statics explained

frames and machines statics

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 a machine has members that take force at three or more places, so those forces no longer follow the line of the bar. The move is to take the structure apart at the pins and write equilibrium for each piece on its own. Kept as one lump the unknowns overflow, and split apart the equations match.

Turn on a third force and the character changes. When a member takes force at three points it can no longer push only along its line, and it bends and takes shear, and that is a multiforce member. Unlike a two-force member, moment and shear both remain.

Take a frame apart at the pins and each member gets its own free-body diagram, and a pin force stands in where the pieces used to meet. As the pieces grow, the equations grow, and the pin forces join the two sides.

At a shared pin the two members push each other with forces of equal size and opposite direction. Switch the member and the pin force flips only its sign, following Newton's 3rd law. Fix one side and the other side is fixed with it.

Drag the load and where the force goes becomes plain. The brace is a two-force member, so one moment equation on the beam gives the brace force, and it grows together with the load P. Collapsing to a single unknown along the line is what the brace is for.

A machine is a device that trades distance for force. Drag the input and the lever grows the force by the ratio of the arm lengths, and that is the mechanical advantage. A large force stands on the short side, and a small force stands on the long side.

In PracticeFrames and machines have multiforce members that take force at three or more points, so unlike a truss the forces do not follow the member line. The solution is dismemberment. Draw a free-body diagram for each member and apply ΣFx=0, ΣFy=0, and ΣM=0, remembering only that a shared pin gives the two members forces of equal size and opposite direction. Spot a two-force member, a brace or a link, and that pin force collapses to one unknown along the line. A frame is built to stand still, and a machine uses the same equilibrium to trade motion for mechanical advantage. Next, distributed load turns these point loads into load spread along a length.

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