Loads falling from the crane’s hook was the most common cause of death and injury in the Californian crane industry between 1997 and 1999, an analysis of crane accidents in a report has shown. The major mode of accidents with mobile cranes, tower cranes and some other types of freestanding cranes was instability the report said. To prevent these accidents in operation manufacturers calculate and test cranes to ensure their stability under such circumstances. All known approaches and codes for calculating stability of cranes resulting from loads falling from the hook are imperfect as they do not have solid theoretical validation.

I propose a new approach for the analytical estimation of a crane’s stability under such circumstances. This approach is entirely based on the analysis of energies, which take part in the process of the crane overturning. This method also allows estimation of the maximum tilt angle of any freestanding crane. Testing an actual crane for stability in a falling load condition is fraught with hazards. To create a safe test method the theory of crane stability under these circumstances should be known. The following is a proposed safe method of crane stability testing under such conditions.

Because cranes lift large objects and move them through the air, their use always entails risks. One of the most fundamental risks that needs to be addressed is the sudden release of a load from the hook. According to the report quoted above there were 158 accidents involving a crane in California between 1 January 1997 and 31 December 1999. Of these, 43 were caused by the load falling and 35 by the crane tipping over. This shows that existing codes for calculation do not provide adequate reliability of cranes against a loss of stability resulting from a sudden release of the load from the hook.

International Standard ISO 4304, specifying conditions to be met when verifying stability by calculation of all crane types except mobile cranes and floating cranes, requires that at sudden release of load from the hook a load equal to 0.2 P’, directed upwards and applied to the crane (to the jib head), shall be taken into account. Here P’ is the rated capacity for the crane as specified by the manufacturer, excluding lifting attachments that are a permanent part of the crane in its operating condition. A similar requirement is given by the German national standard DIN 15019 part 2 for mobile cranes, but here the total lifting load as specified by the manufacturer should be taken into account. For all other crane types DIN 15019 part 1 requires 0.3 P’. International Standard ISO 4305, which specifies the conditions to be taken into account when verifying the stability of mobile cranes, gives no consideration to the sudden release of the load. (Instead of calculation there are some other experimental requirements in ISO 4305 to ensure mobile crane backward stability.)

It should be noted that the requirements given in the above Standards are not proved theoretically. In the solution proposed here the overturning problem is studied by means of a strict but simple analysis of energies that participate in the process of a crane overturning. It results in a procedure for accurate stability assessment.

For different reasons the tilt angle should be restricted to certain values. In considering rail-mounted cranes, for example, it would be desirable to restrict the height, which would not permit any wheel to leave a rail. Therefore, the theory should allow for accurate calculation of the tilt angle under a falling load condition.

Basic relationships

A scheme in Figure 1 (figures and formulae’s currently not available) illustrates the process of a crane overturning as a result of the load falling off the hook. This scheme can be modelled by a nonlinear dynamic system with one degree of freedom. The model assumes the crane to be installed on a firm surface with the allowable slope i. The steady state wind is acting backwards (in an unfavorable direction) and wind gusting is not considered.

The traditional description of the model behavior by differential equations is easy to do but finding analytical solutions to these equations, which could be used for assessment of crane stability, is not a simple task. Another approach considered below is based entirely on analysis of the energies involved in the process of a crane overturning. This approach promotes a better understanding of the overturning process.

The crane overturns after sudden unloading at the expense of work produced by potential energy PQ discharged from elastic elements of the crane (structure, ropes, etc.). This energy is stored in elastic elements in the course of the load Q lifting.

There are two more parts of potential energies which are to be taken into consideration: the potential energy of the stable equilibrium of the crane P0 and the potential energy of unstable equilibrium PU.E.. The energy P0 is determined by the position of the crane where it is at rest or it should appear, when movement is completed. The energy PU.E. establishes the allowable limit up to which the crane can safely move after the load has fallen off the hook. It is also necessary to take into account work produced by the wind shown in Figure 1, and the horizontal component G·i of the vector G at the crane moving.

Hence the stability condition may be expressed algebraically in the following way:

The value of P0 and PU.E. can be found from the following formula (this is potential energy which takes into account only so-called energy of position).

where: ö is an angle shown in Figure 1: ö=0 when the crane is at a position of stable equilibrium; ö=öu.e, when the crane is at a position of unstable equilibrium.

The value of work A can be found from the next formula

Formulae (2) and (3) contain coordinates x,y of the crane’s gravity centre (the point CG in Figure 1) in the coordinate system xOy.

The angle of unstable equilibrium öu.e can be found from an equilibrium equation reflecting all static forces acting about the tipping fulcrum O of Figure 1.

where S=G·y-static moment of the crane weight;

is the sum of the moments from all static loads acting on the crane in a position of rest.

Hence all summands in equation (1) are determined and it is possible to consider the problem to be solved. However, the derived solution is not convenient for practical application, as designers for assessment of crane stability usually compare stabilising and tipping moments from static and dynamic loads acting on the crane.

To obtain the corresponding expression for the crane stability assessment it is necessary to substitute öu.e in inequality (1) and solve it regarding M.

Having done so, putting for the sake of simplicity and sin as ö is a small angle, after rearrangements one can obtain from the expression (1) the following inequality:

which can be directly solved regarding M.

The practical sense has only the positive value of M. Therefore, the final relationships for the stability assessment may be written as follows:

where is the designed stabilising moment;

is the designed tipping moment;

may be called a static equivalent of the tipping moment from dynamic forces.

PQ can be found from well known formula where Q is the weight of the lifted load and z is the elastic vertical displacement of the hook under the action of Q which can be calculated using general methods of structural mechanics.

If for different reasons the angle ö is to be restricted it is necessary to prove additionally that where ömax is the maximum possible tilt angle, which is determined by the crane parameters and value of -DQ ;

öà is an allowable angle which is determined from the next formula where: h is an allowable height of a wheel lift from a rail (e.g. a height of a wheel flange); b is a distance between wheels.

The value of ömax can be determined from the next equation where PP is potential energy when ö=ömax.

Substituting ömax in equation (9) we get the quadratic

The solution of equation (10) having a practical sense is

This solution exists as the stability condition assumes to be fulfilled.

Analysis of the basic relationships

The algebraic relationship (6) is equivalent to a condition of stability (1).

Md is a notional moment as opposed to a physical value; it is a static equivalent moment that reflects the effects of varying in time moments of dynamic forces arising when potential energy PQ is being discharged from elastic elements of a crane. If PQ is not large enough to move the crane beyond an allowable limit the crane turns out to be stable, otherwise the crane loses backward stability. It is important to underline that Md depends not only on PQ but also on S. When PQ turns out to be dangerously large, Md can be reduced at the expense of reduction S. In turn lowering the crane’s centre of gravity can reduce S.

The required value of ömax can be reached by changing coordinates of crane parts.

Safe method of crane stability testing by falling load off the hook

The above theoretical evaluations are marginal. Therefore, for adequate experimental proof of crane stability with a load falling off the hook it is enough to determine experimentally only the value of PQ . The farther proof of stability is carried out by calculation using relevant formulae.

The determination of PQ is reduced to a measurement of z. An example of a method for measurement of z is illustrated in Figure 2 (figures currently not available).

In accordance with this method one should load the crane with the help of a rope attached to the hook through a dynamometer and measure a value of the hook displacement z.

Conclusions

When a load falls off the hook a dynamic process is developing. The movement of a crane within its limits from the initial tilt angle to an allowable tilt angle represents this process. If this process exceeds an allowable limit the crane loses stability.

A sudden unloaded freestanding jib crane disturbed by the potential energy discharged from elastic elements can be modelled by a nonlinear system.

The dynamic process can be described in terms of the energy required to tilt the crane and the work produced at accomplishing that tilt. The stability assessment can be executed as usual using a comparison of stabilising and tipping moments arising from static and dynamic forces acting on a crane in accordance with the relationships (6).

The static equivalent of the dynamic overturning moment is dependent on the energy stored in elastic elements of the crane when it lifts a load. Lowering the crane’s centre of gravity reduces the effect of dynamics caused by discharged potential energy.

For rail mounted jib cranes the allowable angle should be restricted to avoid crane derailment. The maximum value of tilt angle can be assessed in accordance with formula (11).

The theory proposed in this paper gives marginal estimations of stability conditions and of maximum tilt angle. The theory shows that the potential energy stored in elastic elements of the crane is the unique rated component, which defines the accuracy of all calculations. Therefore, the experimental proof of stability and of maximum tilt angle may be limited to measurement of the value of this potential energy, i.e. elastic displacement of the crane hook under the action of the load weight.