Layered Systems for Absorption of Energy from Impulsive Blast Loading on Structures

Abstract

1. Introduction

Blast and impact loading cases are situations where catastrophic damage can be inflicted upon a target. In the case of structures the most unwanted event is the buildings collapse or progressive collapse before occupants can be safely evacuated. Blast and impact loading can range in severity and in causes, such as terrorist attacks like the Oklahoma City bombing or World Trade Centre plane collision, gas explosion accidents like the Ronin Point apartment disaster or even industrially such as the Piper Alpha disaster. Such attacks and accidents are unlikely but evidently possible, creating an immediate concern for the safety of human life present in such situations.

This paper concerns the application of layered systems for the absorption of energy from blast loading. Blast loading characteristically is described as having a high peak overpressure (pressure above atmospheric) upon the blast waves fluid medium colliding with a target, this high pressure will exponentially decay over time and will only have a very short duration, typically a few milliseconds. A true impulsive load is that of infinite pressure and finite duration. For a impulsive loading case, a high loading force acting so quickly upon a target does not allow the entire target to generate a structural resistance as the molecules on the target surface first to meet the blast wave are the only molecules reacting. Only localised resistance will be capable of being generated as the blast/shock wave propagates through the structure. This impulsive loading is what causes the damage to the structure as the structure has had no time to create a resistance. The damaging effect of impulsive loading leads to the purpose of this investigation. By using specific layered protection surrounding a structure of interest an impulsive load colliding with such layer is transformed so the resulting load upon the structure of interest is now a quasi-static load. A quasi-static load is characterised by a finite pressure and infinite duration.

In reality there is no true impulsive or quasi-static load, the goal of this dissertation is to qualitively understand how different properties of protective layers and different arrangements of multi layered systems can transform a high pressure low duration load to a low pressure long duration load, evidently mitigating the damage of blast effects upon a structure.

A quasi-static load from an explosion/impact is desired as the loading is practically a static loading case of a low pressure, this allows the structure of interest to deform appropriately under static structural mechanics (elastically or plastically) and generate its maximum resistive force to the load. Again, pure quasi-static loading is not feasible but a load history as close to it as possible not only allows the structure of interest sufficient time to generate a resistance, but the load the structure has to resist is a considerably lower pressure than the load impinged upon the protective layer.

In order to find what properties create an effective layer the layered system is transformed into a single-degree-of-freedom (SDOF) model where the mass is a lumped and connected to a spring which represents the stiffness of the layer. The response of the layered system for different combinations of mass and stiffness is to be calculated in order to find the most effective layer

properties to transform a high pressure low duration blast/impact load to a low pressure long duration load that will impinge upon the structure.

2. Theoretical Background

In order to help understand the fundamental principles involved with blast loading and the methodology of this project, the scientific background must be discussed and broken down to clarify the specific scope and goals of this project.

2.1.The Blast Wave

Blast waves are generated from either chemical, physical, or nuclear explosions. Chemical explosions involve the rapid oxidation of combustible fuel molecules that are carbon and hydrogen based, causing a highly exothermic reaction. Hydrocarbon fuels do not contain oxygen and must react with the surrounding oxygen in the air in order to combust. Chemical explosives are designed to be detonated remotely and readily contain oxygen so air is not required. Physical explosions occur due to malfunction or natural reaction of physical objects e.g. catastrophic failure of a cylinder of compressed gas, volcanic eruption, or the violent mixing of liquids at high temperature. The most powerful explosions are nuclear. Enormous energy release can be created from either nuclear fusion or fission. In nuclear fusion light atomic nuclei combine (fused) to form heavier nuclei. In nuclear fission the nuclei of heavy atoms are split into smaller parts.

As the only effective protection/prevention of damage from a nuclear explosion is to be more than 20miles away or to be based deep underground to avoid the blast, this dissertation will only concern the explosions caused by chemical and physical events as their magnitude of destruction is much less severe than nuclear events. With this in mind one should remain aware of the catastrophic damage capable from chemical and physical explosions such as the previously mentioned Oklahoma City Bombing and the Piper Alpha disaster. These explosions are highly dangerous but technological means are available to mitigate their effects.

The fundamental characteristic of a blast load is a sudden large yet short release of energy into the surrounding fluid medium (1)(2). Upon detonation of an air blast explosion a sudden energy release occurs from the exothermic reaction. This reaction results in the explosive becoming a hot expanding gas accelerating outward, compressing the immediate air particles surrounding it. The compressed air particles have a knock on effect to subsequent particles transferring the kinetic and heat energy outward in what is called the blast wave. As the blast wave spherically expands away from the point of detonation the blast wave density decreases, deceleration occurs, and the pressure exponentially decays from the initial high peak overpressure due to the blast energy being absorbed by the surrounding air. The term overpressure refers to a pressure which is above atmospheric. In order for the pressure of the gases to return to equilibrium, over-expansion occurs causing the blast wave pressure to drop below atmospheric creating an under-pressure. The under-pressure cannot be greater in magnitude than normal atmospheric pressure but will usually have a

relatively long duration in comparison to the positive phase of the blast overpressure. A graphical example of a blast pressure exponential decay is shown in Figure 1. The decay behaviour follows the ‘Friedlander’ function (1), which mathematically is:

Equation 2.1

P=Po(1− t−t at d )eα ( t−t a

t d )

Where:

P=Pressure

Po=Peak incident overpressure

t=time

t a=Arrival time

t d=Positive blast phaseduration

α=decay coefficient

Figure 1: Typical exponential 'Friedlander' blast pressure decay.

The example in Figure 1 uses a decay coefficient of -1, has zero arrival time, positive phase duration of 0.001 seconds and peak incident overpressure of 100 MPa. When a blast wave emits outward and collides with a structure, the peak incident overpressure is the initial pressure acting upon the structure that will decay exponentially. The overpressure experienced by a target is relative to the strength of the explosive detonated and the distance from the explosive. The strength of an explosive is measured in mass of trinitrotoluene (TNT) and the overpressure experienced is dependent on the scaled distance from the explosive to the target.

Positive Impulse

Negative Impulse

Z= s3√W

Where:

Z=Scaled distance

s=stand−off ¿ target

W=chargemass

It is important to be aware that blast waves will reflect from and diffract around structures, these reflected waves will combine with the later arriving waves creating a potentially larger overpressure than the incident overpressure. This is most important when dealing with confined explosions and numerous regions of increased overpressure can be generated. For the purpose of this dissertation confined explosions and diffraction effects such as clearing are not considered as they nullify the use of certain assumptions creating a much more complicated loading scenario beyond the scope of this dissertation. Keep in mind that a single blast loading situation in itself is highly complex.

2.2.Dynamic Loading on Structures

In classical static mechanics structural elements are designed to provide static equilibrium of forces by satisfying Newton’s laws. Provided the structural element has a high enough ultimate strength, it is capable of sustaining equilibrium with the applied load. Statically the way the load was applied and the response of the structure over time is not an important factor as they are assumed instantaneous. Realistically the load application and material response are vitally important, for example, if a load were applied purely statically; upon loading a structural element will deform by its molecules being forced to rearrange to a certain degree, generating a resistance that opposes the applied load. This occurs over time and is not an instantaneous structural response. Clearly a structure requires some time to generate a sufficient resistance to a statically applied load.

Molecular interaction is responsible for the structure requiring time to develop a resistance. Upon loading statically the molecules in direct contact with the load are the only molecules that are initially subjected to the force from the load. These molecules will deflect due to the load, deflect the molecules around them, and in turn have a knock on effect to subsequent molecules throughout the structure until all molecules have been subject to the applied load, deform appropriately and generate a resistive force. This knock on effect of molecules is a stress wave propagating through the structure.

Static loading will produce small stress waves whereas dynamic loading can produce various waves: elastic, plastic and blast/shock waves. The severity of the stress wave is mostly dependent upon the velocity of the loading and typically with dynamic loads the magnitude of force is significantly higher than that of static loading. Elastic waves are low velocity cases where a material would undergo deformation within its elastic limit and only brittle materials would develop significant deformations. Plastic waves are medium velocity events creating permanent plastic deformation. Blast waves (as

mentioned previously) are high speed, high force events with blast wave speeds up to 1000m/s resulting in catastrophic damage to materials/structures as the stress wave can be so intense that the material behaviour is analogous to that of a fluid.

A structures immediate resistance to a stress wave is its inertial resistance. Inertia is a body’s resistance to a change in momentum; this is quantified by the mass as according to the force and momentum equations:

F=ma

p=mv

Where:

F=force

m=mass

a=acceleration

p=momentum

v=velocity

A large force is required to make a large mass obtain the same acceleration as compared to a smaller mass or the concept can be seen in terms of momentum where a large mass will provide a larger momentum at a specific velocity in comparison to a smaller mass with the same velocity. Thus a large opposing momentum must act in order to influence such mass.

Structural interaction

2.3. The Equivalent Single-Degree-of-Freedom (SDOF) Model

When structures deform under static loads they poses numerous mode shapes of deformation, when subject to dynamic loads of high magnitude and short durations the material response occurs so rapidly that usually a single mode shape predominates to failure. Real structures poses distributed mass, loading, resistance, specific physical constraints, and specific geometry. Equation 2.2 shows the equation of motion without damping of the real values of a structural system if all the elements of mass of the structure experienced the same force and moved uniformly. It must also be assumed the entire mass were to be acting from the structures centre of gravity. These assumptions used to analyse a real structures response create serious errors in the analysis. In order to eliminate errors the spatially distributed properties of structural elements such as beams, columns, and slabs must be transformed into a lumped model. The specific properties to be lumped are the mass, stiffness, and applied load.

Equation 2.2

m z+kz=F (t)

Where:

F ( t )=load as a function of timet

k=stiffness

m=mass

z=displacement

z=acceleration

Using lumped properties, the equivalent SDOF model can be constructed and the response of such model under a dynamic load can be analysed by using Newton’s equations of motion and by satisfying energy and momentum conservation laws (1)(2)(3)(4). The concept of the method is to have equivalent kinetic energy, the equivalent resistance having equal internal strain energy and the equivalent loading having equal external work as the real distributed system. In order to change from the real spatially distributed properties to the equivalent system, transformation factors are used to change the mass, stiffness and loading. Equation 2.3 is the equivalent equation where the “e” subscript denotes equivalent quantities.

Equation 2.3

me z+k e z=Fe(t )

The equivalent quantities are found using the transformation factors:

me=Kmm

k e=K k k

F e=K pF

These factors are calculated so the work done by the distributed forces acting within the real structure is the same as the work done by the respective single force acting in the equivalent system. The transformation factors are calculated by assuming the structure has a constant mode of deformation described by a non-dimensional shape function. As mentioned earlier, when a structure is subject to high magnitude low duration loads the structure will usually deform to failure in a single predominant mode shape. The deformation mode shape due to the dynamic load is usually taken as the same mode shape as if the structure were loaded statically by the dynamic load. At any instance in time the shape function describes the deflection at any point on the structure relative to the static maximum deflection as a function of the distance along the structure’s span/height. Equation 2.4 in conjunction with Figure 2 demonstrates this mathematically and visually.

Equation 2.4

ϕ ( x )= z (x )zmax

Where:

z (x )=displacement at xcoordinate of structure

zmax=maximumdeflection if loadwere applied statically

Figure 2: Deformation shape of a simply supported elastic beam

By integrating the shape function across the span of the beam so that the kinetic energy, internal energy and work done in the equivalent system are the same as the real beam system at any point in time.

Kinetic Energy (mass factor ) Km=∫0

l

ϕ (x )2dx

l

Internal Energy (stiffness factor ) K k=∫0

l

ϕ ( x )dx

l

Work Done (load factor ) K p=∫0

l

ϕ (x )dx

l

Thus:

m z∫0

l

ϕ (x)2dx

l+kz

∫0

l

ϕ (x )dx

l=P(t)

∫0

l

ϕ ( x )dx

l

It is important to design the equivalent system so that its deflection is the same deflection as the most significant point of the real structure in the chosen failure mode, i.e. the real structures point of maximum deflection which results in failure (3).

Table 1: Transformation factors for one-way elements (table 3-12 from TM5-1300)(3)

Pressure-Impulse (iso-damage) curves?

3. Literature ReviewSDOF modelling in general- the good and bad points of its use, limitations, actual applications, comparisons to FEA models, explain why SDOF models are still used.

Material science-elastic response, plastic response, rebounding, stiffening, 1D strain

Multilayered systems

Numerical models

validation

3.1.SDOF modelling

Blast loads on structures are highly complex events, requiring advanced computational power in order to accurately and reliably model them. With the growing technology in the modern world it raises the question: why use SDOF models to analyse blast loads? It is an obvious question, as reliable and accurate results are important in designing against blast loads.

Morison (5) helps make it apparent that non-linear dynamic finite element (FE) analysis only became available in the 1980s. Before FE technology was established a way to analyse blast load scenarios needed to be developed. The initial usage of SDOF blast models was because there were no other ways to analyse structures subject to blast loads other than experimental tests of replica structures, which is highly costly.

In developing SDOF models two methods were created, the modal and equivalent SDOF methods. Within this dissertation only the equivalent SDOF method is discussed and its concept implemented. Morison’s (5) critical review and revision of dynamic response of walls and slabs by SDOF analysis helps to explain why the modal method was pushed out of use.

The modal method first appeared in a 1946 US Manual, “Fundamentals of Protective Design (Non-Nuclear)” EM 1110-345-405, and was re-issued in 1965 as TM5-855-1. Such method assumes that the elastic forced response of a structural member will be approximated by the first mode for free vibration. The natural period of the SDOF model is taken to be the period of the first mode of free vibration of the element with distributed mass. Non-dimensional charts have been created by rigorous analysis of Newton’s equation of motion of a lumped mass-spring system for an idealised elastic-pure plastic resistance function and a range of idealised loading functions, most notably a triangular load history with zero rise time. Figure 3 is one of the charts used to calculate the maximum deflection with the modal SDOF method.

Figure 3: Typical chart from TM5-1300 (3) used to calculate maximum response Xm.

The modal method requires that formulae are available for the member types analysed in order to evaluate the ultimate resistance and elastic limits of the idealised elastic-pure plastic resistance curve, and the natural period of the fundamental mode of vibration.

Despite some success the modal method was taken out of US Air Force Weapons Laboratory manuals (USAFWL) as of 1970 due to two flaws in its method:

1. The requirement of charts of idealised SDOF systems for analysis limited its use and it cannot be used in numerical solutions of general SDOF systems involving more complex loading histories and resistance functions.

2. The method inadequately analyses reaction forces so the SDOF reaction may be a serious underestimate of the reactions at the member supports. This is done by the method only accounting for a forcing function distributed spatially with the same shape as the vibration mode shape. Although this part of a distributed load dominates the displacement it can be less than 25% of the total uniformly distributed load (UDL) on a two-way spanning member. The rest of the loading will make a major contribution to the reaction and hence the SDOF reaction is seriously underestimated.

The equivalent SDOF method as described under the previous heading, was developed after the modal method and published in 1957 in the USACE manual “Design of Structures to Resist the Effects of Atomic Weapons”, EM 1110-345-415 “Principles of Dynamic Analysis and Design”, and in EM 1110-345-416 “Structural Elements Subjected to Dynamic Loads”.

With the eventual development of dynamic FE analysis the equivalent SDOF method still remains a valuable design tool. It is undeniable that FE analyses of blast loads on structures are more accurate than SDOF models as they are capable of including the effects of blast wave clearing, specific structural geometry, blast wave interaction geometry, blast wave rebounding, confinement of blast waves, and even the effects of material and geometric nonlinearities. Despite these clear accuracy advantages such analysis methods require high computational power, specially designed software, extensive man hours, calibration of material properties, and even the geometry of the surrounding environment can require establishing. Due to the complexity of the FE analysis it is usually reserved for research purposes as opposed to an immediate structural design technique.

This is where the readily applicable use of SDOF models can be used in order to design structures to resist blast loads. TM5-1300(3) created by the US Department of the Army uses the SDOF method to analyse structures by substituting structural elements with a stiffness equivalent SDOF structural system and using this model in conjunction with elastic-plastic response spectra to predict the maximum response of the real structural system. Such response spectra were calculated via numerical integration of the equations of motion assuming triangular reverse ramp decay with time of the blast load and the application of average acceleration methods (6). Numerous modern articles of research into blast or impact loading continue to use the SDOF method, either as a preliminary analysis (7) (8), comparison to FE or experimental results (7) (8) (9) (10) (11) (12), or even as the main analysis method of the research (6) (9) (11).

Despite such approximation and application advantages of the SDOF method engineers must adhere to conservative practice and design when using the SDOF method alone. A prime example where without appropriate consideration of loads the SDOF method can fall inaccurate is through the work of Gantes et al (6) titled “Elastic-plastic response spectra for exponential blast loading” where the established elastic-plastic response spectra from TM5-1300 (3) which use a triangular blast load decay were compared with the results of Gantes’ proposed procedure which also used a triangular loading (such comparison showed very close correlation). In turn Gantes compared the triangular loading response spectra with response spectra using the proposed procedure and exponential loading with various decay coefficients.

Table 2: Shows results by Gantes et al (6) comparing the maximum response for triangular and exponential load cases, the variable b here corresponds to α from Equation 2.1Error: Reference source not foundand is a negative value in such equation as it corresponds to decay.

It is frequently found that exponential loading of SDOF models compare much more favourably with experimental results than approximated triangular loads, Gantes (6) concluded that the commonly used assumption of triangular loading can be slightly unconservative, particularly for flexible structural systems, yet can also be significantly over conservative for stiffer structures.

Another example where the

The triangular load assumption used in the SDOF model is an indefinite limitation of the method. Such assumption will incorrectly represent the load a structure is subjected to, subsequently affecting the response found creating an ill-informed design. When considering an impulsive load the high peak overpressure is of most importance therefore the triangular assumption is adapted to ensure the peak overpressure is captured, underestimating the duration of the actual load and over estimating the impulse. In the case of a quasi-static blast load the duration is of most importance, therefore the triangular assumption is adapted to retain the duration of the load history but will exclude a portion of the peak incident overpressure. Again, this falsely represents the pressure time history experienced by a structure and largely over estimates the impulse. With such considerations in mind an exponential blast load time history with a decay coefficient of -1 is to be implemented in the analysis stage of this dissertation.

When concerning failure of a structural element, the SDOF method uses the maximum deflection of the structural element if the load were applied statically.

Another case showing the limitations of the SDOF method is through the work of Yanchao (8) titled “Numerical derivation of pressure-impulse diagrams for the prediction of RC column damage to blast loads”. Physical damage assessment is the key aspect lacking in the SDOF method. A SDOF model cannot convey information about the type of damage a structure has undergone, severity of damage, or the specific location of such damage experienced. Yancho (8) states two main reasons why the P-I diagrams generated by the SDOF method may not give reliable prediction of structure component damage:

1. It is well known that a structure responds to blast load primarily at their local modes. The local modes of the structure may govern the structure damage especially when the blast load is of short duration. The use of SDOF models may not be suitable for structure damage analysis to blast loads and is not suitable to model multi-failure modes of a structural component either. For example, a column might be damaged owing to shear failure initially and subsequently by flexural failure to collapse. Therefore, the P-I diagram generated from analysis of an SDOF system may not give accurate prediction of structural component damage.

2. The deformation-based damage criterion may not be appropriate for the evaluation of local damage of a structural component subjected to blast loads, especially when the damage is caused primarily by shear failure.

The information a SDOF model is capable of conveying which guides an engineer in a damage assessment would be the deflection time history of the structure; different deflections over time would indicate different damages experienced by the structure. Yanchao (8) validates the numerical results by making a numerical analysis of a quarter-scale RC column and comparing them to published test results by Woodson (14) (15). The comparison poses favourably for Yancho et al’s numerical method for deriving P-I diagrams. Yanchao (8) derives the pressure-impulse (P-I) diagrams by using LS-DYNA to perform a numerical analysis of a reinforced concrete (RC) column and compares such P-I diagrams to ones derived using an equivalent SDOF method used by Fallah (13). Such comparison of diagrams is presented in Figure 4. With the positive validation of Yanchao et al’s results it is clear the numerical method proposed by Yancho gives better prediction of P-I digrams than the SDOF approach.

Figure 4 shows how the quasi-static loading region of the P-I diagrams derived using the SDOF approach have a much lower pressure level than the numerical approach, this is where the material idealization and the negligence of strain rate effects in the SDOF approach underestimate the blast-loading resistance capacity of a structural element.

Despite such argument displaying the inability of the SDOF method to convey specific damage and its type; the P-I diagrams obtained through the numerical method proposed by Yancho required powerful computational resources, element and boundary conditions to be established, material model creation, strain rate and bond slip effects to be applied to the model, and validation of results. Yancho et al also state that broad spectrums of experimental results are required to validate such numerical models. Blast load experiments are expensive and not always feasible to be executed.

Figure 4: P-I diagrams for RC column obtained from Yanchao’s proposed formulae and SDOF approach.

The SDOF method helps to establish a convenient, low-cost, readily applicable, and most importantly a conservative design approach that can adequately be used as a preliminary or main design tool to design structures to resist dynamic blast loading events. Figure 4 shows the conservative response created through using the SDOF method. For the scope of this dissertation the SDOF method is appropriate for use in studying the qualitive behaviour of a protective layer to mitigate blast damage to a structure.

Ulrika et al’s comparisons of the numerical to the SDOF model, discuss and explain its implications into this dissertation!

4. Analysis Method

4.1.Concept and Assumptions

The concept of the dynamic SDOF method is to be applied however, equivalent quantities are not needed and only the real physical quantities are considered. Thus the dynamic SDOF model uses the real mass, stiffness, and load to calculate acceleration, velocity and deflection using Newton’s equations of motion. The reason for such analysis is because the physical element analysed is a protective layer, not a structural element. As a physical layer is depicted as being attached to the outside of a structural element, its deformation mode shape is a crushing mode in a simplified one-dimensional SDOF model. Because the straining of the layer occurs in one dimension the outward

straining effects of the layer are ignored. The loading is assumed to be a uniformly distributed, similar to the blast scenario of a far field explosion. Under a uniform load once the layer deflects, every point along the span of the layer is an equally significant point in relating to the deflection of the layer. The effects of the fluid structure interaction as discussed previously in the literature review are to be ignored; hence the structural element protected is considered to not deflect and to be infinitely stiff. Such assumption is necessary as the response of the structure to the blast load is not the concern, the response of the protective layers are. By modelling the protected structural element as infinitely stiff and ignoring its resistance effect to mitigate the blast load, only the protective layers ability to mitigate the blast load is analysed, not the real structures. Clearing effects as discussed in the literature review are to be ignored. Loading complications from confinement are ignored, the analysis is treated as being in an open space, reflections of blast waves from the layers surface are also ignored, this relates to the effects of fluid structure interaction.

ASSUMPTIONS!

Fluid structure interaction is ignored

When analysing the layers the structure does not deflect at any point and is treated as being infinitely stiff in order to transfer resistance forces between layers and towards the structure.

Outward strain effects in a 2nd direction are ignored

Blast load is assumed as a uniformly distributed load and thus related to a single point load on the SDOF model.

Clearing effects are ignored

LAYER

Structure

Uniform Blast Load

Is an open space loading case, zero confinement, the reflecting pressures combining from the blast wave is not considered.

4.2.Equations

The response of the protective layer is calculated by explicit time stepping integration using the Simpson’s rule by assuming a linear variation across each time step.

Taking the equation of motion:

m z+kz=F

Rearranging to make acceleration the subject, acceleration is given as:

zn=Fn−Rn

m

Where:

Fn=applied force of step n

Rn=resistive force of layer∈step n

For the next time step n, the acceleration becomes:

zn+1=Fn+1−Rn

m

Where:

Fnet=Fn+1−Rn

∴ zn+1=Fnet

m

In the equation above the resistance from the previous step must be used because the resistance in the current step requires calculating the acceleration in order to calculate the resistance in the current step; hence the resistance from the current step cannot be found and thus the resistance from the previous time step must be used.

By integrating explicitly using the Simpson’s rule the velocity is obtained:

zn+1= zn+( zn+ zn+1 )∆ t

2

Keep in mind that the initial velocity zn used in the first time step is zero.

Using the Simpson’s rule again the displacement is obtained:

zn+1=zn+( zn+ zn+1 )∆ t

2

Again, the initial displacement zn used in the first time step is zero.

Substituting equations:

zn+1=zn+zn∆ t2

+( zn+ ( zn+ zn+1 )∆ t

2 )∆ t2

zn+1=zn+zn∆ t2

+zn∆ t2

+( zn+ zn+1 )∆ t 2

4

zn+1=zn+ zn∆ t+( zn+ zn+1 ) ∆t 2

4

The acceleration of the current time step uses the resistance calculated from the previous time step to calculate the net force.

P=Fa

A∴Fa=PA

Fa=m z∴ z=Fa

m

Where:

P=blast pressure

Fa=applied force

A=contact area

m=mass

z=acceleration

Fnet=net force

R=resistance forceof layer

5. Results & Discussion

5.1.Single Layer

5.1.1. Elastic Model

5.1.1.1. Effects of Varying StiffnessRealistically, if a protective layer were subject to a considerably large blast load and experienced a relatively high peak overpressure it is most likely that the layers elastic limit would be easily surpassed. Despite this likely outcome the reason for calculating the elastic response results is to clarify its behaviour and how this behaviour changes when using an elastic-plastic response model.

With a blast load incident overpressure of 100MPa and duration of 0.001s (same as Figure 1), Figure 5 to Figure 8 show the SDOF models elastic response of the layer with a set mass of 100kg and stiffness values of:

0, 1e3, 1e4, 5e5, 1e6, 2e6, 4e6, 6e6, 8e6, 1e7, 2e7, 3e7, 4e7, 5e7, 6e7, 7e7, 8e7, 9e7, and 1e8 N/m.

The fixed variables of the SDOF model are:

As the protective layer is loaded by the blast load it will build a resistance, such resistance opposes the blast load and reduces the force acting upon the structure. The load transferred to the structure is the resistance force from the protective layer as due to Newton’s laws every force must be equal and opposite. The equal and opposite resistance force from the protective layer acting on the blast load therefore acts upon the structure, this is the load which is transferred. It is key to realise that despite the force balance being mathematically correct it can be misleading; As the SDOF model is purely a numerical model of a “fake” layer there is no failure deflection specified. If the protective layer reached a failiure deflection while in the positive phase of the blast load, the peak resistance force generated from the layer will be acting on the structure in addition with the remainder overpressure from the blast wave at the time of failure from the protective layer.

Figure 5 shows how the higher the stiffness of the protective layer, the quicker a resistance of such layer can be built, however, the resistance generated is dependent upon the deflection experienced by the layer and is only effective if the layer has not reached failure. Figure 7 shows how the initial peak deflection is smaller for stiffer layers. Noting the deflection in Figure 7 at 0.001s is approximately 0.23m for each stiffness plot, there is a significant difference in resistances of such plots when compared with Figure 5 at 0.001s and in Figure 6 at 0.2m. This relates to the fact that in order for the layer to build a resistance and mitigate the blast load it must first deform, lower deformations are not necessarily good or bad, they must relate to the mass and stiffness of the protective layer.

Again, this must be linked back to a failure deflection of the layer. At 0.23m deflection, where the layer itself is 0.1m thick, it is physically impossible for the layer to deflect in a crushing mode, up to 0.23m. Figure 7 shows the deflection rises very quickly which indicates that the layer has failed quickly and there is no longer a protective layer to mitigate the blast load.

Figure 5: Purely elastic single layer plot of forces for various stiffness values up to 1e8 N/m. The applied load is masked by the net force corresponding to the zero stiffness plot.

Figure 6: Purely elastic single layer resistance plots for various stiffness values up to 1e8 N/m.

1e8 N/m

0 N/m

Figure 7: Purely elastic single layer displacement plots for various stiffness values up to 1e8 N/m.

Figure 8: Purely elastic single layer velocity plots for various stiffness values up to 1e8 N/m.

Figure 9: Comparison of maximum displacements and maximum resistances for various stiffness values up to 1e9 N/m.

Figure 10: Max displacements and max resistances for various stiffness values up to 1e11 N/m.

Figure 11: Ratio of max resistance over max overpressure for various stiffness values up to 1e10 N/m.

The effect of various failure deflections will be analysed when considering the elastic-plastic and strain hardening models, for now the oscillating elastic behaviour is the focus.

Figure 9 and Figure 10 show the effect of various stiffness values on the maximum displacements and maximum resistances generated by a protective layer. Note the different scales on the axis for each plot. Figure 9’s resolution shows how the max displacement decreases for stiffer layers up to a stiffness of 1e9 N/m, yet the max resistance increases. Figure 10’s resolution shows how once past a stiffness of 2e10 N/m the max displacement begins to increase.

5.1.1.2. Effects of Varying Mass

Elastic-plastic model

Strain hardening model

5.2.Double Layer

Discussion & Conclusion

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