Statics · General force systems and equilibrium of the rigid body

The wall-mounted slewing crane

On the right, a trolley carries the load on a horizontal jib. The jib is connected to the wall at ; a tension strut runs from the wall point to the jib end and holds it up. The trolley can be moved — and that is precisely the difference from the previous module: the same load acts quite differently depending on its position on the crane.

Move the trolley with the slider and watch the value box. The tie-rod force grows the further the load travels outwards, even though stays the same. The next chapters develop why.

The section through a rigid body

The procedure stays the same as in the previous module: isolate, draw the free-body diagram, set up the equilibrium conditions, solve. What is new is that the forces now act at different points of the isolated body — so the free-body diagram must show not only which forces act, but also where.

Which body you isolate is again a decision. We are looking for the tie-rod force and the forces at the wall connection — a cut that exposes only one of them does not help.

Hands-on task: Cut it free yourself — first decide which body you isolate, then add the forces at their points of application. For the tie-rod force you may assume either tension or compression; draw the support reactions along the coordinate axes.

Parallel forces and the couple

Two parallel forces can be combined into a resultant: its magnitude is the algebraic sum, , and its line of action lies between them such that the law of the lever of Archimedes is satisfied:

This construction fails in exactly one case — when the two forces are equal in magnitude and opposite in direction. Then and the distances run to infinity. Such a pair is called a couple and it can not be reduced to a single force.

This turning effect is described by the moment of the couple: its magnitude and its sense of rotation, which we draw as a curved arrow. The unit is the newton metre — written with the length written behind the force, so that no confusion with the milli-newton can arise.

One and the same couple has arbitrarily many equivalent representations: the forces may be reduced and the distance increased accordingly — the product stays the same. From this follows the decisive property:

Without changing its effect it can act at any arbitrary point of a rigid body. That is why later one no longer draws a couple at all, but only the symbol with a curved arrow. If several moments act, they are, respecting the sense of rotation, algebraically added:

Moment of a force and lever arm

Along its line of action a force may travel. But what happens if it is moved parallel ? If two opposing forces of magnitude are placed on the new line of action — an equilibrium group that changes nothing — then one of them forms, together with the original force, a couple at the distance . What remains is therefore the force at its new position plus a moment.

This quantity is called the moment of the force with respect to the point ; the superscript names the reference point. The perpendicular distance between the reference point and the line of action is called the lever arm. By convention, a moment counts as positive if it rotates counter-clockwise.

Instead of finding the lever arm geometrically, one usually resolves the force into its Cartesian components and adds their moments:

Behind this stands a statement that carries the whole chapter: The sum of the moments of the individual forces equals the moment of the resultant. It therefore makes no difference whether the forces are added first and the moment taken afterwards, or whether the individual moments are summed. In the literature this statement is also known as the moment theorem; the textbook states it without giving it a name.

In the view on the right you can click the reference point click — , or the trolley. The moment arc of the load moves along, and the cell shows what comes out.

The resultant of plane force systems

Now the general question: how far can an arbitrary plane force group be reduced? One chooses an arbitrary reference point and moves each force parallel to itself until its line of action passes through — adding for each the moment that arises in doing so. What remains is a concurrent force system and a group of moments:

The direction angle is called ; at this point the textbook writes , which in this module already belongs to the inclination of the tie rod.

One more step is possible: instead of stating the resultant through together with the moment , one can state only — but then its line of action must be shifted by the distance

parallel to itself. From this follow four cases to which every plane force group can be reduced: a resultant beside the reference point · a resultant through the reference point () · a pure couple (, in which case the moment is independent of the reference point) · and equilibrium (both zero).

Equilibrium conditions of the rigid body

Equilibrium prevails when the reduction leads to nothing — that is, when both the resultant force and the resultant moment vanish. In components these are three conditions:

Three conditions for three degrees of freedom: in the plane a body can translate in the - and directions and rotate about an axis perpendicular to the drawing plane. Three equations accordingly determine at most three unknowns — for the wall-mounted slewing crane exactly the tie-rod force and the two support reactions.

The moment condition always requires two pieces of information: the reference point and the sense chosen as positive. The textbook therefore marks each balance with a direction arrow — for the horizontal, for the vertical force balance, and for the moment balance about . The derivation below follows this notation.

From this follows the most important practical rule of the chapter: it is usually expedient to choose the reference point so that it lies on the line of action of a force . Its lever arm is then zero and it does not appear in the moment equilibrium at all — one unknown fewer in the equation.

General force systems in space

In the plane there was only a possible rotation — about the axis perpendicular to the drawing plane. In space there are three, and the moment therefore needs three components. It becomes the moment vector:

So that it is not confused with a force in drawings, the moment vector is drawn as a double-headed arrow — as in the view on the right. Its sense follows the corkscrew rule: looking along the vector, a positive moment rotates clockwise.

The procedure does not change: reduce to a resultant force and a resultant moment, then balance. Only now there are six conditions for six degrees of freedom — three translations and three rotations: