0, ΣFy=0, ΣM=0, remembering that a shared pin exerts equal and opposite forces on the two members it joins. Spotting any two-force member (a brace or link) collapses its pin force to a single unknown along its line. A frame is built to stand still; a machine uses the same balance to trade movement for a mechanical advantage. Next, distributed loads replace these point forces with a load spread along a length.

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