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BrowseUploadSign inJoinBooksAudiobooksComicsSheet MusicWelcome to Scribd! Start your free trial and access books, documents and more.Find out morePage 231Chapter 6 Loading from explosions and impact
Alan J.Watson
Commonly, blast and impact loads are of subsecond duration and magnitude tens of times larger than any other loads in the design life of the structure. The maximum positive or rebound negative peaks of stress or displacement are critical for the structure’s survival and subsequent vibrations will only be important if the loads are repetitive. For some industrial structures blast and impact forces are repeated in-service loads and the response must be checked as a serviceability limit state including cracking, vibration and fatigue. The design and construction of structures against accidental or deliberate impact or explosions is now often considered a part of normal design in the ever increasing importance of safety against industrial and transportation accidents or terrorism. Ronan Point (1968), Flixborough (1974), Chemobyl (1986), Piper Alpha (1988), Peterborough (1989), Oklahoma City (1995), and Eschede (1998) all had a profound effect on design philosophy. These accidents highlight the fact that safety is a multi-disciplinary activity and have shown that structural design changes would be beneficial without enormously increasing the cost. If solid abutments had been used instead of columns in the design of the Eschede bridge, or if the rail lines had been given a greater clearance, the bridge would have been more robust. If compartmentalized construction and moment frames had been used in the Oklahoma Federal Building, increasing the total building cost by 2 per cent, the extent of the progressive collapse which followed the explosion would have been reduced. The public inquiry into the collapse of Ronan Point (Griffiths et al., 1968), revealed that the gas explosion produced a peak lateral pressure on the walls of about 42 kN/m2 for a few milliseconds which, aided by the upward explosive pressure on the slab above, displaced the top of the wall removing all support from the floor slab of the flat above. Collapse progressed upwards and impact from the collapsing floor slabs then caused collapse to progress downward. Ronan Point had little restraint against rotational or translational displacements between floor and wall slabs and the blast pressure had been enough to fail the joints designed only for modest wind pressures. A subsequent risk assessment showed that Ronan Point, with 110 flats and a design
Page 232 life of 60 years, had a 2 per cent risk of one of the flats having a structurally damaging explosion in 60 years. The material and human consequences of such incidents are so severe that low risk is not an adequate reason for ignoring the danger. Secondary consequences such as loss of public or business confidence can be equally costly.
6.1.1 Philosophy of design
Since Ronan Point collapsed, the stability of all buildings over four storeys in the UK must be checked with key elements designed for 35 kN/m 2 static loading in the critical direction, or with continuity to limit the area of collapse if a key element fails. The 35 kN/m 2 static load has no statistical significance as an impulsive load, either from blast or impact. The limit state philosophy for structural design uses elastic response for service loads, plastic response for ultimate loads and prevention of overall collapse disproportionate to a local failure. Many buildings have brittle or non-structural elements such as windows and suspended ceilings that are extremely vulnerable to blast pressure and produce hazardous debris, both within and outside a building. The resistance of the fixings and supports of external cladding on a building as well as of the panel itself determines the blast or impact resistance and the interaction with the characteristics of the blast loading function, but there is little blast design guidance available for cladding fixings. Cladding fixings are often hard to inspect but some indication of damage can be obtained from the residual deformations in frames and cladding panels (EPSRC, 1997; Pan and Watson, 1996). Structures must have safe and serviceable paths for all loads, including extreme loads. British Standard Codes of Practice since 1972 have recommended that by using nominal peripheral and horizontal ties buildings would be more robust to resist extreme loads. Explosions and impact loads may differ in both magnitude and direction from static design loads and produce local damage such as cratering of concrete elements or local buckling of steel elements that would reduce the moment or shear capacity locally. Deflections are very similar for structures under distributed static or dynamic loads but not when the load is concentrated (Watson and Ang, 1984).
6.1.2 Diagnosis of extreme loads
The damage to structural elements from extreme loads can be back analysed to find the load parameters, such as the 35 kN/m 2 equivalent static loading from Ronan Point (1968). From an analysis of damaged lamp posts at Flixborough (1974), Roberts and Pritchard (1982) estimated the peak dynamic pressure produced by the explosion. Sadee et al. (1976) estimated overpressure-distance curves from observations of damage to brickwork and concrete structures. The case study in Section 6.4 uses the damage to buildings from an explosion to evaluate the dynamic loads and so assess the cause of damage to other buildings on the site.
Page 233 A compression wave from an explosion in air expands as a three-dimensional blast wave propagating at maximum velocities well above that of low amplitude sound waves. It reflects and refracts from solid surfaces and from atmospheric discontinuities. Explosions also produce high temperatures, which are more locally concentrated than the pressure and decay rapidly, and also produce high velocity fragments from any confining structure, which may impact with a surface before the blast wave arrives. The synergistic effects of blast and fragment impact are not well understood. If an explosion occurs in contact with a solid it produces stress of the same order of magnitude as the elastic modulus of the solid. The air pressure produced at close range has an initial peak, which is orders of magnitude larger than normal atmospheric pressure, but decreasing with distance travelled. Behind the peak the pressure is still above atmospheric but decreasing with time and falls below atmospheric. The potential of this underpressure to produce structural damage is not certain and in part depends on synchronization with the rebound of the structure. An impact produces a localized application of pressure on the surface of the structure, which can only spread into the structure from the point of application. This is in contrast to explosions where the blast pressures rapidly engulf the entire surface of the structure. The important parameters of an impact, for diagnostic or forensic purposes, are the shape, velocity and mass of the impactor, and whether or not the impactor deformed. When pressure is applied very rapidly to the surface of a structure then strain waves are generated which transfer the local dynamic surface deformations into overall structural deformations. An analysis of the transient stress state is necessary when the applied pressure changes more rapidly than the time taken for the strain waves to travel between the boundaries of the structure and establish a state of equilibrium between overall structural resistance and applied pressure effects. During this transition period the transient strain and stress conditions may produce local failures that are decoupled and of different shape from the failures that can occur due to overall structural deformation. The strain waves propagate at characteristic velocities for the material and transfer momentum into the structure by dynamic displacements of the boundary surface. The rates of strain and stress that are produced locally in the material are orders of magnitude greater than those produced in the overall structural deformation, which are again orders of magnitude greater than under slowly applied loading. Most construction materials have enhanced properties at these high rates of strain.
6.2 BLAST PHENOMENA
6.2.1 Explosive sources
A detonation wave travels through high explosives at 5,000–10,000 m/s. At a free air boundary the gaseous products expand at high velocity, pressure and temperature
Figure 6.1 Typical overpressure (above atmospheric po) at a fixed distance from the explosion.
Figure 6.2 Typical overpressure at a fixed time along a radial line from the explosion.
to produce a shock wave with an infinitesimal rise time, producing rapid fluctuations in air pressure and a dynamic wind as it travels from the explosion (Figures 6.1 and 6.2). Air—gas mixtures, dust and vapour clouds release energy by a process of rapid burning known as deflagration. Air shock from a deflagration propagates more slowly and has a longer rise time. For vapour clouds the degree of confinement is critical in determining whether or not there is a detonation or a deflagration. Various forms of organic dust can also produce an explosive reaction. Propane, butane and similar gases in stoichiometric concentrations will explode if there is a source of initiation.
Figure 6.3 Spherical shock wave.
6.2.2 Shock wave parameters
Most explosions, after propagating a short distance, produce a spherical shock wave of surface area A=4π2 (Figure 6.3). r The characteristics of the spherical air shock are as follows: (a) The energy of the shock front/unit surface area decreases with r2 (inverse square law). (b) The peak overpressure ps0 decreases with distance r from the explosion and eventually reduces to a sound wave (Figure 6.4). (c) The velocity of the shock front u is given by:
where us=340 m/s is the sound velocity in air for normal conditions at sea level and atmospheric pressure p0=0.1 N/mm2. The different units used for pressure are 1 N/mm2=1 MPa=145 p.s.i.=10 bar. (d) t dp is the positive duration of the shock wave which increases with distance from the explosion because higher pressures travel faster (Figure 6.5). (e) ps is the overpressure which decays with time at a fixed location depending on ps 0 and tdp (Figure 6.6). (f) v is the velocity of the air particles behind the shock front as they move radially away from the explosion during the positive phase (e.g. for ps0= 3p0 then v=300 m/s) and towards the explosion in the negative phase. (g) pd is the dynamic pressure where ρ density. =air
Explosives differ in the rate at which they detonate and the heat produced and these influence the characteristics of the shock wave in air.2. The peak pressure or impulse define the shock wave but its shape is also distinguished by the rise time.
6.Page 236
Figure 6. positive phase duration or negative phase duration. 1964). decay time. The equivalent weight of TNT is based on peak pressure or impulse and is larger for peak pressure than for impulse.4 Overpressure and dynamic pressure versus range from a I-MT explosion (Biggs. All of these characteristics vary with the distance the shock wave has travelled in air. but of a different size.3 Comparing explosives
The TNT equivalence of an explosive is the weight of TNT which produces a pressure wave in air with one of its characteristics equal to that of the shock wave produced by the explosive at the same distance.2.4 Shock wave sealing
The parameters of the shock wave from one explosive charge can be related to the parameters of the shock wave from a similar shaped charge of the same explosive. (1) Principle of similitude If for two charges of the same shape and the same explosive all the dimensions of the
If the masses of two geometrically similar charges of the same explosive are M1 and M2 then the peak pressures at distances proportional to (M1)1/3 and (M2)1/3. This cube root scaling allows empirical charts to be published from the results of experiments using a wide variety of charge sizes.e. The characteristics vary with the size of the explosive charge and it is experimentally observed that if two spherical charges are made from the same explosive. then the peak pressure ps0 measured at any distance R from the centre of the first charge will be equal to those measured at distance kR from the centre of the second charge. (2) Cube root scaling Since densities are presumed to be equal for the two charges of the same explosive. 1964). corresponding but not necessarily equal times).5 Overpressure and dynamic pressure positive phase duration versus range (Biggs. The +ve impulse. then the peak pressures in the air blast waves produced by these charges will be equal at distances that are in the same ratio as the cube root of the weight of each charge when the atmospheric pressures are the same in the two cases. energy and duration of the second will be k times the corresponding quantities for the first at these related distances.
first are k times those of the second.Page 237
Figure 6. if one is k times larger in its linear dimensions then its mass will be k3 times greater and the principle of similitude can be stated using the cube root of the mass as the scaling factor. respectively will be equal and are said to occur at homologous times (i.
The positive impulses. durations and energy will be proportional to (M1)1/3.6 Overpressure and dynamic decay curves (Biggs. (M1)1/3.Page 238
Figure 6. respectively at those distances and times. That is:
respectively measured at R from W kg of explosive and f. Cube root scaling has been verified by experiment but does not describe the decay of peak pressure with distance. t d are peak pressure. 2 to produce a standard charge using experimental methods.Page 239 where p. Example 1:
Cube root scaling indicates that if a similar charge of mass M2 has a diameter d2=kd 1 then ps0 occurs at R2=kR1 . to obtain shock parameters for any size of explosive charge from those of a standard of the explosive. Ø are unspecified functions. F. (3) Application of scaling laws In practice scaling laws are used: 1. that is:
. I. +ve impulse and duration.
the same peak overpressure and shock wave velocity occurs at 100 m from the 300 kg charge as occurs at 10 m from the 300 g charge. but the +ve duration t d and impulse I are ten times greater.
that is. Example 2 Use cube root scaling to compare the shock wave from a 300 kg explosive charge with the shock wave from a 300 g charge of the same explosive type and shape. the scaled parameters for cube root scaling are:
that is. pressure and velocity are the same for the prototype at homologous times.
.Page 240
7 Weak blast wave reflection. If the plane surface is a rigid protective wall. then at (0. The reflected pressure is of lower amplitude than for the rigid surface. The reflected waves propagate with the same velocity as the incident waves. 2. Providing AXO≤ 35°. ps0 has the same magnification by reflection as when AXO=0°. For greater overpressure or longer +ve duration.5 Interaction of shock waves with plane surfaces
(a) Reflection of weak shocks Spherical shock waves of low overpressure reflect from a plane surface as if the reflected shock waves (Figure 6. At t 2 the real shock covers a circular area of the surface. If the plane surface is the external wall of a normal building. The surface continues to accelerate as long as an overpressure ps exists on one side. If the +ve duration of the shock wave is much longer than the natural period of the surface then surface response is similar to that of a spring instantaneously loaded with a constant load.Page 241
Figure 6.2. The surface may not exceed the limiting elastic deflection if the reflected overpressure is low or +ve duration is short.
. t 1). Influence of surface properties: 1. Peak pressure ps 0(t2) is increased around the circumference of the circle of effect by reflection. the particle velocity v=0 and the peak pressure pr is larger than ps0. radius OA.
6. and on the same perpendicular. t 1) the surface is accelerated and has a velocity and a displacement. it is less than rigid and at (0. from the surface as the real source but on the opposite side of the surface. plastic deformation and possibly collapse may occur.7) came from an imaginary source equidistant.
(b) Reflection of strong shocks Spherical shock waves of high overpressure (ps0>>pv) reflect from rigid or non-rigid plane surfaces in a more complicated way than weak shocks. If the +ve duration of the shock wave is much shorter than Tn then overpressure reduces to zero before any significant deflection occurs and hardly any spring resistance is developed during the +ve phase. At time t1.
Hence peak overpressure ps0 determines the response of a non-rigid surface barrier to shock waves with a relatively long +ve duration and +ve impulse I determines the response to shock waves with a relatively short +ve duration.8). Boundary conditions are v=0 and peak pressure >2ps0(t1).g. the surface overshoots the equilibrium position. The velocity of the reflected shock front R is not constant and so R cannot be drawn on concentric spheres from an imaginary source.
That is. The shock wave system depends on the distance OX (e. and there is only the Mach wave). shock wave I1 reaches the surface at O and reflects. Assuming constant force P and acceleration ÿ. M (triple point) to the surface s.
. if OX=0 no separate reflections are formed. At t>t 1.9). density and velocity very different from normal atmospheric conditions (Figure 6.Page 242
Figure 6. R. is restored by the spring force but once more overshoots and vibrates about equilibrium position at the natural frequency of the spring (Figure 6. the intersection of the incident wave I(t) and reflected wave R(t) is no longer on the surface and a new shock surface M (Mach stem) connects the ring of intersection points of I.8 Spring-mass model. because the reflected shocks are advancing into air with pressure.
10). Resultant pressure >ps0(t) over a clearing period tc=(3Sc)/u where h≥ c≤ (i.e. 1. Initial diffraction: the incident wave reaches F at t0 and is reflected.9 Strong wave reflection.
.Page 243
Figure 6.10 Blast on buildings.6 Blast loading effects on buildings
Consider the building h×b×l with a plane shock wave normal to the wall F.
6.2. The blast loading from the positive overpressure is (Figure 6. after tc reflection effects no S b/2 longer act).
it decays more rapidly with distance and tdd>tdp (Figures 6. The external walls and roof of a building receive the shock wave first from an external explosion.12). back wall B.Page 244
Figure 6. The negative phase of the shock wave is often neglected in assessing blast effects (Figures 6.12Total reflected pressure p r on front wall F. Drag loading: The particle velocity v of the air behind the shock front produces a dynamic wind pressure and a drag pressure Cdpd where Cd=appropriate drag coefficient. Side walls s. Although pd decays less rapidly than pr at a fixed distance from the explosion.4 and 6. General overpressure: acts for as long as the front wall F and the back wall B are subjected to different overpressures.
2. pd on front wall F. There will be a leakage of pressure into the building through openings for as long as there is a positive difference between external and internal
. 3.11 Pressure variations ps. Back face B reaches a steady state pressure at t=(4Sc )/u after the shock wave reaches the back face.
Figure 6.11 and 6. roof R all have negative drag coefficients.5).
(c) Overpressure also depends on friction losses along the tunnel walls. is given as Pn =Ps0(0. In addition to the reflected blast loading. and the internal pressure will decrease. It depends on the charge weight and the distance from the explosion to the tunnel entrance. the following observations are given in TM5–855–1: (a) An increase in overpressure occurs if the cross sectional area of the corridor decreases. In experimental work using tunnels and ducts. For structures where A0 /V0 is small and P<10bar. internal explosions produce a quasi-static pressure which depends on the charge weight to room volume ratio for peak value and on the venting for the quasi-static decay characteristics. depending on the area of the openings A0 and the volume of the structure V0. but does not depend on the dimensions of the normal corridor. (e) Overpressure attenuates as a long duration pulse goes from one corridor into another of larger area A 2. varies within the internal space and is highest close to the leak.Page 245 pressure (P– Pj). in P time Δ msec is: t
where C L is the leakage pressure coefficient given in TM5–855–1 (1986). the average internal pressure increment Δ i . P i. Arrival times of re-reflected shocks can be calculated if a more exact analysis of loading is required.94)n where P s0 is the peak overpressure before the first bend. The peak pressure Pn after n bends of 90° when friction and pressure attenuation between bends is neglected. With internal explosions the transmission of blast waves within the corridors and connected rooms must be analysed. Internal reflections become so complicated that for preliminary analysis re-reflected shocks are neglected. according to the relationships:
. Reflections are also simplified into normal incidence but slant distances are used in determining the reflected pressure. the viscosity and the rate of decay of the shock front. When an explosion occurs inside a building then it is the interior surface of the walls and ceiling of the building which are first loaded by the pressure of the shock wave that reflects and increases the pressure. The internal pressure. (b) A decrease in overpressure occurs if the corridor has sharp turns or bends. If there are openings in the walls or ceiling then there will be venting of pressure out of the building for as long as there is a negative difference in the external and internal pressure (P– P i). (d) Overpressure attenuates with distance into a smooth corridor.
radial or cir-cumferential cracking. shape. overall structural displacements and local deformations. punching and shear failure..
.13 Cracks forming under an impact load on a concrete beam. Local damage includes penetration and perforation by the impactor. Temperature effects of impact are often ignored but may alter the material properties. cratering or depression on the impact face.1 Introduction
The independent variables of impact loading include the mass. Amde et al.3. structure and material properties of the impacting structure.3 IMPACT PHENOMENA
6.13 shows impact on a concrete beam with the cracks formed at 1 msec after impact. or of large mass and much lower velocity such as vehicles with velocities nearer to 10m/sec. 1987). 1996. velocity vector. Impactors can be of low mass and high velocity.000m/sec and fragments of damaged structures. The greater the impulse the more energy there is to absorb and the area of contact determines the distribution of surface pressure. such as bullets with velocities up to 1. may be similar but initially the inertia of the structure produces higher modes of
Figure 6. scabbing or bulging on the distal face. The larger the mass the more likely it is that the impact will cover a large area of contact. In civil engineering the impacted structure is usually stationary and the magnitude of a very short duration impact load is less critical than the impulse or kinetic energy of the impactor which must be absorbed by deformation. Figure 6. Overall deflection from an impact or a static load at the same location. At high rates of loading.Page 246
6. stress waves from the impact and the high strain rate properties of the material determine the location and type of damage (Watson and Chan.
The local response depends on material properties around the impact area. The phenomena requires a stress wave analysis and the strain rate and material constitutive relations are significant influences on the plastic flow and failure criteria.14 where the rigid mass of 1. Velocities over about 500 m/sec produce loading and structural response times measured in µsec. mode of response and the type of damage (Zukas et al. Velocities producing strain rates of about 100 do not enhance the properties of concrete and structural response is primarily elastic with some local plasticity. Figure 6..1 (CEB. Table 6. defined as the kinetic energy per unit area of contact. Figure 6. 1982). Structural response times are measured in msec in the concrete beam. 1982).
deformation causing impact damage such as top face cracking.8kg impacted at 16m/sec. and the local and overall structural response did not occur coincidentally.14 Transient deformation of a concrete beam after impact at midspan. Impact velocities above 2000 m/sec are characteristic of shape charge impact and produce pressures which exceed the material strengths by several orders of magnitude so that solids behave as fluids at the early stages of impact. Alternatively the kinetic energy density of an impactor. The impact velocity determines the strain rate.14 (Watson and Ang. can be used to determine the damaging capability
. which is not typical of static loading.Page 247
Figure 6. In this velocity regime overall global response becomes secondary and is decoupled from the local response. 1988) shows the strain rates from different types of impact.
The impulse on the structure is given by the area under the impact force—time relationship and is equal to the rate of change of momentum of the impacting mass:
. If the impactor has the lower dynamic yield then plastic deformation will be much greater in the impactor and this is a soft impact. and k is determined by its boundary conditions.3. regardless of relative size. If the impactor has the higher dynamic yield. The two masses may also move together. If the dropped mass is relatively large then the two masses may move together after impact. Using conservation of momentum at the instant of impact and conservation of energy for motion after impact. and negative gives the maximum upward or minimum downward deflection of the structure. then much less plastic deformation will occur in the impactor and this is a hard impact. if the impact surfaces are inelastic.
6. structural and material properties. If the dropped mass is small relative to the target mass and both are elastic.2 Modelling impact
Analysis of a mass dropped onto an undamped spring mass system of stiffness k shows that the ratio between the dropped mass m1 with velocity v 1 and the target mass m2 is significant in determining the response. This is a difficult parameter to define when the projectile is irregularly shaped. then the mass may rebound immediately after it impacts. This parameter is considered to be a measure of the impact shear stress produced and Smith and Hetherington (1994) list the transition zones.1 Typical strain rates for types of impact loading. the maximum deflection of the spring when m1 and m2 move together is:
In practice m2 is the effective mass of an impacted structure.Page 248
Table 6. The positive sign gives the maximum downward deflection of the structure.
Traffic Pile driving Aeroplane impact Hard impact Hypervelocity impact
l Strain rate (s−)
6 4 10−–10− 2 10−–100 2 5(10− )–2(100)
10°–5(101 ) 102–106
of a projectile by penetration of the target.
On impact. the maximum deformation is: (6.1) wher:
and v is Poisson’s ratio. (1982) assume the impactor and the target are linear elastic and the duration of the impact is long relative to stress wave transit times. the target and impactor remain in contact and compress for a total distance a at a rate of compression where v1.3 Low velocity impact by low mass projectiles
Cladding of composite sandwich construction is commonly used on buildings and is exposed to impact by small mass projectiles at low velocity: (1) Determination of the magnitude and distribution of surface pressure when an isotropic solid is impacted at normal incidence and low velocity by a spherical impactor. At the instant of impact t=0. E is Young’s modulus. R is radius of spherical impactor. m is mass and subscripts 1 and 2 denote impactor and target parameters.2)
. The maximum contact force is then: (6. The equation of motion when the positive direction of displacement is the same as that of the impulse giving the deflection x at any time t is:
6. Zukas et al.3. respectively. and x=Xω =Impulse/m. Assuming that 3/2 the Hertz law of contact P=nα applies during impact.Page 249 At the instant of impact when the impactor rebounds:
After impact if the mass is in harmonic free vibration at its natural undamped frequency x=X sin ω and x=Xω t cosω t where X is the amplitude of vibration. v2 are the approach velocities of the impactor and target respectively..
6) and using polar co-ordinate r. the magnitude and distribution of the surface pressure is obtained as: (6. area and duration of contact when a flexible cladding panel is impacted by a spherical impactor at normal incidence and low velocity. is given by the Hertz equation for the area of contact when the load P is static: (6. thickness h.4) The Timoshenko pressure distribution over the area of contact is: (6. Plate bending deflection δ is determined p by the force—deflection relationship: . (6. where kp is the stiffness of the plate and is a function of the elastic constants and the boundary conditions..7) This pressure produces internal stresses to compare with the limiting stresses producing failure modes in the target: (2) Determination of the dynamic force. the radius of the area of contact a. When the panel is flexible then local and overall deformation without punching failure is likely at low velocity impact.
. (1982) present an analytical method for determining the response of isotropic and anisotropic laminated panels impacted by a spherical impactor.3)
(6. Young’s modulus E r and Poisson’s ratio=vr with simply supported boundaries. and summing the pressure over the area of contact and equating this to P: (6. isotropic plate of radius R.Page 250 Assuming that when a sphere impacts a flat surface.6) From eqns (6.5) where q0 =pressure at the centre of the contact area x =y=0. Zukas et al. at the boundary of the contact area x 2/a2+y2 /a2=1.y = 0. with a force P. For a circular.2).4)—(6. qx. The local deformation α the Hertzian contact deformation determined is by the force—deformation relationship: .
16.Page 251
Figure 6. Figures 6.
Assuming impact by a rigid impactor on a stationary plate at an approach velocity v =v1. shows that P increases linearly with v but at a reducing rate with h. whether the plate has simply supported or fixed boundaries. plate thickness h decreases). With fixed boundaries the effective masses will be different and dynamic force is greater at a given impact velocity than with simply supported boundaries. the kinetic energy of impact equals the work done on the plate in local and overall deformation.8) Solving the equation for P at a given embed v with known properties of the impactor and composite plate.15 and 6. For a given impact velocity the dynamic force P and area of contact decrease but the contact duration increases as the target flexibility increases (i.15 Free boundaries.e.
Comparisons were made with the modified NDRC (US National Defense Research Committee) empirical formula (Kennedy. The modified NDRC formulae for a semi-infinite target are: (6. Zukas et al.9a) and (6. 1978).Page 252
Figure 6. respectively. perforation and scabbing of reinforced concrete impacted by flat faced cylindrical missiles was investigated by Barr et al. diameter. and the CEA/EDF (French Atomic Energy Commission) empirical formula (Berriaud et al. mass and velocity. 1976).16 Fixed boundaries. M in kg and v in msec−.4 Empirical formulae for low velocity impact on concrete
6. d in metres.9b)
1 where x in metres. and D=M/d3 is calibre
. are missile penetration depth.3. (1982). (1980) using experimental and analytical techniques..
which also serves as the necessary concrete
. The perforation thickness e can be c obtained from:
The CEA/EDF formula gives the perforation velocity as: (6. The structure was designed to resist quasi-static internal pressures.7m thick. in Pa. Accident load cases for the design of a double skin concrete containment structure surrounding a nuclear reactor pressure vessel. 0.10) where p is the concrete density and the range of applicability is:
flat nosed cylindrical missile.Page 253 density.8m to resist a 20 tonne military aircraft crashing at 215m/sec. took the upper bound as catastrophic failure requiring the evacuation of people living outside the reactor site (Eibl. 1993).
6. The outer wall thickness of the containment was 1.4 DESIGN ACTIONS
Concept definition and resistance requirements specify the design parameters. The NDRC formulae were within 30 per cent of the penetration velocity measured in the experiments for different mass and diameter missiles and thickness and strength of concrete. The inner concrete barrier. The CEA/EDF formulae were within 100 per cent of the penetration velocity but the reinforcement and concrete strengths used in the experiments were outside the range of validity. fast dynamic external forces and high temperature from the heat generated in a core melt accident. was designed as a fragment shield against high velocity missiles from bursting plant and equipment and from an assumed hydrogen detonation pressure wave. and extreme loads set the upper bound actions for the ultimate accidental limit state. The inner concrete barrier has an outer metal plate. fast dynamic internal pressures. and σ is concrete compressive strength.
giving the ultimate resistance q (kN/m 2) as: (6. and whether or not the glass had been broken. respectively. a survey of the surrounding damage to windows. The calculated resistance q (kN/m2) for each window. the distance from the explosion. because of the unknowns. 1987. The design method utilized the strain rate sensitivity of the concrete in a Hugoniot curve of hydrostatic pressure against volumetric concrete strain. =a Because of the variability in the strength of glass. d.17. Figure 6. for example. traffic signs and lamp posts. is plotted against r. Analysis of the Ronan Point gas explosion produced the requirement to analyse key elements for 35kN/m2 equivalent static loading.Page 254 reinforcement of this composite structural member. (1976) estimated the overpressure—distance curve from observations of the damage to brickwork and concrete structures. dimensions. Sadee et al. β function of the side lengths L. Windows possibly broken by effects other than overpressure are identified but not used. To obtain a statistical base for characteristic blast pressures the damage to structural elements from an explosion can be back analysed. showing whether or not the glass was broken. This smoothed curve has no broken windows above it if a 50 per cent reduction is made on the theoretical resistance of the broken windows and there are only 10 unbroken windows below it if the 50 per cent tolerance is used. Eyewitness accounts indicated that window panes might have broken either inward or outward. A lower bound estimate of peak overpressure is plotted in Figure 6. thickness and ultimate tensile strength of the glass. Upper and lower limits for this blast overpressure are indicated as (UL) and (LL).
.11) where f kb=ultimate tensile strength of glass. sophisticated analytical techniques are not justified. assuming simple supports on all four sides and uniform pressure on the pane. The resistance of glass to blast pressure depends upon the edge conditions. and the distance travelled by debris. b.17. The resistance of broken and unbroken panes gives an estimate for the blast overpressure at various distances assuming normal incidence. After a gas explosion that destroyed a building in Peterborough. In many cases. and in the degree of fixity to the frame. b= thickness (mm) and short side length (mm). A survey was made of the frame dimensions and glass thickness for all the windows exposed to the direct blast wave. Dragosavic (1973) analysed a rectangular pane of glass. The 1974 Flixborough vapour cloud explosion damaged many structures and from an analysis of lamp posts Roberts and Pritchard (1982) estimated the dynamic pressure produced by the explosion. the calculated results probably do not predict the actual ultimate resistance by better than ±50 per cent (Mainstone. those that could have been broken by flying debris. was used to estimate the characteristics of the explosion (Watson 1994). 1971). assumed to be 84kN/mm2.
Figure 6.17 Explosive peak over pressure estimated from a windo survey.
17.17. T=0. When Biggs’ analysis is applied to an undamaged post at 16 m using a peak pressure of 30 kN/m 2 extrapolated from Figure 6. They claim that current codes specify impact factors that may significantly underestimate the beam response. and using P m=7. The peak pressure predicted is sensitive to the assumed shape of the pressure pulse.4. A is area subjected to the blast pressure (m 2).18 sec. pm is peak blast pressure (kN/m2).1 Idealization of high rate dynamic loads
Design loads or actions are usually computationally manageable idealizations such as the pressure—time idealization for fragment impact.19 (Eibl. As expected it is less than at 60 m. The debris throw distance was compared to that of TNT using an analysis by Kinney and Graham (1985). This showed that 11 kg of TNT would have thrown debris 100 m and 88 kg TNT would have thrown it 200 m. and by simple measurement. If t d<<T it responds to impulse and to peak pressure if td >T. Using the analysis given by Biggs (1964). Figure 6.12) where R m is maximum elastic resistance (kN). and for a hydrogen pressure wave. The post had no visible damage. Figure 6. 1993).
6.093 sec. for impact on shallow buried plates. have idealized impact loads from vehicles moving over simple and continuous beams using impact formulas. Between these limits it responds to both. Chen and Chen (1996) give a load idealization. the duration of the blast pulse td is 0. None had any damage that could be attributed to the explosion and the analysis would therefore be an upper bound estimate of the pressure. The overpressures produced by these quantities of TNT at different ranges are plotted in Figure 6. The building at the centre of the explosion was completely destroyed.18.17. are the peak pressure. Eyewitness accounts and press photographs indicated that debris from the exploded building was thrown up to 200 m from the centre of the explosion. The main characteristics of an idealized impulsive load such as those produced by impact or blast loading. The response to blast pressure depends upon the duration of the blast t d relative to the fundamental natural period of vibration of the post. gives td=0. rise and decay functions and the
. The duration t d was estimated by assuming a triangular pressure time curve with peak pressure pm.5 kN/m 2 from Figure 6. and are very sensitive to range at less than 25 m. Yang and Yau (1997). T is natural period (sec). and assuming a linear resistance—deflection curve: (6.4 sec. indicating that it had not exceeded the elastic limit. Analysing a post at 60 m from the explosion for first mode deformation. If the pulse had a rise time of 16 per cent of the decay time then pm is calculated to be 150 kN/m2 which fits reasonably well with the extrapolated peak overpressure line from the window survey.Page 256 Metal posts close to the explosion provided simple elements for analysis.
for example Craig (1981) have used this as a useful model to show how load and structural parameters interact. can be identified. The response of the structure then depends on how these load parameters relate to the parameters chosen to model the structure. For high rates of loading an undamped springs-mass system is frequently chosen to model the structure and several authors.Page 257
Figure 6.19 Idealized hydrogen detonation pressure.
Figure 6. whether it is local strain or overall structural deflection.
total duration.18 Fragment impact pressure. A potentially damaging condition imposed on a structure by any form of loading. is displacement. which will displace the most. then a Single Degree Of Freedom (SDOF) model can be constructed as an equivalent
. If that part of the structure.
20).22). After several cycles of vibration where the maximum dynamic deflection has reached twice the static deflection (Figure 6. If td≤ n/2 then R(t)<2 in T T the range 0≤t ≤td. 1981).21). This sets the SDOF system into elastic vibration and the dynamic deflection u(t) varies with R(t) as the mass overshoots the equilibrium position.4. A rectangular pulse load with t r → 0 but sets the SDOF t d<<∞system into forced vibration over the time td. the maximum value of R(t) is similar for both the damped and undamped systems and so with only a small error. 2 If t d ≥ n/2 the maximum deflection occurs at t=Tn/2 when R(t)=2.23).Page 258 structure where displacement at the critical point on the structure is given by the extension of the spring.20 Ideal step load.2 Influence of load characteristics on the response of an elastic spring-mass SDOF system
The peak transient deflection of a structure under a specified loading condition is the same as the equivalent SDOF model that can be more easily analysed (Craig. an undamped system can be used to determine R(t)max which occurs at t=Tn/2 where is the undamped natural period. producing an ideal step load with rise time tr→ 0 and duration of load td → ∞(Figure 6. To ease computation the model may be undamped because damping does not significantly alter the magnitude of the first peak of oscillation and it is this deflection which determines whether or not the structure survives extreme loads from explosions or impact. the damped SDOF system comes to rest at R(t)=1. On the first cycle. When t > t d then the mass is in
6. The result gives a response function or dynamic load factor:
For instance the sudden release of the mass m in an SDOF system causes a force P 0=mg to act on the unstretched spring of stiffness k. and it then continues in free vibration (Figure 6.
. The deflection of the mass during the forced vibration 0≤ td is the same as that for an t≤ ideal step load and 0≤ R(t)≤ (Figure 6.
Figure 6.23 Maximum response ratio for a rectangular pulse load.21 Response ratio for ideal step load.Page 259
Figure 6.22 Rectangular pulse load.
free vibration and Craig’s analysis shows that the maximum deflection is given by R(t)max when the SDOF system is undamped.
plastic and recovery deflection for an ideal step load t r → 0. showing the maximum deflection of an elastoplastic undamped SDOF system for a step load . If. an ideal step input). however.5 deflection). the maximum dynamic deflection exceeds the static deflection). the overshoot reduces and small oscillations occur about R(t)=1 . the spring in the SDOF system is taken to be linearly elastic. If the system survives the plastic deflection then it will partially recover and vibrate about a residual deflected position. deflection um will have to satisfy the limiting ductility factor um/u 1.Page 260
Figure 6. as tr/Tn increases. or in the forced td vibration era when td/T n ≥ maximum dynamic deflection reaches twice the static 0.26.4. When this acts on an undamped SDOF system Craig’s analysis shows that the deflection of the mass depends upon the duration of the rise time and the ratio of rise time to the natural period of the system Tn.3 Elastoplastic response of an SDOF system
In all the cases considered above.
R(t) max ≥1 during the free vibration when . and if tr>3Tn then load can be treated as static and dynamic effects ignored.24.17 of this book. and during the forced vibration when t d/Tn ≥ 0. Design charts have been produced from Biggs’ solution.
6. This elastoplastic undamped SDOF system has been analysed by Biggs (1964) to obtain the elastic. the load causes the structure to become plastic.e. then it can be modelled using an undamped SDOF system with an elastoplastic spring with the stiffness function shown in Figure 6.e. one example is given in Figure 2.24 Ramp load. A ramp load of finite rise time t r and duration td → ∞is shown in Figure 6. td → ∞ The maximum .25 (i.
.e. R(t)max=2 in the free vibration era when sin(π /T n)= 1 (i. t d/Tn=0. From Figure 6.25 R(t)max=2 when tr=0 (i.5).
Figure 6.26 Elastoplastic spring stiffness function.Page 261
.25 Response ratio for a ramp load.
alters the constitutive stress— strain relationship for many construction materials and produces strain waves causing local concentrations of stress. These analytical methods can be independent of the rate of loading but high rate loading accelerates the mass of the structure producing inertia forces.5.4). The following section reviews some of the analytical methods available in the literature of dynamic design.
6. Vibrations of this system are damped by an energy absorbing function that relates to the damping in the real structure.2 Methods of design for extreme dynamic loads
6. and assumes that only the first peak amplitude of vibration is significant and that damping does not essentially alter this value. the single mass of the model must be equivalent to the distribution of acceleration on the full mass of the real structure and the resistance of the spring in the model depends upon the stiffness of the real structure for the particular load arrangement.5. there are no inertia forces. This only gives deflection at the centre of loading on the structure and so implies a deformed shape.2. The number of springs equals the number of degrees of freedom the designer has to consider to accurately define the modes of deformation.5. For analysis under dynamic load the real continuous structure is often converted into an equivalent spring—mass system with lumped masses supported by elastic or elastoplastic springs that model the resistance—displacement function of the real structure. For blast and impact design a SDOF is often assumed for the structural element which is then modelled as an undamped system of a single lumped mass on a spring.5 DESIGNED RESPONSE
6.1 Elastic impact factor method This method for concept design assumes structural resistance is that of a massless linear elastic spring of stiffness k. the force-deflection relationship is linear for both static and dynamic loads and energy is conserved.Page 262
6. Since mass relates acceleration to inertia force. A variety of analytical methods have been proposed to predict this structural response and so allow comparisons to be made with the performance limits chosen by the designer for deflection or rotation. including some continuous mass models. produce a response in the structure which depends on how the parameters of the load and the structure relate to each other (see Section 6. A static force W produces deflection us=W/k and impact force F produces a deflection
.1 Principles of design for high rate dynamic loads
Extreme actions due to high rate loading such as come from impact or blast.
When checking whether the structure can survive the first cycle of response to extreme blast or impact loading. 6. (ii) If W moves horizontally and impacts at velocity v then and by substituting for F:
where the impact factor In concept design F is used as an equivalent static load and ud is checked against the ductility ratio.27).Page 263 (i) If force F is produced by mass M falling from height h and static force W=Mg then by conservation of energy of the falling mass and assuming that all KE is transferred into strain energy:
by substituting for F and solving the quadratic equation:
where is an impact factor on deflection and force.5. damping of the SDOF system is often neglected.2 Equivalent systems A more rigorous design converts the structure into an equivalent spring—mass system with lumped masses supported by elastic or elastoplastic springs that model the resistance— displacement function of the real structure. Consider the response of a simply supported beam under a time varying load/unit length of beam w(t) and assume that the deflected shape of the beam is the same as produced by the static application of the load (Baker et al..
for y 0=maximum displacement at midspan and ymax=(5wL4)/(384EI):
. 1983) (Figure 6.2. The number of springs equals the number of degrees of freedom the designer has to consider and although most structural elements have a large number of degrees of freedom their response to dynamic loading can be approximated by a single degree of freedom equivalent system.
then for equivalence y0 is equal in the beam and the model.28.
If the beam is to be represented by an undamped SDOF model with a massless linear spring.28 Equivalent SDOF model.
Figure 6.Page 264
Figure 6.27 Elastic deformation. Figure 6.
29 Plastic deformation.Page 265 External WD=Wy0. A considerable resistance is obtained from the plastic behaviour. Internal strain energy . (1983) have shown that these transformation factors do not change significantly if the beam deforms to the first mode shape thus the higher modes contained in the static deformed shape can be neglected. for several beams and slabs with different support and loading conditions. U and KE
These factors transform an elastic structure into an equivalent SDOF model and are derived and listed by Biggs (1964).
. for the beam and model gives: equating WD.29):
Figure 6. 1983) (Figure 6. If the simply supported beam under a time varying uniformly distributed load w(t) has a plastic hinge formed at midspan then the elastic deformation can be neglected by assuming that the beam has a rigid plastic resistance deformation function (Baker et al. In resisting extreme actions from impact and blast it is uneconomic to design the structure using only the elastic resistance.. Baker et al.
Strain energy U=Ryo for the equivalent plastic SDOF system where R=the plastic resistance of the spring (i. only strain energy differs from the equivalent elastic SDOF. (1983). Equivalence between the structure and the SDOF system is based on deflection. the moment of elemental inertia forces about the left-hand
. yield force).Page 266
For the equivalent system with plastic behaviour. uses the model shown in Figure 6. Km change when yielding occurs.e. Using a free body diagram. not force or stress and dynamic reactions are not given by the spring force.30. for a simply supported beam under a time varying UDL=w(t). for the elastic deflection where a=distance from LH support to the point of action of the resultant of the inertia force:
Note that KL. An analysis by Baker et al.
2.30 Dynamic reaction model. 6.3 Structural response diagrams The maximum deflection of an equivalent SDOF system loaded by blast or impact can be obtained from the solution to the equation of motion. V max occurs when wL is max.5.
support is equated to the moment of the resultant inertia force
For the elastic deformed shape under static load
and the dynamic reaction when the beam remains elastic up to the maximum load is:
When yielding occurs at the maximum load then assuming the beam still has its elastic shape curve and substituting Mp for Mx=L/2
that is.Page 267
Figure 6. Such response
32 for an air blast wave. to a sinusoidal forcing function is plotted in Figure 6.Page 268
Figure 6. P(t)=P0e–t/T. stiffness k.
diagrams can be replotted as pressure-impulse diagrams and used when it is a limit state and not the time history of the structure under transient loads. linear elastic SDOF system.
Figure 6.31 on axes:
where R=Response ratio=dynamic load factor and circular frequency Using Baker’s mathematical approximation Figure 6. t=0. where since e– t/T never reaches zero. that is of interest.32 Approximate air blast wave.
. ÿ=0. For example. T is used as an equivalent duration of loading to solve t/T the equation of motion mÿ+ky=P0 e − for boundary conditions y=0. mass m.31 Response ratio for a sinusoidal forcing function. the response of an undamped.
1983). Quasi-static region when and the structural response y max= (P 0/k) depends on the peak load P 0 and stiffness k. Impulsive region when asymptote:
and the displacement is given by the impulsive where P0 T=Impulse I therefore
is directly proportional to I=P 0T and also depends on the stiffness k and the mass m.33 Response ratio for approximate air blast wave (Baker et al. T T Since two straight line asymptotes can be used to approximate the analytical solution. Transition region when and the displacement
.Page 269
Figure 6.. Baker identifies three different loading regions: 1.
using dy/dt=0. but not on the mass or duration T. the time ωmax at which y(t)=y max is found by trial and error for specific values t of ω and the analytical solution is plotted giving ymax /(P0/k) as a function of ω (Figure 6.33). the applied load P(t) dissipates very little before ymax is reached and the displacement is given by the quasi-static asymptote
2. The applied load drops to 0 before ymax is reached since the duration of load 3. Since the duration .
Figure 6.34 by manipulation of the asymptotes. The load and response time is of the same order of magnitude and ymax depends on the entire loading history since the duration of load Figure 6.
ymax/(P0/k) depends on P0. I which produce the damaging deformation limit ymax in an undamped linear elastic SDOF specified by k and m. For an undamped linear elastic SDOF system the ordinale and abscissa are respectively:
The rectangular hyperbolic curve of the P—I diagram is an isodamage curve defining critical combinations of P0. I. m.34 Response to the pressure-impulse of an air blast wave. equivalent to a specific structure:
.33 is converted into the pressure—impulse (P—I) diagram of Figure 6. k.
●changes in I. but not changes in P 0. The rigid—plastic SDOF system shown in Figure 6. Damage that occurs at a
.35 Response of a rigid—plastic system. An energy balance gives the equation to the quasi-static and impulse asymptotes.4 Isodamage curves Structural response to transient load can be impulsive or quasi-static. as:
6. will move (P0.Page 271
Figure 6.35 uses a Coulomb friction element to model the structural retarding force f where the deformation y=0 if P 0≤ f. ●only changes in P0 will move (P0.2.
●deformation will be larger than the threshold damage if (P0.5. and P− diagrams can be I used for different levels of damage. I) combinations off the quasi-static asymptote. I) combinations off the impulsive asymptote. I) moves to the region above and to the right of the curve. respectively.
1 Energy absorbing crush-up materials
Shock attenuating materials are used to reduce both the peak pressure and the impulse transmitted to structures from air or ground borne shock waves. for example. (2) Plot curves of distance v explosive charge weight using pressure and impulse values from a table of air blast parameters such as the curves for TNT hemi-spherical surface blast from Kingery and Bulmash (1984). An external Shock Mitigating (ESM) system described by Muszynski and Rochefort (1993) uses empty plastic bottles to confine and entrap air in a low-
. which produces the specified mode of damage. External shock mitigating materials must have a low compressive strength with a high compressibility and energy absorption.Page 272 specific displacement ymax can be caused by impulse or by peak pressure and the rectangular hyperbolic curve is then an isodamage contour. So also are materials with a plasto-elastic stress-strain curve such as foamed rubber. When the materials are applied to the external surfaces it is known as ‘backpacking’ and stress waves are attenuated before reaching the structure. so reducing its cost. expanded clay. for ymax=200 mm the impulsive asymptote:
(4) Draw the quasi-static asymptote of the isodamage curve for the mode of damage occurring at ymax=200 mm:
(5) If the transition curve is omitted. Using the following procedure a P− diagram can be constructed for an equivalent elastic I SDOF system where k and m are known. shale and slag. P 0) combination. for example:
(1) On log-log paper mark the ordinate as reflected pressure P0(kPa) and the abscissa as the reflected impulse I (kPa-msec). The strength and thickness of the structure can then be reduced. (3) Draw the impulsive asymptote of the isodamage curve for the mode of damage occurring at ymax. polyurethane foams and cellular concrete are used as shock mitigators. Materials with an elastoplastic stress-strain curve such as rigid polystyrene.6.6 DAMAGE MITIGATION
6. the asymptotes then give a conservative estimate of the (I.
Under X2 loading the Pressure—Crush per cent curve is ideal elastoplastic. however. Static compression tests on these materials demonstrate the importance of confinement on the stress—strain curve.5 per cent. Both ESM systems reduced the transmitted stress on the wall by about 90 per cent of the peak free field stress. there will be a reduction in the kinetic energy transferred to the wall and damage should be reduced. normalized deformation of high density metal honeycombs at initial strain rates from quasi-static to 2. was increased in the impulsive range (i. Under X 1 loading the mean pressure was 0. In the X3 out of plane. when the positive duration of the blast was much less than the natural period of the wall). The minor cell diameters were 4. before testing. axial loading tests there was a 40 per cent difference in the mean crushing pressure between static and impact loading. lateral loading. damage to the wall was increased when an ESM cellular steel panel was used that absorbed energy through plastic crushing of the cells. and the mean pressure is close to the static value. Damage. Another system uses epoxied hollow ceramic beads to form foam. Hulton and MacKenzie (1998) use a qualitative energy analysis to explain why some ESM systems apparently increase the vulnerability of a wall in some circumstances.4 MPa and the Pressure-Crush per cent curve was ideal elastoplastic for all impact loads. 10. and S the minor cell diameter: (6.000/sec has been
. respectively. A dynamic test was performed using ground shock effects of close-in detonations against a concrete basement wall. Experi-mentally it was observed that the ESM reduced the reflected pressure on the wall from the blast wave. 28m/sec.e.7mm and 6. If this corresponds with a reduction in the impulse on the wall.2mm and densities were 130 kg/m3 and 100 kg/m3. Considering the overall energy it was argued that a reduction in the reflected impulse implies a reduction in the energy reflected from the wall which implies an increase in the energy absorbed by the wall and an increase in damage as observed. Zhao and Gary (1998) tested two types of aluminium honeycombs in the three orthogonal directions X1.13) The observed enhancement of the crushing strength is likely to depend more on structural inertia and less on the strain rate sensitivity of aluminium foils. It was concluded from experiments using high explosive air blast against wall panels of lightly reinforced concrete. The mean impact crushing pressure was constant at 5. Wierzbicki (1983) gives the mean crushing pressure pm as a function of the flow stress of honeycomb foils. The impulse was reduced by at least 12. X2 and X3. but for each direction the failure mode was different. h the cell wall thickness. There were no visible differences between the static and impact loaded honeycombs for X 1 and X2 in-plane.09 MPa at all rates of loading and the Presssure-Crush per cent curves were ideal plasto-elastic curves. The specific energy absorbed (J/cm 3) vs.Page 273 density cementitious matrix. under static loading and at impact velocities of 2. that when response was in the impulsive range.
The honeycombs were aluminium with a density 32 per cent that of solid aluminium and stainless steel with a density 37 per cent that of solid steel. Strain at peak stress decreased with increasing strain rate and there was no increase in energy absorption for any of these materials at high strain rate but the honeycomb structure has considerable potential for energy absorption. During the dynamic internal inversion of metal tubes of uniform thickness. increasing the initial crushing stress and the plateau stress. Peaks in the crushing force of ESM systems increase the shock on the structure. The dynamic force pulse was measured. at strain rates of approximately 230/sec. Harrigan et al. It was observed that deformation was distributed more uniformly along the length of the quasi-static specimen but propagated from the impact end in the dynamic test. The plateau stress was the same for crushed and uncrushed specimens at approximately the initial peak stress of the pre-crushed specimens.40 and 1. did not show a distinct difference in the permanent deformation between the quasi-static and impact tests. Strain rate effects were observed on stress in both metals. The compressive dynamic/static strength ratio for the materials tested in a split Hopkinson bar. To assess the dynamic properties of aluminium honeycombs. giving the initial peak stress and the energy absorption characteristics. however. is not expected to be strain rate dependent and the rate effect observed in these tests was considered to be solely caused by a change in collapse mode as the strain rate increased.36. both experimentally and computationally.72 for the surlyn. All specimens had a locking displacement and the specific energy
. Sierakowski and Ross (1993) demonstrated that the properties of novel thermoplastic honeycomb structures manufactured from high impact polystyrene. Aluminium type 5052. The stainless steel honeycomb is likely to have combined material and collapse rate effects. (1998) demonstrated the inertia effects in the performance of energy absorbing materials and structures. The longitudinal wave speed in the polycarbonate was measured to be 500 m/sec. (1998). Some specimens were precrushed to initiate inelastic deformation and some were uncrushed.47 for the polycarbonate and polystyrene and was 3. cylindrical specimens on the end of an instrumented Hopkinson pressure bar were impacted at velocities of 20 m/sec up to 300 m/sec. inertia makes the crushing stress sensitive to impact velocity and modifies the crushing mechanisms at the cell wall. The static load displacement characteristics of the aluminium honeycomb specimens are given in Figure 6. Post-test inspection of the aluminium honeycomb. For cellular aluminium honeycombs. polycarbonate and surlyn. The quasi-static uniaxial load-displacement curves are elastoplastic with an initial peak stress for the pre-crushed specimens typically 60 per cent of that for the initially uncrushed specimens. were strain rate sensitive. was between 1.Page 274 obtained experimentally by Baker et al. The stainless steel honeycomb absorbed almost double the energy absorbed by the aluminium honeycomb at similar deformation. inertia produced an initial peak force in excess of the steady state force and reduced the steady state force compared with its quasi-static value.
Density and strain rate dependence influence the yield or peak initial stress. Polyurethane with density between 0. The ground shock was determined by measuring the flexural stress in the beam at midspan.16 and 0. (1991) tested ESM systems for underground structures by impacting the surface of sand or bentonite clay covering a beam. This was protected from the subsurface shock by an ESM system of either a paper honeycomb of three different strengths. dynamic yield and plateau stress from 1. If lockup occurs before sufficient energy has been absorbed.5 to 2. A reduction of 10 to 30 per cent was discerned at strain rates above 10 3/sec. These experiments have shown that there is a tendency for the lock-up strain to decrease as strain rate increases. It is beneficial that the energy absorption increases significantly with increasing impact velocity.36 The static load-displacement characteristics of an aluminium honeycomb.48 g/cc tested at strain rates from 2×103 to 3×10 3/sec had. The static and dynamic compressive stress—strain curves had the characteristic elastoplastic compaction shape. The strain rate sensitivity of polyurethane foams of different density has been reported by Kuennen and Ross (1991). the plateau stress and the lock-up or compaction strain in polyurethane foams and can have important consequences when they are used for ESM systems.
absorption capacity was determined from the area under the load-displacement curves.Page 275
Figure 6. respectively. The ratio of dynamic to static initial peak crushing stress and plateau stress. The
. Fujimoto et al. or polyethylene foam. Beneficial effects occur when the changes increase the area under the stress strain curve so increasing the energy absorption capacity. increase significantly with impact velocity and would need to be evaluated when designing an ESM system.0 times greater than the static stress at 15 per cent strain. then the ESM system may act to increase the shock loading on the structure. so increasing the transmitted load. The effects are detrimental if lock-up occurs.
It is through the codes of practice and building regulations that Governments bring about changes. Anderson et al. Damage resulting from the failure of an element should not spread disproportionately. bridges and public utilities. Krauthammer et al.7. should be designed for the extreme loading. This section deals only with the recommendations provided in standard codes of practice to improve the robustness of constructed facilities.
.Page 276 static compressive strength—displacement curves for the paper honeycomb was elastoplastic with no compaction curve. Without an effective ESM the beam would need much greater hardness or depth of earth cover to limit the stress in the beam. must be designed to resist extreme loads and if this is impractical. A surface layer with polystyrene beads replacing some of the aggregate. Anderson et al.7 DESIGN CODES
6. was easily perforated although it did reduce the concrete crater. They can be regarded as a minimum requirement and it is possible that no other action needs to be taken against extreme loads when there is only a remote possibility of them occurring. (1995) tested concrete slabs by impact from a 7. but was elastic with a compaction curve. soft materials that mitigate the ground shock will be ineffective in stopping the fragments. changes to improve the robustness of buildings were introduced by an amendment to the Building Regulations 1970 and then to Codes of Practice from 1972.1 General principles of design codes for dynamic loading
Codes of Practice have an important function of reassurance to the general public. a robustness that Ronan Point clearly did not possess.7 MPa to 2. If a subsurface explosion produces fragments. (1992). After the Ronan Point collapse in 1968.
6. for the polyethylene foam. researcher and manufacturer and give legislative control over the standards of the building and construction industry. Most successful was the use of slurry infiltrated concrete (SIFCON) cast as a surface layer on a concrete slab with conventional aggregates. Vulnerable key elements and subframes.75 MPa from an underground explosion. They are the interface between the designer. (1995) reported that a trench filled with soft material reduced the peak stress from 5. The paper honeycomb was the most successful ESM and reduced the stress at midspan to about 30 per cent of the value with no ESM. probably because it compacted early in the deformation. or would have to provide essential recovery services such as hospitals. Shock mitigating systems do not have to be applied to the external surface of an underground structure to reduce the damaging effects of ground shock. then the structure must be provided with ways to limit the collapse following the failure of the element. Facilities which are at risk.62mm armour piercing bullet. The Polyethylene gave hardly any stress reduction in the beam. constructor.
A flow chart of the design procedure for ensuring robustness and the empirical design of ties using partial load factors on loads and material properties is given in Part 2 of the code. combined with dead load and a proportion of the wind and live loads.
.2 Accidental and extreme loading
BS8110: Part 1:1997. This load is applied simultaneously at each floor or roof level and at each level has the value of 1. ●a change in the mode of flexural and longitudinal deformation.7. The load factor takes into account possible increases in load. and shock loading on prestressed concrete beams which could include impact.0 for steel reinforcement. and so does not alter the deformed shape of the structure from that under normal design loads. A structure is robust if it is not disproportionately susceptible to the effects of accidental loading including unexpected and extreme dynamic loading. Dynamic effects include wind loads and vibration. This notional load acts like wind loading or an equivalent horizontal force as a result of frame sway. the steel and concrete yield strength will be enhanced and possibly offset uncertainty in the design load.Page 277
6. ●a change in the material and structural properties with rate of loading. The designer could change the γ factors on load to achieve the same effect.3 for concrete in flexure and 1. These dynamic effects can include: ●support reactions in the same direction as the shock load. Part2:1985 These Codes of Practice for the structural use of concrete recommend ways to improve robustness and structural integrity. ●rapid changes in the distribution of the load across the structural element. ●shock loads opposite in direction to gravity or wind loads. special hazards for flourmills and chemical plants which could include explosions. Structural integrity depends on having a load path for all loads including accidental.5 per cent of the characteristic dead weight of the structure to midheight of the storey above and below. are less than those used in design for the standard static loads. The codes do not relate robustness or structural integrity to any particular loading and a concrete structure of more than four storeys is assumed to have adequate robustness against a general array of accidental actions. reinforcing bars tie the structure together to resist a notional horizontal load. Extreme accidental loads might sever the load path and so redundancy is needed to provide alternative load paths. To make it more robust. If the loads are applied at a rate of straining above 10/sec.05 and materials γ m=1. For these accidental load cases the partial safety factors for loads γ f=1. although the uncertainty of the load and of the material strength is unlikely to be less for accidental load cases. Dynamic loading from impact and explosions produces a different response to that from the normal design loads of gravity and wind. to transfer them from the point of application to a foundation and the ground. and the effects of exceptional loads caused by misuse or accident.
calculated from the total characteristic loads applied to the floor. Flexural reinforcement may also be used to complement the tie steel. Under extreme loading from impact or explosions the serviceability limit states would be exceeded and the feasibility of carrying out repairs would have to be assessed. A wall is able to provide lateral restraint if it is capable of resisting the peripheral tie force F t kN applied horizontally on each metre height of wall. To provide continuous ties in precast concrete construction. The horizontal tie must have a tensile resistance based on the number of storeys. and is constant at 60 kN for buildings above ten storeys and continuous around the edge of the building. must be designed to bridge the increased span when a supporting vertical element on the storey below. the floor to ceiling height or the total design load in the column or wall. They resist the force in the internal ties to provide them with adequate anchorage. and on any horizontal member providing essential lateral support to the key member. In buildings over four storeys.Page 278 Internal ties are reinforcing bars spread evenly within the floor slabs or grouped in walls or beams. Curtailment of flexural longitudinal reinforcement to match the bending moment diagram must not reduce the tie steel at any section. and must be continuous and well anchored. Columns and walls around the perimeter of a building are to be anchored to the structure at each floor and roof level. supported by the key element. is lapped or spliced with anchored bars from the supports. Catenary action may be utilized for this when the necessary horizontal reactions are provided at the adjacent supports. part of the length of each bar in an element. The tie cross sectional area is determined from a prescribed tensile force. An allowance can be made for the strength of the attached element and its connection. Corner columns must be tied in two approximately orthogonal directions at each storey. During construction the risk of imposing extreme loads is present when lifting elements by crane and in collapse of the partially constructed frame. The overall stability of the building
. Key elements in buildings over four storeys are designed for an ultimate load of 34 kN/m2 applied from any direction on the projected area of the member. Peripheral ties are designed to resist a nominal tensile force Ft. The flexural steel is assumed to be acting at its design strength and the tie steel at its characteristic strength. the number of storeys in the structure and the effective span of the slab in the direction of the tie. The benefit of having this reinforcement is clear but it can be omitted if the peripheral tie is within an external wall and horizontal tying anchors the internal ties to the peripheral ties. and on any attached element such as a cladding panel. that depends only upon the number of storeys for buildings up to ten storeys. Vertical ties in each column or wall must be continuous from the lowest to the highest level. Such connections are topped with in situ concrete and the static load in the tie gives the bond or the bearing stress. is removed. beam or slab elements supporting the maximum design ultimate dead and imposed load.
construction and use.25 wf sr La. 0.1: General rules and rules for buildings This recent code is more explicit than many of the BS codes in requiring that a structure shall resist explosions. and avoid disproportionate damage.
. ●tie the structure together. The general recommendations for structural integrity are very similar to those in BS8110:1997 for reinforced concrete buildings. the greater of these or 75 kN at floors and 40 kN at the roof. and La the greatest distance in the direction of the tie between centres of adjacent lines of support. To limit the potential damage. Internal ties. ●select a structural form which has low sensitivity to the hazards. ENV 1993–1–1:1992 Eurocode 3: Design of Steel Structures Part 1. These requirements are met by choosing suitable materials. impact. These ties must have a design tensile resistance not less than 75 kN at floors and 40 kN at the roof. designers could: ●avoid. The force must be: ●for internal ties. there must be not more than 70m 2 or 15 per cent of the area of the storey that collapses under persistent floor loads factored by the ψfactors. edge ties and peripheral ties are all required at each principal floor and roof level to localize accidental damage. or the consequence of human errors. ●for peripheral ties. 0. ●column lengths must have splices with a design tensile resistance not less than of the design vertical load applied to the column from the floor below the splice. by appropriate design and detailing and by specifying control procedures for their production. EC3 differs from BS8110 in specifying that if a single column is removed. Ties in the floors of multi-storey buildings depend upon wf the total design load on the slab. ●for edge ties. The National Application Document for use in the UK with ENV 1993–1–1:1992. st the mean transverse spacing of the ties.5 wf st La. column ties.Page 279 during construction or after accidental damage is likely to depend on these connections. ●select a structural form and design to survive the accidental removal of an individual element. or 1 per cent of the design vertical load in the column at that level. eliminate or reduce the hazards to which the structure is exposed. states that substantial permanent deformation of members and connections is acceptable in achieving these requirements.
Page 280 Key elements are those supporting more than 70 m2 or 15 per cent of the area of the storey. The values γ . To neglect this increase may be conservative for checking the response of the element but not for checking the response of its connections. The design material property is X d=Xk /γ. Actions are classified by their variation in time and impact and blast. This accidental load is assumed to act in combination with the dead and imposed loads using a combination factor. γ and Gj GA. the direction and position of the accidental actions may be different from the permanent and variable actions arising from the normal use of the structure. which allow for variability in a sample from variations in manufacture. j. in which case A d=0. The use of two characteristic values is useful when an accidental action is dynamic with a variable rate of loading. applied in the appropriate directions. Structures under smaller impact loads applied cyclically. The factors ψ and ψ give the quasi-permanent fraction of the variable actions Qk. where γ is the partial safety factor. the main variable action and any accompanying variable actions respectively. Many materials used in construction have an increase in strength and stiffness at high strain rates. When considering an accidental situation. Material properties are characteristic values.05. Qk. These elements must not fail when loaded by an accidental load AK not less than 34 kN/m2 factored by γ A=1. Characteristic values for impact and blast are most likely to be specified by the client and the designer but need to satisfy the minimum AK=34 kN/m 2 specified by the code. EC3 is a limit state code and structural elements must be designed for ultimate limit states when subjected to impact or blast defined as accidental actions.I .I design value can be used after an accidental event for checking the remaining design capacity.I are the characteristic values of the permanent actions.j γ are the partial safety factors for the permanent actions. must be checked from first principles for the serviceability limit state of vibration. The design value of the accidental action is Ad=γAk where γ is the partial safety factor for accidental A A actions. The value of A K is not limited to 34 kN/m 2 but depends upon the importance of the key element and the consequences of failure. design situations and the variable action respectively and are taken as equal to 1. This 1. can often be treated as an equivalent static load if the duration is long relative to the natural period of the structural element.
.1 and Qk.0. the permanent actions in accidental QI.1 2. and their essential lateral restraining elements. for instance by some industrial machinery. M M The design requirement for safety of a structure is that the ultimate limit state design capacity is at least equal to the accidental actions:
where Gk. The reactions from other building components attached to the key element and subjected to AK must also be included but limited by the ultimate strength of the components or connections. or from a close range. Actions are also classified by their spatial variation and this may change rapidly when confined blast pressures are applied. although short relative to other forms of loading.I and Qk.
Accidental explosions are considered to be of the deflagration type from air—gas or air— dust mixtures. The duration is usually longer than that of detonations. during which period it is only approved for provisional application and open for comment. Horizontal static equivalent design forces due to impact of rail traffic on overhead bridges or nearby structures. The bow impact zone is dimensioned above and below the water line but could be altered by the lifting of the bow on impact
. The pressure rise is slow relative to detonations and there is a constant gas pressure phase following the peak. Part 2–7: Actions on structures—Accidental actions due to impact and explosions The three-year period of experimental application for this code began in August 1998. Impact forces on the superstructure of the bridge are not specified. and for heli-copter emergency landing impact when the landing pad is on the roof of a building. These loads are spread over prescribed impact zones. When the consequences are low to medium the static equivalent forces. such as walls and columns. A more advanced analysis is indicated for the most serious consequences. Forces for ship impact are given. then a prescribed horizontal force determined by the clearance is applied to vertical surfaces and a force inclined at 10° to the horizontal acts on the underside of the bridge over the traffic lane. are also specified parallel and perpendicular to the track direction. The maxima are 10. Impact and explosive actions are categorized in terms of injury and death to people.000 kN to 4. and large economic loss to the community. unacceptable change to the environment.000 kN are specified for accidental actions caused by ship impact.500 kN. A structure is considered to be at risk from an accidental action when the probability of the 4 action exceeds 10 − per year. The nominal forces perpendicular to the direction of normal travel vary from 500 kN to 25 kN for these cases. The necessary statistics are not often available and nominal design values are given for use in practice.000 kN and 3.Page 281 ENV 1991–2–7:1998 Eurocode 1: Basis of design and actions on structures. If it is less. are tabulated for the type of road and vehicle.000 kN in the direction of normal travel for a truck on a motorway and the minimum is 40 kN for a car in a parking garage. The maximum force is 1. or prescriptive design and detailing. Horizontal static equivalent design forces due to vehicle impact on the supporting substructure of a bridge. Accidental impact forces from road or rail traffic under bridges or other structures and from vehicles on the bridge are given for the design of structural elements or their protection systems. The acceptable risk level and seriousness of the consequences has to be determined case by case and by public reaction to the cost and disturbance of installing safety measures and reaction after an accident. can be adopted for design of the structural elements and protective systems. depending on the speed of the train and act on a specified area. Static equivalent design forces from 22. respectively. No horizontal forces need to be considered on overhead elements unless the clearance is less than 6 m.
Chen. D.A. S-En.
6..M. paper given at 2nd International Conference on Structures Under Shock and Impact.M. A. Mannheim. pp.E. paper given at 7th International Symposium on Interaction of the Effects of Munitions with Structures. R R. Typical pressures for air—gas and air—dust deflagrations are given as 1. Carter. Cox. Kulesz. McGraw-Hill.Struct.. Berriaud. p.. Dover Publications.000 kN/m 2 but will depend on the size. et al. 181–8.D. Burton. and Kaminskyj.8 REFERENCES
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