Technical mechanics: TestAS practice testlets

Statics studies bodies at rest. A body is in equilibrium if two conditions are met: the sum of all forces acting on it is zero, so it does not start to move, and the sum of all moments about any point is zero, so it does not start to rotate. These two conditions allow engineers to calculate unknown support forces in bridges, shelves, cranes and beams.

ΣF = 0 ΣM = 0 moment M = F·d

Consider a horizontal beam resting on two supports, A at the left end and B at the right end. Loads act downwards, and the supports push upwards with reaction forces F_A and F_B. To find F_B, it is convenient to take moments about A: the unknown force F_A then has no lever arm and drops out of the equation. The moment of F_B about A must balance the moments of all loads about A. Once F_B is known, F_A follows from the force balance: F_A + F_B equals the total load.

The weight of the beam itself also acts as a load. For a uniform beam, it can be treated as a single force acting at the middle of the beam.

As a check, a load placed exactly in the middle of a beam is shared equally between the two supports, and a load placed directly above one support is carried entirely by that support. The closer a load is to a support, the larger the share it places on that support.

Engineers use such calculations to choose supports, bearings and foundations of adequate size and to make sure that no support has to pull rather than push, which a simple resting support cannot do.

The same method applies to more complex structures, such as trusses, which are analysed joint by joint, and to cantilevers, beams fixed at one end only, where the support must provide both a force and a moment. A shelf fixed to a wall with brackets is a simple example: the brackets must resist the turning effect of the load as well as its weight.

A force can make a body rotate about an axis. The turning effect is called the torque, or moment, and it equals the force multiplied by the perpendicular distance between the line of action of the force and the axis, called the lever arm. Torque is measured in newton metres (N·m).

M = F·r balance: sum of clockwise moments = sum of anticlockwise moments

Tools use this relationship. A spanner turns a bolt: the longer the spanner, the smaller the force needed to produce a given torque. Bolts in engines and wheels must be tightened to a specified torque, because too little torque lets the joint loosen and too much can stretch or break the bolt. For this reason mechanics use a torque wrench, which shows or limits the applied torque. A force applied at an angle is less effective: only the component perpendicular to the spanner contributes to the torque.

Cranes illustrate the balance of moments on a large scale. A mobile crane lifts a load at the end of a long boom. The weight of the load produces a moment about the edge on which the crane would tip over, the tipping edge. To prevent tipping, the crane carries a counterweight on the opposite side, and its outriggers are extended to move the tipping edge outwards. The crane is stable as long as the moment of the counterweight and of the crane's own weight about the tipping edge is larger than the moment of the load.

Because the moment of the load grows with its distance from the tipping edge, a crane can lift heavy loads close to its base but only lighter loads at full reach. Every crane therefore has a load chart that states the permitted load for each radius, and modern cranes have a load moment limiter that stops the movement when the limit is approached. In the calculations below, the weight of the crane itself and of the boom is neglected, and a safety margin is not included.