Introduction

Most structures and equipment in the basic industry are designed to operate throughout their useful life within a limit of small elastic deformations. On these occasions, deformed structures return to their original state once the load is removed. This type of deformation occurs in most buildings, tanks, transmission towers, bridges, conveyor belts, silos and other types of machinery and equipment. For this reason, most of the numerical simulations carried out for the design or analysis of these structures, is a so-called linear simulation.

Furthermore, in most of these structures, the loads are considered to be gradually applied and remain constant throughout the analysis time. With these considerations, the analysis does not involve dynamic effects, such as accelerations of the structure or impulses of loads. Although small dynamic effects exist in the actual loads acting on the structure, these are commonly represented by equivalent static loads.

For this reason, the most common type of structural analysis performed is static analysis. Mathematically, this means that only the stiffness matrix is resolved in the system, i.e. every force only generates deformation, not acceleration of the structure. This type of analysis is so common that some authors even refer to structural analysis simply as static analysis.

There are many applications, however, in which the assumptions of static and/or linear analysis are not applicable. In this sense, assessments involving impacts, for example, whether operational or accidental, are far removed from these principles. These studies involve loads that vary rapidly (usually in fractions of a second), generating significant accelerations in the structure (Figure 1). These accelerations generally cause permanent deformations, which no longer correspond to the elastic regime.

 

Figure 1: Non-linear analysis of kick structural failure. SOURCE: Kot Collection.

To assess this type of situation, a more complex and computationally demanding type of analysis is required: non-linear dynamic analysis. This type of analysis generally involves a so-called explicit numerical integration in which, at each instant of time, the acting forces and their effects on the deformation, velocity and acceleration of the structure are calculated.

Kot has a non-linear dynamic analysis team which, for more than a decade, has been helping its clients to obtain reliable solutions in cases involving non-standard structures and loads, where non-linear dynamics is more suitable, or even the only suitable form of structural assessment. Below are some examples of the application of non-linear dynamic analysis.

Operational impacts

In ore crushingfor example, the grates are subjected to the discharge of granular material (Figure 2). In this case, the grate structure has to withstand not only the flow of fine material, but also the impact of larger grain sizes, often weighing tons. When discharged, this material can reach speeds of up to 40 km/h just before colliding with the structure.

 

Figure 2: Unloading material into a feeder. SOURCE: Kot collection.

The value of the stress acting on the structure in these impacts depends on variables such as the location of the impact and the size of the region affected, the time of impact, the acceleration and the level of permanent deformation of the structure. Defining these variables analytically or by means of linear static analysis can often lead to the structure being undersized. As a result, unexpected deformations, cracks and fractures can occur during the useful life of the component, or the need for operational limitations in order to adapt the operation to the resistance of the existing structure.

 

Figure 3: Non-linear analysis of an ore crushing grid. SOURCE: Kot Collection.

Kot has already analyzed several structures of this type, in which non-linear dynamic simulation was used to accurately determine the forces acting on each component of the grid, for each operational requirement. This type of simulation allows for the specific sizing of the grid for each application, instead of the common, and often poor, design.-successful approach of reusing projects that have already been applied in other scenarios.

In some cases, in addition to the structural precaution of impact, there is also concern about the movement of the structure. An example of this was the non-linear analysis of the mobile cover structure of a casting pan (Figure 4). In this situation, there was the possibility of impacts that could not only damage the equipment, but also cause some movement that would dislodge the cover from its support on the ladle, making it unstable and potentially causing it to topple over during operation or lifting.

 

Figure 4: Non-linear impact analysis on a mobile casting pan cover. SOURCE: Kot Collection.

Protective structures

There are structures in which a possible impact is not part of the equipment's normal operating process, but occurs as a result of structural failures or accidents. In such cases, it is necessary to ensure that the impacted structures are able to withstand the stress and contain any fragments. In this way, it is possible to prevent the projected fragments from hitting other components or even people. In these situations, the stresses involved are usually quite significant and the protective structures do not need to survive multiple events, but are replaced after each impact.

In this way, sizing by linear analysis is either unfeasible or excessively conservative, since it does not take into account the portion of energy absorbed in the plastic regime. On the other hand, with non-linear dynamic analysis, the forces involved and the structural response of the protection can be determined precisely, allowing safe and specific sizing for each situation. Kot has experience in developing protections to contain mechanical component failures. Some of the work carried out by the company consists of protections for hydrodynamic coupling explosions, pulley ruptures and protection of vehicle occupants, such as checking the resistance of windows and windshields to impacts.

 

Figure 5: Non-linear analysis of pulley protection. SOURCE: Kot Collection.

 

Figure 6: Non-linear analysis of window impact resistance. SOURCE: Kot Collection.

Due to the possibility of multiphysics being taken into account in the explicit simulation, fluid impacts on structures can also be simulated. In recent activities, for example, it was necessary to develop reinforcements in a receiving chute of an ore feeder. This study was designed to prevent collapse or excessive leakage in the event of internal avalanches in the silo during the feeding process, in which a large mass of material descends uncontrollably, causing serious risks to the operation in the event of accidents.

 

Figure 7: Non-linear analysis of silo unloading. SOURCE: Kot Collection.

Failure analysis and accident reconstruction

In the case of accidents or structural failures, one of the main needs is to thoroughly understand the causes and failure mechanisms involved, in order to prevent similar situations from occurring again. To this end, there is often a need to reconstruct the failure modes that occurred, both to understand the sequence of events and to validate failure hypotheses and determine the loads involved for sizing solutions. In these cases, non-linearity is a fundamental part of the analysis, since in many cases, structures, especially metallic ones, exhibit extensive non-linear behavior immediately prior to failure.

Kot has already been involved in several failure analyses and accident investigations in which linear dynamic simulation has been fundamental for reconstructing what happened, validating hypotheses and defining preventive actions to prevent further occurrences. Even for assessing the severity of possible damage to humans, non-linear dynamic analysis plays a fundamental role in determining the speeds, accelerations and forces involved, which, when associated with biomechanical analysis, allow for a classification of the severity of the impact and its possible consequences.

 

Figure 8: Non-linear failure analysis of a reclaimer: SOURCE: Kot Collection.

Conclusion

In short, it can be said that non-linear dynamic analysis is an advanced computer simulation approach used in engineering to study the behavior of complex systems under the influence of forces and loads that vary over time. In this type of analysis, large deformations, plasticity effects and other non-linear behaviors of structures are taken into account in order to obtain accurate results in situations that static and linear analyses are unable to adequately solve. Non-linear dynamic analysis is particularly useful when dealing with impact/collision problems, structural failures and accidents, fluid-structure interactions, structural checks of mechanical components and any other physical systems in which the responses cannot be adequately represented by linear relationships.

Kot specializes in non-linear dynamic analysis and can help you with reliable, customized solutions, promoting the safety and performance of equipment and structures. Do you need to carry out non-linear analysis on your company's assets? Contact our experts for more information!

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