Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner


  • Jacobo DíazEmail author
  • Miguel Costas
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_223-1



Crashworthiness is the ability of a structure to protect its contents (occupants or cargo) during an impact. The term usually refers to the capacity of a structural system to dissipate kinetic impact energy by itself, by means of a controlled and predictable deformation. Thus, the kinetic energy is transformed into inelastic strain energy, heat, and fracture energy.

Crashworthiness performance is one of the main drivers in automotive design nowadays, and its importance is considerable also in aircraft and railway transportation. The safety of a vehicle is affected by multiple factors, components, and systems. Among them, those involved in preventing and avoiding accidents, or reducing their severity, are encompassed under the category of active safety. On the other hand, passive safety refers to vehicle elements that mitigate the consequences of an accident, such as occupant retention systems, airbags, or some structural elements of the vehicle, like crumple zones or safety cells (Seiffert and Wech, 2003). Therefore, crashworthiness is one of the factors involved in the passive safety of a vehicle. Occupant protection systems like safety belts, airbags, or energy-absorbing trim materials help to reduce the damage and injuries on the passengers. However, they do not pertain to vehicle structure and, hence, do not influence vehicle crashworthiness, but vehicle safety.

Crashworthiness applies to impacts with a duration in the order of milliseconds, with velocities usually below 100 m/s in automobile collisions, considering the addition of horizontal components in a frontal impact of two vehicles. For impacts with the ground or ditching in airplane and helicopter crashes, typical vertical impact velocities are in the range of 5 to 30 m/s; and, in the case of train impacts, they are normally under m/s for high-speed railway vehicles. Therefore, crashworthiness deals with the protection of the occupants at velocities well below typical ballistic impact velocities. This is a different situation to the protection against explosions, or high-velocity impacts, as those events are not related to crashworthiness, and therefore a vehicle equipped to be able to withstand them is not crashworthy, but armored.

Crashworthiness Assessment

The goal of a crashworthy structure is to absorb as much impact energy as possible and, simultaneously, to prevent severe occupant injuries. In that regard, crashworthiness refers to a qualitative structural property, in the same vein as stiffness or strength. However, in order to be considered in a design procedure, quantitative magnitudes have to be associated with the measurement of crashworthiness, in the same way as, for instance, yield stress is related to strength or Young’s modulus is associated to stiffness.

There are several quantitative metrics related to the measurement of energy absorption of a structure, and there are also different biomechanical metrics and injury criteria to evaluate the harm caused to the occupants of a vehicle. Both types of metrics can either be evaluated by means of experimental tests or numerical simulations.

Energy-based crashworthiness metrics are useful in component design, where crush force and displacement can be easily measured through quasi-static and dynamic experimental tests or numerical simulations of isolated components, and energy absorption-related metrics can be readily obtained. However, the evaluation of crashworthiness in larger assemblies or full vehicle structures is not feasible using the aforementioned procedure. In these situations, biomechanical criteria obtained from forces and accelerations measured in critical positions during a crash test are more practical, as they allow to predict and quantify the effect of the impact on the occupants and hence characterize and validate structural crashworthiness.

In what follows, a review of the most relevant crash test methodologies and regulations for crashworthiness certification of different types of vehicles will be presented. Moreover, the appropriate metrics for measuring crash energy absorption and to evaluate the damage to the occupants through biomechanical injury criteria will be explained.

Crash Tests and Regulations

Crashworthiness started to be considered as a requirement for automobile design in the beginning of the second half of the twentieth century, as modern society demanded safer modes of transportation and governments established legislation to guarantee a minimum level of safety in motor vehicles. This required the development of new techniques and methodologies to evaluate, characterize, predict, and improve the crashworthiness of vehicle structures and, hence, the safety of occupants.

Due to the complexity that the analytic or numerical characterization of the structural response of a vehicle in a collision entails, full-scale experimental crash tests were – and still are – regarded as the best way to measure the structural response in a collision, specially in a time when computational simulation was not as developed as nowadays. Vehicle collision surveys show that the most common types of impacts are frontal ones, followed by side, rear, and roll-over crashes (Aarts et al., 2016). Therefore, crash tests try to replicate these scenarios by crashing a vehicle against a barrier at a fixed impact velocity, angle, and overlap. The barrier can be either rigid or deformable, and the vehicle can either impact the barrier, in the case of a frontal crash test, or be impacted by a moving barrier in side or rear impact tests.

To evaluate the damage on the occupants, instrumented full-scale anthropomorphic crash test dummies (ATD) are used to obtain data on the precise body locations which are more prone to suffer a severe damage. Crash test dummies are surrogate models that imitate the size, proportions, stiffness, weight, and articulations of a human body. They can register the deceleration levels and forces that a human body would experiment using accelerometers and load cells. However, as they are built using materials whose mechanical properties are very different from those of biological tissues, they are not capable of registering injuries. Thus, biomechanical injury criteria have to be derived based on the measured accelerations and forces.

ATD are classified according to the crash event that they can measure and also according to their height percentile by age and sex. For instance, the standard dummy for measuring frontal impacts is the Hybrid III (Foster et al., 1977), which is available in several height percentiles, the 50th percentile being the first one to be developed (and the one most commonly used), which represents an average adult male. There are other dummies for side and rear impacts (Moss et al., 2000; Boström et al., 2000) and also other less common crash scenarios.

The configuration of a crash test and the tolerable limits for injury criteria are prescribed in regulations and standards published by government agencies. Currently, there are no global standards for vehicle safety certification, and although regulations follow similar procedures in all territories, they are not strictly equivalent, as some test parameters vary depending on the legislation. The World Forum for Harmonization of Vehicle Regulations, which depends on the United Nations Economic Commission for Europe (UNECE), is responsible for UN Vehicle Regulations No. 94 (2017) and UN Vehicle Regulations No. 95 (2014), dealing with the requirements to be satisfied in the event of a frontal or lateral collision, respectively. These regulations were initially binding for most European UNECE member countries and nowadays apply to all EU members. Since 1995, they have also been signed by other third countries including Japan, Australia, South Africa, New Zealand, and South Korea.

In the United States, the National Highway Traffic Safety Administration (NHTSA) set up the Federal Motor Vehicle Safety Standards (FMVSS) in 1967. These standards are organized in five series, of which the 200-series standards are assigned to crashworthiness regulations. The standards regulating crash tests for frontal and side impact are the FMVSS208 (49 CFR 571.208, 2011) and the FMVSS214 (49 CFR 571.214, 2011), respectively.

In addition to government standards, public and private independent organizations have emerged in order to develop programs to increase consumer awareness of vehicle safety under different crash scenarios. These organizations publish reports with the crashworthiness performance of vehicles, rating them in different crash tests including frontal, side, and pedestrian impact. These tests are based on those used in vehicle certification but have more strict requirements. The publication of these reports, whose fulfillment is not mandatory for vehicles to be sold on the market, has increased the interest of consumers in vehicle safety, which, in turn, increases the motivation of automobile manufacturers to improve crashworthiness.

The first initiative was the New Car Assessment Program (NCAP), created in 1979 by the NHTSA. The Insurance Institute for Highway Safety (IIHS), a nonprofit organization in the United States, started its crash test program in 1995, and the European New Car Assessment Program (Euro-NCAP) was created in 1997. There are equivalent programs in other territories, although the thresholds and requirements vary from one program to another. Table 1 shows an overview of the front and side impact tests prescribed in the main international regulations and assessment programs regarding crash protection.
Table 1

Configuration parameters for some automobile front and side crash impact tests



Velocity (km/h)


Overlap \(\left (\%\right )\)


UN R94


Fixed deformable




48 – 56

Fixed rigid





Fixed deformable





Fixed deformable



UN R95


Moving deformable





Moving deformable





Fixed rigid pole





Moving deformable





Fixed rigid pole


With respect to other modes of transportation, regulations are not as developed and detailed as they are for automobile vehicles. For light airplanes, the first initiatives to assess and enhance their crashworthiness started in the 1950s (De Haven, 1953). Later on, in 1967 the US Army published the Aircraft Crash Survival Design Guide in five volumes, of which volume I (Zimmerman and Merritt, 1989) is a compilation and summary of the rest. Crashworthy airframe structures must protect passengers from excessive accelerations and guarantee the preservation of survivable space inside the cabin area, avoiding the intrusion of large mass components and maintaining emergency egress paths clear. The protection from excessive acceleration injuries is primarily achieved through the subfloor structure and with the implementation of seating systems capable of absorbing the crash energy.

The most relevant full-scale experimental test to certificate crashworthiness in aircraft structures consists of drop testing entire vehicle bodies, in the case of helicopters or small airplanes, or fuselage segments in larger airframes. However, due to the high cost and inherent limitations of these tests, regulations and certification procedures are evolving to allow a combination of component tests and numerical simulations. For large airplanes, the European Aviation Safety Agency (EASA) publishes the certification specifications EASA CS-25 (2018). The equivalent regulation in the United States is the 14 CFR 25 (2011), published by the Federal Aviation Administration (FAA).

For railway vehicles, research and testing has been carried out in recent years in order to evaluate the effect of frontal crash of train vs. train or train against buffer stops or heavy obstacles (Scholes and Lewis, 1993; Sutton, 2002), and, although there is not a corpus of legislation as the one developed for road vehicles, there are some regulations that contain crashworthiness requirements for railway vehicle bodies. In the EU, all new rail vehicles must fulfill the EN 15227:2008+A1:2010 (2010) since 2012, which was first published in 2008. This regulation defines survival spaces for both passengers and crew, which must keep their integrity during the full crushing of the collapse zones at the extremes of the vehicles. They also enforce limits on deceleration in those survival spaces in the most common collision scenarios, considering a speed of 36 km/h for collision of two identical train units and 110 km/h for collision with a heavy obstacle of 9.56: Space between digits. In the International System of Units (SI units), thin, fixed spaces rather than commas are used to mark off groups of three digits, both to the left and to the right of the decimal point. One key aspect of this regulation is that it acknowledges the impracticality of evaluating crash behavior of complete train configurations by experimental testing and authorizes the use of dynamic finite element simulations of the reference collision scenarios to certificate the crashworthiness requirements, although, for areas with large deformations, it requires a full-scale test of vehicle end structures to be conducted in order to verify the numerical model.

In the United States, the Federal Railroad Administration (FRA) is responsible for the 49 CFR 238 (2011), and the American Public Transportation Association (APTA) publishes the APTA PR-CS-S-034-99, Rev 2 (2006), which is a standard for structural and crashworthiness requirements for railroad passenger equipment. These regulations require that energy-absorbing structures with controlled deformation must be used as part of a crash energy management design of vehicles. They also prescribe equivalent static loads to evaluate crashworthiness and set limits on deformation and intrusion into the occupied volume of the vehicle bodies. However, unlike the European regulation, they do not prescribe the combination of test and numerical simulation for crashworthiness certification.

Crash Energy Metrics

As long as the term crashworthiness refers to a global, qualitative, and immeasurable property of a protective structure, some metrics have been typically used to assess and quantify the performance and efficiency of impact-absorbing structural designs. These metrics are used as criteria to judge and compare different designs. The most relevant are presented next.

Fig. 1

Typical force-displacement curve produced in the axial crushing of a thin-walled metal tube. 1: Absorbed energy Ea, 2: peak force Fpeak, 3: mean crushing force Fm, 4: bottoming out at δ = δmax

Absorbed Energy

The absorbed energy represents the amount of kinetic energy that is dissipated by a structure during a crash. This dissipation is due to different transformation mechanisms and physical phenomena, and some of them depend on the materials in the structure. In metals, dissipation typically occurs through plastic folding, strain hardening, ductile fracture, friction, and heating. Composite materials generally exhibit a relatively brittle fracture compared to metals, and dissipation involves mainly inter-laminar and in-plane cracks, delamination, fragmentation, friction, or viscous deformation of the matrix.

For a structure subjected to a crushing force F, the absorbed energy Ea is the work done by the crushing force over the crush length δ and can be obtained as where δmax is the total crush length. Therefore, from the mean value theorem, the mean crush force Fm can be expressed in terms of the absorbed energy as
$$\displaystyle \begin{aligned} F_{\text{m}}=\frac{E_{\text{a}}}{\delta_{\text{max}}}. \end{aligned} $$

Figure 1 shows a typical force-displacement curve obtained in the axial crushing of a thin-walled metal tube, where the absorbed energy calculated through Eq. (1) corresponds with the area under the curve.

Specific Energy Absorption

The specific energy absorption (SEA) is the ratio of the absorbed energy Ea to the mass of a structure m. This parameter can be used to compare the efficiency of different structural designs in terms of the absorbed energy for a given mass:
$$\displaystyle \begin{aligned} \mathrm{SEA}=\frac{E_{\text{a}}}{m}. \end{aligned} $$

It is worth mentioning that a design with a higher SEA value does not necessarily imply that a smaller component can be used to absorb the same amount of energy. As an example, if two identical tubes of aluminum and steel are compared, higher SEA values will be obtained for the aluminum one due to its lower density. However, a longer aluminum tube will be required if the same amount of total energy Ea has to be absorbed.

Peak Force

The peak crushing load Fpeak is the maximum load observed during a crushing test before bottoming out occurs. The latter consists in a rapid increase in the crushing force levels due to the complete compression of the member.

The peak force is usually produced at early stages of the impact, when the collapse starts (Fig. 1). When it comes to the design of energy absorbers, this parameter is particularly interesting for two main reasons. In the first place, it is very important to avoid high force levels at the beginning of the impact, since they could lead to a plastic collapse of other parts of the structure. These parts should collapse only after the previous elements have reached their maximum energy absorption levels. As an illustrative example, in the usual configuration of the frontal part of an automotive structure, where crash boxes are fitted as energy absorbers connecting the bumpers with the front rails of the chassis, if those rails develop a plastic hinge with significant rotation before the crash boxes are crushed, the latter will experience a rigid-body rotation and the crushing will be not axial. The second reason to avoid high peak forces is that the impact force levels are proportional, to some extent, to the acceleration levels the occupants have to withstand. However, an initial high peak is not necessarily harmful, as long as the occupants are properly restrained. Seat belts are usually pretensioned around 25 milliseconds after the collision, so any pulse produced within that time is not transferred to the occupants, and, after they are completely restrained, they will experience only the remaining of the crash deceleration pulse.

Crush Force Efficiency

The crush force efficiency (CFE) measures the uniformity of the crush force, and it is defined as the ratio of the mean crushing force Fm to the peak load Fpeak:
$$\displaystyle \begin{aligned} \mathrm{CFE}=\dfrac {F_{\text{m}}} {F_{\text{peak}}}. \end{aligned} $$

The mean crushing force is the mean value of the force from the beginning of the test until the bottoming out of the component happens (Fig. 1). When it comes to the protection of occupants, the crush force efficiency is specially important because strong variations in the crushing force can be transmitted through the vehicle structure to the passenger compartment and exceed the acceleration tolerances of the occupants. In an ideal energy absorber, the force-displacement curve would follow a straight line with a constant value for the crushing force, and therefore crush force efficiency would be 100%.

With the aim of increasing the CFE of energy absorbers, indentations or triggers are commonly introduced. These consist of induced imperfections in the undeformed configuration of the box, which concentrate the stresses and deformations as the load starts to actuate, thus reducing the initial peak force and increasing the CFE.

Sometimes the uniformity of the crush force is also referred to as the load ratio (LR) or as the load uniformity index (LU), defined as the inverse of the crush force efficiency:
$$\displaystyle \begin{aligned} \mathrm{LR}=\dfrac {1} {\mathrm{CFE}}. \end{aligned} $$

Stroke Efficiency

When a structure with an original length L is axially crushed, folds are developed up to a crushing distance δmax in which there is no more surface available for folding (Fig. 1). At this point, the densification and bottoming out makes the force levels increase rapidly, as the compressed absorber is now elastically loaded in compression. The stroke efficiency (SE) is defined as the ratio of the maximum displacement to the initial length:
$$\displaystyle \begin{aligned} \mathrm{SE}=\dfrac {\delta_{\text{max}}} {L}. \end{aligned} $$

Therefore, the stroke efficiency measures what length of the structural member is actually used up in energy absorption. This becomes a specially useful criterion when there are spatial restrictions, as is usually the case in the front parts of road vehicles. In an ideal energy absorber, stroke efficiency should be 100%, that is, the entire length of the structure is used to absorb crash energy. However, practical values of SE are in the range between 0.4 and 0.8 (Jones, 2011).

Energy-Absorbing Effectiveness Factor

This relatively recent indicator was proposed by Jones (2010) in order to allow comparisons on the effectiveness of the design of energy absorption structures made from different materials. The energy-absorbing effectiveness factor ψ of a system is the ratio of the total energy absorbed by the system to the maximum energy up to failure of a normal tensile specimen made from the same volume of material (or compressive, in the particular case of crushable foams or other cellular materials): where V  is the volume of the energy absorber; σ and ε are the stress and engineering strain, respectively, in a uniaxial tensile test; and εr is the engineering rupture strain of the material the absorber is made of. Therefore, this criterion allows to estimate how efficiently the material in a protective structure is being used in energy absorption.

Biomechanical Injury Criteria

Occupant injuries are classified according to its severity by means of standardized scales. The Abbreviated Injury Scale (AIS) (Gennarelli and Wodzin, 2008), first published in 1971, classifies the injury severity in a range from 0 to 6; 0 meaning no injury and 6 being the most severe injury level, which is virtually unsurvivable.

Accelerations are among the most important factors to consider when it comes to occupant safety. They are related to the likelihood of severe or fatal injuries arising from an impact. The injury level will depend on the deceleration experimented by the occupants, which translates directly into an impact force when the displacement of the body is restricted, either by means of vehicle retention systems or when some part of the body collides with a rigid part of the habitacle. Another important variable affecting injury severity is the amount of time during which the body experiments high deceleration levels, as larger accelerations can be tolerated for short spans. Moreover, there are different biomechanical limits on acceleration and forces for each part of the body, as the regions that suffer the most severe injuries in an accident are, in this order, head, neck, chest, hands, and legs.

Biomechanical injury criteria are semiempirical formulae which include the effect of deceleration and the duration of the deceleration pulse to calculate the damage caused by an accident on the human body. They have been developed based on experimental research carried out with volunteers, cadavers, and animals subjected to high levels of acceleration and impact. These experiments are also used to define the threshold levels for severe injury and to correlate the values of deceleration and the damage on the human body.

Crashworthiness regulations prescribe the injury criteria and the threshold levels that have to be considered in order to certify a vehicle for safe operation. Injury metrics are evaluated from properly filtered force, torque, displacement, and acceleration time records, obtained from a crash test, either experimental or numerically simulated. Some criteria are directly formulated as fixed or cumulative limits over time for these magnitudes in critical body regions, like head, chest, or legs. Examples of this are the femur force criterion (FFC) and the tibia compressive force criterion (TCFC) – both of which limit compressive forces – or the thorax compression criterion (THCC) and the chest deflection criterion (CDC), which limit deformation. In addition to these direct criteria, there are others that are formulated based on calibration from experimental data, the most relevant of which are presented next.

Head Injury Criterion

The head injury criterion (HIC) (Versace, 1971), also known as head performance criterion (HPC), estimates the likelihood of skull fracture or brain injury when there is a head impact, relating the acceleration pulse \(a\left (t\right )\) at the gravity center of the head, in g units, and its duration: where t1 and t2 are the limits of any time interval within the acceleration pulse of duration less than or equal to Δt. HICmax is the maximum value allowed which does not cause injury. For a 50th percentile male, 49 CFR 571.208 (2011) and UN Vehicle Regulations No. 94 (2017) prescribe a time interval Δt of 36 ms and a critical value HICmax of 1000.

Viscous Criterion

The viscous criterion (VC) (Lau and Viano, 1986) evaluates the damage on the soft tissues of the chest. It is calculated as the maximum value of the product of the velocity of deformation of the chest \(V\left (t\right )\) by the normalized chest deformation \(C\left (t\right )\):
$$\displaystyle \begin{aligned} \mathrm{VC}&=\max \left\{k V\left(t\right)C\left(t\right)\right\}\\ &=\max \left\{k \dfrac{\mathrm{d} D\left(t\right)}{\mathrm{d} t} \dfrac{D\left(t\right)}{L_{\mathrm{D}}} \right\} \leq \mathrm{VC}_{\text{max}}, \end{aligned} $$
where \(D\left (t\right )\) is the measured chest deformation over time, LD is the initial chest depth along the direction of the impact, and k is a scale factor to account for the position of the deflection transducer. For the 50th percentile ATD male in a frontal impact, the values of LD and k are 229 mm and 1.3, respectively, and 140 mm and 1.0 in a side impact. In both cases, VCmax is set to 1.0 m/s (UN Vehicle Regulations No. 94, 2017; UN Vehicle Regulations No. 95, 2014), which according to Lau and Viano (1986) corresponds to a 25% probability of severe thoracic injury, with AIS ≥ 4.

Neck Injury Criteria

The neck injury criteria Nij (Klinich et al., 1996) evaluate the combined effect of the axial force Fz and the bending moment My, measured with a load cell at the ATD occipital condyles:
$$\displaystyle \begin{aligned} \mathrm{N}_{\text{ij}}=\dfrac{F_{\text{z}}}{F_{\text{C}}} + \dfrac{M_{\text{y}}}{M_{\text{C}}} \leq 1.0. \end{aligned} $$

Considering the combination of tension and compression values of the axial force Fz and the flexion and extension values of the bending moment, there are four neck injury criteria: NTF, NTE, NCF, and NCE. According to 49 CFR 571.208 (2011), the critical neck force FC is 6806 N in tension and 6160 N in compression. The critical neck bending moment MC is 310 Nm in flexion and 135 Nm in extension. UN Vehicle Regulations No. 94 (2017) do not follow the previous criteria. Instead, they prescribe a direct neck injury criterion (NIC) of cumulative forces in the neck base.

Tibia Index

The tibia index (Mertz, 1993) helps to predict leg injuries through the axial compression Fz and bending moments Mx and My, measured with load cells in the proximal and distal ends of an ATD tibia:
$$\displaystyle \begin{aligned} \mathrm{TI}=\dfrac{F_{\text{z}}}{F_{\text{C}}} + \dfrac{\sqrt{M^2_{\text{x}}+M^2_{\text{y}}}}{M_{\text{C}}} \leq \mathrm{TI}_{\text{max}}, \end{aligned} $$
where FC and MC are the limit values for the axial force and bending moment, respectively. According to Mertz (1984), these values for the 50th percentile male are 35.9 kN and 225 Nm, obtained from quasi-static tests by Yamada (1970). TImax is the threshold above which there is risk of tibia injury. This value is set to 1.3 in the UN Vehicle Regulations No. 94 (2017).

Crashworthiness of Structural Systems

It is desirable for a crashworthy structure to absorb the maximum impact energy possible without significant injuries to vehicle occupants, avoiding at the same time the contact of the occupants with the structure, i.e., limiting the penetration of structural elements into the occupant’s cell.

In order to satisfy the metrics imposed by safety legislation and to improve vehicle performance in assessment programs, vehicle structures have evolved from simple designs, with body and chassis as independent elements, to monocoque structures, which include passenger safety cells, crumple zones, and energy absorption devices.

Safety cells must prevent intrusion into the occupant compartment in frontal, rear, and side impacts and also provide protection for roll-over crashes. Crumple zones are designed to absorb as much of the kinetic energy of a vehicle during an impact by controlled crushing or plastic deformation as possible, so that the remaining energy can be safely transmitted to the occupants and the restraint systems. These progressive crush zones are usually located in the front and rear parts of vehicles, where collisions are more frequent.

Many different engineering solutions have been proposed and implemented throughout the last decades in order to improve the crash performance of vehicles, transportation systems, or structures in general. These solutions range from ad hoc mechanical systems, like hydraulic or pneumatic railway buffers, to structural members designed to withstand structural loads and also to act as shock-absorbing systems in the event of a crash, like the front rails of a car. Although the term crashworthiness comprehends all of the previous, it is more commonly applied to the latter.

The way a structure or component is designed to absorb impact energy is strongly linked to the material’s behavior. The common goal is to achieve a progressive and controlled deformation of the structure so that a major part of the material undergoes significant deformations. The kinetic energy of the impact is then transformed into strain energy, which in turn is usually transformed into heat and eventually dissipated. The design of a structure will determine the amount of strain energy generated during an impact. Thus, an understanding of the different collapse modes is necessary for the designer to make a smart choice.

Behavior of Metallic Impact-Absorbing Members

Energy-absorbing components have been traditionally built in different metals or alloys. The common characteristic of these materials is their ductility: metals are, in general, able to experience large plastic deformations before fracture. Therefore, metal structures intended to meet crashworthiness criteria have to be designed so that a large fraction of the material undergoes large deformations. The typical example of such structures are those known as crash boxes, which consist of a hollow profile intended to progressively crumple, crush, or collapse along its longitudinal axis during an impact. These components can be commonly found behind car bumpers and train buffers, but also in aircraft, at the bottom of elevator shafts, etc. This progressive crushing is achieved by means of kinematically admissible collapse modes of the tube’s walls. For instance, a circular tube subjected to axial crushing will develop either a concertina mode, with ringlike lobes forming during crushing; a diamond mode. with straight edges instead of ring lobes; or a combination of them. Square tubes, on the other hand, are prone to develop alternate straight lobes in their walls, inward and outward (Jones, 2011). More complex cross sections develop different collapse modes, but it is always possible to explain them through a kinematically admissible theoretical model (Lu and Yu, 2003; Chen, 2015). It is important to mention that, in the case of metallic tubes, the SEA is strongly dependent on the collapse mode developed during the crushing. This means that two identical metal tubes will absorb a different amount of energy if different modes are triggered. As an example, the axial crushing of two identical steel cylindrical tubes can be considered. If they are triggered to develop a concertina and diamond modes, respectively, the latter would absorb more energy than the former since larger plastic strains are developed.

As a general rule, the more corners in a profile, the higher the energy absorption. The presence of corners constrains the buckling modes of the walls, reducing the folding length and increasing the absorbed energy. Multi-chamber profiles are a good example of this, where the presence of junctions in the cross section works as a constraint, as long as the walls are thick enough and do not fail at the connections. A quite illustrative example is provided in Fig. 2, where a double-chamber profile subjected to axial crushing is shown. The crushing started with the expected asymmetric collapse mode triggered by the presence of a middle wall. After one or two folds, however, the collapse evolved into a symmetric mode. The cause for this was the failure at the connections between the middle wall and the outer region of the profile, due to insufficient thickness. Once the constraining effect of the middle wall was lost, the profile behaved as two separated parts, both folding independently of each other.

Fig. 2

Transitional collapse mode in a double-chamber profile triggered by the insufficient thickness of the middle wall. The tearing of the middle wall at its corners caused the collapse mode to evolve from an asymmetric into a symmetric pattern. Tested at SIMLab, NTNU

An additional factor that influences the formation of the collapse lobes to a great extent is the ductility of the metal or alloy employed in the construction of the part. The material should be ductile enough to enable the complete folding of the walls without major cracks. Cracks can trigger unexpected changes in the collapse modes and lead to lower levels of energy absorption and repeatability. The ductility requirements are particularly critical when designing parts built in aluminum, since the same alloy can behave in a completely different way if its ageing history changes (temper). Ageing can increase the initial yield stress (and yield stresses in general) but at the same time reduce the ductility. Extensive material and component testing is recommended for crash boxes built in any new – or not enough investigated – aluminum alloy.

The level of work hardening has also a well-reported effect in the progressive collapse of thin-walled structures. The folding length increases with an increased amount of work hardening, and plastic strains trend to be less localized. This is not only true for axial crushing of tubular structures, as it can also be observed in the formation and development of plastic hinges in a structure or component under bending loads. The ageing of an aluminum alloy is again very relevant, as different tempers of the same alloy exhibit a different amount of work hardening. As an example, Fig. 3 shows the change in the folding length in square aluminum profiles of the same alloy with different levels of work hardening. The material in Fig. 3a has a lower initial yield stress and an increased hardening compared to the material in Fig. 3b.

Fig. 3

Differences in the folding length arising from different hardening properties in the same alloy. Tested at SIMLab, NTNU

For some metal alloys, the loading rate can also greatly affect the crashworthiness of a component (see, for instance, Hooputra et al. 2004). Higher stress levels derived for an increase of the loading rate usually lead to a more rapid nucleation of the material, and, hence, the ductility is reduced. Moreover, if the work hardening of the material is similar at low and high strain rates, the increased stress levels make the material more prone to localization, which in turn reduces the ductility. Experiments on aluminum alloys have also reported a more or less inversely linear relationship between the failure strain and the stress level. All these mechanisms can produce behaviors like the ones displayed in Fig. 4, where the collapse mode changes radically under quasi-static and impact axial crushing for the same profile in AA7108-T6. The decrease in ductility due to the higher strain rate causes the profile to tear and to fail catastrophically, which reduces its crashworthiness. It is worth introducing a word of caution here about the risk of extrapolating quasi-static results to an impact scenario if the material is not sufficiently known. Impact experimental tests are always recommended to assess any effect stemming from the strain rate sensitivity of the material.

Fig. 4

Same double-chamber profile in AA7108-T6 alloy subjected to quasi-static (a) and impact axial crushing at 10 m/s (b). Tested at SIMLab, NTNU

As a particular case of metal profiles for energy absorption, top- and double-hat sections are commonly used in automotive body structural components. These sections consist of two cold-formed metal sheets assembled together by means of spot welds or rivets. They are easy to manufacture and very flexible in terms of dimensions and materials, advantages that usually allow to overcome the inconvenience of a slightly lower specific energy absorption compared to extruded profiles. The strength, location, and number of spot welds or self-piercing rivets have been proved to have a significant effect on the crashworthiness of top-hat and double-hat sections. A larger number of rivets or spots welds help to keep the integrity of the profile during deformation but also increase the structure’s weight in the case of rivets. As a rule of thumb, if the spacing between rivets or spot welds is below a 10% of the tube length, there is no additional improvement in the specific energy absorption. A larger spacing can lead to more irregular folding patterns. This rule is however quite generic, and specific studies are recommended for each design. Figure 5 shows the axial crushing of a top-hat steel section where the sheets are joined together using self-piercing rivets. In this case, the spacing and strength of the rivets was enough to maintain a progressive collapse mode of the whole component.

Fig. 5

Axial collapse of a top-hat steel profile joined with self-piercing rivets. Tested at SIMLab, NTNU

Indentations or triggers are usually introduced in components dedicated to impact energy absorption by progressive collapse. These features consist of induced imperfections in the box, which concentrate the stresses and deformations as the load starts to apply, thus making the component to collapse always in the same, predicted way. Introducing triggers has also the effect of reducing the initial peak force, which is also desirable in most cases. However, in the automotive and aerospace industries, there exist what are called repairability criteria. These imply that structures and components should be able to withstand small, low-energy impacts without permanent deformations and, thus, without the need of being repaired or replaced. Therefore, the reduction of the initial peak force with a trigger is bounded by these repairability criteria.

Crashworthiness of Composites and Structural Foams

While the mechanics behind crushing and bending are more or less similar for all metallic structures, the description of the same problem for composite materials varies greatly from one case to another and cannot be generalized.

The specific energy absorption characteristics of most composite materials are superior to those of metals. Thus, they are more suitable for structures where weight is a critical requirement, such as aircraft, aerospace, or high-performance vehicles (Elmarakbi, 2013). Nevertheless, a large number of variables determine the crushing behavior of these elements, and, therefore, it is not a straightforward matter to understand and predict their crash performance. These variables include (but are not limited to) the matrix and fiber mechanical properties, the fiber volume fraction and arrangement, the stacking sequence and orientation in laminated composites, the specimen geometry, the triggering mechanism, and the loading conditions (Abrate, 2005). The main issue here is that, if some specific conditions regarding these variables are not fulfilled, the crash box will develop an unstable or even catastrophic failure mode resulting in very low energy absorption levels.

The ductility of composite materials depends on their fiber content, but in general they show a much more brittle behavior than sheet metals. This is why a correct triggering of the components is essential to guarantee their progressive collapse, even though at a material level their behavior is brittle. It is generally accepted that tubes undergoing this brittle, progressive crushing mode exhibit a combination of two inner degradation mechanisms in the composite walls: the splaying or lamina bending and the fragmentation or transverse shearing crushing modes (Hull, 1991). The former consists of very long interlaminar, intralaminar, and parallel-to-fibers cracks with a minimum or null fracture of axial laminar bundles. The energy is mainly dissipated by the crack growth. The latter exhibits a wedge-shaped end of the laminates, with short interlaminar and longitudinal cracks. The basic energy absorption mechanism here is the fracture of the lamina bundles.

To sum up, fiber-resin composites usually have high strength-to-weight ratios compared to metals, but their crushing is typically dominated by brittle, unstable failure modes.

A relatively recent advent of crashworthiness engineering is the design of externally wrapped fiber-reinforced metal tubes. This way, the ductile metal crushing modes are combined with the high strength and brittle fracture of the fibers with no harm to the predictability or stability of the overall system. These designs have two main advantages compared to a metal-only structure:
  • Composite-metal hybrid structures are intended to reduce the weight of the structural member while keeping or improving the energy absorption and stiffness compared to metal structures.

  • Extreme environmental conditions or minor shocks can cause some problems in metal structures. The composite wrapping acts as a coating and improves the durability of the system.

The general conclusion is that both mean crushing load and specific energy absorption can be increased with this kind of external reinforcement, so that they are even higher than the sum of the two individual contributions of both materials acting separately.

The same conclusion can be applied to structural foams used in energy absorption. These materials consist of a three-dimensional cellular structure which progressively collapses when crushed. They have been used extensively in the packaging industry due to their ability to be crushed with a constant force and their low weight. The predominant types of foam are metal and polymer foams. Metal foams are made commonly by mixing organic beads into the liquid metal in an inert atmosphere. Then the metal cools down and solidifies, while the carbon burns off to leave a cellular matrix. Polymer foams, on the other hand, are usually produced by blowing air bubbles into a hot polymer or a liquid monomer.

Foam-filled metal sections are a solution with potential use in structures dedicated to impact energy absorption. The foam substantially alters the crushing behavior of the outer thin-walled column compared to the empty extrusion, improving the energy absorption levels. This alteration is due to the fact that the foam filling acts as an “elastic” side constraint of the extrusion walls, reducing the buckling length and, therefore, allowing more lobes to be developed during the complete axial crushing. Figure 6 clearly depicts this phenomenon for empty and foam-filled square aluminum extrusions. Furthermore, the presence of the foam filling may also change the deformation mode in circular extrusions.

Fig. 6

Cut of an empty and a foam-filled square extrusions of the same length after quasi-static loading. (Courtesy of SIMLab, NTNU)

In addition to changes in the collapse mode, the presence of foam or any filling material can increase the crushing load much over the sum of both materials acting separately. This is known in the literature as the interaction effect, and some semiempirical formulas have been proposed to predict the average axial crushing load of filled sections accounting for this phenomenon (Hanssen et al., 2000; Costas et al., 2016, 2017).

A natural alternative to synthetic foams can be found in cork, which also presents an inner collapsible cellular structure. Some studies on cork-filled sections can be found, even though the material usually exhibits a lower resistance than that of the polymeric or metal foams. Moreover, the crushing energy is stored as elastic recoverable deformation to a great extent, which is usually not desirable in shock-absorbing structures.

Honeycombs are an engineering solution with a remarkable performance in the field of impact engineering. The honeycomb structure, which can be seen as an extension of multi-cell tubes, is arguably one of the most extensively used energy-absorbing structures in industry. They are placed inside automobile bumper bars, at the front of high-speed train locomotives or in a large number of aircraft structures (Paz et al., 2017). They are commonly made of metal (aluminum and steel being the most common) but also thermoplastics, elastomers, or molded polyolefin. The preferred shape is hexagonal, but a variety of geometries have been (and are being) investigated. The behavior of honeycombs subjected to compressive loading resembles that of foams: an elastic loading followed by a stress plateau and a final densification phase. The plateau phase is produced by the progressive plastic collapse of the cells (metal honeycombs) or by the brittle fracture of the thin walls (rigid, brittle materials like some thermoplastics). The energy-absorbing capability of honeycombs is determined by the cell wall thickness, length, width, and height. The reader is referred to Gibson and Ashby (1997) for a more comprehensive description of this configuration.

Computational Simulation of Structural Crashworthiness

Crashworthiness characterization is a multidisciplinary endeavor that involves mechanics and biology. During an impact there are many nonlinear dynamic phenomena involved: local and global instabilities, contact, large deformations, plastic strains, material damage, and fracture. A precise simulation of a crash event requires computational tools able to incorporate those nonlinear effects and obtain accurate results in a reasonable time.

The automotive industry runs several crash tests every day, each of which has a high cost (more than USD 100,000 per test). This is one of the reasons why finite element simulations are rapidly gaining ground in this field. In the aerospace and railway industries, the proposed approach for crashworthiness certification involves the correlation of numerical models with experimental tests.

Finite element simulations are a versatile and reliable tool to predict the behavior of structures under impact loads. In particular, explicit integration schemes are able to provide accurate results in problems involving large deformations, fracture, or high loading rates (Belytschko et al., 2014). Shell elements are a favorite for simulations of thin-walled metal members subjected to extreme deformations, giving reasonably accurate results in affordable computation times. Figure 7 shows the result of the finite element simulation of the axial crushing of a double-chamber aluminum profile using shell elements. For problems where a higher degree of accuracy is necessary or where the geometries of the parts are not suitable for a shell elements model, solid elements can be used instead, at the expense of much longer analysis times.

Fig. 7

Finite element simulation of the axial crushing of a double-chamber profile using shell elements. The color map represents the value of the damage variable in each element

Good Practice Guidelines for the Finite Element Modelling and Simulation of Structures Under Impact

While finite element courses are present in the study plans of most engineering faculties nowadays, there are some particularities regarding their use in problems involving large deformations or impact loads. This section is intended to provide a few specific recommendations for the modelling and simulation of components or structures subjected to such loads, based on experience. They are included here for indicative purposes only, so they are not compulsory rules in any case.

Geometrical Modelling

  • When using shell elements to model thin-walled parts, filleted corners with small radii can be modelled as sharp corners. This avoids the generation of small elements which can affect the computation time.

  • When a problem is totally symmetric – geometry, loads, and boundary conditions – with respect to a plane, half of the model can be removed and substituted by the pertinent boundary conditions. This can be done for each symmetry plane in the model. For axisymmetric problems, the model can be reduced to the revolved cross section using appropriate elements and boundary conditions.

  • Rigid parts with simple geometries should preferably be modelled analytically (mathematical surfaces) rather than discretized with a mesh, if possible. For more complex geometries, the mesh of the rigid part should be fine enough to enable a realistic computation of the contact forces. This applies for instance to actuators or reaction walls in a crash test.

  • In an experimental crash test, the force measurements may vary depending on the point where they are taken. The same happens in the simulations. It is important that forces are measured exactly at the same location as in the experimental tests. Also, it is recommended that all the parts between the load cell and the interaction surface are included in the finite element model with their right stiffness.

  • If shell elements are used to model collapsing parts, it is important to account for their thickness in the contact formulation. This is particularly important in axially crushed tubes, where the length of the lobes can be affected by an inaccurate contact formulation.

Fig. 8

Variation of the logarithmic strain at failure with different discretizations in a DIC analysis of the same tensile test. (a) εf = 0.337 (b) εf = 0.394 (c) εf = 0.488

Material Modelling

  • For metals, it is important to properly model their work hardening. When using a tabulated stress-strain curve as material input, it is key to extrapolate the curve much beyond the diffuse necking point observed in the tensile test of the material. This is due to the fact that strains in the material can be much higher than the necking strain in a tensile test.

  • When modelling collapsible foams, it can be useful to introduce a locking strain in the constitutive model. This would prevent elements from undergoing extreme local deformations and affect the analysis performance. An example can be foam-filled metal tubes, where the foam elements might get trapped inside the metal folds.

  • Material anisotropy, strain rate sensitivity, and fracture should be included in the model if and only if they are relevant for the material (experimental observations) and the application. These features can greatly increase the computation times. If only small cracks are appreciated in the experimental tests, fracture can be excluded from the material model in exchange for a lower analysis time.

  • Fracture of sheet metals is usually a mesh-dependent problem for most failure criteria, due to strain localization. As an example, Fig. 8 shows how the failure strain changes for different element sizes in a digital image correlation (DIC) of the same tensile test. The same would happen in a finite element simulation. Due to this variation of the failure strain in the elements as a function of the discretization, it is advisable to use the same element size in the component simulation and in the calibration of the material model with the experimental tests. Nevertheless, this effect can be mitigated by using a regularization scheme in the damage model, which can adjust the failure criterion as a function of the element size and/or other criteria (see for instance Costas et al. 2019).

  • When running a simulation of a metal part subjected to an impact, viscoplasticity has to be considered if the strain rate dependency of the material is relevant. But even in those cases where this effect can be neglected, it is a good practice to include viscoplasticity with a small coefficient. It will have almost no effect in the results, but it will provide some damage regularisation.

Element Choice

  • The element discretization should be fine enough to reproduce the curvature of folds and hinges in the collapsing parts. Mesh convergence studies are strongly recommended.

  • Shell elements usually provide an accurate prediction of the collapse of thin-walled metal members with affordable computation times. An even better result can be obtained using solid elements, keeping in mind that a minimum number of elements have to be placed across the thickness. The number of solid elements should be high enough to have at least three integration points across the wall thickness, e.g., three linear elements or one cubic element. Mesh convergence studies are strongly recommended.

  • Elements with reduced integration are commonly preferred for problems involving large deformations. This avoids volumetric locking but usually requires some efficient hourglass control in exchange.

  • Regarding fracture, finite element codes have different default criteria governing element deletion, for example, the failure of the central integration point, the failure of all integration points, etc. This influences the prediction of fracture initiation, giving sometimes important over- or underestimations. It is recommended to check this point if fracture prediction is relevant for the problem.

Analysis Configuration and Post-processing

  • In an explicit analysis, the stable time increment is determined by the smallest element in the mesh if the materials in the model have a similar stiffness. It is usually good practice to invest some time in the meshing to avoid unnecessary small elements that could harm the computational performance of the model.

  • Some finite element codes include an option for sub-cycling explicit analyses. This is useful if there are parts in the model with much shorter or longer stable time increments. The solver splits the model according to the stable time increment of the parts and only applies smaller time increments where they are required.

  • When running a quasi-static simulation in an explicit solver, time or mass scaling are commonly used in order to have reasonable computation times. This must be done in a way such that inertial effects can be neglected. The following three rules are common practice:
    • The loads and the displacement (or velocity) boundary conditions have to be smoothly ramped up at the beginning of the step up to their final values. If they are applied too suddenly, the quasi-static conditions could not be fulfilled.

    • The strain-rate sensitivity of the materials has to be turned off in most cases or scaled by the right factor.

    • The energy balance has to be checked throughout the simulation: the kinetic energy should not exceed 5% of the total energy (internal or external).

  • In simulations involving impact, it is common to filter out the highest frequencies in the force signals. These frequencies correspond to vibrations or stress waves travelling along the components, which could not be of interest.



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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Group of Structural Mechanics, School of Civil EngineeringUniversidade da CoruñaA CoruñaSpain
  2. 2.SFI-CASA, Centre for Advanced Structural Analysis, Department of Structural EngineeringNorwegian University of Science and Technology (NTNU)TrondheimNorway

Section editors and affiliations

  • F. Teixeira-Dias
    • 1
  1. 1.Institute for Infrastructure and Environment (IIE), School of EngineeringThe University of EdinburghEdinburghUK