One of the main tasks in human industry is doing work against the force of gravity, and erecting structures such as bridges and buildings sufficient to withstand the gravitational force imposed on their mass and that of the people they carry. One must have a means of actually building these structures, and one of the most recognizable pieces of machinery for lifting heavy objects in precise ways is the crane.
Long dominating skylines where anything of size is being built, cranes function as levers capable of lifting objects at a distance from the motor and anchor point of the crane. This is done using a boom arm, the length and angle from the ground of which can be varied in accordance with the construction (or de-construction) job at hand.
You may need a lifting calculation formula to determine the lifting capacity of a given crane set-up. This involves mostly basic geometry, but a little understanding of the underlying physics helps too.
Parts and Physics of a Crane
A crane is operated from atop a movable and rotating (but otherwise anchored) platform called an outrigger base, which can be several meters wide. The boom arm extends upward and outward at a given angle (say 30 degrees) for its length, and at the end of this boom arm is an apparatus that lifts the load to be hoisted and moved.
The load (mass times gravity g, or 9.8 m/s2) is (ideally) lifted vertically, so no horizontal forces are in play (windy days play havoc for crane operators). Instead, a tension T (force per unit length) is maintained in the cable when the upward force of the crane (redirected by a pulley at the top of the apparatus) exactly balances the weight of the load. When the motor drives T above this point, the load moves upward, provided the cable is strong enough to withstand the force.
Geometry of a Crane
Viewed from one side, the crane boom, the ground and the vertical cable form a right triangle. The hypotenuse is the boom arm, the long arm of the triangle is the distance r from the outrigger base to the load and the short arm of the hypotenuse is the vertical height h of the boom "tip" above the ground.
The effective radius r has to account for the outrigger base and is thus slightly shortened for calculating lifting capacity; that is, it does not start directly at the motor, where the tip of this de facto right triangle lies.
A Crane in Equilibrium
A plane in equilibrium has no moving parts. This means the sum of the external forces and external torques is zero. Since the load tends to rotate the boom arm downward around its axis at the outrigger base, this torque must be balanced along with balancing the direct downward force exerted by gravity.
- As noted, the sum of the horizontal forces should be zero.
Crane Lifting Capacity Calculation
The standard crane capacity calculation formula is given by
where r is the radius (distance along the ground to the load) and hC is lifting height times capacity. Capacity, in turn, is particular to each boom arm length and angle chosen, and must be looked up in a table such as the one in the Resources.
The final calculation is actually an average, taken by using the value of hC that is maximum for every radius chosen. The points averaged are the minimum radius, r itself, and every exact radius at units of 5.0 meters in between. Thus a complete set of values might look like 1.9, 5.0, 10.0 and 14.2 m, and the average in this case would be the average of four numbers.
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