Because of the wide variety of complex structures and materials in use, this book focuses mainly on basic concepts and methods and simple structural elements and materials. These practical examples are used to illustrate the application of the knowledge presented in the previous chapters.
Vehicle accidents and their consequences .1 Statistics of vehicle accidents
Consequences of vehicle accidents
It is well known that, as with other collision events, a crash of fast moving vehicle(s) occurs within a very short period of time. Regarding the injury and damage to the vehicle's occupants, it is worth noting that in a car accident with rapid deceleration, the occupants'.
Human body’s tolerance to impact
The free-motion head model is a mannequin head separated from the 50th percentile of the Hybrid III male mannequin. The main concern is the magnitude of the hypothetical impact velocity of the passenger into the interior of the vehicle and the maximum average deceleration of the passenger after this impact for 10 ms.
Applications of energy-absorbing structures/materials
- Energy-absorbing structures used to improve vehicles’ crashworthiness
- Energy-absorbing structures used for highway safety
- Energy-absorbing structures used for protection against industrial accidents
- Energy-absorbing structures used for personal safety
- Energy-absorbing structures/materials used for packaging
1.8, the A-pillar, B-pillar and roof rails are designed to maintain the structural integrity of the passenger compartment in the event of a collision accident. For example, when the roof of the driver's cabin is hit by a falling rock (Fig.
Design of energy-absorbing structures and selection of energy-absorbing materials
General principles
1.14(a)) and all the initial kinetic energy of the car is converted into elastic strain energy of the spring. 1.14(b)) and eventually all elastic strain energy will be converted back into kinetic energy of the car.
Idealisation of materials’ behaviour
- Mechanical properties of materials in tension
- Idealised material models
- Moment–curvature relationship for plastic beams
- Plastic hinge and hinge-line
- Mechanical models for materials’ idealisation
- Validity of rigid-plastic idealisation
First, assume that a force Fis is applied quasi-statically to the left end of the elastic-plastic model shown in Fig. Next, consider that a force F is applied quasi-statically to the left end of the rigid plastic model shown in Fig. .
Limit analysis and bound theorems .1 Limit state of perfectly plastic structures
Example: a beam under bending and tension
It is seen from the above derivation that in the case of a plastic hinge the partitioning of the energy dissipation, i.e. it is seen that the influence of the axial force on the energy dissipation at the generalized hinge is very small, provided that the beam is slender (h) < By simply assuming the magnitude of the bending moment MA to be equal to the fully plastic bending moment of the beam Mp, the bending moment diagram shown in Fig. 2.10 (a) A beam with one end clamped and the other simply supported; (b) diagram of elastic bending moment; (c) Bending moment diagram when the moment at both A and B reaches the fully plastic bending moment. That is, |M(x)| £Mp must be satisfied at any diameter over the entire span of the beam. The number of plastic hinges in the collapse mechanism must be equal to the degree of the static indeterminacy of the beam plus one. With the increase of the deflection, the deformed configuration of the beam as sketched in Fig. 2.13(b) that the axial force NA will increase the load-carrying capacity of the beam by adding a term 2NAsinbª2NAbª2NAD/L. It is clear that the contribution of the axial force to the collapse load depends on the slenderness of the beam, 2L/h. With respect to rigid plastic circular plates, Calladine (1968) demonstrated that membrane stresses induced by large deflection will greatly enhance the load-carrying capacity of the plate, even when simply supported. For example, Timoshenko and Woinowsky-Krieger (1959) showed that as long as the central deflection of a simply supported circular plate is equal to the plate thickness, the membrane stress in the midplane of the plate will reach the same order. such as bending stresses. If the applied normal stress is less than the yield stress of the material, Y, then the stress wave is elastic and propagates with a velocity of 2.3(b), then the longitudinal plastic wave will propagate with a velocity of. where EpDenotes the strain hardening modulus of the material. The total displacement of the block is found from the area of the triangle shown in Fig. 2.68 is shown], the ratio of the plastic dissipation to the total input energy is found to be It is obvious that when two spheres are considered as a system, there is a loss of kinetic energy of the system during the collision. It can be seen from this typical example that the inclusion of a velocity-dependent yield moment completely changes the kinematics of the system. Thus, the energy method (i.e. the upper bound method) can be used to determine both the initial collapse load and the initial collapse mechanism for many structural problems. 2.22(b) gives an upper and lower limit of the collapse load; if H >L/2, then the 'global' collapse mechanism shown in Fig. By assuming a large deformation mechanism of the structure under load P, the energy balance leads directly to. 2.86] indicates the variation of the bearing capacity of the beam during its large deflection. It has even been shown in section 2.3.2 that the load-carrying capacity of the beam varies considerably with beam deflection, especially when the ends of the beam are clamped axially; thus the final deflection estimated by Eq. When the upper limit of final displacement is applied, the initial energy of the system, Koin Eq. This chapter introduces the concept and method of dimensional analysis related to the plastic collapse of structures, discusses the similarity requirements for model testing, and presents several experimental techniques for studying the energy absorption of structures. The concepts of physical variables, fundamental dimensions, and dimensional homogeneity form the basis of dimensional analysis, which we discuss in the next section. Second, the number of independent dimensionless groups is only three when the five physical variables are included in the two fundamental dimensions. We are now interested in energy absorption during the entire crushing process - the subject of this book. As a result, their mechanical properties will be different, even if the materials are the same for the model and prototype. This demonstrates that due to the strain rate effect, the model experiences a strain of 85% of its prototype strain. The pendulum's two arms provide a parallelogram action that ensures the impact face remains vertical at all times. In all test methods described, the velocity of the striker after impact is not constant, but varies with displacement until the striker comes to a complete stop. The strain rate is therefore not constant. 3.8 (a) Sketch of a split Hopkinson strip; (b) typical strain signals from the crash bar and transmitter bar; (c) the corresponding dynamic stress-strain curve. The equations enable us to calculate the stress-strain relationship and strain rate for a split Hopkinson bar test. A split Hopkinson bar can also be used to study material properties in tension, torsion, and shear. This chapter presents theoretical and experimental studies of rings and short tubes under in-plane loads. Ring compressed by two point loads The effect of axial force on yield for a rectangular cross section (b¥ h) was discussed in Section 2.2.2. The axial force at the top hinge is zero and at the bottom hinge is P/2. This tensile force is of course lower than in the case where the influence of the axial force was not taken into account. Therefore, for convenience, l can be used as a variable in the calculation of load-displacement curves. which corresponds to the triangular form of PR. Semi-circular arch with an inward load This discrepancy can be accounted for by the strengthening effect, which has two implications here. This essentially revealed that the length of the effective moment arm decreases with deflection, in addition to increasing the bending moment. High values of mR correspond to a relatively flat load-deflection curve, as predicted by de Runtz and Hodge (1963) for a rigid-perfectly plastic material. It can also be noted that by choosing materials with different strain hardening effects, the force-deflection curve can be adjusted to be close to the ideal rectangular force-deflection relationship for an energy absorber as described in Chapter 1. The first two equations are conservation of momentum for the distorted part of the system and ring i respectively. Accounting for elastic waves explains the observed plastic deformation at the distal end of the system. Note that for hexagonally packed arrays this number alternates between 9 and 10 due to the packing configuration.). For hexagonally packed arrays, each tube is subjected to six equally and evenly distributed forces. Other ring/tube systems In this chapter we discuss cases of tubes under local transverse buckling by a bush or a wedge and also the bending of rectangular and square tubes, as well as the bending of thin-walled members of channel and corner sections. This calculation took into account the stretching effect of the interface, as well as the large change in geometry. Empirically, Stronge (1985) suggested that the energy corresponding to the ballistic limit, defined as the velocity when the projectile either sticks to the target or exits with negligible velocity, for a spherical missile with a diameter of 6.35 mm £d . The fully plastic bending moment of the central section for the current configuration is easy to determine; it is. As deformation continues, the fully plastic bending moment at the central part decreases due to the change in geometry [Eq. These tests used a plug to clamp one end of the pipe, using a cantilevered arrangement. Often, however, a bulge forms on the pressure surface, replacing much of the triangular area. The above analysis can be applied to the case where both ends of the pole are fixed. In the strips discussed above, the return line of the central plastic hinge is perpendicular to the direction of the axial force P. If we neglect the torsion of the flange due to the variable angular deformation in the longitudinal direction, the total plastic energy dissipation is . For each increasing value of f, the corresponding value of l can be solved numerically, resulting in curvature k(Equation [5,60]) and Dq(Equation [5,64]). Other loading systems and comments Axial collapse modes and typical force–displacement curves The deforming tube wall has been found to bend in a meridian direction instead of in a straight line (Abramowicz, 1983; Abramowicz and Jones, 1984b and 1986). Here we present a simple estimate for the mean circumferential strain rate, which is assumed to be representative of the problem. Our next task is to evaluate the energy dissipation for each of the four types of plastic zones. Those for the two cylindrical shells are easy to perform, similar to the analysis in the previous chapters. Abramowicz and Jones (1984a), asymmetric collapse mode Abramowicz and Jones (1984a), symmetric collapse mode Wierzbicki et al. Test points fall within the limits of the two formulas, for each hat section type. 10.35] of the relationship between the plateau stress and the relative density is the foam crushing force. To estimate the value of the degree of deformation, we assume that the foam is uniformly compressed into a fold of length 2H. Similarly, the improvement of the plateau tension of foam due to the strain-rate effect can be considered. However, there is a limit to the maximum value of this plateau stress: if the foam is dense and therefore has a high level of plateau stress, a tube may have a strong tendency to undergo Euler-type buckling, which significantly reduces energy absorption. In the following it is assumed that the contact surfaces of the two bodies are smooth and the contact areas will develop asymmetrically. The maximum contact pressure is found. acting in the center of the contact circle.Statically admissible stress field and lower bound theorem
Kinematically admissible velocity/displacement field and upper bound theorem
Effects of large deformation .1 Background
Analysis of an illustrative example
Various aspects of effects of large deformation
Concluding remarks
Effects of dynamic loading
Stress wave propagation and its effects on energy absorption
Inertia and its effects on energy absorption
Strain-rate and its effects on energy absorption
Energy method
Energy method used in determining incipient collapse load and mechanism
Energy method used in case of large deformation
Energy method used in case of dynamic deformation
Dimensional analysis
Physical variables, fundamental dimensions and dimensional homogeneity
Dimensional analysis
Small-scale structural models .1 Similarity requirements
Quantities difficult to scale exactly
Experimental techniques .1 Universal testing machine
Drop hammer, sled and pendulum
Split Hopkinson pressure bar
Gas guns and other techniques
Ring pulled by two point loads
Built-in semi-circular arch under point loads
Semi-circular arch with an outwards load
Ring compressed by two flat plates
Laterally constrained tubes
One-dimensional ring system under end impact
Lateral crushing of arrays of circular tubes
Concluding remarks
Circular tube under point loading
Indentation of a circular tube by a blunt wedge
Bending collapse of thin-walled members .1 Square and rectangular sections
Circular tubular sections
Bending collapse of channel sections
Bending of angle sections
Circular tubes
Theoretical models
Square tubes
Top-hat and double-hat sections
Effect of foam filling
Further remarks
Local deformation of structures due to impact .1 Kinetics of a direct central collision between
Indentation caused by contact force
