1

COMPUTATIONAL ANALYSIS OF PROTECTIVE STRUCTURES

Tiong Jackson1, Wang Yanxiao Austin1, Kang Kok Wei2 1Catholic High School, 9 Bishan Street 22 Singapore 579767

2Defence Science and Technology Agency, 1 Depot Road, Singapore 109679

ABSTRACT

The usage of Finite Element Method for the prediction of the weapon effects on concrete

structures was studied in this research. It was found that fragment loadings from the

breakup of cased charges have a more significant impact on concrete structures than the

blast wave. Various parameters such as the element size, loading segment methods and

erosion has also been studied to observe the influences on the structural response

predictions. Lastly, concrete strength, rebar diameter and rebar arrangements are all

important factors that contribute to the resistance of a concrete structure against weapon

effects.

INTRODUCTION

The ability to accurately assess weapons effect on buildings is vital in the design of

protective structures. Weapons effect refers primarily to the combined blast and fragment

loading. Blast refers to the thermal energy converted in the explosive and fragment loading

refers to the kinetic energy imparted into the shattered casing of the bomb, which inflicts

damage and may cause structural failure. Through numerical modelling, one is able to

evaluate how a specific structure will react to a certain weapon load without the cost

implications of physical experimental testing.

BACKGROUND

Numerical modelling is a branch of computer simulation which consists of mathematical

models using numerical time-stepping procedures to obtain the model’s behaviour over

time [1]. The finite element method (FEM) numerical technique can be used for the

modelling of weapons effect to obtain a precise, simulated result. FEM allows the accurate

representation of complex geometry, inclusion of dissimilar material properties, as well as

capturing and representing local effects [2].

In this paper, the effect of modelling parameters when using FEM on assessing damage to

reinforced concrete slabs from bare and cased charges detonating near such slabs will be

discussed. The main questions that arise are:

a. How does the result of combined blast and fragment loading differ from the individual

loading of blast and fragment?

b. What are the parameters to most accurately and efficiently model structural response

to weapon effects?

2

c. How can protective structures be designed to resist combined blast and fragment

loading?

METHODOLOGY

The FEM program, LS-DYNA[3][4] is used in this research for studying the usage of FEM

in simulating the response of concrete structures to combined blast and fragment loadings.

FINITE ELEMENT METHOD

The concrete slab is modelled as a half model along the XZ plane using LS-PrePost1 and

is dimensioned as 3x0.5x0.25m (length x breadth x height). The block is meshed using

hexahedral SOLID elements. Reinforcement bars (rebars) are placed in the model 0.05 m

from the top and bottom surfaces and at least 0.05 m from the sides of the slab. They are

spaced 0.10 m apart. The rebars are modelled with a circular cross sections of either 0.012m

or 0.025m diameter, depending on the orientation and assigned as a BEAM section. The

nodes of the rebars are merged with the nodes of the slab that share the same point in the

model, “attaching” them to the slab. (see Fig A1 in ANNEX)

Fig 1 - Fragment Generation Program

1 LS-Prepost is an LS-DYNA software for modeling of the structure in preprocessing and

reviewing the results in post processing

3

Material Properties

For the concrete material that is assigned to the slab, the Karagozian & Case (K&C)

Concrete Model - Release III is used. The material model is a three-invariant model, uses

three shear failure surfaces, includes damage and strain-rate effects. A mass density of 2300

kg/m3 and a concrete strength of 35 MPa is assigned to the material.

A load curve is also assigned to the material for strain-rate effects, namely strength

enhancement versus effective strain rate for the concrete (see Table A1 in ANNEX). Other

material parameters for concrete are generated by the software based on the concrete

strength.

For the steel material that is assigned to all the rebars and shear links, Material Type 3 also

known as *MAT_003 or *MAT_PLASTIC_KINEMATIC is used as it is suited to model

isotropic and kinematic hardening plasticity with the option of including rate effects. A

mass density of 7850 kg/m3, Young’s modulus of 210 GPa, Poisson's ratio of 0.3, yield

stress of 500 MPa, and failure strain of 0.2 is assigned to the material.

For certain models, erosion model *MAT_ADD_EROSION is applied to the concrete

material. Erosion automatically removes elements with nodes that rapidly gained huge

amounts of displacement and lead to great deformation of the model. This is to avoid the

long calculation times due to excessive deformation of these elements and facilitate result

generation. The only user entered parameter for this material is a shear strain at failure of

0.5.

Segment Generation

Segments are applied onto the surface of the model for blast and fragment loadings and

multiple segments are generated across the surface in order to constrain the loading of each

fragment to a specific area on the surface. The load of each segment is unique. (see Fig A2

and A5 in ANNEX)

To generate the inputs specifying the various segments, an external code (see Fig 1) is

written. Each segment has a segment ID on the first row followed by the information of

each element that makes up the segment on each row below it. Each element code contains

the ID of its nodes arranged such that they would run either clockwise or anti-clockwise

around the element when read from left to right in the code. A clockwise arrangement

means a normal contact force pointing in the positive direction in the z axis of the software

while the opposite is true for an anti-clockwise arrangement.

Blast Loading

The blast used in this study is defined by the detonation of a 10kg mass equivalent of TNT

at a stand-off distance of 1m from the surface central to the half slab. The blast source type

used is that of a spherical air burst, with no amplification of initial shock wave due to

interaction with the ground surface. The blast wave

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reaches the slab at 0.29 milliseconds. The blast is defined in LSDYNA under

*LOAD_BLAST_ENHANCED.

The blast effect is loaded onto all the segments on the top side of the slab where the blast

is above through the use of *LOAD_BLAST_SEGMENT_SET in the software.

Fragment Loading

The x and y positional coordinates, the x and z component velocities2, mass and arrival

time of the fragments are provided. This information would be converted into pressure-

time histories which will be applied to each segment. This is done through an externally

written code.

Calculating Pressure And Duration Of Impact Generated By Each Fragment

The depth that a fragment would penetrate into the concrete material used is first calculated

based on the penetration curve obtained in [3]:

𝑑(𝜙, 𝑣) = 𝐴1𝜙𝑣2 + 𝐴2𝜙2𝑣 + 𝐴3𝜙𝑣 + 𝐴4

A1=2.1237E-6, A2=1.8472E-5, A3=2.1417E-4 and A4=1.2947E-4

d = penetration distance, 𝜙 = diameter of fragment, v = velocity of fragment

The volume of a fragment is calculated from its mass by assuming a uniform density of

7680 kg/m3 and the fragment diameter is obtained by assuming a perfectly spherical

fragment shape. The velocity of a fragment is taken through vector addition of its x and z

component velocities.

With the penetration depth and impact velocity known, duration of the impact can be

obtained as we assume that the fragment experiences linear deceleration throughout the

impact:

𝑡 =2𝑑

𝑣

t = duration of impact

The pressure exerted onto a segment is found assuming it is uniform throughout the

segment:

𝑝 = 𝑚𝑣

𝑡𝐴

2 The y component of the fragment velocity is not accounted for as the cased charge is

assumed to be cylindrical in shape and lies parallel to the y-axis, hence most of fragment

velocities would be directed perpendicular to the y-axis (see Fig A7 in ANNEX for

orientation of shell).

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p = pressure exerted onto segment, m = mass of fragment, A = area of segment

There are cases whereby the fragment impacts on the edge or corner of a segment that is

shared with other segments. In such cases, the loading by that fragment would be applied

onto all the segments sharing that point of impact, however, the pressure of the load will

be divided by the number of segments loaded by that fragment.

Summing Pressure-Time Graphs Of All Fragment Impact On A Segment

Since it is assumed that a fragment experience linear deceleration throughout the impact,

the pressure-time graph caused by a single fragment would be a rectangular pulse, with a

constant pressure over a time period.

The pulse caused by every fragment on a

single segment would need to be summed

up and presented in a form recognised by

LS-DYNA. Each curve input is named

*DEFINE_CURVE and consists of the

segment ID in the first row, followed by

the pressure and time coordinates of each

point of the graph, on the left and right

respectively, for every row down the

column in order of increasing time. Also,

each segment has to point to a load curve

in the input *LOAD_SEGMENT_SET.

Finding out the fragments that impact a

segment and the generation of the curve

inputs of each segment is done with an

external program (refer to Fig 2).

Fig 2 - Fragment Generation Program

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Other Parameters

Other user defined parameters are as listed below:

a. Boundaries are defined for XYZ directions for the nodes comprising of the front and

back surface, and only in the Y direction for the left surface. (refer Fig A3 and A4 in

ANNEX);

b. Termination time under *CONTROL at 0.09 seconds;

c. Hourglass control is considered in certain models to ensure computational stability. The

user inputs for the *HOURGLASS card for these models are defined as having an

hourglass control type of 1 (standard LS-DYNA viscous form), an hourglass coefficient

of 0.1, and a quadratic bulk viscosity coefficient of 1.5.

Data Collection and Comparison

Two result files are generated to show the results, namely d3plot and blstfor. Both files are

generated with a time interval for output of 0.001 seconds with a total of 92 frames for the

0.09 second model. The main data compared in this report will be vertical displacement of

the central node at the middle of the top edge of the slab that is lying on the plane of

symmetry and the fringe component of damage on the bottom side of the slab to assess the

damage as a result of the individual loadings and parameter differences.

RESULTS & DISCUSSION

Blast, Fragment, and Combined Blast and Fragment Loading

To compare the differences of blast, fragment, and their combined effect on reinforced

concrete slabs, a reinforced concrete slab was modelled with an element size of 0.05m and

segment size 0.1m. The slab was then subject to the aforementioned loadings. The results

are as follows:

a. Blast Loading

When loaded with blast only, the slab showed a maximum vertical deflection of 0.00752m.

The response was largely elastic, indicating the blast inflicted relatively little damage on

the reinforced concrete slab.

b. Fragment Loading

When loaded with fragment only, the slab showed a maximum vertical deflection of

0.04949m. Plastic response was observed in this case and a significant amount of damage

to the underside of the slab. (see Fig A9 in ANNEX)

c. Combined Blast and Fragment Loading

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When loaded with the combined effect of blast and fragments, the slab showed a maximum

vertical deflection of 0.086m. The combined loading also resulted in a similarly large

amount of plastic response as the model which has been subjected to fragment loading.

Hence, it can be observed from this comparison that the majority of the damage caused by

explosives to reinforced concrete walls come from the fragmentation effect of the casing.

The fragments apply a significantly higher load on the structure, causing the structure to

deflect more and hence, sustain greater damage as a result. When both blast and fragments

are loaded onto a structure, the resulting load is also much greater than just the sum of the

individual loadings.

Effects of Modelling Parameters on Results

To ensure the accuracy of the model, a study on the effect of varying the sizes of segments

and elements and the presence of erosion was conducted.

a. Segment size

To deduce the significance of segment sizes, the same half model of a reinforced concrete

slab as described in preceding section was used. The slab was then assigned 3 different

segment sizes of 0.1 m, 0.2 m and 0.4 m. 3 different loadings, namely blast, fragment, and

combined blast and fragment were applied and the results are as follows.

Fig 4 - (above) Vertical-displacement of the central node against time for models of

element size 0.05m and segment sizes of 0.1 metre (S100), 0.2 metre (S200), and 0.4

metre (S400) with blast load only.

b. Blast Only

When only blast loading is applied, the deflection curves are similar. This can be attributed

to the fact that blast loading is applied to all defined surfaces on the side of the slab where

the blast propagates, hence it is not affected by the segment sizes even though it is loaded

with the *LOAD_SEGMENT_SET card. All the models yielded a maximum central node

displacement of 0.00752 m (refer to Fig 3.0).

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0 0.01 0.02 0.03 0.04 0.05

Ver

ical

-dis

pla

cem

ent/

m

Time/s

S100 S200 S400

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Fig 3 - (above) Vertical-displacement of the central node against time.

c. Fragment only

With regards to fragment loading, as segment size decreases, the displacement of the

central node increases. This can be attributed to the fact that loading, or pressure on the

specified segment, is calculated by force over area. Hence, reduction in segment area

resulted in a higher pressure on each segment, which translated to a higher load on the

segment. (refer to Fig 3.1)

Combine Blast and Fragment

The differences in fragment loading as a result of varying segment sizes is amplified when

combined with blast. The 0.4 m segment size model yielded a maximum central node

displacement of 0.05634 m, while the 0.2 m segment size yielded a displacement of

0.05717 m, and the 0.1 m segment size yielding a displacement of 0.08630 m. (refer to Fig

3.2)

This study showed that a smaller segment size used would generally yield a greater

maximum vertical-displacement only when fragments are loaded. This is because the

fragment loading is inversely proportional to the size of the segments in the calculations

for the input. When a smaller segment size is used, the fragment loading is spread out over

a smaller area, hence exerting a greater pressure on the segments.

a. Element Size

To deduce the significance of element sizes, a half model of a reinforced concrete slab was

modelled with 2 different element sizes of 0.01 m, and 0.05 m, all with a segment size of

0.1 m. A combined loading of blast and fragment was applied, with the results as follows.

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Fig 5 - (above) Vertical-displacement of the central node against time for models of

element size 0.05m and segment sizes of 0.01 metre (S10), 0.025 metre (S25), and 0.05

metre (S50) with fragment load only.

Fig 6 - (above) Vertical-displacement of the central node against time for models of

element size 0.05m and segment sizes of 0.01

b. Vertical-Displacement

When comparing the 0.05 m and 0.01 m element size models, the variation of element size

has not caused the maximum displacement of the central node to change much, with

maximum displacements of 0.08630 m and 0.08931 m respectively. It is also to be noted

that the 0.05 m element size model becomes mostly plastic after reaching maximum

displacement, but the 0.01 m element size model still retains some form of elasticity and is

hence able to rebound more than the 0.05 m element size model.

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0 0.01 0.02 0.03 0.04 0.05

Ver

tica

l-d

isp

lace

men

t/m

Time/s

S100 S200 S400

-0.1

-0.08

-0.06

-0.04

-0.02

0

0 0.01 0.02 0.03 0.04 0.05

Vet

ical

-dis

pla

cem

ent/

m

Time/s

S100 S200 S400

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Fig 7 - Vertical-displacement of the central nodes against time.

c. Damage

The damage of the models are relatively similar for both top and bottom surfaces, with the

boundaries of the damaged areas becoming more defined as the element sizes get smaller.

This can be attributed to the finer mesh generated as a result of smaller element sizes,

allowing the resulting strain map to be displayed more accurately. (See Fig A14 to A19 in

ANNEX)

d. Erosion

To deduce the significance of erosion, a half model of a reinforced concrete slab was

modelled with element and segment size of 0.01 m. 2 cases were ran with identical

fragment and blast loadings but the first case does not have *MAT_ADD_EROSION

applied to it while the second case has.

With erosion, the computation time of the simulation drastically decreased to reasonable

levels as compared to no erosion. This is because the small segment size of the model

resulted in extremely high amounts of pressure applied onto the impacted areas of the

model, causing the nodes in those area to experience great displacement and the extreme

deformation of the structure results in a long computing time. Erosion automatically

remove the elements of these nodes to avoid the computation time that they incur. (See Fig

A31 and A32 in ANNEX)

Design of Protective Structures

To define what loading parameters will cause a structure to fail, there is a failure criterion

that the structure has to meet. For reinforced concrete structures, this failure criteria is a

support rotation of 2 in order to protect personnel behind the structure. By using

Pythagoras’ theorem, we have deduced that for a slab 3 m in length, the maximum

displacement of the centre point from its original position under load while still keeping

within the 2 limit of support rotation is 0.052 m. Hence, the variations in simulation

parameters are to find the minimum parameter values (namely concrete strength, rebar

diameter, and rebar placement) in order to meet such a criterion.

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a. Concrete Strength

The strength of materials used is the main factor when it comes to designing a weapons

effect resistant structure, with the material in this case being concrete. A reinforced

concrete slab model with a 0.05 m element size and 0.1 m segment size was chosen for this

simulation. The default concrete strength, set at 35 MPa, which will result in the central

node displacing 0.086 m in the Z-axis, thus failing to protect personnel behind the structure.

In order to keep displacement to 0.052 m or less, the concrete strength has to be increased

to 59.8 MPa or higher.

Fig 8 - Vertical-displacement of the central node against time for concrete strength of

35MPa and 59.8MPa.

b. Rebar Diameter

The diameter of the rebars can also affect how a structure like a concrete slab responds to

a blast and fragment loading. The tension in the rebars when pulled help to reduce the

movement of the structure under load. A rebar with a larger diameter, will result in greater

tension for a given y displacement and hence cause the structure to move less. A reinforced

concrete slab model with a 0.05 m element size and 0.1 m segment size was chosen for this

simulation. The default diameter of the lengthwise (3-m-long) rebars is 0.025 m, which

will result in a maximum central node displacement of 0.086 m in the Z-axis. By increasing

the diameter of lengthwise rebars to 0.05539 m, a maximum displacement of 0.0514 m, is

achieved. However, in reality, this is not a viable option as the maximum diameters of

rebars available commercially is 0.04m.

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Fig 9 - Vertical-displacement of the central node against time for rebar diameter of

0.025m and 0.0514m.

c. Rebar Placement

Another factor affecting the concrete response is the placement of the rebars. For the study

of this factor, 3 different rebar placements were arranged for 3 different simulations and all

had element sizes of 0.01m and segment sizes of 0.1m with blast and fragment loadings

and identical rebar length and diameters.

Lengthwise Rebar Placement (RebarL)

Rebars parallel to the length of the structure are placed near the top and bottom of the

structure (see Fig A23 in ANNEX).

Lengthwise and Breadthwise Rebar Placement (BASE)

Same rebar arrangement as lengthwise rebar placement but rebars parallel to the breadth of

the structure are also placed near the top and bottom of the structure. (see Fig A24 in

ANNEX).

Fig 10 - Vertical-displacement of the central node against time rebar placement variation.

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Lengthwise and Breadthwise Rebar Placement with Shear Links (Shear-Linked)

Same rebar arrangement as lengthwise and breadthwise rebar placement but rebars parallel

to the height of the structure (shear links) are placed throughout it.(see Fig A25 in

ANNEX).

From the displacement time graph in Fig 5.2, we can see that the maximum vertical-

displacement of the models with lengthwise and breadthwise rebar placement and

breadthwise rebar placement only does not differ significantly. This may be attributed to

the fact that the structure does not bend much along its breadth due to the restriction of the

boundary condition placed along the breadth of the models.

We can also see that the placement of the shear links greatly reduce the vertical-

displacement of the model. This is because while the structure only bends significantly

horizontally, it also experiences much damage in the form of expansion in the vertical axis

as it bends, which concrete are weak to. Placement of the shear links, which are resistant

to expansion, strengthens the structure in the vertical axis, causing it to experience less

strain and hence has a lower vertical-displacement (see Fig A24 and A25 in ANNEX).

LIMITATIONS AND FURTHER WORK

Simulation of the effects of cased charges with FEM has its own fundamental limitations

as it would be practically impossible to perfectly account for all kinds of variable from the

physical world to give a perfect model on a computer. For example, many effects of an

explosion of cased charges remain unpredictable and random, such as the number of

fragments created and the momentum and position of each and every fragment. We can

only make simulations that closely approximate these effects.

As the element sizes decreases one should expect the result of the simulation to become

more and more accurate as the slab becomes less discrete. However, this was not what we

found. Instead, as the element size decreased from 0.05m to 0.025m, there was a large

decrease in the vertical-displacement whereas a decrease in element size from 0.025m to

0.01m showed a large increase in vertical-displacement. This is something that can be

studied in future research. (see Fig A13 in ANNEX)

CONCLUSION

Blast and Fragment loading effects

We have seen that in a cased charge explosion, the blast pressure is not the only factor that

we must account for. In fact, this research showed that the fragment loading having

significantly greater impact on a structure than blast loading.

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Effects of Modelling Parameters on results

We found that decreasing the segment sizes increased the maximum vertical-displacement

of the structures only when fragments were loaded. This is due to the fact that the loading

pressure of a fragment is directly related to the size of the segments in our calculations. We

also found changes in the vertical-displacements of the structure when the element sizes

were varied. However, there were no trend observed in the changes in vertical-

displacements as it is observed that the 0.025 m element size model did not produce results

similar to those of the other element sizes despite having the same loading, modelling

parameters, and material properties. (see Fig A13 in ANNEX) We also learnt of how

erosion can speed up the computation time when numerous extremely small elements are

computed, however this process removes entire elements making the simulation less

accurate. Hence it is discouraged.

Design of Explosion Resistant Structures

Lastly, we learnt how rebars play an important role in building a concrete structure capable

of withstanding weapon effects and we can make a concrete structure more resistant by

increasing the rebar diameter and strategic placement of rebars parallel to the length of the

structure both near the top and bottom of the structure and vertical rebars connecting the

top and bottom lengthwise rebars. We also learnt that a greater concrete strength leads to a

lower vertical displacement of the structure.

ACKNOWLEDGEMENTS

We would like to express our sincere gratitude to our project mentor from DSTA, Doctor

Kang Kok Wei, for the invaluable guidance throughout this Research@YDSP journey and

everyone else who has supported us in one way or another. We would also like to extend

our thanks to DSTA for the opportunity to embark on this research project.

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REFERENCES

1. Dr.ir. J. Mediavilla-Varas and Ir. J.C.A.M. van Doormaal, 2009, Klotz Group

Engineering Tool - Effect of cased ammunition. WP3: Modelling damage and

response of Kasun structure, p197-257, In: TNO report

2. Dr. Jesus Mediavilla Varas MSc., Dr. Jaap Weerheijm MSc., 2011, Simulations of

damage and response of KASUN Houses WP3: The effect of cased ammunition,

p110 to 167, In: TNO report

3. LS-DYNA Keyword User’s Manual Volume I

4. LS-DYNA Keyword User’s Manual Volume II Material Models

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ANNEX

Fig A1 Dimensions of Slab and Rebars

Fig A2 Displaying a single segment and the fragments applied onto the slab.

Fig A3 - XYZ directions constrained front and back surface nodes

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Fig A4 - Y direction constrained left surface nodes.

14

Fig A5, A6 and A7 - (top to bottom) Typical pressure pulse histories, diagram of

relative momentum of fragments (bubble area) and their position on the surface of

the slab in metres and diagram of the location of the cased charge.

15

Fig A8 - Vertical-displacement of the central node against time between blast only,

fragment only, and combined blast and fragment effect for element sizes of 0.05m and

segment sizes of 0.1m.

16

Fig A9 - (top to bottom) Damage of blast, fragment and combined loading.

17

Fig A10 - Vertical-displacement of the central node against time for blast only load

between segment sizes of 0.1metre, 0.2metre and 0.4metre and element size of 0.01metre.

Fig A10 - Vertical-displacement of the central node against time for fragment only load

between segment sizes of 0.1metre, 0.2metre and 0.4metre and element size of 0.01metre.

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0 0.01 0.02 0.03 0.04 0.05

Ver

ical

-dis

pla

cem

ent/

m

Time/s

S100 S200 S400

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0 0.01 0.02 0.03 0.04 0.05

Ver

tica

l-d

isp

lace

men

t/m

Time/sS100 S200 S400

18

Fig A12 - Vertical-displacement of the central node against time for combined blast and

fragment loads between segment sizes of 0.1metre, 0.2metre and 0.4metre and element

size of 0.01metre.

Fig A13 - Vertical-displacement of the central nodes against time between models of

element sizes of 0.01 m (E10), 0.025 m (E25), and 0.05 m (E50) and segment size of

0.1m.

-0.1

-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

Ver

tica

l-d

isp

lace

men

t/m

Time/s

S100 S200 S400

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Fig A14 (above) and Fig A15 (below) - Damage of 0.01 m element size, 0.1 m segment

size reinforced concrete slab when subject to combined blast and fragment loading shown

from the top and bottom respectively.

20

Fig A16 (above) and Fig A17 (below) - Damage of 0.025 m element size, 0.1 m segment

size reinforced concrete slab when subject to combined blast and fragment loading shown

from top and bottom respectively.

21

Fig 18 (above) and Fig A19 (below) - Damage of 0.05 m element size, 0.1 m segment

size reinforced concrete slab when subject to combined blast and fragment loading for top

and bottom respectively.

22

Fig A20 - Vertical-displacement of the central node against time for concrete strength

variation.

Fig A21 - Vertical-displacement of the central node against time rebar diameter variation.

Fig A22 - Vertical-displacement of the central node against time rebar placement variation.

23

Fig A23 – Lengthwise rebar arrangement

Fig A24 - Lengthwise + breadthwise rebar arrangements.

Fig A25 - Shear linked rebar arrangements

24

Fig A26 and A27(top to bottom)(above) Slab loading with shear links(top) and without

shear links(bottom)

25

Fig A28 Flowchart for segment generation

26

Fig A29 Summary of segment generation code

27

Fig A30 Summary of fragment generation code

28

Fig A31 and A32 (top to bottom) Effect of erosion on model

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Strain-Rate (1/ms) Enhancement

-3.0E+01 9.70

-3.0E-01 9.70

-1.0E-01 6.72

-3.0E-02 4.50

-1.0E-02 3.12

-3.0E-03 2.09

-1.0E-03 1.45

-1.0E-04 1.36

-1.0E-05 1.28

-1.0E-06 1.20

-1.0E-07 1.13

-1.0E-08 1.06

0.0E+00 1.00

3.0E-08 1.00

1.0E-07 1.03

1.0E-06 1.08

1.0E-05 1.14

1.0E-04 1.20

1.0E-03 1.26

3.0E-03 1.29

1.0E-02 1.33

3.0E-02 1.36

1.0E-01 2.04

3.0E-01 2.94

3.0E+01 2.94

Table A1Enhancement versus effective strain rate for 45.4 MPa concrete