handbook engineering section12 transportation ch83
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Khalil, T. B. Safety Analysis
The Engineering Handbook.
Ed. Richard C. Dorf
Boca Raton: CRC Press LLC, 2000
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83Safety Analysis
83.1 Mathematical ModelsLumped Parameter Models Hybrid Models Finite Element Models
83.2 Summary
Tawfik B. KhalilGeneral Motors Corp.
Safety of transportation systems, including land, water, and space vehicles, can be defined as theability of the vehicle structure to provide sufficient protection to mitigate occupants' harm and to
reduce cargo damage in the event of a crash. This goal is typically achieved by a combination of
structural crashworthiness to manage the crash energy and by a system of restraints within the
passenger compartment to minimize the impact forces on the human body during the second
collision. Crash energy management is viewed here as absorption of the crash kinetic energy of the
vehicle while maintaining sufficient resistance to sustain the passenger compartment
integrity.
Safety studies for land motor vehicles, the subject discussed in this chapter, are accomplished by
a combination of experimental and analytical techniques. Experimental techniques involve sled
tests, in which mechanical surrogates of humans (anthropomorphic test devices, or "dummies")
are subjected to dynamic loads similar to a vehicle decelerationtime pulse to study occupant
response, in either frontal or lateral impact modes. The measured dummy kinematics and
associated loads (moments) provide a measure of the impact severity and the effectiveness of the
restraint system in reducing loads transferred to the occupant. Another type of test typically run to
ensure total vehicle structural integrity (crashworthiness) and compliance with
government-mandated regulations is the full-scale frontal vehicle to barrier impact. In this test a
fully instrumented vehicle with a dummy in the driver seat is launched to impact a rigid barrier
from an initial velocity of 30 mph. Other tests include side impacts, rear impacts, and rollover
simulations.
Such experimental studies are not only time consuming but also expensive, particularly at the
early stages of design, where only prototype vehicles are available. The need to simulate the crashevent by an analytical procedure is obvious. This chapter addresses the use of analytical techniques
in design of transportation systems, with particular emphasis on motor
vehicles.
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Developing mathematical models for structural crashworthiness and occupant response to impact is
conceptually easy; it involves solving a set of partial differential equations that govern the response
of structures to dynamic loads, subject to initial and boundary conditions. In practice, however, the
problem is complicated due to the following factors:
The crash process is a dynamic event persisting for a short duration of approximately
100200 milliseconds.
Vehicle structures are typically complex in geometry, manufactured from metallic and
composite shell components, and assembled by various fastening techniques.
Biomechanical simulation of human or mechanical surrogates to impact requires extensive
knowledge of human anthropometry, biological tissue properties, and human tolerance to
impact.
The governing equations are highly nonlinear due to large deformations, large rotations,
buckling, elastic-plastic rate-dependent material response, and contact and folding in the shell
structures.
Given the aforementioned factors, it is not surprising that analytical simulations of vehicle
collisions and occupant response to impact have been evolving over the last 25 years. Three types
of models are used in safety simulations: lumped parameter models (LP), hybrid models (HM),
andfinite element (FE) models.
Lumped Parameter Models
The first vehicle structure LP model was developed [Kamal, 1970] by using lumped spring-mass
components. It simulated the vehicle response to frontal impact into a rigid wall by a
unidimensional model consisting of three masses, which represented the inertia of the vehiclebody, engine, transmission, and drive shafts. The masses are interconnected by nonlinear springs to
simulate the compliance of the vehicle structure. The force-deformation characteristics of the
springs are determined by quasi-static crush of vehicle components, which incidentally require
significant experience on the behavior of thin sheet metal structural components subject to large
deformations and various end conditions (e.g., fixed, hinged, or free). This type of model is still
widely used by crash engineers because of its simplicity and surprisingly relative accuracy in
comparison with test data. In fact, this modeling approach has been extended to simulate side
impact collisions between two vehicles. It is important to note, however, that developing such
models relies on experimental data and extensive experience on structural behavior in crash
environments. Further, translating model parameters into design data is not immediately
obvious.
Two-dimensional and three-dimensional LP models [Prasad and Chou, 1989] are also developed
to simulate occupant response to deceleration pulses generated by vehicle structures. These models
consist of a group of masses that simulate the inertia and CG locations of anthropomorphic
dummies used in crash testing. The dummy segments are connected by joints with appropriate
moment-rotation and force-deformation characteristics to represent biomechanical human
articulation. Interactions between the dummy model and the passenger interior compartment are
achieved by specifying force-deformation curves between the dummy segment and the potential
83.1 Mathematical Models
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contact target. Figure 83.1shows an example of a three-dimensional average size dummy model
used in crash simulations. Similar to LP models of vehicle structure, the occupant models are
relatively simple to develop and not computationally demanding. In fact, all LP models can be run
in minutes on an engineering workstation or a personal computer.
Figure 83.1 Three-dimensional LP model of a seated dummy, represented by ellipsoidal rigid bodiesinterconnected by appropriate joints.
Hybrid Models
Hybrid models were developed to remedy the limitations inherent in the LP spring-mass models.
The modeling technique, simply, recommended calculation of the force-deformation component
response from structural mechanics equations, instead of testing, which would subsequently beused as the spring property in the LP model. The recommended components initially were generic
S-shaped hollow beams built from thin sheet metal [Ni, 1981]. Two beams typically represent the
lower or mid-rails (also referred to as torque boxes ) of the vehicle and represent the main
longitudinal load-path carriers from the bumper to the vehicle body. The analysis was
accomplished by a finite difference solution, which treated the shell structure by a series of beams
with appropriate geometry and material models. Plastic behavior at the hinges was accomplished
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by a moment-rotation curve, derived experimentally. Although the component response can be
calculated in three dimensions, due to the inability of the LP technology to superimpose
path-dependent plastic deformations, the analysis allowed only uniaxial deformations. Further, the
boundary conditions at the spring ends cannot be accurately represented. These limitations
rendered the technique approximate. Yet, it is commonly used in vehicle design due to itssimplicity and to its low requirements of computer resources.
Finite Element Models
There are two types of FE models, discussed in the following sections, that are used in structural
crashworthiness:
Heuristic Beam Models
Heuristic models (semianalytical models) are formulated by complementing the equations of
mechanics with experimental data and empirical information. These models are developed to
provide design guidelines for vehicle structure at the early stages in vehicle conception. Four typesof models are constructed and analyzed in parallel to investigate the synergy between the major
collision modes, namely, front, rear, side, and rollover impacts. At the early stage in vehicle
design, crashworthiness is considered in parallel with other design requirements, such as
packaging, vehicle dynamics, noise and vibrations, and so on. Accordingly, a computationally
efficient scheme along with a fast process to build models is necessary. This led to the
development of FE beam models [Mahmood and Paluzeny, 1986], which define all major
components of the vehicle skeleton by means of beam elements. With skill and experience, the
influence of connecting panels can be included in the analysis.
The basic building blocks of these models are structural members that are referred to as columns
when subjected to uniaxial compression and as beams when subjected to bending deformations. In
either case these components are manufactured from stamped thin sheet metal. Column membersare typically exposed to axial or slightly off-angle loads, which can produce progressive
(accordion) crush, as shown in Fig. 83.2 . This type of progressively stable collapse, highly desired
in energy absorption, requires a compressive load smaller than the Euler buckling load and larger
than the plastic yield of the column plates. Unstable column collapse, which can include some
folding, is less efficient in absorbing energy.
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Beam bending is the dominant mode of collapse in many vehicle structures, due to its need for
the least amount of energy. Collapse by pure bending is rare. In most vehicle crashes, structural
deformations involve a combination of axial compression, bending, and torsion. Component
collapse will initiate where the compressive stress exceeds the material yield/local-buckling
strength by forming a plastic hinge. The structure cannot continue to support an increasing load at
the hinge and stress redistribution occurs, followed by the formation of more plastic hinges. This
continues until eventually the structure evolves into a kinematically movable framework, as shown
inFig. 83.3. Therefore, it is important that the model captures the plastic hinge formations and
subsequent linkage kinematics effects.
Figure 83.2 Schematic of axial compression of a straight column, showing progressive formation ofaccordion folds.
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Approximate formulas, based on a force method or a displacement method, are derived
[Mahmood and Paluszny, 1986] to determine the peak and average crush loads on the basis of
local buckling and plastic yielding of thin-walled columns and beams. The component geometry is
subdivided into plate elements, which are joined by nodes. A computer program is developed to
determine the maximum load-carrying capacity of vehicle structures and subsequent energy
absorption following large deformation collapse of the structure. The computation can account for
compression and biaxial bending deformations.
Analytically Based FE Models
All previously cited models require some prior knowledge of the potential failure mechanism of
the structure, in addition to experimental data as input to the model. It has always been the desire
of safety engineers to develop analytically based models for vehicle crash and occupant dynamics
simulations. These models should be based on the physical process involved in the crash
eventgeometry of the structure, basic stress-strain response of the material, initial conditions of
impact and boundary/constraint conditions. This type of analysisknown in mechanics as
initial-boundary value problemrequires the solution of a nonlinear, coupled system of partial
differential equations that can only be applied to extremely simple geometries. Accordingly,
classical closed form solutions are nearly inapplicable to real-world structural mechanicsproblems.
FE technology was introduced in the early 1960s for linear structural analysis, in which the
geometry of the structure can be discretized into a set of idealized substructures, called elements.
Several Fortran computer codes were developed, for both research and commercial applications.
Application of the FE technology to crashworthiness analysis did not gain serious momentum until
the mid-1980s, due to accelerated advances in explicit time-integration nonlinear FE technology
Figure 83.3 Schematic of S-rail bending deformations, showing plastic hinge formations and subsequent
linkage-like kinematics.
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[Liu et al., 1986] and due to the development of FE codes with reduced integrated elements for
spatial discretization and versatile contact algorithms [Goudreau and Hallquist, 1982]. In addition,
the introduction of supercomputers provided the necessary impetus for applying the technology to
practical problems.
FE crash codes are based on updated Lagrangian mechanics. The equations of motion areobtained from stating the balance of linear momentum in an integral form and introducing spatial
discretization by linear isoparametric elements. The semidiscretized second-order set of equations
of motion can be written as
Ma = P(x; t) Q(x; t)
where M is the diagonal mass matrix, a is the acceleration vector, P is the external force vector,
Qis the nodal internal force vector,x is a spatial coordinate, and tis time. The solution of the
previous set of equations in time is accomplished by the explicit central difference technique. The
integration scheme, though conditionally stable, has the advantage of avoiding implicit integration
and iterative solution of the stiffness matrix.Initially, this technology was applied to analyze generic components (columns and S-rails) with
emphasis on analytically capturing the plastic hinge formation, peak load, sustained collapse load,
and associated energy absorption. Following this, a number of simulations modeled structural
components manufactured from thin sheet metal and assembled by spot welding [Khalil and
Vander Lugt, 1989].Figure 83.4shows the initial and deformed configurations of the front-end
structure of an experimental vehicle launched to impact a rigid wall from an initial velocity of 50
km/h. The model simulated the structure by predominantly quadrilateral shell elements, which
allows for both bending and membrane deformations. Elastic-plastic material properties with
appropriate strain hardening and rate effects were assigned to the shell elements. Constraint
conditions were used to tie the shell nodes where spot welds were used. A single-surface contact
definition was specified for the frontal part of the structure to allow for sheet metal stacking
without penetration. The predicted peak force from this impact was 250 kN, which agreed quite
well with test data.
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After advances in the ability of FE technology to simulate subsystem impact response, the
methodology was extended to simulate full-scale vehicle collisions. Current simulations include
frontal vehicle collision with a rigid barrier, commonly conducted for compliance with federal
safety standards.Figure 83.5shows an FE model of a vehicle structure before and after impact
with a rigid barrier from an initial velocity of 50 km/h [Johnson and Skynar, 1989]. Other models
of vehicle structures simulate a movable deformable barrier impacting the side of a stationary
vehicle. Also, simulations of vehicle-to-vehicle frontal impact as well as rear impact have been
successfully attempted.
Figure 83.4 FE model of a vehicle structure front end, launched to impact a rigid barrier from an initial
velocity = 50 km/h.
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Figure 83.5 FE model of frontal vehicle collision with a rigid barrier, initial velocity = 50 km/h.
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Recently, the technology has been extended to simulate air bag inflation and deployment of
driver-side and passenger-side air bags.Figure 83.6shows an isolated, folded driver-side air bag in
its initial configuration and its shape subsequent to inflation. Of particular interest here, from the
FE simulation point of view, is the technology's ability to simulate the bag fabric material, which
can sustain tension and no compression; the gas dynamics and their interaction with the bag toallow for pressurization and controlled leakage; and, finally, the contact and interactions among the
bag layers, which should allow for deployment without penetration.
Figure 83.6 FE simulation of driver-side air bag inflation.
The limitation of using LP models in simulating occupants has been recognized for some time.
With the success demonstrated in simulating vehicle structures, analysts were encouraged to
extend FE analysis to simulate occupant interactions with interior passenger compartments [Khalil
and Lin, 1991].Figure 83.7shows an FE model of a dummy used to simulate an occupant in crash
testing.
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83.2 Summary
Several analytical techniques are used in safety analysis to simulate vehicle structural response and
occupant behavior in a crash environment. Early models include lumped-parameter, hybrid, and
FE heuristic beam models. These models are characterized by gross geometric approximations,
and, consequently, they are quick to develop and require minimum computer resources that can be
provided by a PC. In the past seven years, detailed representations of vehicle structures by FE
models have evolved in size and complexity from geometries represented by 2800 shell elements
with one or two material models to current models simulated by over 50 000 shell, solid, and beamelements. These models also include several material representations for metallic and nonmetallic
components. Currently, vehicle models exist in the open literature for frontal, side, and rear-impact
simulations. Also, modeling of occupant interactions with inflatable restraint systems has recently
been published and discussed. It is anticipated that in the near future (within five years) system
models representing vehicle structures, occupants, and restraint systems will be in the
neighborhood of 100 000 elements and will become a routine design tool in the transportation
Figure 83.7 FE representation of a seated dummy with three-point belt harness.
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industry. However, this increase in model sizecoupled with demands for lighter vehicles
manufactured from materials such as aluminum and compositeswill require new developments
in FE technology and hardware architecture to allow for reducing model development effort and
computation time.
Defining Terms
Anthropomorphic: Describes a mechanical manikin that possesses geometric, inertial, and
material characteristics similar to a human's.
Crashworthiness: The ability of a vehicle structure to absorb mechanically energy resulting from
collision with another object while maintaining integrity of the passenger compartment.
Heuristic model: A model formulated from discrete deformable elements with built-in empirical
knowledge.
Hybrid model: A lumped parameter model in which the discrete springs are replaced by
deformable components.
Lumped parameter model: A mechanical system model that represents a continuum structureby discrete masses, springs, and dampers.
References
Goudreau, G. L. and Hallquist, J. O. 1982. Recent developments in large-scale finite element
Lagrangian hydrocode technology. Comp. Methods Appl. Mech. Eng. 33: 725757.
Johnson, J. P. and Skynar, M. J. 1989. Automotive crash analysis. In Crashworthiness and
Occupant Protection in Transportation Systems, ed. T. B. Khalil and A. I. King, p. 2733.
AMD-Vol. 106, BED-Vol. 13. ASME, New York.
Kamal, M. M. 1970. Analysis and simulation of vehicle to barrier impact. SAE.
700414:14981503.Khalil, T. B. and Lin, K. H. 1991. Hybrid III thoracic impact of self-aligning steering wheel by
finite element analysis and mini-sled tests. In 35th Stapp Car Crash Conference
Proceedings. Paper No. 912894. SAE, Warrendale, PA.
Khalil, T. B. and Vander Lugt, D. A. 1989. Identification of vehicle front structure
crashworthiness by experiments and finite element analysis. In Crashworthiness and
Occupant Protection in Transportation Systems, ed. T. B. Khalil and A. I. King, p. 4153.
AMD-Vol. 106, BED-Vol. 13. ASME, New York.
Liu, W. K., Belytschko, T., and Chang, H. 1986. An arbitrary Lagrangian-Eulerian finite element
method for path-dependent materials. Comp. Methods Appl. Mech. Eng. 58: 227245.
Mahmood, H. F. and Paluzeny, A. 1986. Analytical technique for simulating crash response ofvehicle structures composed of beam elements. In Sixth International Conference on Vehicle
Structural Mechanics. Paper No. 860820. SAE, Warrendale, PA.
Ni, C. M. 1981. A general purpose technique for nonlinear dynamic response of integrated
structures. In Fourth International Conference on Vehicle Structural Mechanics. SAE,
Warrendale, PA.
Prasad, P. and Chou, C. C. 1989. A review of mathematical occupant simulation models. In
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Crashworthiness and Occupant Protection in Transportation Systems, ed. T. B. Khalil and
A. I. King, p. 95113. AMD-Vol. 106. ASME, New York.
Further Information
ASME Winter Annual Meeting Proceedings: published annually by the Applied Mechanics
Division.
Vehicle Structures Mechanics Conference: published biannually by the Society of Automotive
Engineers (SAE).
U.S. Department of Transportation, International Conference on Experimental Safety Vehicles:
published biannually by the National Highway Traffic Safety Administration.
Stapp Car Crash Conference: published annually by the Society of Automotive Engineers (SAE).